From 7c9fbc32d30e69325c6cc2ab22e276ea2f0c5070 Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Sun, 23 Apr 2023 16:32:05 +0100 Subject: [PATCH 01/50] Add commented sans serif fonts --- packages.sty | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/packages.sty b/packages.sty index f897fe3..eba83f2 100644 --- a/packages.sty +++ b/packages.sty @@ -28,8 +28,10 @@ % Serif and math fonts \usepackage{newtxtext,newtxmath} % Sans-serif font -\usepackage[scaled=.95]{FiraSans} -% \usepackage[eulergreek]{sansmath} +\usepackage[scaled=0.95]{FiraSans} +% \usepackage[oldstyle,scale=0.95]{opensans} +% \usepackage{roboto} +% \usepackage{noto} \usepackage{csquotes} % When loading babel with biblatex \usepackage{anyfontsize} % For custom font sizes From 345a0320da87a58011b72f48ed9bacb12a57de6d Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Sun, 23 Apr 2023 16:32:16 +0100 Subject: [PATCH 02/50] Simplify caption --- chapters/glitch.tex | 2 +- references.bib | 38 +++++++++++++++++++------------------- 2 files changed, 20 insertions(+), 20 deletions(-) diff --git a/chapters/glitch.tex b/chapters/glitch.tex index 86d1f76..c7dca8c 100644 --- a/chapters/glitch.tex +++ b/chapters/glitch.tex @@ -248,7 +248,7 @@ \subsubsection{The Effect of Helium Abundance on \(\gamma\)} \begin{figure} \centering \includegraphics{figures/adiabatic-ionisation-temp.pdf} - \caption[Temperature-density profile for a Sun-like star.]{Temperature-density profile for a Sun-like star. \emph{Top:} The first adiabatic exponent \(\gamma\) as a function of temperature and density, calculated using Eq. 53 of \citet{Houdayer.Reese.ea2021} for a helium mass fraction, \(Y=0.25\). \emph{Bottom:} The change in \(\gamma\) induced by a change in helium abundance, \(\Delta Y = 0.1\). In both panels, the dashed line shows the temperature-density profile of model S.\\--- \emph{This figure is a recreation of Fig. 5 in \citet{Houdayer.Reese.ea2021}.}} + \caption[Temperature-density profile for a Sun-like star.]{Temperature-density profile for a Sun-like star. \emph{Top:} The first adiabatic exponent \(\gamma\) as a function of temperature and density, calculated using Eq. 53 of \citet{Houdayer.Reese.ea2021} for a helium mass fraction, \(Y=0.25\). \emph{Bottom:} The change in \(\gamma\) induced by a change in helium abundance, \(\Delta Y = 0.1\). In both panels, the dashed line shows the temperature-density profile of model S. \emph{This figure is a recreation of Fig. 5 in \citet{Houdayer.Reese.ea2021}.}} \label{fig:gamma-temp-density} \end{figure} diff --git a/references.bib b/references.bib index 35057e2..737de53 100644 --- a/references.bib +++ b/references.bib @@ -773,7 +773,7 @@ @article{Bouchy.Carrier2001 author = {Bouchy, F. and Carrier, F.}, year = {2001}, month = jul, - journal = {A\&A}, + journal = {\aap}, volume = {374}, number = {1}, pages = {L5-L8}, @@ -843,7 +843,7 @@ @article{Brewer.Stello2009 author = {Brewer, B. J. and Stello, D.}, year = {2009}, month = jun, - journal = {Monthly Notices of the Royal Astronomical Society}, + journal = {\mnras}, volume = {395}, number = {4}, pages = {2226--2233}, @@ -1098,7 +1098,7 @@ @article{Chaplin.Bedding.ea2011 author = {Chaplin, W. J. and Bedding, T. R. and Bonanno, A. and Broomhall, A.-M. and Garc{\'i}a, R. A. and Hekker, S. and Huber, D. and Verner, G. A. and Basu, S. and Elsworth, Y. and Houdek, G. and Mathur, S. and Mosser, B. and New, R. and Stevens, I. R. and Appourchaux, T. and Karoff, C. and Metcalfe, T. S. and {Molenda-{\.Z}akowicz}, J. and Monteiro, M. J. P. F. G. and Thompson, M. J. and {Christensen-Dalsgaard}, J. and Gilliland, R. L. and Kawaler, S. D. and Kjeldsen, H. and Ballot, J. and Benomar, O. and Corsaro, E. and Campante, T. L. and Gaulme, P. and Hale, S. J. and Handberg, R. and Jarvis, E. and R{\'e}gulo, C. and Roxburgh, I. W. and Salabert, D. and Stello, D. and Mullally, F. and Li, J. and Wohler, W.}, year = {2011}, month = apr, - journal = {ApJL}, + journal = {\apjl}, volume = {732}, number = {1}, pages = {L5}, @@ -1174,7 +1174,7 @@ @article{Charbonneau2021 author = {Charbonneau, Rebecca}, year = {2021}, month = dec, - journal = {American Indian Culture and Research Journal}, + journal = {Am. Indian Cult. Res. J.}, volume = {45}, number = {1}, pages = {71--94}, @@ -1473,7 +1473,7 @@ @article{Crameri.Shephard.ea2020 author = {Crameri, Fabio and Shephard, Grace E. and Heron, Philip J.}, year = {2020}, month = oct, - journal = {Nat Commun}, + journal = {Nat. Commun.}, volume = {11}, number = {1}, pages = {5444}, @@ -1904,7 +1904,7 @@ @article{Feuillet.Bovy.ea2016 author = {Feuillet, Diane K. and Bovy, Jo and Holtzman, Jon and Girardi, L{\'e}o and MacDonald, Nick and Majewski, Steven R. and Nidever, David L.}, year = {2016}, month = jan, - journal = {The Astrophysical Journal}, + journal = {\apj}, volume = {817}, pages = {40}, issn = {0004-637X}, @@ -1981,7 +1981,7 @@ @article{Frankel.Sanders.ea2020 author = {Frankel, Neige and Sanders, Jason and Ting, Yuan-Sen and Rix, Hans-Walter}, year = {2020}, month = jun, - journal = {The Astrophysical Journal}, + journal = {\apj}, volume = {896}, pages = {15}, issn = {0004-637X}, @@ -2349,7 +2349,7 @@ @article{Goupil2019 author = {Goupil, MarieJo}, year = {2019}, month = dec, - journal = {Bulletin de la Societe Royale des Sciences de Liege}, + journal = {Bull. Soc. R. Sci. Liege}, volume = {88}, pages = {115--146}, urldate = {2021-02-11}, @@ -2601,7 +2601,7 @@ @article{Hatt.Nielsen.ea2023 author = {Hatt, Emily and Nielsen, Martin B. and Chaplin, William J. and Ball, Warrick H. and Davies, Guy R. and Bedding, Timothy R. and Buzasi, Derek L. and Chontos, Ashley and Huber, Daniel and Kayhan, Cenk and Li, Yaguang and White, Timothy R. and Cheng, Chen and Metcalfe, Travis S. and Stello, Dennis}, year = {2023}, month = jan, - journal = {A\&A}, + journal = {\aap}, volume = {669}, pages = {A67}, issn = {0004-6361, 1432-0746}, @@ -2786,7 +2786,7 @@ @article{Hon.Bellinger.ea2020 author = {Hon, Marc and Bellinger, Earl P. and Hekker, Saskia and Stello, Dennis and Kuszlewicz, James S.}, year = {2020}, month = sep, - journal = {Monthly Notices of the Royal Astronomical Society}, + journal = {\mnras}, volume = {499}, pages = {2445--2461}, issn = {0035-8711}, @@ -2802,7 +2802,7 @@ @article{Hon.Huber.ea2021 author = {Hon, Marc and Huber, Daniel and Kuszlewicz, James S. and Stello, Dennis and Sharma, Sanjib and Tayar, Jamie and Zinn, Joel C. and Vrard, Mathieu and Pinsonneault, Marc H.}, year = {2021}, month = oct, - journal = {ApJ}, + journal = {\apj}, volume = {919}, number = {2}, pages = {131}, @@ -3548,7 +3548,7 @@ @article{Li.Davies.ea2022 author = {Li, Tanda and Davies, Guy R and Lyttle, Alexander J and Ball, Warrick H and Carboneau, Lindsey M and Garc{\'i}a, Rafael A}, year = {2022}, month = apr, - journal = {Monthly Notices of the Royal Astronomical Society}, + journal = {\mnras}, volume = {511}, number = {4}, pages = {5597--5610}, @@ -3802,7 +3802,7 @@ @article{Maury.Pickering1897 } @article{Mazumdar.Michel.ea2012, - title = {Seismic Detection of Acoustic Sharp Features in the {{CoRoT}} Target {$<$}{{ASTROBJ}}{$>$}{{HD}} 49933{$<$}/{{ASTROBJ}}{$>$}}, + title = {Seismic Detection of Acoustic Sharp Features in the {{CoRoT}} Target {{HD}} 49933}, author = {Mazumdar, A. and Michel, E. and Antia, H. M. and Deheuvels, S.}, year = {2012}, month = apr, @@ -3867,7 +3867,7 @@ @article{Metcalfe.Creevey.ea2014 author = {Metcalfe, T. S. and Creevey, O. L. and Do{\u g}an, G. and Mathur, S. and Xu, H. and Bedding, T. R. and Chaplin, W. J. and {Christensen-Dalsgaard}, J. and Karoff, C. and Trampedach, R. and Benomar, O. and Brown, B. P. and Buzasi, D. L. and Campante, T. L. and {\c C}elik, Z. and Cunha, M. S. and Davies, G. R. and Deheuvels, S. and Derekas, A. and Mauro, M. P. Di and Garc{\'i}a, R. A. and Guzik, J. A. and Howe, R. and MacGregor, K. B. and Mazumdar, A. and Montalb{\'a}n, J. and Monteiro, M. J. P. F. G. and Salabert, D. and Serenelli, A. and Stello, D. and Ste{\c}{\'s}licki, M. and Suran, M. D. and Y{\i}ld{\i}z, M. and Aksoy, C. and Elsworth, Y. and Gruberbauer, M. and Guenther, D. B. and Lebreton, Y. and Molaverdikhani, K. and Pricopi, D. and Simoniello, R. and White, T. R.}, year = {2014}, month = oct, - journal = {ApJS}, + journal = {\apjs}, volume = {214}, number = {2}, pages = {27}, @@ -4027,7 +4027,7 @@ @article{Monteiro.Christensen-Dalsgaard.ea1994 author = {Monteiro, M. J. P. F. G. and {Christensen-Dalsgaard}, J. and Thompson, M. J.}, year = {1994}, month = mar, - journal = {Astronomy and Astrophysics}, + journal = {\aap}, volume = {283}, pages = {247--262}, issn = {0004-6361}, @@ -4164,7 +4164,7 @@ @article{Mosser.Belkacem.ea2010 author = {Mosser, B. and Belkacem, K. and Goupil, M.-J. and Miglio, A. and Morel, T. and Barban, C. and Baudin, F. and Hekker, S. and Samadi, R. and Ridder, J. De and Weiss, W. and Auvergne, M. and Baglin, A.}, year = {2010}, month = jul, - journal = {A\&A}, + journal = {\aap}, volume = {517}, pages = {A22}, publisher = {{EDP Sciences}}, @@ -4746,7 +4746,7 @@ @article{Rendle.Buldgen.ea2019 author = {Rendle, Ben M and Buldgen, Ga{\"e}l and Miglio, Andrea and Reese, Daniel and Noels, Arlette and Davies, Guy R and Campante, Tiago L and Chaplin, William J and Lund, Mikkel N and Kuszlewicz, James S and Scott, Laura J A and Scuflaire, Richard and Ball, Warrick H and Smetana, Jiri and Nsamba, Benard}, year = {2019}, month = mar, - journal = {Monthly Notices of the Royal Astronomical Society}, + journal = {\mnras}, volume = {484}, number = {1}, pages = {771--786}, @@ -5011,7 +5011,7 @@ @article{Sandquist.Stello.ea2020 author = {Sandquist, Eric L. and Stello, Dennis and Arentoft, Torben and Brogaard, Karsten and Grundahl, Frank and Vanderburg, Andrew and Hedlund, Anne and DeWitt, Ryan and Ackerman, Taylor R. and Aguilar, Miguel and Buckner, Andrew J. and Juarez, Christian and Ortiz, Arturo J. and Richarte, David and Rivera, Daniel I. and Schlapfer, Levi}, year = {2020}, month = mar, - journal = {The Astronomical Journal}, + journal = {\aj}, volume = {159}, pages = {96}, issn = {0004-6256}, @@ -5256,7 +5256,7 @@ @article{SilvaAguirre.Stello.ea2020 author = {Silva Aguirre, V{\'i}ctor and Stello, Dennis and Stokholm, Amalie and Mosumgaard, Jakob R. and Ball, Warrick H. and Basu, Sarbani and Bossini, Diego and Bugnet, Lisa and Buzasi, Derek and Campante, Tiago L. and Carboneau, Lindsey and Chaplin, William J. and Corsaro, Enrico and Davies, Guy R. and Elsworth, Yvonne and Garc{\'i}a, Rafael A. and Gaulme, Patrick and Hall, Oliver J. and Handberg, Rasmus and Hon, Marc and Kallinger, Thomas and Kang, Liu and Lund, Mikkel N. and Mathur, Savita and Mints, Alexey and Mosser, Benoit and Orhan, Zeynep {\c C}elik and Rodrigues, Tha{\'i}se S. and Vrard, Mathieu and Y{\i}ld{\i}z, Mutlu and Zinn, Joel C. and {\"O}rtel, Sibel and Beck, Paul G. and Bell, Keaton J. and Guo, Zhao and Jiang, Chen and Kuszlewicz, James S. and Kuehn, Charles A. and Li, Tanda and Lundkvist, Mia S. and Pinsonneault, Marc and Tayar, Jamie and Cunha, Margarida S. and Hekker, Saskia and Huber, Daniel and Miglio, Andrea and Monteiro, Mario J. P. F. G. and Slumstrup, Ditte and Winther, Mark L. and Angelou, George and Benomar, Othman and B{\'o}di, Attila and Moura, Bruno L. De and Deheuvels, S{\'e}bastien and Derekas, Aliz and Mauro, Maria Pia Di and Dupret, Marc-Antoine and Jim{\'e}nez, Antonio and Lebreton, Yveline and Matthews, Jaymie and Nardetto, Nicolas and do Nascimento, Jose D. and Pereira, Filipe and D{\'i}az, Luisa F. Rodr{\'i}guez and Serenelli, Aldo M. and Spitoni, Emanuele and Stonkut{\.e}, Edita and Su{\'a}rez, Juan Carlos and Szab{\'o}, Robert and Eylen, Vincent Van and Ventura, Rita and Verma, Kuldeep and Weiss, Achim and Wu, Tao and Barclay, Thomas and {Christensen-Dalsgaard}, J{\o}rgen and Jenkins, Jon M. and Kjeldsen, Hans and Ricker, George R. and Seager, Sara and Vanderspek, Roland}, year = {2020}, month = jan, - journal = {ApJL}, + journal = {\apjl}, volume = {889}, number = {2}, pages = {L34}, From 13d8325ab56e790fefe5b968f8886fd2430bbf8c Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Sun, 23 Apr 2023 22:34:30 +0100 Subject: [PATCH 03/50] Update introductory paragraphs --- chapters/introduction.tex | 10 +++--- references.bib | 69 +++++++++++++++++++++++++++++++++++++++ 2 files changed, 74 insertions(+), 5 deletions(-) diff --git a/chapters/introduction.tex b/chapters/introduction.tex index 0977104..06f6a0a 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -16,9 +16,9 @@ \chapter{Introduction} % \citet{Hertzsprung1909} made early estimates of absolute magnitude from the widths of the spectral lines -Early efforts to understand the stars began by finding relations between their spectral classification and brightness in a given photometric band (magnitude) on what was later called a Hertzsprung-Russell (HR) diagram \citep[e.g.][]{Russell1914}. An HR diagram shows the absolute magnitude (or luminosity, \(L\)) of a star against its spectral class (or effective temperature, \(\teff\)). Astronomers found that stars were not uniformly distributed on the HR diagram, but instead lay in distinct sequences. For example, the `main sequence' was named as the region where most stars were found and was later inferred to be where stars spend most of their life. +Early efforts to understand the stars began by finding relations between their spectral classification and magnitude in a given photometric band on what was later called a Hertzsprung-Russell (HR) diagram \citep[e.g.][]{Russell1914}. An HR diagram shows the absolute magnitude (or luminosity, \(L\)) of a star against its spectral class (or effective temperature, \(\teff\)). Astronomers found that stars were not uniformly distributed on the HR diagram, but were instead grouped in distinct sequences. For example, the region where most stars were found was called the \emph{main sequence}. -Initial insight into stellar evolution came about when astronomers ordered open clusters on the HR diagram \needcite. Assuming clusters formed at the same time with similar chemical abundances, the only expected difference between stars were their mass and multiplicity. Using stars of known mass (e.g. from orbital solutions to binary systems), astronomers could trace stars of similar mass from young to old clusters \needcite. This provided an early approximation of a stellar evolutionary track. Deriving the stellar radius (\(R\)) from the relation \(L \propto R^2 \teff^4\), they inferred that stars start on the main sequence and then get brighter and larger before rapidly cooling and ascending the red giant branch. +Initial insight into stellar evolution came about when astronomers studied open clusters on the HR diagram \needcite. These were groups of stars found at a similar distance and close together on the sky. Assuming clusters formed at the same time with similar chemical abundances, the only expected difference between stars were their mass and multiplicity. Using stars of known mass (e.g. from orbital solutions to binary systems), astronomers could trace lines of constant mass from younger to older clusters \needcite. This provided an early approximation of a stellar evolutionary track --- the path a star takes on the HR diagram during its evolution. Deriving the stellar radius (\(R\)) from the relation \(L \propto R^2 \teff^4\), they inferred that stars started on the left-hand edge of the main sequence and became brighter and larger throughout most of their lifetime. At some point, stars would leave the main sequence, rapidly cool and expand, and then ascend a region of the HR diagram known as the \emph{red giant branch}. % \citet{Kuiper1938} found an empirical mass-luminosity relation. @@ -28,11 +28,11 @@ \chapter{Introduction} % The luminosity and effective temperature could be estimated from the magnitude and colour of the stars. From luminosity and temperature, we could derive the radius of stars. Early stellar mass estimates came from visual and spectroscopic binaries. Spectroscopy provides abundances of chemical species ionised in the stellar atmosphere. However, except for the Sun, stellar age and helium abundance has no model-independence. The latter ionisations at temperatures and densities higher than the surface of stars like the Sun. -The question of what stars were made of and how they evolved still remained. Spectroscopy revealed abundances of elements excited in stellar atmospheres, but gave little insight on how these elements were created. \citet{Payne1925} proposed that stars were comprised of mostly hydrogen and helium. The idea was radical at the time, but was later followed up with plausible estimates of solar helium abundance \citep[e.g.][]{Schwarzschild1946}. At the same time, advancements in nuclear science spawned the theory of stellar nucleosynthesis \citep{Hoyle1946}. Stars produced elements heavier than hydrogen and helium through nuclear fusion reactions. This discovery only furthered interest in stellar evolution, because the production of these elements were tied to the formation of planets and origin of life in the universe. +The question of what stars were made of and how they evolved still remained. Spectroscopy revealed relative abundances of elements excited in stellar atmospheres, but determining their absolute abundances was difficult. \citet{Payne1925} proposed that stars were comprised of mostly hydrogen and helium, later reinforced by estimates of helium content in the Sun \citep[e.g.][]{Schwarzschild1946}. The idea was radical at the time because it did not match the abundance of elements found on Earth. Meanwhile, advancements in nuclear science spawned the theory of stellar nucleosynthesis \citep{Hoyle1946}. Stars produced elements heavier than hydrogen and helium through nuclear fusion reactions. This discovery explained the production of many elements in the universe, tying the formation and evolution of stars to planetary formation and the conditions for life in the universe. -In the second half of the 20th century, the theory of stellar evolution advanced. With this came computational methods for simulating stars \citep[e.g.][]{Kippenhahn.Weigert.ea1967}. Astronomers could start to compare observations and empirical relations with simulated stars. We call this process `modelling stars'. +In the second half of the 20th century, the theory of stellar evolution advanced. With this came computational methods for simulating stars \citep[e.g.][]{Kippenhahn.Weigert.ea1967}. Astronomers could start to compare observations and empirical relations with simulated stars. We call this process \emph{modelling stars}. Early models of the Sun benefited from independent age estimates from geology and neutrino-production rates from observations of cosmic rays. With these constraints, and the Sun as a calibrator, researchers could start to refine their models and test them on other stars. \todo{example?} -On a similar timescale, a new field called asteroseismology emerged to help further constrain stellar models. Asteroseismology allowed astronomers to effectively `look' inside stars. While the field spans a wide range of oscillating stars, we choose to focus on stars which oscillate like the Sun. In Section \ref{sec:seismo} we give a brief history and theory of these solar-like oscillators. +On a similar timescale, a new field emerged which gave astronomers a model-independent way of studying the inside of stars. Identifying regular perturbations of the surface of the Sun, researchers realised that they pertained to high-order, stochastically-driven spherical harmonic oscillations. Measuring these oscillation modes, they could study the Sun in a similar way to how seismologists study the Earth. Named asteroseismology, this new field was able to study many non-radial overtones which probed different depths of the star, thus distinguishing itself from the well-know study of radially pulsating stars \citep[e.g. Cepheid variables;][]{Leavitt1908}. In Section \ref{sec:seismo} we give a brief history and theory of the asteroseismology of stars like the Sun. % Approximations for the equation of state by \citet{Eggleton.Faulkner.ea1973}. diff --git a/references.bib b/references.bib index 737de53..d53e46d 100644 --- a/references.bib +++ b/references.bib @@ -150,6 +150,20 @@ @article{Albert2020 keywords = {Astrophysics - Instrumentation and Methods for Astrophysics} } +@article{Alpher.Bethe.ea1948, + title = {The {{Origin}} of {{Chemical Elements}}}, + author = {Alpher, R. A. and Bethe, H. and Gamow, G.}, + year = {1948}, + month = apr, + journal = {Phys. Rev.}, + volume = {73}, + pages = {803--804}, + issn = {1536-6065}, + doi = {10.1103/PhysRev.73.803}, + urldate = {2023-04-23}, + annotation = {ADS Bibcode: 1948PhRv...73..803A} +} + @article{Anderson.Hogg.ea2018, ids = {Anderson.Hogg.ea2018a}, title = {Improving {{Gaia Parallax Precision}} with a {{Data-driven Model}} of {{Stars}}}, @@ -1462,6 +1476,19 @@ @article{Corsaro.DeRidder.ea2015 keywords = {asteroseismology,helium,statistics} } +@article{Cowling1966, + title = {The {{Development}} of the {{Theory}} of {{Stellar Structure}}}, + author = {Cowling, T. G.}, + year = {1966}, + month = jan, + journal = {\qjras}, + volume = {7}, + pages = {121}, + issn = {0035-8738}, + urldate = {2023-04-23}, + annotation = {ADS Bibcode: 1966QJRAS...7..121C} +} + @book{Cox.Giuli1968, title = {Principles of Stellar Structure}, author = {Cox, J. P. and Giuli, R. T.}, @@ -3452,6 +3479,18 @@ @article{Kuszlewicz.Chaplin.ea2019 keywords = {asteroseismology,methods: data analysis,methods: statistical,notion} } +@article{Leavitt1908, + title = {1777 Variables in the {{Magellanic Clouds}}}, + author = {Leavitt, Henrietta S.}, + year = {1908}, + month = jan, + journal = {Ann. Harv. Coll. Obs.}, + volume = {60}, + pages = {87-108.3}, + urldate = {2023-04-23}, + annotation = {ADS Bibcode: 1908AnHar..60...87L} +} + @article{Lebreton.Goupil.ea2014, title = {How Accurate Are Stellar Ages Based on Stellar Models?: {{I}}. {{The}} Impact of Stellar Models Uncertainties}, shorttitle = {How Accurate Are Stellar Ages Based on Stellar Models?}, @@ -5143,6 +5182,21 @@ @article{Scutt.Murphy.ea2023 annotation = {ADS Bibcode: 2023arXiv230211025S} } +@article{Sears1964, + title = {Helium {{Content}} and {{Neutrino Fluxes}} in {{Solar Models}}.}, + author = {Sears, R. L.}, + year = {1964}, + month = aug, + journal = {\apj}, + volume = {140}, + pages = {477}, + issn = {0004-637X}, + doi = {10.1086/147942}, + urldate = {2023-04-23}, + abstract = {A variety of evolutionary sequences of models for the solar interior has been computed, corresponding to variations in input data, to obtain some idea of the uncertainties involved in predicting a solar neutrino flux. It is concluded that the neutrino flux can be estimated to within a factor of 2, the primary uncertainty being the initial homogeneous solar composition; detailed results are given. With a preferred value of the heavy-element-to-hydrogen ratio Z/X = 0.028, the helium content necessary to fit a model to the observed solar luminosity is found to be Y = 0.27.}, + annotation = {ADS Bibcode: 1964ApJ...140..477S} +} + @article{Serenelli.Basu.ea2009, title = {New {{Solar Composition}}: {{The Problem}} with {{Solar Models Revisited}}}, shorttitle = {New {{Solar Composition}}}, @@ -5907,6 +5961,21 @@ @article{Vrard.Mosser.ea2015 keywords = {asteroseismology,helium} } +@article{Wagoner.Fowler.ea1967, + title = {On the {{Synthesis}} of {{Elements}} at {{Very High Temperatures}}}, + author = {Wagoner, Robert V. and Fowler, William A. and Hoyle, F.}, + year = {1967}, + month = apr, + journal = {\apj}, + volume = {148}, + pages = {3}, + issn = {0004-637X}, + doi = {10.1086/149126}, + urldate = {2023-04-23}, + abstract = {A detailed calculation of element production in the early stages of a homogeneous and isotropic expanding universe as well as within imploding-exploding supermassive stars has been made. If the recently measured microwave background radiation is due to primeval photons, then significant quantities of only D, He3, He4, and Li7 can be produced in the universal fireball. Reasonable agreement with solar- system abundances for these nuclei is obtained if the present temperature is 3 K and if the present density is 2 X 10-" gm cm', corresponding to a deceleration parameter qo 5 X 10-'. However, massive stars "bouncing" at temperatures 10 K can convert the universal D and He3 into C, N, 0, Ne, Mg, and some heavier elements in amounts observed in the oldest stars. The mass gaps at A = 5 and 8 are bridged by the reactions He' (He4, )Be7(He4, ) C11. Bounces at higher temperatures bridge the mass gaps through 3 He4 H C12 and mainly produce metals of the iron group, plus a small amount of heavier elements synthesized by a new kind of r-process (rapid neutron capture). It is found that very low abundances of He4, as recently observed in some stars, can be produced in a universe in which the electron neutrinos are degenerate.}, + annotation = {ADS Bibcode: 1967ApJ...148....3W} +} + @article{Weiss.Schlattl2008, title = {{{GARSTEC}}\textemdash the {{Garching Stellar Evolution Code}}. {{The}} Direct Descendant of the Legendary {{Kippenhahn}} Code}, author = {Weiss, Achim and Schlattl, Helmut}, From d3dffca7375c5f8a3a0b83ee67298888a9463658 Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Sun, 23 Apr 2023 22:50:15 +0100 Subject: [PATCH 04/50] Add alternative sans serif titles commented out --- packages.sty | 10 ++++++++++ 1 file changed, 10 insertions(+) diff --git a/packages.sty b/packages.sty index eba83f2..e5c99e0 100644 --- a/packages.sty +++ b/packages.sty @@ -76,17 +76,27 @@ % \titleformat*{\subsubsection}{\sffamily\bfseries} % modify subsubsection headings separately % \titleformat*{\paragraph}{\sffamily\bfseries} +% OLD STYLE % \titleformat{\chapter}[display] % {\sffamily\bfseries\fontsize{64}{72}\selectfont} % Careful with custom font if change size % {\filleft\thechapter} % {20pt} % {\filright\huge} + +% CENTERED NORMAL FONT \titleformat{\chapter}[display] {\normalfont\LARGE\centering} % Careful with custom font if change size {\chaptertitlename\ \thechapter} {20pt} {\MakeUppercase} +% CENTERED SANS SERIF +% \titleformat{\chapter}[display] +% {\sffamily\bfseries\LARGE\centering} % Careful with custom font if change size +% {\chaptertitlename\ \thechapter} +% {20pt} +% {\MakeUppercase} + %% HYPERLINKS % \usepackage{url} % Nice URL typesetting and line breaks From 56d5f32381af07030c8031a964b4f847bdaacb8a Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Mon, 24 Apr 2023 13:29:03 +0100 Subject: [PATCH 05/50] Chnage sans serif font --- packages.sty | 11 ++++++++++- 1 file changed, 10 insertions(+), 1 deletion(-) diff --git a/packages.sty b/packages.sty index e5c99e0..f98aee4 100644 --- a/packages.sty +++ b/packages.sty @@ -28,10 +28,12 @@ % Serif and math fonts \usepackage{newtxtext,newtxmath} % Sans-serif font -\usepackage[scaled=0.95]{FiraSans} +% \usepackage[scaled=0.95,book]{FiraSans} +% \renewcommand*\oldstylenums[1]{{\firaoldstyle #1}} % \usepackage[oldstyle,scale=0.95]{opensans} % \usepackage{roboto} % \usepackage{noto} +\usepackage[scaled=0.9]{inter} \usepackage{csquotes} % When loading babel with biblatex \usepackage{anyfontsize} % For custom font sizes @@ -97,6 +99,13 @@ % {20pt} % {\MakeUppercase} +% CENTERED NORMAL FONT + SANS +% \titleformat{\chapter}[display] +% {\sffamily\bfseries\LARGE\centering} % Careful with custom font if change size +% {\chaptertitlename\ \thechapter} +% {20pt} +% {\MakeUppercase} + %% HYPERLINKS % \usepackage{url} % Nice URL typesetting and line breaks From bce38c5b0565ed3fb55e3089e61e198d3cc978ac Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Mon, 24 Apr 2023 13:29:15 +0100 Subject: [PATCH 06/50] Add comment for sans serif title --- uobthesis.cls | 2 ++ 1 file changed, 2 insertions(+) diff --git a/uobthesis.cls b/uobthesis.cls index 482a639..217343f 100644 --- a/uobthesis.cls +++ b/uobthesis.cls @@ -71,12 +71,14 @@ \vfill\LARGE% \MakeUppercase{\@title}% + % {\sffamily\bfseries\MakeUppercase{\@title}}% \vspace{0.5cm}\normalsize% by\LARGE%% Immediately switch back to large before new line \vspace{0.5cm}% \MakeUppercase{\@author}% + % {\sffamily\bfseries\MakeUppercase{\@author}}% \vfill\normalsize% A thesis submitted to the University of Birmingham for the degree of\\ From 6b6e239205aaa7c278555d22ce19d583ae78744b Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Mon, 24 Apr 2023 13:29:34 +0100 Subject: [PATCH 07/50] Remove line numbers --- thesis.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/thesis.tex b/thesis.tex index 98667f2..fbb01f0 100644 --- a/thesis.tex +++ b/thesis.tex @@ -21,7 +21,7 @@ % Packages and setup are in packages.sty % Use 'print' option for print version (reduced color) % Use 'editor' option for line numbers -\usepackage[editor]{packages} +\usepackage{packages} % AAS macro package \usepackage{aas_macros} From e706021d7a234b2157760025bc04e52b0e959004 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Mon, 24 Apr 2023 13:29:50 +0100 Subject: [PATCH 08/50] Add chapter intro text --- chapters/conclusion.tex | 6 +++++- chapters/glitch-gp.tex | 4 ++++ chapters/glitch.tex | 3 +++ chapters/hbm.tex | 4 ++++ chapters/introduction.tex | 6 +++++- 5 files changed, 21 insertions(+), 2 deletions(-) diff --git a/chapters/conclusion.tex b/chapters/conclusion.tex index 3fde0db..fd3515c 100644 --- a/chapters/conclusion.tex +++ b/chapters/conclusion.tex @@ -10,7 +10,11 @@ % all distributions of LaTeX version 2005/12/01 or later. % % -\chapter{Conclusion} +\chapter{Conclusion and Future Prospects} + +\textit{We conclude this thesis by providing a summary of the work herein and reflect upon key results. Then, we consider possible improvements to our hierarchical model and method for emulating stellar simulations. Finally, we discuss future prospects for applying our method to data from current and upcoming missions.} + +\section{Thesis Summary} We demonstrated that a hierarchical Bayesian model (HBM) can improve the inference of stellar radii, masses and ages. Introducing the concept of an HBM in Chapter \ref{chap:hbm}, we showed how we can include population-level distributions to inform star-level parameter in a Bayesian model. Then, we applied an HBM to model a well-studied sample of oscillating dwarf and subgiant stars in Chapter \ref{chap:hmd}. While accounting for uncertainty in helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)), we were still able to achieve statistical uncertainties of 1.2 per cent in radius, 2.5 per cent in mass and 12 per cent in age. This provided a framework for modelling large populations of stars at the same time and making the most out of noisy data. diff --git a/chapters/glitch-gp.tex b/chapters/glitch-gp.tex index 8375fb0..78a5eef 100644 --- a/chapters/glitch-gp.tex +++ b/chapters/glitch-gp.tex @@ -12,6 +12,10 @@ % \chapter[Modelling Acoustic Glitches with a Gaussian Process]{Modelling Acoustic Glitch Signatures in Stellar Oscillations with a Gaussian Process}\label{chap:glitch-gp} +\textit{In this chapter, we apply a new method for modelling acoustic glitch signatures in the radial mode frequencies of solar-like oscillators. We compare our method with another using a model star with different levels of noise. Then, we apply both methods to the star 16 Cyg A to provide a real-world working example. We show that our method can be used to find the strength and location of glitches caused by the second ionisation of helium and the base of the convective zone. We also demonstrate that our method improves modelling the smoothly varying component of the mode frequencies.} + +\section{Introduction} + Let us consider a non-rotating star which oscillates at frequencies \(\nu_{nl}\), where \(n\) and \(l\) are the radial order and angular degree of the modes. We may model the modes as the sum of a smoothly-varying component, \(\tilde{f}(n, l)\), and quickly-varying, small change in frequency, \(\delta\nu\), arising from glitches in the stellar structure, % \begin{equation} diff --git a/chapters/glitch.tex b/chapters/glitch.tex index c7dca8c..a59b1a5 100644 --- a/chapters/glitch.tex +++ b/chapters/glitch.tex @@ -12,6 +12,9 @@ % \chapter[Acoustic Glitches in Solar-Like Oscillators]{Acoustic Glitches in Solar-Like Oscillators as a Signature of Helium Abundance}\label{chap:glitch} +\textit{Having demonstrated a hierarchical model over initial stellar helium abundance, we explore an asteroseismic signature of helium abundance which could provide more observational constraint. In this chapter, we introduce the concept of a glitch in the structure of a star producing a measurable signal in its observed oscillation modes. We start with a simple one-dimensional example in Section \ref{sec:1d-glitch}. Then, we introduce glitches due to helium ionisation and the base of the convective zone in solar-like oscillators. We provide background on the glitches and their effect on stellar oscillation modes in advance of Chapter \ref{chap:glitch-gp}, where we apply a novel method for modelling these glitch signatures.} + +\section{Introduction} % \epigraph{\singlespacing``Ideals are like stars: you will not succeed in touching them with your hands, but like the seafaring man on the ocean desert of waters, you choose them as your guides, and following them, you reach your destiny.''}{\emph{Carl Schurz}} % So far in this thesis, we have shown that a hierarchical Bayesian model can be used to infer the helium abundance distribution in a stellar population. We also found how this improves the inference of fundamental stellar parameters. However, there is limited information about helium abundance in the stellar observables used (e.g. \(L, \teff, \Delta\nu\)). diff --git a/chapters/hbm.tex b/chapters/hbm.tex index c931079..8b1e480 100644 --- a/chapters/hbm.tex +++ b/chapters/hbm.tex @@ -12,6 +12,10 @@ % \chapter{Hierarchical Bayesian Models}\label{chap:hbm} +\textit{In this chapter, we introduce the concept of a hierarchical Bayesian model in the context of determining stellar parameters. We use a simplified analogy for measuring distances to stars in an open cluster to demonstrate the advantages and usage of a hierarchical model in Section \ref{sec:hbm-dist}. Finally, we identify parameters which could be treated hierarchically when modelling the physical properties of many stars, priming the reader for Chapter \ref{chap:hmd}.} + +\section{Introduction} + When modelling a single star, we can use a variant of Bayes' theorem (Equation \ref{eq:bayes}) to estimate the posterior probability of its parameters. We choose priors to represent our current knowledge. For example, if we observe a star in the Milky Way at random, we expect its mass to belong to an Initial Mass Function \citep[IMF; e.g.][]{Chabrier2003} without other prior knowledge. In fact, an IMF prior is an input option for BASTA \citep{AguirreBorsen-Koch.Rorsted.ea2022}. Other prior assumptions include a helium enrichment law which determines the star's initial helium abundance from its metallicity, or that its age must not be older than the universe. Let us consider modelling a large population of stars simultaneously. For example, we may want to create a catalogue of stellar parameters to use in exoplanet and galactic research. We could treat the parameters for each star independently, repeating the modelling procedure for each star in the sample. However, we know that stars belong to population distributions like the IMF. In some cases, these population distributions are not well understood and assuming one exactly can introduce bias into our model. Instead, it could be better to let the data inform such population priors. diff --git a/chapters/introduction.tex b/chapters/introduction.tex index 06f6a0a..54255d7 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -10,12 +10,16 @@ % all distributions of LaTeX version 2005/12/01 or later. % % -\chapter{Introduction} +\chapter{Introduction to Modelling Stars with Asteroseismology} + +\textit{In this chapter, we introduce the current state of modelling stars with asteroseismology and the types of stars being studied in this work. We start with a brief history of understanding the stars spanning the last century. In Section \ref{sec:seismo}, we introduce asteroseismology of stars which oscillate like the Sun. Then, we provide examples of asteroseismology being used to model large samples of dwarf and subgiant stars in Section \ref{sec:many-stars}. Finally, we introduce the concept of modelling stars the `Bayesian way' with some examples of current methods and their limitations.} % Since the late 19th century, astronomers have been trying to understand the physical structure and evolution of the Sun and other stars. They wanted to determine whether other stars were like the Sun and whether they changed with time. Answers to these questions could tell us where we came from and what the future holds for the solar system. While little was known of this at the time, astronomers started by gathering data in an effort to map the night sky. The invention of the spectrograph by \citet{Draper1874} allowed researchers to systematically classify stars by their brightness in different wavelengths of light \citep{Maury.Pickering1897}. This was an pivotal early step towards the large-scale stellar surveys we are used to today. % \citet{Hertzsprung1909} made early estimates of absolute magnitude from the widths of the spectral lines +\section{Understanding the Stars}\label{sec:stars} + Early efforts to understand the stars began by finding relations between their spectral classification and magnitude in a given photometric band on what was later called a Hertzsprung-Russell (HR) diagram \citep[e.g.][]{Russell1914}. An HR diagram shows the absolute magnitude (or luminosity, \(L\)) of a star against its spectral class (or effective temperature, \(\teff\)). Astronomers found that stars were not uniformly distributed on the HR diagram, but were instead grouped in distinct sequences. For example, the region where most stars were found was called the \emph{main sequence}. Initial insight into stellar evolution came about when astronomers studied open clusters on the HR diagram \needcite. These were groups of stars found at a similar distance and close together on the sky. Assuming clusters formed at the same time with similar chemical abundances, the only expected difference between stars were their mass and multiplicity. Using stars of known mass (e.g. from orbital solutions to binary systems), astronomers could trace lines of constant mass from younger to older clusters \needcite. This provided an early approximation of a stellar evolutionary track --- the path a star takes on the HR diagram during its evolution. Deriving the stellar radius (\(R\)) from the relation \(L \propto R^2 \teff^4\), they inferred that stars started on the left-hand edge of the main sequence and became brighter and larger throughout most of their lifetime. At some point, stars would leave the main sequence, rapidly cool and expand, and then ascend a region of the HR diagram known as the \emph{red giant branch}. From 8566527fa9cc835512c2ff8ee77e4be8c55d7fb7 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Mon, 24 Apr 2023 14:23:20 +0100 Subject: [PATCH 09/50] Fix seismo section --- chapters/introduction.tex | 28 ++++++++++++++-------------- 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/chapters/introduction.tex b/chapters/introduction.tex index 54255d7..164121c 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -58,7 +58,7 @@ \subsection{A Brief History of Asteroseismology} Several decades ago, 5-minute oscillations in the radial velocity of the solar surface were observed by \citet{Leighton.Noyes.ea1962}, leading to the inference of acoustic waves trapped beneath the solar photosphere \citep{Ulrich1970}. A further decade of study culminated in the measurement of regular patterns of individual oscillation modes in the Doppler radial velocity \citep{Claverie.Isaak.ea1979} and total irradiance \citep{Woodard.Hudson1983a} of the Sun. Initially thought to be short-lived irregularities on the surface, these modes were found to be compatible with stochastically excited standing waves penetrating deep into the Sun. Later, \citet{Deubner.Gough1984} introduced the word \emph{helioseismology} (analogous to geo-seismology) to describe the study of the solar interior using observations of these modes. Helioseismology was soon responsible for breakthrough solar research, from measuring differential rotation \citep{Deubner.Ulrich.ea1979} to solving the mismatch between predicted and measured solar neutrino production \citep{Bahcall.Ulrich1988}. -Astronomers initially debated the mechanism driving solar oscillations in the form of standing pressure waves (or \emph{p modes}). \citet{Goldreich.Keeley1977} suggested what became the prevailing theory, that the p modes were stochastically excited by near-surface convection. Hence, we might expect solar-like oscillations to be present in other stars which have a convective envelope similar to the Sun. Shortly thereafter, \citet{Christensen-Dalsgaard1984} introduced the term \emph{asteroseismology} --- the study of the internal structure of stars with many observable modes of oscillation. Subsequently, solar-like oscillations were discovered in a few bright stars. Among the first were Procyon and \(\alpha\) Cen A \citep{Gelly.Grec.ea1986}, with individual modes later resolved by \citet{Martic.Schmitt.ea1999} and \citet{Bouchy.Carrier2001} respectively. +Astronomers initially debated the mechanism driving the modes of standing pressure waves (or \emph{p modes}) in the Sun. \citet{Goldreich.Keeley1977} suggested what became the prevailing theory, that the p modes were stochastically excited by near-surface convection. Hence, we might expect solar-like oscillations to be present in other stars which have a convective envelope similar to the Sun. Shortly thereafter, \citet{Christensen-Dalsgaard1984} introduced the term \emph{asteroseismology} --- the study of the internal structure of stars with many observable modes of oscillation. Subsequently, solar-like oscillations were discovered in a few bright stars. Among the first were Procyon and \(\alpha\) Cen A \citep{Gelly.Grec.ea1986}, with individual modes later resolved by \citet{Martic.Schmitt.ea1999} and \citet{Bouchy.Carrier2001} respectively. Instrumental and atmospheric noise limited the progress of asteroseismology with ground-based equipment to studies of small number of bright dwarf stars. Asteroseismology requires high-cadence (\(\sim \SIrange{1}{10}{\minute}\)) brightness observations over long time-periods (\(\sim \SI{1}{\year}\)) with precisions of \todo{precision}. The first dedicated space-based missions which met these requirements arrived in the late 2000s, accelerating progress in the field. Initially, the \emph{CoRoT} mission \citep{Baglin.Auvergne.ea2006} detected solar-like oscillations in thousands of red giant stars \citep{DeRidder.Barban.ea2009,Mosser.Belkacem.ea2010}. Then, the \emph{Kepler} mission \citep{Borucki.Koch.ea2010} yielded oscillations in thousands more red giants \citep{Pinsonneault.Elsworth.ea2014} and hundreds of main sequence stars similar to the Sun \citep{Serenelli.Johnson.ea2017}. Most recently, \emph{TESS} \citep{Ricker.Winn.ea2015} added thousands more dwarf and giant stars to the roster of solar-like oscillators \citep{Hon.Huber.ea2021,SilvaAguirre.Stello.ea2020,Hatt.Nielsen.ea2023}. @@ -71,7 +71,7 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \label{fig:spherical-harmonics} \end{figure} -Oscillations on the surface of a star can be approximated by spherical harmonic functions with angular degree \(l\) and azimuthal order \(m\). The angular degree is the number of nodes on the surface of the star. We show a representation of the surface spherical harmonics for the first four angular degrees in Figure \ref{fig:spherical-harmonics}. For each \(l\), there exists \(2l+1\) solutions with different azimuthal order (\(m\)) corresponding to the different orientations of the nodes over the spherical surface. Additionally, the oscillation modes have unique frequency solutions for different radial orders (\(n\)), proportional to the number of wave nodes radially throughout the star. +Oscillations on the surface of a star can be approximated by spherical harmonic functions with angular degree \(l\) and azimuthal order \(m\). The angular degree is the number of nodes on the surface of the star. We show a representation of the surface spherical harmonics for the first four angular degrees in Figure \ref{fig:spherical-harmonics}. For each \(l\), there exists \(2l+1\) solutions with different azimuthal order (\(m\)) corresponding to the different orientations of the nodes over the spherical surface. Additionally, the oscillation modes have unique frequency solutions for different radial orders (\(n\)), proportional to the number of nodes radially throughout the star. \begin{figure}[tb] \centering @@ -82,25 +82,25 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} In solar-like oscillators, p modes are stochastically excited by near-surface convection. Typically, the timescale of this process drives high-order modes in main sequence stars \needcite[\(n \sim 20\)]. We can identify these modes in a frequency-power spectrum derived from photometric or radial velocity time series observations. For instance, both stars in the 16 Cyg system are solar-like oscillators with similar properties to the Sun \needcite. Using 16 Cyg A as an example, we downloaded the power spectrum determined by the \emph{Kepler} Asteroseismic Science Operations Centre (KASOC) using \emph{Kepler} observations\footnote{\url{https://kasoc.phys.au.dk}}. Shown in Figure \ref{fig:seismo-psd}, the power spectrum of 16 Cyg A has a distinct power excess around \SI{2000}{\micro\hertz}. -We call the location of the peak in power excess the `frequency at maximum power', \(\numax\). Dependent on near-surface conditions, \citet{Brown.Gilliland.ea1991} suggested \(\numax\) scales with the acoustic cut-off frequency --- the highest frequency at which acoustic waves can reflect near the stellar surface. Subsequently, \citet{Kjeldsen.Bedding1995} found that \(\numax \propto g\teff^{\,-1/2}\) where \(g\) and \(\teff\) are the near-surface gravitational field strength and temperature. The power excess also has a Gaussian-like shape around \(\numax\). We expect this shape to come from the distribution in convection timescale near the surface responsible for mode excitation \needcite. +The power excess has a Gaussian-like shape around frequencies compatible with the near-surface convective timescale responsible for mode excitation. We call the location of this Gaussian the `frequency at maximum power', \(\numax\). Dependent on near-surface conditions, \citet{Brown.Gilliland.ea1991} suggested \(\numax\) scales with the acoustic cut-off frequency --- the highest frequency at which acoustic waves can reflect near the stellar surface. Subsequently, \citet{Kjeldsen.Bedding1995} found that \(\numax \propto g\teff^{\,-1/2}\) where \(g\) and \(\teff\) are the near-surface gravitational field strength and temperature. -Looking closely at the power excess in Figure \ref{fig:seismo-psd}, we can see a comb of approximately equally spaced peaks. Each peak corresponds to one or more oscillation modes, with its central frequency and shape providing information about the internal stellar structure. Naturally, higher frequency modes correspond to higher \(n\). However, the angular degree and azimuthal order are harder to identify. We saw in Figure \ref{fig:spherical-harmonics} how modes of higher \(l\) have more anti-nodes on the surface. Therefore, the overall effect of the oscillations cancel out when integrating over the observed surface. Consequentially, observed mode amplitude decreases with \(l\) when observing total stellar irradiance, leaving only \(l \lesssim 3\) detectable \needcite. With this assumption, we can assume the tallest peaks are \(l=0,1\), and the smaller peaks are \(l=2,3\), all modulated by the wider Gaussian-like envelope. +Looking closely at the power excess in Figure \ref{fig:seismo-psd}, we can see a comb of approximately equally spaced peaks. Each peak corresponds to one or more oscillation modes, with their central frequencies and frequency differences providing information about the internal stellar structure. Naturally, higher frequency modes correspond to higher \(n\). However, the angular degree and azimuthal order are harder to identify. We saw in Figure \ref{fig:spherical-harmonics} how modes of higher \(l\) have more anti-nodes on the surface. Therefore, the overall effect of the oscillations cancel out when measuring irradiance integrated over the stellar surface. Consequentially, observed mode amplitude decreases with \(l\), leaving only \(l \lesssim 3\) detectable \needcite. From this, we can assume the tallest peaks are \(l=0,1\), and the smaller peaks are \(l=2,3\), all modulated by the wider Gaussian-like envelope. -As an aside, the observed mode frequencies will split for different \(m\) via the Doppler effect in the case of a rotating star. Measuring this splitting can constrain the rotation rate of the star. This has lead to breakthrough studies into gyrochronology and\dots \citep[e.g.][]{Hall.Davies.ea2021}. However, we will hereafter consider the case of a slowly rotating, spherically symmetric star, such that solutions of different \(m\) are approximately the same frequency. +For a spherically symmetric, non-rotating star, modes with different \(m\) oscillate at the same frequency and cannot be distinguished. However, the observed mode frequencies will split for different \(m\) via the Doppler effect in the case of a rotating or distorted (asymmetric) star. Measuring this splitting can constrain the rotation rate of the star. This has lead to breakthrough studies into gyrochronology and\dots \citep[e.g.][]{Hall.Davies.ea2021}. Hereafter, we will consider only the case of a slowly rotating, spherically symmetric star. -If we consider an acoustic wave in a one-dimensional homogeneous medium, then we would expect each mode of oscillation to be an integer multiple of the fundamental mode. While the case for a star is more complicated, we can also approximate the frequencies for different modes as a multiple of some characteristic frequency. \citet{Tassoul1980} found that the modes could be approximated by assuming the asymptotic limit where \(l/n \rightarrow 0\), giving the following expression \citep[cf.][]{Gough1986}, +If we consider an acoustic wave in a one-dimensional homogeneous medium, then we would expect each mode of oscillation to be an integer multiple of the fundamental mode. While the case for a star is more complicated, we can also approximate the frequencies for different modes as a multiple of a characteristic frequency. \citet{Tassoul1980} found that the modes could be approximated by assuming the asymptotic limit where \(l/n \rightarrow 0\), giving the following expression \citep[cf.][]{Gough1986}, % \begin{equation} \nu_{nl} \simeq \left(n + \frac{l}{2} + \varepsilon\right) \nu_0 + O(\nu_{nl}^{-1}), \label{eq:asy} \end{equation} % -where \(\varepsilon\) is some constant offset and \(O\) represents higher order terms. The characteristic frequency, \(\nu_0\), is the inverse of the acoustic diameter, +where \(\varepsilon\) is some constant offset and \(O(\nu_{nl}^{-1})\) represents higher order terms. The characteristic frequency, \(\nu_0\), is the inverse of the acoustic diameter, % \begin{equation} \nu_0 = \left(2 \int_{0}^{R} \frac{\dd r}{c(r)}\right)^{-1}, \end{equation} % -where \(c(r)\) is the sound speed as a function of radius, \(r\), and \(R\) is the stellar radius. Similarly to other variable stars, \citet{Ulrich1986} found that this characteristic frequency relates to the mean density by \(\nu_0 \propto \overline{\rho}^{\,1/2}\). While \(\nu_0\) is not directly detectable in solar-like oscillators, we can approximate it by taking the difference between consecutive modes of the same angular degree, \(\Delta\nu_{nl} = \nu_{nl} - \nu_{n-1\,l}\). Thus, estimates of a global (or average) large frequency separation, \(\Delta\nu \simeq \nu_0\), can provide information about the density of a star, leading to independent constraint on its mass and radius. +where \(c(r)\) is the sound speed as a function of radius (\(r\)) and \(R\) is the radius of the star. Similarly to other variable stars, \citet{Ulrich1986} found that this characteristic frequency relates to the mean density by \(\nu_0 \propto \overline{\rho}^{\,1/2}\). While \(\nu_0\) is not directly detectable in solar-like oscillators, we can approximate it by taking the difference between consecutive modes of the same angular degree, \(\Delta\nu_{nl} = \nu_{nl} - \nu_{n-1\,l}\). Thus, estimates of a global (or average) large frequency separation, \(\Delta\nu \simeq \nu_0\), can provide information about the density of a star, leading to independent constraint on its mass and radius \needcite. \begin{figure}[tb] \centering @@ -109,7 +109,7 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \label{fig:seismo-echelle} \end{figure} -The asymptotic expression helps us identify modes in a star. If the first term of Equation \ref{eq:asy} was exact, we would expect odd and even modes to be grouped together and separated by \(\dnu/2\). To see this, we revisit the power spectrum of 16 Cyg A, now with an estimate of the noise divided out. We can see the regular pattern predicted by Equation \ref{eq:asy} in the left panel of Figure \ref{fig:seismo-echelle}. Every other mode is approximately separated by \(\dnu\). To see this effect over the wider spectrum, we created an \emph{echelle} plot in the right panel. Folding the spectrum by an estimate of \(\dnu\) reveals a sequence of ridges corresponding to modes of different angular degree. Odd and even angular degree are grouped together, although do not lie on top of each other. The small difference between modes of different \(l\) is described by the higher order terms neglected from Equation \ref{eq:asy}. A faint ridge corresponding to \(l=3\) modes is also visible next to the \(l=1\) ridge. However, 16 Cyg A represents one of the highest signal-to-noise dwarf stars observed by \emph{Kepler}, so the \(l=3\) ridge is usually not visible. +The asymptotic expression helps us identify modes in a star. If the first term of Equation \ref{eq:asy} was exact, we would expect odd and even modes to be grouped together and separated by \(\dnu/2\). To show this, we revisit the power spectrum of 16 Cyg A and estimate a signal-to-noise ratio (SNR) by dividing out a moving median in steps of \SI{0.005}{\dex}. We can see the regular pattern predicted by Equation \ref{eq:asy} in the left panel of Figure \ref{fig:seismo-echelle}. Every other mode is approximately separated by \(\dnu\). To see this effect over the wider spectrum, we created an \emph{echelle} plot in the right panel. Folding the spectrum by an estimate of \(\dnu\) reveals a sequence of ridges corresponding to modes of different angular degree. Odd and even angular degree are grouped together, although do not lie on top of each other. The small difference between modes of different \(l\) is described by the higher order terms neglected from Equation \ref{eq:asy}. A faint ridge corresponding to \(l=3\) modes is also visible next to the \(l=1\) ridge. However, 16 Cyg A represents one of the highest SNR dwarf stars observed by \emph{Kepler}, and the \(l=3\) ridge is otherwise not usually visible. % Once we have identified a solar-like oscillator, what information is there to gain from asteroseismology? We have discussed how parameters \(\numax\) and \(\dnu\) scale with global stellar properties. Scaling these parameters with respect to the Sun, we can obtain relations for the radius and mass of the star, % % @@ -126,11 +126,11 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \section[Modelling Stars with Asteroseismology]{Modelling Many Stars with Asteroseismology}\label{sec:many-stars} -Recent space-based missions like \emph{Kepler} and \emph{TESS} enable asteroseismology with many stars. They allow astronomers to study regions of the HR diagram in more detail, from main sequence dwarf stars to helium burning red clump stars. We draw particular attention to \emph{Kepler}, the first of the two missions. \emph{Kepler}'s primary mission observed a single patch of sky for around 4 years \citep{Borucki.Koch.ea2010}, cut short by a mechanical failure which spawned the \emph{K2} secondary mission \citep{Howell.Sobeck.ea2014}. The mission took photometric measurements in short (\SI{60}{\second}) and long (\SI{30}{\minute}) cadences. During the primary mission, large asteroseismic datasets were gathered. The long baseline of 4 years allowed for high frequency precision not yet seen by \emph{K2} or \emph{TESS}. +Recent space-based missions like \emph{Kepler} and \emph{TESS} enable asteroseismology with many stars. Like \emph{CoRoT}, these missions were primarily designed to detect exoplanets via the transit method. However, their instrumental requirements were also well-suited to asteroseismology. This allowed astronomers to study regions of the HR diagram in more detail, from main sequence dwarf stars red giants. We draw particular attention to \emph{Kepler}, the first of the two missions. \emph{Kepler}'s primary mission observed a single \SI{15}{\degree} diameter patch of sky for around 4 years \citep{Borucki.Koch.ea2010}, cut short by a mechanical failure which spawned the secondary \emph{K2} mission \citep{Howell.Sobeck.ea2014}. The mission took photometric measurements in short (\SI{60}{\second}) and long (\SI{30}{\minute}) cadences. During the primary mission, large asteroseismic datasets were gathered. The long baseline of 4 years allowed for high frequency precision not yet seen by \emph{K2} or \emph{TESS}. % In this section, we introduce the sample of stars being studied in this thesis relative to the broader astronomical picture. -Of the asteroseismic targets found by \emph{Kepler}, we focus on dwarf and subgiant solar-like oscillators. These are stars with a similar mass to the Sun either on the main sequence or post-main sequence before reaching the red giant branch. From an asteroseismic perspective, the power spectra of these stars are relatively simple. As stars approach the red giant branch, their modes can `mix' with gravity-modes in the core, leading to irregular patterns which are difficult to model and identify \needcite. Hence, these simple asteroseismic stars provide a good, consistent place to start modelling populations of stars. +Of the asteroseismic targets found by \emph{Kepler}, we focus on dwarf and subgiant solar-like oscillators. These are stars with a similar mass to the Sun either on the main sequence or post-main sequence before reaching the red giant branch. From an asteroseismic perspective, the power spectra of these stars are relatively simple. As they approach the red giant branch, their modes begin to couple with buoyancy-driven modes in the core, leading to irregular patterns which are comparably difficult to identify and model \needcite. Hence, these simple asteroseismic stars provide a good, consistent place to start modelling populations of stars. Furthermore, the majority of exoplanet host stars have so far been found around dwarf stars, making these solar-like oscillators a target for large-scale stellar characterisation. % Modelling many stars with asteroseismology is complemented by other recent large-scale stellar surveys. High-precision astrometry from the \emph{Gaia} mission \citep{GaiaCollaboration.Prusti.ea2016} has provided improved distances and orbital solutions. The APOGEE large-scale spectroscopic survey \citep{Majewski.Schiavon.ea2017} has also yielded precise chemical abundances. These surveys have enabled studies of the assembly history of our galaxy. For example, \citet{Helmi.Babusiaux.ea2018} discovered a merger between the Milky Way and Gaia-Enceladus by analysing the motions and abundances from \emph{Gaia} and APOGEE. Asteroseismology has accompanied this work by helping determine the ages of stars tied to the merger \citep{Chaplin.Serenelli.ea2020,Montalban.Mackereth.ea2021}. @@ -148,11 +148,11 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \label{fig:hr-diagram} \end{figure} -In Figure \ref{fig:hr-diagram}, we show a color-magnitude diagram made using magnitudes and parallaxes from \emph{Gaia} Data Release 3 \citep[DR3;][]{GaiaCollaboration.Vallenari.ea2022}. For this illustrative plot, we have neglected effects from extinction. The background distribution shows solar-neighbourhood \emph{Gaia} sources with a parallax greater than \SI{5}{\milli\aarcsec} for context. Over-plot is the distribution of \emph{Kepler} objects cross-matched with \emph{Gaia} DR3\footnote{\url{https://gaia-kepler.fun}}. The densest region lies in the low- to intermediate-mass main sequence (\SIrange{0.8}{1.2}{\solarmass}) but we can also see a clear red giant branch and red clump. The inset plot draws attention to the region occupied by dwarf and subgiant solar-like oscillators where we give some examples. +In Figure \ref{fig:hr-diagram}, we show a color-magnitude diagram made using magnitudes and parallaxes from \emph{Gaia} Data Release 3 \citep[DR3;][]{GaiaCollaboration.Vallenari.ea2022}. For this illustrative plot, we have neglected the effect of extinction. The background distribution shows solar-neighbourhood \emph{Gaia} sources with a parallax greater than \SI{5}{\milli\aarcsec} for context. Over-plot is the distribution of \emph{Kepler} objects cross-matched with \emph{Gaia} DR3\footnote{The cross-matched dataset was obtained from \url{https://gaia-kepler.fun}}. The densest region lies in the low- to intermediate-mass main sequence (\SIrange{0.8}{1.2}{\solarmass}) but we can also see a clear red giant branch and red clump in the upper-right of the distribution. The inset plot draws attention to the region occupied by dwarf and subgiant solar-like oscillators, where we give some examples. -\citet{Chaplin.Kjeldsen.ea2011} identified the first catalogue of \(\sim 500\) dwarf and subgiant solar-like oscillators (black circles in Figure \ref{fig:hr-diagram}) by measuring \(\dnu\) and \(\numax\) in \emph{Kepler} data. Later, \citet{Chaplin.Basu.ea2014} determined ages, masses and radii for these stars using \(\dnu\) and \(\numax\) complemented by photometry and ground-based spectroscopy where available. The subsequent arrival of APOGEE spectroscopy allowed \citet{Serenelli.Johnson.ea2017} to revisit this sample with a more consistent set of \(\teff\) and metallicity. By comparing observations to models of stellar evolution, they found radii, masses and ages with uncertainties of around 3, 5 and 20 per cent respectively. +\citet{Chaplin.Kjeldsen.ea2011} identified the first large catalogue of \(\sim 500\) dwarf and subgiant solar-like oscillators (black circles in Figure \ref{fig:hr-diagram}) by measuring \(\dnu\) and \(\numax\) in \emph{Kepler} data. Later, \citet{Chaplin.Basu.ea2014} determined ages, masses and radii for these stars using \(\dnu\) and \(\numax\) complemented by photometry and ground-based spectroscopy where available. The subsequent arrival of APOGEE spectroscopy allowed \citet{Serenelli.Johnson.ea2017} to revisit this sample with a more consistent set of \(\teff\) and metallicity. By comparing observations to models of stellar evolution, they found radii, masses and ages with uncertainties of around 3, 5 and 20 per cent respectively. -For a subset of these stars, the SNR was low enough to identify many individual modes. \citet{Appourchaux.Chaplin.ea2012} were among the first to publish individual mode frequencies (\(\nu_{nl}\)) for around 60 stars. These were later modelled by \citet{Metcalfe.Creevey.ea2014} who found modelling individual modes doubled precision of radii, masses and ages over using \(\dnu\) and \(\numax\) alone. Around the same time, the number of confirmed exoplanets was increasing rapidly \needcite. This motivated a more detailed study of 35 exoplanet host stars by \citet{SilvaAguirre.Davies.ea2015} with modes identified by \citet{Davies.SilvaAguirre.ea2016}. We can see these as yellow triangles in Figure \ref{fig:hr-diagram}. The remaining best targets, referred to as the LEGACY sample, were modelled by \citet{SilvaAguirre.Lund.ea2017} using modes identified by \citet{Lund.SilvaAguirre.ea2017}. We show these as magenta squares in Figure \ref{fig:hr-diagram}. +For a subset of these stars, the SNR was high enough to identify many individual modes. \citet{Appourchaux.Chaplin.ea2012} were among the first to publish individual mode frequencies (\(\nu_{nl}\)) for around 60 of these stars. This sample was later modelled by \citet{Metcalfe.Creevey.ea2014} who found observing individual modes doubled precision of radii, masses and ages over using \(\dnu\) and \(\numax\) alone. Around the same time, the number of confirmed exoplanets was increasing rapidly \needcite. This motivated a more detailed study of 35 exoplanet host stars by \citet{SilvaAguirre.Davies.ea2015} with modes identified by \citet{Davies.SilvaAguirre.ea2016}. We can see these as yellow triangles in Figure \ref{fig:hr-diagram}. The remaining best targets, referred to as the LEGACY sample, were modelled by \citet{SilvaAguirre.Lund.ea2017} using modes identified by \citet{Lund.SilvaAguirre.ea2017}. We show these as magenta squares in Figure \ref{fig:hr-diagram}. % These targets are still being studied extensively today helium glitch \citet{Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2019}. From 02d786da301e04cc1665e383a73bed231e5fbf39 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Mon, 24 Apr 2023 15:24:27 +0100 Subject: [PATCH 10/50] Add Daft reference --- chapters/lyttle21.tex | 2 +- references.bib | 9 +++++++++ 2 files changed, 10 insertions(+), 1 deletion(-) diff --git a/chapters/lyttle21.tex b/chapters/lyttle21.tex index e33541c..fe7db74 100644 --- a/chapters/lyttle21.tex +++ b/chapters/lyttle21.tex @@ -202,7 +202,7 @@ \subsubsection{No-Pooled (NP) Model}\label{sec:np} \begin{figure}[tb] \centering \includegraphics{figures/partial_pool_pgm.png} - \caption[A probabilistic graphical model of the partially-pooled hierarchical model.]{A probabilistic graphical model (PGM) of the partially-pooled (PP) hierarchical model. Nodes outside of the grey rectangle represent the hyperparameters in the model. Nodes inside the grey rectangle represent individual stellar parameters. Dark grey nodes represent observables which each have their respective observational uncertainties given by the solid black nodes. The direction of the arrows represent the dependencies in the generative model.} + \caption[A probabilistic graphical model of the partially-pooled hierarchical model.]{A probabilistic graphical model (PGM) of the partially-pooled (PP) hierarchical model. Nodes outside of the grey rectangle represent the hyperparameters in the model. Nodes inside the grey rectangle represent individual stellar parameters. Dark grey nodes represent observables which each have their respective observational uncertainties given by the solid black nodes. The direction of the arrows represent the dependencies in the generative model. \emph{This diagram was made using \textsc{Daft} \citep{Foreman-Mackey.Hogg.ea2021}.}} \label{fig:pgm} \end{figure} diff --git a/references.bib b/references.bib index d53e46d..72014e0 100644 --- a/references.bib +++ b/references.bib @@ -2002,6 +2002,15 @@ @article{Foreman-Mackey.Hogg.ea2013 keywords = {python,statistical analysis} } +@misc{Foreman-Mackey.Hogg.ea2021, + title = {Daft-Dev/Daft: Daft v0.1.2}, + author = {{Foreman-Mackey}, Dan and Hogg, David W. and Fulford, David S. and {daft-bot} and Dobos, L{\'a}szl{\'o} and McFee, Brian and Murphy, Kevin P and Lindemann, Oliver and Gerold, Pierre and Agrawal, Varun}, + year = {2021}, + month = mar, + doi = {10.5281/zenodo.4615289}, + howpublished = {Zenodo} +} + @article{Frankel.Sanders.ea2020, title = {Keeping {{It Cool}}: {{Much Orbit Migration}}, yet {{Little Heating}}, in the {{Galactic Disk}}}, shorttitle = {Keeping {{It Cool}}}, From a700f70b9dc249d0fd2e729daf3b6ce233ee867f Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Mon, 24 Apr 2023 15:24:48 +0100 Subject: [PATCH 11/50] Check off main feedback --- chapters/hbm.tex | 35 +++++++++++++++++++---------------- 1 file changed, 19 insertions(+), 16 deletions(-) diff --git a/chapters/hbm.tex b/chapters/hbm.tex index 8b1e480..e2e3517 100644 --- a/chapters/hbm.tex +++ b/chapters/hbm.tex @@ -29,22 +29,24 @@ \section{Introduction} In this section, we use the example of measuring distances to stars in an open cluster to demonstrate a hierarchical Bayesian model (HBM). This example is loosely based on the work of \citet{Leistedt.Hogg2017}, which presents a hierarchical model of the colour-magnitude diagram to improve distances from \emph{Gaia} \citep{GaiaCollaboration.Prusti.ea2016}. However, instead of considering the population distributions over magnitude and colour, we build a hierarchy over the stellar distances. -We created a dataset analogous to an open cluster of \(N_\mathrm{stars}\) stars. We gave each \(i\)-th star a dimensionless distance (\(d_i\)) from the observer drawn randomly from a normal distribution with a mean of 10 and standard deviation of 0.1. Then, we converted these to dimensionless parallax using the relation \(\varpi = 1/d\). We also gave each star an absolute visual magnitude (\(\absmag_i\)) drawn randomly from a standard normal distribution. To get a quantity proportional to apparent magnitude (\(\appmag_i\)) for each star, we used the relation \(\appmag_i = \absmag_i + 5 \log_{10} d_i\). For the purpose of this example, we ignored additional real-world effects such as extinction and reddening. +We created a dataset analogous to an open cluster of \(N_\mathrm{stars}=20\) stars. We gave each \(i\)-th star a dimensionless distance (\(d_i\)) from the observer drawn randomly from a normal distribution with a mean of 10 and standard deviation of 0.1. Then, we converted these to dimensionless parallax using the relation \(\varpi = 1/d\). We also gave each star an absolute visual magnitude (\(\absmag_i\)) drawn randomly from a standard normal distribution. To get a quantity proportional to apparent magnitude (\(\appmag_i\)) for each star, we used the relation \(\appmag_i = \absmag_i + 5 \log_{10} d_i\). For the purpose of this example, we ignored additional real-world effects such as extinction and reddening. -\begin{table}[tb] - \centering - \caption{Simulated dimensionless distance, magnitudes and parallax for \(N_\mathrm{stars}=20\) belonging to an open cluster analogue.} - \label{tab:hbm-data} - \input{tables/hbm-data} -\end{table} +% \begin{table}[tb] +% \centering +% \caption{Simulated dimensionless distance, magnitudes and parallax for \(N_\mathrm{stars}=20\) belonging to an open cluster analogue.} +% \label{tab:hbm-data} +% \input{tables/hbm-data} +% \end{table} -We simulated noisy observations of \(\varpi_i\) and \(\appmag_i\) by randomly drawing from a normal distribution centred on their true values with standard deviations of \(\sigma_{\appmag,i} = 0.1\) and \(\sigma_{\varpi,i} = 0.01\) respectively. We repeated this for \(N_\mathrm{stars}=20\) stars and present the true values and observables in Table \ref{tab:hbm-data}. For real-world context, the uncertainties on \(\varpi\) from \emph{Gaia} \citep{GaiaCollaboration.Prusti.ea2016} Data Release 3 \citep[][]{GaiaCollaboration.Vallenari.ea2022} are approximately \SI{0.02}{\milli\aarcsec} for stars with \emph{Gaia} G-band magnitudes of less than 15. Therefore, if our choice of \(\sigma_{\varpi,i}\) was representative of \emph{Gaia} uncertainties, then the distance to our stellar cluster would be \(\sim \SI{5}{\kilo\parsec}\). An example open cluster at this distance is NGC 6791, at \SI{4}{\kilo\parsec} \citep{Brogaard.Bruntt.ea2011}. We note that by the same comparison, our chosen spread in dimensionless distances of 0.1 corresponds to an order of magnitude more than typical cluster sizes of a few parsecs. +We simulated noisy observations of \(\varpi_i\) and \(\appmag_i\) by randomly drawing from a normal distribution centred on their true values with standard deviations of \(\sigma_{\appmag,i} = 0.1\) and \(\sigma_{\varpi,i} = 0.01\) respectively. +% We repeated this for \(N_\mathrm{stars}=20\) stars and present the true values and observables in Table \ref{tab:hbm-data}. +For real-world context, the uncertainties on \(\varpi\) from \emph{Gaia} \citep{GaiaCollaboration.Prusti.ea2016} Data Release 3 \citep[][]{GaiaCollaboration.Vallenari.ea2022} are approximately \SI{0.02}{\milli\aarcsec} for stars with \emph{Gaia} G-band magnitudes of less than 15. Therefore, if our choice of \(\sigma_{\varpi,i}\) was representative of \emph{Gaia} uncertainties, then the distance to our stellar cluster would be \(\sim \SI{5}{\kilo\parsec}\). An example open cluster at this distance is NGC 6791, at \SI{4}{\kilo\parsec} \citep{Brogaard.Bruntt.ea2011}. We note that by the same comparison, our chosen spread in dimensionless distances of 0.1 corresponds to an order of magnitude more than typical cluster sizes of a few parsecs. However, we chose to exaggerate the distance spread to make it easier to detect for the purpose of this example. In Section \ref{sec:simple-model}, we describe a simple Bayesian model for determining distances and absolute magnitudes of stars in the cluster. Then, we define the HBM in Section \ref{sec:hbm-model} which incorporates a population-level distribution over distance. We outline our method for inferring model parameters in Section \ref{sec:hbm-inf} and then compare results from the models in Section \ref{sec:hbm-comp}. Finally, we explore how the HBM scales with the number of stars observed in Section \ref{sec:hbm-scale}. \subsection{Simple Model}\label{sec:simple-model} -We started with a simxple model which treats each star independently. Using Bayes' theorem, we write the posterior probability density of the model parameters, \(d_i, \absmag_i\) given the observed parameters \(\varpi_i, \appmag_i\) as, +We started with a simple model which treats each star independently. Using Bayes' theorem, we write the posterior probability density of the model parameters, \(d_i, \absmag_i\) given the observed parameters \(\varpi_i, \appmag_i\) as, % \begin{equation} p(d_i, \absmag_i \mid \varpi_i, \appmag_i) \propto p(\varpi_i, \appmag_i \mid d_i, \absmag_i) \, p(d_i, \absmag_i). @@ -60,18 +62,19 @@ \subsection{Simple Model}\label{sec:simple-model} % where \(\mathcal{N}(x \,|\, \mu, \sigma^2)\) is a normal distribution over \(x\) with a mean of \(\mu\) and variance of \(\sigma^2\). -We assumed stars in the cluster were equally likely to be between a distance of 0 and 20. We also assumed the absolute magnitudes were likely to be normally distributed centred on 0 and scaled by 10. Therefore, the prior probability of the model parameters is, +We assumed stars in the cluster were equally likely to be between a distance of 0 and 20. We also assumed the absolute magnitudes were likely to be normally distributed centred on 0 and scaled by 10. Therefore, the prior probability of the model parameters was, % \begin{equation} p(d_i, \absmag_i) = \mathcal{U}(d_i \mid 0, 20) \, \mathcal{N}(\absmag_i \mid 0, 100), \end{equation} % -where \(\mathcal{U}(x \,|\, a, b)\) is a uniform distribution over \(x\) from \(a\) to \(b\). We chose weakly-informative priors on the parameters, given that we know the true values for this example. However, these priors are fairly unrealistic and should be adapted to represent our expectation in real-world cases. For example, the exponential prior from \citet{Bailer-Jones.Rybizki.ea2018} would me more appropriate when observing stars radially outward from the galactic disk. +where \(\mathcal{U}(x \,|\, a, b)\) is a uniform distribution over \(x\) from \(a\) to \(b\). We chose wide, weakly-informative priors here because the focus of this example is on the hierarchical component introduced in the next section. However, the priors should be chosen more carefully in a real-world application. +% However, these priors are fairly unrealistic and should be adapted to represent our expectation in real-world cases. For example, the exponential prior from \citet{Bailer-Jones.Rybizki.ea2018} would me more appropriate when observing stars radially outward from the galactic disk. \begin{figure}[tb] \centering \includegraphics{figures/simple-pgm.pdf} - \caption[Probabilistic graphical model for the simple (non-hierarchical) model]{Probabilistic graphical model for the simple (non-hierarchical) model. Parameters are given by circular nodes and connected by arrows showing the direction of dependency. Observed parameters are shaded, and fixed parameters are given by filled dots. The box represents a set of parameters belonging to the \(i\)-th star.} + \caption[Probabilistic graphical model for the simple (non-hierarchical) model]{Probabilistic graphical model for the simple (non-hierarchical) model. Parameters are given by circular nodes and connected by arrows showing the direction of dependency. Observed parameters are shaded, and fixed parameters are given by filled dots. The box represents a set of parameters belonging to the \(i\)-th star. \emph{This diagram was made using \textsc{Daft} \citep{Foreman-Mackey.Hogg.ea2021}.}} \label{fig:simple-pgm} \end{figure} @@ -79,7 +82,7 @@ \subsection{Simple Model}\label{sec:simple-model} \subsection{Hierarchical Model}\label{sec:hbm-model} -In this section, we present an HBM which includes the known correlation between distances to the stars in this open cluster analogue. We assumed the stars are all members of the same open cluster. Therefore, their distances can be modelled by some distribution. For this example, we assumed that each distance is drawn from a normal distribution characterised by new \emph{hyperparameters} \(\mu_d\) and \(\sigma_d\), +In this section, we present an HBM which includes the known correlation between distances to the stars in this open cluster analogue. We assumed the stars are all members of the same open cluster. Therefore, their distances can be modelled by some tight distribution. For this example, we assumed that each distance is drawn from a normal distribution characterised by new \emph{hyperparameters} \(\mu_d\) and \(\sigma_d\), % \begin{equation} d_i \sim \mathcal{N}(\mu_d, \sigma_d^2). @@ -87,7 +90,7 @@ \subsection{Hierarchical Model}\label{sec:hbm-model} % The hyperparameters are so-called because they take a single value for the population of stars. Hence, the hierarchy of the model arises as some parameters represent how individual parameters are distributed in the population. -Each stellar distance, \(d_i\), is no longer independent. Hence, we modified the posterior probability distribution to account for this new correlation, +Each stellar distance, \(d_i\), is no longer treated independently by the model. Hence, we modified the posterior probability distribution to account for this correlation, % \begin{equation} p(\mu_d, \sigma_d, \vect{d}, \vect{\absmag} \mid \vect{\varpi}, \vect{\appmag}) \propto p(\vect{\varpi}, \vect{\appmag} \mid \vect{d}, \vect{\absmag}) \, p(\vect{d} \mid \mu_d, \sigma_d) \, p(\mu_d, \sigma_d, \vect{\absmag}), @@ -128,7 +131,7 @@ \subsection{Inferring the Model Parameters}\label{sec:hbm-inf} To infer the model parameters, we need to calculate the marginalised posterior distributions for each parameter. We could obtain these analytically by integrating the full posterior distribution over all model parameters except for the parameter of interest. Alternatively, we can approximate the marginalised posterior using a Markov Chain Monte Carlo (MCMC) sampling algorithm. We chose the latter approach because it is scalable to more complicated models where the marginalisation is not analytically solvable. -We used the No U-Turn Sampler \citep[NUTS;][]{Hoffman.Gelman2014} as implemented in the \textsc{NumPyro} Python package \citep{Phan.Pradhan.ea2019,Bingham.Chen.ea2019} to sample from the approximate posterior distribution for both models. We took 1500 samples, with the first 500 samples discarded as `warmup' steps, repeated for 10 MCMC chains. We increased the target accept probability from 0.8 to 0.98 for the HBM, to minimise the number of divergences encountered during sampling. The resulting marginalised posterior samples amounted to \num{10000} per parameter. +We used the No U-Turn Sampler \citep[NUTS;][]{Hoffman.Gelman2014} as implemented in the \textsc{NumPyro} Python package \citep{Phan.Pradhan.ea2019,Bingham.Chen.ea2019} to sample from the approximate posterior distribution for both models. We ran the sampler for 1000 steps following 500 `warmup' steps (used to adapt the sampling procedure) and repeated for 10 MCMC chains. To reduce the number of divergences encountered during sampling, we increased the target accept probability from 0.8 to 0.98 for the HBM. The resulting marginalised posterior samples amounted to \num{10000} per parameter. \subsection{Comparing the Models}\label{sec:hbm-comp} @@ -139,7 +142,7 @@ \subsection{Comparing the Models}\label{sec:hbm-comp} \label{fig:hbm-results} \end{figure} -We compared the difference between model parameters and their true values in Figure \ref{fig:hbm-results}. The HBM showed clear improvement over the simple model in all parameters except for apparent magnitude. Pooling the distances shrank their standard deviations by about one third and converged on their true values. This had the effect of removing the observational noise on \(\varpi\), as we can see the posterior predictions for parallax also approached their true values. Since apparent magnitude depended on both \(d\) and \(\absmag\), better distances also improved the absolute magnitudes. +We compared the difference between model parameters and their true values in Figure \ref{fig:hbm-results}. The HBM showed clear improvement over the simple model in all parameters except for apparent magnitude. Pooling the distances reduced their standard deviations by about one third and converged on their true values. This had the effect of reducing the observational noise on \(\varpi\), as we can see the posterior predictions for parallax also approached their true values. Since apparent magnitude depended on both \(d\) and \(\absmag\), better distances also improved the absolute magnitudes. % \todo{Hand-wavy why apparent magnitude saw no improvement was because the observational noise on parallax had a larger impact on the distance than with apparent magnitude. \(\appmag \propto \log_{10}d\), whereas \(\varpi = d^{-1}\). Additionally, the fractional uncertainty on parallax was greater than that of apparent magnitude. The noise budget is dominated by the parallax, so this is the first to be improved by better distances. \(\sigma_\varpi/\varpi = \sigma_d \varpi\), but \(\sigma_\appmag / \appmag \propto \sigma_d \varpi / \appmag\)} From a1571aac535fb3c730f3ac4b05c6c65c9b0385b7 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Mon, 24 Apr 2023 15:44:32 +0100 Subject: [PATCH 12/50] Change title --- chapters/introduction.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/chapters/introduction.tex b/chapters/introduction.tex index 164121c..d199cde 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -10,7 +10,7 @@ % all distributions of LaTeX version 2005/12/01 or later. % % -\chapter{Introduction to Modelling Stars with Asteroseismology} +\chapter[Introduction to Inferring Stellar Properties]{Introduction to Inferring Stellar Properties with Asteroseismology} \textit{In this chapter, we introduce the current state of modelling stars with asteroseismology and the types of stars being studied in this work. We start with a brief history of understanding the stars spanning the last century. In Section \ref{sec:seismo}, we introduce asteroseismology of stars which oscillate like the Sun. Then, we provide examples of asteroseismology being used to model large samples of dwarf and subgiant stars in Section \ref{sec:many-stars}. Finally, we introduce the concept of modelling stars the `Bayesian way' with some examples of current methods and their limitations.} From 489ab5cafc55b9c4fc3e9ec3365a78d27dc5921d Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Mon, 24 Apr 2023 17:14:09 +0100 Subject: [PATCH 13/50] Start restructure --- chapters/conclusion.tex | 27 ++++++++++++++++++++++----- 1 file changed, 22 insertions(+), 5 deletions(-) diff --git a/chapters/conclusion.tex b/chapters/conclusion.tex index fd3515c..a2163aa 100644 --- a/chapters/conclusion.tex +++ b/chapters/conclusion.tex @@ -16,13 +16,30 @@ \chapter{Conclusion and Future Prospects} \section{Thesis Summary} -We demonstrated that a hierarchical Bayesian model (HBM) can improve the inference of stellar radii, masses and ages. Introducing the concept of an HBM in Chapter \ref{chap:hbm}, we showed how we can include population-level distributions to inform star-level parameter in a Bayesian model. Then, we applied an HBM to model a well-studied sample of oscillating dwarf and subgiant stars in Chapter \ref{chap:hmd}. While accounting for uncertainty in helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)), we were still able to achieve statistical uncertainties of 1.2 per cent in radius, 2.5 per cent in mass and 12 per cent in age. This provided a framework for modelling large populations of stars at the same time and making the most out of noisy data. +In this thesis, we built a hierarchical Bayesian model (HBM) to improve the inference of stellar parameters with asteroseismology. Introducing the concept of an HBM in Chapter \ref{chap:hbm}, we showed how simultaneously modelling population-level distributions over stellar parameters can inform parameters for individual stars. We found that pooling parameters this way reduced their uncertainties by up to a factor of \(\sqrt{N}\) where \(N\) is the number of stars in the population. + +We built an HBM to model a well-studied sample of dwarf and subgiant solar-like oscillators in Chapter \ref{chap:hmd}. We parametrised a linear helium enrichment law as the mean of a population prior over initial stellar helium abundance (\(Y\)). While accounting for uncertainty in helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)), we were still able to achieve statistical uncertainties of 1.2 per cent in radius, 2.5 per cent in mass and 12 per cent in age. This provided a framework for modelling large populations of stars at the same time. In our HBM, we assumed a linear helium enrichment law as the mean of a population distribution in initial stellar helium abundance. We marginalised over the uncertainty in the parameters of this law, improving upon other work which assume a fixed parametrisation of the law calibrated to the Sun \citep[e.g.][]{Serenelli.Johnson.ea2017}. We found the slope of this law (\(\Delta Y/\Delta Z\)) to be \(\approx 1\) and \(\approx 1.6\), with and without including the Sun-as-a-star in our population. Although these values of \(\Delta Y/\Delta Z\) were within 2-\(\sigma\) of each other and agreed with the literature, including the Sun had a clear effect on both \(Y\) and \(\mlt\). This offset may have been a result in our choice of \(\teff\) scale, suggesting an additional systematic we could add to the model. With some improvements to the HBM, we may be able to further break the degeneracy between \(\mlt\) and \(Y\). +The HBM required a function to map fundamental parameters to observables. We built an emulator to approximate 1D numerical models of stellar evolution. Training a neural network on MESA stellar simulations, we could generate observable parameters (\(\teff, \dnu, L, [\mathrm{M/H}]\)) with typical precisions of less than \(\sim 0.1\) per cent (see Appendix \ref{apx:hmd}). This provided a simple, continuous and differentiable model well suited to modern, gradient-based MCMC algorithms. One advantage to using a neural network emulator was its scalability. The linear algebra involved allowed for fast predictions for large numbers of stars in parallel. Furthermore, the neural network can be scaled up to higher input and output dimensions with little performance impact, making the method transferable to other kinds of stars. For example, we recently trained a neural network to emulate the regularly spaced mode frequencies as a part of a Bayesian stellar model of \(\delta\) Scuti-type oscillators \citep{Scutt.Murphy.ea2023}. + +In Chapter \ref{chap:glitch}, we demonstrated that glitches in stellar structure cause a periodic signal, \(\delta\nu\) in the mode frequencies. One such glitch arises from the second ionisation of helium, with the amplitude of \(\delta\nu_\helium\) correlating with helium abundance. Modelling this glitch was difficult because there is uncertainty. In Chapter \ref{chap:glitch-gp}, we developed a new method for measuring the glitch parameters using a Gaussian process. We + +% In Chapter \ref{chap:glitch}, we recalled that p mode frequencies carry information about acoustic glitches inside a star. However, the exact functional form of the modes with radial order is not known. We showed that a Gaussian Process (GP) could be employed to marginalise over the uncertainty in this functional form and improve detection of the helium glitch signature. Our method showed promise compared to those which have come before \citep[e.g.][]{Verma.Raodeo.ea2019}. We found the GP method was better able to find the true acoustic depth of He\,\textsc{ii} ionisation in our model star than the alternative, motivating a more quantitative comparison in the future. We hope to build a more informed prior on the model parameters and publish this method soon with more examples. + \section{Improving the Hierarchical Model} -In Chapter \ref{chap:glitch}, we recalled that p mode frequencies carry information about acoustic glitches inside a star. However, the exact functional form of the modes with radial order is not known. We showed that a Gaussian Process (GP) could be employed to marginalise over the uncertainty in this functional form and improve detection of the helium glitch signature. Our method showed promise compared to those which have come before \citep[e.g.][]{Verma.Raodeo.ea2019}. We found the GP method was better able to find the true acoustic depth of He\,\textsc{ii} ionisation in our model star than the alternative, motivating a more quantitative comparison in the future. We hope to build a more informed prior on the model parameters and publish this method soon with more examples. +Future find a way to get this into the HBM + +Emulate more models - red giants and other physics + +Building more learnt priors + +\section{Current and Future Data} + +Apply to more stars from current and future missions. + The helium glitch parameters for a given star correlate with its near-surface helium abundance \citep{Houdek.Gough2007}. Therefore, a natural next step would be to include helium glitch parameters in our HBM. Our GP glitch model can be applied to both observed and modelled mode frequencies, providing extra parameters to include in our stellar model emulator (see Section \ref{sec:conc-nn}). Including these should improve inference of helium abundance for stars with individual modes identified \citep[e.g.][]{Davies.SilvaAguirre.ea2016,Lund.SilvaAguirre.ea2017}. Since our HBM models the population distribution of helium, even a small number of stars with good helium constraint will in-turn improve helium estimates for the rest of the population. This introduces the possibility of testing more complex models of helium enrichment. @@ -32,15 +49,15 @@ \section{Improving the Hierarchical Model} % Once we extend the model to red giants, we can also consider including population distributions on stellar clusters. For example, \emph{Kepler} observed open clusters NGC... which all include solar-like oscillators \needcite. We can -\section{Emulating Stellar Models}\label{sec:conc-nn} +% \section{Emulating Stellar Models}\label{sec:conc-nn} -We built an emulator which approximated 1D numerical models of stellar evolution to use in our HBM. Training a neural network on MESA stellar simulations, we could predict observables with typical precision of less than \(\sim 0.1\) per cent (see Appendix \ref{apx:hmd}). This provided a simple, continuous and differentiable model well suited to modern, gradient-based MCMC algorithms. We found one advantage to using a neural network emulator was its scalability. The linear algebra involved allowed for fast predictions for a large numbers of stars in parallel. Furthermore, the neural network can be scaled up to higher input and output dimensions with little performance impact, making the method transferable to other kinds of stars. For example, we recently trained a neural network to emulate the regularly spaced mode frequencies as a part of a Bayesian stellar model of \(\delta\) Scuti-type oscillators \citep{Scutt.Murphy.ea2023}. +% We built an emulator which approximated 1D numerical models of stellar evolution to use in our HBM. Training a neural network on MESA stellar simulations, we could predict observables with typical precision of less than \(\sim 0.1\) per cent (see Appendix \ref{apx:hmd}). This provided a simple, continuous and differentiable model well suited to modern, gradient-based MCMC algorithms. We found one advantage to using a neural network emulator was its scalability. The linear algebra involved allowed for fast predictions for a large numbers of stars in parallel. Furthermore, the neural network can be scaled up to higher input and output dimensions with little performance impact, making the method transferable to other kinds of stars. For example, we recently trained a neural network to emulate the regularly spaced mode frequencies as a part of a Bayesian stellar model of \(\delta\) Scuti-type oscillators \citep{Scutt.Murphy.ea2023}. We also expect our emulation method to scale to red giant solar-like oscillators for which observations are abundant (see Section \ref{sec:conc-future}). We trained the emulator on a grid of stellar models from the zero-age main sequence to the base of the red giant branch for masses from \SIrange{0.8}{1.2}{\solarmass}. The upper mass limit was motivated by the diminishing outer convective envelope (responsible for driving solar-like oscillators) in these stars on the main sequence. However, extending the emulator to model red giant solar-like oscillators would require expanding the grid up to \(\sim\SI{2.0}{\solarmass}\). We would need to compute thrice as many evolutionary tracks and evolve existing models further. However, stars with \(M \gtrsim \SI{1.1}{\solarmass}\) have a convective core on the main sequence which introduces an additional model uncertainty from mixing at its boundary. Parametrising this would further multiply the number of input tracks. To handle more dimensions, we should research ways of selectively computing stellar models or augmenting the grid \citep[e.g.][]{Li.Davies.ea2022} where the neural network error is large. % Currently, we compute a grid of models where inputs are spaced regularly. However, the neural network may perform better in some regions and worse in others. We could create an algorithm which computes more stellar evolutionary tracks in regions of parameter space where the neural network performs poorly. This way we could start with a sparse grid of training data, then generate more tracks where the neural network error is high and retrain. -\section{Current and Future Data}\label{sec:conc-future} +% \section{Current and Future Data}\label{sec:conc-future} We tested the HBM on stars observed by \emph{Kepler}, but there are a few current and upcoming missions from which we can increase our sample size. Recently, \citet{Hatt.Nielsen.ea2023} identified a sample of \(\sim 4000\) solar-like oscillators in 120- and 20-second cadence \emph{TESS} data. Of these, around 50 \todo{check} are dwarf and subgiant stars which we could include in a future iteration of the HBM. With larger sample sizes, we can further increase the precision of pooled parameters and better characterise their spread in the population distribution. However, we anticipate much bigger improvement with future observing missions expected to launch in a few years time. From 70d58562be6d6b0f4699cc7cd6adf94fe578355c Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Mon, 24 Apr 2023 22:37:11 +0100 Subject: [PATCH 14/50] Finalise new conclusion structure --- chapters/conclusion.tex | 33 ++++++++++++++++----------------- 1 file changed, 16 insertions(+), 17 deletions(-) diff --git a/chapters/conclusion.tex b/chapters/conclusion.tex index a2163aa..010059c 100644 --- a/chapters/conclusion.tex +++ b/chapters/conclusion.tex @@ -14,38 +14,37 @@ \chapter{Conclusion and Future Prospects} \textit{We conclude this thesis by providing a summary of the work herein and reflect upon key results. Then, we consider possible improvements to our hierarchical model and method for emulating stellar simulations. Finally, we discuss future prospects for applying our method to data from current and upcoming missions.} -\section{Thesis Summary} +\section*{Summary} -In this thesis, we built a hierarchical Bayesian model (HBM) to improve the inference of stellar parameters with asteroseismology. Introducing the concept of an HBM in Chapter \ref{chap:hbm}, we showed how simultaneously modelling population-level distributions over stellar parameters can inform parameters for individual stars. We found that pooling parameters this way reduced their uncertainties by up to a factor of \(\sqrt{N}\) where \(N\) is the number of stars in the population. +In this thesis, we built a hierarchical Bayesian model (HBM) to improve the inference of stellar parameters with asteroseismology. Introducing the concept of an HBM in Chapter \ref{chap:hbm}, we showed how population-level distributions can be used as a prior over individual stellar parameters. We found that pooling parameters this way reduced their uncertainties by up to a factor of \(\sqrt{N}\) where \(N\) is the number of stars in the population. -We built an HBM to model a well-studied sample of dwarf and subgiant solar-like oscillators in Chapter \ref{chap:hmd}. We parametrised a linear helium enrichment law as the mean of a population prior over initial stellar helium abundance (\(Y\)). While accounting for uncertainty in helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)), we were still able to achieve statistical uncertainties of 1.2 per cent in radius, 2.5 per cent in mass and 12 per cent in age. This provided a framework for modelling large populations of stars at the same time. +In Chapter \ref{chap:hmd}, we built an HBM to estimate the masses, radii,and ages for a well-studied sample of \(\sim 60\) dwarf and subgiant solar-like oscillators. Limited by observational noise, existing modelling techniques typically struggle to account for the uncertainty in initial helium abundance (\(Y\)) and mixing-length theory parameter (\(\mlt\)) for these stars. We showed that applying a hierarchical prior over \(Y\) and \(\mlt\) allowed us to simultaneously marginalise over their uncertainty and model their distribution in the population. Pooling \(Y\) and \(\mlt\) in this way, we were still able to achieve statistical uncertainties of 1.2 per cent in radius, 2.5 per cent in mass and 12 per cent in age. In the best cases, our HBM halved the uncertainty in stellar mass compared to the same model without parameter pooling. This provided a scalable and reproducible framework for modelling large populations of stars at the same time. -In our HBM, we assumed a linear helium enrichment law as the mean of a population distribution in initial stellar helium abundance. We marginalised over the uncertainty in the parameters of this law, improving upon other work which assume a fixed parametrisation of the law calibrated to the Sun \citep[e.g.][]{Serenelli.Johnson.ea2017}. We found the slope of this law (\(\Delta Y/\Delta Z\)) to be \(\approx 1\) and \(\approx 1.6\), with and without including the Sun-as-a-star in our population. Although these values of \(\Delta Y/\Delta Z\) were within 2-\(\sigma\) of each other and agreed with the literature, including the Sun had a clear effect on both \(Y\) and \(\mlt\). This offset may have been a result in our choice of \(\teff\) scale, suggesting an additional systematic we could add to the model. With some improvements to the HBM, we may be able to further break the degeneracy between \(\mlt\) and \(Y\). +% In our HBM, we assumed a linear helium enrichment law as the mean of a population distribution over \(Y\). We marginalised over the uncertainty in the parameters of this law, improving upon other work which assume a fixed parametrisation of the law calibrated to the Sun \citep[e.g.][]{Serenelli.Johnson.ea2017}. We found the slope of this law (\(\Delta Y/\Delta Z\)) to be \(\approx 1\) and \(\approx 1.6\), with and without including the Sun-as-a-star in our population. Although these values of \(\Delta Y/\Delta Z\) were within 2-\(\sigma\) of each other and agreed with the literature, including the Sun had a clear effect on both \(Y\) and \(\mlt\). This offset may have been a result in our choice of \(\teff\) scale, suggesting an additional systematic we could add to the model. With some improvements to the HBM, we may be able to further break the degeneracy between \(\mlt\) and \(Y\). -The HBM required a function to map fundamental parameters to observables. We built an emulator to approximate 1D numerical models of stellar evolution. Training a neural network on MESA stellar simulations, we could generate observable parameters (\(\teff, \dnu, L, [\mathrm{M/H}]\)) with typical precisions of less than \(\sim 0.1\) per cent (see Appendix \ref{apx:hmd}). This provided a simple, continuous and differentiable model well suited to modern, gradient-based MCMC algorithms. One advantage to using a neural network emulator was its scalability. The linear algebra involved allowed for fast predictions for large numbers of stars in parallel. Furthermore, the neural network can be scaled up to higher input and output dimensions with little performance impact, making the method transferable to other kinds of stars. For example, we recently trained a neural network to emulate the regularly spaced mode frequencies as a part of a Bayesian stellar model of \(\delta\) Scuti-type oscillators \citep{Scutt.Murphy.ea2023}. +The HBM required a function to map fundamental parameters to observables. We built an emulator to approximate 1D numerical models of stellar evolution. Training a neural network on MESA stellar simulations, we could generate observable parameters (\(\teff, \dnu, L, [\mathrm{M/H}]\)) with typical precisions of less than \(\sim 0.1\) per cent (see Appendix \ref{apx:hmd}). This provided a simple, continuous and differentiable model well suited to modern, gradient-based MCMC algorithms. One advantage to using a neural network emulator was its scalability. The basic linear algebra involved allowed fast evaluations for large numbers of stars in parallel. Furthermore, the neural network could be scaled up to higher input and output dimensions with little performance impact, making the method transferable to other kinds of stars. For example, we recently trained a neural network to emulate the regularly spaced mode frequencies as a part of a Bayesian stellar model of \(\delta\) Scuti-type oscillators \citep{Scutt.Murphy.ea2023}. -In Chapter \ref{chap:glitch}, we demonstrated that glitches in stellar structure cause a periodic signal, \(\delta\nu\) in the mode frequencies. One such glitch arises from the second ionisation of helium, with the amplitude of \(\delta\nu_\helium\) correlating with helium abundance. Modelling this glitch was difficult because there is uncertainty. In Chapter \ref{chap:glitch-gp}, we developed a new method for measuring the glitch parameters using a Gaussian process. We +In Chapter \ref{chap:glitch}, we demonstrated that glitches in stellar structure cause a periodic signal, \(\delta\nu\) in p mode frequencies. One such glitch arises from the second ionisation of helium, with the amplitude of \(\delta\nu_\helium\) correlating with helium abundance. -% In Chapter \ref{chap:glitch}, we recalled that p mode frequencies carry information about acoustic glitches inside a star. However, the exact functional form of the modes with radial order is not known. We showed that a Gaussian Process (GP) could be employed to marginalise over the uncertainty in this functional form and improve detection of the helium glitch signature. Our method showed promise compared to those which have come before \citep[e.g.][]{Verma.Raodeo.ea2019}. We found the GP method was better able to find the true acoustic depth of He\,\textsc{ii} ionisation in our model star than the alternative, motivating a more quantitative comparison in the future. We hope to build a more informed prior on the model parameters and publish this method soon with more examples. - -\section{Improving the Hierarchical Model} +In Chapter \ref{chap:glitch-gp}, we presented a new method for modelling the glitch signature using a Gaussian process (GP). Past methods for measuring the glitch in the mode frequencies (\(\nu_{nl}\)) used a polynomial in \(n\) to approximate the smooth functional form of the frequencies, over which the periodic glitch signature could be modelled \citep[e.g.][]{Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2019}. We replaced the polynomial with a GP characterised by a kernel describing our prior belief of the function's smoothness and flexibility. We applied this method and compared it to the polynomial method to model the glitch signature in radial mode frequencies for a test star and 16 Cyg A. The GP allowed us to marginalise over our uncertainty in the functional form of \(\nu_{n\,0}\) with \(n\). We found that the polynomial method was too restrictive and unable to account for the uncertainty in our model. On the other hand, the GP -Future find a way to get this into the HBM +% In Chapter \ref{chap:glitch}, we recalled that p mode frequencies carry information about acoustic glitches inside a star. However, the exact functional form of the modes with radial order is not known. We showed that a Gaussian Process (GP) could be employed to marginalise over the uncertainty in this functional form and improve detection of the helium glitch signature. Our method showed promise compared to those which have come before \citep[e.g.][]{Verma.Raodeo.ea2019}. We found the GP method was better able to find the true acoustic depth of He\,\textsc{ii} ionisation in our model star than the alternative, motivating a more quantitative comparison in the future. We hope to build a more informed prior on the model parameters and publish this method soon with more examples. -Emulate more models - red giants and other physics +\section*{Improving the Hierarchical Model} -Building more learnt priors +The helium glitch parameters for a given star correlate with its near-surface helium abundance. Therefore, a natural next step would be to include helium glitch parameters as an additional observable in our HBM. Our GP glitch model can be applied to both observed and modelled mode frequencies, providing extra parameters to include in our stellar model emulator. Including these should improve inference of helium abundance for stars with individual modes identified \citep[e.g.][]{Davies.SilvaAguirre.ea2016,Lund.SilvaAguirre.ea2017}. Since our HBM models the population distribution of helium, even a small number of stars with good helium constraint will in-turn improve helium estimates for the rest of the population. This introduces the possibility of testing more complex models of helium enrichment. -\section{Current and Future Data} +We also expect our emulation method to scale to red giant solar-like oscillators for which observations are abundant. We trained the emulator on a grid of stellar models from the zero-age main sequence to the base of the red giant branch for masses from \SIrange{0.8}{1.2}{\solarmass}. The upper mass limit was motivated by the diminishing outer convective envelope (responsible for driving solar-like oscillators) in these stars on the main sequence. However, extending the emulator to model red giant solar-like oscillators would require expanding the grid up to \(\sim\SI{2.0}{\solarmass}\). We would need to compute thrice as many evolutionary tracks and evolve existing models further. However, stars with \(M \gtrsim \SI{1.1}{\solarmass}\) have a convective core on the main sequence which introduces an additional model uncertainty from mixing at its boundary. Parametrising this would further multiply the number of input tracks. To handle more dimensions, we should research ways of selectively computing stellar models or augmenting the grid \citep[e.g.][]{Li.Davies.ea2022} where the neural network error is large. -Apply to more stars from current and future missions. +There are a few additional systematic uncertainties we could also include in the HBM. In Chapter \ref{chap:hmd}, we did not consider the effect of uncertain atmospheric physics which effects the mode frequencies. Surface correction methods exist \citep[e.g.][]{Ball.Gizon2014,Kjeldsen.Bedding.ea2008} but vary across the HR diagram when compared with 3D hydrodynamical simulations \cite{Sonoi.Samadi.ea2015}. \citet{Compton.Bedding.ea2018} found a range of surface corrections can shift modelled frequencies at \(\numax\) by up to \(\sim 0.5\) per cent. This would amount to a systematic effect on \(\dnu\) which we would expect to correlate with other stellar parameters. Although \citet{Nsamba.Campante.ea2018} found the surface correction to have a small effect on inferred stellar parameters, when doing population inference this effect is likely to scale up. Therefore, a future iteration of the HBM should account for the surface term systematic. +\section*{Current and Future Data} -The helium glitch parameters for a given star correlate with its near-surface helium abundance \citep{Houdek.Gough2007}. Therefore, a natural next step would be to include helium glitch parameters in our HBM. Our GP glitch model can be applied to both observed and modelled mode frequencies, providing extra parameters to include in our stellar model emulator (see Section \ref{sec:conc-nn}). Including these should improve inference of helium abundance for stars with individual modes identified \citep[e.g.][]{Davies.SilvaAguirre.ea2016,Lund.SilvaAguirre.ea2017}. Since our HBM models the population distribution of helium, even a small number of stars with good helium constraint will in-turn improve helium estimates for the rest of the population. This introduces the possibility of testing more complex models of helium enrichment. +% The helium glitch parameters for a given star correlate with its near-surface helium abundance. Therefore, a natural next step would be to include helium glitch parameters in our HBM. Our GP glitch model can be applied to both observed and modelled mode frequencies, providing extra parameters to include in our stellar model emulator (see Section \ref{sec:conc-nn}). Including these should improve inference of helium abundance for stars with individual modes identified \citep[e.g.][]{Davies.SilvaAguirre.ea2016,Lund.SilvaAguirre.ea2017}. Since our HBM models the population distribution of helium, even a small number of stars with good helium constraint will in-turn improve helium estimates for the rest of the population. This introduces the possibility of testing more complex models of helium enrichment. % We can also extend the hierarchical aspect of the model. There are other parameters which we expect to correlate in a population of stars. Binary star systems are likely to share common ages and chemical abundances. Some examples are 16 Cyg A and B with an age of \(\sim\SI{7}{\giga\year}\) \citep{Davies.Chaplin.ea2015,Metcalfe.Chaplin.ea2012}, and \(\alpha\) Cen A and B \citep{Kjeldsen.Bedding.ea2005,Bouchy.Carrier2002} with ages \(\sim\SI{6}{\giga\year}\) \citep{Bazot.Bourguignon.ea2012}. Since the masses of these systems are constrained independency of the models, they act as additional points of calibration within the model. -There are a few additional systematic uncertainties we could also include in the HBM. In Chapter \ref{chap:hmd}, we did not consider the effect of uncertain atmospheric physics which effects the mode frequencies. Surface correction methods exist \citep[e.g.][]{Ball.Gizon2014,Kjeldsen.Bedding.ea2008} but vary across the HR diagram when compared with 3D hydrodynamical simulations \cite{Sonoi.Samadi.ea2015}. \citet{Compton.Bedding.ea2018} found a range of surface corrections can shift modelled frequencies at \(\numax\) by up to \(\sim 0.5\) per cent. This would amount to a systematic effect on \(\dnu\) which we would expect to correlate with other stellar parameters. Although \citet{Nsamba.Campante.ea2018} found the surface correction to have a small effect on inferred stellar parameters, when doing population inference this effect is likely to scale up. Therefore, a future iteration of the HBM should account for the surface term systematic. +% There are a few additional systematic uncertainties we could also include in the HBM. In Chapter \ref{chap:hmd}, we did not consider the effect of uncertain atmospheric physics which effects the mode frequencies. Surface correction methods exist \citep[e.g.][]{Ball.Gizon2014,Kjeldsen.Bedding.ea2008} but vary across the HR diagram when compared with 3D hydrodynamical simulations \cite{Sonoi.Samadi.ea2015}. \citet{Compton.Bedding.ea2018} found a range of surface corrections can shift modelled frequencies at \(\numax\) by up to \(\sim 0.5\) per cent. This would amount to a systematic effect on \(\dnu\) which we would expect to correlate with other stellar parameters. Although \citet{Nsamba.Campante.ea2018} found the surface correction to have a small effect on inferred stellar parameters, when doing population inference this effect is likely to scale up. Therefore, a future iteration of the HBM should account for the surface term systematic. % Once we extend the model to red giants, we can also consider including population distributions on stellar clusters. For example, \emph{Kepler} observed open clusters NGC... which all include solar-like oscillators \needcite. We can @@ -53,7 +52,7 @@ \section{Current and Future Data} % We built an emulator which approximated 1D numerical models of stellar evolution to use in our HBM. Training a neural network on MESA stellar simulations, we could predict observables with typical precision of less than \(\sim 0.1\) per cent (see Appendix \ref{apx:hmd}). This provided a simple, continuous and differentiable model well suited to modern, gradient-based MCMC algorithms. We found one advantage to using a neural network emulator was its scalability. The linear algebra involved allowed for fast predictions for a large numbers of stars in parallel. Furthermore, the neural network can be scaled up to higher input and output dimensions with little performance impact, making the method transferable to other kinds of stars. For example, we recently trained a neural network to emulate the regularly spaced mode frequencies as a part of a Bayesian stellar model of \(\delta\) Scuti-type oscillators \citep{Scutt.Murphy.ea2023}. -We also expect our emulation method to scale to red giant solar-like oscillators for which observations are abundant (see Section \ref{sec:conc-future}). We trained the emulator on a grid of stellar models from the zero-age main sequence to the base of the red giant branch for masses from \SIrange{0.8}{1.2}{\solarmass}. The upper mass limit was motivated by the diminishing outer convective envelope (responsible for driving solar-like oscillators) in these stars on the main sequence. However, extending the emulator to model red giant solar-like oscillators would require expanding the grid up to \(\sim\SI{2.0}{\solarmass}\). We would need to compute thrice as many evolutionary tracks and evolve existing models further. However, stars with \(M \gtrsim \SI{1.1}{\solarmass}\) have a convective core on the main sequence which introduces an additional model uncertainty from mixing at its boundary. Parametrising this would further multiply the number of input tracks. To handle more dimensions, we should research ways of selectively computing stellar models or augmenting the grid \citep[e.g.][]{Li.Davies.ea2022} where the neural network error is large. +% We also expect our emulation method to scale to red giant solar-like oscillators for which observations are abundant (see Section \ref{sec:conc-future}). We trained the emulator on a grid of stellar models from the zero-age main sequence to the base of the red giant branch for masses from \SIrange{0.8}{1.2}{\solarmass}. The upper mass limit was motivated by the diminishing outer convective envelope (responsible for driving solar-like oscillators) in these stars on the main sequence. However, extending the emulator to model red giant solar-like oscillators would require expanding the grid up to \(\sim\SI{2.0}{\solarmass}\). We would need to compute thrice as many evolutionary tracks and evolve existing models further. However, stars with \(M \gtrsim \SI{1.1}{\solarmass}\) have a convective core on the main sequence which introduces an additional model uncertainty from mixing at its boundary. Parametrising this would further multiply the number of input tracks. To handle more dimensions, we should research ways of selectively computing stellar models or augmenting the grid \citep[e.g.][]{Li.Davies.ea2022} where the neural network error is large. % Currently, we compute a grid of models where inputs are spaced regularly. However, the neural network may perform better in some regions and worse in others. We could create an algorithm which computes more stellar evolutionary tracks in regions of parameter space where the neural network performs poorly. This way we could start with a sparse grid of training data, then generate more tracks where the neural network error is high and retrain. From e3fd260369e3aecc8df26870a91209e7bddedfce Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Mon, 24 Apr 2023 22:41:09 +0100 Subject: [PATCH 15/50] Change title back to Introduction --- chapters/introduction.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/chapters/introduction.tex b/chapters/introduction.tex index d199cde..bf0858d 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -10,7 +10,7 @@ % all distributions of LaTeX version 2005/12/01 or later. % % -\chapter[Introduction to Inferring Stellar Properties]{Introduction to Inferring Stellar Properties with Asteroseismology} +\chapter[Introduction]{Introduction} \textit{In this chapter, we introduce the current state of modelling stars with asteroseismology and the types of stars being studied in this work. We start with a brief history of understanding the stars spanning the last century. In Section \ref{sec:seismo}, we introduce asteroseismology of stars which oscillate like the Sun. Then, we provide examples of asteroseismology being used to model large samples of dwarf and subgiant stars in Section \ref{sec:many-stars}. Finally, we introduce the concept of modelling stars the `Bayesian way' with some examples of current methods and their limitations.} From 7b910c14903b5ec6d96fdf29e9663da87f7f2993 Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Mon, 24 Apr 2023 22:42:49 +0100 Subject: [PATCH 16/50] Fix typo --- chapters/glitch.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/chapters/glitch.tex b/chapters/glitch.tex index a59b1a5..e4ad197 100644 --- a/chapters/glitch.tex +++ b/chapters/glitch.tex @@ -12,7 +12,7 @@ % \chapter[Acoustic Glitches in Solar-Like Oscillators]{Acoustic Glitches in Solar-Like Oscillators as a Signature of Helium Abundance}\label{chap:glitch} -\textit{Having demonstrated a hierarchical model over initial stellar helium abundance, we explore an asteroseismic signature of helium abundance which could provide more observational constraint. In this chapter, we introduce the concept of a glitch in the structure of a star producing a measurable signal in its observed oscillation modes. We start with a simple one-dimensional example in Section \ref{sec:1d-glitch}. Then, we introduce glitches due to helium ionisation and the base of the convective zone in solar-like oscillators. We provide background on the glitches and their effect on stellar oscillation modes in advance of Chapter \ref{chap:glitch-gp}, where we apply a novel method for modelling these glitch signatures.} +\textit{Having demonstrated a hierarchical model over initial stellar helium abundance, we now explore an asteroseismic signature of helium abundance which could provide more observational constraint. In this chapter, we introduce the concept of a glitch in the structure of a star producing a measurable signal in its observed oscillation modes. We start with a simple one-dimensional example in Section \ref{sec:1d-glitch}. Then, we introduce glitches due to helium ionisation and the base of the convective zone in solar-like oscillators. We provide background on the glitches and their effect on stellar oscillation modes in advance of Chapter \ref{chap:glitch-gp}, where we apply a novel method for modelling these glitch signatures.} \section{Introduction} % \epigraph{\singlespacing``Ideals are like stars: you will not succeed in touching them with your hands, but like the seafaring man on the ocean desert of waters, you choose them as your guides, and following them, you reach your destiny.''}{\emph{Carl Schurz}} From 10b26e342f1be207bb818320d1e23b107ab11777 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Tue, 25 Apr 2023 11:34:04 +0100 Subject: [PATCH 17/50] Proof reading intro --- chapters/glitch.tex | 16 +++++++--------- 1 file changed, 7 insertions(+), 9 deletions(-) diff --git a/chapters/glitch.tex b/chapters/glitch.tex index e4ad197..6089859 100644 --- a/chapters/glitch.tex +++ b/chapters/glitch.tex @@ -12,34 +12,32 @@ % \chapter[Acoustic Glitches in Solar-Like Oscillators]{Acoustic Glitches in Solar-Like Oscillators as a Signature of Helium Abundance}\label{chap:glitch} -\textit{Having demonstrated a hierarchical model over initial stellar helium abundance, we now explore an asteroseismic signature of helium abundance which could provide more observational constraint. In this chapter, we introduce the concept of a glitch in the structure of a star producing a measurable signal in its observed oscillation modes. We start with a simple one-dimensional example in Section \ref{sec:1d-glitch}. Then, we introduce glitches due to helium ionisation and the base of the convective zone in solar-like oscillators. We provide background on the glitches and their effect on stellar oscillation modes in advance of Chapter \ref{chap:glitch-gp}, where we apply a novel method for modelling these glitch signatures.} +\textit{Having demonstrated a hierarchical model over initial stellar helium abundance, we now explore an asteroseismic signature of helium abundance which could provide more observational constraint. In this chapter, we introduce the concept of a glitch in the structure of a star producing a measurable signal in its observable oscillation modes. We start with a simple one-dimensional example in Section \ref{sec:1d-glitch}. Then, in Section \ref{sec:glitch-star} we introduce glitches due to helium ionisation and the base of the convective zone in solar-like oscillators. We review theoretical background on the glitches and their effect on stellar oscillation modes in advance of Chapter \ref{chap:glitch-gp}, where we apply a novel method for modelling these glitch signatures.} \section{Introduction} % \epigraph{\singlespacing``Ideals are like stars: you will not succeed in touching them with your hands, but like the seafaring man on the ocean desert of waters, you choose them as your guides, and following them, you reach your destiny.''}{\emph{Carl Schurz}} % So far in this thesis, we have shown that a hierarchical Bayesian model can be used to infer the helium abundance distribution in a stellar population. We also found how this improves the inference of fundamental stellar parameters. However, there is limited information about helium abundance in the stellar observables used (e.g. \(L, \teff, \Delta\nu\)). -An acoustic glitch is a sharp variation of the sound speed inside a medium. The presence of a glitch induces a sinusoidal signature in consecutive standing pressure wave modes. The reason for this is not entirely intuitive, so we go through a simple, one-dimensional example in Section \ref{sec:1d-glitch}. +An acoustic glitch is a rapid variation of the sound speed inside a medium. The presence of a glitch induces a periodic signature in consecutive p mode frequencies. Acoustic glitches in Sun-like stars arise from sharp variations in their structure, such as the base of the convection zone (BCZ) and the first and second ionisation of helium (He\,\textsc{i} and He\,\textsc{ii}). -Acoustic glitches in Sun-like stars arise from sharp variations in their structure, such as the base of the convection zone (BCZ) and helium ionisation zones (He\,\textsc{i} and He\,\textsc{ii}). Early work identified glitches in solar oscillation modes by analysing their second differences, \(\Delta_2\nu_{nl} \equiv \nu_{n-1\,l} - 2\nu_{nl} + \nu_{n+1\,l}\). Second and higher-order differences remove some smoothly varying components to the mode frequencies and effectively amplify glitch signatures. Assuming a sharp localised discontinuity in sound speed at He\,\textsc{ii} ionisation and the BCZ, \citet{Basu.Antia.ea1994,Basu1997} were able to model these glitches and measure the extent of convective overshoot below the BCZ. +Early work identified glitches in solar p modes by analysing their second differences, \(\Delta_2\nu_{nl} \equiv \nu_{n-1\,l} - 2\nu_{nl} + \nu_{n+1\,l}\). Second and higher-order differences remove some smoothly varying components of the mode frequencies and amplify faster varying signals. Assuming a sharp localised discontinuity in sound speed at He\,\textsc{ii} ionisation and the BCZ, \citet{Basu.Antia.ea1994,Basu1997} modelled glitch signatures in the Sun to constrain the extent of convective overshoot below the BCZ. \citet{Monteiro.Thompson1998} further developed a model of the He\,\textsc{ii} ionisation glitch signature by accounting for its finite width in the star, later also applying it to study the helium ionisation zone of the Sun \citep{Monteiro.Thompson2005}. Around the same time, \citet{Basu.Mazumdar.ea2004} showed that the amplitude of the He\,\textsc{ii} glitch signature correlated with the fractional helium abundance (\(Y\)) near the surface of Sun-like stars. A few years later, \citet{Houdek.Gough2007} proposed a closer physical approximation of the He\,\textsc{ii} glitch. They derived the glitch signature which was later used in many studies of solar-like oscillators. Since helium ionises below the stellar atmosphere for these stars, asteroseismology was able to probe helium abundance where spectroscopy could not. -\citet{Monteiro.Thompson1998} further developed a model of the He\,\textsc{ii} ionisation glitch signature by accounting for its finite width in the star, later applying it to study the Sun \citep{Monteiro.Thompson2005}. Around the same time, \citet{Basu.Mazumdar.ea2004} showed the amplitude of the He\,\textsc{ii} glitch signature correlated with the fractional helium abundance (\(Y\)) near the surface of Sun-like stars. Since helium ionises below the stellar atmosphere for these stars, asteroseismology was able to probe where spectroscopy could not. A few years later, \citet{Houdek.Gough2007} proposed a closer physical approximation of the He\,\textsc{ii} glitch. They derived the glitch signature which was later used in many studies of solar-like oscillators. +Glitches could be further studied in stars other than the Sun with the advent of space-based missions. \citet{Miglio.Montalban.ea2010,Mazumdar.Michel.ea2012} were among the first to measure glitch signatures in other stars after \emph{CoRoT} provided evidence of solar-like oscillations in red giants. Then, studies of glitches in main sequence stars observed by \emph{Kepler} compared fitting to the modes directly against using second differences \citep{Mazumdar.Monteiro.ea2012,Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2017}. More recently, \citet{Verma.Raodeo.ea2019} used measurements of the glitch to determine the helium abundance of the same sample of stars. Using a variety of stellar models, they investigated helium enrichment with metallicity. -The glitch could be further studied in stars other than the Sun with the advent of space-based missions. With \emph{CoRoT} providing evidence of solar-like oscillations in red giants, \citet{Miglio.Montalban.ea2010,Mazumdar.Michel.ea2012} found evidence of glitch in red giants. Sample of \emph{Kepler} stars \citep{Mazumdar.Monteiro.ea2012,Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2017} compare fitting directly and to second differences. For example, 16 Cyg A and B \citet{Verma.Faria.ea2014}. Then \citet{Verma.Raodeo.ea2019} extended this to model to determine helium abundances. Using a variety of stellar models they were not able to find a consistent helium enrichment law. - -In Section \ref{sec:glitch-star}, we demonstrate how glitches in stellar structure due to helium ionisation and the BCZ affect the mode frequencies of solar-like oscillators. Starting with the variational principle \citep{Chandrasekhar1964}, we show how to get to the He\,\textsc{ii} glitch signature formula from \citet{Houdek.Gough2007}. Hence, this chapter serves as a primer for Chapter \ref{chap:glitch-gp}, where we apply a new model for measuring acoustic glitches in \(\nu_{nl}\) using a Gaussian process. +The reason that acoustic glitches cause a periodic signal in the mode frequencies is not obvious. Hence, we go through a simple, one-dimensional example in Section \ref{sec:1d-glitch}. Then, in Section \ref{sec:glitch-star}, we demonstrate how glitches in stellar structure due to helium ionisation and the BCZ affect the mode frequencies of solar-like oscillators. Starting with the variational principle \citep{Chandrasekhar1964}, we show how to get to the He\,\textsc{ii} glitch signature formula from \citet{Houdek.Gough2007}. \section[1D Glitch Example]{A One-Dimensional Example of a Glitch}\label{sec:1d-glitch} \newcommand*{\glitch}{\ensuremath{{\mathrm{g}}}} -A rapid variation in the structure of a medium induces a periodic perturbation (\(\delta\omega\)) to the eigenfrequencies. To demonstrate this, we will explore a simple one-dimensional example \citep[e.g][]{Verner2005}. Consider a medium bound from \(x=0\) to \(x=L\) in which pressure waves can propagate at constant speed \(c\). The longitudinal displacement of the wave \(\xi\) must obey the wave equation, +A rapid variation in the structure of a medium induces a periodic perturbation (\(\delta\omega\)) to the eigenfrequencies. To demonstrate this, we will explore a simple one-dimensional example \citep[similar to that of][]{Verner2005}. Consider a medium bound from \(x=0\) to \(x=L\) in which pressure waves can propagate at constant speed (\(c\)). The longitudinal displacement of the wave (\(\xi\)) must obey the wave equation, % \begin{equation} \frac{\partial^2\xi(x, t)}{\partial t^2} = c^2 \frac{\partial^2\xi(x, t)}{\partial x^2}, \end{equation} % -at a given position \(x\) and time \(t\). A general solution to the wave may be written as a sum of right- and left-travelling waves. In terms of the angular frequency \(\omega\), wave number \(k\), and complex coefficients \((A, B)\), +at a given position (\(x\)) and time (\(t\)). A general solution to the wave may be written as a sum of right- and left-travelling waves. In terms of the angular frequency (\(\omega\)), wave number (\(k\)), and complex coefficients \((A, B)\), % \begin{equation} \xi(x, t) = A \ee^{i (\omega t - k x)} + B \ee^{i (\omega t + k x)}, From 793bdfdd79907cf433699972f6e37237cd1177f3 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Tue, 25 Apr 2023 16:55:53 +0100 Subject: [PATCH 18/50] Finish proof reading glitch chapter --- chapters/glitch.tex | 146 +++++++++++++++++++++++--------------------- 1 file changed, 77 insertions(+), 69 deletions(-) diff --git a/chapters/glitch.tex b/chapters/glitch.tex index 6089859..406c254 100644 --- a/chapters/glitch.tex +++ b/chapters/glitch.tex @@ -12,18 +12,18 @@ % \chapter[Acoustic Glitches in Solar-Like Oscillators]{Acoustic Glitches in Solar-Like Oscillators as a Signature of Helium Abundance}\label{chap:glitch} -\textit{Having demonstrated a hierarchical model over initial stellar helium abundance, we now explore an asteroseismic signature of helium abundance which could provide more observational constraint. In this chapter, we introduce the concept of a glitch in the structure of a star producing a measurable signal in its observable oscillation modes. We start with a simple one-dimensional example in Section \ref{sec:1d-glitch}. Then, in Section \ref{sec:glitch-star} we introduce glitches due to helium ionisation and the base of the convective zone in solar-like oscillators. We review theoretical background on the glitches and their effect on stellar oscillation modes in advance of Chapter \ref{chap:glitch-gp}, where we apply a novel method for modelling these glitch signatures.} +\textit{Having demonstrated a hierarchical model over initial stellar helium abundance, we now explore an asteroseismic signature of helium which could provide more observational constraint. In this chapter, we introduce the concept of a glitch in the structure of a star producing a measurable signal in its observable oscillation modes. We start with a simple one-dimensional example in Section \ref{sec:1d-glitch}. Then, in Section \ref{sec:glitch-star} we introduce glitches due to helium ionisation and the base of the convective zone in solar-like oscillators. We review theoretical background on the glitches and their effect on stellar oscillation modes in advance of Chapter \ref{chap:glitch-gp}, where we apply a novel method for modelling these glitch signatures.} \section{Introduction} % \epigraph{\singlespacing``Ideals are like stars: you will not succeed in touching them with your hands, but like the seafaring man on the ocean desert of waters, you choose them as your guides, and following them, you reach your destiny.''}{\emph{Carl Schurz}} % So far in this thesis, we have shown that a hierarchical Bayesian model can be used to infer the helium abundance distribution in a stellar population. We also found how this improves the inference of fundamental stellar parameters. However, there is limited information about helium abundance in the stellar observables used (e.g. \(L, \teff, \Delta\nu\)). -An acoustic glitch is a rapid variation of the sound speed inside a medium. The presence of a glitch induces a periodic signature in consecutive p mode frequencies. Acoustic glitches in Sun-like stars arise from sharp variations in their structure, such as the base of the convection zone (BCZ) and the first and second ionisation of helium (He\,\textsc{i} and He\,\textsc{ii}). +An acoustic glitch is a rapid variation of the sound speed inside a medium. The presence of a glitch induces a periodic signature in consecutive p mode frequencies. Acoustic glitches in Sun-like stars arise from sharp variations in their structure, such as the base of the convection zone (BCZ) and the first and second ionisation of helium (He\,\textsc{i} and He\,\textsc{ii}). There have been several attempts to measure this effect in the Sun and other stars over the past few decades. -Early work identified glitches in solar p modes by analysing their second differences, \(\Delta_2\nu_{nl} \equiv \nu_{n-1\,l} - 2\nu_{nl} + \nu_{n+1\,l}\). Second and higher-order differences remove some smoothly varying components of the mode frequencies and amplify faster varying signals. Assuming a sharp localised discontinuity in sound speed at He\,\textsc{ii} ionisation and the BCZ, \citet{Basu.Antia.ea1994,Basu1997} modelled glitch signatures in the Sun to constrain the extent of convective overshoot below the BCZ. \citet{Monteiro.Thompson1998} further developed a model of the He\,\textsc{ii} ionisation glitch signature by accounting for its finite width in the star, later also applying it to study the helium ionisation zone of the Sun \citep{Monteiro.Thompson2005}. Around the same time, \citet{Basu.Mazumdar.ea2004} showed that the amplitude of the He\,\textsc{ii} glitch signature correlated with the fractional helium abundance (\(Y\)) near the surface of Sun-like stars. A few years later, \citet{Houdek.Gough2007} proposed a closer physical approximation of the He\,\textsc{ii} glitch. They derived the glitch signature which was later used in many studies of solar-like oscillators. Since helium ionises below the stellar atmosphere for these stars, asteroseismology was able to probe helium abundance where spectroscopy could not. +Early work identified glitches in solar p modes by analysing their second differences, \(\Delta_2\nu_{nl} \equiv \nu_{n-1\,l} - 2\nu_{nl} + \nu_{n+1\,l}\). Second and higher-order differences remove some slowly varying components of the mode frequencies and amplify faster varying components. Assuming a sharp localised discontinuity in sound speed at He\,\textsc{ii} ionisation and the BCZ, \citet{Basu.Antia.ea1994,Basu1997} modelled glitch signatures in the Sun to constrain the extent of convective overshoot at the BCZ. \citet{Monteiro.Thompson1998} further developed a model of the He\,\textsc{ii} ionisation glitch signature by accounting for its finite width in the star, later applying it to study the helium ionisation zone of the Sun \citep{Monteiro.Thompson2005}. Around the same time, \citet{Basu.Mazumdar.ea2004} showed that the amplitude of the He\,\textsc{ii} glitch signature correlated with the fractional helium abundance (\(Y\)) near the surface of Sun-like stars. A few years later, \citet{Houdek.Gough2007} proposed a closer physical approximation of the He\,\textsc{ii} glitch. They derived the glitch signature which was later used in several studies of solar-like oscillators. -Glitches could be further studied in stars other than the Sun with the advent of space-based missions. \citet{Miglio.Montalban.ea2010,Mazumdar.Michel.ea2012} were among the first to measure glitch signatures in other stars after \emph{CoRoT} provided evidence of solar-like oscillations in red giants. Then, studies of glitches in main sequence stars observed by \emph{Kepler} compared fitting to the modes directly against using second differences \citep{Mazumdar.Monteiro.ea2012,Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2017}. More recently, \citet{Verma.Raodeo.ea2019} used measurements of the glitch to determine the helium abundance of the same sample of stars. Using a variety of stellar models, they investigated helium enrichment with metallicity. +Since helium ionises below the stellar atmosphere for cool stars (\(\teff \sim \SI{e5}{\kelvin}\)), asteroseismology was able to probe helium abundance where spectroscopy could not. Glitches could be further studied in stars other than the Sun with the advent of space-based missions. \citet{Miglio.Montalban.ea2010,Mazumdar.Michel.ea2012} were among the first to measure glitch signatures in other stars after \emph{CoRoT} provided evidence of solar-like oscillations in red giants. Then, studies of glitches in main sequence stars observed by \emph{Kepler} compared fitting to the modes frequencies directly with using second differences \citep{Mazumdar.Monteiro.ea2012,Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2017}. More recently, \citet{Verma.Raodeo.ea2019} used measurements of the glitch to estimate the helium abundance for the LEGACY sample of stars \citet{Lund.SilvaAguirre.ea2017}. The reason that acoustic glitches cause a periodic signal in the mode frequencies is not obvious. Hence, we go through a simple, one-dimensional example in Section \ref{sec:1d-glitch}. Then, in Section \ref{sec:glitch-star}, we demonstrate how glitches in stellar structure due to helium ionisation and the BCZ affect the mode frequencies of solar-like oscillators. Starting with the variational principle \citep{Chandrasekhar1964}, we show how to get to the He\,\textsc{ii} glitch signature formula from \citet{Houdek.Gough2007}. @@ -31,25 +31,25 @@ \section{Introduction} \newcommand*{\glitch}{\ensuremath{{\mathrm{g}}}} -A rapid variation in the structure of a medium induces a periodic perturbation (\(\delta\omega\)) to the eigenfrequencies. To demonstrate this, we will explore a simple one-dimensional example \citep[similar to that of][]{Verner2005}. Consider a medium bound from \(x=0\) to \(x=L\) in which pressure waves can propagate at constant speed (\(c\)). The longitudinal displacement of the wave (\(\xi\)) must obey the wave equation, +A rapid variation in the structure of a medium induces a periodic perturbation (\(\delta\omega\)) to the eigenfrequencies. To demonstrate this, we will explore a simple one-dimensional example \citep[similar to that of][]{Verner2005}. Consider a medium bound from \(x=0\) to \(x=L\) in which pressure waves can propagate at constant speed, \(c\). The longitudinal displacement of the wave (\(\xi\)) obeys the wave equation, % \begin{equation} \frac{\partial^2\xi(x, t)}{\partial t^2} = c^2 \frac{\partial^2\xi(x, t)}{\partial x^2}, \end{equation} % -at a given position (\(x\)) and time (\(t\)). A general solution to the wave may be written as a sum of right- and left-travelling waves. In terms of the angular frequency (\(\omega\)), wave number (\(k\)), and complex coefficients \((A, B)\), +at a given position (\(x\)) and time (\(t\)). We may write a general solution to the wave as a sum of right- and left-travelling waves. In terms of the angular frequency (\(\omega\)), wave number (\(k\)), and complex coefficients \((A, B)\), % \begin{equation} \xi(x, t) = A \ee^{i (\omega t - k x)} + B \ee^{i (\omega t + k x)}, \end{equation} % -where \(\omega\) and \(k\) satisfy \(\omega = c k\). Solving for the boundary condition \(\xi(0, t) = 0\) we find \(B = - A\). Substituting Euler's formula, \(A = (r/2) \ee^{i\phi}\), we can write the real solution for \(\xi\) as, +where \(\omega\) and \(k\) satisfy \(\omega = c k\). Solving for the boundary condition, \(\xi(0, t) = 0\), we find \(B = - A\). Substituting Euler's formula, \(A = (r/2) \ee^{i\phi}\), we can represent the physical component of the wave by its real solution, % \begin{equation} \real\left[\xi(x, t)\right] = r \sin k x \sin(\omega t + \phi), \end{equation} % -representing the physical component of the wave, where \(r\) and \(\phi\) are the amplitude and temporal phase respectively. Solutions for \(\omega\) which satisfy \(\xi(L, t)=0\) may then be found, +where \(r\) and \(\phi\) are the amplitude and temporal phase respectively. Solutions for \(\omega\) which satisfy \(\xi(L, t)=0\) may then be found, % \begin{equation} \omega_n = c \frac{n \pi}{L}, \label{eq:omega-n} @@ -65,7 +65,7 @@ \section{Introduction} \label{fig:1d-diagram} \end{figure} -Now, let us suppose there is a small structural perturbation (or glitch) in the medium at position \(x_\glitch\) with half-width \(\delta x\). Figure \ref{fig:1d-diagram} shows this system divided into 3 regions, with region 2 containing the glitch. In region 2, the speed of sound is \(c + \delta c\) and the corresponding wave number is \(k + \delta k\). We want to find the frequencies which correspond to standing waves in this system and compare them to that of the homogeneous medium above. We will show that the resulting perturbation to the eigenfrequencies (\(\delta\omega\)) is periodic, with an amplitude and period that relates to the properties of the glitch. +Now, let us suppose there is a small structural perturbation (or glitch) in the medium at position \(x_\glitch\) with half-width \(\delta x\). Figure \ref{fig:1d-diagram} shows this system divided into 3 regions, with region 2 containing the glitch. The speed of sound is \(c + \delta c\) in region 2, where we let the corresponding wave number be \(k + \delta k\). We want to find the frequencies which correspond to standing waves in this system and compare them to that of the homogeneous medium above. We will show that the resulting perturbation to the eigenfrequencies (\(\delta\omega\)) is periodic, with an amplitude and period that relates to the properties of the glitch. Firstly, we propose solutions to the wave for each region by considering reflection and transmission at each boundary. Initially ignoring the wave superposed by a reflection at \(x=L\), % @@ -75,16 +75,18 @@ \section{Introduction} \xi_3(x, t) &= D \ee^{i(\omega t - k x)}, \label{eq:xi3-r} \end{align} % -where complex coefficients \(A\) and \(C\) represent reflections, and \(B\) and \(D\) represent transmissions, at \(x_\glitch \pm \delta x\) respectively. Later, we will substitute the left-travelling wave (\(- \xi\{-k, -\delta k\}\)) after determining the values of the coefficients. +where complex coefficients \(A\) and \(C\) represent reflections, and \(B\) and \(D\) represent transmissions at \(x_\glitch \pm \delta x\) respectively. Later, we will substitute the left-travelling wave (\(- \xi\{-k, -\delta k\}\)) after determining the values of the coefficients before solving for \(\omega\). -The internal boundary conditions for this system are found by enforcing spacial continuity at \(x - \delta x\) and \(x + \delta x\), +The internal boundary conditions for this system are given by enforcing spacial continuity at \(x_\glitch \pm \delta x\), % -\begin{align*} - \xi_1(x_\glitch - \delta x, t) &= \xi_2(x_\glitch - \delta x, t), \\ - \xi_2(x_\glitch + \delta x, t) &= \xi_3(x_\glitch + \delta x, t), \\ - \frac{\partial \xi_1}{\partial x}(x_\glitch - \delta x, t) &= \frac{\partial \xi_2}{\partial x}(x_\glitch - \delta x, t), \\ - \frac{\partial \xi_2}{\partial x}(x_\glitch + \delta x, t) &= \frac{\partial \xi_3}{\partial x}(x_\glitch + \delta x, t). -\end{align*} +\begin{equation} + \begin{split} + \xi_1(x_\glitch - \delta x, t) &= \xi_2(x_\glitch - \delta x, t), \\ + \xi_2(x_\glitch + \delta x, t) &= \xi_3(x_\glitch + \delta x, t), \\ + \frac{\partial \xi_1}{\partial x}(x_\glitch - \delta x, t) &= \frac{\partial \xi_2}{\partial x}(x_\glitch - \delta x, t), \\ + \frac{\partial \xi_2}{\partial x}(x_\glitch + \delta x, t) &= \frac{\partial \xi_3}{\partial x}(x_\glitch + \delta x, t). + \end{split} +\end{equation} % Solving these simultaneously with the Python package \textsc{sympy} gives the following equations for the complex coefficients\footnote{The code for these derivations are available at \url{\gitremote/tree/\gitbranch/notebooks}}, % @@ -107,20 +109,22 @@ \section{Introduction} \xi_1(x, t) = \ee^{i \omega t} \left[ \frac{r_A}{2} \left( \ee^{i(kx + \phi_A)} - \ee^{-i(kx + \phi_A)} \right) - \left( \ee^{ikx} - \ee^{-ikx} \right) \right], \label{eq:xi1} \end{equation} % -where its real component is, +where its real component in trigonometric form is, \begin{equation} \real\left[\xi_1(x, t)\right] = \sin \omega t \left[2 \sin kx - r_A \sin(kx + \phi_A)\right]. \end{equation} % -However, this does not satisfy the outer boundary condition that the displacement is always zero at \(x=0\); in other words, \(\xi_1(0, t) = - r_A \sin \omega t \sin(\phi_A) \neq 0\) everywhere. To fix this, we introduce a small displacement phase \(\epsilon\) caused by the glitch, and let \(x \rightarrow x + \epsilon\). We will determine \(\epsilon\) shortly. In the meantime, superposing the right-travelling wave and substituting Euler's formula into Equations \ref{eq:xi2-r} and \ref{eq:xi3-r}, we can write the real components of the wave functions, +However, this equation does not satisfy the outer boundary condition that the displacement is always zero at \(x=0\); in other words, \(\xi_1(0, t) = - r_A \sin \omega t \sin(\phi_A) \neq 0\), everywhere. To fix this, we introduce a small phase displacement \(\epsilon\) caused by the glitch and let \(x \rightarrow x + \epsilon\). + +Superposing the right-travelling wave and substituting Euler's formula into Equations \ref{eq:xi2-r} and \ref{eq:xi3-r}, we write the real components of the wave functions in trigonometric as, % \begin{align} - \real[\xi_1(x, t)] &= \sin \omega t \left\{2 \sin[k (x + \epsilon)] - r_A \sin[k(x + \epsilon) - \phi_A]\right\} \label{eq:xi1-real} \\ - \real[\xi_2(x, t)] &= \sin \omega t \left\{ r_B \sin[(k + \delta k)(x + \epsilon) - \phi_B] - r_C \sin[(k + \delta k)(x + \epsilon) - \phi_C]\right\} \\ - \real[\xi_3(x, t)] &= \sin \omega t \left\{r_D \sin[k(x + \epsilon) - \phi_D]\right\} \label{eq:xi3-real} + \real[\xi_1(x, t)] &= \sin \omega t \left\{2 \sin[k (x + \epsilon)] - r_A \sin[k(x + \epsilon) - \phi_A]\right\}, \label{eq:xi1-real} \\ + \real[\xi_2(x, t)] &= \sin \omega t \left\{ r_B \sin[(k + \delta k)(x + \epsilon) - \phi_B] - r_C \sin[(k + \delta k)(x + \epsilon) - \phi_C]\right\}, \\ + \real[\xi_3(x, t)] &= \sin \omega t \left\{r_D \sin[k(x + \epsilon) - \phi_D]\right\}. \label{eq:xi3-real} \end{align} % -Imposing the boundary condition \(\xi_1(0, t) = 0\), we solve Equation \ref{eq:xi1-real} for \(\epsilon\) at \(x=0\), +We can see that Equation \ref{eq:xi1-real} now satisfies the condition that \(\xi_1(0, t) = 0\). Imposing this boundary condition, we solve Equation \ref{eq:xi1-real} for \(\epsilon\) at \(x=0\), % \begin{align} \epsilon &= \frac{1}{k} \tan^{-1}\left( \frac{(r_A / 2) \sin(\phi_A)}{1 - (r_A/2) \cos(\phi_A)} \right), \notag\\ @@ -128,7 +132,7 @@ \section{Introduction} \end{align} % -Finally, we can impose the boundary condition \(\xi_3(L, t) = 0\) to solve for \(\omega\). Setting Equation \ref{eq:xi3-real} to zero, we can rewrite it in terms of the real and imaginary components of \(D\), +Finally, we use the boundary condition \(\xi_3(L, t) = 0\) to solve for \(\omega\). Setting Equation \ref{eq:xi3-real} to zero, we rewrite it in terms of the real and imaginary components of \(D\), % \begin{align} \sin \omega t \left\{r_D \sin[k(L + \epsilon) - \phi_D]\right\} &= 0, \quad (\div \sin \omega t) \notag \\ @@ -136,23 +140,23 @@ \section{Introduction} \real[D] \sin[k(L + \epsilon)] - \imag[D] \cos[k(L + \epsilon)] &=0. \label{eq:1d-glitch-sol} \end{align} % -The glitch affects the amplitude and phase of the wave. If we set \(\epsilon = 0\) and \(D = 1\) we recover the homogeneous frequency solutions (Equation \ref{eq:omega-n}). Unfortunately, solving Equation \ref{eq:1d-glitch-sol} for \(\omega\) is not possible analytically. However, we can find individual roots (or oscillation modes) numerically. +The glitch affects the amplitude and phase of the wave, because if we set \(\epsilon = 0\) and \(D = 1\) we recover the homogeneous frequency solutions (Equation \ref{eq:omega-n}). Unfortunately, solving Equation \ref{eq:1d-glitch-sol} for \(\omega\) is not possible analytically. However, we can find individual roots (or oscillation modes) numerically. \begin{figure} \centering \includegraphics{figures/glitch-1d-example-results.pdf} - \caption[The change in mode frequency induced by a rapid change in sound speed for the 1D example.]{The change in mode frequency induced by a change in sound speed of \(\delta c\) from \(x_\glitch - \delta x\) to \(x_\glitch + \delta x\) in a one-dimensional medium, bound such that \(x \in [0, 1]\) (see Figure \ref{fig:1d-diagram}). Outside of the perturbation the speed of sound, \(c=1\). - The frequencies in the top panel are offset by \(\omega_0\). - Points are joined by straight lines to guide the eye. + \caption[The change in mode frequency induced by a rapid change in sound speed for the 1D example.]{The change in mode frequency induced by a change in sound speed of \(\delta c\) from \(x_\glitch - \delta x\) to \(x_\glitch + \delta x\) in a one-dimensional medium, bound such that \(x \in [0, 1]\) (see Figure \ref{fig:1d-diagram}). Outside of the perturbation the speed of sound, \(c=1\). The frequency perturbations are offset by \(\omega_0\) given in the legend of the top panel. Points are joined by straight lines to guide the eye but do not represent real solutions. } \label{fig:1d-results} \end{figure} -Let us use \(\omega'_n\) to denote the solutions to Equation \ref{eq:1d-glitch-sol}, where \(n\) is a positive integer. We find \(\omega'_n\) by solving Equation \ref{eq:1d-glitch-sol} using Newton's method for \(n = 1,\dots,50\). Using dimensionless units of length and time, we set \(c=1\), \(L=1\), and test several values of \(x_\glitch\), \(\delta x\), and \(\delta c\). Initial guesses for \(\omega'_n\) are obtained from the homogeneous medium solutions in Equation \ref{eq:omega-n}. The difference between the solutions for \(\omega'_n\) and those from the homogeneous medium, \(\delta \omega_n = \omega'_n - \omega_n\), are shown in Figure \ref{fig:1d-results}. We can see a periodic component of \(\delta\omega\) induced by the glitch. Physically, this arises from the change in phase required to satisfy the boundary conditions of the glitch region. As the wave nodes pass in and out of the region with changing \(n\), the sensitivity of the wave to the glitch oscillates. The overall sensitivity to the glitch depends on how much the wave changes inside the glitch, hence why low \(n\) modes have smaller \(\delta\omega\). +Let us use \(\omega'_n\) to denote the solutions to Equation \ref{eq:1d-glitch-sol}, where \(n\) is a positive integer. We find \(\omega'_n\) by solving Equation \ref{eq:1d-glitch-sol} using Newton's method for \(n = 1,\dots,50\) modes. Using dimensionless units of length and time, we set \(c=1\), \(L=1\), and test several values of \(x_\glitch\), \(\delta x\), and \(\delta c\). Initial guesses for \(\omega'_n\) are obtained from the homogeneous medium solutions (\(\omega_n\)) in Equation \ref{eq:omega-n}. We show the difference between the glitch solutions and those from the homogeneous medium, \(\delta \omega_n = \omega'_n - \omega_n\), in Figure \ref{fig:1d-results}. We can see a periodic component to \(\delta\omega\) induced by the glitch. As the wave nodes pass in and out of the region with changing \(n\), the sensitivity of the wave to the glitch varies periodically. The overall sensitivity to the glitch depends on how much the wave changes inside the glitch, hence why low \(n\) modes have smaller \(\delta\omega\). + +% Physically, this arises from the change in phase required to satisfy the boundary conditions of the glitch region. -The functional form of \(\delta\omega\) appears to have a linear component and a short period oscillation modulated by a longer period. As the location of the glitch (\(x_\glitch\)) gets smaller, the short period of \(\delta\omega\) decreases. If we imagine the spacial distribution of nodes in the system as a function of \(n\), the density of nodes is larger towards the centre of the system. The periodicity arises from the nodes passing in and out of the glitch region with changing \(n\). Therefore, where the density of wave nodes is higher, we expect the short period of \(\delta\omega\) to decrease. Similarly, as the the half-width of the glitch (\(\delta x\)) increases, the longer period increases. +The functional form of \(\delta\omega\) appears to have a linear component and a short periodicity modulated by a longer periodicity. As the location of the glitch (\(x_\glitch\)) gets smaller, the short period of \(\delta\omega\) decreases. If we imagine the spacial distribution of nodes in the system as a function of \(n\), the density of nodes is larger towards the centre of the system. The periodicity arises from the nodes passing in and out of the glitch region with changing \(n\). Therefore, where the density of wave nodes is higher, we expect the short period of \(\delta\omega\) to decrease. Similarly, as the the half-width of the glitch (\(\delta x\)) increases, the longer period increases. -Furthermore, an increasing change in sound speed (\(\delta c\)) increases the amplitude of \(\delta\omega\). This result is intuitive, as we expect a perturbation in \(c\) to be proportional to a perturbation in \(\omega\). Finally, increasing both \(\delta x\) and \(\delta c\) increases the slope of \(\delta\omega\). The sensitivity of a mode to the glitch increases with \(n\) and depends on how much the wave changes in the glitch region. A larger glitch allows modes of smaller \(n\) to `see' the glitch region, thus increasing the linear slope of \(\delta\omega\). +Furthermore, an increasing change in sound speed (\(\delta c\)) increases the amplitude of \(\delta\omega\). This result is intuitive, as we expect a larger change in \(c\) to correspond to a large change in \(\omega\). Finally, increasing both \(\delta x\) and \(\delta c\) increases the slope of \(\delta\omega\). The sensitivity of a mode to the glitch increases with \(n\) and depends on how much the wave changes in the glitch region. Modes of smaller \(n\) become more sensitive to the glitch region as \(\delta x\) and \(\delta c\) increase, thus increasing the linear slope of \(\delta\omega\). %This may be interpreted as the nodes of each standing wave passing in and out of region 2 with increasing \(n\). Where there is a node, the wave is least sensitive to a change in structure, and @@ -163,7 +167,7 @@ \section{Introduction} \label{fig:1d-phase} \end{figure} -The small phase offset \(\epsilon\) in Equation \ref{eq:1d-phase} is required for the wave function to satisfy the boundary conditions at \(x = 0\). However, adding \(\epsilon\) shifts the effective location of \(x_\glitch\) --- it changes the scale of the \(x\)-axis by a factor of \((1 + \epsilon)\). We plot \(\epsilon\) against \(\omega\) in Figure \ref{fig:1d-phase} and show that its magnitude is \(\sim 10^{-4}\), much smaller than the location and size of region 2. The periodicity caused by the glitch also shows up in Figure \ref{fig:1d-phase}, with its properties affected in a similar way to Figure \ref{fig:1d-results}. +The small phase offset \(\epsilon\) in Equation \ref{eq:1d-phase} is required for the wave function to satisfy the boundary conditions at \(x = 0\). However, adding \(\epsilon\) shifts the effective location of \(x_\glitch\) --- it changes the scale of the \(x\)-axis by a factor of \((1 + \epsilon)\). We plot \(\epsilon\) against \(\omega\) in Figure \ref{fig:1d-phase} and show that its magnitude is \(\sim 10^{-4}\), much smaller than the location and size of region 2. The periodicity caused by the glitch also shows up in Figure \ref{fig:1d-phase}, with its properties affected in a similar way to Figure \ref{fig:1d-results}. This is because the more a mode is affected by the glitch, the greater the phase offset required to satisfy the boundary conditions. Finding an approximate solution for \(\delta\omega\) is beyond the scope of this example. However, we can show that by modelling \(\delta\omega\), we can recover information about the structural glitch. Let us build a model \(\delta\omega = f(\omega)\). Looking at Figure \ref{fig:1d-results}, we propose a form for \(f\), % @@ -171,7 +175,7 @@ \section{Introduction} f(\omega) = a_1 \omega - a_2 \sin (\tau_1 \omega) \cos (\tau_2 \omega), \label{eq:1d-domega-func} \end{equation} % -where \(a_1\) and \(a_2\) are coefficients which are both functions of \(\delta x\) and \(\delta c\). Parameters \(\tau_1\) and \(\tau_2\) are the `frequencies' (with dimensionless units of time\footnote{An angular frequency in angular frequency space has units of time.}) of the periodic component to \(\delta\omega\), which are functions of \(\delta x\) and \(x_\glitch\) respectively. +where \(a_1\) and \(a_2\) are coefficients which are both functions of \(\delta x\) and \(\delta c\). Parameters \(\tau_1\) and \(\tau_2\) are the `frequencies' (with dimensionless units of time\footnote{An angular frequency in angular frequency space has units of time.}) of the periodic component to \(\delta\omega\). Given Figure \ref{fig:1d-results}, we expect \(\tau_1\) and \(\tau_2\) to be related to \(\delta x\) and \(x_\glitch\) respectively. \begin{figure}[tb] \centering @@ -180,11 +184,11 @@ \section{Introduction} \label{fig:1d-fit} \end{figure} -We fit Equation \ref{eq:1d-domega-func} to \(\delta\omega_n\) obtained from a glitch located at \(x_\glitch = 0.9\), with half-width \(\delta x = 0.02\), and change in sound speed \(\delta c = 0.03\). The best fitting line is shown in Figure \ref{fig:1d-fit}. We found \(\tau_1 \approx \num{0.0389}\) and \(\tau_2 \approx \num{0.199}\). This corresponded to \(\tau_1 \simeq 2\delta\tau\), where \(\delta\tau\) is half the acoustic width of the glitch --- the time at which sound takes to transverse the region. Since the speed of sound \(c \approx \num{1}\) throughout the medium, the true value of \(\delta\tau = 0.02\). We also found the second `frequency' \(\tau_2 \simeq 2\tau_\glitch\), where \(\tau_\glitch\) is the sound travel time from the nearest edge to the centre of the glitch (in this case \num{0.1}). Referring back to Figure \ref{fig:1d-results}, we can infer that the relation between parameters of Equation \ref{eq:1d-domega-func} and the glitch should approximately hold for different values of \(x_\glitch\), \(\delta x\), and \(\delta c\). +We fit Equation \ref{eq:1d-domega-func} to \(\delta\omega_n\) obtained from a glitch located at \(x_\glitch = 0.9\), with half-width \(\delta x = 0.02\), and change in sound speed \(\delta c = 0.03\). The best fitting line is shown in Figure \ref{fig:1d-fit}. We found \(\tau_1 \approx \num{0.0389}\) and \(\tau_2 \approx \num{0.199}\). The former corresponds to \(\tau_1 \simeq 2\delta\tau\), where \(\delta\tau\) is half the acoustic width of the glitch --- the time at which sound takes to transverse the region. Since the speed of sound \(c \approx \num{1}\) throughout the medium, the true value of \(\delta\tau \approx 0.02\). The latter `frequency' corresponds to \(\tau_2 \simeq 2\tau_\glitch\), where \(\tau_\glitch\) is the sound travel time from the nearest edge to the centre of the glitch (in this case \(\tau_\glitch \approx 0.1\)). Referring back to Figure \ref{fig:1d-results}, we can see how these relations should hold for different values of \(x_\glitch\), \(\delta x\), and \(\delta c\) providing the glitch region is small. % NOTE: Could take this further to show a1 = 2 dc dtau / L and a2 = dc / L -It would be tempting to test this further, but we have shown that modelling the signature of a glitch (\(\delta\omega\)) can help characterise its properties. Although the structure of a star is more complicated than this example, we can extend this principle to find glitch signatures in the mode frequencies of solar-like oscillators. Therefore, we will build upon this analogy and explore acoustic glitches in stars in the next section. +It would be tempting to test this further, but we have shown that the glitch signature in \(\omega'_n\) relates to the properties of the glitch. Although the structure of a star is more complicated than this example, we can extend this principle to find glitch signatures in the mode frequencies of solar-like oscillators. Therefore, we will build upon this analogy in the next section where we review some examples of acoustic glitches in stars. % NOTES: Go slowly through this. The next step is to fit a simple mode to theses oscillations and show that we can find x0 and dx. And show how the amplitude scales with dc. @@ -198,17 +202,17 @@ \section{Introduction} c^2 = \gamma \frac{P}{\rho},\label{eq:sound} \end{equation} % -where \(\gamma \equiv \Gamma_1\) is the first adiabatic exponent, +where \(\gamma\) is the first adiabatic exponent, % \begin{equation} \gamma = \left( \frac{\partial \ln P}{\partial \ln \rho} \right)_S, \end{equation} % -at constant entropy, \(S\). \citet{Chandrasekhar1939} introduced three adiabatic exponents (\(\Gamma_1,\Gamma_2,\Gamma_3\)) to describe the non-ideal gas inside a star. However, in this chapter we do not use the other two and hence refer the first as \(\gamma\). +at constant entropy, \(S\). \citet{Chandrasekhar1939} introduced three adiabatic exponents (\(\Gamma_1,\Gamma_2,\Gamma_3\)) to describe the non-ideal gas inside a star. However, in this chapter we do not use the other two and hence refer the first as \(\gamma \equiv \Gamma_1\). -For the most part, \(\gamma\), \(P\), and \(\rho\) change smoothly with radius inside a star. However, a small structural glitch in these quantities would lead to a sudden change in sound speed. In the previous section, we showed how such a perturbation can lead to a periodic perturbation in the eigenfrequencies of pressure waves in a homogeneous medium. Characterising this signal allowed us to measure the properties of the glitch. If similar glitches were present in a star, then we might be able to do the same. In this section, we explore the origins of glitches inside a solar-like star. Then, we see what effect these have on the eigenfrequencies, a quantity we can measure through asteroseismology. +For the most part, \(\gamma\), \(P\), and \(\rho\) change smoothly with radius inside a star. However, a rapid structural glitch in these quantities would lead to a sudden change in sound speed. In the previous section, we showed how such a perturbation can lead to a periodicity in the eigenfrequencies for a homogeneous medium. Characterising this signal allowed us to measure the properties of the glitch. If similar glitches were present in a star, then we might be able to do the same. In this section, we explore the origins of glitches inside a solar-like star. Then, we see what effect these have on the eigenfrequencies, a quantity we can measure through asteroseismology. -Firstly, let us consider the sound speed profile of a Sun-like star. Particularly, we want to see how the sound speed changes on the timescale of a pressure wave moving through the star. As discovered in Section \ref{sec:1d-glitch}, a convenient timescale to work with is the acoustic depth, \(\tau\). This is not to be confused with the symbol for the age of the star in Chapter \ref{chap:hmd}. Here, we define \(\tau\) as the time taken for a pressure wave to travel from the surface (\(R\)) to some radius \(r\) in a star under the assumption of spherical symmetry, +Firstly, let us consider the sound speed profile of a Sun-like star. Particularly, we want to see how the sound speed changes on the timescale of a pressure wave travelling through the star. As discovered in Section \ref{sec:1d-glitch}, a convenient timescale to work with is the acoustic depth, \(\tau\). This is not to be confused with the symbol for stellar age in Chapter \ref{chap:hmd}. Here, we define \(\tau\) as the time taken for a pressure wave to travel from the surface (\(R\)) to some radius (\(r\)) in a star under the assumption of spherical symmetry, % \begin{equation} \tau(r) = \tau_0 - \int_0^{r} \frac{\dd r'}{c(r')},\label{eq:tau} @@ -223,17 +227,15 @@ \section{Introduction} \label{fig:sound-speed-gradient} \end{figure} -In Figure \ref{fig:sound-speed-gradient}, we show the sound speed gradient with respect to \(\tau\) for a Sun-like stellar model (model S; see Section \ref{sec:model-s}). We see how the speed of sound changes smoothly throughout the star. In the convective envelope, where p-modes propagate, there is a noticeable wiggle around \SI{700}{\second} and a sharp change in direction at its base. The first is caused by the ionisation of helium, which we will explore further in Section \ref{sec:helium-glitch}. The second is due to a discontinuity in the second temperature gradient as the stellar interior goes from convectively unstable to radiative. This will be discussed in Section \ref{sec:bcz-glitch}. +In Figure \ref{fig:sound-speed-gradient}, we show the sound speed gradient with respect to \(\tau\) for a Sun-like stellar model (model S; see Section \ref{sec:model-s}). We see how the speed of sound changes smoothly throughout the star. In the convective zone, there is a noticeable wiggle around \SI{700}{\second} and a sharp change in direction at its base. The first is caused by the ionisation of helium, which we will explore further in Section \ref{sec:helium-glitch}. The second is due to a discontinuity in the temperature gradient as the stellar interior goes from unstable due to convection to radiative. This will be discussed in Section \ref{sec:bcz-glitch}. \subsection{Sun-Like Model Star}\label{sec:model-s} -We computed a representative Sun-like model star (hereafter model S) using MESA \citep[version 12115;][]{Paxton.Bildsten.ea2011,Paxton.Cantiello.ea2013,Paxton.Marchant.ea2015,Paxton.Schwab.ea2018,Paxton.Smolec.ea2019,Jermyn.Bauer.ea2023}. The model was computed with an initial mass of \SI{1.0}{\solarmass} to a central fractional hydrogen abundance of 0.6. We used initial fractional helium and heavy-element abundances of 0.28 and 0.02 respectively. The mixing-length was parameterised by \(\alpha_\mlt=1.9\) and a turbulent pressure factor of 1. We evolved the star using element diffusion with MESA's default coefficients \citep{Stanton.Murillo2016}. In addition, we used MESA's default \citet{Grevesse.Sauval1998} solar chemical composition and opacity tables.We also output pulsation profile data in FGONG format to later use when computing oscillation modes in Chapter \ref{chap:glitch-gp}. Our resulting model S had an age of \SI{4.073}{\giga\year}, \(\teff \approx \SI{5682}{\kelvin}\), and \(R \approx \SI{1.014}{\solarradius}\). Since the model was evolved with element diffusion, the surface fractional helium and heavy-element were approximately \num{0.2515} and \num{0.01810} respectively. +We computed a representative Sun-like model star (hereafter `model S') using MESA \citep[version 12115;][]{Paxton.Bildsten.ea2011,Paxton.Cantiello.ea2013,Paxton.Marchant.ea2015,Paxton.Schwab.ea2018,Paxton.Smolec.ea2019,Jermyn.Bauer.ea2023}. The model was computed with a mass of \SI{1.0}{\solarmass} to a central fractional hydrogen abundance of \(X_c=0.6\). We used initial fractional helium and heavy-element abundances of \(Y_\mathrm{init} = 0.28\) and \(Z_\mathrm{init}=0.02\) respectively. The mixing-length was parametrised by \(\alpha_\mlt=1.9\) and a turbulent pressure factor of 1. We evolved the star using element diffusion with MESA's default coefficients \citep{Stanton.Murillo2016}. In addition, we used MESA's default \citet{Grevesse.Sauval1998} solar chemical composition and opacity tables. We also output pulsation profile data in FGONG format to later use when computing oscillation modes in Chapter \ref{chap:glitch-gp}. Our resulting model S had an age of \SI{4.073}{\giga\year}, \(\teff \approx \SI{5682}{\kelvin}\), and \(R \approx \SI{1.014}{\solarradius}\). Since the model was evolved with element diffusion, the surface fractional helium and heavy-element were approximately \(Y_\mathrm{surf} \approx 0.2515\) and \(Z_\mathrm{surf} \approx 0.01810\) respectively. \subsection{Helium Ionisation Glitch}\label{sec:helium-glitch} -In this section, we will first show how the sound speed inside a solar-like star is affected by the ionisation of hydrogen and helium. Then, we will see that an increase in helium abundance (\(Y\)) increases the effect of helium ionisation on the speed of sound. Starting with the variational principle, we derive a commonly used form of the glitch signature found in the mode frequencies. In the process, we show that the p modes are sensitive to rapid changes in sound speed near the surface. - -\subsubsection{The Effect of Helium Abundance on \(\gamma\)} +In this section, we will first show how the sound speed inside a solar-like star is affected by the ionisation of hydrogen and helium. Then, we will see that an increase in helium abundance (\(Y\)) increases the effect of helium ionisation on the speed of sound. Starting with the variational principle, we derive the frequently used equation for the glitch signature \(\delta\nu\) induced by the second ionisation of helium from \citet{Houdek.Gough2007}. In the outer convective envelope of Sun-like stars, the temperature (\(\SIrange{e4}{e5}{\kelvin}\)) and density (\(\SIrange{e-5}{e-3}{\gram\per\centi\metre\cubed}\)) is suitable for ionising hydrogen and helium \citep{Eggleton.Faulkner.ea1973}. As a given chemical species in the star ionises, the number of particles and hence chemical potential of the species changes. This induces a gradient in the thermal free energy of the gas which relates to the pressure and entropy of the gas. Thus, we expect ionisation to cause a change in the pressure-density gradient at constant entropy, \(\gamma\). Since \(\gamma\) relates to the sound speed from Equation \ref{eq:sound}, any sudden change in \(\gamma\) will induce a change in \(c\). @@ -244,7 +246,9 @@ \subsubsection{The Effect of Helium Abundance on \(\gamma\)} \label{fig:gamma-zones} \end{figure} -In Figure \ref{fig:gamma-zones}, we show \(\gamma\) for model S against fractional acoustic depth, shaded by regions of ionisation. For an ideal monatmoic gas, \(\gamma=5/3\), but \(\gamma < 5/3\) in regions where helium and hydrogen ionise. Close to the surface of the star, hydrogen ionisation has the largest effect on \(\gamma\) because it makes up the majority of stellar matter. The first (He\,\textsc{i}) and second (He\,\textsc{ii}) ionisations of helium occur deeper in the star. We can see that the second ionisation of helium has a greater affect on \(\gamma\) than the first. The effect of the second He ionisation causes a rapid changing in \(\gamma\) over a few per cent in \(\tau\) (equivalent to \(\sim \SI{50}{\second}\) in model S). +In Figure \ref{fig:gamma-zones}, we show \(\gamma\) for model S against fractional acoustic depth, shaded by regions of ionisation. For an ideal monatmoic gas, \(\gamma=5/3\), but we see that \(\gamma < 5/3\) in regions where helium and hydrogen ionise. Close to the surface of the star, hydrogen ionisation has the largest effect on \(\gamma\) because it makes up the majority of stellar matter. The first (He\,\textsc{i}) and second (He\,\textsc{ii}) ionisations of helium occur deeper in the star. We can see that the second ionisation of helium has a greater affect on \(\gamma\) than the first. The effect of the He\,\textsc{ii} ionisation causes a rapid change in \(\gamma\) over a few per cent in \(\tau\) (equivalent to \(\sim \SI{100}{\second}\) in model S). + +\subsubsection{The Effect of Helium Abundance on \(\gamma\)} \begin{figure} \centering @@ -253,7 +257,7 @@ \subsubsection{The Effect of Helium Abundance on \(\gamma\)} \label{fig:gamma-temp-density} \end{figure} -We were able to isolate the contributions to \(\gamma\) from ionisation using a recent derivation by \citet{Houdayer.Reese.ea2021}. They obtained an approximate formula for \(\gamma\) as a function of temperature (\(T\)) and density valid for the outer convective zone of Sun-like stars. Using their formula, we plot \(\gamma\) for a range of \(T\) and \(\rho\) in Figure \ref{fig:gamma-temp-density}. The first panel illustrates the three ionisation regions of hydrogen and helium. The value of \(\gamma\) decreases when the ionisation reaction is occurring. The magnitude of this effect depends on the temperature-density profile of the star which we can see over-plot for model S. We can imagine a hotter star shifting this line such that ionisation occurs closer to the surface. In the second panel, we see that an increase in helium abundance by 0.1 decreases \(\gamma\) by up to \(\sim 0.03\) in the He\,\textsc{ii} ionisation region along the model S profile. This shows that helium abundance correlates with the size of the glitch in \(\gamma\). However, there is clearly still some dependence on other stellar quantities. +We were able to isolate the contributions to \(\gamma\) from ionisation using a recent derivation by \citet{Houdayer.Reese.ea2021}. They obtained an approximate formula for \(\gamma\) as a function of temperature (\(T\)) and density valid for the outer convective zone of Sun-like stars. Using their formula, we plot \(\gamma\) for a range of \(T\) and \(\rho\) in Figure \ref{fig:gamma-temp-density}. The first panel illustrates the three ionisation regions of hydrogen and helium. The value of \(\gamma\) decreases when the ionisation reaction is occurring. The magnitude of this effect depends on the intersection with the temperature-density profile of the star (over-plot for model S). We can imagine a hotter star shifting this line such that ionisation occurs closer to the surface. Along the model S profile, we see that an increase in helium abundance by 0.1 decreases \(\gamma\) by up to \(\sim 0.03\) in the He\,\textsc{ii} ionisation region. This shows that helium abundance correlates with the depth of the glitch in \(\gamma\). However, there is clearly still some dependence on \(T\) and \(\rho\) which are governed by mass, chemical composition, and evolutionary phase. % The exact effect of ionisation on \(\gamma\) is not known analytically. However, \citet{Houdayer.Reese.ea2021} recently approximated \(\gamma\) for the convective zone of a Sun-like star in their study of the helium ionisation glitch. In this subsection, we combine the equations derived in their work to help us understand the relation between ionisation and \(\gamma\). Let us consider an \(M\) mass star with fractional hydrogen and helium abundances of \(X\) and \(Y\) respectively. From \citet{Houdayer.Reese.ea2021}, we approximate the first adiabatic exponent as, % % @@ -291,8 +295,6 @@ \subsubsection{The Effect of Helium Abundance on \(\gamma\)} % % % where \(q = 2\) for hydrogen, \(q = 1\) for helium, \(\overline{m} = M/N\) is the mean mass, and \(g_\mathbb{X}^i\) is the ground-state degeneracy of ionisation state \(i\). We used these equations to produce Figures \ref{fig:gamma-zones} and \ref{fig:gamma-temp-density}. -\subsubsection{The Effect of \(\gamma\) on p Mode Frequencies} - \begin{figure} \centering \includegraphics{figures/helium-ionisation-sound-speed.pdf} @@ -300,7 +302,9 @@ \subsubsection{The Effect of \(\gamma\) on p Mode Frequencies} \label{fig:gamma-sound-speed} \end{figure} -To show the effect of small changes in \(Y\) on the sound speed, we plot the sound speed gradient in Figure \ref{fig:gamma-sound-speed}. The three models shown were evolved to the same central temperature as model S with initial helium abundances of 0.26, 0.28, and, 0.3. The dominant effect of helium abundance appears as a Gaussian-like depression in \(\gamma\) around the second ionisation of helium. We see how larger helium abundance increases the width and depression in \(\gamma\), which is reflected in the sound speed gradient. A larger \(Y\) also leads to a relative reduction in hydrogen abundance (\(X\)) which shrinks the width of the hydrogen ionisation region. +To show the effect of small changes in \(Y\) on the sound speed, we plot the sound speed gradient in Figure \ref{fig:gamma-sound-speed}. The three models shown were evolved to the same \(X_c = 0.6\) as model S with initial helium abundances of 0.26, 0.28, and, 0.3. The dominant effect of helium abundance appears as a Gaussian-like depression in \(\gamma\) around the second ionisation of helium. We see how larger helium abundance increases the width and depression in \(\gamma\), which is reflected in the sound speed gradient. A larger \(Y\) also leads to a relative reduction in hydrogen abundance (\(X\)) which shrinks the width of the hydrogen ionisation region. + +\subsubsection{The Effect of a Change in \(\gamma\) on p Mode Frequencies} To see how a change in \(\gamma\) affects the mode frequencies, we explore the sensitivity of a mode to rapid structural changes in the star. Starting with the variational principle, we can approximate the characteristic frequencies of a spherically symmetric, slowly rotating star by \citep{Chandrasekhar1964}, % @@ -314,7 +318,7 @@ \subsubsection{The Effect of \(\gamma\) on p Mode Frequencies} % \mathcal{E} = \int_0^R \left[\gamma P (\dive{\vect{\xi}})^2 + 2(\vect{\xi}\cdot \nabla P) \dive{\vect{\xi}} + (\vect{\xi}\cdot \nabla P) (\vect{\xi}\cdot\nabla\ln\rho)\right] r^2 \, \dd r. % \end{equation} % -to its inertia (\(\mathcal{E}\)). The mode inertia is, +to its inertia (\(\mathcal{I}\)). The mode inertia is given by, % \begin{equation} \mathcal{I} = \int_0^R \vect{\xi} \cdot \vect{\xi} \, \rho r^2 \, \dd r. @@ -342,7 +346,7 @@ \subsubsection{The Effect of \(\gamma\) on p Mode Frequencies} \frac{\delta\omega}{\omega} = \int_0^R \left(\mathcal{K}_{c^2,\rho} \frac{\delta c^2}{c^2} + \mathcal{K}_{\rho,c^2} \frac{\delta \rho}{\rho} \right) \dd r.\label{eq:kernels} \end{equation} % -where \(\mathcal{K}_{a, b}\) gives the relative effect on \(\omega\) due to a perturbation in a state variable \(a\) at fixed \(b\), at a given \(r\). The kernels, defined fully in \citet{Gough.Thompson1991}, can show the sensitivity of the wave to a change in state in the star. We plot \(\mathcal{K}_{c^2,\rho}\) and \(\mathcal{K}_{\rho,c^2}\) in Figure \ref{fig:kernels} for a few radial oscillation modes. Both decay through the star, meaning that the modes are less sensitive to structural changes deeper in the star. The kernels also oscillate at different frequencies, meaning that some modes will be more or less sensitive to a glitch than others. +where \(\mathcal{K}_{a, b}\) gives the relative effect on \(\omega\) at a given \(r\) due to a perturbation in a state variable \(a\) at fixed \(b\). The kernels, defined fully in \citet{Gough.Thompson1991}, can show the sensitivity of the wave to a change in state in the star. We plot \(\mathcal{K}_{c^2,\rho}\) and \(\mathcal{K}_{\rho,c^2}\) in Figure \ref{fig:kernels} for a few radial oscillation modes. Both decay through the star, meaning that the modes are less sensitive to structural changes deeper in the star. The kernels also oscillate at different frequencies corresponding to the radial order, meaning the amount they intersect with a glitch changes with \(\omega\). We can evaluate Equation \ref{eq:kernels} to find the effect on \(\omega\) due to a change in \(\gamma\) from helium ionisation. The first term can be easily rewritten in terms of a change in \(\gamma\) because \(c^2 \propto \gamma\), hence \(\mathcal{K}_{\gamma,\rho} \delta \gamma / \gamma \equiv \mathcal{K}_{c^2,\rho} \delta c^2 / c^2\). Helium ionisation has a negligible effect on density, so we can assume \(\delta\rho/\rho \approx 0\) in the ionisation region. Therefore, we may rewrite Equation \ref{eq:kernels} due to a change in \(\gamma\) as, % @@ -357,47 +361,51 @@ \subsubsection{The Effect of \(\gamma\) on p Mode Frequencies} \end{equation} % -Firstly, we expand the mode inertia, \(\mathcal{I}\). The Lagrangian perturbation vector can be written in terms of a radial and horizontal component, \(\vect{\xi} = \xi_r \hat{r} + \vect{\xi}_h\). In the high-order limit where \(l/n \rightarrow 0\), the horizontal component \(\vect{\xi}_h\) is negligible. Therefore, we can rewrite the mode inertia as, +\newcommand*{\propconst}{\ensuremath{{\mathcal{A}}}} + +In order to solve Equation \ref{eq:delta-omega}, we first expand the mode inertia, \(\mathcal{I}\). The Lagrangian perturbation vector can be written in terms of a radial and horizontal component, \(\vect{\xi} = \xi_r \hat{r} + \vect{\xi}_h\). In the high-order limit where \(l/n \rightarrow 0\), the horizontal component \(\vect{\xi}_h\) is negligible. Therefore, we can rewrite the mode inertia as, % \begin{equation} \mathcal{I} \simeq \int_0^R \xi_r^2 \rho r^2 \, \dd r. \end{equation} % -Then, we use an approximation of \(\xi_r\) from \citet{Gough1993}, +Then, we use a suitable approximation of \(\xi_r\) from \citet{Gough1993}, % \begin{equation} - \xi_r \simeq \frac{\psi_0}{r}\sqrt{\frac{K}{\rho}} \cos\psi,\label{eq:xi-r} + \xi_r \simeq \frac{\propconst}{r}\sqrt{\frac{K}{\rho}} \cos\psi,\label{eq:xi-r} \end{equation} % -where \(\psi_0\) is a proportionality constant and the radial wave number \(K \simeq \omega / c\) for high-order acoustic modes. The phase term \(\psi \simeq \omega \tau + \epsilon\) when \(\tau\) is not close to the upper turning point (where the wave is reflected near the stellar surface). The small offset \(\epsilon\) is a slowly varying function of \(\tau\). Substituting Equation \ref{eq:xi-r} into the mode inertia gives, +where \(\propconst\) is a proportionality constant. The radial wave number \(K \simeq \omega / c\) for high-order acoustic modes. +% Because the speed of sound changes slowly between nodes? +Assuming \(\tau\) is not close to the upper turning point (where the wave is reflected near the stellar surface), the phase term \(\psi \simeq \omega \tau + \epsilon\). The small offset \(\epsilon\) is a slowly varying function of \(\tau\). Substituting Equation \ref{eq:xi-r} into the mode inertia gives, % \begin{align} - \mathcal{I} &\simeq \psi_0^2 \int_0^R K \cos^2\psi \, \dd r \notag \\ - &= \frac12 \omega \psi_0^2 \int_0^R (1 + \cos 2 \psi) \frac{\dd r}{c}. + \mathcal{I} &\simeq \propconst^2 \int_0^R K \cos^2\psi \, \dd r, \notag \\ + &= \frac12 \omega \propconst^2 \int_0^R (1 + \cos 2 \psi) \frac{\dd r}{c}. \end{align} % Finally, changing to an integral over the acoustic depth, \(\tau\) we can evaluate the integral for high-order modes where \(\omega_n \ll \tau_0^{\,-1}\), % \begin{equation} - \mathcal{I} \simeq \frac12 \omega \psi_0^2 \int_0^{\tau_0} [1 + \cos 2 (\omega\tau + \epsilon)] \, \dd \tau \simeq \frac12 \omega \psi_0^2 \tau_0. + \mathcal{I} \simeq \frac12 \omega \propconst^2 \int_0^{\tau_0} [1 + \cos 2 (\omega\tau + \epsilon)] \, \dd \tau \simeq \frac12 \omega \propconst^2 \tau_0. \label{eq:inertia} \end{equation} % Secondly, we substitute an approximation for \((\dive{\vect{\xi}})^2\) from \citet{Gough1993}, % \begin{equation} - (\dive{\vect{\xi}})^2 \simeq \frac{\psi_0^2 \omega^3}{\gamma P c r^2} \sin^2\psi, \label{eq:div-xi} + (\dive{\vect{\xi}})^2 \simeq \frac{\propconst^2 \omega^3}{\gamma P c r^2} \sin^2\psi, \label{eq:div-xi} \end{equation} % -into Equation \ref{gamma-kernel} to get, +into Equation \ref{eq:gamma-kernel} to get, % \begin{align} \left.\frac{\delta\omega}{\omega}\right|_\gamma &\simeq \frac{1}{2\omega^2\mathcal{I}} \int_0^{R} \delta\gamma P (\dive{\vect{\xi}})^2 r^2 \, \dd r, \notag \\ - &\simeq \frac{\omega \psi_0^2}{2 \mathcal{I}} \int_0^{R} \frac{\delta\gamma}{\gamma}\sin^2\psi\frac{\dd r}{c}, \notag \\ - &= \frac{\omega \psi_0^2}{4 \mathcal{I}} \int_0^{R} \frac{\delta\gamma}{\gamma}(1 - \cos 2\psi)\frac{\dd r}{c}. + &\simeq \frac{\omega \propconst^2}{2 \mathcal{I}} \int_0^{R} \frac{\delta\gamma}{\gamma}\sin^2\psi\frac{\dd r}{c}, \notag \\ + &= \frac{\omega \propconst^2}{4 \mathcal{I}} \int_0^{R} \frac{\delta\gamma}{\gamma}(1 - \cos 2\psi)\frac{\dd r}{c}. \end{align} % -We can see that \(\delta\omega\) is split into a smooth and a periodic component. The smooth component may be treated later, but for now we focus on the oscillating part of \(\delta\omega\). Substituting for \(\mathcal{I}\) and \(\psi\), changing to an integral over acoustic depth we get, +We can see that \(\delta\omega\) is split into a smooth and a periodic component. The smooth component may be treated later, but for now we focus on the oscillating part of \(\delta\omega\). Substituting \(\mathcal{I}\) from Equation \ref{eq:inertia} and changing to an integral over acoustic depth we get, % \begin{equation} \left.\frac{\delta\omega}{\omega}\right|_{\gamma,\mathrm{osc}} \simeq - \frac{1}{2\tau_0} \int_0^{\tau_0} \frac{\delta\gamma}{\gamma} \cos 2 (\omega\tau + \epsilon) \, \dd \tau. \label{eq:omega-osc} @@ -406,7 +414,7 @@ \subsubsection{The Effect of \(\gamma\) on p Mode Frequencies} \subsubsection{A Functional Form of the Helium Glitch Signature} -We have shown that a change in \(\gamma\) can induce a periodicity in \(\omega\). The functional from of this periodicity depends on \(\delta\gamma/\gamma\). As shown in Figures \ref{fig:gamma-zones} and \ref{fig:gamma-temp-density}, the dominant perturbation due to a change in helium abundance is from the second ionisation of helium. There have been different attempts to approximate \(\delta\gamma/\gamma\,|_\heII\) in the literature, for example using a Dirac delta function or a triangular function \citep{Monteiro.Christensen-Dalsgaard.ea1994, Monteiro.Thompson2005}. In recent years, work modelling the glitch has used the formulation from \citet{Houdek.Gough2007} where the change in \(\gamma\) due to He\,\textsc{ii} ionisation is modelled with a Gaussian shape, +We have shown that a change in \(\gamma\) can induce a periodicity in \(\omega\). The functional from of this periodicity depends on \(\delta\gamma/\gamma\). As shown in Figures \ref{fig:gamma-zones} and \ref{fig:gamma-temp-density}, the dominant change in \(\gamma\) due to a change in helium abundance is from the second ionisation of helium. There have been different attempts to approximate \(\delta\gamma/\gamma\,|_\heII\) in the literature, for example using a Dirac delta function or a triangular function \citep{Monteiro.Christensen-Dalsgaard.ea1994,Monteiro.Thompson2005}. In recent years, work modelling the glitch has used the formulation from \citet{Houdek.Gough2007} where the change in \(\gamma\) due to He\,\textsc{ii} ionisation is modelled with a Gaussian shape, % \begin{equation} \left.\frac{\delta\gamma}{\gamma}\right|_\heII \simeq - \frac{\Gamma_\heII}{\Delta_\heII \sqrt{2\pi}} \, \ee^{- \frac12{(\tau - \tau_\heII)^2}/{\Delta_\heII^2} }, \label{eq:he-gamma} @@ -414,7 +422,7 @@ \subsubsection{A Functional Form of the Helium Glitch Signature} % where \(\Gamma_\heII\) is the area, \(\Delta_\heII\) is the characteristic width, and \(\tau_\heII\) is the center of the ionisation region. -Here, we independetly verify result for \(\delta\omega\) due to He\,\textsc{ii} ionisation from \citet{Houdek.Gough2007}. Substituting Equation \ref{eq:he-gamma} into Equation \ref{eq:omega-osc} with a change of variables to \(x = (\tau - \tau_\heII)/\Delta_\heII\), we get, +Here, we verify the result for \(\delta\omega\) due to He\,\textsc{ii} ionisation from \citet{Houdek.Gough2007}. Substituting Equation \ref{eq:he-gamma} into Equation \ref{eq:omega-osc} with a change of variables to \(x = (\tau - \tau_\heII)/\Delta_\heII\), we get, % \begin{equation} \left.\frac{\delta\omega}{\omega}\right|_{\heII, \mathrm{osc}} \simeq \frac{\Gamma_\heII}{2\sqrt{2\pi} \, \tau_0} \, \int_{-\infty}^\infty \ee^{- x^2/2} \cos 2 (\Delta_\heII \omega x + \widetilde{\epsilon_\heII}) \, \dd x \label{eq:omega-ii-osc} @@ -439,7 +447,7 @@ \subsubsection{A Functional Form of the Helium Glitch Signature} % \begin{equation} \begin{split} - I(a) = I(0) \ee^{- 2 \Delta_\heII^2 \omega^2 a^2}, \qquad I(0) &= \int_{-\infty}^\infty \ee^{- x^2/2 } \cos 2 \widetilde{\epsilon_\heII} \, \dd x, \\ + I(a) = I(0) \ee^{- 2 \Delta_\heII^2 \omega^2 a^2}; \qquad I(0) &= \int_{-\infty}^\infty \ee^{- x^2/2 } \cos 2 \widetilde{\epsilon_\heII} \, \dd x, \\ &= \sqrt{2\pi} \, \cos 2 \widetilde{\epsilon_\heII}, \end{split} \end{equation} @@ -471,7 +479,7 @@ \subsection{Base of the Convective Zone Glitch}\label{sec:bcz-glitch} % Overshoot actually causes discontinuity in first derivative, not second \citep{Zahn1991}. -We could take a similar approach to the previous section but now considering the second kernel \(\mathcal{K}_{\rho,c^2}\) in Equation \ref{eq:kernels} since the discontinuity affects \(\rho\). However, \citet{Houdek.Gough2007} take another approach by considering the discontinuity as it appears in the acoustic cut-off frequency, \(\omega_{ac}\). The details of this derivation are beyond the scope of this work. Instead, we quote their result for the change in frequency induced by a density discontinuity at the base of the convective zone located at an acoustic depth of \(\tau_\bcz\) \citep[cf.][Eq. 17]{Houdek.Gough2007}, +We could take a similar approach to the previous section, now considering the second kernel \(\mathcal{K}_{\rho,c^2}\) since the discontinuity affects \(\rho\). However, \citet{Houdek.Gough2007} take another approach by considering the discontinuity as it appears in the acoustic cut-off frequency, \(\omega_{ac}\). The details of this derivation are beyond the scope of this work. Instead, we quote their result for the change in frequency induced by the BCZ at an acoustic depth of \(\tau_\bcz\) \citep[cf.][Eq. 17]{Houdek.Gough2007}, % \begin{equation} \left.\frac{\delta\omega}{\omega}\right|_{\bcz,\mathrm{osc}} \simeq \frac{c_\bcz^2\Delta_\bcz}{8\tau_0 \omega^3} \left(1 + {1}/{4\tau_0^2\omega^2}\right)^{-1/2} \cos\left[2(\tau_\bcz \omega + \epsilon_\bcz) + \tan^{-1}(2\tau_0\omega)\right] @@ -483,7 +491,7 @@ \subsection{Base of the Convective Zone Glitch}\label{sec:bcz-glitch} \Delta_\bcz = \left[\frac{\dd^2 \ln\rho}{\dd r^2}\right]_{-r_\bcz}^{+r_\bcz}, \end{equation} % -is the change in second derivative of density at the base of the convective zone (\(r = r_\bcz\)). +is the change in second density derivative at the BCZ (\(r = r_\bcz\)). For high-order modes, \(\tau_0 \omega \gg 1\) and thus \(\tan^{-1}(2\tau_0\omega) \simeq \pi/2\), and \((1 + {1}/{4\tau_0^2\omega^2})^{-1/2} \simeq 1\). Therefore, we can simplify this and write it in a more familiar form in terms of cyclic frequency, % @@ -493,4 +501,4 @@ \subsection{Base of the Convective Zone Glitch}\label{sec:bcz-glitch} % where \(\phi_\bcz\) is some approximately constant phase term. -Now we have equations for the effect of He\,\textsc{ii} ionisation and the BCZ for a given mode frequency \(\nu\), we can use them to model the glitch signatures as they appear in observed mode frequencies, \(\nu_{nl}\). Naturally, these equations are not exact. Several assumptions were made during their derivations. Therefore, if we want to model the glitch signature in \(\nu_{nl}\), we need some way of accounting for the uncertainty in the above equations. In the next chapter, we apply this model to characterise the glitch for a test star and real star. Although such models have been applied to other Sun-like stars before \citep[e.g.][]{Mazumdar.Monteiro.ea2014,Verma.Faria.ea2014}, we propose a new statistical model which uses a Gaussin process to account for the uncertainty in the functional form of \(\nu_{nl}\). +Now we have equations for the effect of He\,\textsc{ii} ionisation and the BCZ for a given mode frequency \(\nu\), we can use them to model the glitch signatures as they appear in observed mode frequencies, \(\nu_{nl}\). Naturally, these equations are not exact. Several assumptions were made during their derivations. Therefore, if we want to model the glitch signature in \(\nu_{nl}\), we need some way of accounting for the uncertainty in the above equations. In the next chapter, we apply this model to characterise the glitch for some example stars. Although such models have been applied to other Sun-like stars before \citep[e.g.][]{Mazumdar.Monteiro.ea2014,Verma.Faria.ea2014}, we propose a new statistical model to account for the uncertainty in the functional form of \(\nu_{nl}\) with \(n\). From a04785de8ed19eaa29cd25286d432256feee1a81 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Tue, 25 Apr 2023 17:33:58 +0100 Subject: [PATCH 19/50] Start proof reading glitch GP chapter --- chapters/glitch-gp.tex | 46 +++++++++++++++++++++--------------------- 1 file changed, 23 insertions(+), 23 deletions(-) diff --git a/chapters/glitch-gp.tex b/chapters/glitch-gp.tex index 78a5eef..359a897 100644 --- a/chapters/glitch-gp.tex +++ b/chapters/glitch-gp.tex @@ -12,7 +12,7 @@ % \chapter[Modelling Acoustic Glitches with a Gaussian Process]{Modelling Acoustic Glitch Signatures in Stellar Oscillations with a Gaussian Process}\label{chap:glitch-gp} -\textit{In this chapter, we apply a new method for modelling acoustic glitch signatures in the radial mode frequencies of solar-like oscillators. We compare our method with another using a model star with different levels of noise. Then, we apply both methods to the star 16 Cyg A to provide a real-world working example. We show that our method can be used to find the strength and location of glitches caused by the second ionisation of helium and the base of the convective zone. We also demonstrate that our method improves modelling the smoothly varying component of the mode frequencies.} +\textit{In this chapter, we apply a new method for modelling acoustic glitch signatures in the radial mode frequencies of solar-like oscillators. We compare our method with another using a model star with different levels of noise. Then, we apply both methods to the star 16 Cyg A to provide a real-world working example. We show that our method can be used to find the strength and location of glitches caused by the second ionisation of helium and the base of the convective zone. We also demonstrate that our method improves the treatment of the smoothly varying component of the mode frequencies.} \section{Introduction} @@ -24,14 +24,14 @@ \section{Introduction} % where \(\delta\nu\) is some function of frequency which may, for example, be evaluated at \(\tilde{f}(n, l)\). -In principle, \(\delta\nu\) could arise from any glitches expected for a particular star. For this work, we consider glitches in main sequence solar-like oscillators. Previously, we derived approximations for \(\delta\nu\) arising from acoustic glitches caused by the second ionisation of helium and the BCZ. Here, we choose to ignore contributions to \(\delta\nu\) from the first ionisation of helium. As a result, we let \(\delta\nu = \delta\nu_\helium + \delta\nu_\bcz\) where each component depends on parameters relating to properties of the glitches (cf. Equations \ref{eq:he-osc} and \ref{eq:bcz-osc}), +In principle, \(\delta\nu\) could arise from any glitches expected in a particular star. For this work, we consider glitches in main sequence solar-like oscillators. In Chapter \ref{chap:glitch}, we derived approximations for \(\delta\nu\) arising from acoustic glitches caused by the second ionisation of helium and the BCZ. For this work, we choose to ignore contributions to \(\delta\nu\) from the first ionisation of helium. As a result, we let \(\delta\nu = \delta\nu_\helium + \delta\nu_\bcz\) where each component depends on parameters relating to properties of the glitches (cf. Equations \ref{eq:he-osc} and \ref{eq:bcz-osc}), % \begin{align} \delta\nu_\helium &= \alpha_\helium \nu_0 \nu \, \ee^{-\beta_\helium \nu^2} \sin(4\pi\tau_\helium\nu + \phi_\helium), \label{eq:he-glitch}\\ \delta\nu_\bcz &= \alpha_\bcz \nu_0 \nu^{-2} \, \sin(4\pi\tau_\bcz\nu + \phi_\bcz). \label{eq:bcz-glitch} \end{align} % -The parameters \(\alpha_\helium \cong \Gamma_\heII\) and \(\beta_\helium \propto \Delta_\heII^2\) relate to the area and variance of the Gaussian-like depression in \(\gamma\) caused by the second ionisation of helium. The amplitude parameter for the BCZ glitch, \(\alpha_\bcz \propto \Delta_{\bcz}\) is proportional to the difference in the second density derivative at the base of the convective zone and has units of frequency squared. The approximate acoustic depths of second helium ionisation and the BCZ are given by \(\tau_\helium\) and \(\tau_\bcz\) respectively, and \(\phi\) represents an arbitrary phase constant. +The parameters \(\alpha_\helium \simeq \Gamma_\heII\) and \(\beta_\helium \propto \Delta_\heII^2\) relate to the area and variance of the Gaussian-like depression in \(\gamma\) caused by the second ionisation of helium. The amplitude parameter for the BCZ glitch, \(\alpha_\bcz \propto \Delta_{\bcz}\), is proportional to the difference in the second density derivative at the base of the convective zone and has units of frequency squared. The approximate acoustic depths of second helium ionisation and the BCZ are given by \(\tau_\helium\) and \(\tau_\bcz\) respectively, and \(\phi_\helium\) and \(\phi_\bcz\) are arbitrary phase constants. Providing \(\tilde{f}(n, l)\) is a good approximation of the mode frequencies, we can calculate \(\delta\nu\) at \(\nu = \tilde{f}(n, l)\) to predict \(f(n, l)\). For example, \(\tilde{f}(n,l)\) could be a \(K\)-th order polynomial in \(n\) with coefficients \(a_{lk}\) \citep[e.g.][]{Kjeldsen.Bedding.ea2005,Ulrich1986}, % @@ -39,11 +39,11 @@ \section{Introduction} \tilde{f}(n, l) = \nu_0 \sum_{k=0}^{K} a_{lk} n^k. \label{eq:poly} \end{equation} % -The linear component of this is equivalent to the asymptotic expression (cf. Equation \ref{eq:asy}) and the remaining terms describe curvature in the mode frequencies. However, there are drawbacks to using a polynomial for \(\tilde{f}(n, l)\). Whilst a polynomial with \(K = \infty\) can represent any function, this is impractical here. If \(K\) is too low, then it will not be flexible enough, biasing \(\delta\nu_\helium\) and \(\delta\nu_\bcz\). Yet, if \(K\) is too high, then it will over-fit and the glitch component may be missed. Regularising the polynomial is one solution to over-fitting, but this adds extra parameters to tune. Additionally, a finite polynomial represents only a small fraction of function space, leading to our model to be systematically biased to a particular functional form of \(\tilde{f}\). +The linear component of this is equivalent to the asymptotic expression (see Equation \ref{eq:asy}) and the remaining terms describe curvature in the mode frequencies. However, there are drawbacks to using a polynomial for \(\tilde{f}(n, l)\). Whilst a polynomial with \(K = \infty\) can represent any function, this is impractical here. If \(K\) is too low, then it will not be flexible enough, biasing the inference of \(\delta\nu_\helium\) and \(\delta\nu_\bcz\). Yet, if \(K\) is too high, then the model will over-fit and the glitch component may be missed. Regularising the polynomial is one solution to over-fitting, but this adds extra parameters to tune. Additionally, a finite polynomial represents only a small fraction of function space, leading to our model to be systematically biased to a particular functional form of \(\tilde{f}\). \defcitealias{Verma.Raodeo.ea2019}{V19} -Directly fitting the glitch the above way has been done before \citep[e.g.][]{Verma.Faria.ea2014,Verma.Raodeo.ea2017,Mazumdar.Monteiro.ea2014}. In this work, we will compare the most recent version of this method \citep[][hereafter V19]{Verma.Raodeo.ea2019} with a new method defined in this work. Our method will make use of a Gaussian Process to marginalise over our uncertainty in the functional form of \(f\). In the next section, we introduce the data used in comparing the two methods. Then, we define both modelling methods being compared in Section \ref{sec:glitch-methods}. Finally, we apply the methods to the data and present the results and discussion in Sections \ref{sec:glitch-results} and \ref{sec:glitch-disc}. +Directly fitting the glitch the above way has been done before \citep[e.g.][]{Verma.Faria.ea2014,Verma.Raodeo.ea2017,Mazumdar.Monteiro.ea2014}. In this work, we will compare the most recent version of this method \citep[][hereafter V19]{Verma.Raodeo.ea2019} with a new method. Our method will make use of a Gaussian Process (GP) to marginalise over our uncertainty in the functional form of \(f\). In the next section, we introduce the data used in comparing the two methods. Then, we define both modelling methods being compared in Section \ref{sec:glitch-methods}. Finally, we apply the methods to the data and present the results and discussion in Sections \ref{sec:glitch-results} and \ref{sec:glitch-disc}. \section{Data}\label{sec:glitch-data} @@ -63,9 +63,9 @@ \section{Data}\label{sec:glitch-data} \label{fig:glitch-test-obs} \end{figure} -\paragraph{Test Star} We created three sets of test data for worst-, better-, and best-case scenarios using stellar model S. We recall that stellar model S is similar to the Sun, with surface parameters of \(\teff = \SI{5682}{\kelvin}\), \(\log g = 4.426\) and \([\mathrm{Fe/H}] = 0.03\), and bulk parameters of \(M = \SI{1.00}{\solarmass}\), \(R = \SI{1.01}{\solarmass}\) and \(\mathrm{Age} = \SI{4.07}{\giga\year}\). We calculated radial order modes (\(l=0\)) for the test star using the \textsc{GYRE} oscillation code \citep{Townsend.Teitler2013}. We then selected different numbers of modes (\(N\)) symmetrically about a reference frequency, \(\nu_\mathrm{ref} = \SI{2900}{\micro\hertz}\) (close to the expected frequency at maximum power of the star). For each test case, we added differing amounts of Gaussian noise (scaled by \(\sigma_\obs\)) to the frequencies. The parameters and mode frequencies for each test case are shown in Table \ref{tab:glitch-obs} and the modes are plot in Figure \ref{fig:glitch-test-obs}. We can see the effect of the helium glitch in the echelle plots, where there is a large `wiggle' at low frequency. The effect of the glitch is visible in the better case and clearest in the best case scenario. +\paragraph{Test Star} We created three sets of test data for worst-, better-, and best-case scenarios using stellar model S (defined in Section \ref{sec:model-s}). We recall that model S is similar to the Sun, with surface parameters of \(\teff = \SI{5682}{\kelvin}\), \(\log g = 4.426\) and \([\mathrm{Fe/H}] = 0.03\), and bulk parameters of \(M = \SI{1.00}{\solarmass}\), \(R = \SI{1.01}{\solarmass}\) and \(t_\star = \SI{4.07}{\giga\year}\). We calculated radial order modes (\(l=0\)) for the test star using the \textsc{GYRE} oscillation code \citep{Townsend.Teitler2013}. We then selected different numbers of modes (\(N\)) symmetrically about a reference frequency, \(\nu_\mathrm{ref} = \SI{2900}{\micro\hertz}\) (close to the expected frequency at maximum power of the star). For each test case, we added differing amounts of Gaussian noise (scaled by \(\sigma_\obs\)) to the frequencies. The parameters and mode frequencies for each test case are shown in Table \ref{tab:glitch-obs}. We also plotted the modes on an echelle diagram in Figure \ref{fig:glitch-test-obs}. We can see the effect of the helium glitch in the echelle diagrams, where there is a large `wiggle' at low frequency. The glitch signature is visible in the better case and clearest in the best-case scenario. -\paragraph{16 Cyg A} We used the asteroseismic benchmark star 16 Cyg A as an example real star to test the methods. We adopted values for 16 radial mode frequencies identified by \citet{Lund.SilvaAguirre.ea2017} using observations from \emph{Kepler} \citep[][KIC 12069424]{Borucki.Koch.ea2010}. The mode frequencies and their associated uncertainties are given in Table \ref{tab:glitch-obs}. The glitch has been previously studied for 16 Cyg A with its binary companion 16 Cyg B in \citet{Verma.Faria.ea2014}, making it a useful subject for comparison. Similarly to model S, this target is a solar analogue. However, it is slightly hotter, more evolved and more metal-rich, with \(\teff = \SI{5825(50)}{\kelvin}\), \(\log g = \SI{4.33(7)}{\dex}\) and \([\mathrm{Fe/H}] = \SI{0.10(3)}{\dex}\) \citep{Ramirez.Melendez.ea2009}. Its bulk stellar parameters are \(M \approx \SI{1.1}{\solarmass}\), \(R \approx \SI{1.2}{\solarradius}\) and \(\mathrm{Age} \approx \SI{7}{\giga\year}\) \citep{SilvaAguirre.Lund.ea2017}. +\paragraph{16 Cyg A} We used the asteroseismic benchmark star 16 Cyg A as an example real star to test both methods. We adopted values for 16 radial mode frequencies identified by \citet{Lund.SilvaAguirre.ea2017} using observations from \emph{Kepler} \citep[][KIC 12069424]{Borucki.Koch.ea2010}. The mode frequencies and their associated uncertainties are given in Table \ref{tab:glitch-obs}. The glitch has been previously studied for 16 Cyg A with its binary companion 16 Cyg B in \citet{Verma.Faria.ea2014}, making it a useful subject for comparison. Similarly to model S, this target is a solar analogue. However, it is slightly hotter, more evolved and more metal-rich, with \(\teff = \SI{5825(50)}{\kelvin}\), \(\log g = \SI{4.33(7)}{\dex}\) and \([\mathrm{Fe/H}] = \SI{0.10(3)}{\dex}\) \citep{Ramirez.Melendez.ea2009}. Its bulk stellar parameters are \(M \approx \SI{1.1}{\solarmass}\), \(R \approx \SI{1.2}{\solarradius}\) and \(t_\star \approx \SI{7}{\giga\year}\) \citep{SilvaAguirre.Lund.ea2017}. \section{Methods}\label{sec:glitch-methods} @@ -79,7 +79,7 @@ \subsection{The V19 Method} f_A(n) = \tilde{f}_A(n) + \delta\nu_\helium + \delta\nu_\bcz; \quad \tilde{f}_A(n) = \sum_{k=0}^{4} b_k n^k, \end{equation} % -\sloppy where \(b_k \equiv a_{0k} \nu_0\) from Equation \ref{eq:poly}. The model parameters are given by \(\vect{\theta}_A = (b_0, \dots, b_4, a_\helium, \beta_\helium, \tau_\helium, \phi_\helium, a_\bcz, \tau_\bcz, \phi_\bcz)\), where the glitch amplitude parameters are modified to include \(\nu_0\) such that \(a_i \equiv \alpha_i\nu_0\). The \(\nu_0\) parameter is not explicitly included in the \citetalias{Verma.Raodeo.ea2019} model, but we find it useful to keep the scaling in mind because \(\nu_0\) is included in the GP model. +\sloppy where \(b_k \equiv a_{0k} \nu_0\) from Equation \ref{eq:poly}. The model parameters are given by \(\vect{\theta}_A = (b_0, \dots, b_4, a_\helium, \beta_\helium, \tau_\helium, \phi_\helium, a_\bcz, \tau_\bcz, \phi_\bcz)\), where the glitch amplitude parameters are modified to include \(\nu_0\) such that \(a_i \equiv \alpha_i\nu_0\). The \(\nu_0\) parameter is not explicitly included in the \citetalias{Verma.Raodeo.ea2019} model, but we find it useful to keep the scaling in mind and we include \(\nu_0\) in the GP model. The model parameters are optimised by minimising a \(\chi^2\) cost function with a regularisation term, % @@ -89,17 +89,17 @@ \subsection{The V19 Method} % where \(\nu_n^\obs\) and \(\sigma_n^\obs\) are the observed mode and its uncertainty at radial order \(n\), and \(\lambda\) is the regularisation parameter. The regularisation was introduced to avoid the polynomial over-fitting and absorbing the glitch terms. -We fitted the model using the \textsc{GlitchPy} code\footnote{\url{https://github.com/alexlyttle/GlitchPy}, adapted from \url{https://github.com/kuldeepv89/GlitchPy}.}. The fitting method is described in \citet{Verma.Raodeo.ea2019}, in which we adopted the same value for \(\lambda=7\) and bounds for the selection of initial parameters. We chose the initial parameters randomly within their bounds and optimised them using a BFGS minimisation of \(\chi^2\) \citep{Fletcher1987}, repeated 200 times until a global minimum was found. We repeated this for 1000 realisations of the observed \(\nu_n\) with Gaussian noise scaled by \(\sigma_n^\obs\) to obtain a range of possible solutions. +We fitted the model using the \textsc{GlitchPy} code\footnote{\url{https://github.com/alexlyttle/GlitchPy}, adapted from \url{https://github.com/kuldeepv89/GlitchPy}.}. The fitting method is described in \citet{Verma.Raodeo.ea2019}, in which we adopted the same value for \(\lambda=7\) and bounds for the selection of initial parameters. We chose the initial parameters randomly within their bounds and optimised them using a BFGS minimisation of \(\chi^2\) \citep{Fletcher1987}, repeated 200 times until a global minimum was found. We further repeated this for 1000 realisations of the observed \(\nu_n\) with Gaussian noise scaled by \(\sigma_n^\obs\) to obtain a range of possible solutions. \subsection{The GP Method} -In the second method, we used a Gaussian Process (GP) instead of a high-order polynomial to model the smooth varying component of the mode frequencies. The GP represents a probability distribution over function space, meaning that we can quantify the uncertainty associated with the functional form of $f$. We write our model for a set of modes \(\vect{n} = \{n_i\}_{i=1}^N\) as a random draw from a GP, +In the second method, we used a Gaussian Process (GP) instead of a high-order polynomial to model the smoothly varying component of the mode frequencies. The GP represents a probability distribution in function-space, meaning that we can quantify the uncertainty associated with the functional form of $f$. We write our model for a set of modes \(\vect{n} = \{n_i\}_{i=1}^N\) as a random draw from a GP, % \begin{equation} - f_{B}(\vect{n}) \sim \mathcal{GP}\left[ m(\vect{n}), k(\vect{n}, \vect{n}') \right],\text{\footnotemark} + f_{B}(\vect{n}) \sim \mathcal{GP}\left[ m(\vect{n}), k(\vect{n}, \vect{n}') \right],%\text{\footnotemark} \end{equation} % -\footnotetext{In this section we use the convention that some random variable \(y\) drawn from a distribution \(q\) given parameters \(x\) may be written as \(y \sim q(x)\).}% +% \footnotetext{In this section we use the convention that some random variable \(y\) drawn from a distribution \(q\) given parameters \(x\) may be written as \(y \sim q(x)\).}% where \(m\) and \(k\) are the mean and kernel functions, and \(\vect{n}'\) is any other set of radial orders, \(\vect{n}' = \{n'_j\}_{j=1}^M\). The mean function describes where we expect \(\nu_n\) to be given some approximate physical reasoning. Therefore, we define our mean function for any \(n\)-th order mode as, % \begin{equation} @@ -114,15 +114,15 @@ \subsection{The GP Method} k(n_i, n'_j) = \alpha_k \nu_0 \, \ee^{- (n_i - n'_j)^2 / 2\lambda_k^2}, \end{equation} % -where \(\alpha_k\) is a dimensionless amplitude scale factor and \(\lambda_k\) is the length scale (in units of radial order). Both kernel parameters control the flexibility of the GP. The amplitude parameter scales the covariance and \(\lambda_k\) describes the breadth of correlation between different modes. As \(\lambda_k \rightarrow 0\), the off-diagonal terms of the kernel approach zero producing Gaussian noise with a variance of \(\alpha_k\nu_0\). We found values of \(\alpha_k = 0.5\) and \(\lambda_k = 5\) predicted smoothly varying functions compatible with our expectation. +where \(\alpha_k\) is a dimensionless amplitude scale factor and \(\lambda_k\) is the length scale (in units of radial order). Both kernel parameters control the flexibility of the GP. The amplitude parameter scales the covariance and \(\lambda_k\) describes the breadth of correlation between different modes. As \(\lambda_k \rightarrow 0\), the off-diagonal terms of the kernel approach zero producing Gaussian noise with a variance of \(\alpha_k\nu_0\). We found values of \(\alpha_k = 0.5\) and \(\lambda_k = 5\) predicted smoothly varying functions compatible with our prior expectation. -The GP likelihood for some set of observations \(\vect{\nu}_\obs = \{\nu_{n_i}^\obs\}_{i=1}^N\) is a multivariate normal distribution centred on the mean function with covariance provided by the kernel function. We added Gaussian noise terms to the diagonal of the covariance matrix, +The GP likelihood for some set of observations \(\vect{\nu}_\obs = \{\nu_{n_i}^\obs\}_{i=1}^N\) is a multivariate normal distribution centred on the mean function with covariance provided by the kernel function. We also added Gaussian noise terms to the diagonal of the covariance matrix, % \begin{equation} \vect{\nu}_\obs \sim \mathcal{N}\left( \vect{\mu}, \, \vect{\Sigma} \right); \quad \vect{\Sigma} = \vect{K} + \mathrm{diag}(\sigma^2 + \vect{\sigma}_\obs^2), \label{eq:gp-like} \end{equation} % -where \(\vect{\mu} = m(\vect{n})\) and \(\vect{K} = k(\vect{n}, \vect{n})\). The scales of Gaussian noise in the model and observations are \(\sigma\) and \(\vect{\sigma}_\obs\) respectively. We included \(\sigma\) to account for noise in the model, for example from the difference between \(\delta\nu\) evaluated at \(\tilde{f}_B(n)\) and at the `true' mode frequencies. +where \(\vect{\mu} = m(\vect{n})\) and \(\vect{K} = k(\vect{n}, \vect{n})\). The scales of Gaussian noise in the model and observations are \(\sigma\) and \(\vect{\sigma}_\obs\) respectively. We included \(\sigma\) to account for uncorrelated noise in the model, for example from the difference between \(\delta\nu\) evaluated at \(\tilde{f}_B(n)\) and at the `true' mode frequencies. To make noiseless predictions of mode frequencies \(\vect{\nu}_\star\) at new radial orders \(\vect{n}_\star\), we drew from the following multivariate normal distribution \citep{Rasmussen.Williams2006}, % @@ -138,7 +138,7 @@ \subsection{The GP Method} % % This satisfies the consistency requirement that GP must specify any mode \(\nu_n \sim \mathcal{N}(\mu_n, K_{nn})\) where \(K_{nn}\) is the sub-matrix of the covariance of -To estimate the probability of our model parameters, \(\vect{\theta}_B = (\nu_0, \varepsilon, \alpha_\helium, \beta_\helium, \tau_\helium, \phi_\helium, a_\bcz, \tau_\bcz, \phi_\bcz, \sigma)\), given observations of mode frequencies, we used Bayes' theorem. Hence, we write the posterior probability density as, +We used Bayes' theorem (see Equation \ref{eq:bayes}) to estimate the probability of our model parameters, \(\vect{\theta}_B = (\nu_0, \varepsilon, \alpha_\helium, \beta_\helium, \tau_\helium, \phi_\helium, a_\bcz, \tau_\bcz, \phi_\bcz, \sigma)\), given observations of mode frequencies. Hence, we write the posterior probability density as, % \begin{equation} p(\vect{\theta}_B \mid \vect{\nu}_\obs) = \frac{p(\vect{\nu}_\obs \mid \vect{\theta}_B)\,p(\vect{\theta}_B)}{p(\vect{\nu}_\obs)} \equiv \frac{\mathcal{L}(\vect{\theta}_B)\,p(\vect{\theta}_B)}{\mathcal{Z}}, @@ -149,12 +149,12 @@ \subsection{The GP Method} Following Equation \ref{eq:gp-like}, we defined the log-likelihood of the model as, % \begin{equation} - \ln\mathcal{L}_B = - \frac12 \left[ {(\vect{\nu}_\obs - \vect{\mu})^\mathsf{T} \vect{\Sigma}^{\,-1} (\vect{\nu}_\obs - \vect{\mu})} + \ln(2 \pi | \vect{\Sigma} | ) \right], + \ln\mathcal{L}_B = - \frac12 \left[ {(\vect{\nu}_\obs - \vect{\mu})^\mathsf{T} \vect{\Sigma}^{\,-1} (\vect{\nu}_\obs - \vect{\mu})} + \ln(| \vect{\Sigma} |) + N\ln(2 \pi) \right], \end{equation} % -where \(\vect{\mu}\) and \(\vect{\Sigma}\) depend on \(\vect{\theta}_B\). The prior transform for \(\vect{\theta}_B\) is the inverse cumulative distribution function associated with the prior distribution \(p(\vect{\theta}_B)\). We defined the prior independently for each of \(\vect{\theta}_B\) such that the total prior is the product of prior distributions for each parameter, \(p(\vect{\theta}_B) = \prod_j p(\theta_{j})\). In the following paragraphs, we specify our choice of prior distributions for each model parameter. +where \(\vect{\mu}\) and \(\vect{\Sigma}\) depend on \(\vect{\theta}_B\). The prior transform for \(\vect{\theta}_B\) is the inverse cumulative distribution function associated with the prior distribution \(p(\vect{\theta}_B)\). We defined the prior independently for each of \(\vect{\theta}_B\) such that the total prior is the product of prior distributions for each parameter, \(p(\vect{\theta}_B) = \prod_j p(\theta_{j})\). In the following paragraphs, we specify our choice of prior distributions for each model parameter. % We also summarise the prior distributions and their parameters in Table \todo{Table summary} -Starting with the parameters for \(\tilde{f}_B(n)\), we chose to sample them from normal distributions, +\paragraph{Linear Background} Starting with the parameters for \(\tilde{f}_B(n)\), we chose to sample them from normal distributions, % \begin{gather*} \nu_0 \sim \mathcal{N}(\overline{\nu}_0, s_{\nu_0}^2), \quad \varepsilon \sim \mathcal{N}(\overline{\varepsilon}, s_\varepsilon^2),%\\ @@ -165,14 +165,14 @@ \subsection{The GP Method} % Priors for the following parameters follow a log-normal distribution to ensure they are positive. We also exploit the property that the scale of a normal distribution in natural log-space is approximately the scale in real-space as a fraction of the distribution mean. -We derived a prior on \(\tau_\helium \cong \tau_\heII\) and \(\tau_\bcz\) by observing how they scale with acoustic radius \(\tau_0\) in the grid of stellar models from \citet{Lyttle.Davies.ea2021} and in \citet{Verma.Rorsted.ea2022}. Typically, the fraction depth of He\,\textsc{ii} ionisation \(\tau_\heII/\tau_0 \approx 0.2\) and BCZ \(\tau_\bcz/\tau_0 \approx 0.6\) for main sequence solar-like oscillators (see, e.g. Figure \ref{fig:gamma-sound-speed}). Therefore, we form our prior distributions from these relations using the relation \({\tau}_0 = (2\nu_0)^{-1}\), with an additional spread of 20 per cent to account for variance with stellar properties. To ensure the parameters remain positive, we defined their priors in natural log-space, +\paragraph{Acoustic Depths} We derived a prior on \(\tau_\helium \cong \tau_\heII\) and \(\tau_\bcz\) by observing how they scale with acoustic radius \(\tau_0\) in the grid of stellar models from \citet{Lyttle.Davies.ea2021} and in \citet{Verma.Rorsted.ea2022}. Typically, the fraction depth of He\,\textsc{ii} ionisation \(\tau_\heII/\tau_0 \approx 0.2\) and BCZ \(\tau_\bcz/\tau_0 \approx 0.6\) for main sequence solar-like oscillators (see, e.g. Figure \ref{fig:gamma-sound-speed}). Therefore, we form our prior distributions from these relations using the relation \({\tau}_0 = (2\nu_0)^{-1}\), with an additional spread of 20 per cent to account for variance with stellar properties. To ensure the parameters remain positive, we defined their priors in natural log-space, % \begin{gather*} \ln\tau_\helium \sim \mathcal{N}\left[ \ln(\overline{\tau}_\helium), \, s_{\ln\tau}^2 \right], \quad \ln\tau_\bcz \sim \mathcal{N}\left[\ln(3\overline{\tau}_\helium), \, s_{\ln\tau}^2 \right],\\ \overline{\tau}_\helium = (10 \overline{\nu}_0)^{-1}, \quad s_{\ln\tau}^2 = (1/5)^2 + (s_{\nu_0}/\overline{\nu}_0)^2, \end{gather*} -A prior on the glitch amplitude parameters is less trivial. By observing \(\gamma\) profiles in the grid of stellar models, we assume that the width of the helium ionisation region is about 8 per cent of the acoustic depth of the region, \(\Delta_\heII/\tau_\heII \approx 0.08\). Thus, we centre the prior on \(\overline{\beta}_\helium\) obtained from this assumption and the relation \(\beta_\helium = 8\pi^2\Delta_\heII^2\). We then propagate the variance from the prior on \(\tau_\helium\), +\paragraph{Helium Glitch Amplitude} A prior on the glitch amplitude parameters is less trivial. By observing \(\gamma\) profiles in the grid of stellar models, we assume that the width of the helium ionisation region is about 8 per cent of the acoustic depth of the region, \(\Delta_\heII/\tau_\heII \approx 0.08\). Thus, we centre the prior on \(\overline{\beta}_\helium\) obtained from this assumption and the relation \(\beta_\helium = 8\pi^2\Delta_\heII^2\). We then propagate the variance from the prior on \(\tau_\helium\), % \begin{equation*} \ln\beta_\helium \sim \mathcal{N}\left[ \ln(\overline{\beta}_\helium), \, 4 s_{\ln\tau}^2 \right], \quad \overline{\beta}_\helium = \frac{32}{625} \, \pi^2 \overline{\tau}_\helium^2. @@ -186,7 +186,7 @@ \subsection{The GP Method} % We verified that the priors on \(\alpha_\helium\) and \(\beta_\helium\) are appropriate by checking that the prior amplitude at \(\nu_\mathrm{ref}\) peaks between \SIrange{0}{1}{\micro\hertz}, decaying thereafter. -We chose to formulate the prior on \(\alpha_\bcz\) such that the BCZ glitch amplitude is approximately \SI{0.1}{\micro\hertz} at \(\nu_\mathrm{ref}\) (an order of magnitude smaller than the helium glitch). This occurs when \(\alpha_\bcz/\nu_0 \approx \SI{30}{\micro\hertz}\). We scaled the distribution by 80 per cent to account for uncertainty in our subjective choice of prior, +\paragraph{BCZ Glitch Amplitude} We chose to formulate the prior on \(\alpha_\bcz\) such that the BCZ glitch amplitude is approximately \SI{0.1}{\micro\hertz} at \(\nu_\mathrm{ref}\) (an order of magnitude smaller than the helium glitch). This occurs when \(\alpha_\bcz/\nu_0 \approx \SI{30}{\micro\hertz}\). We scaled the distribution by 80 per cent to account for uncertainty in our subjective choice of prior, % \begin{equation*} \ln\alpha_\bcz \sim \mathcal{N}\left[ \ln 30\overline{\nu}_0, \, (4/5)^2 + (s_{\nu_0}/\overline{\nu}_0)^2\right]. @@ -200,7 +200,7 @@ \subsection{The GP Method} \input{tables/glitch-test-prior.tex} \end{table} -The mean and variance for the aforementioned prior distributions are summarised in Table \ref{tab:glitch-prior} for the test star and 16 Cyg A. Prior distributions for the remaining parameters were the same for all stars. For example, the prior on the phase parameters \(\phi_\helium\) and \(\phi_\bcz\) was uniformly distributed from 0 to \(2\pi\). We also used an uninformative prior on the model uncertainty, \(\ln\sigma \sim \mathcal{N}( - \ln 100, 4)\), centred on an uncertainty of \SI{0.01}{\micro\hertz}. +The mean and variance for the aforementioned prior distributions are summarised in Table \ref{tab:glitch-prior} for the test star and 16 Cyg A. Prior distributions for the remaining parameters were the same for all stars. For example, the prior on the phase parameters \(\phi_\helium\) and \(\phi_\bcz\) was uniformly distributed from 0 to \(2\pi\). We also used an uninformative prior on the model uncertainty, \(\ln\sigma \sim \mathcal{N}( - \ln 100, 4)\), centred on an uncertainty of \SI{0.01}{\micro\hertz}. \todo{Add these to table also?} We sampled the posterior distribution using the nested sampling package \textsc{Dynesty} \citep{Speagle2020,Koposov.Speagle.ea2023}. We applied the multi-ellipsoid bounding method \citep{Feroz.Hobson.ea2009} with 500 live points and the random walk sampling method \citep{Skilling2006} with a minimum of 50 steps before proposing a new live point. We enabled periodic boundary conditions for \(\phi_\helium\) and \(\phi_\bcz\) with the prior transform projecting to periodic space using \(\phi = \phi'\,\mathrm{mod}\,2\pi\), where \(\phi'\) is unconstrained. Otherwise, the nested sampler ran with its default parameters. We used \textsc{jax} \citep{Bradbury.Frostig.ea2018} to make use of accelerated linear algebra (XLA) and just-in-time (JIT) compilation, and the \textsc{tinygp} package\footnote{\url{https://github.com/dfm/tinygp}} for building the GP. For analysis and comparison with the \citetalias{Verma.Raodeo.ea2019} method, we drew 1000 points randomly from the posterior samples according to their estimated weights. From b1683d1d7363b93a4331607b946b94e225b60fd9 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Wed, 26 Apr 2023 11:18:28 +0100 Subject: [PATCH 20/50] Add restructured prior table and fix tense --- chapters/glitch-gp.tex | 50 ++++++++++++++++++++-------------- references.bib | 32 +++++++++++----------- tables/glitch-test-prior-2.tex | 17 ++++++++++++ 3 files changed, 63 insertions(+), 36 deletions(-) create mode 100644 tables/glitch-test-prior-2.tex diff --git a/chapters/glitch-gp.tex b/chapters/glitch-gp.tex index 359a897..53d89e9 100644 --- a/chapters/glitch-gp.tex +++ b/chapters/glitch-gp.tex @@ -93,7 +93,7 @@ \subsection{The V19 Method} \subsection{The GP Method} -In the second method, we used a Gaussian Process (GP) instead of a high-order polynomial to model the smoothly varying component of the mode frequencies. The GP represents a probability distribution in function-space, meaning that we can quantify the uncertainty associated with the functional form of $f$. We write our model for a set of modes \(\vect{n} = \{n_i\}_{i=1}^N\) as a random draw from a GP, +In the second method, we used a Gaussian Process \citep[GP; see][for a recent review]{Aigrain.Foreman-Mackey2022} instead of a high-order polynomial to model the smoothly varying component of the mode frequencies. The GP represents a probability distribution in function-space, meaning that we can quantify the uncertainty associated with the functional form of $f$. We write our model for a set of modes \(\vect{n} = \{n_i\}_{i=1}^N\) as a random draw from a GP, % \begin{equation} f_{B}(\vect{n}) \sim \mathcal{GP}\left[ m(\vect{n}), k(\vect{n}, \vect{n}') \right],%\text{\footnotemark} @@ -146,63 +146,73 @@ \subsection{The GP Method} % where \(\mathcal{L}(\vect{\theta}_B)\) is the likelihood of the data given the model, \(p(\vect{\theta}_B)\) is the prior probability density of the model parameters, and \(\mathcal{Z}\) is the `evidence' (or normalisation). The true posterior density function cannot be derived analytically, so we estimated it with the `nested sampling' algorithm \citep{Skilling2004}. By evaluating the prior volume at contours of constant likelihood, nested sampling estimates \(\mathcal{Z}\) to weight samples according to their posterior probability. This method requires functions for the log-likelihood and a transform which maps random samples in the unit hypercube to prior parameter space. +\begin{table} + \centering + % \caption{The mean and variance for the prior normal distributions on each parameter where values are not explicitly given in the text.} + \caption[The GP model parameters and their prior distributions.]{The GP model parameters and their prior distributions. Normal distributions are parametrised by the mean and variance (i.e. \(\mathcal{N}(\mu,\sigma^2)\)). Uniform distributions are parametrised by their lower and uppower bounds.} + \label{tab:glitch-prior} + \input{tables/glitch-test-prior-2.tex} +\end{table} + Following Equation \ref{eq:gp-like}, we defined the log-likelihood of the model as, % \begin{equation} \ln\mathcal{L}_B = - \frac12 \left[ {(\vect{\nu}_\obs - \vect{\mu})^\mathsf{T} \vect{\Sigma}^{\,-1} (\vect{\nu}_\obs - \vect{\mu})} + \ln(| \vect{\Sigma} |) + N\ln(2 \pi) \right], \end{equation} % -where \(\vect{\mu}\) and \(\vect{\Sigma}\) depend on \(\vect{\theta}_B\). The prior transform for \(\vect{\theta}_B\) is the inverse cumulative distribution function associated with the prior distribution \(p(\vect{\theta}_B)\). We defined the prior independently for each of \(\vect{\theta}_B\) such that the total prior is the product of prior distributions for each parameter, \(p(\vect{\theta}_B) = \prod_j p(\theta_{j})\). In the following paragraphs, we specify our choice of prior distributions for each model parameter. % We also summarise the prior distributions and their parameters in Table \todo{Table summary} +where \(\vect{\mu}\) and \(\vect{\Sigma}\) depend on \(\vect{\theta}_B\). The prior transform for \(\vect{\theta}_B\) is the inverse cumulative distribution function associated with the prior distribution \(p(\vect{\theta}_B)\). We defined the prior independently for each of \(\vect{\theta}_B\). The total prior distribution is the product of prior distributions for each parameter, \(p(\vect{\theta}_B) = \prod_j p(\theta_{j})\). We give the prior distributions and their shape parameters in Table \ref{tab:glitch-prior}. In the following paragraphs, we justify our choice of prior distributions based on approximate scaling relations between the parameters. % We also summarise the prior distributions and their parameters in Table \todo{Table summary} -\paragraph{Linear Background} Starting with the parameters for \(\tilde{f}_B(n)\), we chose to sample them from normal distributions, +\paragraph{Smooth Component} Starting with the parameters for \(\tilde{f}_B(n)\), we chose to sample them from normal distributions, % \begin{gather*} \nu_0 \sim \mathcal{N}(\overline{\nu}_0, s_{\nu_0}^2), \quad \varepsilon \sim \mathcal{N}(\overline{\varepsilon}, s_\varepsilon^2),%\\ % \phi_\helium, \phi_\bcz \sim \mathcal{U}\left(0, 2\pi\right), \end{gather*} % -centred on \(\overline{\nu}_0\) and \(\overline{\varepsilon}\) and scaled by \(s_{\nu_0}\) and \(s_\varepsilon\). In practice, the location and scale parameters could come from global estimates of \(\langle\Delta\nu_n\rangle\) or a linear fit to the modes. For the test stars, we determined \(\overline{\nu}_0\) and \(\overline{\varepsilon}\) from a linear fit to the true mode frequencies and added representative uncertainties of 10 and 5 per cent respectively. For 16 Cyg A, we used measurements of \(\langle\Delta\nu_n\rangle\) and \(\nu_{\max}\) from \citet{Lund.SilvaAguirre.ea2017} to estimate \(\overline{\nu}_0\) and \(\overline{\varepsilon}\). +centred on \(\overline{\nu}_0\) and \(\overline{\varepsilon}\) and scaled by \(s_{\nu_0}\) and \(s_\varepsilon\). For example, the location and scale parameters could come from global estimates of \(\langle\Delta\nu_n\rangle\) or a linear fit to the modes. For the test stars, we determined \(\overline{\nu}_0\) and \(\overline{\varepsilon}\) from a linear fit to the true mode frequencies and added representative uncertainties of 10 and 5 per cent respectively. For 16 Cyg A, we used measurements of \(\langle\Delta\nu_n\rangle\) and \(\nu_{\max}\) from \citet{Lund.SilvaAguirre.ea2017} to estimate \(\overline{\nu}_0\) and \(\overline{\varepsilon}\). % Priors for the following parameters follow a log-normal distribution to ensure they are positive. We also exploit the property that the scale of a normal distribution in natural log-space is approximately the scale in real-space as a fraction of the distribution mean. -\paragraph{Acoustic Depths} We derived a prior on \(\tau_\helium \cong \tau_\heII\) and \(\tau_\bcz\) by observing how they scale with acoustic radius \(\tau_0\) in the grid of stellar models from \citet{Lyttle.Davies.ea2021} and in \citet{Verma.Rorsted.ea2022}. Typically, the fraction depth of He\,\textsc{ii} ionisation \(\tau_\heII/\tau_0 \approx 0.2\) and BCZ \(\tau_\bcz/\tau_0 \approx 0.6\) for main sequence solar-like oscillators (see, e.g. Figure \ref{fig:gamma-sound-speed}). Therefore, we form our prior distributions from these relations using the relation \({\tau}_0 = (2\nu_0)^{-1}\), with an additional spread of 20 per cent to account for variance with stellar properties. To ensure the parameters remain positive, we defined their priors in natural log-space, +\paragraph{Acoustic Depths} We derived a prior on \(\tau_\helium\) and \(\tau_\bcz\) by observing how they scale with acoustic radius \(\tau_0\) in the grid of stellar models from \citet{Lyttle.Davies.ea2021} and in \citet{Verma.Rorsted.ea2022}. Typically, the fractional depth of He\,\textsc{ii} ionisation \(\tau_\heII/\tau_0 \approx 0.2\) and BCZ \(\tau_\bcz/\tau_0 \approx 0.6\) for main sequence solar-like oscillators (see, e.g. Figure \ref{fig:gamma-sound-speed}). We formed the prior distributions from these relations, estimating the acoustic radius from \({\tau}_0 = (2\nu_0)^{-1}\). To account for possible variance with stellar properties, we added a spread of 20 per cent. We defined their priors in natural log-space to ensure they remained positive, % \begin{gather*} - \ln\tau_\helium \sim \mathcal{N}\left[ \ln(\overline{\tau}_\helium), \, s_{\ln\tau}^2 \right], \quad \ln\tau_\bcz \sim \mathcal{N}\left[\ln(3\overline{\tau}_\helium), \, s_{\ln\tau}^2 \right],\\ - \overline{\tau}_\helium = (10 \overline{\nu}_0)^{-1}, \quad s_{\ln\tau}^2 = (1/5)^2 + (s_{\nu_0}/\overline{\nu}_0)^2, + \ln\tau_\helium \sim \mathcal{N}\left[ \ln(\overline{\tau}_\helium), \, s_{\ln\tau}^2 \right], \quad \ln\tau_\bcz \sim \mathcal{N}\left[\ln(3\,\overline{\tau}_\helium), \, s_{\ln\tau}^2 \right],\\ + \overline{\tau}_\helium = (10\,\overline{\nu}_0)^{-1}, \quad s_{\ln\tau}^2 = (1/5)^2 + (s_{\nu_0}/\overline{\nu}_0)^2, \end{gather*} -\paragraph{Helium Glitch Amplitude} A prior on the glitch amplitude parameters is less trivial. By observing \(\gamma\) profiles in the grid of stellar models, we assume that the width of the helium ionisation region is about 8 per cent of the acoustic depth of the region, \(\Delta_\heII/\tau_\heII \approx 0.08\). Thus, we centre the prior on \(\overline{\beta}_\helium\) obtained from this assumption and the relation \(\beta_\helium = 8\pi^2\Delta_\heII^2\). We then propagate the variance from the prior on \(\tau_\helium\), +\paragraph{Helium Glitch Amplitude} The prior on the glitch amplitude parameters was less trivial. By observing \(\gamma\) profiles in the grid of stellar models, we assumed that the width of the helium ionisation region was about 8 per cent of the acoustic depth of the region, \(\Delta_\heII/\tau_\heII \approx 0.08\). Thus, we centred the prior on \(\overline{\beta}_\helium\) obtained from this assumption and used the relation \(\beta_\helium = 8\pi^2\Delta_\heII^2\) from comparison to Equation \ref{eq:he-osc}. We then propagated the variance from the prior on \(\tau_\helium\), % \begin{equation*} \ln\beta_\helium \sim \mathcal{N}\left[ \ln(\overline{\beta}_\helium), \, 4 s_{\ln\tau}^2 \right], \quad \overline{\beta}_\helium = \frac{32}{625} \, \pi^2 \overline{\tau}_\helium^2. \end{equation*} % -Then, we centre the prior for \(\alpha_\helium\) to satisfy a depth of 0.1 in \(\delta\gamma/\gamma\) caused by helium ionisation using Equation \ref{eq:he-gamma}, +Then, we centred the prior for \(\alpha_\helium\) to satisfy a depth of 0.1 in \(\delta\gamma/\gamma\) caused by helium ionisation using Equation \ref{eq:he-gamma}, % \begin{gather*} \ln\alpha_\helium \sim \mathcal{N}\left[ 1/2 \, \ln({\overline{\beta}_\helium}/{400\pi}), \, s_{\ln\tau}^2 \right]. \end{gather*} % -We verified that the priors on \(\alpha_\helium\) and \(\beta_\helium\) are appropriate by checking that the prior amplitude at \(\nu_\mathrm{ref}\) peaks between \SIrange{0}{1}{\micro\hertz}, decaying thereafter. +We verified that the priors on \(\alpha_\helium\) and \(\beta_\helium\) were appropriate by checking that the prior amplitude at \(\nu_\mathrm{ref}\) peaks from \SIrange{0}{1}{\micro\hertz}, decaying thereafter. This was compatible with our prior belief for the helium glitch amplitude. -\paragraph{BCZ Glitch Amplitude} We chose to formulate the prior on \(\alpha_\bcz\) such that the BCZ glitch amplitude is approximately \SI{0.1}{\micro\hertz} at \(\nu_\mathrm{ref}\) (an order of magnitude smaller than the helium glitch). This occurs when \(\alpha_\bcz/\nu_0 \approx \SI{30}{\micro\hertz}\). We scaled the distribution by 80 per cent to account for uncertainty in our subjective choice of prior, +\paragraph{BCZ Glitch Amplitude} We constructed the prior on \(\alpha_\bcz\) such that the BCZ glitch amplitude was approximately \SI{0.1}{\micro\hertz} at \(\nu_\mathrm{ref}\) (an order of magnitude smaller than the helium glitch). This occurred when \(\alpha_\bcz/\nu_0 \approx \SI{30}{\micro\hertz}\). We also scaled the distribution by an additional 80 per cent, % \begin{equation*} - \ln\alpha_\bcz \sim \mathcal{N}\left[ \ln 30\overline{\nu}_0, \, (4/5)^2 + (s_{\nu_0}/\overline{\nu}_0)^2\right]. + \ln\alpha_\bcz \sim \mathcal{N}\left[ \ln(30\,\overline{\nu}_0), \, (4/5)^2 + (s_{\nu_0}/\overline{\nu}_0)^2\right]. \end{equation*} % -\begin{table} - \centering - \caption{The mean and variance for the prior normal distributions on each parameter where values are not explicitly given in the text.} - \label{tab:glitch-prior} - \input{tables/glitch-test-prior.tex} -\end{table} +% \begin{table} +% \centering +% % \caption{The mean and variance for the prior normal distributions on each parameter where values are not explicitly given in the text.} +% \caption[The GP model parameters and their prior distributions.]{The GP model parameters and their prior distributions. Normal distributions are parametrised by the mean and variance (i.e. \(\mathcal{N}(\mu,\sigma^2)\)). Uniform distributions are parametrised by their lower and uppower bounds.} +% \label{tab:glitch-prior} +% \input{tables/glitch-test-prior-2.tex} +% \end{table} -The mean and variance for the aforementioned prior distributions are summarised in Table \ref{tab:glitch-prior} for the test star and 16 Cyg A. Prior distributions for the remaining parameters were the same for all stars. For example, the prior on the phase parameters \(\phi_\helium\) and \(\phi_\bcz\) was uniformly distributed from 0 to \(2\pi\). We also used an uninformative prior on the model uncertainty, \(\ln\sigma \sim \mathcal{N}( - \ln 100, 4)\), centred on an uncertainty of \SI{0.01}{\micro\hertz}. \todo{Add these to table also?} +% The mean and variance for the aforementioned prior distributions are summarised in Table \ref{tab:glitch-prior} for the test star and 16 Cyg A. +Prior distributions for the remaining parameters were the same for both stars. For example, the prior on the phase parameters \(\phi_\helium\) and \(\phi_\bcz\) was uniformly distributed from 0 to \(2\pi\). We also used a weakly informative prior on the model uncertainty, \(\ln\sigma \sim \mathcal{N}( - \ln 100, 4)\), centred on an uncertainty of \SI{0.01}{\micro\hertz}. -We sampled the posterior distribution using the nested sampling package \textsc{Dynesty} \citep{Speagle2020,Koposov.Speagle.ea2023}. We applied the multi-ellipsoid bounding method \citep{Feroz.Hobson.ea2009} with 500 live points and the random walk sampling method \citep{Skilling2006} with a minimum of 50 steps before proposing a new live point. We enabled periodic boundary conditions for \(\phi_\helium\) and \(\phi_\bcz\) with the prior transform projecting to periodic space using \(\phi = \phi'\,\mathrm{mod}\,2\pi\), where \(\phi'\) is unconstrained. Otherwise, the nested sampler ran with its default parameters. We used \textsc{jax} \citep{Bradbury.Frostig.ea2018} to make use of accelerated linear algebra (XLA) and just-in-time (JIT) compilation, and the \textsc{tinygp} package\footnote{\url{https://github.com/dfm/tinygp}} for building the GP. For analysis and comparison with the \citetalias{Verma.Raodeo.ea2019} method, we drew 1000 points randomly from the posterior samples according to their estimated weights. +We sampled the posterior distribution using the nested sampling package \textsc{Dynesty} \citep{Speagle2020,Koposov.Speagle.ea2023}. We applied the multi-ellipsoid bounding method \citep{Feroz.Hobson.ea2009} with 500 live points and the random walk sampling method \citep{Skilling2006} with a minimum of 50 steps before proposing a new live point. In addition, we enabled periodic boundary conditions for \(\phi_\helium\) and \(\phi_\bcz\) with the prior transform projecting to periodic space using \(\phi = \phi'\,\mathrm{mod}\,2\pi\), where \(\phi'\) is unconstrained. Otherwise, the nested sampler ran with its default parameters. We used \textsc{jax} \citep{Bradbury.Frostig.ea2018} to make use of accelerated linear algebra (XLA) and just-in-time (JIT) compilation, and \textsc{tinygp} \citep{Foreman-Mackey.Yadav.ea2022} to build the GP. For analysis and comparison with the \citetalias{Verma.Raodeo.ea2019} method, we drew 1000 points randomly from the posterior samples according to their estimated weights. % Rearranged into dimensionless quantities, \(f = \nu/\nu_0\), \(t = \tau/\tau_0\), \(a_\helium = \nu_0\alpha_\helium\), \(b_\helium = \nu_0 \beta_\helium\), and \(a_\bcz = \alpha_\bcz/\nu_0^2\), % % diff --git a/references.bib b/references.bib index 72014e0..01ed4f4 100644 --- a/references.bib +++ b/references.bib @@ -146,7 +146,6 @@ @article{Albert2020 adsnote = {Provided by the SAO/NASA Astrophysics Data System}, adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv201215286A}, archiveprefix = {arxiv}, - eid = {arXiv:2012.15286}, keywords = {Astrophysics - Instrumentation and Methods for Astrophysics} } @@ -1397,7 +1396,6 @@ @article{Clevert.Unterthiner.ea2015 abstract = {We introduce the "exponential linear unit" (ELU) which speeds up learning in deep neural networks and leads to higher classification accuracies. Like rectified linear units (ReLUs), leaky ReLUs (LReLUs) and parametrized ReLUs (PReLUs), ELUs alleviate the vanishing gradient problem via the identity for positive values. However, ELUs have improved learning characteristics compared to the units with other activation functions. In contrast to ReLUs, ELUs have negative values which allows them to push mean unit activations closer to zero like batch normalization but with lower computational complexity. Mean shifts toward zero speed up learning by bringing the normal gradient closer to the unit natural gradient because of a reduced bias shift effect. While LReLUs and PReLUs have negative values, too, they do not ensure a noise-robust deactivation state. ELUs saturate to a negative value with smaller inputs and thereby decrease the forward propagated variation and information. Therefore, ELUs code the degree of presence of particular phenomena in the input, while they do not quantitatively model the degree of their absence. In experiments, ELUs lead not only to faster learning, but also to significantly better generalization performance than ReLUs and LReLUs on networks with more than 5 layers. On CIFAR-100 ELUs networks significantly outperform ReLU networks with batch normalization while batch normalization does not improve ELU networks. ELU networks are among the top 10 reported CIFAR-10 results and yield the best published result on CIFAR-100, without resorting to multi-view evaluation or model averaging. On ImageNet, ELU networks considerably speed up learning compared to a ReLU network with the same architecture, obtaining less than 10\% classification error for a single crop, single model network.}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, archiveprefix = {arxiv}, - eid = {arXiv:1511.07289}, keywords = {Computer Science - Machine Learning} } @@ -1412,7 +1410,6 @@ @article{Coc.Uzan.ea2013 abstract = {Primordial or Big Bang nucleosynthesis (BBN) is one of the three historical strong evidences for the Big-Bang model together with the expansion of the Universe and the Cosmic Microwave Background radiation (CMB). The recent results by the Planck mission have slightly changed the estimate of the baryonic density Omega\_b, compared to the previous WMAP value. This article updates the BBN predictions for the light elements using the new value of Omega\_b determined by Planck, as well as an improvement of the nuclear network and new spectroscopic observations. While there is no major modification, the error bars of the primordial D/H abundance (2.67+/-0.09) x 10\^\{-5\} are narrower and there is a slight lowering of the primordial Li/H abundance (4.89\^+0.41\_-0.39) x 10\^\{-10\}. However, this last value is still -0.5ex\textasciitilde 3 times larger than its observed spectroscopic abundance in halo stars of the Galaxy. Primordial Helium abundance is now determined to be Y\_p = 0.2463+/-0.0003.}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, archiveprefix = {arxiv}, - eid = {arXiv:1307.6955}, keywords = {Astrophysics - Cosmology and Extragalactic Astrophysics} } @@ -1669,7 +1666,6 @@ @article{Dillon.Langmore.ea2017 abstract = {The TensorFlow Distributions library implements a vision of probability theory adapted to the modern deep-learning paradigm of end-to-end differentiable computation. Building on two basic abstractions, it offers flexible building blocks for probabilistic computation. Distributions provide fast, numerically stable methods for generating samples and computing statistics, e.g., log density. Bijectors provide composable volume-tracking transformations with automatic caching. Together these enable modular construction of high dimensional distributions and transformations not possible with previous libraries (e.g., pixelCNNs, autoregressive flows, and reversible residual networks). They are the workhorse behind deep probabilistic programming systems like Edward and empower fast black-box inference in probabilistic models built on deep-network components. TensorFlow Distributions has proven an important part of the TensorFlow toolkit within Google and in the broader deep learning community.}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, archiveprefix = {arxiv}, - eid = {arXiv:1711.10604}, keywords = {Computer Science - Artificial Intelligence,Computer Science - Machine Learning,Computer Science - Programming Languages,Statistics - Machine Learning} } @@ -2003,7 +1999,7 @@ @article{Foreman-Mackey.Hogg.ea2013 } @misc{Foreman-Mackey.Hogg.ea2021, - title = {Daft-Dev/Daft: Daft v0.1.2}, + title = {{{daft-dev/daft: daft v0.1.2}}}, author = {{Foreman-Mackey}, Dan and Hogg, David W. and Fulford, David S. and {daft-bot} and Dobos, L{\'a}szl{\'o} and McFee, Brian and Murphy, Kevin P and Lindemann, Oliver and Gerold, Pierre and Agrawal, Varun}, year = {2021}, month = mar, @@ -2011,6 +2007,18 @@ @misc{Foreman-Mackey.Hogg.ea2021 howpublished = {Zenodo} } +@misc{Foreman-Mackey.Yadav.ea2022, + title = {{{dfm/tinygp: v0.2.3}}}, + shorttitle = {Dfm/Tinygp}, + author = {{Foreman-Mackey}, Dan and Yadav, Sachin and {theorashid} and Fowlie, Andrew and Tronsgaard, Ren{\'e} and Schmerler, Steve and Killestein, Thomas}, + year = {2022}, + month = oct, + doi = {10.5281/zenodo.7269074}, + urldate = {2023-04-26}, + abstract = {What's Changed Fixing \#87 by @dfm in https://github.com/dfm/tinygp/pull/88 found two typos by @theorashid in https://github.com/dfm/tinygp/pull/92 Updating notebook execution config to remove warnings by @dfm in https://github.com/dfm/tinygp/pull/93 Minor Typos by @yadav-sachin in https://github.com/dfm/tinygp/pull/99 fix typo in docs by @andrewfowlie in https://github.com/dfm/tinygp/pull/107 Update tree\_map -{$>$} tree\_util.tree\_map to avoid FutureWarning by @tkillestein in https://github.com/dfm/tinygp/pull/114 Checking tree structure and shapes of X\_test input to condition by @dfm in https://github.com/dfm/tinygp/pull/119 Removing deprecation warning and jitting predict by @dfm in https://github.com/dfm/tinygp/pull/120 Fixing the gradient of the L2 distance at the origin by @dfm in https://github.com/dfm/tinygp/pull/121 Adding check for unsorted input coordinates when using QuasisepSolver by @dfm in https://github.com/dfm/tinygp/pull/123 Fixing behavior of DotProduct kernel on scalar inputs by @dfm in https://github.com/dfm/tinygp/pull/124 Release candidate v0.2.3 by @dfm in https://github.com/dfm/tinygp/pull/125 New Contributors @andrewfowlie made their first contribution in https://github.com/dfm/tinygp/pull/107 @tkillestein made their first contribution in https://github.com/dfm/tinygp/pull/114 Full Changelog: https://github.com/dfm/tinygp/compare/v0.2.2...v0.2.3}, + howpublished = {Zenodo} +} + @article{Frankel.Sanders.ea2020, title = {Keeping {{It Cool}}: {{Much Orbit Migration}}, yet {{Little Heating}}, in the {{Galactic Disk}}}, shorttitle = {Keeping {{It Cool}}}, @@ -2336,7 +2344,7 @@ @incollection{Gough1993 title = {Linear Adiabatic Stellar Pulsation.}, booktitle = {Astrophysical {{Fluid Dynamics}}. {{Les Houches Session LXVII}}}, author = {Gough, D. O.}, - editor = {Zahn, J. -P. and {Zinn-Justin}, J.}, + editor = {Zahn, J.-P. and {Zinn-Justin}, J.}, year = {1993}, month = jan, series = {Lecture {{Notes}} of the {{Les Houches Summer School}}}, @@ -2764,7 +2772,6 @@ @article{Hogg.Bovy.ea2010 abstract = {We go through the many considerations involved in fitting a model to data, using as an example the fit of a straight line to a set of points in a two-dimensional plane. Standard weighted least-squares fitting is only appropriate when there is a dimension along which the data points have negligible uncertainties, and another along which all the uncertainties can be described by Gaussians of known variance; these conditions are rarely met in practice. We consider cases of general, heterogeneous, and arbitrarily covariant two-dimensional uncertainties, and situations in which there are bad data (large outliers), unknown uncertainties, and unknown but expected intrinsic scatter in the linear relationship being fit. Above all we emphasize the importance of having a "generative model" for the data, even an approximate one. Once there is a generative model, the subsequent fitting is non-arbitrary because the model permits direct computation of the likelihood of the parameters or the posterior probability distribution. Construction of a posterior probability distribution is indispensible if there are "nuisance parameters" to marginalize away.}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, archiveprefix = {arxiv}, - eid = {arXiv:1008.4686}, keywords = {Astrophysics - Instrumentation and Methods for Astrophysics,Physics - Data Analysis,Statistics and Probability} } @@ -2813,7 +2820,6 @@ @article{Hogg2012 abstract = {In this pedagogical text aimed at those wanting to start thinking about or brush up on probabilistic inference, I review the rules by which probability distribution functions can (and cannot) be combined. I connect these rules to the operations performed in probabilistic data analysis. Dimensional analysis is emphasized as a valuable tool for helping to construct non-wrong probabilistic statements. The applications of probability calculus in constructing likelihoods, marginalized likelihoods, posterior probabilities, and posterior predictions are all discussed.}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, archiveprefix = {arxiv}, - eid = {arXiv:1205.4446}, keywords = {Astrophysics - Instrumentation and Methods for Astrophysics,Physics - Data Analysis,Statistics and Probability} } @@ -3313,7 +3319,6 @@ @article{Kingma.Ba2014 abstract = {We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and/or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and/or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, archiveprefix = {arxiv}, - eid = {arXiv:1412.6980}, keywords = {Computer Science - Machine Learning} } @@ -3417,7 +3422,7 @@ @article{Kjeldsen.Bedding1995 } @misc{Koposov.Speagle.ea2023, - title = {Joshspeagle/Dynesty: V2.1.0}, + title = {{{joshspeagle/dynesty: v2.1.0}}}, shorttitle = {Joshspeagle/Dynesty}, author = {Koposov, Sergey and Speagle, Josh and Barbary, Kyle and Ashton, Gregory and Bennett, Ed and Buchner, Johannes and Scheffler, Carl and Cook, Ben and Talbot, Colm and Guillochon, James and Cubillos, Patricio and Ramos, Andr{\'e}s Asensio and Johnson, Ben and Lang, Dustin and Ilya and Dartiailh, Matthieu and Nitz, Alex and McCluskey, Andrew and Archibald, Anne and Deil, Christoph and {Foreman-Mackey}, Dan and Goldstein, Danny and Tollerud, Erik and Leja, Joel and Kirk, Matthew and Pitkin, Matt and Sheehan, Patrick and Cargile, Phillip and Patel, Ruskin and Angus, Ruth}, year = {2023}, @@ -3816,7 +3821,6 @@ @article{Masters.Luschi2018 abstract = {Modern deep neural network training is typically based on mini-batch stochastic gradient optimization. While the use of large mini-batches increases the available computational parallelism, small batch training has been shown to provide improved generalization performance and allows a significantly smaller memory footprint, which might also be exploited to improve machine throughput. In this paper, we review common assumptions on learning rate scaling and training duration, as a basis for an experimental comparison of test performance for different mini-batch sizes. We adopt a learning rate that corresponds to a constant average weight update per gradient calculation (i.e., per unit cost of computation), and point out that this results in a variance of the weight updates that increases linearly with the mini-batch size \$m\$. The collected experimental results for the CIFAR-10, CIFAR-100 and ImageNet datasets show that increasing the mini-batch size progressively reduces the range of learning rates that provide stable convergence and acceptable test performance. On the other hand, small mini-batch sizes provide more up-to-date gradient calculations, which yields more stable and reliable training. The best performance has been consistently obtained for mini-batch sizes between \$m = 2\$ and \$m = 32\$, which contrasts with recent work advocating the use of mini-batch sizes in the thousands.}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, archiveprefix = {arxiv}, - eid = {arXiv:1804.07612}, keywords = {Computer Science - Computer Vision and Pattern Recognition,Computer Science - Machine Learning,Statistics - Machine Learning} } @@ -4193,7 +4197,7 @@ @article{Morton.Winn2014 } @misc{Morton2015a, - title = {Isochrones: {{Stellar}} Model Grid Package}, + title = {{{isochrones: Stellar model grid package}}}, author = {Morton, Timothy D.}, year = {2015}, month = mar, @@ -4566,7 +4570,6 @@ @article{Peimbert2008 abstract = {I present a brief review on the determination of the primordial helium abundance by unit mass, Yp. I discuss the importance of the primordial helium abundance in: (a) cosmology, (b) testing the standard big bang nucleosynthesis, (c) studying the physical conditions in H II regions, (d) providing the initial conditions for stellar evolution models, and (e) testing the galactic chemical evolution models.}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, archiveprefix = {arxiv}, - eid = {arXiv:0811.2980}, keywords = {Astrophysics} } @@ -4600,7 +4603,6 @@ @article{Phan.Pradhan.ea2019 adsnote = {Provided by the SAO/NASA Astrophysics Data System}, adsurl = {https://ui.adsabs.harvard.edu/abs/2019arXiv191211554P}, archiveprefix = {arxiv}, - eid = {arXiv:1912.11554}, keywords = {Computer Science - Artificial Intelligence,Computer Science - Machine Learning,Computer Science - Programming Languages,G.3,I.2.5,Statistics - Machine Learning} } @@ -4710,7 +4712,6 @@ @article{Ramachandran.Zoph.ea2017 abstract = {The choice of activation functions in deep networks has a significant effect on the training dynamics and task performance. Currently, the most successful and widely-used activation function is the Rectified Linear Unit (ReLU). Although various hand-designed alternatives to ReLU have been proposed, none have managed to replace it due to inconsistent gains. In this work, we propose to leverage automatic search techniques to discover new activation functions. Using a combination of exhaustive and reinforcement learning-based search, we discover multiple novel activation functions. We verify the effectiveness of the searches by conducting an empirical evaluation with the best discovered activation function. Our experiments show that the best discovered activation function, \$f(x) = x \textbackslash cdot \textbackslash text\{sigmoid\}(\textbackslash beta x)\$, which we name Swish, tends to work better than ReLU on deeper models across a number of challenging datasets. For example, simply replacing ReLUs with Swish units improves top-1 classification accuracy on ImageNet by 0.9\textbackslash\% for Mobile NASNet-A and 0.6\textbackslash\% for Inception-ResNet-v2. The simplicity of Swish and its similarity to ReLU make it easy for practitioners to replace ReLUs with Swish units in any neural network.}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, archiveprefix = {arxiv}, - eid = {arXiv:1710.05941}, keywords = {Computer Science - Computer Vision and Pattern Recognition,Computer Science - Machine Learning,Computer Science - Neural and Evolutionary Computing} } @@ -4939,7 +4940,6 @@ @article{Ruder2016 abstract = {Gradient descent optimization algorithms, while increasingly popular, are often used as black-box optimizers, as practical explanations of their strengths and weaknesses are hard to come by. This article aims to provide the reader with intuitions with regard to the behaviour of different algorithms that will allow her to put them to use. In the course of this overview, we look at different variants of gradient descent, summarize challenges, introduce the most common optimization algorithms, review architectures in a parallel and distributed setting, and investigate additional strategies for optimizing gradient descent.}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, archiveprefix = {arxiv}, - eid = {arXiv:1609.04747}, keywords = {Computer Science - Machine Learning} } diff --git a/tables/glitch-test-prior-2.tex b/tables/glitch-test-prior-2.tex new file mode 100644 index 0000000..823c23a --- /dev/null +++ b/tables/glitch-test-prior-2.tex @@ -0,0 +1,17 @@ +\begin{tabular}{lll} +\toprule +Parameter & \multicolumn{2}{c}{Prior} \\ +\midrule + & Test Star & 16 Cyg A \\ +\midrule +$\nu_0/\si{\micro\hertz}$ & $\mathcal{N}(132.8,0.01)$ & $\mathcal{N}(103.28,0.0025)$ \\ +$\varepsilon$ & $\mathcal{N}(1.4,0.0025)$ & $\mathcal{N}(1.45,0.0025)$ \\ +$\ln(\alpha_\mathrm{cz}/\si{\micro\hertz\squared})$ & $\mathcal{N}(8.29,0.64)$ & $\mathcal{N}(8.04,0.64)$ \\ +$\ln(\alpha_\mathrm{He}/\si{\mega\second})$ & $\mathcal{N}(-11.1,0.04)$ & $\mathcal{N}(-10.85,0.04)$ \\ +$\ln(\beta_\mathrm{He}/\si{\mega\second\squared})$ & $\mathcal{N}(-15.07,0.16)$ & $\mathcal{N}(-14.56,0.16)$ \\ +$\ln(\tau_\mathrm{cz}/\si{\mega\second})$ & $\mathcal{N}(-6.09,0.04)$ & $\mathcal{N}(-5.84,0.04)$ \\ +$\ln(\tau_\mathrm{He}/\si{\mega\second})$ & $\mathcal{N}(-7.19,0.04)$ & $\mathcal{N}(-6.94,0.04)$ \\ +$\ln(\sigma/\si{\micro\hertz})$ & $\mathcal{N}(- 4.605, 4)$ & $\mathcal{N}(- 4.605, 4)$ \\ +$\phi_\helium, \phi_\bcz$ & $\mathcal{U}(0, 2\pi)$ & $\mathcal{U}(0, 2\pi)$ \\ +\bottomrule +\end{tabular} From b7147b9a38a2872d53f5de3e8b4e019d66a99735 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Wed, 26 Apr 2023 13:22:09 +0100 Subject: [PATCH 21/50] Remove glossaries --- thesis.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/thesis.tex b/thesis.tex index fbb01f0..389189e 100644 --- a/thesis.tex +++ b/thesis.tex @@ -133,8 +133,8 @@ \tableofcontents \listoffigures % Figures \listoftables % Tables - \printglossary[type=\acronymtype,title={List of Acronyms}] % Acronyms - \printglossary[title={List of Terms}] % Terms + % \printglossary[type=\acronymtype,title={List of Acronyms}] % Acronyms + % \printglossary[title={List of Terms}] % Terms \mainmatter From 4d90a4598da68074269362f5f0adfb1e1d50ccb3 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Wed, 26 Apr 2023 13:22:26 +0100 Subject: [PATCH 22/50] Tidy up discussion and conclusion --- chapters/glitch-gp.tex | 29 +++++++++++++++-------------- 1 file changed, 15 insertions(+), 14 deletions(-) diff --git a/chapters/glitch-gp.tex b/chapters/glitch-gp.tex index 53d89e9..ab9abdf 100644 --- a/chapters/glitch-gp.tex +++ b/chapters/glitch-gp.tex @@ -238,7 +238,7 @@ \subsection{Test Star} \label{fig:glitch-test-signal} \end{figure} -In Figure \ref{fig:glitch-test-signal}, we plot the predicted glitch component, \(\delta\nu\), using 50 draws from the posterior distribution on the model parameters for both methods. We subtracted the median background component from the observed \(\nu_n\) over-plot. For each scenario, both methods predicted similar median glitch components. However, the \citetalias{Verma.Raodeo.ea2019} method showed more extreme multimodality in all cases. For example, the better case showed a high-amplitude (\(\sim \SI{2}{\micro\hertz}\)) BCZ glitch solution which is not present with the GP method. Such a large signal would indicate a highly unlikely density discontinuity at the BCZ. Additionally, the two methods differed by \(\sim \SI{1}{\micro\hertz}\) at low frequency in the best case. Despite this, the \citetalias{Verma.Raodeo.ea2019} method appeared to be more confident in its glitch solutions compared to the GP method. +In Figure \ref{fig:glitch-test-signal}, we plot the predicted total glitch component (\(\delta\nu\)) using 50 draws from the posterior distribution for both methods. We subtracted the median smooth component (\(\tilde{f}(n)\)) from the observed \(\nu_n\) over-plot. For each scenario, both methods predicted similar glitch components. However, the \citetalias{Verma.Raodeo.ea2019} method showed more extreme multimodality in all cases. For example, the better-case showed a high-amplitude (\(\sim \SI{2}{\micro\hertz}\)) BCZ glitch solution which is not present with the GP method. Such a large BCZ signature would be unlikely. Additionally, the two methods differed by \(\sim \SI{1}{\micro\hertz}\) at low frequency in the best case. Despite this, the \citetalias{Verma.Raodeo.ea2019} method appeared to be more confident in its glitch solutions compared to the GP method. \begin{figure}[!tb] \centering @@ -247,9 +247,10 @@ \subsection{Test Star} \label{fig:glitch-test-tau} \end{figure} -We plotted posterior distributions of the glitch acoustic depths, \(\tau_\helium\) and \(\tau_\bcz\), in Figure \ref{fig:glitch-test-tau} and compared them to the sound speed gradient of the test star from Figure \ref{fig:sound-speed-gradient}. We expect the acoustic depths to approximately line up with the sharp structural changes. For the worst case, both methods gave broad distributions for the acoustic depths, compatible with their respective initial guesses and priors. The \citetalias{Verma.Raodeo.ea2019} method initial guesses appeared to underestimate \(\tau_\bcz\), whereas the GP method prior was broad enough to encompass a wide range of possible \(\tau_\bcz\). In the better and best cases, we found that the \citetalias{Verma.Raodeo.ea2019} had strong multimodality. For example, the better case found solutions for \(\tau_\helium\) far deeper into the star than we would expect, at around \SI{1500}{\second} and \SI{2500}{\second}. +In Figure \ref{fig:glitch-test-tau}, we plotted posterior distributions for the glitch acoustic depths, \(\tau_\helium\) and \(\tau_\bcz\), and compared them to the sound speed gradient of the test star from Figure \ref{fig:sound-speed-gradient}. We expect the acoustic depths to approximately line up with the sharp structural changes. For the worst case, both methods gave broad distributions for the acoustic depths, compatible with their respective initial guesses and priors. The \citetalias{Verma.Raodeo.ea2019} method initial guesses appeared to underestimate \(\tau_\bcz\), whereas the GP method prior was broad enough to encompass a wide range of possible \(\tau_\bcz\). In the better and best cases, we found that the \citetalias{Verma.Raodeo.ea2019} solutions were multimodal. For example, the better-case found solutions for \(\tau_\helium\) far deeper into the star than we would expect, at around \SI{1500}{\second} and \SI{2500}{\second}. -As predicted by \citet{Houdek.Gough2007} and shown in \citet{Verma.Faria.ea2014}, the the values for \(\tau_\helium\) obtained were under-predicted compared to the location of the trough due to the second ionisation of helium. This is because \(\delta\nu_\helium\) does not include the smaller glitch component due to the first ionisation of helium, located at a smaller \(\tau\). We can see this for the best star fit with the \citetalias{Verma.Raodeo.ea2019} method which finds \(\tau_\helium = \SI{619(15)}{\second}\). The depression in \(\gamma\) due to He\,\textsc{ii} ionisation in the respective stellar model is located at \SI{733}{\second}. On the other hand, the GP method was closer with \(\tau_\helium = \SI{696(19)}{\second}\). +% As predicted by \citet{Houdek.Gough2007} and shown in \citet{Verma.Faria.ea2014}... This is because \(\delta\nu_\helium\) does not include the smaller glitch component due to the first ionisation of helium, located at a smaller \(\tau\). +The the values for \(\tau_\helium\) obtained were under-predicted compared to the location of the trough due to the second ionisation of helium. We can see this for the best star fit with the \citetalias{Verma.Raodeo.ea2019} method which finds \(\tau_\helium = \SI{619(15)}{\second}\). The depression in \(\gamma\) due to He\,\textsc{ii} ionisation in the respective stellar model is located at \SI{733}{\second}. On the other hand, the GP method was closer with \(\tau_\helium = \SI{696(19)}{\second}\). \begin{figure}[tb] \centering @@ -258,7 +259,7 @@ \subsection{Test Star} \label{fig:glitch-test-amplitude} \end{figure} -We compared helium glitch amplitudes at \(\nu_\mathrm{ref}\) from both methods in Figure \ref{fig:glitch-test-amplitude}. The \citetalias{Verma.Raodeo.ea2019} method preferred low-amplitude solutions for the worst and better cases than the best case compared to the GP method. On the other hand, the GP method reflects our prior in the worst case, with the width of the distribution shrinking as the data improves. The GP method does show some bi-modality in the better case, with a solution at \(A_\helium^\mathrm{ref} \approx 0.7\) which was not found by the \citetalias{Verma.Raodeo.ea2019} method. In the best case, the \citetalias{Verma.Raodeo.ea2019} method obtained \(A_\helium^\mathrm{ref} = 0.347_{-0.005}^{+0.006}\), whereas the GP method found \(A_\helium^\mathrm{ref} = 0.296_{-0.036}^{+0.042}\). +We compared helium glitch amplitudes at \(\nu_\mathrm{ref}\) from both methods in Figure \ref{fig:glitch-test-amplitude}. The \citetalias{Verma.Raodeo.ea2019} method preferred low-amplitude solutions for the worst and better cases than the best case compared to the GP method. On the other hand, the GP method reflects our prior in the worst case, with the width of the distribution shrinking as the data improves. The GP method does show some bi-modality in the better case, with higher solutions at \(A_\helium^\mathrm{ref} \approx 0.7\) which were not found by the \citetalias{Verma.Raodeo.ea2019} method. In the best case, the \citetalias{Verma.Raodeo.ea2019} method obtained \(A_\helium^\mathrm{ref} = 0.347_{-0.005}^{+0.006}\), whereas the GP method found \(A_\helium^\mathrm{ref} = 0.296_{-0.036}^{+0.042}\). % Why not use delta gamma / gamma as the probe of helium abundance? That is proportional to alpha / sqrt(beta). Then, an update to this method can use this and beta as a parameter and then work out alpha from that, since beta should scale with tau and the depth is a signature of helium abundance. Be careful as number of modes correlates with delta nu value of fit. @@ -266,9 +267,9 @@ \subsection{16 Cyg A} We compared the results from both methods applied to the 16 Cyg A data in Figure \ref{fig:glitch-16cyga}. We found that the \citetalias{Verma.Raodeo.ea2019} method predicted extreme solutions for the glitch where the GP method did not. There was also a difference of about \SI{1}{\micro\hertz} at the low frequency end such that the \citetalias{Verma.Raodeo.ea2019} predicted a larger glitch amplitude. Otherwise, the two methods gave similar predictions for the glitch function. -Both methods found similar values for \(\tau_\helium\) but relatively different distributions for \(\tau_\bcz\). Regarding the helium glitch, the \citetalias{Verma.Raodeo.ea2019} method found \(\tau_\helium = 917_{-53}^{+50} \, \mathrm{s}\), and our GP method obtained \(\tau_\helium = 931_{-88}^{+59} \, \mathrm{s}\), within 1-\(\sigma\) of each other. As seen in the previous section, our method found a slightly larger value of \(\tau_\helium\) than the \citetalias{Verma.Raodeo.ea2019} method, although not significantly in this case. \citet{Verma.Faria.ea2014} fit the glitch with \(l=0,1,2\) modes and found an acoustic depth of \SI{930(14)}{\second}, within 1-\(\sigma\) of both methods in this work. +Both methods found similar values for \(\tau_\helium\) but relatively different distributions for \(\tau_\bcz\). Regarding the helium glitch, the \citetalias{Verma.Raodeo.ea2019} method found \(\tau_\helium = 917_{-53}^{+50} \, \mathrm{s}\), and our GP method obtained \(\tau_\helium = 931_{-88}^{+59} \, \mathrm{s}\), within 1-\(\sigma\) of each other. Similarly to with the test star, the GP method found a slightly larger value of \(\tau_\helium\) than the \citetalias{Verma.Raodeo.ea2019} method, although not significantly in this case. \citet{Verma.Faria.ea2014} fit the glitch with \(l=0,1,2\) modes and found an acoustic depth of \SI{930(14)}{\second}, within 1-\(\sigma\) of both methods in this work. -We calculated the helium glitch amplitude at a reference frequency of \SI{2188.5}{\micro\hertz}, equivalent to the value of \(\nu_{\max}\) obtained by \citet{Lund.SilvaAguirre.ea2017}. Samples from both posteriors are shown in the bottom panel of Figure \ref{fig:glitch-16cyga}. We found \(A_\helium^\mathrm{ref} = 0.260_{-0.065}^{+0.050}\) for the \citetalias{Verma.Raodeo.ea2019} method, and \(A_\helium^\mathrm{ref} = 0.333_{-0.073}^{+0.081}\) for the GP method. Both values were about 1-\(\sigma\) apart. Additionally, both methods found \(A_\bcz^\mathrm{ref} \sim 0.1\), with the GP method favouring a smaller value. The \citetalias{Verma.Raodeo.ea2019} method found some solutions with \(A_\bcz^\mathrm{ref}\) larger than \(A_\helium^\mathrm{ref}\), something we would not expect because the modes are less sensitive to structural changes deeper in the star. +We calculated the helium glitch amplitude at a reference frequency of \SI{2188.5}{\micro\hertz}, equivalent to the value of \(\nu_{\max}\) obtained by \citet{Lund.SilvaAguirre.ea2017}. Samples from both posteriors are shown in the bottom panel of Figure \ref{fig:glitch-16cyga}. We found \(A_\helium^\mathrm{ref} = 0.260_{-0.065}^{+0.050}\,\si{\micro\hertz}\) for the \citetalias{Verma.Raodeo.ea2019} method, and \(A_\helium^\mathrm{ref} = 0.333_{-0.073}^{+0.081}\,\si{\micro\hertz}\) for the GP method. Both values were about 1-\(\sigma\) apart. Additionally, both methods found \(A_\bcz^\mathrm{ref} \sim 0.1\,\si{\micro\hertz}\), with the GP method favouring a smaller value. The \citetalias{Verma.Raodeo.ea2019} method found some solutions with \(A_\bcz^\mathrm{ref}\) larger than \(A_\helium^\mathrm{ref}\), something we would not expect because the modes are less sensitive to structural changes deeper in the star. \begin{figure}[!tb] \centering @@ -288,7 +289,7 @@ \section{Discussion}\label{sec:glitch-disc} \label{fig:best-smooth} \end{figure} -Throughout this work, we found a smaller \(\delta\nu\) amplitude at low frequency with the GP method than with the \citetalias{Verma.Raodeo.ea2019} method. This was particularly visible in the best case and in 16 Cyg A. We expected this was a result of the different smooth background models. In Figure \ref{fig:best-smooth} we plotted the smooth component of each model extended to lower order, unobserved modes. We found the GP background component has a turning point at \(\nu \approx \SI{1900}{\micro\hertz}\), higher than the \citetalias{Verma.Raodeo.ea2019} method at \(\nu \approx \SI{1500}{\micro\hertz}\). The smooth component of the \citetalias{Verma.Raodeo.ea2019} method was confidently incorrect outside of the observed frequencies. Conversely, the GP method predicted closer to the truth with increasing uncertainty further from the observations. Therefore, it appears that the GP provided a more accurate representation of the true underlying function than the polynomial. +Throughout this work, we found a smaller \(\delta\nu\) amplitude at low frequency with the GP method than with the \citetalias{Verma.Raodeo.ea2019} method. This was particularly visible in the best case and in 16 Cyg A. We expected this was a result of the different smooth background models. In Figure \ref{fig:best-smooth} we plotted the smooth component of each model extended to lower order, unobserved modes. We found the GP background component had a turning point at \(\nu \approx \SI{1900}{\micro\hertz}\) which was higher than the \citetalias{Verma.Raodeo.ea2019} method at \(\nu \approx \SI{1500}{\micro\hertz}\). The smooth component of the \citetalias{Verma.Raodeo.ea2019} method was confidently incorrect outside of the observed frequencies. Conversely, the GP method predicted closer to the truth with increasing uncertainty further from the observations. It appeared that the GP provided a more accurate representation of the underlying function than the polynomial. \begin{figure}[!tb] \centering @@ -297,23 +298,23 @@ \section{Discussion}\label{sec:glitch-disc} \label{fig:smooth-res} \end{figure} -In the test star's best case and 16 Cyg A, the GP method found a higher \(\tau_\helium\) than the \citetalias{Verma.Raodeo.ea2019} method. The GP was possibly better able to distinguish between He\,\textsc{i} and He\,\textsc{ii} ionisation compared to the polynomial. To verify this, we plotted the difference between each model's predictions with the glitch subtracted in Figure \ref{fig:smooth-res}. We saw a clear oscillatory signal in the differences. The period of this signal in the test star corresponded to an acoustic depth of \(\sim \SI{500}{\second}\), matching the location of He\,\textsc{i} ionisation in model S. The oscillatory signal was less pronounced for 16 Cyg A, but corresponded to a plausible He\,\textsc{i} acoustic depth of \(\sim \SI{700}{\second}\). The polynomial was not flexible enough to pick up this signal, thus lowering its inferred value for \(\tau_\helium\). As a result, the GP method more accurately determined the acoustic depth of the second helium ionisation zone. +In the test star's best case and 16 Cyg A, the GP method found a higher \(\tau_\helium\) than the \citetalias{Verma.Raodeo.ea2019} method. This could be because the GP was better able to distinguish between the He\,\textsc{ii} glitch and the smaller He\,\textsc{i} glitch. To explore this, we plotted the difference between each model's predictions for the smooth component in Figure \ref{fig:smooth-res}. We saw a clear periodic signal in the differences. The period of this signal in the test star corresponded to an acoustic depth of \(\sim \SI{500}{\second}\), matching the location of He\,\textsc{i} ionisation in model S. The periodic signal was less pronounced for 16 Cyg A, but corresponded to a plausible He\,\textsc{i} acoustic depth of \(\sim \SI{700}{\second}\). We expect that the polynomial was not flexible enough to pick up the He\,\textsc{i} ionisation signature, hence lowering its mean value for \(\tau_\helium\). % Discuss the prior -The \citetalias{Verma.Raodeo.ea2019} method found several extreme solutions for \(\delta\nu\) whereas GP method did not. We expected this because the GP method used a prior on the model parameters. We tested relaxing the prior on \(\vect{\theta}_B\) and recovered similar solutions to the \citetalias{Verma.Raodeo.ea2019} method, showing that the prior helped eliminate unrealistic solutions. Therefore, care should be taken over the choice of prior on \(\vect{\theta}_B\) to realistically reflect our expectation. If some of our prior assumptions are incorrect they may bias the results. For example, the outer convective region gets shallower as stars get hotter (approaching \(\teff \approx \SI{7000}{\kelvin}\)) making the assumption that \(\tau_\bcz/\tau_0 \approx 0.6\) an overestimate in these cases. We accommodate for this with a wide prior on \(\tau_\bcz\), but this could be improved. There is a notable temperature dependence to \(\tau_\bcz/\tau_0\) observed in the grid of stellar models which could be exploited in the future when constructing the prior. +The \citetalias{Verma.Raodeo.ea2019} method found several extreme solutions for \(\delta\nu\) whereas GP method did not. We expected this because the GP method used a prior over the model parameters. We tested relaxing the prior on \(\vect{\theta}_B\) and found similar multimodal posteriors to the \citetalias{Verma.Raodeo.ea2019} method. This showed that the prior helped eliminate unrealistic solutions. However, care should be taken over the choice of prior on \(\vect{\theta}_B\) to realistically reflect our expectation. If some of our prior assumptions are incorrect they may bias the results. For example, the outer convective region gets shallower as stars get hotter (approaching \(\teff \approx \SI{7000}{\kelvin}\)) making the assumption that \(\tau_\bcz/\tau_0 \approx 0.6\) an overestimate in these cases. We accommodate for this with a wide prior on \(\tau_\bcz\), but our prior could be more informed. There is a notable temperature dependence to \(\tau_\bcz/\tau_0\) observed in the grid of stellar models which could be exploited in the future when constructing the prior. -Additionally, the samples for \(\alpha_\helium\) and \(\beta_\helium\) were correlated in both methods. This was expected, because larger values of \(\beta_\helium\) can be compensated for with a larger amplitude factor \(\alpha_\helium\). In the GP method, we did not include this expected correlation in our prior. Hence, we found that having broader priors on \(\alpha_\helium\) and \(\beta_\helium\) lead to the prior predicting unrealistic glitches. In future work, we could devise a multivariate prior for the amplitude parameters to account for this. However, we note that our approach still improves on the \citetalias{Verma.Raodeo.ea2019} method by using a prior. +Additionally, the joint posteriors for \(\alpha_\helium\) and \(\beta_\helium\) were correlated in both methods. This was expected, because larger values of \(\beta_\helium\) can be compensated for with a larger amplitude factor \(\alpha_\helium\). In the GP method, we did not include this expected correlation in our prior. Hence, we found that having broader priors on \(\alpha_\helium\) and \(\beta_\helium\) lead to the prior predicting unrealistic glitches. We could improve upon this by using multivariate prior for the amplitude parameters to account for this. Despite this, our approach still improves on the \citetalias{Verma.Raodeo.ea2019} method by using a prior in the first place. -A potential limitation of the GP method is that we calculate the glitch at \(\tilde{f}_B(n)\) from the linear asymptotic equation (Equation \ref{eq:asy}). In regions where the gradient of \(\delta(\nu)\) is high, the difference between \(\tilde{f}_B(n)\) and the true frequency can be up to (\(\sim \SI{0.1}{\micro\hertz}\)). The GP kernel function cannot absorb this difference because is varies on a short length-scale. Instead, we accounted for this uncertainty by adding Gaussian noise to the model parametrised by \(\sigma\). However, for the best test star, we found \(\sigma \approx 0.05\), which was larger than the observational uncertainty, \(\sigma_\obs = 0.01\). This could limit our method's inference ability for the best asteroseismic targets. One solution is to replace \(\tilde{f}_B(n)\) with a quadratic \citep[e.g.][]{Nielsen.Davies.ea2021} which better approximates \(\nu_n\). However, the GP kernel would need to be adjusted to account for this. We leave this for the next iteration of this method. +A potential limitation of the GP method is that we calculate the glitch at \(\tilde{f}_B(n)\) from the linear asymptotic equation (Equation \ref{eq:asy}). In regions where the gradient of \(\delta\nu\) is high, the difference between \(\tilde{f}_B(n)\) and the true frequency can be up to \(\sim \SI{0.1}{\micro\hertz}\). Our choice of kernel function cannot absorb this difference because is varies on a short length-scale. Instead, we accounted for this uncertainty by adding Gaussian noise to the model parametrised by \(\sigma\). However, for the best test star, we found \(\sigma \approx 0.05\), which was larger than the observational uncertainty, \(\sigma_\obs = 0.01\). This could limit our method's inference ability for the best asteroseismic targets. One solution is to replace \(\tilde{f}_B(n)\) with a quadratic \citep[e.g.][]{Nielsen.Davies.ea2021} which better approximates \(\nu_n\). However, the GP kernel would need to be adjusted to account for this. % Using the glitch parameters as a helium diagnostic could involve measuring the amplitude of the helium glitch at a reference frequency such as \(\nu_{\max}\). We should expect solutions for this reference amplitude (\(A_\mathrm{ref}\)) to converge on a value as the data quality increases from worst to best. It is also important that the uncertainty of the reference amplitude is accurate if using it to constrain helium abundance in a population of stars. If using V19 method on a population of stars, this could bias inference towards lower values of helium abundance. \section{Conclusion} We introduced a new method for modelling acoustic glitches in solar-like oscillators using a Gaussian Process. Testing the method on a model star, we found that it more accurately characterised the underlying, smoothly-varying functional form of the radial modes than the -\citetalias{Verma.Raodeo.ea2019} method. Furthermore, our method was able to absorb the glitch component from He\,\textsc{i} ionisation, for which the polynomial was not flexible enough. +\citetalias{Verma.Raodeo.ea2019} method. Furthermore, our method appeared able to absorb the glitch component from He\,\textsc{i} ionisation, for which the polynomial was not flexible enough. However, this raises the question of whether He\,\textsc{i} ionisation glitch should be included in the model. -Additionally, the GP method provided more believable uncertainties on the glitch parameters, whereas the \citetalias{Verma.Raodeo.ea2019} method was over-confident with the best data and under-confident with the worst. This would become a problem if using the results to make further inference about helium enrichment. For example, the GP modelled correlated `noise' in the mode frequencies arising from He\,\textsc{i} ionisation, not possible with the polynomial in the \citetalias{Verma.Raodeo.ea2019} method. However, this raises the question of whether He\,\textsc{i} ionisation glitch should be included in the model --- this depends how much helium abundance is to be gained from its parameters. +Additionally, the GP method provided more believable uncertainties on the glitch parameters, whereas the \citetalias{Verma.Raodeo.ea2019} method was over-confident with the best data and under-confident with the worst. Robust uncertainties are important when using the results to make further inference about helium enrichment. In this case, the GP marginalised over correlated noise in the model, not possible with the polynomial in the \citetalias{Verma.Raodeo.ea2019} method. -Future development of the method could involve building a prior for the glitch parameters. For example, we could start with fitting the model to simulated stars and using the results to build an empirical prior. Then, we could run the model on the full LEGACY asteroseismic sample of main sequence stars \citep{Lund.SilvaAguirre.ea2017} and compare our results to those from \citet{Verma.Raodeo.ea2019}. Consequently, we can add parameters carrying information about helium abundance to the hierarchical model introduced in \citet{Lyttle.Davies.ea2021}. Ultimately, our goal is to scale this method to the \(\sim \num{1000}\) high signal-to-noise solar-like oscillators expected to be observed by \emph{PLATO} \citep{Rauer.Catala.ea2014}. +Future development of the method could involve building a prior for the glitch parameters. For example, we could start with fitting the model to simulated stars and using the results to build an empirical prior. Then, we could run the model on a larger asteroseismic sample of main sequence stars \citep[e.g.][]{Lund.SilvaAguirre.ea2017,Davies.SilvaAguirre.ea2016} and compare our results to those from \citet{Verma.Raodeo.ea2019}. Additionally, we could add parameters from the GP model to the hierarchical model introduced in \citet{Lyttle.Davies.ea2021}. Ultimately, our goal is to scale this method in anticipation of the \(\sim 10^5\) solar-like oscillators expected to be observed by \emph{PLATO} \citep{Rauer.Catala.ea2014}. From 1664896c7cfe6961a2cdc78d96fb1ec932987a41 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Wed, 26 Apr 2023 16:03:55 +0100 Subject: [PATCH 23/50] Finish conclusion restructure. --- chapters/conclusion.tex | 22 ++++++++++------------ references.bib | 15 +++++++++++++++ 2 files changed, 25 insertions(+), 12 deletions(-) diff --git a/chapters/conclusion.tex b/chapters/conclusion.tex index 010059c..288ab3d 100644 --- a/chapters/conclusion.tex +++ b/chapters/conclusion.tex @@ -12,31 +12,29 @@ % \chapter{Conclusion and Future Prospects} -\textit{We conclude this thesis by providing a summary of the work herein and reflect upon key results. Then, we consider possible improvements to our hierarchical model and method for emulating stellar simulations. Finally, we discuss future prospects for applying our method to data from current and upcoming missions.} +\textit{We conclude this thesis by providing a summary of the work herein and reflecting upon key results. Then, we consider possible improvements to our hierarchical model and method for emulating stellar simulations. Finally, we discuss future prospects for applying our method to data from current and upcoming missions.} \section*{Summary} In this thesis, we built a hierarchical Bayesian model (HBM) to improve the inference of stellar parameters with asteroseismology. Introducing the concept of an HBM in Chapter \ref{chap:hbm}, we showed how population-level distributions can be used as a prior over individual stellar parameters. We found that pooling parameters this way reduced their uncertainties by up to a factor of \(\sqrt{N}\) where \(N\) is the number of stars in the population. -In Chapter \ref{chap:hmd}, we built an HBM to estimate the masses, radii,and ages for a well-studied sample of \(\sim 60\) dwarf and subgiant solar-like oscillators. Limited by observational noise, existing modelling techniques typically struggle to account for the uncertainty in initial helium abundance (\(Y\)) and mixing-length theory parameter (\(\mlt\)) for these stars. We showed that applying a hierarchical prior over \(Y\) and \(\mlt\) allowed us to simultaneously marginalise over their uncertainty and model their distribution in the population. Pooling \(Y\) and \(\mlt\) in this way, we were still able to achieve statistical uncertainties of 1.2 per cent in radius, 2.5 per cent in mass and 12 per cent in age. In the best cases, our HBM halved the uncertainty in stellar mass compared to the same model without parameter pooling. This provided a scalable and reproducible framework for modelling large populations of stars at the same time. +In Chapter \ref{chap:hmd}, we built an HBM to estimate the masses, radii,and ages for a well-studied sample of \(\sim 60\) dwarf and subgiant solar-like oscillators. Limited by observational noise, existing modelling techniques typically struggle to account for the uncertainty in initial helium abundance (\(Y\)) and mixing-length theory parameter (\(\mlt\)) for these stars. We showed that applying a hierarchical prior over \(Y\) and \(\mlt\) allowed us to simultaneously marginalise over their uncertainty and model their distribution in the population. Pooling \(Y\) and \(\mlt\) in this way, we were still able to achieve statistical uncertainties of 1.2 per cent in radius, 2.5 per cent in mass, and 12 per cent in age. Notably, our HBM halved the uncertainty in stellar mass compared to the same model without parameter pooling. This provided a scalable and reproducible framework for modelling large populations of stars at the same time. % In our HBM, we assumed a linear helium enrichment law as the mean of a population distribution over \(Y\). We marginalised over the uncertainty in the parameters of this law, improving upon other work which assume a fixed parametrisation of the law calibrated to the Sun \citep[e.g.][]{Serenelli.Johnson.ea2017}. We found the slope of this law (\(\Delta Y/\Delta Z\)) to be \(\approx 1\) and \(\approx 1.6\), with and without including the Sun-as-a-star in our population. Although these values of \(\Delta Y/\Delta Z\) were within 2-\(\sigma\) of each other and agreed with the literature, including the Sun had a clear effect on both \(Y\) and \(\mlt\). This offset may have been a result in our choice of \(\teff\) scale, suggesting an additional systematic we could add to the model. With some improvements to the HBM, we may be able to further break the degeneracy between \(\mlt\) and \(Y\). -The HBM required a function to map fundamental parameters to observables. We built an emulator to approximate 1D numerical models of stellar evolution. Training a neural network on MESA stellar simulations, we could generate observable parameters (\(\teff, \dnu, L, [\mathrm{M/H}]\)) with typical precisions of less than \(\sim 0.1\) per cent (see Appendix \ref{apx:hmd}). This provided a simple, continuous and differentiable model well suited to modern, gradient-based MCMC algorithms. One advantage to using a neural network emulator was its scalability. The basic linear algebra involved allowed fast evaluations for large numbers of stars in parallel. Furthermore, the neural network could be scaled up to higher input and output dimensions with little performance impact, making the method transferable to other kinds of stars. For example, we recently trained a neural network to emulate the regularly spaced mode frequencies as a part of a Bayesian stellar model of \(\delta\) Scuti-type oscillators \citep{Scutt.Murphy.ea2023}. +The HBM required a function to map model parameters to observables. We built an emulator to approximate 1D numerical models of stellar evolution. Training a neural network on MESA stellar simulations, we were able to predict observable parameters (\(\teff, \dnu, L, [\mathrm{M/H}]\)) with typical precisions of less than \(\sim 0.1\) per cent (see Appendix \ref{apx:hmd}). This provided a simple, continuous and differentiable model well suited to modern, gradient-based MCMC algorithms. Another advantage to using a neural network emulator was its scalability. The simple matrix algebra involved is well suited to fast evaluations on a graphics processing unit (GPU) for large numbers of stars at the same time. Furthermore, the neural network could be scaled up to higher input and output dimensions with little performance impact, making the method transferable to other kinds of stars. For example, we recently trained a neural network to emulate the regularly spaced mode frequencies as a part of a Bayesian stellar model of \(\delta\) Scuti-type oscillators \citep{Scutt.Murphy.ea2023}. -In Chapter \ref{chap:glitch}, we demonstrated that glitches in stellar structure cause a periodic signal, \(\delta\nu\) in p mode frequencies. One such glitch arises from the second ionisation of helium, with the amplitude of \(\delta\nu_\helium\) correlating with helium abundance. - -In Chapter \ref{chap:glitch-gp}, we presented a new method for modelling the glitch signature using a Gaussian process (GP). Past methods for measuring the glitch in the mode frequencies (\(\nu_{nl}\)) used a polynomial in \(n\) to approximate the smooth functional form of the frequencies, over which the periodic glitch signature could be modelled \citep[e.g.][]{Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2019}. We replaced the polynomial with a GP characterised by a kernel describing our prior belief of the function's smoothness and flexibility. We applied this method and compared it to the polynomial method to model the glitch signature in radial mode frequencies for a test star and 16 Cyg A. The GP allowed us to marginalise over our uncertainty in the functional form of \(\nu_{n\,0}\) with \(n\). We found that the polynomial method was too restrictive and unable to account for the uncertainty in our model. On the other hand, the GP +In Chapter \ref{chap:glitch}, we recalled that glitches in stellar structure cause a periodic signal, \(\delta\nu\) in p mode frequencies. One such glitch arises from the second ionisation of helium, with the amplitude of \(\delta\nu_\helium\) correlating with helium abundance. Subsequently, we presented a new method for modelling the glitch signature using a Gaussian process (GP) in Chapter \ref{chap:glitch-gp}. Past methods for measuring the glitch in the mode frequencies (\(\nu_{nl}\)) used a polynomial in \(n\) to approximate the smooth functional form of the frequencies, over which the periodic glitch signature could be modelled \citep[e.g.][]{Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2019}. We replaced the polynomial with a GP characterised by a kernel describing our prior belief of the function's smoothness and flexibility. We applied this method to model the glitch signature in radial mode frequencies for a fake, Sun-like star and 16 Cyg A. The GP allowed us to marginalise over our uncertainty in the functional form of \(\nu_{n\,0}\) with \(n\). In comparison, we found that the polynomial method was too restrictive and unable to marginalise over uncertainty in the model. % In Chapter \ref{chap:glitch}, we recalled that p mode frequencies carry information about acoustic glitches inside a star. However, the exact functional form of the modes with radial order is not known. We showed that a Gaussian Process (GP) could be employed to marginalise over the uncertainty in this functional form and improve detection of the helium glitch signature. Our method showed promise compared to those which have come before \citep[e.g.][]{Verma.Raodeo.ea2019}. We found the GP method was better able to find the true acoustic depth of He\,\textsc{ii} ionisation in our model star than the alternative, motivating a more quantitative comparison in the future. We hope to build a more informed prior on the model parameters and publish this method soon with more examples. \section*{Improving the Hierarchical Model} -The helium glitch parameters for a given star correlate with its near-surface helium abundance. Therefore, a natural next step would be to include helium glitch parameters as an additional observable in our HBM. Our GP glitch model can be applied to both observed and modelled mode frequencies, providing extra parameters to include in our stellar model emulator. Including these should improve inference of helium abundance for stars with individual modes identified \citep[e.g.][]{Davies.SilvaAguirre.ea2016,Lund.SilvaAguirre.ea2017}. Since our HBM models the population distribution of helium, even a small number of stars with good helium constraint will in-turn improve helium estimates for the rest of the population. This introduces the possibility of testing more complex models of helium enrichment. +The helium glitch parameters for a given star correlate with its near-surface helium abundance. Therefore, a natural next step would be to include helium glitch parameters as an additional observable in our HBM. Our GP glitch model can be applied to both observed and modelled mode frequencies, providing extra parameters to include in our stellar model emulator. Adding these should improve inference of helium abundance for stars with individual modes identified \citep[e.g.][]{Davies.SilvaAguirre.ea2016,Lund.SilvaAguirre.ea2017}. Since our HBM simultaneously models the population distribution of helium, even a small number of stars with good helium constraint will in-turn improve helium estimates for the rest of the population. This introduces the possibility of testing more complex models of helium enrichment. -We also expect our emulation method to scale to red giant solar-like oscillators for which observations are abundant. We trained the emulator on a grid of stellar models from the zero-age main sequence to the base of the red giant branch for masses from \SIrange{0.8}{1.2}{\solarmass}. The upper mass limit was motivated by the diminishing outer convective envelope (responsible for driving solar-like oscillators) in these stars on the main sequence. However, extending the emulator to model red giant solar-like oscillators would require expanding the grid up to \(\sim\SI{2.0}{\solarmass}\). We would need to compute thrice as many evolutionary tracks and evolve existing models further. However, stars with \(M \gtrsim \SI{1.1}{\solarmass}\) have a convective core on the main sequence which introduces an additional model uncertainty from mixing at its boundary. Parametrising this would further multiply the number of input tracks. To handle more dimensions, we should research ways of selectively computing stellar models or augmenting the grid \citep[e.g.][]{Li.Davies.ea2022} where the neural network error is large. +We also expect the HBM to scale to red giant solar-like oscillators for which observations are abundant. We trained the emulator on a grid of stellar models from the zero-age main sequence to the base of the red giant branch for masses from \SIrange{0.8}{1.2}{\solarmass}. The upper mass limit was motivated by the diminishing outer convective envelope (responsible for driving solar-like oscillators) in these stars on the main sequence. However, extending the emulator to model red giant solar-like oscillators would require expanding the grid up to \(\sim\SI{2.0}{\solarmass}\). We would need to compute thrice as many evolutionary tracks and evolve existing models further. Stars with \(M \gtrsim \SI{1.1}{\solarmass}\) have a convective core on the main sequence which introduces an additional model uncertainty from mixing at its boundary. Parametrising this process would further multiply the number of input tracks, increasing dimensionality and grid computation time. Therefore, we should research ways of selectively computing stellar models. For example, we could upsample the grid \citep[e.g.][]{Li.Davies.ea2022} where the neural network error is large. -There are a few additional systematic uncertainties we could also include in the HBM. In Chapter \ref{chap:hmd}, we did not consider the effect of uncertain atmospheric physics which effects the mode frequencies. Surface correction methods exist \citep[e.g.][]{Ball.Gizon2014,Kjeldsen.Bedding.ea2008} but vary across the HR diagram when compared with 3D hydrodynamical simulations \cite{Sonoi.Samadi.ea2015}. \citet{Compton.Bedding.ea2018} found a range of surface corrections can shift modelled frequencies at \(\numax\) by up to \(\sim 0.5\) per cent. This would amount to a systematic effect on \(\dnu\) which we would expect to correlate with other stellar parameters. Although \citet{Nsamba.Campante.ea2018} found the surface correction to have a small effect on inferred stellar parameters, when doing population inference this effect is likely to scale up. Therefore, a future iteration of the HBM should account for the surface term systematic. +There are a few additional systematic uncertainties we could also include in the HBM. For example, in Chapter \ref{chap:hmd} we did not consider the inaccuracies of near-surface physics which effect modelled mode frequencies. So-called `surface correction' methods exist \citep[e.g.][]{Ball.Gizon2014,Kjeldsen.Bedding.ea2008} but vary across the HR diagram when compared with 3D hydrodynamical simulations \citep{Sonoi.Samadi.ea2015}. \citet{Compton.Bedding.ea2018} found a range of surface corrections can shift modelled frequencies at \(\numax\) by up to \(\sim 0.5\) per cent. This would amount to a systematic effect on \(\dnu\) which we would expect to correlate with other stellar parameters. Therefore, a future iteration of the HBM should account for the surface term systematic. \section*{Current and Future Data} @@ -58,13 +56,13 @@ \section*{Current and Future Data} % \section{Current and Future Data}\label{sec:conc-future} -We tested the HBM on stars observed by \emph{Kepler}, but there are a few current and upcoming missions from which we can increase our sample size. Recently, \citet{Hatt.Nielsen.ea2023} identified a sample of \(\sim 4000\) solar-like oscillators in 120- and 20-second cadence \emph{TESS} data. Of these, around 50 \todo{check} are dwarf and subgiant stars which we could include in a future iteration of the HBM. With larger sample sizes, we can further increase the precision of pooled parameters and better characterise their spread in the population distribution. However, we anticipate much bigger improvement with future observing missions expected to launch in a few years time. +We tested the HBM on stars observed by \emph{Kepler}, but there are a few current and upcoming missions which we can utilise to increase our sample size. With larger sample sizes, we can further increase the precision of pooled parameters and better characterise their spread in the population distribution. Recently, \citet{Hatt.Nielsen.ea2023} identified a sample of \(\sim 4000\) solar-like oscillators in 120- and 20-second cadence \emph{TESS} data. Of these, around 50 are dwarf and subgiant stars which we could include in a future iteration of the HBM. However, we anticipate much bigger improvement with future missions expected to launch in a few years time. % This has been exceptionally useful with galactic archaology with the \(\sim 150,000\) oscillating red giants detected by \citet{Hon.Huber.ea2021}. -Towards the end of the 2020s, the upcoming \emph{PLATO} mission will observe tens of thousands of dwarf and subgiant solar-like oscillators \citep{Rauer.Catala.ea2014}. \emph{PLATO} aims to discover hundreds more exoplanets orbiting solar-type stars across a wider proportion of the sky when compared to \emph{Kepler}. Among its targets are around \num{20000} bright (V < 11) oscillating F-K dwarf stars to be observed over a baseline of around 2 years \citep{Goupil2017}. Using our HBM method on a sample this size could see a reduction in uncertainty on helium abundance from 0.01 to 0.0005. While this is the maximum expected uncertainty reduction (as discussed in Chapter \ref{chap:hbm}), it shows that we can start to consider more complex population distributions in helium and other stellar parameters. +Towards the end of the 2020s, the \emph{PLATO} mission will observe tens of thousands of dwarf and subgiant solar-like oscillators \citep{Rauer.Catala.ea2014}. \emph{PLATO} aims to discover hundreds of exoplanets orbiting solar-type stars across a wider proportion of the sky than observed by \emph{Kepler}. Among its targets are around \num{20000} bright (V < 11) oscillating F-K dwarf stars to be observed over a baseline of around 2 years \citep{Goupil2017}. Using our HBM method on a sample this size could see a reduction in uncertainty (\(\sigma\)) on helium abundance from 0.01 to 0.0005. While this is the maximum expected uncertainty reduction (as discussed in Chapter \ref{chap:hbm}), it shows that we can start to consider more complex population distributions in helium and other stellar parameters. -While \emph{PLATO} will offer unprecedented numbers of main sequence solar-like oscillators, we already have large samples of more evolved asteroseismic stars to include in a future HBM. Combined, \emph{Kepler}, \emph{K2}, and \emph{TESS} have yielded \(\sim 150,000\) red giant solar-like oscillators to date \citep{Hon.Huber.ea2021,Yu.Huber.ea2018}. Providing that we can extend our stellar model emulator to more evolved stars (see Section \ref{sec:conc-nn}), this will enable more extensions to the HBM. For example, since \emph{TESS} is an all-sky survey, we could test including kinematics and galactic positions in the helium enrichment law. Additionally, observations of open clusters introduce more population distributions on age, distance and chemical abundances. +While \emph{PLATO} will offer unprecedented numbers of main sequence solar-like oscillators, we already have large samples of more evolved asteroseismic stars to include in a future HBM. Combined, \emph{Kepler}, \emph{K2}, and \emph{TESS} have yielded \(\sim 150,000\) red giant solar-like oscillators to date \citep{Hon.Huber.ea2021,Yu.Huber.ea2018}. Providing that we can extend our stellar model emulator to more these stars, expanding our dataset will allow us to test more complex population-distributions. For example, since \emph{TESS} is an all-sky survey, we could include kinematics and galactic positions in the helium enrichment law. Additionally, observations of open clusters and binary star systems introduce more population distributions over age, distance and chemical abundances. % The number of dwarf and subgiant solar-like oscillators expected from \emph{PLATO} will be comparable to the number of red giant oscillators already found with \emph{Kepler} and \emph{TESS} \needcite. In the meantime we could test extending our method to red giant stars to make use of the abundance of data. This comes with additional challenges. Oscillating red giants include masses \(\gtrsim \SI{1.2}{\solarmass}\) which would have had a convective core during their hydrogen-burning phase of evolution. In this case, we would have to consider overshooting at the convective core boundary. This is an approximation of the physics to simulate mixing at the boundary bringing fresh hydrogen fuel into the core and extending the main sequence lifetime. diff --git a/references.bib b/references.bib index 01ed4f4..14e88e1 100644 --- a/references.bib +++ b/references.bib @@ -4058,6 +4058,21 @@ @article{Mohan.Scaife.ea2022 annotation = {ADS Bibcode: 2022MNRAS.511.3722M} } +@article{Molaro2023, + title = {On {{Galileo}}'s Self-Portrait {{Mentioned}} by {{Thomas Salusbury}}}, + author = {Molaro, Paolo}, + editor = {Campion, Nicholas and Impey, Chris}, + year = {2018}, + journal = {Imagining Worlds Explor. Astron. Cult.}, + pages = {233--244}, + publisher = {{Sophia Centre Press}}, + doi = {10.48550/arXiv.2304.12320}, + urldate = {2023-04-26}, + abstract = {An intriguing reference to the existence of a self-portrait by Galileo Galilei is contained in the biography of the scientist by Thomas Salusbury dated ca. 1665, of which only one incomplete and inaccessible copy exists. Galileo grew up in a Renaissance atmosphere, acquiring an artistic touch. He was a musician, a writer and also a painter, as reported by Viviani and documented by his watercolours of the Moon and drawings of solar spots. Recently a new portrait with a remarkable similarity to the portraits of Galileo Galilei by Santi di Tito (1601), Domenico Tintoretto (ca. 1604), and Furini (ca. 1612) has been found and examined using sophisticated face recognition techniques. If the identity could be confirmed, other elements, such as the young age of Galileo or the seam in the canvas revealed by infrared and X-ray analysis, may suggest a possible link with the self-portrait mentioned by Salusbury.}, + keywords = {Astrophysics - Instrumentation and Methods for Astrophysics,Physics - History and Philosophy of Physics}, + annotation = {ADS Bibcode: 2023arXiv230412320M} +} + @article{Montalban.Mackereth.ea2021, title = {Chronologically Dating the Early Assembly of the {{Milky Way}}}, author = {Montalb{\'a}n, Josefina and Mackereth, J. Ted and Miglio, Andrea and Vincenzo, Fiorenzo and Chiappini, Cristina and Buldgen, Gael and Mosser, Beno{\^i}t and Noels, Arlette and Scuflaire, Richard and Vrard, Mathieu and Willett, Emma and Davies, Guy R. and Hall, Oliver J. and Nielsen, Martin Bo and Khan, Saniya and Rendle, Ben M. and {van Rossem}, Walter E. and Ferguson, Jason W. and Chaplin, William J.}, From d981cef14e59dc38df86d83f4e86909e68e86da1 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Wed, 26 Apr 2023 18:01:36 +0100 Subject: [PATCH 24/50] Proof introduction --- chapters/introduction.tex | 14 ++--- references.bib | 118 ++++++++++++++++++++++++++++++++++++++ 2 files changed, 125 insertions(+), 7 deletions(-) diff --git a/chapters/introduction.tex b/chapters/introduction.tex index bf0858d..8807992 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -22,7 +22,7 @@ \section{Understanding the Stars}\label{sec:stars} Early efforts to understand the stars began by finding relations between their spectral classification and magnitude in a given photometric band on what was later called a Hertzsprung-Russell (HR) diagram \citep[e.g.][]{Russell1914}. An HR diagram shows the absolute magnitude (or luminosity, \(L\)) of a star against its spectral class (or effective temperature, \(\teff\)). Astronomers found that stars were not uniformly distributed on the HR diagram, but were instead grouped in distinct sequences. For example, the region where most stars were found was called the \emph{main sequence}. -Initial insight into stellar evolution came about when astronomers studied open clusters on the HR diagram \needcite. These were groups of stars found at a similar distance and close together on the sky. Assuming clusters formed at the same time with similar chemical abundances, the only expected difference between stars were their mass and multiplicity. Using stars of known mass (e.g. from orbital solutions to binary systems), astronomers could trace lines of constant mass from younger to older clusters \needcite. This provided an early approximation of a stellar evolutionary track --- the path a star takes on the HR diagram during its evolution. Deriving the stellar radius (\(R\)) from the relation \(L \propto R^2 \teff^4\), they inferred that stars started on the left-hand edge of the main sequence and became brighter and larger throughout most of their lifetime. At some point, stars would leave the main sequence, rapidly cool and expand, and then ascend a region of the HR diagram known as the \emph{red giant branch}. +Initial insight into stellar evolution came about when astronomers studied open clusters on the HR diagram \citep{Trumpler1930}. These are groups of stars found at a similar distance and close together on the sky. Assuming clusters formed at the same time with similar chemical abundances, the only expected difference between stars is their mass and multiplicity. Using stars of known mass (e.g. from orbital solutions to binary systems) and luminosity functions, astronomers could trace lines of similar mass and composition from younger to older clusters \citep[e.g.][]{Sandage1957}. Coupled with early arguments from stellar physics \citep{Chandrasekhar1939}, scientists formed stellar evolutionary tracks --- the path a star takes on the HR diagram during its evolution. Deriving the stellar radius (\(R\)) from the relation \(L \propto R^2 \teff^4\), they inferred that stars started on the left-hand edge of the main sequence and became brighter and larger throughout most of their lifetime. At some point, stars would leave the main sequence, rapidly cool and expand, and then ascend a region of the HR diagram known as the \emph{red giant branch}. % \citet{Kuiper1938} found an empirical mass-luminosity relation. @@ -32,11 +32,11 @@ \section{Understanding the Stars}\label{sec:stars} % The luminosity and effective temperature could be estimated from the magnitude and colour of the stars. From luminosity and temperature, we could derive the radius of stars. Early stellar mass estimates came from visual and spectroscopic binaries. Spectroscopy provides abundances of chemical species ionised in the stellar atmosphere. However, except for the Sun, stellar age and helium abundance has no model-independence. The latter ionisations at temperatures and densities higher than the surface of stars like the Sun. -The question of what stars were made of and how they evolved still remained. Spectroscopy revealed relative abundances of elements excited in stellar atmospheres, but determining their absolute abundances was difficult. \citet{Payne1925} proposed that stars were comprised of mostly hydrogen and helium, later reinforced by estimates of helium content in the Sun \citep[e.g.][]{Schwarzschild1946}. The idea was radical at the time because it did not match the abundance of elements found on Earth. Meanwhile, advancements in nuclear science spawned the theory of stellar nucleosynthesis \citep{Hoyle1946}. Stars produced elements heavier than hydrogen and helium through nuclear fusion reactions. This discovery explained the production of many elements in the universe, tying the formation and evolution of stars to planetary formation and the conditions for life in the universe. +The chemical composition, energy sources and evolutionary physics of stars were still uncertain. Spectroscopy revealed relative abundances of elements excited in stellar atmospheres, but determining their absolute abundances was difficult. \citet{Payne1925} proposed that stars were comprised of mostly hydrogen and helium, later reinforced by estimates of helium content in the Sun \citep[e.g.][]{Schwarzschild1946}. The idea was radical at the time because it did not match the abundance of elements found on Earth. Meanwhile, advancements in nuclear science spawned the theory of stellar nucleosynthesis \citep{Hoyle1946}. Stars produce elements heavier than hydrogen and helium through nuclear fusion reactions. This discovery explained the production of many elements in the universe, tying the formation and evolution of stars to planetary formation and the conditions for life in the universe. -In the second half of the 20th century, the theory of stellar evolution advanced. With this came computational methods for simulating stars \citep[e.g.][]{Kippenhahn.Weigert.ea1967}. Astronomers could start to compare observations and empirical relations with simulated stars. We call this process \emph{modelling stars}. Early models of the Sun benefited from independent age estimates from geology and neutrino-production rates from observations of cosmic rays. With these constraints, and the Sun as a calibrator, researchers could start to refine their models and test them on other stars. \todo{example?} +In the second half of the 20th century, the theory of stellar evolution advanced. With this came computational methods for simulating stars \citep[e.g.][]{Kippenhahn.Weigert.ea1967}. Astronomers could start to compare observations of stars and clusters with simulated stars. We call this process \emph{modelling stars}. Early models of the Sun benefited from independent age estimates from geology and neutrino-production rates from observing cosmic rays. With these constraints, and the Sun as a calibrator, researchers could start to refine their models and test them on other stars. For example, \citet{Iben1967} tested stellar models on main sequence stars with similar masses to the Sun using observations of the clusters M67 and NGC 188. -On a similar timescale, a new field emerged which gave astronomers a model-independent way of studying the inside of stars. Identifying regular perturbations of the surface of the Sun, researchers realised that they pertained to high-order, stochastically-driven spherical harmonic oscillations. Measuring these oscillation modes, they could study the Sun in a similar way to how seismologists study the Earth. Named asteroseismology, this new field was able to study many non-radial overtones which probed different depths of the star, thus distinguishing itself from the well-know study of radially pulsating stars \citep[e.g. Cepheid variables;][]{Leavitt1908}. In Section \ref{sec:seismo} we give a brief history and theory of the asteroseismology of stars like the Sun. +On a similar timescale, a new field emerged which gave astronomers a model-independent way of studying the inside of stars. Identifying regular perturbations of the surface of the Sun, researchers realised that they pertained to high-order, stochastically-driven spherical harmonic oscillations. Measuring these oscillation modes, they could study the Sun in a similar way to how seismologists study the Earth. This new field, called asteroseismology, studied multiple non-radial oscillations, distinguishing itself from the well-know study of radially pulsating stars \citep[e.g. Cepheid variables;][]{Leavitt1908}. In Section \ref{sec:seismo} we give a brief history and theory of the asteroseismology for stars like the Sun. % Approximations for the equation of state by \citet{Eggleton.Faulkner.ea1973}. @@ -52,15 +52,15 @@ \section{Understanding the Stars}\label{sec:stars} \section[Solar-Like Oscillators]{Asteroseismology of Solar-Like Oscillators}\label{sec:seismo} -Like many aspects of stellar physics, asteroseismology began with the Sun. In the next section, we recall the origins of asteroseismology. Then, we give some theoretical background on solar-like oscillations in Section \ref{sec:seismo-theory}. We introduce terminology commonly used in asteroseismology to be able to understand the rest of this thesis. However, we recommend the recent review by \citet{Aerts2021} or the lecture notes by \citet{Christensen-Dalsgaard2014} for a more detailed understanding of the field. +In this section, we provide a brief history and theory of asteroseismology to provide a foundation from which to understand this thesis. However, we recommend the recent review by \citet{Aerts2021} or the lecture notes by \citet{Christensen-Dalsgaard2014} for a more detailed understanding of the field. \subsection{A Brief History of Asteroseismology} Several decades ago, 5-minute oscillations in the radial velocity of the solar surface were observed by \citet{Leighton.Noyes.ea1962}, leading to the inference of acoustic waves trapped beneath the solar photosphere \citep{Ulrich1970}. A further decade of study culminated in the measurement of regular patterns of individual oscillation modes in the Doppler radial velocity \citep{Claverie.Isaak.ea1979} and total irradiance \citep{Woodard.Hudson1983a} of the Sun. Initially thought to be short-lived irregularities on the surface, these modes were found to be compatible with stochastically excited standing waves penetrating deep into the Sun. Later, \citet{Deubner.Gough1984} introduced the word \emph{helioseismology} (analogous to geo-seismology) to describe the study of the solar interior using observations of these modes. Helioseismology was soon responsible for breakthrough solar research, from measuring differential rotation \citep{Deubner.Ulrich.ea1979} to solving the mismatch between predicted and measured solar neutrino production \citep{Bahcall.Ulrich1988}. -Astronomers initially debated the mechanism driving the modes of standing pressure waves (or \emph{p modes}) in the Sun. \citet{Goldreich.Keeley1977} suggested what became the prevailing theory, that the p modes were stochastically excited by near-surface convection. Hence, we might expect solar-like oscillations to be present in other stars which have a convective envelope similar to the Sun. Shortly thereafter, \citet{Christensen-Dalsgaard1984} introduced the term \emph{asteroseismology} --- the study of the internal structure of stars with many observable modes of oscillation. Subsequently, solar-like oscillations were discovered in a few bright stars. Among the first were Procyon and \(\alpha\) Cen A \citep{Gelly.Grec.ea1986}, with individual modes later resolved by \citet{Martic.Schmitt.ea1999} and \citet{Bouchy.Carrier2001} respectively. +Astronomers initially debated the mechanism driving the modes of standing pressure waves (or \emph{p modes}) in the Sun. \citet{Goldreich.Keeley1977} suggested what became the prevailing theory, that the p modes were stochastically excited by near-surface convection. Hence, we might expect solar-like oscillations to be present in other stars which have a convective envelope similar to the Sun. Shortly thereafter, \citet{Christensen-Dalsgaard1984} was among those to introduce the term \emph{asteroseismology} --- the study of the internal structure of stars with many observable oscillation modes. Subsequently, solar-like oscillations were discovered in a few bright stars. Among the first were Procyon and \(\alpha\) Cen A \citep{Gelly.Grec.ea1986}, with individual modes later resolved by \citet{Martic.Schmitt.ea1999} and \citet{Bouchy.Carrier2001} respectively. -Instrumental and atmospheric noise limited the progress of asteroseismology with ground-based equipment to studies of small number of bright dwarf stars. Asteroseismology requires high-cadence (\(\sim \SIrange{1}{10}{\minute}\)) brightness observations over long time-periods (\(\sim \SI{1}{\year}\)) with precisions of \todo{precision}. The first dedicated space-based missions which met these requirements arrived in the late 2000s, accelerating progress in the field. Initially, the \emph{CoRoT} mission \citep{Baglin.Auvergne.ea2006} detected solar-like oscillations in thousands of red giant stars \citep{DeRidder.Barban.ea2009,Mosser.Belkacem.ea2010}. Then, the \emph{Kepler} mission \citep{Borucki.Koch.ea2010} yielded oscillations in thousands more red giants \citep{Pinsonneault.Elsworth.ea2014} and hundreds of main sequence stars similar to the Sun \citep{Serenelli.Johnson.ea2017}. Most recently, \emph{TESS} \citep{Ricker.Winn.ea2015} added thousands more dwarf and giant stars to the roster of solar-like oscillators \citep{Hon.Huber.ea2021,SilvaAguirre.Stello.ea2020,Hatt.Nielsen.ea2023}. +Instrumental and atmospheric noise limited the progress of asteroseismology with ground-based equipment to studies of small number of bright dwarf stars. Asteroseismology of solar-like oscillators requires high-cadence (\(\sim \SIrange{1}{10}{\minute}\)) brightness observations over long time-periods (\(\sim \SI{1}{\year}\)) with precision of, for example \(\lesssim \SI{100}{ppm\per\hour}\) for \(I\)-band magnitude \(\lesssim 10\) \citep{Schofield.Chaplin.ea2019}. The first dedicated space-based missions which met these requirements arrived in the late 2000s, accelerating progress in the field. Initially, the \emph{CoRoT} mission \citep{Baglin.Auvergne.ea2006} detected solar-like oscillations in thousands of red giant stars \citep{DeRidder.Barban.ea2009,Mosser.Belkacem.ea2010}. Then, the \emph{Kepler} mission \citep{Borucki.Koch.ea2010} yielded oscillations in thousands more red giants \citep{Pinsonneault.Elsworth.ea2014} and hundreds of main sequence stars similar to the Sun \citep{Serenelli.Johnson.ea2017}. Most recently, \emph{TESS} \citep{Ricker.Winn.ea2015} added thousands more dwarf and giant stars to the roster of solar-like oscillators \citep{Hon.Huber.ea2021,SilvaAguirre.Stello.ea2020,Hatt.Nielsen.ea2023}. \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} diff --git a/references.bib b/references.bib index 14e88e1..2093400 100644 --- a/references.bib +++ b/references.bib @@ -2110,6 +2110,21 @@ @article{GaiaCollaboration.Vallenari.ea2022 annotation = {ADS Bibcode: 2022arXiv220800211G} } +@article{Gamow1938, + title = {Nuclear {{Energy Sources}} and {{Stellar Evolution}}}, + author = {Gamow, G.}, + year = {1938}, + month = apr, + journal = {Phys. Rev.}, + volume = {53}, + pages = {595--604}, + issn = {1536-6065}, + doi = {10.1103/PhysRev.53.595}, + urldate = {2023-04-26}, + abstract = {The behavior of a star with a thermonuclear energy source consistent with our present knowledge about nuclear reactions is studied in relation to the problems of stellar evolution and interpretation of the Hertzsprung-Russell diagram and mass-luminosity relation. It is found that in the case of ordinary thermonuclear reactions, with the absence of selective temperature effects (nuclear resonance) the central temperature and luminosity of the star (with constant mass) will rapidly increase in the process of evolution. If, however, such selective effects are present, the energy-production at the center of the star will cease, beginning with the stage when the central temperature reaches the selective value, and energy will be produced only in a spherical shell around the center. This shell will have exactly the selective temperature value corresponding to the thermonuclear reaction in question, and its radius will slowly increase in the process of evolution causing a very slow increase of luminosity. Finally, in the third stage of evolution, when all hydrogen necessary for thermonuclear reactions has been consumed, the star will start a rapid contraction and, passing through the high density stage, will end its life as a cool body. It is also indicated that the star model with a shell source does not possess the property of "super-stability" characteristic for the point-source models.}, + annotation = {ADS Bibcode: 1938PhRv...53..595G} +} + @article{Garcia.Ballot2019, title = {Asteroseismology of Solar-Type Stars}, author = {Garc{\'i}a, Rafael A. and Ballot, J{\'e}r{\^o}me}, @@ -3080,6 +3095,21 @@ @article{Iben.Ehrman1962 abstract = {Abstract image available at: http://adsabs.harvard.edu/abs/1962ApJ...135..770I} } +@article{Iben1967, + title = {Stellar {{Evolution}}.{{VI}}. {{Evolution}} from the {{Main Sequence}} to the {{Red-Giant Branch}} for {{Stars}} of {{Mass}} 1 {{M}}\_\{sun\}, 1.25 {{M}}\_\{sun\}, and 1.5 {{M}}\_\{sun\}}, + author = {Iben, Jr., Icko}, + year = {1967}, + month = feb, + journal = {\apj}, + volume = {147}, + pages = {624}, + issn = {0004-637X}, + doi = {10.1086/149040}, + urldate = {2023-04-26}, + abstract = {Evolution from the main sequence to the red-giant phase is discussed for Population I stars of mass M/Mo = 1,1.25, and 1 5. Prior to the giant phase in all three stars, energy generation by the p-p chain reactions dominates over energy generation by the CN cycle reactions. During the main hydrogen- burning phase, a convective core is not present in the 1 Mo star but does occur in both the 1.25 Mo and 1.5 Mo stars. Over a major portion of the main-sequence phase, the convective core grows in both the 1.25 Mo and 1.5 Mo stars as the CN cycle reactions become increasingly important. As each star rises along the red-giant branch, the CN cycle reactions eventually become the major source of energy in the hydrogen-burning shell. The time spent by a star burning hydrogen in a thick shell relative to the time spent in the core hydrogen-burning phase is found to be a strongly decreasing function of stellar mass. Partially responsible for this mass dependence are three factors: (1) for more massive stars, the mass fraction in the convective core, and hence the mass over which hydrogen is exhausted at the end of the core hydrogen-burning phase, is closer to the effective -Chandrasekhar limit; (2) with decreasing stellar mass, the variation of hydrogen through the shell becomes more gradual, and (3) with decreasing mass, electron degeneracy becomes more important in the hydrogen-exhausted core during the thick shell-burning phase. The latter two factors have the effect of increasing the -Chandrasekhar limit. In contrast with the case of more massive stars discussed in earlier papers of this series, electron degeneracy is responsible for a major fraction of the pressure and electron conduction is the major mode of energy flow in the hydrogen-exhausted core of all three stars during the giant phase The result is that all three stars possess a nearly isothermal core along the giant branch. During and following the shell-narrowing phase in all three stars, the surface lithium abundance decreases regularly with decreasing surface temperature as envelope convection extends deeper and deeper into the star. Such is the case also for the surface ratio of C12 to N14. Comparatively large amounts of He3 are made in all three stars, and it is suggested that a large fraction of the He3 in the galactic disk was perhaps formed in ordinary stars. Comparison of individual tracks with cluster diagrams for NGC 1.88 and M67 provides evidence for the qualitative correctness of several characteristic features of the theoretical tracks. The pronounced and rapid change in luminosity and in surface temperature during the phase of over-all contraction is related to a "gap" in M67; the decrease with decreasing mass in the magnitude of the luminosity drop during the shell-narrowing phase is related to the change with cluster age in the slope of the subgiant branch; for a given luminosity, the decrease with decreasing mass in surface temperature along the giant branch is related to the decrease with increasing cluster age in surface temperature along the cluster giant branch. The ages of NGC 188 and M67 are estimated to be (11 + 2) x 1O yr and (5.5 + 1) X 1O yr, respectively.}, + annotation = {ADS Bibcode: 1967ApJ...147..624I} +} + @article{Izotov.Thuan2010, title = {The {{Primordial Abundance}} of {{4He}}: {{Evidence}} for {{Non-Standard Big Bang Nucleosynthesis}}}, shorttitle = {The {{Primordial Abundance}} of {{4He}}}, @@ -3112,6 +3142,21 @@ @article{Jermyn.Bauer.ea2023 annotation = {ADS Bibcode: 2023ApJS..265...15J} } +@article{Johnson.Hiltner1956, + title = {Observational {{Confirmation}} of a {{Theory}} of {{Stellar Evolution}}.}, + author = {Johnson, H. L. and Hiltner, W. A.}, + year = {1956}, + month = mar, + journal = {\apj}, + volume = {123}, + pages = {267}, + issn = {0004-637X}, + doi = {10.1086/146159}, + urldate = {2023-04-26}, + abstract = {A "standard" main sequence for age zero has been computed from the observed Hyades, Praesepe, and Pleiades main sequences. The computations were made on the basis of a recent theory of stellar evolution. This "standard" main sequence turns out to be the lower envelope of the near-by stars in the color-magnitude diagram. New distance moduli are given for three clusters. They were obtained by fitting the cluster main sequences to our standard main sequence. A comparison of the luminosities of the brighter cluster stars with the values for these same stars that were obtained from other investigations confirms the evolutionary corrections to the main sequence.}, + annotation = {ADS Bibcode: 1956ApJ...123..267J} +} + @article{Johnson.Penny.ea2020, title = {Predictions of the {{Nancy Grace Roman Space Telescope Galactic Exoplanet Survey}}. {{II}}. {{Free-floating Planet Detection Rates}}}, author = {Johnson, Samson A. and Penny, Matthew and Gaudi, B. Scott and Kerins, Eamonn and Rattenbury, Nicholas J. and Robin, Annie C. and Calchi Novati, Sebastiano and Henderson, Calen B.}, @@ -5052,6 +5097,21 @@ @article{Samadi.Deru.ea2019 annotation = {ADS Bibcode: 2019A\&A...624A.117S} } +@article{Sandage1957, + title = {Observational {{Approach}} to {{Evolution}}. {{III}}. {{Semiempirical Evolution Tracks}} for {{M67}} and {{M3}}.}, + author = {Sandage, Allan}, + year = {1957}, + month = sep, + journal = {\apj}, + volume = {126}, + pages = {326}, + issn = {0004-637X}, + doi = {10.1086/146405}, + urldate = {2023-04-26}, + abstract = {A method is presented for obtaining the tracks of evolution for individual stars in the subgiant and giant regions of color-magnitude diagrams of star clusters. The method utilizes the evolutionary information contained in the observed luminosity functions and color-magnitude diagrams of clusters. It is applied to the galactic cluster M67 and to the globular cluster M3. The evolutionary tracks, the time scale for evolution along these tracks, and the fraction of the total mass exhausted of hydrogen have been computed for both clusters, and the results are tabulated in Tables 3-10. The computed fraction of the mass exhausted of hydrogen for stars in M3 is compared with the theoretical predictions of the HoyleSchwarzschild (H-S) models. Fair agreemenL is obtained It is shown that the H-S models are capable of predicting nearly the correct luminosity function for M3 except at the very top of the giant sequence. The time taken for stars to evolve along the horizontal branch in M3 from B - V = 0.50 to B - V = -0.10 is found to be 2 3 X 108 years. The lifetime of the RR Lyrae phase of the evolution is 8 X 10 years. The expected rate of change of the period of the RR Lyrae stars due to this evolution is At/t = 2.4 X 10-li, or 0.1 second per century, which is about a factor of 5 below the limit of detectability with the available data. The observed period changes for RR Lyrae stars in M3 average twenty times this value and are believed to be due to causes other than evolution.}, + annotation = {ADS Bibcode: 1957ApJ...126..326S} +} + @article{Sandquist.Jessen-Hansen.ea2016, title = {The {{Age}} and {{Distance}} of the {{Kepler Open Cluster NGC}} 6811 from an {{Eclipsing Binary}}, {{Turnoff Star Pulsation}}, and {{Giant Asteroseismology}}}, author = {Sandquist, Eric L. and {Jessen-Hansen}, J. and Shetrone, Matthew D. and Brogaard, Karsten and Meibom, S{\o}ren and Leitner, Marika and Stello, Dennis and Bruntt, Hans and Antoci, Victoria and Orosz, Jerome A. and Grundahl, Frank and Frandsen, S{\o}ren}, @@ -5130,6 +5190,22 @@ @article{Schmitt.Rosner.ea1984 annotation = {ADS Bibcode: 1984ApJ...282..316S} } +@article{Schofield.Chaplin.ea2019, + title = {The {{Asteroseismic Target List}} for {{Solar-like Oscillators Observed}} in 2 Minute {{Cadence}} with the {{Transiting Exoplanet Survey Satellite}}}, + author = {Schofield, Mathew and Chaplin, William J. and Huber, Daniel and Campante, Tiago L. and Davies, Guy R. and Miglio, Andrea and Ball, Warrick H. and Appourchaux, Thierry and Basu, Sarbani and Bedding, Timothy R. and {Christensen-Dalsgaard}, J{\o}rgen and Creevey, Orlagh and Garc{\'i}a, Rafael A. and Handberg, Rasmus and Kawaler, Steven D. and Kjeldsen, Hans and Latham, David W. and Lund, Mikkel N. and Metcalfe, Travis S. and Ricker, George R. and Serenelli, Aldo and Silva Aguirre, Victor and Stello, Dennis and Vanderspek, Roland}, + year = {2019}, + month = mar, + journal = {\apjs}, + volume = {241}, + pages = {12}, + issn = {0067-0049}, + doi = {10.3847/1538-4365/ab04f5}, + urldate = {2023-04-26}, + abstract = {We present the target list of solar-type stars to be observed in short-cadence (2 minute) for asteroseismology by the NASA Transiting Exoplanet Survey Satellite (TESS) during its 2 year nominal survey mission. The solar-like Asteroseismic Target List (ATL) is comprised of bright, cool main-sequence and subgiant stars and forms part of the larger target list of the TESS Asteroseismic Science Consortium. The ATL uses the Gaia Data Release 2 and the Extended Hipparcos Compilation (XHIP) to derive fundamental stellar properties, to calculate detection probabilities, and to produce a rank-ordered target list. We provide a detailed description of how the ATL was produced and calculate expected yields for solar-like oscillators based on the nominal photometric performance by TESS. We also provide a publicly available source code that can be used to reproduce the ATL, thereby enabling comparisons of asteroseismic results from TESS with predictions from synthetic stellar populations.}, + keywords = {Astrophysics - Solar and Stellar Astrophysics,catalogs,space vehicles: instruments,stars: fundamental parameters,stars: oscillations,surveys}, + annotation = {ADS Bibcode: 2019ApJS..241...12S} +} + @article{Schwamb.Orosz.ea2013, title = {Planet {{Hunters}}: {{A Transiting Circumbinary Planet}} in a {{Quadruple Star System}}}, shorttitle = {Planet {{Hunters}}}, @@ -5553,6 +5629,33 @@ @article{Stevens.Bellstedt.ea2020 annotation = {ADS Bibcode: 2020NatAs...4..843S} } +@article{Stromgren1933, + title = {On the {{Interpretation}} of the {{Hertzsprung-Russell-Diagram}}. {{Mit}} 4 {{Abbildungen}}.}, + author = {Str{\"o}mgren, Bengt}, + year = {1933}, + month = jan, + journal = {Z. Astrophys.}, + volume = {7}, + pages = {222}, + issn = {0372-8331}, + urldate = {2023-04-26}, + annotation = {ADS Bibcode: 1933ZA......7..222S} +} + +@article{Stromgren1952, + title = {Evolution of Stars.}, + author = {Stromgren, B.}, + year = {1952}, + month = jan, + journal = {\aj}, + volume = {57}, + pages = {65--83}, + issn = {0004-6256}, + doi = {10.1086/106713}, + urldate = {2023-04-26}, + annotation = {ADS Bibcode: 1952AJ.....57...65S} +} + @article{Stumpe.Smith.ea2012, title = {Kepler {{Presearch Data Conditioning I}}\textemdash{{Architecture}} and {{Algorithms}} for {{Error Correction}} in {{Kepler Light Curves}}}, author = {Stumpe, Martin C. and Smith, Jeffrey C. and Van Cleve, Jeffrey E. and Twicken, Joseph D. and Barclay, Thomas S. and Fanelli, Michael N. and Girouard, Forrest R. and Jenkins, Jon M. and Kolodziejczak, Jeffery J. and McCauliff, Sean D. and Morris, Robert L.}, @@ -5735,6 +5838,21 @@ @article{Trampedach.Stein.ea2014 keywords = {convection,mixing-length theory,stars: atmospheres,stars: evolution,stellar convection} } +@article{Trumpler1930, + title = {Preliminary Results on the Distances, Dimensions and Space Distribution of Open Star Clusters}, + author = {Trumpler, Robert Julius}, + year = {1930}, + month = jan, + journal = {Lick Obs. Bull.}, + volume = {420}, + pages = {154--188}, + issn = {0075-9317}, + doi = {10.5479/ADS/bib/1930LicOB.14.154T}, + urldate = {2023-04-26}, + keywords = {STARS: DISTRIBUTION,STARS: OPEN CLUSTERS}, + annotation = {ADS Bibcode: 1930LicOB..14..154T} +} + @article{Ulrich1970, title = {The {{Five-Minute Oscillations}} on the {{Solar Surface}}}, author = {Ulrich, Roger K.}, From 270567acd02798e014dec41086bb79472f5d03d6 Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Wed, 26 Apr 2023 22:25:14 +0100 Subject: [PATCH 25/50] Fix order of magnitude --- chapters/glitch-gp.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/chapters/glitch-gp.tex b/chapters/glitch-gp.tex index ab9abdf..c914533 100644 --- a/chapters/glitch-gp.tex +++ b/chapters/glitch-gp.tex @@ -317,4 +317,4 @@ \section{Conclusion} Additionally, the GP method provided more believable uncertainties on the glitch parameters, whereas the \citetalias{Verma.Raodeo.ea2019} method was over-confident with the best data and under-confident with the worst. Robust uncertainties are important when using the results to make further inference about helium enrichment. In this case, the GP marginalised over correlated noise in the model, not possible with the polynomial in the \citetalias{Verma.Raodeo.ea2019} method. -Future development of the method could involve building a prior for the glitch parameters. For example, we could start with fitting the model to simulated stars and using the results to build an empirical prior. Then, we could run the model on a larger asteroseismic sample of main sequence stars \citep[e.g.][]{Lund.SilvaAguirre.ea2017,Davies.SilvaAguirre.ea2016} and compare our results to those from \citet{Verma.Raodeo.ea2019}. Additionally, we could add parameters from the GP model to the hierarchical model introduced in \citet{Lyttle.Davies.ea2021}. Ultimately, our goal is to scale this method in anticipation of the \(\sim 10^5\) solar-like oscillators expected to be observed by \emph{PLATO} \citep{Rauer.Catala.ea2014}. +Future development of the method could involve building a prior for the glitch parameters. For example, we could start with fitting the model to simulated stars and using the results to build an empirical prior. Then, we could run the model on a larger asteroseismic sample of main sequence stars \citep[e.g.][]{Lund.SilvaAguirre.ea2017,Davies.SilvaAguirre.ea2016} and compare our results to those from \citet{Verma.Raodeo.ea2019}. Additionally, we could add parameters from the GP model to the hierarchical model introduced in \citet{Lyttle.Davies.ea2021}. Ultimately, our goal is to scale this method in anticipation of the \(\sim 10^4\) solar-like oscillators expected to be observed by \emph{PLATO} \citep{Rauer.Catala.ea2014}. From c95bd3ec3106b7262bac79a7143cbca65e299137 Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Wed, 26 Apr 2023 22:25:47 +0100 Subject: [PATCH 26/50] Proof intro --- chapters/introduction.tex | 20 +++--- references.bib | 145 ++++++++++++++++++++++++++++++++++++++ 2 files changed, 156 insertions(+), 9 deletions(-) diff --git a/chapters/introduction.tex b/chapters/introduction.tex index 8807992..e4b15ab 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -60,7 +60,7 @@ \subsection{A Brief History of Asteroseismology} Astronomers initially debated the mechanism driving the modes of standing pressure waves (or \emph{p modes}) in the Sun. \citet{Goldreich.Keeley1977} suggested what became the prevailing theory, that the p modes were stochastically excited by near-surface convection. Hence, we might expect solar-like oscillations to be present in other stars which have a convective envelope similar to the Sun. Shortly thereafter, \citet{Christensen-Dalsgaard1984} was among those to introduce the term \emph{asteroseismology} --- the study of the internal structure of stars with many observable oscillation modes. Subsequently, solar-like oscillations were discovered in a few bright stars. Among the first were Procyon and \(\alpha\) Cen A \citep{Gelly.Grec.ea1986}, with individual modes later resolved by \citet{Martic.Schmitt.ea1999} and \citet{Bouchy.Carrier2001} respectively. -Instrumental and atmospheric noise limited the progress of asteroseismology with ground-based equipment to studies of small number of bright dwarf stars. Asteroseismology of solar-like oscillators requires high-cadence (\(\sim \SIrange{1}{10}{\minute}\)) brightness observations over long time-periods (\(\sim \SI{1}{\year}\)) with precision of, for example \(\lesssim \SI{100}{ppm\per\hour}\) for \(I\)-band magnitude \(\lesssim 10\) \citep{Schofield.Chaplin.ea2019}. The first dedicated space-based missions which met these requirements arrived in the late 2000s, accelerating progress in the field. Initially, the \emph{CoRoT} mission \citep{Baglin.Auvergne.ea2006} detected solar-like oscillations in thousands of red giant stars \citep{DeRidder.Barban.ea2009,Mosser.Belkacem.ea2010}. Then, the \emph{Kepler} mission \citep{Borucki.Koch.ea2010} yielded oscillations in thousands more red giants \citep{Pinsonneault.Elsworth.ea2014} and hundreds of main sequence stars similar to the Sun \citep{Serenelli.Johnson.ea2017}. Most recently, \emph{TESS} \citep{Ricker.Winn.ea2015} added thousands more dwarf and giant stars to the roster of solar-like oscillators \citep{Hon.Huber.ea2021,SilvaAguirre.Stello.ea2020,Hatt.Nielsen.ea2023}. +Instrumental and atmospheric noise limited the progress of asteroseismology with ground-based equipment to studies of small number of bright dwarf stars. Asteroseismology of solar-like oscillators requires high-cadence (\(\sim \SIrange{1}{10}{\minute}\)) brightness observations over long baselines (\(\sim \SI{1}{\year}\)) with noise of, for example \(\lesssim \SI{100}{ppm\per\hour}\) for \(I\)-band magnitude \(\lesssim 10\) \citep{Schofield.Chaplin.ea2019}. The first dedicated space-based missions which met these requirements arrived in the late 2000s, accelerating progress in the field. Initially, the \emph{CoRoT} mission \citep{Baglin.Auvergne.ea2006} detected solar-like oscillations in thousands of red giant stars \citep{DeRidder.Barban.ea2009,Mosser.Belkacem.ea2010}. Then, the \emph{Kepler} mission \citep{Borucki.Koch.ea2010} yielded oscillations in thousands more red giants \citep{Pinsonneault.Elsworth.ea2014} and hundreds of main sequence stars similar to the Sun \citep{Serenelli.Johnson.ea2017}. Most recently, \emph{TESS} \citep{Ricker.Winn.ea2015} added thousands more dwarf and giant stars to the roster of solar-like oscillators \citep{Hon.Huber.ea2021,SilvaAguirre.Stello.ea2020,Hatt.Nielsen.ea2023}. \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} @@ -80,11 +80,13 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \label{fig:seismo-psd} \end{figure} -In solar-like oscillators, p modes are stochastically excited by near-surface convection. Typically, the timescale of this process drives high-order modes in main sequence stars \needcite[\(n \sim 20\)]. We can identify these modes in a frequency-power spectrum derived from photometric or radial velocity time series observations. For instance, both stars in the 16 Cyg system are solar-like oscillators with similar properties to the Sun \needcite. Using 16 Cyg A as an example, we downloaded the power spectrum determined by the \emph{Kepler} Asteroseismic Science Operations Centre (KASOC) using \emph{Kepler} observations\footnote{\url{https://kasoc.phys.au.dk}}. Shown in Figure \ref{fig:seismo-psd}, the power spectrum of 16 Cyg A has a distinct power excess around \SI{2000}{\micro\hertz}. +In solar-like oscillators, p modes are stochastically excited by near-surface convection. Typically, the timescale of this process drives high-order modes in main sequence stars (\(n \sim 20\)). We can identify these modes in a frequency-power spectrum derived from photometric or radial velocity time series observations. For instance, both stars in the 16 Cygni planet hosting system are solar-like oscillators with similar properties to the Sun \citep{Metcalfe.Chaplin.ea2012,Davies.Chaplin.ea2015,Metcalfe.Creevey.ea2014}. Using 16 Cyg A as an example, we downloaded the power spectrum determined by the \emph{Kepler} Asteroseismic Science Operations Centre (KASOC) using \emph{Kepler} observations\footnote{\url{https://kasoc.phys.au.dk}}. Shown in Figure \ref{fig:seismo-psd}, the power spectrum of 16 Cyg A has a distinct power excess around \SI{2000}{\micro\hertz}. -The power excess has a Gaussian-like shape around frequencies compatible with the near-surface convective timescale responsible for mode excitation. We call the location of this Gaussian the `frequency at maximum power', \(\numax\). Dependent on near-surface conditions, \citet{Brown.Gilliland.ea1991} suggested \(\numax\) scales with the acoustic cut-off frequency --- the highest frequency at which acoustic waves can reflect near the stellar surface. Subsequently, \citet{Kjeldsen.Bedding1995} found that \(\numax \propto g\teff^{\,-1/2}\) where \(g\) and \(\teff\) are the near-surface gravitational field strength and temperature. +The power excess has a Gaussian-like shape around frequencies compatible with the near-surface convective timescale responsible for mode excitation. We call the location of this Gaussian the \emph{frequency at maximum power}, \(\numax\). Dependent on near-surface conditions, \citet{Brown.Gilliland.ea1991} suggested \(\numax\) scales with the acoustic cut-off frequency --- the highest frequency at which acoustic waves can reflect near the stellar surface. Subsequently, \citet{Kjeldsen.Bedding1995} found that \(\numax \propto g\teff^{\,-1/2}\) where \(g\) and \(\teff\) are the near-surface gravitational field strength and temperature. -Looking closely at the power excess in Figure \ref{fig:seismo-psd}, we can see a comb of approximately equally spaced peaks. Each peak corresponds to one or more oscillation modes, with their central frequencies and frequency differences providing information about the internal stellar structure. Naturally, higher frequency modes correspond to higher \(n\). However, the angular degree and azimuthal order are harder to identify. We saw in Figure \ref{fig:spherical-harmonics} how modes of higher \(l\) have more anti-nodes on the surface. Therefore, the overall effect of the oscillations cancel out when measuring irradiance integrated over the stellar surface. Consequentially, observed mode amplitude decreases with \(l\), leaving only \(l \lesssim 3\) detectable \needcite. From this, we can assume the tallest peaks are \(l=0,1\), and the smaller peaks are \(l=2,3\), all modulated by the wider Gaussian-like envelope. +Looking closely at the power excess in Figure \ref{fig:seismo-psd}, we can see a comb of approximately equally spaced peaks. Each peak corresponds to one or more oscillation modes, with their central frequencies and frequency differences providing information about the internal stellar structure. Naturally, higher frequency modes correspond to higher \(n\). However, the angular degree and azimuthal order are harder to identify. We saw in Figure \ref{fig:spherical-harmonics} how modes of higher \(l\) have more anti-nodes on the surface. Therefore, the overall effect of the oscillations cancel out when measuring irradiance integrated over the stellar surface. Consequentially, observed mode amplitude decreases with \(l\), leaving only \(l \lesssim 3\) detectable. +% Not Section 2.1 of JCD Lecture Notes (2014) derives this +We can assume the tallest peaks are \(l=0,1\), and the smaller peaks are \(l=2,3\), all modulated by the wider Gaussian-like envelope. For a spherically symmetric, non-rotating star, modes with different \(m\) oscillate at the same frequency and cannot be distinguished. However, the observed mode frequencies will split for different \(m\) via the Doppler effect in the case of a rotating or distorted (asymmetric) star. Measuring this splitting can constrain the rotation rate of the star. This has lead to breakthrough studies into gyrochronology and\dots \citep[e.g.][]{Hall.Davies.ea2021}. Hereafter, we will consider only the case of a slowly rotating, spherically symmetric star. @@ -100,7 +102,7 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \nu_0 = \left(2 \int_{0}^{R} \frac{\dd r}{c(r)}\right)^{-1}, \end{equation} % -where \(c(r)\) is the sound speed as a function of radius (\(r\)) and \(R\) is the radius of the star. Similarly to other variable stars, \citet{Ulrich1986} found that this characteristic frequency relates to the mean density by \(\nu_0 \propto \overline{\rho}^{\,1/2}\). While \(\nu_0\) is not directly detectable in solar-like oscillators, we can approximate it by taking the difference between consecutive modes of the same angular degree, \(\Delta\nu_{nl} = \nu_{nl} - \nu_{n-1\,l}\). Thus, estimates of a global (or average) large frequency separation, \(\Delta\nu \simeq \nu_0\), can provide information about the density of a star, leading to independent constraint on its mass and radius \needcite. +where \(c(r)\) is the sound speed as a function of radius (\(r\)) and \(R\) is the radius of the star. Similarly to other variable stars, \citet{Ulrich1986} found that this characteristic frequency relates to the mean density by \(\nu_0 \propto \overline{\rho}^{\,1/2}\). While \(\nu_0\) is not directly detectable in solar-like oscillators, we can approximate it by taking the difference between consecutive modes of the same angular degree, \(\Delta\nu_{nl} = \nu_{nl} - \nu_{n-1\,l}\). Thus, estimates of a global (or average) \emph{large frequency separation}, \(\Delta\nu \simeq \nu_0\), can provide information about the density of a star, leading to independent constraint on its mass and radius \needcite. \begin{figure}[tb] \centering @@ -130,7 +132,7 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} % In this section, we introduce the sample of stars being studied in this thesis relative to the broader astronomical picture. -Of the asteroseismic targets found by \emph{Kepler}, we focus on dwarf and subgiant solar-like oscillators. These are stars with a similar mass to the Sun either on the main sequence or post-main sequence before reaching the red giant branch. From an asteroseismic perspective, the power spectra of these stars are relatively simple. As they approach the red giant branch, their modes begin to couple with buoyancy-driven modes in the core, leading to irregular patterns which are comparably difficult to identify and model \needcite. Hence, these simple asteroseismic stars provide a good, consistent place to start modelling populations of stars. Furthermore, the majority of exoplanet host stars have so far been found around dwarf stars, making these solar-like oscillators a target for large-scale stellar characterisation. +Of the asteroseismic targets found by \emph{Kepler}, we focus on dwarf and subgiant solar-like oscillators. These are stars with a similar mass to the Sun either on the main sequence or post-main sequence before ascending the red giant branch. From an asteroseismic perspective, the power spectra of dwarf stars are relatively simple. As solar-like oscillators approach the red giant branch, their modes begin to couple with buoyancy-driven modes in the core \citep{Bedding.Mosser.ea2011,Mosser.Barban.ea2011}. This leads to irregular patterns which can be used to separate evolutionary state and study properties of the core \citep[e.g.][]{Mosser.Vrard.ea2015}. On the other hand, the relative simplicity of main sequence oscillators make them a suitable place to start. Furthermore, the work in this thesis anticipates the upcoming \emph{PLATO} mission which aims to observe \(\sim 10^4\) dwarf and subgiant oscillators \citep{Rauer.Aerts.ea2016}. % Modelling many stars with asteroseismology is complemented by other recent large-scale stellar surveys. High-precision astrometry from the \emph{Gaia} mission \citep{GaiaCollaboration.Prusti.ea2016} has provided improved distances and orbital solutions. The APOGEE large-scale spectroscopic survey \citep{Majewski.Schiavon.ea2017} has also yielded precise chemical abundances. These surveys have enabled studies of the assembly history of our galaxy. For example, \citet{Helmi.Babusiaux.ea2018} discovered a merger between the Milky Way and Gaia-Enceladus by analysing the motions and abundances from \emph{Gaia} and APOGEE. Asteroseismology has accompanied this work by helping determine the ages of stars tied to the merger \citep{Chaplin.Serenelli.ea2020,Montalban.Mackereth.ea2021}. @@ -152,7 +154,7 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \citet{Chaplin.Kjeldsen.ea2011} identified the first large catalogue of \(\sim 500\) dwarf and subgiant solar-like oscillators (black circles in Figure \ref{fig:hr-diagram}) by measuring \(\dnu\) and \(\numax\) in \emph{Kepler} data. Later, \citet{Chaplin.Basu.ea2014} determined ages, masses and radii for these stars using \(\dnu\) and \(\numax\) complemented by photometry and ground-based spectroscopy where available. The subsequent arrival of APOGEE spectroscopy allowed \citet{Serenelli.Johnson.ea2017} to revisit this sample with a more consistent set of \(\teff\) and metallicity. By comparing observations to models of stellar evolution, they found radii, masses and ages with uncertainties of around 3, 5 and 20 per cent respectively. -For a subset of these stars, the SNR was high enough to identify many individual modes. \citet{Appourchaux.Chaplin.ea2012} were among the first to publish individual mode frequencies (\(\nu_{nl}\)) for around 60 of these stars. This sample was later modelled by \citet{Metcalfe.Creevey.ea2014} who found observing individual modes doubled precision of radii, masses and ages over using \(\dnu\) and \(\numax\) alone. Around the same time, the number of confirmed exoplanets was increasing rapidly \needcite. This motivated a more detailed study of 35 exoplanet host stars by \citet{SilvaAguirre.Davies.ea2015} with modes identified by \citet{Davies.SilvaAguirre.ea2016}. We can see these as yellow triangles in Figure \ref{fig:hr-diagram}. The remaining best targets, referred to as the LEGACY sample, were modelled by \citet{SilvaAguirre.Lund.ea2017} using modes identified by \citet{Lund.SilvaAguirre.ea2017}. We show these as magenta squares in Figure \ref{fig:hr-diagram}. +For a subset of these stars, the SNR was high enough to identify many individual modes. \citet{Appourchaux.Chaplin.ea2012} were among the first to publish individual mode frequencies (\(\nu_{nl}\)) for around 60 of these stars. This sample was later modelled by \citet{Metcalfe.Creevey.ea2014} who found observing individual modes doubled precision of radii, masses and ages over using \(\dnu\) and \(\numax\) alone. Around the same time, the number of confirmed exoplanets was increasing rapidly \citep{Burke.Bryson.ea2014}. This motivated a more detailed study of 35 exoplanet host stars by \citet{SilvaAguirre.Davies.ea2015} with modes identified by \citet{Davies.SilvaAguirre.ea2016}. We can see these as yellow triangles in Figure \ref{fig:hr-diagram}. The remaining best targets, referred to as the LEGACY sample, were modelled by \citet{SilvaAguirre.Lund.ea2017} using modes identified by \citet{Lund.SilvaAguirre.ea2017}. We show these as magenta squares in Figure \ref{fig:hr-diagram}. % These targets are still being studied extensively today helium glitch \citet{Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2019}. @@ -192,7 +194,7 @@ \section{Modelling Stars the Bayesian Way}\label{sec:modelling-stars} We can also predict oscillation mode frequencies (\(\nu_{nl}\)) with simulations. For example, the GYRE code developed by \citet{Townsend.Teitler2013} uses the output of MESA to compute oscillation modes for a given \(n\) and \(l\). While the physics of p mode propagation in the star is relatively well-known, our understanding of the atmospheric boundary conditions are not. As such, there is a known discrepancy between the simulated and observed p modes. The nature of this can lead to a systematic bias on modelled \(\dnu\) and \(\nu_{nl}\). Corrections for this effect exist \citep[e.g.][]{Ball.Gizon2014} but are still not fully understood. -The complexity of stellar models means that the marginalised posterior distributions are not analytically derivable. Therefore, we use numerical methods like Markov Chain Monte Carlo (MCMC) to estimate the posterior \needcite. Typically, this involves exploring parameter space with multiple calls to \(f\) for different values of \(\vect{\theta}\). In the case of MCMC-based algorithms like Hamiltonian Monte Carlo (HMC) and the No U-Turn Sampler (NUTS), the gradient of \(f\) is also required. There are several open-source software packages widely used to implement these algorithms including \textsc{PyMC} \citep{Salvatier.Wiecki.ea2016} and \textsc{NumPyro} \citep{Phan.Pradhan.ea2019}. +The complexity of stellar models means that the marginalised posterior distributions are not analytically derivable. Therefore, we use numerical methods like Markov Chain Monte Carlo (MCMC) to estimate the posterior. Typically, this involves exploring parameter space with multiple calls to \(f\) for different values of \(\vect{\theta}\). In the case of MCMC-based algorithms like Hamiltonian Monte Carlo (HMC) and the No U-Turn Sampler (NUTS), the gradient of \(f\) is also required. There are several open-source software packages widely used to implement these algorithms including \textsc{PyMC} \citep{Salvatier.Wiecki.ea2016} and \textsc{NumPyro} \citep{Phan.Pradhan.ea2019}. There are some existing methods for determining stellar parameters using this Bayesian approach. \citet{Bazot.Bourguignon.ea2008} used the MCMC algorithm to sample model parameters with on-the-fly stellar model calculation. While this method can be tailored to individual stars, it is very computationally expensive. Each proposed set of \(\vect{\theta}\) spawns a stellar simulation which evolves to a given age. Steps prior to this age may be discarded and the simulation can take minutes to hours for each set of \(\vect{\theta}\). This is not a viable solution for modelling large numbers of stars. @@ -200,7 +202,7 @@ \section{Modelling Stars the Bayesian Way}\label{sec:modelling-stars} Alternatively, we could approximate \(f\) by interpolating a grid of stellar models. This is done in the Asteroseismic Inference on a Massive Scale (AIMS) pipeline by \citet{Lund.Reese2018,Rendle.Buldgen.ea2019}. Interpolation methods can be slow and scale poorly with dimensionality and number of points on the grid. A small subset of the grid can be used to mitigate this issue. However, we want a solution which can be extended to model many stars at once. -Many of these methods do not inherently account for systematic uncertainty from approximations of stellar physics and other assumptions. Asteroseismology has allowed us to model stars more precisely. However, this has exposed systematic uncertainties in common assumptions used when computing these models \citep[e.g.][]{Tayar.Claytor.ea2022}. For example, fractional helium abundance (\(Y\)), which cannot be determined spectroscopically in cool stars, is often assumed to follow a linear enrichment law \needcite. This law assumes that helium is enriched in the interstellar medium linearly with metallicity. \citet{Lebreton.Goupil.ea2014} found that varying \(Y\) by \(\pm\,0.03\) can have \(\gtrsim \mp\,20\) per cent effect on stellar age \citep{Lebreton.Goupil.ea2014}. +Many of these methods do not inherently account for systematic uncertainty from approximations of stellar physics and other assumptions. Asteroseismology has allowed us to model stars more precisely. However, this has exposed systematic uncertainties in common assumptions used when computing these models \citep[e.g.][]{Tayar.Claytor.ea2022}. For example, fractional helium abundance (\(Y\)), which cannot be determined spectroscopically in cool stars, is often assumed to follow a linear enrichment law \citep{Chiosi.Matteucci1982,Ribas.Jordi.ea2000,Casagrande.Flynn.ea2007}. This law assumes that helium is enriched in the interstellar medium linearly with metallicity. \citet{Lebreton.Goupil.ea2014} found that varying \(Y\) by \(\pm\,0.03\) can have \(\gtrsim \mp\,20\) per cent effect on stellar age \citep{Lebreton.Goupil.ea2014}. Another assumption is the value of the mixing-length theory parameter (\(\mlt\)). This parametrises a common approximation of convective mixing used in stellar models \citep{Gough1977}. Many of the aforementioned methods assume a value of \(\mlt\) calibrated to the Sun. However, 3D hydrodynamical models have shown different values of \(\mlt\) do a better job of approximating convection for different stars \citep{Magic.Weiss.ea2015}. diff --git a/references.bib b/references.bib index 2093400..003a537 100644 --- a/references.bib +++ b/references.bib @@ -549,6 +549,22 @@ @article{Basu1997 annotation = {ADS Bibcode: 1997MNRAS.288..572B} } +@article{Batalha2014, + title = {Exploring Exoplanet Populations with {{NASA}}'s {{Kepler Mission}}}, + author = {Batalha, Natalie M.}, + year = {2014}, + month = sep, + journal = {Proc. Natl. Acad. Sci.}, + volume = {111}, + pages = {12647--12654}, + issn = {0027-8424}, + doi = {10.1073/pnas.1304196111}, + urldate = {2023-04-26}, + abstract = {The Kepler Mission is exploring the diversity of planets and planetary systems. Its legacy will be a catalog of discoveries sufficient for computing planet occurrence rates as a function of size, orbital period, star type, and insolation flux. The mission has made significant progress toward achieving that goal. Over 3,500 transiting exoplanets have been identified from the analysis of the first 3 y of data, 100 planets of which are in the habitable zone. The catalog has a high reliability rate (85-90\% averaged over the period/radius plane), which is improving as follow-up observations continue. Dynamical (e.g., velocimetry and transit timing) and statistical methods have confirmed and characterized hundreds of planets over a large range of sizes and compositions for both single- and multiple-star systems. Population studies suggest that planets abound in our galaxy and that small planets are particularly frequent. Here, I report on the progress Kepler has made measuring the prevalence of exoplanets orbiting within one astronomical unit of their host stars in support of the National Aeronautics and Space Administration's long-term goal of finding habitable environments beyond the solar system.}, + keywords = {Astrophysics - Earth and Planetary Astrophysics}, + annotation = {ADS Bibcode: 2014PNAS..11112647B} +} + @article{Bazot.Bourguignon.ea2008, title = {Estimation of Stellar Parameters Using {{Monte Carlo Markov Chains}}}, author = {Bazot, M. and Bourguignon, S. and {Christensen-Dalsgaard}, J.}, @@ -581,6 +597,22 @@ @article{Bazot.Bourguignon.ea2012 annotation = {ADS Bibcode: 2012MNRAS.427.1847B} } +@article{Bedding.Mosser.ea2011, + title = {Gravity Modes as a Way to Distinguish between Hydrogen- and Helium-Burning Red Giant Stars}, + author = {Bedding, Timothy R. and Mosser, Benoit and Huber, Daniel and Montalb{\'a}n, Josefina and Beck, Paul and {Christensen-Dalsgaard}, J{\o}rgen and Elsworth, Yvonne P. and Garc{\'i}a, Rafael A. and Miglio, Andrea and Stello, Dennis and White, Timothy R. and De Ridder, Joris and Hekker, Saskia and Aerts, Conny and Barban, Caroline and Belkacem, Kevin and Broomhall, Anne-Marie and Brown, Timothy M. and Buzasi, Derek L. and Carrier, Fabien and Chaplin, William J. and {di Mauro}, Maria Pia and Dupret, Marc-Antoine and Frandsen, S{\o}ren and Gilliland, Ronald L. and Goupil, Marie-Jo and Jenkins, Jon M. and Kallinger, Thomas and Kawaler, Steven and Kjeldsen, Hans and Mathur, Savita and Noels, Arlette and Silva Aguirre, Victor and Ventura, Paolo}, + year = {2011}, + month = mar, + journal = {\nat}, + volume = {471}, + pages = {608--611}, + issn = {0028-0836}, + doi = {10.1038/nature09935}, + urldate = {2023-04-26}, + abstract = {Red giants are evolved stars that have exhausted the supply of hydrogen in their cores and instead burn hydrogen in a surrounding shell. Once a red giant is sufficiently evolved, the helium in the core also undergoes fusion. Outstanding issues in our understanding of red giants include uncertainties in the amount of mass lost at the surface before helium ignition and the amount of internal mixing from rotation and other processes. Progress is hampered by our inability to distinguish between red giants burning helium in the core and those still only burning hydrogen in a shell. Asteroseismology offers a way forward, being a powerful tool for probing the internal structures of stars using their natural oscillation frequencies. Here we report observations of gravity-mode period spacings in red giants that permit a distinction between evolutionary stages to be made. We use high-precision photometry obtained by the Kepler spacecraft over more than a year to measure oscillations in several hundred red giants. We find many stars whose dipole modes show sequences with approximately regular period spacings. These stars fall into two clear groups, allowing us to distinguish unambiguously between hydrogen-shell-burning stars (period spacing mostly \textasciitilde 50seconds) and those that are also burning helium (period spacing \textasciitilde 100 to 300 seconds).}, + keywords = {Astrophysics - Solar and Stellar Astrophysics}, + annotation = {ADS Bibcode: 2011Natur.471..608B} +} + @article{Bedding.Murphy.ea2020, title = {Very Regular High-Frequency Pulsation Modes in Young Intermediate-Mass Stars}, author = {Bedding, Timothy R. and Murphy, Simon J. and Hey, Daniel R. and Huber, Daniel and Li, Tanda and Smalley, Barry and Stello, Dennis and White, Timothy R. and Ball, Warrick H. and Chaplin, William J. and Colman, Isabel L. and Fuller, Jim and Gaidos, Eric and Harbeck, Daniel R. and Hermes, J. J. and Holdsworth, Daniel L. and Li, Gang and Li, Yaguang and Mann, Andrew W. and Reese, Daniel R. and Sekaran, Sanjay and Yu, Jie and Antoci, Victoria and Bergmann, Christoph and Brown, Timothy M. and Howard, Andrew W. and Ireland, Michael J. and Isaacson, Howard and Jenkins, Jon M. and Kjeldsen, Hans and McCully, Curtis and Rabus, Markus and Rains, Adam D. and Ricker, George R. and Tinney, Christopher G. and Vanderspek, Roland K.}, @@ -990,6 +1022,23 @@ @article{Buldgen.Salmon.ea2016 keywords = {asteroseismology,stars: fundamental parameters,stars: interiors,stars: oscillations} } +@article{Burke.Bryson.ea2014, + title = {Planetary {{Candidates Observed}} by {{Kepler IV}}: {{Planet Sample}} from {{Q1-Q8}} (22 {{Months}})}, + shorttitle = {Planetary {{Candidates Observed}} by {{Kepler IV}}}, + author = {Burke, Christopher J. and Bryson, Stephen T. and Mullally, F. and Rowe, Jason F. and Christiansen, Jessie L. and Thompson, Susan E. and Coughlin, Jeffrey L. and Haas, Michael R. and Batalha, Natalie M. and Caldwell, Douglas A. and Jenkins, Jon M. and Still, Martin and Barclay, Thomas and Borucki, William J. and Chaplin, William J. and Ciardi, David R. and Clarke, Bruce D. and Cochran, William D. and Demory, Brice-Olivier and Esquerdo, Gilbert A. and Gautier, III, Thomas N. and Gilliland, Ronald L. and Girouard, Forrest R. and Havel, Mathieu and Henze, Christopher E. and Howell, Steve B. and Huber, Daniel and Latham, David W. and Li, Jie and Morehead, Robert C. and Morton, Timothy D. and Pepper, Joshua and Quintana, Elisa and Ragozzine, Darin and Seader, Shawn E. and Shah, Yash and Shporer, Avi and Tenenbaum, Peter and Twicken, Joseph D. and Wolfgang, Angie}, + year = {2014}, + month = feb, + journal = {\apjs}, + volume = {210}, + pages = {19}, + issn = {0067-0049}, + doi = {10.1088/0067-0049/210/2/19}, + urldate = {2023-04-26}, + abstract = {We provide updates to the Kepler planet candidate sample based upon nearly two years of high-precision photometry (i.e., Q1-Q8). From an initial list of nearly 13,400 threshold crossing events, 480 new host stars are identified from their flux time series as consistent with hosting transiting planets. Potential transit signals are subjected to further analysis using the pixel-level data, which allows background eclipsing binaries to be identified through small image position shifts during transit. We also re-evaluate Kepler Objects of Interest (KOIs) 1-1609, which were identified early in the mission, using substantially more data to test for background false positives and to find additional multiple systems. Combining the new and previous KOI samples, we provide updated parameters for 2738 Kepler planet candidates distributed across 2017 host stars. From the combined Kepler planet candidates, 472 are new from the Q1-Q8 data examined in this study. The new Kepler planet candidates represent \textasciitilde 40\% of the sample with R P \textasciitilde{} 1 R {$\oplus$} and represent \textasciitilde 40\% of the low equilibrium temperature (T eq {$<$} 300 K) sample. We review the known biases in the current sample of Kepler planet candidates relevant to evaluating planet population statistics with the current Kepler planet candidate sample.}, + keywords = {Astrophysics - Earth and Planetary Astrophysics,catalogs,eclipses,planetary systems,space vehicles}, + annotation = {ADS Bibcode: 2014ApJS..210...19B} +} + @article{Campante.Lund.ea2016, title = {Spin-{{Orbit Alignment}} of {{Exoplanet Systems}}: {{Ensemble Analysis Using Asteroseismology}}}, shorttitle = {Spin-{{Orbit Alignment}} of {{Exoplanet Systems}}}, @@ -2838,6 +2887,21 @@ @article{Hogg2012 keywords = {Astrophysics - Instrumentation and Methods for Astrophysics,Physics - Data Analysis,Statistics and Probability} } +@article{Holman.Touma.ea1997, + title = {Chaotic Variations in the Eccentricity of the Planet Orbiting 16 {{Cygni B}}}, + author = {Holman, Matthew and Touma, Jihad and Tremaine, Scott}, + year = {1997}, + month = mar, + journal = {\nat}, + volume = {386}, + pages = {254--256}, + issn = {0028-0836}, + doi = {10.1038/386254a0}, + urldate = {2023-04-26}, + abstract = {The planet recently discovered1 orbiting the star 16 Cyg B has the largest eccentricity (e= 0.67) of any known planet. Planets that form in circumstellar disks are expected to have nearly circular orbits, although gravitational interactions in a system of two or more planets could generate high-eccentricity orbits2,3. Here we suggest that the eccentric orbit of 16 Cyg Bb arises from gravitational interactions with the distant companion star, 16 Cyg A. Assuming that 16 Cyg Bb formed in a nearly circular orbit, with the orbital plane inclined between 45\textdegree{} and 135\textdegree{} to the orbital plane of 16 Cyg A, and that there are no other planets with a mass similar to that of Jupiter within 30 astronomical units (AU, the average distance between the Earth and the Sun), then 16 Cyg Bb will oscillate between low-eccentricity and high-eccentricity orbits. The transitions between these orbits should occur every 107-109 years, with the planet spending up to 35 per cent of its lifetime with an eccentricity e{$>$} 0.6. These results imply that planetary orbits in binary stellar systems commonly experience periods of high eccentricity and dynamical chaos, and that such planets may occasionally collide with the primary star.}, + annotation = {ADS Bibcode: 1997Natur.386..254H} +} + @article{Hon.Bellinger.ea2020, title = {Asteroseismic Inference of Subgiant Evolutionary Parameters with Deep Learning}, author = {Hon, Marc and Bellinger, Earl P. and Hekker, Saskia and Stello, Dennis and Kuszlewicz, James S.}, @@ -3229,6 +3293,22 @@ @article{Kahn1961 abstract = {Abstract image available at: http://adsabs.harvard.edu/abs/1961ApJ...134..343K} } +@article{Karakas2014, + title = {Helium Enrichment and Carbon-Star Production in Metal-Rich Populations}, + author = {Karakas, Amanda I.}, + year = {2014}, + month = nov, + journal = {\mnras}, + volume = {445}, + pages = {347--358}, + issn = {0035-8711}, + doi = {10.1093/mnras/stu1727}, + urldate = {2023-04-26}, + abstract = {We present new theoretical stellar evolutionary models of metal-rich asymptotic giant branch (AGB) stars. Stellar models are evolved with initial masses between 1 and 7 M{$\odot$} at Z = 0.007, and 1 and 8 M{$\odot$} at Z = 0.014 (solar) and at Z = 0.03. We evolve models with a canonical helium abundance and with helium-enriched compositions (Y = 0.30, 0.35, and 0.40) at Z = 0.014 and 0.03. The efficiency of third dredge-up and the mass range of carbon stars decreases with an increase in metallicity. We predict carbon stars form from initial masses between 1.75 and 7 M{$\odot$} at Z = 0.007 and between 2 and 4.5 M{$\odot$} at solar metallicity. At Z = 0.03, the mass range for C-star production is narrowed to 3.25-4 M{$\odot$}. The third dredge-up is reduced when the helium content of the model increases owing to the reduced number of thermal pulses on the AGB. A small increase of {$\Delta$}Y = 0.05 is enough to prevent the formation of C stars at Z = 0.03, depending on the mass-loss rate, whereas at Z = 0.014, an increase of {$\Delta$}Y {$\greaterequivlnt$} 0.1 is required to prevent the formation of C stars. We speculate that the probability of finding C stars in a stellar population depends as much on the helium abundance as on the metallicity. To explain the paucity of C stars in the inner region of M31, we conclude that the observed stars have Y {$\greaterequivlnt$} 0.35 or that the stellar metallicity is higher than [Fe/H] {$\approx$} 0.1.}, + keywords = {Astrophysics - Solar and Stellar Astrophysics,galaxies: abundances,Galaxy: abundances,Galaxy: bulge,stars: abundances,stars: AGB and post-AGB,stars: carbon}, + annotation = {ADS Bibcode: 2014MNRAS.445..347K} +} + @article{Karamanis.Beutler.ea2021, title = {Zeus: A {{PYTHON}} Implementation of Ensemble Slice Sampling for Efficient {{Bayesian}} Parameter Inference}, shorttitle = {Zeus}, @@ -3976,6 +4056,22 @@ @article{Metcalfe.Creevey.ea2014 langid = {english} } +@article{Metcalfe.Creevey.ea2015, + title = {Asteroseismic {{Modeling}} of 16 {{Cyg A}} \& {{B}} Using the {{Complete Kepler Data Set}}}, + author = {Metcalfe, Travis S. and Creevey, Orlagh L. and Davies, Guy R.}, + year = {2015}, + month = oct, + journal = {\apj}, + volume = {811}, + pages = {L37}, + issn = {0004-637X}, + doi = {10.1088/2041-8205/811/2/L37}, + urldate = {2023-04-26}, + abstract = {Asteroseismology of bright stars with well-determined properties from parallax measurements and interferometry can yield precise stellar ages and meaningful constraints on the composition. We substantiate this claim with an updated asteroseismic analysis of the solar-analog binary system 16 Cyg A \& B using the complete 30-month data sets from the Kepler space telescope. An analysis with the Asteroseismic Modeling Portal, using all of the available constraints to model each star independently, yields the same age (t = 7.0 {$\pm$} 0.3 Gyr) and composition (Z = 0.021 {$\pm$} 0.002, Yi = 0.25 {$\pm$} 0.01) for both stars, as expected for a binary system. We quantify the accuracy of the derived stellar properties by conducting a similar analysis of a Kepler-like data set for the Sun, and we investigate how the reliability of asteroseismic inference changes when fewer observational constraints are available or when different fitting methods are employed. We find that our estimates of the initial helium mass fraction are probably biased low by 0.02-0.03 from neglecting diffusion and settling of heavy elements, and we identify changes to our fitting method as the likely source of small shifts from our initial results in 2012. We conclude that in the best cases reliable stellar properties can be determined from asteroseismic analysis even without independent constraints on the radius and luminosity.}, + keywords = {Astrophysics - Earth and Planetary Astrophysics,Astrophysics - Solar and Stellar Astrophysics,HD 186427,stars: individual: HD 186408,stars: interiors,stars: oscillations,stars: solar-type}, + annotation = {ADS Bibcode: 2015ApJ...811L..37M} +} + @article{Miglio.Chiappini.ea2017, title = {{{PLATO}} as It Is : {{A}} Legacy Mission for {{Galactic}} Archaeology}, shorttitle = {{{PLATO}} as It Is}, @@ -4271,6 +4367,22 @@ @misc{Morton2015a keywords = {Software} } +@article{Mosser.Barban.ea2011, + title = {Mixed Modes in Red-Giant Stars Observed with {{CoRoT}}}, + author = {Mosser, B. and Barban, C. and Montalb{\'a}n, J. and Beck, P. G. and Miglio, A. and Belkacem, K. and Goupil, M. J. and Hekker, S. and De Ridder, J. and Dupret, M. A. and Elsworth, Y. and Noels, A. and Baudin, F. and Michel, E. and Samadi, R. and Auvergne, M. and Baglin, A. and Catala, C.}, + year = {2011}, + month = aug, + journal = {\aap}, + volume = {532}, + pages = {A86}, + issn = {0004-6361}, + doi = {10.1051/0004-6361/201116825}, + urldate = {2023-04-26}, + abstract = {Context. The CoRoT mission has provided thousands of red-giant light curves. The analysis of their solar-like oscillations allows us to characterize their stellar properties. Aims: Up to now, the global seismic parameters of the pressure modes have been unable to distinguish red-clump giants from members of the red-giant branch. As recently done with Kepler red giants, we intend to analyze and use the so-called mixed modes to determine the evolutionary status of the red giants observed with CoRoT. We also aim at deriving different seismic characteristics depending on evolution. Methods: The complete identification of the pressure eigenmodes provided by the red-giant universal oscillation pattern allows us to aim at the mixed modes surrounding the {$\mathscr{l}$} = 1 expected eigenfrequencies. A dedicated method based on the envelope autocorrelation function is proposed to analyze their period separation. Results: We have identified the mixed-mode signature separation thanks to their pattern that is compatible with the asymptotic law of gravity modes. We have shown that, independent of any modeling, the g-mode spacings help to distinguish the evolutionary status of a red-giant star. We then report the different seismic and fundamental properties of the stars, depending on their evolutionary status. In particular, we show that high-mass stars of the secondary clump present very specific seismic properties. We emphasize that stars belonging to the clump were affected by significant mass loss. We also note significant population and/or evolution differences in the different fields observed by CoRoT. The CoRoT space mission, launched 2006 December 27, was developed and is operated by the CNES, with participation of the Science Programs of ESA, ESA\v{S}s RSSD, Austria, Belgium, Brazil, Germany, and Spain.Apeendix A is available in electronic form at http://www.aanda.org}, + keywords = {Astrophysics - Solar and Stellar Astrophysics,methods: data analysis,stars: interiors,stars: mass-loss,stars: oscillations}, + annotation = {ADS Bibcode: 2011A\&A...532A..86M} +} + @article{Mosser.Belkacem.ea2010, title = {Red-Giant Seismic Properties Analyzed with {{CoRoT}}}, author = {Mosser, B. and Belkacem, K. and Goupil, M.-J. and Miglio, A. and Morel, T. and Barban, C. and Baudin, F. and Hekker, S. and Samadi, R. and Ridder, J. De and Weiss, W. and Auvergne, M. and Baglin, A.}, @@ -5807,6 +5919,23 @@ @article{Thoul.Bahcall.ea1994 keywords = {Abundance,Computerized Simulation,Diffusion,Flow Equations,Heavy Elements,Helium,Solar Interior,Stellar Composition,Stellar Evolution,Stellar Models,Subroutines} } +@article{Tognelli.DellOmodarme.ea2021, + title = {Bayesian Calibration of the Mixing Length Parameter {{$\alpha$ML}} and of the Helium-to-Metal Enrichment Ratio {{$\Delta$Y}}/{{$\Delta$Z}} with Open Clusters: The {{Hyades}} Test-Bed}, + shorttitle = {Bayesian Calibration of the Mixing Length Parameter {{$\alpha$ML}} and of the Helium-to-Metal Enrichment Ratio {{$\Delta$Y}}/{{$\Delta$Z}} with Open Clusters}, + author = {Tognelli, E. and Dell'Omodarme, M. and Valle, G. and Prada Moroni, P. G. and Degl'Innocenti, S.}, + year = {2021}, + month = jan, + journal = {\mnras}, + volume = {501}, + pages = {383--397}, + issn = {0035-8711}, + doi = {10.1093/mnras/staa3686}, + urldate = {2023-04-26}, + abstract = {We tested the capability of a Bayesian procedure to calibrate both the helium abundance and the mixing length parameter ({$\alpha$}ML), using precise photometric data for main-sequence (MS) stars in a cluster with negligible reddening and well-determined distance. The method has been applied first to a mock data set generated to mimic Hyades MS stars and then to the real Hyades cluster. We tested the impact on the results of varying the number of stars in the sample, the photometric errors, and the estimated [Fe/H]. The analysis of the synthetic data set shows that {$\alpha$}ML is recovered with a very good precision in all the analysed cases (with an error of few percent), while [Fe/H] and the helium-to-metal enrichment ratio {$\Delta$}Y/{$\Delta$}Z are more problematic. If spectroscopic determinations of [Fe/H] are not available and thus [Fe/H] has to be recovered alongside with {$\Delta$}Y/{$\Delta$}Z and {$\alpha$}ML, the well-known degeneracy between [Fe/H]-{$\Delta$}Y/{$\Delta$}Z-{$\alpha$}ML could result in a large uncertainty on the recovered parameters, depending on the portion of the MS used for the analysis. On the other hand, the prior knowledge of an accurate [Fe/H] value puts a strong constraint on the models, leading to a more precise parameters recovery. Using the current set of PISA models, the most recent [Fe/H] value and the Gaia photometry and parallaxes for the Hyades cluster, we obtained the average values {$<\alpha$}ML{$>$} = 2.01 {$\pm$} 0.05 and {$<\Delta$}Y/{$\Delta$}Z{$>$} = 2.03 {$\pm$} 0.33, sensitively reducing the uncertainty in these important parameters.}, + keywords = {Astrophysics - Astrophysics of Galaxies,Astrophysics - Solar and Stellar Astrophysics,methods: numerical,methods: statistical,stars: abundances,stars: evolution,stars: fundamental parameters,stars: low-mass}, + annotation = {ADS Bibcode: 2021MNRAS.501..383T} +} + @article{Townsend.Teitler2013, title = {{{GYRE}}: An Open-Source Stellar Oscillation Code Based on a New {{Magnus Multiple Shooting}} Scheme}, shorttitle = {{{GYRE}}}, @@ -6177,6 +6306,22 @@ @article{Willett.Lintott.ea2013 annotation = {ADS Bibcode: 2013MNRAS.435.2835W} } +@article{Winn.Fabrycky2015, + title = {The {{Occurrence}} and {{Architecture}} of {{Exoplanetary Systems}}}, + author = {Winn, Joshua N. and Fabrycky, Daniel C.}, + year = {2015}, + month = aug, + journal = {\araa}, + volume = {53}, + pages = {409--447}, + issn = {0066-4146}, + doi = {10.1146/annurev-astro-082214-122246}, + urldate = {2023-04-26}, + abstract = {The basic geometry of the Solar System-the shapes, spacings, and orientations of the planetary orbits-has long been a subject of fascination as well as inspiration for planet-formation theories. For exoplanetary systems, those same properties have only recently come into focus. Here we review our current knowledge of the occurrence of planets around other stars, their orbital distances and eccentricities, the orbital spacings and mutual inclinations in multiplanet systems, the orientation of the host star's rotation axis, and the properties of planets in binary-star systems.}, + keywords = {Astrophysics - Earth and Planetary Astrophysics}, + annotation = {ADS Bibcode: 2015ARA\&A..53..409W} +} + @article{Wong.Gabrie.ea2022, title = {{{flowMC}}: {{Normalizing-flow}} Enhanced Sampling Package for Probabilistic Inference in {{Jax}}}, shorttitle = {{{flowMC}}}, From 2d5076a7d3de46d2054ee6d55c25ccb513627880 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Thu, 27 Apr 2023 12:17:26 +0100 Subject: [PATCH 27/50] Add typewriter font --- packages.sty | 2 ++ 1 file changed, 2 insertions(+) diff --git a/packages.sty b/packages.sty index f98aee4..3f6fa1d 100644 --- a/packages.sty +++ b/packages.sty @@ -34,6 +34,8 @@ % \usepackage{roboto} % \usepackage{noto} \usepackage[scaled=0.9]{inter} +% Typewriter font +\usepackage[zerostyle=b]{newtxtt} \usepackage{csquotes} % When loading babel with biblatex \usepackage{anyfontsize} % For custom font sizes From 85b7bb93c190e32ec65b57610a8a3e03a11aa519 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Thu, 27 Apr 2023 12:17:38 +0100 Subject: [PATCH 28/50] Add software section --- chapters/software.tex | 11 +++++++++++ thesis.tex | 29 +++++++++++++---------------- 2 files changed, 24 insertions(+), 16 deletions(-) create mode 100644 chapters/software.tex diff --git a/chapters/software.tex b/chapters/software.tex new file mode 100644 index 0000000..3e8f773 --- /dev/null +++ b/chapters/software.tex @@ -0,0 +1,11 @@ +\chapter*{Data Availability \& Software} +\addcontentsline{toc}{chapter}{Data Availability \& Software} + +The data and code underlying this thesis are available in the online supplementary material of \citet{Lyttle.Davies.ea2021}, and in the Zenodo database at \url{https://dx.doi.org/10.5281/zenodo.4746353}. The code used to produce the remainder of this work will be made public at \url{https://github.com/alexlyttle/thesis} and in the Zenodo database upon publication. This thesis includes data collected by the \emph{Kepler} mission. Funding for the \emph{Kepler} mission is provided by the NASA Science Mission directorate This work has also used data from the European Space Agency (ESA) mission +\emph{Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} +Data Processing and Analysis Consortium (DPAC, +\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC +has been provided by national institutions, in particular the institutions +participating in the \emph{Gaia} Multilateral Agreement. To query other published datasets, this research used the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in \citet{Ochsenbein.Bauer.ea2000}. Finally, this work used the \emph{Gaia}-\emph{Kepler} crossmatch database at \url{https://gaia-kepler.fun} created by Megan Bedell. + +I acknowledge use of the \textsc{Python} programming language (Python Software Foundation, \url{https://www.python.org}) for the majority of code written for this work. Specific Python packages used are referenced in-text with the exception of: \texttt{matplotlib} \citep[v3.6.2;][]{Caswell.Lee.ea2022,Hunter2007} and \texttt{seaborn} \citep{Waskom2021} for creating plots; \texttt{scipy} \citep{Virtanen.Gommers.ea2020} for general scientific computational methods; \texttt{astropy} \citep{AstropyCollaboration.Price-Whelan.ea2022} for reading and writing astronomical data; \texttt{astroquery} \citep{Ginsburg.Sipocz.ea2019} for querying astronomy databases; and \texttt{lightkurve} \citep{LightkurveCollaboration.Cardoso.ea2018} for processing \emph{Kepler} light curves. Finally, this thesis was compiled directly to PDF from \LaTeX~source code based on the \texttt{uob-thesis-template} template (\url{https://github.com/alexlyttle/uob-thesis-template}). diff --git a/thesis.tex b/thesis.tex index 389189e..cabac94 100644 --- a/thesis.tex +++ b/thesis.tex @@ -46,6 +46,7 @@ chapters/glitch, chapters/glitch-gp, chapters/conclusion, + chapters/software, appendices/lyttle21 } @@ -89,32 +90,27 @@ %% ABSTRACT ------------------------------------------------------------------- % \abstract{% -As high-precision asteroseismology modelling of stars continues to advance, it is becoming increasingly important to account for the systematic effects that arise from our assumptions of the stellar helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)). To address this issue, we present a novel approach for improving the inference of the fundamental parameters of a sample of Kepler dwarfs and subgiants with a narrow mass range of \(0.8 < M < 1.2 \mathrm{M}_{\odot}\). Our method includes a statistical treatment of \(Y\) and \(\mlt\) and employs a hierarchical Bayesian model that incorporates information about the distribution of these parameters in the population. Specifically, we fit a linear helium enrichment law that includes an intrinsic scatter and a normal distribution in \(\mlt\). We explore various levels of pooling parameters, with and without the Sun as a calibrator, and report \(\Delta Y/\Delta Z = 1.05^{+0.28}_{-0.25}\) and \(\mu_\alpha = 1.90^{+0.10}_{-0.09}\) when the Sun is included. Despite accounting for the uncertainties in \(Y\) and \(\mlt\), we are able to report statistical uncertainties of 2.5 per cent in mass, 1.2 per cent in radius, and 12 per cent in age. Moreover, our approach can be extended to larger samples, which would enable us to obtain more precise constraints on the fundamental parameters of individual stars and improve our understanding of the population-level inference. -} + As high-precision asteroseismology modelling of stars continues to advance, it is becoming increasingly important to account for the systematic effects that arise from our assumptions of the stellar helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)). To address this issue, we present a novel approach for improving the inference of the fundamental parameters of a sample of Kepler dwarfs and subgiants with a narrow mass range of \(0.8 < M < 1.2 \mathrm{M}_{\odot}\). Our method includes a statistical treatment of \(Y\) and \(\mlt\) and employs a hierarchical Bayesian model that incorporates information about the distribution of these parameters in the population. Specifically, we fit a linear helium enrichment law that includes an intrinsic scatter and a normal distribution in \(\mlt\). We explore various levels of pooling parameters, with and without the Sun as a calibrator, and report \(\Delta Y/\Delta Z = 1.05^{+0.28}_{-0.25}\) and \(\mu_\alpha = 1.90^{+0.10}_{-0.09}\) when the Sun is included. Despite accounting for the uncertainties in \(Y\) and \(\mlt\), we are able to report statistical uncertainties of 2.5 per cent in mass, 1.2 per cent in radius, and 12 per cent in age. Moreover, our approach can be extended to larger samples, which would enable us to obtain more precise constraints on the fundamental parameters of individual stars and improve our understanding of the population-level inference. + + We present a new method for modelling acoustic glitch signatures in the radial mode frequencies of solar-like oscillators using a Gaussian Process. We compare our approach with another method using a model star with varying levels of noise and apply both methods to the star 16 Cyg A to provide a real-world example. Our results show that our method accurately determines the strength and location of glitches caused by the second ionisation of helium and the base of the convective zone. Moreover, we demonstrate that our Gaussian Process approach improves the treatment of the smoothly varying component of the mode frequencies, outperforming a 4th order polynomial in this regard. We also find that our method finds less multimodality in the posterior for acoustic depth by using a prior to inform the glitch parameters. However, the inclusion of the He\,\textsc{i} ionisation glitch in the model remains a question. Overall, our study suggests that the Gaussian Process method shows promise as a tool for modelling acoustic glitches in solar-like oscillators, with potential applications in asteroseismology and studies of stellar structure and evolution.} %% ACKNOWLEDGEMENTS ----------------------------------------------------------- % \acknowledgements{% +I am tremendously grateful to my PhD supervisor Guy R. Davies for his support and excellent mentorship throughout my postgraduate research. I also thank my first co-supervisor Andrea Miglio for his guidence before his move to the University of Bologna. I extend my thanks to Amaury Triaud, who took over from Andrea as my co-supervisor and has since been an inspiring mentor and leader of our research group. Additionally, I would like to thank in advance my PhD examiners Daniel Reese and Chris Moore, and viva chairperson Annelies Mortier. + \begin{CJK*}{UTF8}{gbsn} % INTERNAL SUPPORT - I am tremendously grateful to my PhD supervisor Guy R. Davies for his support and excellent mentorship throughout my postgraduate research. I also thank my first co-supervisor Andrea Miglio for his guidence before his move to the University of Bologna. I extend my thanks to Amaury Triaud, who took over from Andrea as my co-supervisor and has since been an inspiring mentor and leader of our research group. It has been a pleasure to work with the Sun, Stars, and Exoplanets research group here at the University of Birmignham. Particularly, I would like to thank all of my fellow current and former PhD students with whom I have exchanged ideas, advice, laughs, and stories: T. Baycroft, L. Carboneau, Y. Davies, A. Dixon, G. Dransfield, A. Freckleton, O. Hall, E. Hatt, V. Hod\v{z}i\'{c}, S. Khan, E. Ross, O. Scutt, M. Standing, W. Van Rossem, and E. Willett. I also appreciate the support of other postdoctoral researchers in our group, notably M. Nielsen, Tanda Li (李坦达), and W. Ball. I extend my thanks to the support staff in the School of Physics and Astronomy, particularly our office secretary Lou for her unwavering support over a challenging few years. + It has been a pleasure to work with the Sun, Stars, and Exoplanets research group here at the University of Birmingham. I would like to thank all of my fellow current and former PhD students with whom I have exchanged ideas, advice, laughs, and stories. + % : T. Baycroft, L. Carboneau, Y. Davies, A. Dixon, G. Dransfield, A. Freckleton, O. Hall, E. Hatt, V. Hod\v{z}i\'{c}, S. Khan, E. Ross, O. Scutt, M. Standing, W. Van Rossem, and E. Willett. + I also appreciate the support of the Head of School Bill Chaplin, and other postdoctoral researchers in our group, notably M. Nielsen, Tanda Li (李坦达), and W. Ball. I extend my thanks to the support staff in the School of Physics and Astronomy, particularly our office secretary Lou for her unwavering support over a challenging few years. I also thank my external collaborators, for example Nick Saunders at the University of Hawaii and Simon Murphy at the University of Southern Queensland, with whom I have worked during my graduate studies. \end{CJK*} -% EXTERNAL COLLABORATORS -I thank my external collaborators. Nick Saunders at the University of Hawaii. Simon Murphy at the University of Southern Queensland. Co-authors on my research publications. +Last but not least, I am eternally grateful to the support of my loving family and friends. Particularly, I thank my fianc\'{e} Hannah, my parents Katy and Paul, and my brother Dom. % FUNDING -I acknowledge the support of the public who have indirectly funded the research in this work via various public agencies. Particularly, I thank the Science and Technology Facilities Council who funded my PhD. I also acknowledge that this work is a part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (CartographY; grant agreement ID 804752). Finally, I am grateful for the financial and academic support from the Alan Turing Enrichment scheme. This scheme provided me the opportunity to learn from and collaborate with the wider world of machine learning in science. - -% DATA AND SOFTWARE -The use of data and software. Open source community. This work made use of the \href{https://gaia-kepler.fun}{gaia-kepler.fun} crossmatch database created by Megan Bedell. - -% FAMILY AND FRIENDS -Support of my loving family and friends. - -% THESIS EXAMINERS -Finally, I would like to thank my PhD examiners Daniel Reese and Chris Moore, and viva chairperson Annelies Mortier. +I acknowledge the support of the public who have indirectly funded the research in this work via various publicly funded agencies. Specifically, I thank the Science and Technology Facilities Council (STFC) who funded my PhD. I also acknowledge that this work is a part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (CartographY; grant agreement ID 804752). Finally, I am grateful for the financial and academic support from the Alan Turing Enrichment scheme. This scheme provided me the opportunity to learn from and collaborate with the wider world of machine learning in science. } @@ -147,7 +143,8 @@ \include{chapters/glitch} \include{chapters/glitch-gp} \include{chapters/conclusion} - + \include{chapters/software} + %% BIBLIOGRAPHY ------------------------------------------------------------- % \begin{refcontext}[sorting=nyt] From f0f7c367de505d891d748e81d37c30da39e92729 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Thu, 27 Apr 2023 12:18:06 +0100 Subject: [PATCH 29/50] Fix software font style --- chapters/conclusion.tex | 2 +- chapters/glitch-gp.tex | 2 +- chapters/glitch.tex | 2 +- chapters/hbm.tex | 4 +- chapters/introduction.tex | 6 +-- chapters/lyttle21.tex | 2 +- references.bib | 98 +++++++++++++++++++++++++++++++++++++++ 7 files changed, 107 insertions(+), 9 deletions(-) diff --git a/chapters/conclusion.tex b/chapters/conclusion.tex index 288ab3d..31c3e35 100644 --- a/chapters/conclusion.tex +++ b/chapters/conclusion.tex @@ -10,7 +10,7 @@ % all distributions of LaTeX version 2005/12/01 or later. % % -\chapter{Conclusion and Future Prospects} +\chapter{Conclusion \& Future Prospects} \textit{We conclude this thesis by providing a summary of the work herein and reflecting upon key results. Then, we consider possible improvements to our hierarchical model and method for emulating stellar simulations. Finally, we discuss future prospects for applying our method to data from current and upcoming missions.} diff --git a/chapters/glitch-gp.tex b/chapters/glitch-gp.tex index c914533..15fbaf9 100644 --- a/chapters/glitch-gp.tex +++ b/chapters/glitch-gp.tex @@ -212,7 +212,7 @@ \subsection{The GP Method} % The mean and variance for the aforementioned prior distributions are summarised in Table \ref{tab:glitch-prior} for the test star and 16 Cyg A. Prior distributions for the remaining parameters were the same for both stars. For example, the prior on the phase parameters \(\phi_\helium\) and \(\phi_\bcz\) was uniformly distributed from 0 to \(2\pi\). We also used a weakly informative prior on the model uncertainty, \(\ln\sigma \sim \mathcal{N}( - \ln 100, 4)\), centred on an uncertainty of \SI{0.01}{\micro\hertz}. -We sampled the posterior distribution using the nested sampling package \textsc{Dynesty} \citep{Speagle2020,Koposov.Speagle.ea2023}. We applied the multi-ellipsoid bounding method \citep{Feroz.Hobson.ea2009} with 500 live points and the random walk sampling method \citep{Skilling2006} with a minimum of 50 steps before proposing a new live point. In addition, we enabled periodic boundary conditions for \(\phi_\helium\) and \(\phi_\bcz\) with the prior transform projecting to periodic space using \(\phi = \phi'\,\mathrm{mod}\,2\pi\), where \(\phi'\) is unconstrained. Otherwise, the nested sampler ran with its default parameters. We used \textsc{jax} \citep{Bradbury.Frostig.ea2018} to make use of accelerated linear algebra (XLA) and just-in-time (JIT) compilation, and \textsc{tinygp} \citep{Foreman-Mackey.Yadav.ea2022} to build the GP. For analysis and comparison with the \citetalias{Verma.Raodeo.ea2019} method, we drew 1000 points randomly from the posterior samples according to their estimated weights. +We sampled the posterior distribution using the nested sampling \textsc{Python} package \texttt{dynesty} \citep{Speagle2020,Koposov.Speagle.ea2023}. We applied the multi-ellipsoid bounding method \citep{Feroz.Hobson.ea2009} with 500 live points and the random walk sampling method \citep{Skilling2006} with a minimum of 50 steps before proposing a new live point. In addition, we enabled periodic boundary conditions for \(\phi_\helium\) and \(\phi_\bcz\) with the prior transform projecting to periodic space using \(\phi = \phi'\,\mathrm{mod}\,2\pi\), where \(\phi'\) is unconstrained. Otherwise, the nested sampler ran with its default parameters. We used the \textsc{Python} packages \texttt{jax} \citep{Bradbury.Frostig.ea2018} to make use of accelerated linear algebra (XLA) and just-in-time (JIT) compilation, and \texttt{tinygp} \citep{Foreman-Mackey.Yadav.ea2022} to build the GP. For analysis and comparison with the \citetalias{Verma.Raodeo.ea2019} method, we drew 1000 points randomly from the posterior samples according to their estimated weights. % Rearranged into dimensionless quantities, \(f = \nu/\nu_0\), \(t = \tau/\tau_0\), \(a_\helium = \nu_0\alpha_\helium\), \(b_\helium = \nu_0 \beta_\helium\), and \(a_\bcz = \alpha_\bcz/\nu_0^2\), % % diff --git a/chapters/glitch.tex b/chapters/glitch.tex index 406c254..f4733d8 100644 --- a/chapters/glitch.tex +++ b/chapters/glitch.tex @@ -88,7 +88,7 @@ \section{Introduction} \end{split} \end{equation} % -Solving these simultaneously with the Python package \textsc{sympy} gives the following equations for the complex coefficients\footnote{The code for these derivations are available at \url{\gitremote/tree/\gitbranch/notebooks}}, +Solving these simultaneously with the \textsc{Python} package \texttt{sympy} \citep{Meurer.Smith.ea2017} gives the following equations for the complex coefficients\footnote{The code for these derivations are available at \url{\gitremote/tree/\gitbranch/notebooks}}, % \begin{align} A &= \delta k (2k + \delta k) (1 - \ee^{4i \delta x (k + \delta k)}) \ee^{- 2i k (x_\glitch - \delta x)} \alpha^{-1}, \\ diff --git a/chapters/hbm.tex b/chapters/hbm.tex index e2e3517..5140918 100644 --- a/chapters/hbm.tex +++ b/chapters/hbm.tex @@ -74,7 +74,7 @@ \subsection{Simple Model}\label{sec:simple-model} \begin{figure}[tb] \centering \includegraphics{figures/simple-pgm.pdf} - \caption[Probabilistic graphical model for the simple (non-hierarchical) model]{Probabilistic graphical model for the simple (non-hierarchical) model. Parameters are given by circular nodes and connected by arrows showing the direction of dependency. Observed parameters are shaded, and fixed parameters are given by filled dots. The box represents a set of parameters belonging to the \(i\)-th star. \emph{This diagram was made using \textsc{Daft} \citep{Foreman-Mackey.Hogg.ea2021}.}} + \caption[Probabilistic graphical model for the simple (non-hierarchical) model]{Probabilistic graphical model for the simple (non-hierarchical) model. Parameters are given by circular nodes and connected by arrows showing the direction of dependency. Observed parameters are shaded, and fixed parameters are given by filled dots. The box represents a set of parameters belonging to the \(i\)-th star. \emph{This diagram was made using the \textsc{Python} package \texttt{daft} \citep{Foreman-Mackey.Hogg.ea2021}.}} \label{fig:simple-pgm} \end{figure} @@ -131,7 +131,7 @@ \subsection{Inferring the Model Parameters}\label{sec:hbm-inf} To infer the model parameters, we need to calculate the marginalised posterior distributions for each parameter. We could obtain these analytically by integrating the full posterior distribution over all model parameters except for the parameter of interest. Alternatively, we can approximate the marginalised posterior using a Markov Chain Monte Carlo (MCMC) sampling algorithm. We chose the latter approach because it is scalable to more complicated models where the marginalisation is not analytically solvable. -We used the No U-Turn Sampler \citep[NUTS;][]{Hoffman.Gelman2014} as implemented in the \textsc{NumPyro} Python package \citep{Phan.Pradhan.ea2019,Bingham.Chen.ea2019} to sample from the approximate posterior distribution for both models. We ran the sampler for 1000 steps following 500 `warmup' steps (used to adapt the sampling procedure) and repeated for 10 MCMC chains. To reduce the number of divergences encountered during sampling, we increased the target accept probability from 0.8 to 0.98 for the HBM. The resulting marginalised posterior samples amounted to \num{10000} per parameter. +We used the No U-Turn Sampler \citep[NUTS;][]{Hoffman.Gelman2014} as implemented in the \textsc{Python} package \texttt{numpyro} \citep{Phan.Pradhan.ea2019,Bingham.Chen.ea2019} to sample from the approximate posterior distribution for both models. We ran the sampler for 1000 steps following 500 `warmup' steps (used to adapt the sampling procedure) and repeated for 10 MCMC chains. To reduce the number of divergences encountered during sampling, we increased the target accept probability from 0.8 to 0.98 for the HBM. The resulting marginalised posterior samples amounted to \num{10000} per parameter. \subsection{Comparing the Models}\label{sec:hbm-comp} diff --git a/chapters/introduction.tex b/chapters/introduction.tex index e4b15ab..efdc3bd 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -88,7 +88,7 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} % Not Section 2.1 of JCD Lecture Notes (2014) derives this We can assume the tallest peaks are \(l=0,1\), and the smaller peaks are \(l=2,3\), all modulated by the wider Gaussian-like envelope. -For a spherically symmetric, non-rotating star, modes with different \(m\) oscillate at the same frequency and cannot be distinguished. However, the observed mode frequencies will split for different \(m\) via the Doppler effect in the case of a rotating or distorted (asymmetric) star. Measuring this splitting can constrain the rotation rate of the star. This has lead to breakthrough studies into gyrochronology and\dots \citep[e.g.][]{Hall.Davies.ea2021}. Hereafter, we will consider only the case of a slowly rotating, spherically symmetric star. +For a spherically symmetric, non-rotating star, modes with different \(m\) oscillate at the same frequency and cannot be distinguished. However, the observed mode frequencies will split for different \(m\) via the Doppler effect in the case of a rotating or distorted (asymmetric) star. Measuring this splitting can constrain the rotation rate of the star. This has lead to breakthrough studies into gyrochronology and \todo{finish} \citep[e.g.][]{Hall.Davies.ea2021}. Hereafter, we will consider only the case of a slowly rotating, spherically symmetric star. If we consider an acoustic wave in a one-dimensional homogeneous medium, then we would expect each mode of oscillation to be an integer multiple of the fundamental mode. While the case for a star is more complicated, we can also approximate the frequencies for different modes as a multiple of a characteristic frequency. \citet{Tassoul1980} found that the modes could be approximated by assuming the asymptotic limit where \(l/n \rightarrow 0\), giving the following expression \citep[cf.][]{Gough1986}, % @@ -102,7 +102,7 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \nu_0 = \left(2 \int_{0}^{R} \frac{\dd r}{c(r)}\right)^{-1}, \end{equation} % -where \(c(r)\) is the sound speed as a function of radius (\(r\)) and \(R\) is the radius of the star. Similarly to other variable stars, \citet{Ulrich1986} found that this characteristic frequency relates to the mean density by \(\nu_0 \propto \overline{\rho}^{\,1/2}\). While \(\nu_0\) is not directly detectable in solar-like oscillators, we can approximate it by taking the difference between consecutive modes of the same angular degree, \(\Delta\nu_{nl} = \nu_{nl} - \nu_{n-1\,l}\). Thus, estimates of a global (or average) \emph{large frequency separation}, \(\Delta\nu \simeq \nu_0\), can provide information about the density of a star, leading to independent constraint on its mass and radius \needcite. +where \(c(r)\) is the sound speed as a function of radius (\(r\)) and \(R\) is the radius of the star. Similarly to other variable stars, \citet{Ulrich1986} found that this characteristic frequency relates to the mean density by \(\nu_0 \propto \overline{\rho}^{\,1/2}\). While \(\nu_0\) is not directly detectable in solar-like oscillators, we can approximate it by taking the difference between consecutive modes of the same angular degree, \(\Delta\nu_{nl} = \nu_{nl} - \nu_{n-1\,l}\). Thus, estimates of a global (or average) \emph{large frequency separation}, \(\Delta\nu \simeq \nu_0\), can provide information about the density of a star, leading to independent constraint on its mass and radius (e.g. from scaling relations to the Sun). \begin{figure}[tb] \centering @@ -194,7 +194,7 @@ \section{Modelling Stars the Bayesian Way}\label{sec:modelling-stars} We can also predict oscillation mode frequencies (\(\nu_{nl}\)) with simulations. For example, the GYRE code developed by \citet{Townsend.Teitler2013} uses the output of MESA to compute oscillation modes for a given \(n\) and \(l\). While the physics of p mode propagation in the star is relatively well-known, our understanding of the atmospheric boundary conditions are not. As such, there is a known discrepancy between the simulated and observed p modes. The nature of this can lead to a systematic bias on modelled \(\dnu\) and \(\nu_{nl}\). Corrections for this effect exist \citep[e.g.][]{Ball.Gizon2014} but are still not fully understood. -The complexity of stellar models means that the marginalised posterior distributions are not analytically derivable. Therefore, we use numerical methods like Markov Chain Monte Carlo (MCMC) to estimate the posterior. Typically, this involves exploring parameter space with multiple calls to \(f\) for different values of \(\vect{\theta}\). In the case of MCMC-based algorithms like Hamiltonian Monte Carlo (HMC) and the No U-Turn Sampler (NUTS), the gradient of \(f\) is also required. There are several open-source software packages widely used to implement these algorithms including \textsc{PyMC} \citep{Salvatier.Wiecki.ea2016} and \textsc{NumPyro} \citep{Phan.Pradhan.ea2019}. +The complexity of stellar models means that the marginalised posterior distributions are not analytically derivable. Therefore, we use numerical methods like Markov Chain Monte Carlo (MCMC) to estimate the posterior. Typically, this involves exploring parameter space with multiple calls to \(f\) for different values of \(\vect{\theta}\). In the case of MCMC-based algorithms like Hamiltonian Monte Carlo (HMC) and the No U-Turn Sampler (NUTS), the gradient of \(f\) is also required. There are several open-source \textsc{Python} packages widely used to implement these algorithms including \texttt{pymc} \citep{Salvatier.Wiecki.ea2016} and \texttt{numpyro} \citep{Phan.Pradhan.ea2019}. There are some existing methods for determining stellar parameters using this Bayesian approach. \citet{Bazot.Bourguignon.ea2008} used the MCMC algorithm to sample model parameters with on-the-fly stellar model calculation. While this method can be tailored to individual stars, it is very computationally expensive. Each proposed set of \(\vect{\theta}\) spawns a stellar simulation which evolves to a given age. Steps prior to this age may be discarded and the simulation can take minutes to hours for each set of \(\vect{\theta}\). This is not a viable solution for modelling large numbers of stars. diff --git a/chapters/lyttle21.tex b/chapters/lyttle21.tex index fe7db74..5d9dea6 100644 --- a/chapters/lyttle21.tex +++ b/chapters/lyttle21.tex @@ -202,7 +202,7 @@ \subsubsection{No-Pooled (NP) Model}\label{sec:np} \begin{figure}[tb] \centering \includegraphics{figures/partial_pool_pgm.png} - \caption[A probabilistic graphical model of the partially-pooled hierarchical model.]{A probabilistic graphical model (PGM) of the partially-pooled (PP) hierarchical model. Nodes outside of the grey rectangle represent the hyperparameters in the model. Nodes inside the grey rectangle represent individual stellar parameters. Dark grey nodes represent observables which each have their respective observational uncertainties given by the solid black nodes. The direction of the arrows represent the dependencies in the generative model. \emph{This diagram was made using \textsc{Daft} \citep{Foreman-Mackey.Hogg.ea2021}.}} + \caption[A probabilistic graphical model of the partially-pooled hierarchical model.]{A probabilistic graphical model (PGM) of the partially-pooled (PP) hierarchical model. Nodes outside of the grey rectangle represent the hyperparameters in the model. Nodes inside the grey rectangle represent individual stellar parameters. Dark grey nodes represent observables which each have their respective observational uncertainties given by the solid black nodes. The direction of the arrows represent the dependencies in the generative model. \emph{This diagram was made using the \textsc{Python} package \texttt{daft} \citep{Foreman-Mackey.Hogg.ea2021}.}} \label{fig:pgm} \end{figure} diff --git a/references.bib b/references.bib index 003a537..a851595 100644 --- a/references.bib +++ b/references.bib @@ -281,6 +281,23 @@ @article{AstropyCollaboration.Price-Whelan.ea2018 keywords = {astronomy,methods: data analysis,methods: miscellaneous,methods: statistical,python,reference systems} } +@article{AstropyCollaboration.Price-Whelan.ea2022, + title = {The {{Astropy Project}}: {{Sustaining}} and {{Growing}} a {{Community-oriented Open-source Project}} and the {{Latest Major Release}} (v5.0) of the {{Core Package}}}, + shorttitle = {The {{Astropy Project}}}, + author = {{Astropy Collaboration} and {Price-Whelan}, Adrian M. and Lim, Pey Lian and Earl, Nicholas and Starkman, Nathaniel and Bradley, Larry and Shupe, David L. and Patil, Aarya A. and Corrales, Lia and Brasseur, C. E. and N{\"o}the, Maximilian and Donath, Axel and Tollerud, Erik and Morris, Brett M. and Ginsburg, Adam and Vaher, Eero and Weaver, Benjamin A. and Tocknell, James and Jamieson, William and {van Kerkwijk}, Marten H. and Robitaille, Thomas P. and Merry, Bruce and Bachetti, Matteo and G{\"u}nther, H. Moritz and Aldcroft, Thomas L. and {Alvarado-Montes}, Jaime A. and Archibald, Anne M. and B{\'o}di, Attila and Bapat, Shreyas and Barentsen, Geert and Baz{\'a}n, Juanjo and Biswas, Manish and Boquien, M{\'e}d{\'e}ric and Burke, D. J. and Cara, Daria and Cara, Mihai and Conroy, Kyle E. and Conseil, Simon and Craig, Matthew W. and Cross, Robert M. and Cruz, Kelle L. and D'Eugenio, Francesco and Dencheva, Nadia and Devillepoix, Hadrien A. R. and Dietrich, J{\"o}rg P. and Eigenbrot, Arthur Davis and Erben, Thomas and Ferreira, Leonardo and {Foreman-Mackey}, Daniel and Fox, Ryan and Freij, Nabil and Garg, Suyog and Geda, Robel and Glattly, Lauren and Gondhalekar, Yash and Gordon, Karl D. and Grant, David and Greenfield, Perry and Groener, Austen M. and Guest, Steve and Gurovich, Sebastian and Handberg, Rasmus and Hart, Akeem and {Hatfield-Dodds}, Zac and Homeier, Derek and Hosseinzadeh, Griffin and Jenness, Tim and Jones, Craig K. and Joseph, Prajwel and Kalmbach, J. Bryce and Karamehmetoglu, Emir and Ka{\l}uszy{\'n}ski, Miko{\l}aj and Kelley, Michael S. P. and Kern, Nicholas and Kerzendorf, Wolfgang E. and Koch, Eric W. and Kulumani, Shankar and Lee, Antony and Ly, Chun and Ma, Zhiyuan and MacBride, Conor and Maljaars, Jakob M. and Muna, Demitri and Murphy, N. A. and Norman, Henrik and O'Steen, Richard and Oman, Kyle A. and Pacifici, Camilla and Pascual, Sergio and {Pascual-Granado}, J. and Patil, Rohit R. and Perren, Gabriel I. and Pickering, Timothy E. and Rastogi, Tanuj and Roulston, Benjamin R. and Ryan, Daniel F. and Rykoff, Eli S. and Sabater, Jose and Sakurikar, Parikshit and Salgado, Jes{\'u}s and Sanghi, Aniket and Saunders, Nicholas and Savchenko, Volodymyr and Schwardt, Ludwig and {Seifert-Eckert}, Michael and Shih, Albert Y. and Jain, Anany Shrey and Shukla, Gyanendra and Sick, Jonathan and Simpson, Chris and Singanamalla, Sudheesh and Singer, Leo P. and Singhal, Jaladh and Sinha, Manodeep and Sip{\H o}cz, Brigitta M. and Spitler, Lee R. and Stansby, David and Streicher, Ole and {\v S}umak, Jani and Swinbank, John D. and Taranu, Dan S. and Tewary, Nikita and Tremblay, Grant R. and {de Val-Borro}, Miguel and Van Kooten, Samuel J. and Vasovi{\'c}, Zlatan and Verma, Shresth and {de Miranda Cardoso}, Jos{\'e} Vin{\'i}cius and Williams, Peter K. G. and Wilson, Tom J. and Winkel, Benjamin and {Wood-Vasey}, W. M. and Xue, Rui and Yoachim, Peter and Zhang, Chen and Zonca, Andrea and {Astropy Project Contributors}}, + year = {2022}, + month = aug, + journal = {\apj}, + volume = {935}, + pages = {167}, + issn = {0004-637X}, + doi = {10.3847/1538-4357/ac7c74}, + urldate = {2023-04-27}, + abstract = {The Astropy Project supports and fosters the development of open-source and openly developed Python packages that provide commonly needed functionality to the astronomical community. A key element of the Astropy Project is the core package astropy, which serves as the foundation for more specialized projects and packages. In this article, we summarize key features in the core package as of the recent major release, version 5.0, and provide major updates on the Project. We then discuss supporting a broader ecosystem of interoperable packages, including connections with several astronomical observatories and missions. We also revisit the future outlook of the Astropy Project and the current status of Learn Astropy. We conclude by raising and discussing the current and future challenges facing the Project.}, + keywords = {1855,1858,1866,Astronomy data analysis,Astronomy software,Astrophysics - Instrumentation and Methods for Astrophysics,Open source software}, + annotation = {ADS Bibcode: 2022ApJ...935..167A} +} + @article{AstropyCollaboration.Robitaille.ea2013, title = {Astropy: {{A}} Community {{Python}} Package for Astronomy}, shorttitle = {Astropy}, @@ -1085,6 +1102,18 @@ @article{Casagrande.Ramirez.ea2010 keywords = {infrared: stars,stars: abundances,stars: atmospheres,stars: fundamental parameters,techniques: photometric} } +@misc{Caswell.Lee.ea2022, + title = {Matplotlib/Matplotlib: {{REL}}: V3.6.2}, + shorttitle = {Matplotlib/Matplotlib}, + author = {Caswell, Thomas A. and Lee, Antony and Droettboom, Michael and de Andrade, Elliott Sales and Hoffmann, Tim and Klymak, Jody and Hunter, John and Firing, Eric and Stansby, David and Varoquaux, Nelle and Nielsen, Jens Hedegaard and Root, Benjamin and May, Ryan and Elson, Phil and Sepp{\"a}nen, Jouni K. and Dale, Darren and Lee, Jae-Joon and Gustafsson, Oscar and McDougall, Damon and {hannah} and Straw, Andrew and Hobson, Paul and Lucas, Greg and Gohlke, Christoph and Vincent, Adrien F. and Yu, Tony S. and Ma, Eric and Silvester, Steven and Moad, Charlie and Kniazev, Nikita}, + year = {2022}, + month = nov, + doi = {10.5281/zenodo.7275322}, + urldate = {2023-04-27}, + abstract = {This is the second bugfix release of the 3.6.x series. This release contains several bug-fixes and adjustments: Avoid mutating dictionaries passed to subplots Fix bbox\_inches='tight' on a figure with constrained layout enabled Fix auto-scaling of ax.hist density with histtype='step' Fix compatibility with PySide6 6.4 Fix evaluating colormaps on non-NumPy arrays Fix key reporting in pick events Fix thread check on PyPy 3.8 Handle input to ax.bar that is all NaN Make rubber band more visible on Tk and Wx backends Restore (and warn on) seaborn styles in style.library Restore get\_renderer function in deprecated tight\_layout nb/webagg: Fix resize handle on WebKit browsers (e.g., Safari)}, + howpublished = {Zenodo} +} + @article{Chabrier2003, title = {Galactic {{Stellar}} and {{Substellar Initial Mass Function}}}, author = {Chabrier, Gilles}, @@ -3146,6 +3175,22 @@ @article{Huber.Zinn.ea2017 keywords = {parallaxes,stars: distances,stars: fundamental parameters,stars: late-type,stars: oscillations,techniques: photometric} } +@article{Hunter2007, + title = {Matplotlib: {{A 2D Graphics Environment}}}, + shorttitle = {Matplotlib}, + author = {Hunter, John D.}, + year = {2007}, + month = may, + journal = {Comput. Sci. Eng.}, + volume = {9}, + number = {3}, + pages = {90--95}, + issn = {1558-366X}, + doi = {10.1109/MCSE.2007.55}, + abstract = {Matplotlib is a 2D graphics package used for Python for application development, interactive scripting,and publication-quality image generation across user interfaces and operating systems}, + keywords = {application development,Computer languages,Equations,Graphical user interfaces,Graphics,Image generation,Interpolation,Operating systems,Packaging,Programming profession,Python,scientific programming,scripting languages,User interfaces} +} + @article{Iben.Ehrman1962, title = {The {{Internal Structure}} of {{Middle Main-Sequence Stars}}.}, author = {Iben, Jr., Icko and Ehrman, John R.}, @@ -4072,6 +4117,20 @@ @article{Metcalfe.Creevey.ea2015 annotation = {ADS Bibcode: 2015ApJ...811L..37M} } +@article{Meurer.Smith.ea2017, + title = {{{SymPy}}: Symbolic Computing in {{Python}}}, + author = {Meurer, Aaron and Smith, Christopher P. and Paprocki, Mateusz and {\v C}ert{\'i}k, Ond{\v r}ej and Kirpichev, Sergey B. and Rocklin, Matthew and Kumar, {\relax Am}iT and Ivanov, Sergiu and Moore, Jason K. and Singh, Sartaj and Rathnayake, Thilina and Vig, Sean and Granger, Brian E. and Muller, Richard P. and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J. and Terrel, Andy R. and Rou{\v c}ka, {\v S}t{\v e}p{\'a}n and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith and Cimrman, Robert and Scopatz, Anthony}, + year = {2017}, + month = jan, + journal = {PeerJ Comput. Sci.}, + volume = {3}, + pages = {e103}, + issn = {2376-5992}, + doi = {10.7717/peerj-cs.103}, + abstract = {SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.}, + keywords = {Computer algebra system,Python,Symbolics} +} + @article{Miglio.Chiappini.ea2017, title = {{{PLATO}} as It Is : {{A}} Legacy Mission for {{Galactic}} Archaeology}, shorttitle = {{{PLATO}} as It Is}, @@ -4585,6 +4644,22 @@ @article{Nsamba.Monteiro.ea2018 annotation = {ADS Bibcode: 2018MNRAS.479L..55N} } +@article{Ochsenbein.Bauer.ea2000, + title = {The {{VizieR}} Database of Astronomical Catalogues}, + author = {Ochsenbein, F. and Bauer, P. and Marcout, J.}, + year = {2000}, + month = apr, + journal = {\aaps}, + volume = {143}, + pages = {23--32}, + issn = {0365-0138}, + doi = {10.1051/aas:2000169}, + urldate = {2023-04-27}, + abstract = {VizieR is a database grouping in an homogeneous way thousands of astronomical catalogues gathered for decades by the Centre de Donn\'ees de Strasbourg (CDS) and participating institutes. The history and current status of this large collection is briefly presented, and the way these catalogues are being standardized to fit in the VizieR system is described. The architecture of the database is then presented, with emphasis on the management of links and of accesses to very large catalogues. Several query interfaces are currently available, making use of the ASU protocol, for browsing purposes or for use by other data processing systems such as visualisation tools.}, + keywords = {ASTRONOMICAL DATA BASES: MISCELLANEOUS,Astrophysics,CATALOGS}, + annotation = {ADS Bibcode: 2000A\&AS..143...23O} +} + @article{Onehag.Gustafsson.ea2014, title = {Abundances and Possible Diffusion of Elements in {{M}} 67 Stars}, author = {{\"O}nehag, Anna and Gustafsson, Bengt and Korn, Andreas J.}, @@ -6198,6 +6273,17 @@ @article{Villante.Serenelli.ea2014 keywords = {neutrinos,Sun: abundances,Sun: helioseismology,Sun: interior} } +@article{Virtanen.Gommers.ea2020, + title = {{{SciPy}} 1.0: {{Fundamental}} Algorithms for Scientific Computing in Python}, + author = {Virtanen, Pauli and Gommers, Ralf and Oliphant, Travis E. and Haberland, Matt and Reddy, Tyler and Cournapeau, David and Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and Bright, Jonathan and {van der Walt}, St{\'e}fan J. and Brett, Matthew and Wilson, Joshua and Millman, K. Jarrod and Mayorov, Nikolay and Nelson, Andrew R. J. and Jones, Eric and Kern, Robert and Larson, Eric and Carey, C J and Polat, {\.I}lhan and Feng, Yu and Moore, Eric W. and VanderPlas, Jake and Laxalde, Denis and Perktold, Josef and Cimrman, Robert and Henriksen, Ian and Quintero, E. A. and Harris, Charles R. and Archibald, Anne M. and Ribeiro, Ant{\^o}nio H. and Pedregosa, Fabian and {van Mulbregt}, Paul and {SciPy 1.0 Contributors}}, + year = {2020}, + journal = {Nat. Methods}, + volume = {17}, + pages = {261--272}, + doi = {10.1038/s41592-019-0686-2}, + adsurl = {https://rdcu.be/b08Wh} +} + @article{vonHippel.Jefferys.ea2006, title = {Inverting {{Color}}-{{Magnitude Diagrams}} to {{Access Precise Star Cluster Parameters}}: {{A Bayesian Approach}}}, shorttitle = {Inverting {{Color}}-{{Magnitude Diagrams}} to {{Access Precise Star Cluster Parameters}}}, @@ -6247,6 +6333,18 @@ @article{Wagoner.Fowler.ea1967 annotation = {ADS Bibcode: 1967ApJ...148....3W} } +@article{Waskom2021, + title = {Seaborn: Statistical Data Visualization}, + author = {Waskom, Michael L.}, + year = {2021}, + journal = {J. Open Source Softw.}, + volume = {6}, + number = {60}, + pages = {3021}, + publisher = {{The Open Journal}}, + doi = {10.21105/joss.03021} +} + @article{Weiss.Schlattl2008, title = {{{GARSTEC}}\textemdash the {{Garching Stellar Evolution Code}}. {{The}} Direct Descendant of the Legendary {{Kippenhahn}} Code}, author = {Weiss, Achim and Schlattl, Helmut}, From 137cfcc1a6e215024367138b5795823612300b41 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Thu, 27 Apr 2023 15:30:01 +0100 Subject: [PATCH 30/50] Finish abstract --- thesis.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/thesis.tex b/thesis.tex index cabac94..f374ba8 100644 --- a/thesis.tex +++ b/thesis.tex @@ -90,9 +90,9 @@ %% ABSTRACT ------------------------------------------------------------------- % \abstract{% - As high-precision asteroseismology modelling of stars continues to advance, it is becoming increasingly important to account for the systematic effects that arise from our assumptions of the stellar helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)). To address this issue, we present a novel approach for improving the inference of the fundamental parameters of a sample of Kepler dwarfs and subgiants with a narrow mass range of \(0.8 < M < 1.2 \mathrm{M}_{\odot}\). Our method includes a statistical treatment of \(Y\) and \(\mlt\) and employs a hierarchical Bayesian model that incorporates information about the distribution of these parameters in the population. Specifically, we fit a linear helium enrichment law that includes an intrinsic scatter and a normal distribution in \(\mlt\). We explore various levels of pooling parameters, with and without the Sun as a calibrator, and report \(\Delta Y/\Delta Z = 1.05^{+0.28}_{-0.25}\) and \(\mu_\alpha = 1.90^{+0.10}_{-0.09}\) when the Sun is included. Despite accounting for the uncertainties in \(Y\) and \(\mlt\), we are able to report statistical uncertainties of 2.5 per cent in mass, 1.2 per cent in radius, and 12 per cent in age. Moreover, our approach can be extended to larger samples, which would enable us to obtain more precise constraints on the fundamental parameters of individual stars and improve our understanding of the population-level inference. + Modelling stars to understand stellar physics and characterise their ages and masses is cucial for studying exoplanetary systems and the evolution of the Milky Way. As stellar modelling advances with the advent of high-precision asteroseismology, it is becoming increasingly important to account for the systematic effects that arise from our physical assumptions. In this thesis, I present a novel approach for improving the inference of fundamental stellar parameters using a hierarchical Bayesian model. I introduce a statistical treatment which `pools' helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)) to incorporate information about their distributions in the population. Specifically, I model \(Y\) as a distribution with a mean from a linear helium enrichment law parametrised by \(\Delta Y/\Delta Z\). I test our method on a sample of dwarfs and subgiants observed by \emph{Kepler} with a narrow mass range of \(0.8 < M/\mathrm{M}_{\odot} < 1.2\). Exploring various levels of pooling parameters, with and without the Sun as a calibrator, I report \(\Delta Y/\Delta Z = 1.05^{+0.28}_{-0.25}\) when the Sun is included in the sample. Despite marginalising over uncertainties in \(Y\) and \(\mlt\), I am able to report statistical uncertainties of 2.5 per cent in mass, 1.2 per cent in radius, and 12 per cent in age. Moreover, my approach can be extended to larger samples, enabling further uncertainty reduction in fundamental parameters and an improved characterisation of population-level distributions. - We present a new method for modelling acoustic glitch signatures in the radial mode frequencies of solar-like oscillators using a Gaussian Process. We compare our approach with another method using a model star with varying levels of noise and apply both methods to the star 16 Cyg A to provide a real-world example. Our results show that our method accurately determines the strength and location of glitches caused by the second ionisation of helium and the base of the convective zone. Moreover, we demonstrate that our Gaussian Process approach improves the treatment of the smoothly varying component of the mode frequencies, outperforming a 4th order polynomial in this regard. We also find that our method finds less multimodality in the posterior for acoustic depth by using a prior to inform the glitch parameters. However, the inclusion of the He\,\textsc{i} ionisation glitch in the model remains a question. Overall, our study suggests that the Gaussian Process method shows promise as a tool for modelling acoustic glitches in solar-like oscillators, with potential applications in asteroseismology and studies of stellar structure and evolution.} + There is additional information on \(Y\) to be gained from detailed asteroseismology. Acoustic glitches, which arise from rapid changes in stellar structure (e.g. from helium ionisation), leave a periodic signature in the mode frequencies (\(\nu_{nl}\)) in solar-like oscillators. I present a new method for modelling acoustic glitch signatures in the radial mode frequencies using a Gaussian Process (GP). The GP provides a statistical treatment of uncertainty in the smooth component of our model for \(\nu_{nl}\) as a function of radial order, \(n\). Using a model star and 16 Cyg A, I compare this approach to another method which models the smooth component with a 4th-order polynomial. My results show that the GP method accurately determines the strength and location of acoustic glitches caused by He\,\textsc{ii} ionisation and the base of the convective zone. I find that using a prior to inform the glitch parameters in my method reduces the occurrence of extreme, unrealistic solutions in the posterior. Furthermore, I demonstrate that the GP approach outperforms the polynomial by absorbing the lesser signature of He\,\textsc{i} ionisation in the modes. However, the inclusion of the He\,\textsc{i} ionisation glitch in the model remains a question. Overall, my results suggest that the GP method should be further tested on more solar-like oscillators and then integrated into the hierarchical model presented in this work.} %% ACKNOWLEDGEMENTS ----------------------------------------------------------- From b3d79366f96afdbedd7247117ce10faf43f3989e Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Thu, 27 Apr 2023 15:30:16 +0100 Subject: [PATCH 31/50] Capitalise GP --- chapters/conclusion.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/chapters/conclusion.tex b/chapters/conclusion.tex index 31c3e35..f6c3d67 100644 --- a/chapters/conclusion.tex +++ b/chapters/conclusion.tex @@ -24,7 +24,7 @@ \section*{Summary} The HBM required a function to map model parameters to observables. We built an emulator to approximate 1D numerical models of stellar evolution. Training a neural network on MESA stellar simulations, we were able to predict observable parameters (\(\teff, \dnu, L, [\mathrm{M/H}]\)) with typical precisions of less than \(\sim 0.1\) per cent (see Appendix \ref{apx:hmd}). This provided a simple, continuous and differentiable model well suited to modern, gradient-based MCMC algorithms. Another advantage to using a neural network emulator was its scalability. The simple matrix algebra involved is well suited to fast evaluations on a graphics processing unit (GPU) for large numbers of stars at the same time. Furthermore, the neural network could be scaled up to higher input and output dimensions with little performance impact, making the method transferable to other kinds of stars. For example, we recently trained a neural network to emulate the regularly spaced mode frequencies as a part of a Bayesian stellar model of \(\delta\) Scuti-type oscillators \citep{Scutt.Murphy.ea2023}. -In Chapter \ref{chap:glitch}, we recalled that glitches in stellar structure cause a periodic signal, \(\delta\nu\) in p mode frequencies. One such glitch arises from the second ionisation of helium, with the amplitude of \(\delta\nu_\helium\) correlating with helium abundance. Subsequently, we presented a new method for modelling the glitch signature using a Gaussian process (GP) in Chapter \ref{chap:glitch-gp}. Past methods for measuring the glitch in the mode frequencies (\(\nu_{nl}\)) used a polynomial in \(n\) to approximate the smooth functional form of the frequencies, over which the periodic glitch signature could be modelled \citep[e.g.][]{Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2019}. We replaced the polynomial with a GP characterised by a kernel describing our prior belief of the function's smoothness and flexibility. We applied this method to model the glitch signature in radial mode frequencies for a fake, Sun-like star and 16 Cyg A. The GP allowed us to marginalise over our uncertainty in the functional form of \(\nu_{n\,0}\) with \(n\). In comparison, we found that the polynomial method was too restrictive and unable to marginalise over uncertainty in the model. +In Chapter \ref{chap:glitch}, we recalled that glitches in stellar structure cause a periodic signal, \(\delta\nu\) in p mode frequencies. One such glitch arises from the second ionisation of helium, with the amplitude of \(\delta\nu_\helium\) correlating with helium abundance. Subsequently, we presented a new method for modelling the glitch signature using a Gaussian Process (GP) in Chapter \ref{chap:glitch-gp}. Past methods for measuring the glitch in the mode frequencies (\(\nu_{nl}\)) used a polynomial in \(n\) to approximate the smooth functional form of the frequencies, over which the periodic glitch signature could be modelled \citep[e.g.][]{Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2019}. We replaced the polynomial with a GP characterised by a kernel describing our prior belief of the function's smoothness and flexibility. We applied this method to model the glitch signature in radial mode frequencies for a fake, Sun-like star and 16 Cyg A. The GP allowed us to marginalise over our uncertainty in the functional form of \(\nu_{n\,0}\) with \(n\). In comparison, we found that the polynomial method was too restrictive and unable to marginalise over uncertainty in the model. % In Chapter \ref{chap:glitch}, we recalled that p mode frequencies carry information about acoustic glitches inside a star. However, the exact functional form of the modes with radial order is not known. We showed that a Gaussian Process (GP) could be employed to marginalise over the uncertainty in this functional form and improve detection of the helium glitch signature. Our method showed promise compared to those which have come before \citep[e.g.][]{Verma.Raodeo.ea2019}. We found the GP method was better able to find the true acoustic depth of He\,\textsc{ii} ionisation in our model star than the alternative, motivating a more quantitative comparison in the future. We hope to build a more informed prior on the model parameters and publish this method soon with more examples. From 0d2d61723b3a7b789b82f21397cf54b64a7a74d6 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Thu, 27 Apr 2023 16:04:10 +0100 Subject: [PATCH 32/50] Quick-look spell check --- chapters/conclusion.tex | 2 +- chapters/glitch-gp.tex | 4 ++-- chapters/glitch.tex | 6 +++--- chapters/introduction.tex | 2 +- chapters/lyttle21.tex | 2 +- thesis.tex | 2 +- 6 files changed, 9 insertions(+), 9 deletions(-) diff --git a/chapters/conclusion.tex b/chapters/conclusion.tex index f6c3d67..4554ae6 100644 --- a/chapters/conclusion.tex +++ b/chapters/conclusion.tex @@ -58,7 +58,7 @@ \section*{Current and Future Data} We tested the HBM on stars observed by \emph{Kepler}, but there are a few current and upcoming missions which we can utilise to increase our sample size. With larger sample sizes, we can further increase the precision of pooled parameters and better characterise their spread in the population distribution. Recently, \citet{Hatt.Nielsen.ea2023} identified a sample of \(\sim 4000\) solar-like oscillators in 120- and 20-second cadence \emph{TESS} data. Of these, around 50 are dwarf and subgiant stars which we could include in a future iteration of the HBM. However, we anticipate much bigger improvement with future missions expected to launch in a few years time. -% This has been exceptionally useful with galactic archaology with the \(\sim 150,000\) oscillating red giants detected by \citet{Hon.Huber.ea2021}. +% This has been exceptionally useful with galactic archaeology with the \(\sim 150,000\) oscillating red giants detected by \citet{Hon.Huber.ea2021}. Towards the end of the 2020s, the \emph{PLATO} mission will observe tens of thousands of dwarf and subgiant solar-like oscillators \citep{Rauer.Catala.ea2014}. \emph{PLATO} aims to discover hundreds of exoplanets orbiting solar-type stars across a wider proportion of the sky than observed by \emph{Kepler}. Among its targets are around \num{20000} bright (V < 11) oscillating F-K dwarf stars to be observed over a baseline of around 2 years \citep{Goupil2017}. Using our HBM method on a sample this size could see a reduction in uncertainty (\(\sigma\)) on helium abundance from 0.01 to 0.0005. While this is the maximum expected uncertainty reduction (as discussed in Chapter \ref{chap:hbm}), it shows that we can start to consider more complex population distributions in helium and other stellar parameters. diff --git a/chapters/glitch-gp.tex b/chapters/glitch-gp.tex index 15fbaf9..4d12e6b 100644 --- a/chapters/glitch-gp.tex +++ b/chapters/glitch-gp.tex @@ -149,7 +149,7 @@ \subsection{The GP Method} \begin{table} \centering % \caption{The mean and variance for the prior normal distributions on each parameter where values are not explicitly given in the text.} - \caption[The GP model parameters and their prior distributions.]{The GP model parameters and their prior distributions. Normal distributions are parametrised by the mean and variance (i.e. \(\mathcal{N}(\mu,\sigma^2)\)). Uniform distributions are parametrised by their lower and uppower bounds.} + \caption[The GP model parameters and their prior distributions.]{The GP model parameters and their prior distributions. Normal distributions are parametrised by the mean and variance (i.e. \(\mathcal{N}(\mu,\sigma^2)\)). Uniform distributions are parametrised by their lower and upper bounds.} \label{tab:glitch-prior} \input{tables/glitch-test-prior-2.tex} \end{table} @@ -204,7 +204,7 @@ \subsection{The GP Method} % \begin{table} % \centering % % \caption{The mean and variance for the prior normal distributions on each parameter where values are not explicitly given in the text.} -% \caption[The GP model parameters and their prior distributions.]{The GP model parameters and their prior distributions. Normal distributions are parametrised by the mean and variance (i.e. \(\mathcal{N}(\mu,\sigma^2)\)). Uniform distributions are parametrised by their lower and uppower bounds.} +% \caption[The GP model parameters and their prior distributions.]{The GP model parameters and their prior distributions. Normal distributions are parametrised by the mean and variance (i.e. \(\mathcal{N}(\mu,\sigma^2)\)). Uniform distributions are parametrised by their lower and upper bounds.} % \label{tab:glitch-prior} % \input{tables/glitch-test-prior-2.tex} % \end{table} diff --git a/chapters/glitch.tex b/chapters/glitch.tex index f4733d8..3ddad30 100644 --- a/chapters/glitch.tex +++ b/chapters/glitch.tex @@ -242,11 +242,11 @@ \subsection{Helium Ionisation Glitch}\label{sec:helium-glitch} \begin{figure}[tb] \centering \includegraphics{figures/adiabatic-ionisation-regions.pdf} - \caption[The depressions in the first adiabatic exponent caused by the ionisation of hydrogen and helium.]{The depressions in the first adiabatic exponent (\(\gamma\)) caused by the ionisation of hydrogen (H), and the first and second ionisations of helium (He\,\textsc{i} and He\,\textsc{ii}). The horizonal axis is the fractional acoustic depth from the surface of the star, \(\tau/\tau_0\).} + \caption[The depressions in the first adiabatic exponent caused by the ionisation of hydrogen and helium.]{The depressions in the first adiabatic exponent (\(\gamma\)) caused by the ionisation of hydrogen (H), and the first and second ionisations of helium (He\,\textsc{i} and He\,\textsc{ii}). The horizontal axis is the fractional acoustic depth from the surface of the star, \(\tau/\tau_0\).} \label{fig:gamma-zones} \end{figure} -In Figure \ref{fig:gamma-zones}, we show \(\gamma\) for model S against fractional acoustic depth, shaded by regions of ionisation. For an ideal monatmoic gas, \(\gamma=5/3\), but we see that \(\gamma < 5/3\) in regions where helium and hydrogen ionise. Close to the surface of the star, hydrogen ionisation has the largest effect on \(\gamma\) because it makes up the majority of stellar matter. The first (He\,\textsc{i}) and second (He\,\textsc{ii}) ionisations of helium occur deeper in the star. We can see that the second ionisation of helium has a greater affect on \(\gamma\) than the first. The effect of the He\,\textsc{ii} ionisation causes a rapid change in \(\gamma\) over a few per cent in \(\tau\) (equivalent to \(\sim \SI{100}{\second}\) in model S). +In Figure \ref{fig:gamma-zones}, we show \(\gamma\) for model S against fractional acoustic depth, shaded by regions of ionisation. For an ideal monatomic gas, \(\gamma=5/3\), but we see that \(\gamma < 5/3\) in regions where helium and hydrogen ionise. Close to the surface of the star, hydrogen ionisation has the largest effect on \(\gamma\) because it makes up the majority of stellar matter. The first (He\,\textsc{i}) and second (He\,\textsc{ii}) ionisations of helium occur deeper in the star. We can see that the second ionisation of helium has a greater affect on \(\gamma\) than the first. The effect of the He\,\textsc{ii} ionisation causes a rapid change in \(\gamma\) over a few per cent in \(\tau\) (equivalent to \(\sim \SI{100}{\second}\) in model S). \subsubsection{The Effect of Helium Abundance on \(\gamma\)} @@ -428,7 +428,7 @@ \subsubsection{A Functional Form of the Helium Glitch Signature} \left.\frac{\delta\omega}{\omega}\right|_{\heII, \mathrm{osc}} \simeq \frac{\Gamma_\heII}{2\sqrt{2\pi} \, \tau_0} \, \int_{-\infty}^\infty \ee^{- x^2/2} \cos 2 (\Delta_\heII \omega x + \widetilde{\epsilon_\heII}) \, \dd x \label{eq:omega-ii-osc} \end{equation} % -where \(\widetilde{\epsilon_\heII} = \omega\tau_\heII + \epsilon_\heII\) and the phase \(\epsilon=\epsilon_\heII\) is assumed constant accross the glitch region. We can solve the above integral analytically using differentiation under the integral sign by introducing an arbitrary variable \(a\), +where \(\widetilde{\epsilon_\heII} = \omega\tau_\heII + \epsilon_\heII\) and the phase \(\epsilon=\epsilon_\heII\) is assumed constant across the glitch region. We can solve the above integral analytically using differentiation under the integral sign by introducing an arbitrary variable \(a\), % \begin{equation} I(a) = \int_{-\infty}^\infty \ee^{- x^2/2 } \cos 2 (\Delta_\heII \omega x a + \widetilde{\epsilon_\heII}) \, \dd x diff --git a/chapters/introduction.tex b/chapters/introduction.tex index efdc3bd..b6558b4 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -88,7 +88,7 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} % Not Section 2.1 of JCD Lecture Notes (2014) derives this We can assume the tallest peaks are \(l=0,1\), and the smaller peaks are \(l=2,3\), all modulated by the wider Gaussian-like envelope. -For a spherically symmetric, non-rotating star, modes with different \(m\) oscillate at the same frequency and cannot be distinguished. However, the observed mode frequencies will split for different \(m\) via the Doppler effect in the case of a rotating or distorted (asymmetric) star. Measuring this splitting can constrain the rotation rate of the star. This has lead to breakthrough studies into gyrochronology and \todo{finish} \citep[e.g.][]{Hall.Davies.ea2021}. Hereafter, we will consider only the case of a slowly rotating, spherically symmetric star. +Modes with different \(m\) cannot be distinguished for a spherically symmetric, non-rotating star. In the case of a rotating or distorted (asymmetric) star, the observed mode frequencies will split for different \(m\) via the Doppler effect. Measuring this splitting can constrain the rotation rate of the star \citep[e.g.][]{Davies.Chaplin.ea2015,Garcia.Ceillier.ea2014,Deheuvels.Garcia.ea2012}. Such studies have lead to breakthrough research on the rotational evolution of stars \citep[e.g.][]{Angus.Aigrain.ea2015,Hall.Davies.ea2021,vanSaders.Ceillier.ea2016}. However, we will hereafter consider only the case of a slowly rotating, spherically symmetric star. If we consider an acoustic wave in a one-dimensional homogeneous medium, then we would expect each mode of oscillation to be an integer multiple of the fundamental mode. While the case for a star is more complicated, we can also approximate the frequencies for different modes as a multiple of a characteristic frequency. \citet{Tassoul1980} found that the modes could be approximated by assuming the asymptotic limit where \(l/n \rightarrow 0\), giving the following expression \citep[cf.][]{Gough1986}, % diff --git a/chapters/lyttle21.tex b/chapters/lyttle21.tex index 5d9dea6..d39a4ae 100644 --- a/chapters/lyttle21.tex +++ b/chapters/lyttle21.tex @@ -13,7 +13,7 @@ \chapter[Hierarchically Modelling Dwarf and Subgiant Stars]{Hierarchically Modelling \emph{Kepler} Dwarfs and Subgiants to Improve Inference of Stellar Properties with Asteroseismology}\label{chap:hmd} \textit{% - This chapter is taken verbatim from \citet{Lyttle.Davies.ea2021}. I led the project and wrote all of the text except for the description of stellar models and input physics in Section \ref{sec:grid} which was written by Tanda Li\footnote{Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China}. In this work, we present a hierarchical Bayesian model which encodes information about the distribution of helium abundance (\(Y\)) and mixing-length-theory parameter (\(\mlt\)) in a population of dwarf and subgiant solar-like oscillators. + This chapter is taken verbatim from \citet{Lyttle.Davies.ea2021}. I led the project and wrote all of the text except for the description of stellar models and input physics in Section \ref{sec:grid} which was written by Tanda Li\footnote{Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China}. In this work, we present a hierarchical Bayesian model which encodes information about the distribution of helium abundance (\(Y\)) and mixing-length-theory parameter (\(\mlt\)) in a population of dwarf and subgiant solar-like oscillators. } \section{Introduction} diff --git a/thesis.tex b/thesis.tex index f374ba8..00a6be6 100644 --- a/thesis.tex +++ b/thesis.tex @@ -69,7 +69,7 @@ % % Optional license, uncomment this if you want to use custom license text % Default text is 'All Rights Reserved.' -\license{\doclicenseThis} % This example uses the doclicense package +\license{\doclicenseThis} % This example uses the 'doclicense' package %% DEDICATION ----------------------------------------------------------------- From 8cea3b2a2439b57f23e46a9a18dea9c4b42671f1 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Thu, 27 Apr 2023 16:34:56 +0100 Subject: [PATCH 33/50] Detailed spell check --- thesis.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/thesis.tex b/thesis.tex index 00a6be6..baf6e77 100644 --- a/thesis.tex +++ b/thesis.tex @@ -90,7 +90,7 @@ %% ABSTRACT ------------------------------------------------------------------- % \abstract{% - Modelling stars to understand stellar physics and characterise their ages and masses is cucial for studying exoplanetary systems and the evolution of the Milky Way. As stellar modelling advances with the advent of high-precision asteroseismology, it is becoming increasingly important to account for the systematic effects that arise from our physical assumptions. In this thesis, I present a novel approach for improving the inference of fundamental stellar parameters using a hierarchical Bayesian model. I introduce a statistical treatment which `pools' helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)) to incorporate information about their distributions in the population. Specifically, I model \(Y\) as a distribution with a mean from a linear helium enrichment law parametrised by \(\Delta Y/\Delta Z\). I test our method on a sample of dwarfs and subgiants observed by \emph{Kepler} with a narrow mass range of \(0.8 < M/\mathrm{M}_{\odot} < 1.2\). Exploring various levels of pooling parameters, with and without the Sun as a calibrator, I report \(\Delta Y/\Delta Z = 1.05^{+0.28}_{-0.25}\) when the Sun is included in the sample. Despite marginalising over uncertainties in \(Y\) and \(\mlt\), I am able to report statistical uncertainties of 2.5 per cent in mass, 1.2 per cent in radius, and 12 per cent in age. Moreover, my approach can be extended to larger samples, enabling further uncertainty reduction in fundamental parameters and an improved characterisation of population-level distributions. + Modelling stars to understand stellar physics and characterise their ages and masses is crucial for studying exoplanetary systems and the evolution of the Milky Way. As stellar modelling advances with the advent of high-precision asteroseismology, it is becoming increasingly important to account for the systematic effects that arise from our physical assumptions. In this thesis, I present a novel approach for improving the inference of fundamental stellar parameters using a hierarchical Bayesian model. I introduce a statistical treatment which `pools' helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)) to incorporate information about their distributions in the population. Specifically, I model \(Y\) as a distribution with a mean from a linear helium enrichment law parametrised by \(\Delta Y/\Delta Z\). I test our method on a sample of dwarfs and subgiants observed by \emph{Kepler} with a narrow mass range of \(0.8 < M/\mathrm{M}_{\odot} < 1.2\). Exploring various levels of pooling parameters, with and without the Sun as a calibrator, I report \(\Delta Y/\Delta Z = 1.05^{+0.28}_{-0.25}\) when the Sun is included in the sample. Despite marginalising over uncertainties in \(Y\) and \(\mlt\), I am able to report statistical uncertainties of 2.5 per cent in mass, 1.2 per cent in radius, and 12 per cent in age. Moreover, my approach can be extended to larger samples, enabling further uncertainty reduction in fundamental parameters and an improved characterisation of population-level distributions. There is additional information on \(Y\) to be gained from detailed asteroseismology. Acoustic glitches, which arise from rapid changes in stellar structure (e.g. from helium ionisation), leave a periodic signature in the mode frequencies (\(\nu_{nl}\)) in solar-like oscillators. I present a new method for modelling acoustic glitch signatures in the radial mode frequencies using a Gaussian Process (GP). The GP provides a statistical treatment of uncertainty in the smooth component of our model for \(\nu_{nl}\) as a function of radial order, \(n\). Using a model star and 16 Cyg A, I compare this approach to another method which models the smooth component with a 4th-order polynomial. My results show that the GP method accurately determines the strength and location of acoustic glitches caused by He\,\textsc{ii} ionisation and the base of the convective zone. I find that using a prior to inform the glitch parameters in my method reduces the occurrence of extreme, unrealistic solutions in the posterior. Furthermore, I demonstrate that the GP approach outperforms the polynomial by absorbing the lesser signature of He\,\textsc{i} ionisation in the modes. However, the inclusion of the He\,\textsc{i} ionisation glitch in the model remains a question. Overall, my results suggest that the GP method should be further tested on more solar-like oscillators and then integrated into the hierarchical model presented in this work.} @@ -98,7 +98,7 @@ %% ACKNOWLEDGEMENTS ----------------------------------------------------------- % \acknowledgements{% -I am tremendously grateful to my PhD supervisor Guy R. Davies for his support and excellent mentorship throughout my postgraduate research. I also thank my first co-supervisor Andrea Miglio for his guidence before his move to the University of Bologna. I extend my thanks to Amaury Triaud, who took over from Andrea as my co-supervisor and has since been an inspiring mentor and leader of our research group. Additionally, I would like to thank in advance my PhD examiners Daniel Reese and Chris Moore, and viva chairperson Annelies Mortier. +I am tremendously grateful to my PhD supervisor Guy R. Davies for his support and excellent mentorship throughout my postgraduate research. Additionally, I thank my first co-supervisor Andrea Miglio for his guidence before his move to the University of Bologna. I extend my thanks to Amaury Triaud, who took over from Andrea as my co-supervisor and has since been an inspiring mentor and leader of our research group. I would like to thank in advance my PhD examiners Daniel Reese and Chris Moore, and viva chairperson Annelies Mortier. \begin{CJK*}{UTF8}{gbsn} % INTERNAL SUPPORT From a11db4b86e3e7cb9f41d32e9cbd512fc32d96db4 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Thu, 27 Apr 2023 17:10:32 +0100 Subject: [PATCH 34/50] Grammer check --- chapters/lyttle21.tex | 26 +++++++++++++------------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/chapters/lyttle21.tex b/chapters/lyttle21.tex index d39a4ae..0035205 100644 --- a/chapters/lyttle21.tex +++ b/chapters/lyttle21.tex @@ -13,7 +13,7 @@ \chapter[Hierarchically Modelling Dwarf and Subgiant Stars]{Hierarchically Modelling \emph{Kepler} Dwarfs and Subgiants to Improve Inference of Stellar Properties with Asteroseismology}\label{chap:hmd} \textit{% - This chapter is taken verbatim from \citet{Lyttle.Davies.ea2021}. I led the project and wrote all of the text except for the description of stellar models and input physics in Section \ref{sec:grid} which was written by Tanda Li\footnote{Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China}. In this work, we present a hierarchical Bayesian model which encodes information about the distribution of helium abundance (\(Y\)) and mixing-length-theory parameter (\(\mlt\)) in a population of dwarf and subgiant solar-like oscillators. + This chapter is taken verbatim from \citet{Lyttle.Davies.ea2021}. I confirm that all writing is my own except for the description of stellar models and input physics in Section \ref{sec:grid} which was written by Tanda Li\footnote{Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China}. In this work, we present a hierarchical Bayesian model which encodes information about the distribution of helium abundance (\(Y\)) and mixing-length-theory parameter (\(\mlt\)) in a population of dwarf and subgiant solar-like oscillators. } \section{Introduction} @@ -30,9 +30,9 @@ \section{Introduction} The use of HBMs has been demonstrated in other areas of astrophysics to reduce individual parameter uncertainties by encoding prior information about the distribution of the parameter in a population. For example, HBMs have been used with data from \emph{Gaia}, improving distance measures \citep{Leistedt.Hogg2017, Anderson.Hogg.ea2018} and calibrating the red clump as a standard candle \citep{Hawkins.Leistedt.ea2017, Chan.Bovy2020} using asteroseismology \citep{Hall.Davies.ea2019}. In other instances, HBMs have been used to infer binary-star and exoplanet eccentricities \citep{Hogg.Myers.ea2010}, obliquity of transit systems \citep{Morton.Winn2014}, stellar inclination with asteroseismology \citep{Campante.Lund.ea2016, Kuszlewicz.Chaplin.ea2019}, and the age-metallicity relation of the solar neighbourhood \citep{Feuillet.Bovy.ea2016}. -To describe the distribution of $Y$ in this work, we assume a linear helium enrichment law characterised by freely varied population parameters: the gradient given by $\Delta Y / \Delta Z$, the primordial helium abundance at $Z=0$ ($Y_P$) and an intrinsic spread in helium ($\sigma_Y$). There have been many studies to determine a linear enrichment law, from modelling eclipsing binaries \citep{Ribas.Jordi.ea2000} to spectroscopy of galactic H\textsc{ii} regions \citep{Balser2006}. Values of $\Delta Y / \Delta Z$ have also been determined for samples of main sequence (MS) stars \citep{Casagrande.Flynn.ea2007}, open clusters \citep{Brogaard.VandenBerg.ea2012}, and more recently with asteroseismology using individual oscillation frequencies \citep{SilvaAguirre.Lund.ea2017} and the glitch due to the second helium ionisation zone \citep{Verma.Raodeo.ea2019}. In these studies the value of the enrichment ratio was typically inferred to be $1 \lesssim \Delta Y / \Delta Z \lesssim 3$. However, helium enrichment is unlikely to be exactly linear with metallicity and may depend on other chemical abundances \citep{West.Heger2013} or the location of the star in the Milky Way \citep{Frebel2010}. Therefore, we account for some deviation from the linear law by introducing $\sigma_Y$. Moreover, our method may be adapted to different helium enrichment priors in future work. +To describe the distribution of $Y$ in this work, we assume a linear helium enrichment law characterised by freely varied population parameters: the gradient given by $\Delta Y / \Delta Z$, the primordial helium abundance at $Z=0$ ($Y_P$) and an intrinsic spread in helium ($\sigma_Y$). There have been many studies to determine a linear enrichment law, from modelling eclipsing binaries \citep{Ribas.Jordi.ea2000} to spectroscopy of galactic H\,\textsc{ii} regions \citep{Balser2006}. Values of $\Delta Y / \Delta Z$ have also been determined for samples of main sequence (MS) stars \citep{Casagrande.Flynn.ea2007}, open clusters \citep{Brogaard.VandenBerg.ea2012}, and more recently with asteroseismology using individual oscillation frequencies \citep{SilvaAguirre.Lund.ea2017} and the glitch due to the second helium ionisation zone \citep{Verma.Raodeo.ea2019}. In these studies the value of the enrichment ratio was typically inferred to be $1 \lesssim \Delta Y / \Delta Z \lesssim 3$. However, helium enrichment is unlikely to be exactly linear with metallicity and may depend on other chemical abundances \citep{West.Heger2013} or the location of the star in the Milky Way \citep{Frebel2010}. Therefore, we account for some deviation from the linear law by introducing $\sigma_Y$. Moreover, our method may be adapted to different helium enrichment priors in future work. -The widely used mixing-length theory (MLT) of convection, parametrised by $\mlt$, has been tested throughout the Hertzsprung-Russell (HR) diagram with 3D hydrodynamical simulations \citep{Trampedach.Stein.ea2014, Magic.Weiss.ea2015} and asteroseismology \citep{Tayar.Somers.ea2017, Viani.Basu.ea2018, Li.Bedding.ea2018} with values in the range $0.8 \lesssim \mlt/\alpha_{\mathrm{MLT}, \odot} \lesssim 1.2$ where $\alpha_{\mathrm{MLT}, \odot}$ is the value calibrated to the Sun. However, in many grids of stellar models, a constant value calibrated to reproduce the Sun is assumed. This can lead to systematic uncertainties in stellar ages due to the effects of variable mixing depending on the mixing length. The MLT approximates convective mixing and thus we expect the value of $\mlt$ to vary from star-to-star due to various effects, from changes in chemical composition to other sources of mixing described poorly by MLT. The investigation of more complex $\mlt$ distributions is beyond the scope of this work. Instead, we experiment with two prior assumptions for $\mlt$ in the population. The first assumes $\mlt$ is normally distributed in our sample with a spread ($\sigma_\alpha$) to account for the aforementioned variation in $\mlt$. The second assumes $\mlt$ is constant throughout the sample. +The widely used mixing-length theory (MLT) of convection, parametrised by $\mlt$, has been tested throughout the Hertzsprung-Russell (HR) diagram with 3D hydrodynamical simulations \citep{Trampedach.Stein.ea2014, Magic.Weiss.ea2015} and asteroseismology \citep{Tayar.Somers.ea2017, Viani.Basu.ea2018, Li.Bedding.ea2018} with values in the range $0.8 \lesssim \mlt/\alpha_{\mathrm{MLT}, \odot} \lesssim 1.2$ where $\alpha_{\mathrm{MLT}, \odot}$ is the value calibrated to the Sun. However, in many grids of stellar models, a constant value calibrated to reproduce the Sun is assumed. This can lead to systematic uncertainties in stellar ages due to the effects of variable mixing depending on the mixing length. The MLT approximates convective mixing, and thus we expect the value of $\mlt$ to vary from star-to-star due to various effects, from changes in chemical composition to other sources of mixing described poorly by MLT. The investigation of more complex $\mlt$ distributions is beyond the scope of this work. Instead, we experiment with two prior assumptions for $\mlt$ in the population. The first assumes $\mlt$ is normally distributed in our sample with a spread ($\sigma_\alpha$) to account for the aforementioned variation in $\mlt$. The second assumes $\mlt$ is constant throughout the sample. Our HBM requires a way to map from the stellar initial (or bulk) properties to predict observables. We can achieve this with a large grid of stellar evolutionary models. However, a discrete grid can produce inaccurate posterior distributions, limited to the grid resolution. Increasing the resolution is computationally demanding, especially when scaling to higher input dimensions. One solution is to interpolate the stellar models, as is common in the isochrone-fitting method \citep[see e.g.][]{Berger.Huber.ea2020}. However, interpolation can become computationally expensive at high input dimensions and grid size, and evaluating the likelihood using modern Bayesian sampling techniques is slow. Therefore, we use machine learning to map stellar model inputs to observables to provide a fast way to sample the HBM. In this work, we train an artificial neural network (ANN) on a large grid of stellar models. There have been similar applications of ANNs in asteroseismology \citep{Verma.Hanasoge.ea2016, Bellinger.Angelou.ea2016, Hon.Stello.ea2017, Hon.Stello.ea2018a, Hendriks.Aerts2019} but not yet in the context of an HBM. Using the machine learning speed-up, we demonstrate a scalable method for obtaining fundamental stellar parameters. @@ -130,7 +130,7 @@ \subsubsection{Stellar Models and Input Physics}\label{subsec:stellar_model} Atomic diffusion of helium and heavy elements was also taken into account. \textsc{MESA} calculates particle diffusion and gravitational settling by solving Burger's equations using the method and diffusion coefficients of \citet{Thoul.Bahcall.ea1994}. We considered eight elements (${}^1{\rm H}, {}^3{\rm He}, {}^4{\rm He}, {}^{12}{\rm C}, {}^{14}{\rm N}, {}^{16}{\rm O}, {}^{20}{\rm Ne}$, and ${}^{24}{\rm Mg}$) -for diffusion calculations, and had the charge calculated by the \textsc{MESA} ionization module, which estimates the typical ionic charge as a function of $T$, $\rho$, and free electrons per nucleon from \citet{Paquette.Pelletier.ea1986}. +for diffusion calculations, and had the charge calculated by the \textsc{MESA} ionisation module, which estimates the typical ionic charge as a function of $T$, $\rho$, and free electrons per nucleon from \citet{Paquette.Pelletier.ea1986}. \subsubsection{Oscillation Models and Asteroseismic Quantities}\label{subsec:seismo_model} @@ -202,7 +202,7 @@ \subsubsection{No-Pooled (NP) Model}\label{sec:np} \begin{figure}[tb] \centering \includegraphics{figures/partial_pool_pgm.png} - \caption[A probabilistic graphical model of the partially-pooled hierarchical model.]{A probabilistic graphical model (PGM) of the partially-pooled (PP) hierarchical model. Nodes outside of the grey rectangle represent the hyperparameters in the model. Nodes inside the grey rectangle represent individual stellar parameters. Dark grey nodes represent observables which each have their respective observational uncertainties given by the solid black nodes. The direction of the arrows represent the dependencies in the generative model. \emph{This diagram was made using the \textsc{Python} package \texttt{daft} \citep{Foreman-Mackey.Hogg.ea2021}.}} + \caption[A probabilistic graphical model of the partially-pooled hierarchical model.]{A probabilistic graphical model (PGM) of the partially-pooled (PP) hierarchical model. Nodes outside the grey rectangle represent the hyperparameters in the model. Nodes inside the grey rectangle represent individual stellar parameters. Dark grey nodes represent observables which each have their respective observational uncertainties given by the solid black nodes. The direction of the arrows represent the dependencies in the generative model. \emph{This diagram was made using the \textsc{Python} package \texttt{daft} \citep{Foreman-Mackey.Hogg.ea2021}.}} \label{fig:pgm} \end{figure} @@ -212,7 +212,7 @@ \subsubsection{No-Pooled (NP) Model}\label{sec:np} p(\boldsymbol{\Theta} | \boldsymbol{D}) = \prod_{i=1}^{N_{\mathrm{stars}}} p(\boldsymbol{\theta}_i | \boldsymbol{d}_i), \end{equation} % -where $\boldsymbol{\Theta}$ is the matrix of model parameters and $\boldsymbol{D}$ is the matrix of observables. A probabilistic graphical model (PGM) of the NP model can be seen inside the grey box of Fig. \ref{fig:pgm}, without the arrow connecting $Z_\mathrm{init}$ to $Y_\mathrm{init}$. We ignore the nodes outside the box because these correspond the the PP model described next. +where $\boldsymbol{\Theta}$ is the matrix of model parameters and $\boldsymbol{D}$ is the matrix of observables. A probabilistic graphical model (PGM) of the NP model can be seen inside the grey box of Fig. \ref{fig:pgm}, without the arrow connecting $Z_\mathrm{init}$ to $Y_\mathrm{init}$. We ignore the nodes outside the box because these correspond to the PP model described next. \subsubsection{Partial-Pooled (PP) Model}\label{sec:pp} @@ -258,7 +258,7 @@ \subsubsection{Partial-Pooled (PP) Model}\label{sec:pp} \end{equation} % -We gave all of the hyperparameters weakly informative priors, with the exception of $Y_P$ for which we adopt a recent measurement of the primordial helium abundance from big bang nucleosynthesis (BBN) as the mean \citep{Pitrou.Coc.ea2018}, with a standard deviation representative of the range of values in the literature \citep{Aver.Olive.ea2015, Peimbert.Peimbert.ea2016, Cooke.Fumagalli2018}. Hence, we assumed priors on the hyperparameters as follows, +We gave all the hyperparameters weakly informative priors, except for $Y_P$ for which we adopt a recent measurement of the primordial helium abundance from big bang nucleosynthesis (BBN) as the mean \citep{Pitrou.Coc.ea2018}, with a standard deviation representative of the range of values in the literature \citep{Aver.Olive.ea2015, Peimbert.Peimbert.ea2016, Cooke.Fumagalli2018}. Hence, we assumed priors on the hyperparameters as follows, % \begin{align*} {\Delta Y}/{\Delta Z} &\sim 4.0\cdot\mathcal{B}(1.2, 1.2),\\ @@ -275,7 +275,7 @@ \subsubsection{Partial-Pooled (PP) Model}\label{sec:pp} \end{equation} % -We produced a PGM for the model, depicted in Fig. \ref{fig:pgm}. The hyperparameters are shown outside of the grey box containing the individual stellar parameters to represent the hierarchical aspect of the model. +We produced a PGM for the model, depicted in Fig. \ref{fig:pgm}. The hyperparameters are shown outside the grey box containing the individual stellar parameters to represent the hierarchical aspect of the model. \subsubsection{Max-Pooled (MP) Model}\label{sec:mp} @@ -283,7 +283,7 @@ \subsubsection{Max-Pooled (MP) Model}\label{sec:mp} %%%%%%%%%%%%%%%% MAX-POOLED MODEL %%%%%%%%%%%%%%%% -We built another hierarchical model similar to the PP model except that $\mlt$ is max-pooled (MP). In this model, we assumed that $\mlt$ must be the same value for every star in the sample, but still allowed it to freely vary on a population-level. Thus the hyperparameters are now, $\boldsymbol{\phi} = \{\Delta Y/\Delta Z, Y_P, \sigma_Y, \mlt\}$. The posterior distribution of the model takes the same form as in Equation \ref{eq:hbmbayes} except that the MLT parameter for the $i$-th star is, +We built another hierarchical model similar to the PP model except that $\mlt$ is max-pooled (MP). In this model, we assumed that $\mlt$ must be the same value for every star in the sample, but still allowed it to freely vary on a population-level. Thus, the hyperparameters are now, $\boldsymbol{\phi} = \{\Delta Y/\Delta Z, Y_P, \sigma_Y, \mlt\}$. The posterior distribution of the model takes the same form as in Equation \ref{eq:hbmbayes} except that the MLT parameter for the $i$-th star is, % \begin{equation} \alpha_{\mathrm{MLT}, i} = \mlt, @@ -298,7 +298,7 @@ \subsubsection{Max-Pooled (MP) Model}\label{sec:mp} \subsection{The Sun as a Star}\label{sec:sun} -Pooling parameters in an HBM allows us to use the Sun as a calibrator in a unique way. Rather than calibrating our model physics to the Sun and then assuming the calibrated parameters across our sample, we can include the Sun as a part of the same population as our sample of stars. If we assume $Y_\mathrm{init}$ and $\mlt$ for the Sun are a part of the same prior distribution as for the rest of the sample, then we can simply add solar observables to our model inputs. +Pooling parameters in an HBM allows us to use the Sun as a calibrator uniquely. Rather than calibrating our model physics to the Sun and then assuming the calibrated parameters across our sample, we can include the Sun as a part of the same population as our sample of stars. If we assume $Y_\mathrm{init}$ and $\mlt$ for the Sun are a part of the same prior distribution as for the rest of the sample, then we can simply add solar observables to our model inputs. For both the PP and MP models, we iterated with and without data for the Sun included in the population, referred to as PPS and MPS respectively. We adopted the solar data in Table \ref{tab:sun} with uncertainties conservatively limited to the accuracy of the ANN for $R$, $L$ and representative of variation in the literature for $\teff$. We also adopted $\dnu=\SI{135.1(2)}{\micro\hertz}$ with a central value from \citet{Huber.Bedding.ea2011} and a standard deviation representative of variations in measurements of the solar $\dnu$ \citep{Broomhall.Chaplin.ea2011}. @@ -319,7 +319,7 @@ \subsection{Sampling}\label{sec:sampling} Since our initial sample was chosen based on masses from \citetalias{Serenelli.Johnson.ea2017}, we expected some stars to lie outside (or near the boundary) of the observational parameter space provided by our grid of stellar models. We used an initial run of the NP model to catch and remove these stars. During the initial run, we dropped 16 of the 81 stars from the sample. Of the removed stars, we found the posteriors in $M$ for 6 skewed towards the prior upper mass limit of \SI{1.2}{\solarmass}. The remaining 10 removed stars suffered poor convergence during sampling ($\hat{r} >> 1.04$) which could be because of poor step-size tuning and sampling at the prior boundary. -Out of the remaining 65 stars with results from the NP model, 2 stars were dropped from the PP model. A consequence of partially pooling parameters is that a population spread, $\sigma$ allows for individual parameters to vary outside of the prior range given to the population mean, $\mu$. In this case, individual stellar $\mlt$ was allowed to vary outside the range for which the ANN was valid if $\sigma_\alpha$ was large. The 2 removed stars happened to have high likelihoods outside of the valid $\mlt$ range. We found that removing the same 2 stars from the other models made negligible difference to the results, so we leave a solution to this problem to future work. Naturally, we did not see the same issue in the MP model, so we proceeded with modelling the same 65 stars as with the NP model. +Out of the remaining 65 stars with results from the NP model, 2 stars were dropped from the PP model. A consequence of partially pooling parameters is that a population spread, $\sigma$ allows for individual parameters to vary outside the prior range given to the population mean, $\mu$. In this case, individual stellar $\mlt$ was allowed to vary outside the range for which the ANN was valid if $\sigma_\alpha$ was large. The 2 removed stars happened to have high likelihoods outside the valid $\mlt$ range. We found that removing the same 2 stars from the other models made negligible difference to the results, so we leave a solution to this problem to future work. Naturally, we did not see the same issue in the MP model, so we proceeded with modelling the same 65 stars as with the NP model. \section{Results}\label{sec:res} @@ -462,7 +462,7 @@ \subsection{Comparison with APOKASC Results}\label{sec:comp} Secondly, our choice of \citet{Asplund.Grevesse.ea2009} solar chemical mixture differs from the \citet{Grevesse.Sauval1998} mixtures adopted by \citetalias{Serenelli.Johnson.ea2017}. The former leads to a solar heavy-element to hydrogen ratio of $(Z/X)_\odot = 0.0181$, and the latter, $(Z/X)_\odot = 0.0230$. Typically, \citet{Grevesse.Sauval1998} abundances are favoured in asteroseismic modelling because they are better able to reproduce measurements of helium in the Sun from helioseismology \citep{Serenelli.Basu.ea2009}. An effect of using the \citet{Asplund.Grevesse.ea2009} abundances, is that it favours lower $Z_\mathrm{init}$ for a given $\metallicity_\mathrm{surf}$. As a result, models using \citet{Grevesse.Sauval1998} abundances on average underestimate radii and mass compared to those without by about 1 and 0.5 per cent respectively \citep{Nsamba.Campante.ea2018}. -Although updated, much of our observable data is comparable to that of \citetalias{Serenelli.Johnson.ea2017}, with the exception of $\teff$. The preferred results from \citetalias{Serenelli.Johnson.ea2017} were determined using a photometric $\teff$ scale which we found to be on average $\sim \SI{170}{\kelvin}$ greater than our spectroscopic scale from DR14. In \citetalias{Serenelli.Johnson.ea2017}, they saw a similar offset between the DR13 $\teff$ available at the time. They found a median difference in mass, radius, and age of approximately $-6$, $-2$, and $+35$ per cent respectively with results from the photometric $\teff$ scale subtracted from the spectroscopic scale. +Although updated, much of our observable data is comparable to that of \citetalias{Serenelli.Johnson.ea2017}, except for $\teff$. The preferred results from \citetalias{Serenelli.Johnson.ea2017} were determined using a photometric $\teff$ scale which we found to be on average $\sim \SI{170}{\kelvin}$ greater than our spectroscopic scale from DR14. In \citetalias{Serenelli.Johnson.ea2017}, they saw a similar offset between the DR13 $\teff$ available at the time. They found a median difference in mass, radius, and age of approximately $-6$, $-2$, and $+35$ per cent respectively with results from the photometric $\teff$ scale subtracted from the spectroscopic scale. % \begin{figure} % \centering @@ -489,7 +489,7 @@ \subsection{Comparison with APOKASC Results}\label{sec:comp} \label{fig:unc-comp} \end{figure} -In Fig. \ref{fig:unc-comp} we compare our statistical uncertainties for $M$, $R$, and $\tau$ with those for the equivalent stars from \citetalias{Serenelli.Johnson.ea2017}. We found that the NP model yielded comparable uncertainties to \citetalias{Serenelli.Johnson.ea2017} but note that these are likely underestimated due the influence of the prior boundaries for $Y_\mathrm{init}$ and $\mlt$. We expected larger uncertainties because we included additional free parameters ($Y_\mathrm{init}$ and $\mlt$) over the work of \citetalias{Serenelli.Johnson.ea2017}. However, when we treat these parameters hierarchically, we saw a reduction in uncertainties from all of the pooled models. This is because our prior assumptions about the population allows for the sharing of information between the stars. This uncertainty reduction scales with the number of stars in our sample, demonstrated by the results for the synthetic stars in Fig. \ref{fig:shrinkage}. Thus, hierarchically modelling our population resulted in improved statistical uncertainties in stellar fundamental parameters. +In Fig. \ref{fig:unc-comp} we compare our statistical uncertainties for $M$, $R$, and $\tau$ with those for the equivalent stars from \citetalias{Serenelli.Johnson.ea2017}. We found that the NP model yielded comparable uncertainties to \citetalias{Serenelli.Johnson.ea2017} but note that these are likely underestimated due the influence of the prior boundaries for $Y_\mathrm{init}$ and $\mlt$. We expected larger uncertainties because we included additional free parameters ($Y_\mathrm{init}$ and $\mlt$) over the work of \citetalias{Serenelli.Johnson.ea2017}. However, when we treat these parameters hierarchically, we saw a reduction in uncertainties from all the pooled models. This is because our prior assumptions about the population allows for the sharing of information between the stars. This uncertainty reduction scales with the number of stars in our sample, demonstrated by the results for the synthetic stars in Fig. \ref{fig:shrinkage}. Thus, hierarchically modelling our population resulted in improved statistical uncertainties in stellar fundamental parameters. % \begin{figure} % \centering From 92a9df52e73811cea79561f2835b0e10e99e572d Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Thu, 27 Apr 2023 17:41:26 +0100 Subject: [PATCH 35/50] Grammar checks --- chapters/glitch.tex | 12 ++-- chapters/hbm.tex | 14 ++--- chapters/introduction.tex | 10 ++-- references.bib | 114 ++++++++++++++++++++++++++++++++++++++ 4 files changed, 132 insertions(+), 18 deletions(-) diff --git a/chapters/glitch.tex b/chapters/glitch.tex index 3ddad30..d218d67 100644 --- a/chapters/glitch.tex +++ b/chapters/glitch.tex @@ -145,7 +145,7 @@ \section{Introduction} \begin{figure} \centering \includegraphics{figures/glitch-1d-example-results.pdf} - \caption[The change in mode frequency induced by a rapid change in sound speed for the 1D example.]{The change in mode frequency induced by a change in sound speed of \(\delta c\) from \(x_\glitch - \delta x\) to \(x_\glitch + \delta x\) in a one-dimensional medium, bound such that \(x \in [0, 1]\) (see Figure \ref{fig:1d-diagram}). Outside of the perturbation the speed of sound, \(c=1\). The frequency perturbations are offset by \(\omega_0\) given in the legend of the top panel. Points are joined by straight lines to guide the eye but do not represent real solutions. + \caption[The change in mode frequency induced by a rapid change in sound speed for the 1D example.]{The change in mode frequency induced by a change in sound speed of \(\delta c\) from \(x_\glitch - \delta x\) to \(x_\glitch + \delta x\) in a one-dimensional medium, bound such that \(x \in [0, 1]\) (see Figure \ref{fig:1d-diagram}). Outside the perturbation the speed of sound, \(c=1\). The frequency perturbations are offset by \(\omega_0\) given in the legend of the top panel. Points are joined by straight lines to guide the eye but do not represent real solutions. } \label{fig:1d-results} \end{figure} @@ -154,7 +154,7 @@ \section{Introduction} % Physically, this arises from the change in phase required to satisfy the boundary conditions of the glitch region. -The functional form of \(\delta\omega\) appears to have a linear component and a short periodicity modulated by a longer periodicity. As the location of the glitch (\(x_\glitch\)) gets smaller, the short period of \(\delta\omega\) decreases. If we imagine the spacial distribution of nodes in the system as a function of \(n\), the density of nodes is larger towards the centre of the system. The periodicity arises from the nodes passing in and out of the glitch region with changing \(n\). Therefore, where the density of wave nodes is higher, we expect the short period of \(\delta\omega\) to decrease. Similarly, as the the half-width of the glitch (\(\delta x\)) increases, the longer period increases. +The functional form of \(\delta\omega\) appears to have a linear component and a short periodicity modulated by a longer periodicity. As the location of the glitch (\(x_\glitch\)) gets smaller, the short period of \(\delta\omega\) decreases. If we imagine the spacial distribution of nodes in the system as a function of \(n\), the density of nodes is larger towards the centre of the system. The periodicity arises from the nodes passing in and out of the glitch region with changing \(n\). Therefore, where the density of wave nodes is higher, we expect the short period of \(\delta\omega\) to decrease. Similarly, as the half-width of the glitch (\(\delta x\)) increases, the longer period increases. Furthermore, an increasing change in sound speed (\(\delta c\)) increases the amplitude of \(\delta\omega\). This result is intuitive, as we expect a larger change in \(c\) to correspond to a large change in \(\omega\). Finally, increasing both \(\delta x\) and \(\delta c\) increases the slope of \(\delta\omega\). The sensitivity of a mode to the glitch increases with \(n\) and depends on how much the wave changes in the glitch region. Modes of smaller \(n\) become more sensitive to the glitch region as \(\delta x\) and \(\delta c\) increase, thus increasing the linear slope of \(\delta\omega\). @@ -167,7 +167,7 @@ \section{Introduction} \label{fig:1d-phase} \end{figure} -The small phase offset \(\epsilon\) in Equation \ref{eq:1d-phase} is required for the wave function to satisfy the boundary conditions at \(x = 0\). However, adding \(\epsilon\) shifts the effective location of \(x_\glitch\) --- it changes the scale of the \(x\)-axis by a factor of \((1 + \epsilon)\). We plot \(\epsilon\) against \(\omega\) in Figure \ref{fig:1d-phase} and show that its magnitude is \(\sim 10^{-4}\), much smaller than the location and size of region 2. The periodicity caused by the glitch also shows up in Figure \ref{fig:1d-phase}, with its properties affected in a similar way to Figure \ref{fig:1d-results}. This is because the more a mode is affected by the glitch, the greater the phase offset required to satisfy the boundary conditions. +The small phase offset \(\epsilon\) in Equation \ref{eq:1d-phase} is required for the wave function to satisfy the boundary conditions at \(x = 0\). However, adding \(\epsilon\) shifts the effective location of \(x_\glitch\) --- it changes the scale of the \(x\)-axis by a factor of \((1 + \epsilon)\). We plot \(\epsilon\) against \(\omega\) in Figure \ref{fig:1d-phase} and show that its magnitude is \(\sim 10^{-4}\), much smaller than the location and size of region 2. The periodicity caused by the glitch also shows up in Figure \ref{fig:1d-phase}, with its properties affected similarly to Figure \ref{fig:1d-results}. This is because the more a mode is affected by the glitch, the greater the phase offset required to satisfy the boundary conditions. Finding an approximate solution for \(\delta\omega\) is beyond the scope of this example. However, we can show that by modelling \(\delta\omega\), we can recover information about the structural glitch. Let us build a model \(\delta\omega = f(\omega)\). Looking at Figure \ref{fig:1d-results}, we propose a form for \(f\), % @@ -246,7 +246,7 @@ \subsection{Helium Ionisation Glitch}\label{sec:helium-glitch} \label{fig:gamma-zones} \end{figure} -In Figure \ref{fig:gamma-zones}, we show \(\gamma\) for model S against fractional acoustic depth, shaded by regions of ionisation. For an ideal monatomic gas, \(\gamma=5/3\), but we see that \(\gamma < 5/3\) in regions where helium and hydrogen ionise. Close to the surface of the star, hydrogen ionisation has the largest effect on \(\gamma\) because it makes up the majority of stellar matter. The first (He\,\textsc{i}) and second (He\,\textsc{ii}) ionisations of helium occur deeper in the star. We can see that the second ionisation of helium has a greater affect on \(\gamma\) than the first. The effect of the He\,\textsc{ii} ionisation causes a rapid change in \(\gamma\) over a few per cent in \(\tau\) (equivalent to \(\sim \SI{100}{\second}\) in model S). +In Figure \ref{fig:gamma-zones}, we show \(\gamma\) for model S against fractional acoustic depth, shaded by regions of ionisation. For an ideal monatomic gas, \(\gamma=5/3\), but we see that \(\gamma < 5/3\) in regions where helium and hydrogen ionise. Close to the surface of the star, hydrogen ionisation has the largest effect on \(\gamma\) because it makes up the majority of stellar matter. The first (He\,\textsc{i}) and second (He\,\textsc{ii}) ionisations of helium occur deeper in the star. We can see that the second ionisation of helium has a greater effect on \(\gamma\) than the first. The effect of the He\,\textsc{ii} ionisation causes a rapid change in \(\gamma\) over a few per cent in \(\tau\) (equivalent to \(\sim \SI{100}{\second}\) in model S). \subsubsection{The Effect of Helium Abundance on \(\gamma\)} @@ -321,7 +321,7 @@ \subsubsection{The Effect of a Change in \(\gamma\) on p Mode Frequencies} to its inertia (\(\mathcal{I}\)). The mode inertia is given by, % \begin{equation} - \mathcal{I} = \int_0^R \vect{\xi} \cdot \vect{\xi} \, \rho r^2 \, \dd r. + \mathcal{I} = \int_0^R \vect{\xi} \cdot \vect{\xi} \, \rho r^2 \, \dd r, \end{equation} % where the amplitude and direction of a pressure wave at a given point in a star is given by the so-called Lagrangian perturbation vector \(\vect{\xi}\). @@ -420,7 +420,7 @@ \subsubsection{A Functional Form of the Helium Glitch Signature} \left.\frac{\delta\gamma}{\gamma}\right|_\heII \simeq - \frac{\Gamma_\heII}{\Delta_\heII \sqrt{2\pi}} \, \ee^{- \frac12{(\tau - \tau_\heII)^2}/{\Delta_\heII^2} }, \label{eq:he-gamma} \end{equation} % -where \(\Gamma_\heII\) is the area, \(\Delta_\heII\) is the characteristic width, and \(\tau_\heII\) is the center of the ionisation region. +where \(\Gamma_\heII\) is the area, \(\Delta_\heII\) is the characteristic width, and \(\tau_\heII\) is the centre of the ionisation region. Here, we verify the result for \(\delta\omega\) due to He\,\textsc{ii} ionisation from \citet{Houdek.Gough2007}. Substituting Equation \ref{eq:he-gamma} into Equation \ref{eq:omega-osc} with a change of variables to \(x = (\tau - \tau_\heII)/\Delta_\heII\), we get, % diff --git a/chapters/hbm.tex b/chapters/hbm.tex index 5140918..4c6237b 100644 --- a/chapters/hbm.tex +++ b/chapters/hbm.tex @@ -20,7 +20,7 @@ \section{Introduction} Let us consider modelling a large population of stars simultaneously. For example, we may want to create a catalogue of stellar parameters to use in exoplanet and galactic research. We could treat the parameters for each star independently, repeating the modelling procedure for each star in the sample. However, we know that stars belong to population distributions like the IMF. In some cases, these population distributions are not well understood and assuming one exactly can introduce bias into our model. Instead, it could be better to let the data inform such population priors. -Hierarchical (or multi-level) Bayesian models parametrise population-level prior distributions on individual stellar parameters. These population-parameters (\emph{hyperparameters}) which govern the population must themselves have prior distributions as per the Bayesian formalisation. The hierarchical aspect comes from the distinction between hyperparameters which take one value across a population and those which vary from star-to-star. In the next section, we go through an simple example hierarchical model. Then, we discuss a few parameters which could be given the hierarchical treatment in Section \ref{sec:hbm-phys}. +Hierarchical (or multi-level) Bayesian models parametrise population-level prior distributions on individual stellar parameters. These population-parameters (\emph{hyperparameters}) which govern the population must themselves have prior distributions as per the Bayesian formalisation. The hierarchical aspect comes from the distinction between hyperparameters which take one value across a population and those which vary from star-to-star. In the next section, we go through a simple example hierarchical model. Then, we discuss a few parameters which could be given the hierarchical treatment in Section \ref{sec:hbm-phys}. \section[Stellar Distances]{Stellar Distances in an Open Cluster Analogue}\label{sec:hbm-dist} @@ -62,7 +62,7 @@ \subsection{Simple Model}\label{sec:simple-model} % where \(\mathcal{N}(x \,|\, \mu, \sigma^2)\) is a normal distribution over \(x\) with a mean of \(\mu\) and variance of \(\sigma^2\). -We assumed stars in the cluster were equally likely to be between a distance of 0 and 20. We also assumed the absolute magnitudes were likely to be normally distributed centred on 0 and scaled by 10. Therefore, the prior probability of the model parameters was, +We assumed stars in the cluster were equally likely to be between a distance of 0 and 20. Moreover, we assumed the absolute magnitudes were likely to be normally distributed centred on 0 and scaled by 10. Therefore, the prior probability of the model parameters was, % \begin{equation} p(d_i, \absmag_i) = \mathcal{U}(d_i \mid 0, 20) \, \mathcal{N}(\absmag_i \mid 0, 100), @@ -121,17 +121,17 @@ \subsection{Hierarchical Model}\label{sec:hbm-model} \begin{figure}[tb] \centering \includegraphics{figures/hbm-pgm.pdf} - \caption[Probabilistic graphical model for the hierarchical model]{Probabilistic graphical model extension of Figure \ref{fig:simple-pgm} but for the HBM. The hyperparameters exist outside of the box to show that they are the same across the population of stars.} + \caption[Probabilistic graphical model for the hierarchical model]{Probabilistic graphical model extension of Figure \ref{fig:simple-pgm} but for the HBM. The hyperparameters exist outside the box to show that they are the same across the population of stars.} \label{fig:hbm-pgm} \end{figure} -We show the probabilistic graphical model of the HBM in Figure \ref{fig:hbm-pgm}. Here, we see how all individual stellar parameters depend on \(\mu_d\) and \(\sigma_d\). This is a simple extension of Figure \ref{fig:simple-pgm}, but with parameters existing outside of the box to illustrate the model hierarchy. We can imagine extending this framework to multiple levels, or adding additional hyperparameters. +We show the probabilistic graphical model of the HBM in Figure \ref{fig:hbm-pgm}. Here, we see how all individual stellar parameters depend on \(\mu_d\) and \(\sigma_d\). This is a simple extension of Figure \ref{fig:simple-pgm}, but with parameters existing outside the box to illustrate the model hierarchy. We can imagine extending this framework to multiple levels, or adding additional hyperparameters. \subsection{Inferring the Model Parameters}\label{sec:hbm-inf} To infer the model parameters, we need to calculate the marginalised posterior distributions for each parameter. We could obtain these analytically by integrating the full posterior distribution over all model parameters except for the parameter of interest. Alternatively, we can approximate the marginalised posterior using a Markov Chain Monte Carlo (MCMC) sampling algorithm. We chose the latter approach because it is scalable to more complicated models where the marginalisation is not analytically solvable. -We used the No U-Turn Sampler \citep[NUTS;][]{Hoffman.Gelman2014} as implemented in the \textsc{Python} package \texttt{numpyro} \citep{Phan.Pradhan.ea2019,Bingham.Chen.ea2019} to sample from the approximate posterior distribution for both models. We ran the sampler for 1000 steps following 500 `warmup' steps (used to adapt the sampling procedure) and repeated for 10 MCMC chains. To reduce the number of divergences encountered during sampling, we increased the target accept probability from 0.8 to 0.98 for the HBM. The resulting marginalised posterior samples amounted to \num{10000} per parameter. +To sample from the approximate posterior distribution for both models, we used the No U-Turn Sampler \citep[NUTS;][]{Hoffman.Gelman2014} as implemented in the \textsc{Python} package \texttt{numpyro} \citep{Phan.Pradhan.ea2019,Bingham.Chen.ea2019}. We ran the sampler for 1000 steps following 500 warm-up steps (used to adapt the sampling procedure) and repeated for 10 MCMC chains. To reduce the number of divergences encountered during sampling, we increased the target accept probability from 0.8 to 0.98 for the HBM. The resulting marginalised posterior samples amounted to \num{10000} per parameter. \subsection{Comparing the Models}\label{sec:hbm-comp} @@ -162,7 +162,7 @@ \subsection{Comparing the Models}\label{sec:hbm-comp} \label{fig:hbm-global} \end{figure} -One consequence of the HBM was that it parametrised the population mean (\(\mu_d\)) and standard deviation (\(\sigma_d\)) of the distances to stars in the cluster independently from the observed noise. For the simple model, we estimated \(\mu_d\) and \(\sigma_d\) by taking the sample mean and standard deviation of distances in the cluster for each posterior sample. We compared the resulting posterior distributions for \(\mu_d\) and \(\sigma_d\) from the two models in Figure \ref{fig:hbm-global}. We found the mean distance from the simple model was \(\mu_d = 10.50 \pm 0.27\), whereas the HBM was more accurate with \(\mu_d = 10.01 \pm 0.24\). We also found the simple model massively overestimated the standard deviation of cluster distances with \(\sigma_d = 1.815_{-0.293}^{+0.360}\), compared to the HBM's more accurate value of \(\sigma_d = 0.131_{-0.088}^{+0.280}\). The simple model cannot easily distinguish between the uncertainty on individual distances and the spread of the population. +One consequence of the HBM was that it parametrised the population mean (\(\mu_d\)) and standard deviation (\(\sigma_d\)) of the distances to stars in the cluster separately from the observed noise. For the simple model, we estimated \(\mu_d\) and \(\sigma_d\) by taking the sample mean and standard deviation of distances in the cluster for each posterior sample. We compared the resulting posterior distributions for \(\mu_d\) and \(\sigma_d\) from the two models in Figure \ref{fig:hbm-global}. The mean distance from the simple model was \(\mu_d = 10.50 \pm 0.27\), whereas the HBM was more accurate with \(\mu_d = 10.01 \pm 0.24\). The simple model massively overestimated the standard deviation of cluster distances with \(\sigma_d = 1.815_{-0.293}^{+0.360}\), compared to the HBM's more accurate value of \(\sigma_d = 0.131_{-0.088}^{+0.280}\). The simple model could not distinguish between the uncertainty on individual distances and the spread of the population. \subsection{Scaling with the Number of Stars}\label{sec:hbm-scale} @@ -183,7 +183,7 @@ \subsection{Scaling with the Number of Stars}\label{sec:hbm-scale} \paragraph{Mass} As mentioned in the introduction to this chapter, an IMF may be used as a prior on the mass of a star. If modelling stars with masses spanning a couple of orders of magnitude, it makes sense to draw their masses from an IMF. To make this hierarchical, we would parametrise the IMF as a conditional distribution which informs individual stellar masses. This way, the population of stars constrains the IMF while in-turn sharing information. In this thesis, we only consider stars in a narrow mass range (\SIrange{0.8}{1.2}{\solarmass}) where the IMF does not change much. However, a mass hierarchy like this may be useful in future work. -\paragraph{Age} We do not expect a star to be older than the universe. This makes the age of the universe a natural upper limit to a prior on stellar age. There are also populations of stars where we expect the age to be tightly related. For example, star systems like binaries and clusters are expected to have been formed at a similar time. Similarly to the distances in Section \ref{sec:hbm-dist}, a hierarchical model in age would parametrise the mean and variance of ages in these systems and in-turn improve other connected model parameters. This sort of analysis could be extended to find mixtures of age distributions in the galaxy which could indicate mergers \citep[e.g. \emph{Gaia}-Enceladus;][]{Helmi.Babusiaux.ea2018}. However, we do not further consider a HBM in age this in this work. +\paragraph{Age} We do not expect a star to be older than the universe. This makes the age of the universe a natural upper limit to a prior on stellar age. There are also populations of stars where we expect the age to be tightly related. For example, star systems like binaries and clusters are expected to have been formed at a similar time. Similarly to the distances in Section \ref{sec:hbm-dist}, a hierarchical model in age would parametrise the mean and variance of ages in these systems and in-turn improve other connected model parameters. This sort of analysis could be extended to find mixtures of age distributions in the galaxy which could indicate mergers \citep[e.g. \emph{Gaia}-Enceladus;][]{Helmi.Babusiaux.ea2018}. However, we do not further consider an HBM in age this in this work. \paragraph{Chemical Abundances} Unlike mass and age, surface abundances of stars can be measured directly with spectroscopy. However, not all abundances are easily measured. For example, helium ionises below the surface of the cool stars being studied in this work. Also, diffusion and settling of elements during stellar evolution means surface abundances differ from formation to time of observation. Similarly to age, we can expect clusters of stars to share initial abundances. However, the distribution of abundances in the Milky Way may also be parametrised. As stars evolve they convert hydrogen into helium and heavier elements. Supernovae enrich the galaxy with these elements monotonically, which go on to constitute new stars. A hierarchical model could include this assumption to tie the abundance of helium to heavier elements. diff --git a/chapters/introduction.tex b/chapters/introduction.tex index b6558b4..3d990c5 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -32,11 +32,11 @@ \section{Understanding the Stars}\label{sec:stars} % The luminosity and effective temperature could be estimated from the magnitude and colour of the stars. From luminosity and temperature, we could derive the radius of stars. Early stellar mass estimates came from visual and spectroscopic binaries. Spectroscopy provides abundances of chemical species ionised in the stellar atmosphere. However, except for the Sun, stellar age and helium abundance has no model-independence. The latter ionisations at temperatures and densities higher than the surface of stars like the Sun. -The chemical composition, energy sources and evolutionary physics of stars were still uncertain. Spectroscopy revealed relative abundances of elements excited in stellar atmospheres, but determining their absolute abundances was difficult. \citet{Payne1925} proposed that stars were comprised of mostly hydrogen and helium, later reinforced by estimates of helium content in the Sun \citep[e.g.][]{Schwarzschild1946}. The idea was radical at the time because it did not match the abundance of elements found on Earth. Meanwhile, advancements in nuclear science spawned the theory of stellar nucleosynthesis \citep{Hoyle1946}. Stars produce elements heavier than hydrogen and helium through nuclear fusion reactions. This discovery explained the production of many elements in the universe, tying the formation and evolution of stars to planetary formation and the conditions for life in the universe. +The chemical composition, energy sources and evolutionary physics of stars were still uncertain. Spectroscopy revealed relative abundances of elements excited in stellar atmospheres, but determining their absolute abundances was difficult. \citet{Payne1925} proposed that stars comprised mostly hydrogen and helium, later reinforced by estimates of helium content in the Sun \citep[e.g.][]{Schwarzschild1946}. The idea was radical at the time because it did not match the abundance of elements found on Earth. Meanwhile, advancements in nuclear science spawned the theory of stellar nucleosynthesis \citep{Hoyle1946}. Stars produce elements heavier than hydrogen and helium through nuclear fusion reactions. This discovery explained the production of many elements in the universe, tying the formation and evolution of stars to planetary formation and the conditions for life in the universe. In the second half of the 20th century, the theory of stellar evolution advanced. With this came computational methods for simulating stars \citep[e.g.][]{Kippenhahn.Weigert.ea1967}. Astronomers could start to compare observations of stars and clusters with simulated stars. We call this process \emph{modelling stars}. Early models of the Sun benefited from independent age estimates from geology and neutrino-production rates from observing cosmic rays. With these constraints, and the Sun as a calibrator, researchers could start to refine their models and test them on other stars. For example, \citet{Iben1967} tested stellar models on main sequence stars with similar masses to the Sun using observations of the clusters M67 and NGC 188. -On a similar timescale, a new field emerged which gave astronomers a model-independent way of studying the inside of stars. Identifying regular perturbations of the surface of the Sun, researchers realised that they pertained to high-order, stochastically-driven spherical harmonic oscillations. Measuring these oscillation modes, they could study the Sun in a similar way to how seismologists study the Earth. This new field, called asteroseismology, studied multiple non-radial oscillations, distinguishing itself from the well-know study of radially pulsating stars \citep[e.g. Cepheid variables;][]{Leavitt1908}. In Section \ref{sec:seismo} we give a brief history and theory of the asteroseismology for stars like the Sun. +On a similar timescale, a new field emerged which gave astronomers a model-independent way of studying the inside of stars. Identifying regular perturbations of the surface of the Sun, researchers realised that they pertained to high-order, stochastically-driven spherical harmonic oscillations. Measuring these oscillation modes, they could study the Sun similarly to how seismologists study the Earth. This new field, called asteroseismology, studied multiple non-radial oscillations, distinguishing itself from the well-know study of radially pulsating stars \citep[e.g. Cepheid variables;][]{Leavitt1908}. In Section \ref{sec:seismo} we give a brief history and theory of the asteroseismology for stars like the Sun. % Approximations for the equation of state by \citet{Eggleton.Faulkner.ea1973}. @@ -150,7 +150,7 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \label{fig:hr-diagram} \end{figure} -In Figure \ref{fig:hr-diagram}, we show a color-magnitude diagram made using magnitudes and parallaxes from \emph{Gaia} Data Release 3 \citep[DR3;][]{GaiaCollaboration.Vallenari.ea2022}. For this illustrative plot, we have neglected the effect of extinction. The background distribution shows solar-neighbourhood \emph{Gaia} sources with a parallax greater than \SI{5}{\milli\aarcsec} for context. Over-plot is the distribution of \emph{Kepler} objects cross-matched with \emph{Gaia} DR3\footnote{The cross-matched dataset was obtained from \url{https://gaia-kepler.fun}}. The densest region lies in the low- to intermediate-mass main sequence (\SIrange{0.8}{1.2}{\solarmass}) but we can also see a clear red giant branch and red clump in the upper-right of the distribution. The inset plot draws attention to the region occupied by dwarf and subgiant solar-like oscillators, where we give some examples. +In Figure \ref{fig:hr-diagram}, we show a colour-magnitude diagram made using magnitudes and parallaxes from \emph{Gaia} Data Release 3 \citep[DR3;][]{GaiaCollaboration.Vallenari.ea2022}. For this illustrative plot, we have neglected the effect of extinction. The background distribution shows solar-neighbourhood \emph{Gaia} sources with a parallax greater than \SI{5}{\milli\aarcsec} for context. Over-plot is the distribution of \emph{Kepler} objects cross-matched with \emph{Gaia} DR3\footnote{The cross-matched dataset was obtained from \url{https://gaia-kepler.fun}}. The densest region lies in the low- to intermediate-mass main sequence (\SIrange{0.8}{1.2}{\solarmass}) but we can also see a clear red giant branch and red clump in the upper-right of the distribution. The inset plot draws attention to the region occupied by dwarf and subgiant solar-like oscillators, where we give some examples. \citet{Chaplin.Kjeldsen.ea2011} identified the first large catalogue of \(\sim 500\) dwarf and subgiant solar-like oscillators (black circles in Figure \ref{fig:hr-diagram}) by measuring \(\dnu\) and \(\numax\) in \emph{Kepler} data. Later, \citet{Chaplin.Basu.ea2014} determined ages, masses and radii for these stars using \(\dnu\) and \(\numax\) complemented by photometry and ground-based spectroscopy where available. The subsequent arrival of APOGEE spectroscopy allowed \citet{Serenelli.Johnson.ea2017} to revisit this sample with a more consistent set of \(\teff\) and metallicity. By comparing observations to models of stellar evolution, they found radii, masses and ages with uncertainties of around 3, 5 and 20 per cent respectively. @@ -180,7 +180,7 @@ \section{Modelling Stars the Bayesian Way}\label{sec:modelling-stars} p(\vect{\theta} \mid \vect{y}) = \frac{p(\vect{y} \mid \vect{\theta})\,p(\vect{\theta})}{p(\vect{y})}, \end{equation} % -where \(p(\vect{y} \mid \vect{\theta})\) is the \emph{likelihood}, \(p(\vect{\theta})\) is the \emph{prior}, and \(p(\vect{y})\) is the \emph{evidence}. The prior encodes our expectation for \(\vect{\theta}\). If the observations are bad (i.e. have a small likelihood) then the prior dominates and we return our current belief. However, if the observations are good enough, the likelihood has more influence on the posterior and our belief is updated. As a result, we can use the Bayesian methodology to systematically update our beliefs. This provides a statistical scheme from which to build our stellar models. +where \(p(\vect{y} \mid \vect{\theta})\) is the \emph{likelihood}, \(p(\vect{\theta})\) is the \emph{prior}, and \(p(\vect{y})\) is the \emph{evidence}. The prior encodes our expectation for \(\vect{\theta}\). The prior dominates if the observations are bad (i.e. have a small likelihood) and we return our current belief. However, if the observations are good enough, the likelihood has more influence on the posterior and our belief is updated. As a result, we can use the Bayesian methodology to systematically update our beliefs. This provides a statistical scheme from which to build our stellar models. Let us consider a stellar model with three parameters, \(\vect{\theta} = t_\star, M, Z\). For instance, we might want the age of a star to help date a galactic merger, or its mass to characterise an exoplanetary system. To get the probability over age given our observations, we must \emph{marginalise} over the joint posterior with respect to all other parameters. The marginal posterior probability distribution for \(t_\star\) is, % @@ -196,7 +196,7 @@ \section{Modelling Stars the Bayesian Way}\label{sec:modelling-stars} The complexity of stellar models means that the marginalised posterior distributions are not analytically derivable. Therefore, we use numerical methods like Markov Chain Monte Carlo (MCMC) to estimate the posterior. Typically, this involves exploring parameter space with multiple calls to \(f\) for different values of \(\vect{\theta}\). In the case of MCMC-based algorithms like Hamiltonian Monte Carlo (HMC) and the No U-Turn Sampler (NUTS), the gradient of \(f\) is also required. There are several open-source \textsc{Python} packages widely used to implement these algorithms including \texttt{pymc} \citep{Salvatier.Wiecki.ea2016} and \texttt{numpyro} \citep{Phan.Pradhan.ea2019}. -There are some existing methods for determining stellar parameters using this Bayesian approach. \citet{Bazot.Bourguignon.ea2008} used the MCMC algorithm to sample model parameters with on-the-fly stellar model calculation. While this method can be tailored to individual stars, it is very computationally expensive. Each proposed set of \(\vect{\theta}\) spawns a stellar simulation which evolves to a given age. Steps prior to this age may be discarded and the simulation can take minutes to hours for each set of \(\vect{\theta}\). This is not a viable solution for modelling large numbers of stars. +There are some existing methods for determining stellar parameters using this Bayesian approach. \citet{Bazot.Bourguignon.ea2008} used the MCMC algorithm to sample model parameters with on-the-fly stellar model calculation. While this method can be tailored to individual stars, it is very computationally expensive. Each proposed set of \(\vect{\theta}\) spawns a stellar simulation which evolves to a given age. Steps prior to this age may be discarded, and the simulation can take minutes to hours for each set of \(\vect{\theta}\). This is not a viable solution for modelling large numbers of stars. A more efficient solution is to sample a discrete grid of models \citep{Gruberbauer.Guenther.ea2012,Gruberbauer.Guenther.ea2013}. A recent tool which does this is the BAyesian STellar algorithm (BASTA) developed by \citet{AguirreBorsen-Koch.Rorsted.ea2022}. The authors pre-compute a grid of simulated stars corresponding to a set of relevant input parameters. They weight the grid points according to the prior and then compute the marginalised posterior for each model parameter. diff --git a/references.bib b/references.bib index a851595..f63f216 100644 --- a/references.bib +++ b/references.bib @@ -192,6 +192,22 @@ @article{Ando.Osaki1975 keywords = {Acoustic Instability,Astrophysics,Eddington Approximation,Hydrogen Ions,Magnetoacoustic Waves,Nonadiabatic Theory,Nonstabilized Oscillation,Radiative Transfer,Solar Atmosphere,Stellar Envelopes} } +@article{Angus.Aigrain.ea2015, + title = {Calibrating Gyrochronology Using {{Kepler}} Asteroseismic Targets}, + author = {Angus, Ruth and Aigrain, Suzanne and {Foreman-Mackey}, Daniel and McQuillan, Amy}, + year = {2015}, + month = jun, + journal = {\mnras}, + volume = {450}, + pages = {1787--1798}, + issn = {0035-8711}, + doi = {10.1093/mnras/stv423}, + urldate = {2023-04-27}, + abstract = {Among the available methods for dating stars, gyrochronology is a powerful one because it requires knowledge of only the star's mass and rotation period. Gyrochronology relations have previously been calibrated using young clusters, with the Sun providing the only age dependence, and are therefore poorly calibrated at late ages. We used rotation period measurements of 310 Kepler stars with asteroseismic ages, 50 stars from the Hyades and Coma Berenices clusters and 6 field stars (including the Sun) with precise age measurements to calibrate the gyrochronology relation, whilst fully accounting for measurement uncertainties in all observable quantities. We calibrated a relation of the form P = An \texttimes{} (B - V - c)b, where P is rotation period in days, A is age in Myr, B and V are magnitudes and a, b and n are the free parameters of our model. We found a = 0.40\^\{+0.3\}\_\{-0.05\}, b = 0.31\^\{+0.05\}\_\{-0.02\} and n = 0.55\^\{+0.02\}\_\{-0.09\}. Markov Chain Monte Carlo methods were used to explore the posterior probability distribution functions of the gyrochronology parameters and we carefully checked the effects of leaving out parts of our sample, leading us to find that no single relation between rotation period, colour and age can adequately describe all the subsets of our data. The Kepler asteroseismic stars, cluster stars and local field stars cannot all be described by the same gyrochronology relation. The Kepler asteroseismic stars may be subject to observational biases; however, the clusters show unexpected deviations from the predicted behaviour, providing concerns for the overall reliability of gyrochronology as a dating method.}, + keywords = {Astrophysics - Earth and Planetary Astrophysics,Astrophysics - Solar and Stellar Astrophysics,methods: statistical,stars: evolution,stars: fundamental parameters,stars: oscillations,stars: rotation,stars: solar-type}, + annotation = {ADS Bibcode: 2015MNRAS.450.1787A} +} + @article{Antia.Basu1994, title = {Measuring the {{Helium Abundance}} in the {{Solar Envelope}}: {{The Role}} of the {{Equation}} of {{State}}}, shorttitle = {Measuring the {{Helium Abundance}} in the {{Solar Envelope}}}, @@ -492,6 +508,39 @@ @article{Barber.Dobkin.ea1996 keywords = {convex hull,Delaunay triangulation,halfspace intersection,Voronoi diagram} } +@article{Barnes2003, + title = {On the {{Rotational Evolution}} of {{Solar-}} and {{Late-Type Stars}}, {{Its Magnetic Origins}}, and the {{Possibility}} of {{Stellar Gyrochronology}}}, + author = {Barnes, Sydney A.}, + year = {2003}, + month = mar, + journal = {\apj}, + volume = {586}, + pages = {464--479}, + issn = {0004-637X}, + doi = {10.1086/367639}, + urldate = {2023-04-27}, + abstract = {We propose a simple interpretation of the rotation period data for solar- and late-type stars. The open cluster and Mount Wilson star observations suggest that rotating stars lie primarily on two sequences, initially called I and C. Some stars lie in the intervening gap. These sequences, and the fractional numbers of stars on each sequence, evolve systematically with cluster age, enabling us to construct crude rotational isochrones allowing ``stellar gyrochronology,'' a procedure, on improvement, likely to yield ages for individual field stars. The age and color dependences of the sequences allow the identification of the underlying mechanism, which appears to be primarily magnetic. The majority of solar- and late-type stars possess a dominant Sun-like, or interface, magnetic field, which connects the convective envelope to both the radiative interior of the star and the exterior, where winds can drain off angular momentum. These stars spin down Skumanich style. An age-decreasing fraction of young G, K, and M stars, which are rapid rotators, possess only a convective field, which is not only inefficient in depleting angular momentum but also incapable of coupling the surface convection zone to the inner radiative zone, so that only the outer zone is spun down, and on an exponential timescale. These stars do not yet possess large-scale dynamos. The large-scale magnetic field associated with the dynamo, apparently created by the shear between the decoupled radiative and convective zones, (re)couples the convective and radiative zones and drives a star from the convective to the interface sequence through the gap on a timescale that increases as stellar mass decreases. Fully convective stars do not possess such an interface, cannot generate an interface dynamo, and hence can never make such a transition. Helioseismic results for the present-day Sun agree with this scheme, which also explains the rotational bimodality observed by Herbst and collaborators among pre-main-sequence stars and the termination of this bimodality when stars become fully convective. This paradigm also provides a new basis for understanding stellar X-ray and chromospheric activity, light-element abundances, and perhaps other stellar phenomena that depend on rotation. This is Paper 13 of the WIYN Open Cluster Study (WOCS).}, + keywords = {Astrophysics,Galaxy: Open Clusters and Associations: General,Stars: Evolution,Stars: Interiors,Stars: Late-Type,Stars: Magnetic Fields,Stars: Rotation}, + annotation = {ADS Bibcode: 2003ApJ...586..464B} +} + +@article{Barnes2007, + title = {Ages for {{Illustrative Field Stars Using Gyrochronology}}: {{Viability}}, {{Limitations}}, and {{Errors}}}, + shorttitle = {Ages for {{Illustrative Field Stars Using Gyrochronology}}}, + author = {Barnes, Sydney A.}, + year = {2007}, + month = nov, + journal = {\apj}, + volume = {669}, + pages = {1167--1189}, + issn = {0004-637X}, + doi = {10.1086/519295}, + urldate = {2023-04-27}, + abstract = {We here develop an improved way of using a rotating star as a clock, set it using the Sun, and demonstrate that it keeps time well. This technique, called gyrochronology, derives ages for low-mass main-sequence stars using only their rotation periods and colors. The technique is developed here and used to derive ages for illustrative groups of nearby field stars with measured rotation periods. We first demonstrate the reality of the interface sequence, the unifying feature of the rotational observations of cluster and field stars that makes the technique possible, and extend it beyond the proposal of Skumanich by specifying the mass dependence of rotation for these stars. We delineate which stars it cannot currently be used on. We then calibrate the age dependence using the Sun. The errors are propagated to understand their dependence on color and period. Representative age errors associated with the technique are estimated at \textasciitilde 15\% (plus possible systematic errors) for late F, G, K, and early M stars. Gyro ages for the Mount Wilson stars are shown to be in good agreement with chromospheric ages for all but the bluest stars, and probably superior. Gyro ages are then calculated for each of the active main-sequence field stars studied by Strassmeier and collaborators. These are shown to have a median age of 365 Myr. The sample of single field stars assembled by Pizzolato and collaborators is then assessed and shown to have gyro ages ranging from under 100 Myr to several Gyr, with a median age of 1.2 Gyr. Finally, we demonstrate that the individual components of the three wide binaries {$\xi$} Boo AB, 61 Cyg AB, and {$\alpha$} Cen AB yield substantially the same gyro ages.}, + keywords = {Astrophysics,Galaxy: Open Clusters and Associations: General,Stars: Activity,Stars: Evolution,Stars: Late-Type,Stars: Magnetic Fields,Stars: Rotation}, + annotation = {ADS Bibcode: 2007ApJ...669.1167B} +} + @article{Basu.Antia.ea1994, title = {Helioseismic Measurement of the Extent of Overshoot below the Solar Convection Zone}, author = {Basu, Sarbani and Antia, H. M. and Narasimha, D.}, @@ -1685,6 +1734,22 @@ @article{Davies.SilvaAguirre.ea2016 keywords = {asteroseismology,planetary systems,planets and satellites: fundamental parameters,stars: evolution,stars: fundamental parameters,stars: oscillations} } +@article{Deheuvels.Garcia.ea2012, + title = {Seismic {{Evidence}} for a {{Rapidly Rotating Core}} in a {{Lower-giant-branch Star Observed}} with {{Kepler}}}, + author = {Deheuvels, S. and Garc{\'i}a, R. A. and Chaplin, W. J. and Basu, S. and Antia, H. M. and Appourchaux, T. and Benomar, O. and Davies, G. R. and Elsworth, Y. and Gizon, L. and Goupil, M. J. and Reese, D. R. and Regulo, C. and Schou, J. and Stahn, T. and Casagrande, L. and {Christensen-Dalsgaard}, J. and Fischer, D. and Hekker, S. and Kjeldsen, H. and Mathur, S. and Mosser, B. and Pinsonneault, M. and Valenti, J. and Christiansen, J. L. and Kinemuchi, K. and Mullally, F.}, + year = {2012}, + month = sep, + journal = {\apj}, + volume = {756}, + pages = {19}, + issn = {0004-637X}, + doi = {10.1088/0004-637X/756/1/19}, + urldate = {2023-04-27}, + abstract = {Rotation is expected to have an important influence on the structure and the evolution of stars. However, the mechanisms of angular momentum transport in stars remain theoretically uncertain and very complex to take into account in stellar models. To achieve a better understanding of these processes, we desperately need observational constraints on the internal rotation of stars, which until very recently was restricted to the Sun. In this paper, we report the detection of mixed modes\textemdash i.e., modes that behave both as g modes in the core and as p modes in the envelope\textemdash in the spectrum of the early red giant KIC 7341231, which was observed during one year with the Kepler spacecraft. By performing an analysis of the oscillation spectrum of the star, we show that its non-radial modes are clearly split by stellar rotation and we are able to determine precisely the rotational splittings of 18 modes. We then find a stellar model that reproduces very well the observed atmospheric and seismic properties of the star. We use this model to perform inversions of the internal rotation profile of the star, which enables us to show that the core of the star is rotating at least five times faster than the envelope. This will shed new light on the processes of transport of angular momentum in stars. In particular, this result can be used to place constraints on the angular momentum coupling between the core and the envelope of early red giants, which could help us discriminate between the theories that have been proposed over the last few decades.}, + keywords = {Astrophysics - Solar and Stellar Astrophysics,stars: evolution,stars: individual: KIC 7341231,stars: interiors,stars: oscillations}, + annotation = {ADS Bibcode: 2012ApJ...756...19D} +} + @article{DeRidder.Barban.ea2009, title = {Non-Radial Oscillation Modes with Long Lifetimes in Giant Stars}, author = {De Ridder, Joris and Barban, Caroline and Baudin, Fr{\'e}d{\'e}ric and Carrier, Fabien and Hatzes, Artie P. and Hekker, Saskia and Kallinger, Thomas and Weiss, Werner W. and Baglin, Annie and Auvergne, Michel and Samadi, R{\'e}za and Barge, Pierre and Deleuil, Magali}, @@ -2217,6 +2282,22 @@ @article{Garcia.Ballot2019 keywords = {Asteroseismology,Solar analogs,Stellar oscillations} } +@article{Garcia.Ceillier.ea2014, + title = {Rotation and Magnetism of {{Kepler}} Pulsating Solar-like Stars. {{Towards}} Asteroseismically Calibrated Age-Rotation Relations}, + author = {Garc{\'i}a, R. A. and Ceillier, T. and Salabert, D. and Mathur, S. and {van Saders}, J. L. and Pinsonneault, M. and Ballot, J. and Beck, P. G. and Bloemen, S. and Campante, T. L. and Davies, G. R. and {do Nascimento}, Jr., J. -D. and Mathis, S. and Metcalfe, T. S. and Nielsen, M. B. and Su{\'a}rez, J. C. and Chaplin, W. J. and Jim{\'e}nez, A. and Karoff, C.}, + year = {2014}, + month = dec, + journal = {\aap}, + volume = {572}, + pages = {A34}, + issn = {0004-6361}, + doi = {10.1051/0004-6361/201423888}, + urldate = {2023-04-27}, + abstract = {Kepler ultra-high precision photometry of long and continuous observations provides a unique dataset in which surface rotation and variability can be studied for thousands of stars. Because many of these old field stars also have independently measured asteroseismic ages, measurements of rotation and activity are particularly interesting in the context of age-rotation-activity relations. In particular, age-rotation relations generally lack good calibrators at old ages, a problem that this Kepler sample of old-field stars is uniquely suited to address. We study the surface rotation and photometric magnetic activity of a subset of 540 solar-like stars on the main-sequence and the subgiant branch for which stellar pulsations have been measured. The rotation period was determined by comparing the results from two different analysis methods: i) the projection onto the frequency domain of the time-period analysis, and ii) the autocorrelation function of the light curves. Reliable surface rotation rates were then extracted by comparing the results from two different sets of calibrated data and from the two complementary analyses. General photometric levels of magnetic activity in this sample of stars were also extracted by using a photometric activity index, which takes into account the rotation period of the stars. We report rotation periods for 310 out of 540 targets (excluding known binaries and candidate planet-host stars); our measurements span a range of 1 to 100 days. The photometric magnetic activity levels of these stars were computed, and for 61.5\% of the dwarfs, this level is similar to the range, from minimum to maximum, of the solar magnetic activity. We demonstrate that hot dwarfs, cool dwarfs, and subgiants have very different rotation-age relationships, highlighting the importance of separating out distinct populations when interpreting stellar rotation periods. Our sample of cool dwarf stars with age and metallicity data of the highest quality is consistent with gyrochronology relations reported in the literature. Full Table 3 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/572/A34}, + keywords = {asteroseismology,Astrophysics - Solar and Stellar Astrophysics,stars: activity,stars: evolution,stars: oscillations,stars: rotation,stars: solar-type}, + annotation = {ADS Bibcode: 2014A\&A...572A..34G} +} + @article{Garcia.Mathur.ea2010, title = {{{CoRoT Reveals}} a {{Magnetic Activity Cycle}} in a {{Sun-Like Star}}}, author = {Garc{\'i}a, Rafael A. and Mathur, Savita and Salabert, David and Ballot, J{\'e}r{\^o}me and R{\'e}gulo, Clara and Metcalfe, Travis S. and Baglin, Annie}, @@ -3870,6 +3951,23 @@ @article{Ludwig.Freytag.ea1999 keywords = {CONVECTION,HYDRODYNAMICS,STARS: EVOLUTION,STARS: LATE-TYPE} } +@article{Lund.Miesch.ea2014, + title = {Differential {{Rotation}} in {{Main-sequence Solar-like Stars}}: {{Qualitative Inference}} from {{Asteroseismic Data}}}, + shorttitle = {Differential {{Rotation}} in {{Main-sequence Solar-like Stars}}}, + author = {Lund, Mikkel N. and Miesch, Mark S. and {Christensen-Dalsgaard}, J{\o}rgen}, + year = {2014}, + month = aug, + journal = {\apj}, + volume = {790}, + pages = {121}, + issn = {0004-637X}, + doi = {10.1088/0004-637X/790/2/121}, + urldate = {2023-04-27}, + abstract = {Understanding differential rotation of Sun-like stars is of great importance for insight into the angular momentum transport in these stars. One means of gaining such information is that of asteroseismology. By a forward modeling approach we analyze in a qualitative manner the impact of different differential rotation profiles on the splittings of p-mode oscillation frequencies. The optimum modes for inference on differential rotation are identified along with the best value of the stellar inclination angle. We find that in general it is not likely that asteroseismology can be used to make an unambiguous distinction between a rotation profile such as a conical Sun-like profile and a cylindrical profile. In addition, it seems unlikely that asteroseismology of Sun-like stars will result in inferences on the radial profile of the differential rotation, such as can be done for red giants. At best, one could possibly obtain the sign of the radial differential rotation gradient. Measurements of the extent of the latitudinal differential from frequency splitting are, however, more promising. One very interesting aspect that could likely be tested from frequency splittings is whether the differential rotation is solar-like or anti-solar-like in nature, in the sense that a solar-like profile has an equator rotating faster than the poles.}, + keywords = {asteroseismology,Astrophysics - Solar and Stellar Astrophysics,methods: analytical,stars: oscillations,stars: rotation,stars: solar-type}, + annotation = {ADS Bibcode: 2014ApJ...790..121L} +} + @article{Lund.Reese2018, title = {Tutorial: {{Asteroseismic Stellar Modelling}} with {{AIMS}}}, shorttitle = {Tutorial}, @@ -6145,6 +6243,22 @@ @article{VanderPlas2018 keywords = {methods: data analysis,methods: statistical} } +@article{vanSaders.Ceillier.ea2016, + title = {Weakened Magnetic Braking as the Origin of Anomalously Rapid Rotation in Old Field Stars}, + author = {{van Saders}, Jennifer L. and Ceillier, Tugdual and Metcalfe, Travis S. and Silva Aguirre, Victor and Pinsonneault, Marc H. and Garc{\'i}a, Rafael A. and Mathur, Savita and Davies, Guy R.}, + year = {2016}, + month = jan, + journal = {\nat}, + volume = {529}, + pages = {181--184}, + issn = {0028-0836}, + doi = {10.1038/nature16168}, + urldate = {2023-04-27}, + abstract = {A knowledge of stellar ages is crucial for our understanding of many astrophysical phenomena, and yet ages can be difficult to determine. As they become older, stars lose mass and angular momentum, resulting in an observed slowdown in surface rotation. The technique of `gyrochronology' uses the rotation period of a star to calculate its age. However, stars of known age must be used for calibration, and, until recently, the approach was untested for old stars (older than 1\,gigayear, Gyr). Rotation periods are now known for stars in an open cluster of intermediate age (NGC 6819; 2.5\,Gyr old), and for old field stars whose ages have been determined with asteroseismology. The data for the cluster agree with previous period-age relations, but these relations fail to describe the asteroseismic sample. Here we report stellar evolutionary modelling, and confirm the presence of unexpectedly rapid rotation in stars that are more evolved than the Sun. We demonstrate that models that incorporate dramatically weakened magnetic braking for old stars can\textemdash unlike existing models\textemdash reproduce both the asteroseismic and the cluster data. Our findings might suggest a fundamental change in the nature of ageing stellar dynamos, with the Sun being close to the critical transition to much weaker magnetized winds. This weakened braking limits the diagnostic power of gyrochronology for those stars that are more than halfway through their main-sequence lifetimes.}, + keywords = {Astrophysics - Solar and Stellar Astrophysics}, + annotation = {ADS Bibcode: 2016Natur.529..181V} +} + @article{Verma.Faria.ea2014, title = {Asteroseismic {{Estimate}} of {{Helium Abundance}} of a {{Solar Analog Binary System}}}, author = {Verma, Kuldeep and Faria, Jo{\~a}o P. and Antia, H. M. and Basu, Sarbani and Mazumdar, Anwesh and Monteiro, M{\'a}rio J. P. F. G. and Appourchaux, Thierry and Chaplin, William J. and Garc{\'i}a, Rafael A. and Metcalfe, Travis S.}, From 2df79bf3c90dcff4c7ca8676a6fb1f653274a671 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Thu, 27 Apr 2023 18:03:37 +0100 Subject: [PATCH 36/50] More grammatical checks --- appendices/lyttle21.tex | 12 ++++++------ chapters/conclusion.tex | 2 +- chapters/glitch-gp.tex | 16 ++++++++-------- 3 files changed, 15 insertions(+), 15 deletions(-) diff --git a/appendices/lyttle21.tex b/appendices/lyttle21.tex index 67f52d0..d43114f 100644 --- a/appendices/lyttle21.tex +++ b/appendices/lyttle21.tex @@ -24,7 +24,7 @@ \section{Artificial Neural Network}\label{sec:ann} Once we constructed our grid of models, we needed a way in which we could continuously sample the grid for use in our statistical model. We opted to train an Artificial Neural Network (ANN). The ANN is advantageous over interpolation because it scales well with dimensionality, training and evaluation is fast, and gradient evaluation is easy due to its roots in linear algebra \citep{Haykin2007}. We trained an ANN on the data generated by the grid of stellar models to map fundamentals to observables. Firstly, we split the grid into a \emph{train} and \emph{validation} dataset for tuning the ANN, as described in Appendix \ref{sec:train}. We then tested a multitude of ANN configurations and training data inputs, repeatedly evaluating them with the validation dataset in Appendix \ref{sec:opt}. In Appendix \ref{sec:test}, we reserved a set of randomly generated, off-grid stellar models as our final \emph{test} dataset to evaluate the approximation ability of the best-performing ANN independently from our train and validation data. Here, we briefly describe the theory and motivation behind the ANN. -An ANN is a network of artificial \emph{neurons} which each transform some input vector, $\boldsymbol{x}$ based on trainable weights, $\boldsymbol{w}$ and a bias, $b$. The weights are represented by the connections between neurons and the bias is a unique scalar associated with each neuron. A multi-layered ANN is where neurons are arranged into a series of layers such that any neuron in layer $j-1$ is connected to at least one of the neurons in layer $j$. +An ANN is a network of artificial \emph{neurons} which each transform an input vector, $\boldsymbol{x}$ based on trainable weights, $\boldsymbol{w}$ and a bias, $b$. The weights are represented by the connections between neurons and the bias is a unique scalar associated with each neuron. A multi-layered ANN is where neurons are arranged into a series of layers such that any neuron in layer $j-1$ is connected to at least one of the neurons in layer $j$. \begin{figure} \centering @@ -55,7 +55,7 @@ \section{Artificial Neural Network}\label{sec:ann} To fit the ANN, we used a set of training data, $\boldsymbol{\mathbb{D}}_\mathrm{train} = \{\boldsymbol{\mathbb{X}}_i, \boldsymbol{\mathbb{Y}}_i\}_{i=1}^{N_\mathrm{train}}$ comprising $N_\mathrm{train}$ input-output pairs. We split the training data into pseudo-random batches, $\boldsymbol{\mathbb{D}}_\mathrm{batch}$ because this has been shown to improve ANN stability and computational efficiency \citep{Masters.Luschi2018}. The set of predictions made for each batch is evaluated using a \emph{loss} function which primarily comprises an error function, $E(\boldsymbol{\mathbb{D}}_\mathrm{batch})$ to quantify the difference between the training data outputs ($\boldsymbol{\mathbb{Y}}$) and predictions ($\widetilde{\boldsymbol{\mathbb{Y}}}$). We also considered an additional term to the loss called \emph{regularisation} which helps reduce over-fitting \citep{Goodfellow.Bengio.ea2016}. During fitting, the weights are updated after each batch using an algorithm called the \emph{optimizer}, back-propagating the error with the goal of minimising the loss such that $\widetilde{\boldsymbol{\mathbb{Y}}} \approx \boldsymbol{\mathbb{Y}}$ \citep[see e.g.][]{Rumelhart.Hinton.ea1986}. -We trained the ANN using \textsc{TensorFlow} \citep{Abadi.Barham.ea2016}. We varied the architecture, number of batches, choice of loss function, optimizer, and regularisation during the optimisation phase. For each set of ANN parameters, we initialised the ANN with a random set of weights and biases and minimized the loss over a given number of \emph{epochs}. An epoch is defined as one iteration through the entire training dataset, $\boldsymbol{\mathbb{D}}_\mathrm{train}$. We tracked the loss for each ANN using an independent validation dataset to determine the most effective choice of ANN parameters (see Appendix \ref{sec:opt}). +We trained the ANN using \textsc{TensorFlow} \citep{Abadi.Barham.ea2016}. We varied the architecture, number of batches, choice of loss function, optimizer, and regularisation during the optimisation phase. For each set of ANN parameters, we initialised the ANN with a random set of weights and biases and minimised the loss over a given number of \emph{epochs}. An epoch is defined as one iteration through the entire training dataset, $\boldsymbol{\mathbb{D}}_\mathrm{train}$. We tracked the loss for each ANN using an independent validation dataset to determine the most effective choice of ANN parameters (see Appendix \ref{sec:opt}). \subsection{Train, Validation, and Test Data}\label{sec:train} @@ -73,7 +73,7 @@ \subsection{Tuning}\label{sec:opt} %%%%%%%%%%%%%%%%%% OPTIMIZATION %%%%%%%%%%%%%%%%%% -We trained an ANN to reproduce stellar observables according to our choice of physics with greater accuracy than typical observational precisions. We experimented with a variety of ANN parameter choices, such as the architecture, activation function, optimization algorithm, and loss function. We tuned the ANN parameters by varying them in both a grid-based and heuristic approach, each time evaluating the accuracy using the validation dataset. +We trained an ANN to reproduce stellar observables according to our choice of physics with greater accuracy than typical observational precisions. We experimented with a variety of ANN parameter choices, such as the architecture, activation function, optimisation algorithm, and loss function. We tuned the ANN parameters by varying them in both a grid-based and heuristic approach, each time evaluating the accuracy using the validation dataset. During initial tuning, we found that having stellar age as an input was unstable, because it varied heavily with the other input parameters. We mitigated this by introducing an input to describe the fraction of time a star had spent in a given evolutionary phase, $f_\mathrm{evol}$. % @@ -94,7 +94,7 @@ \subsection{Tuning}\label{sec:opt} We also observed that the ANN trained poorly in areas with a high rate of change in observables, likely because of poor grid coverage in those areas. To bias training to such areas, we calculated the gradient in $\teff$ and $\log g$ between each point for each stellar evolutionary track and used them as optional weights to the loss during tuning. These weights multiplied the difference between the ANN prediction and the training data in our chosen loss function. -After preliminary tuning, we chose the ANN input and output parameters to be $\boldsymbol{\mathbb{X}} = \{f_\mathrm{evol}, M, \mlt, Y_\mathrm{init}, Z_\mathrm{init}\}$ and $\boldsymbol{\mathbb{Y}} = \{\log(\tau), \teff, R, \dnu, \metallicity_\mathrm{surf}\}$ respectively. A generalised form of our neural network is depicted in Fig. \ref{fig:net}. The inputs corresponded to initial conditions in the stellar modelling code and the outputs corresponded to surface conditions throughout the lifetime of the star, with the exception of age which is mapped from $f_\mathrm{evol}$. +After preliminary tuning, we chose the ANN input and output parameters to be $\boldsymbol{\mathbb{X}} = \{f_\mathrm{evol}, M, \mlt, Y_\mathrm{init}, Z_\mathrm{init}\}$ and $\boldsymbol{\mathbb{Y}} = \{\log(\tau), \teff, R, \dnu, \metallicity_\mathrm{surf}\}$ respectively. A generalised form of our neural network is depicted in Fig. \ref{fig:net}. The inputs corresponded to initial conditions in the stellar modelling code and the outputs corresponded to surface conditions throughout the lifetime of the star, except for age which is mapped from $f_\mathrm{evol}$. We standardised the training dataset by subtracting the median, $\mu_{1/2}$ and dividing by the standard deviation, $\sigma$ for each input and output parameter. We found that the ANN performed better when the training data was scaled in this way as opposed to no scaling at all. We present the parameters used to standardise the training dataset in Table \ref{tab:std}. @@ -109,7 +109,7 @@ \subsection{Tuning}\label{sec:opt} We evaluated the performance of three activation functions: the hyperbolic-tangent, the rectified linear unit \citep[ReLU;][]{Hahnloser.Sarpeshkar.ea2000, Glorot.Bordes.ea2011} and the exponential linear unit \citep[ELU;][]{Clevert.Unterthiner.ea2015}. Although the ReLU activation function out-performed the other two in speed and accuracy, the resulting ANN output was not smooth. The discontinuity in the ReLU function, $f(x) = \max(0, x)$ in turn caused the output to be discontinuous. This made the ANN difficult to sample for our choice of statistical model (see Section \ref{sec:hbm}). Out of the remaining activation functions, ELU performed the best, providing a smooth output which was well-suited to our probabilistic sampling methods. -We compared the performance of two optimisers: Adam \citep{Kingma.Ba2014} and stochastic gradient descent \citep[SGD; see e.g.][]{Ruder2016} with and without momentum \citep{Qian1999}. Both optimizers required a choice of \emph{learning rate} which determined the rate at which the weights were adjusted during training. We found that Adam performed well but the validation loss was noisy as a function of epochs as it struggled to converge. The SGD optimizer was less noisy than Adam, but it was difficult to tune the learning rate. However, SGD with momentum allowed for more adaptive weight updates and out-performed the other configurations. +We compared the performance of two optimisers: Adam \citep{Kingma.Ba2014} and stochastic gradient descent \citep[SGD; see e.g.][]{Ruder2016} with and without momentum \citep{Qian1999}. Both optimizers required a choice of \emph{learning rate} which determined the rate at which the weights were adjusted during training. We found that Adam performed well, but the validation loss was noisy as a function of epochs as it struggled to converge. The SGD optimizer was less noisy than Adam, but it was difficult to tune the learning rate. However, SGD with momentum allowed for more adaptive weight updates and out-performed the other configurations. There are several ways to reduce over-fitting, from minimising the complexity of the architecture, to increasing the size and coverage of the training dataset. One alternative is to introduce weight regularisation. So-called L2 regularisation adds a term, $\sim \lambda_k \sum_i w_{i, k}^2$ to the loss function for a given hidden layer, $k$ which acts to keep the weights small. We varied the magnitude of $\lambda_k$ and found that if it was too large it would dominate the loss function, but if it was too small then performance on the validation dataset was poorer. @@ -148,7 +148,7 @@ \subsection{Testing}\label{sec:test} \input{tables/test_random.tex} \end{table} -To represent the accuracy of the ANN, we present the median, $\mu_{1/2}$ and MAD estimator, $\sigma_\mathrm{MAD} = 1.4826\cdot\mathrm{median}(|E(x) - \mu_{1/2}|)$ of the error ($E(x)$) in Table \ref{tab:test}. The median is close to zero for all parameters, showing little systematic bias in the ANN. The MAD is also lower than observational uncertainties quoted in Section \ref{sec:data}. The spread in error for $\dnu$ of $\SI{0.06}{\mu\Hz}$ is comparable to a small number of observations with the best signal-to-noise. However, the error in $\dnu$ predictions is also comparable to other systematic uncertainties in $\dnu$ discussed in Section \ref{subsec:seismo_model}. Therefore, a robust model which takes account of systematic uncertainties pertaining to $\dnu$, including those from the ANN, will be explored in future work (Carboneau et al. in preparation). +To represent the accuracy of the ANN, we present the median, $\mu_{1/2}$ and MAD estimator, $\sigma_\mathrm{MAD} = 1.4826\cdot\mathrm{median}(|E(x) - \mu_{1/2}|)$ of the error ($E(x)$) in Table \ref{tab:test}. The median is close to zero for all parameters, showing little systematic bias in the ANN. The MAD is also lower than observational uncertainties quoted in Section \ref{sec:data}. The spread in error for $\dnu$ of $\SI{0.06}{\mu\Hz}$ is comparable to that of observations with the best signal-to-noise. However, the error in $\dnu$ predictions is also comparable to other systematic uncertainties in $\dnu$ discussed in Section \ref{subsec:seismo_model}. Therefore, a robust model which takes account of systematic uncertainties pertaining to $\dnu$, including those from the ANN, will be explored in future work (Carboneau et al. in preparation). \section{Prior Distributions}\label{sec:beta} diff --git a/chapters/conclusion.tex b/chapters/conclusion.tex index 4554ae6..66601e2 100644 --- a/chapters/conclusion.tex +++ b/chapters/conclusion.tex @@ -30,7 +30,7 @@ \section*{Summary} \section*{Improving the Hierarchical Model} -The helium glitch parameters for a given star correlate with its near-surface helium abundance. Therefore, a natural next step would be to include helium glitch parameters as an additional observable in our HBM. Our GP glitch model can be applied to both observed and modelled mode frequencies, providing extra parameters to include in our stellar model emulator. Adding these should improve inference of helium abundance for stars with individual modes identified \citep[e.g.][]{Davies.SilvaAguirre.ea2016,Lund.SilvaAguirre.ea2017}. Since our HBM simultaneously models the population distribution of helium, even a small number of stars with good helium constraint will in-turn improve helium estimates for the rest of the population. This introduces the possibility of testing more complex models of helium enrichment. +The helium glitch parameters for a given star correlate with its near-surface helium abundance. Therefore, a natural next step would be to include helium glitch parameters as an additional observable in our HBM. Our GP glitch model can be applied to both observed and modelled mode frequencies, providing extra parameters to include in our stellar model emulator. Adding these should improve inference of helium abundance for stars with individual modes identified \citep[e.g.][]{Davies.SilvaAguirre.ea2016,Lund.SilvaAguirre.ea2017}. Since our HBM simultaneously models the population distribution of helium, even a few stars with good helium constraint will in-turn improve helium estimates for the rest of the population. This introduces the possibility of testing more complex models of helium enrichment. We also expect the HBM to scale to red giant solar-like oscillators for which observations are abundant. We trained the emulator on a grid of stellar models from the zero-age main sequence to the base of the red giant branch for masses from \SIrange{0.8}{1.2}{\solarmass}. The upper mass limit was motivated by the diminishing outer convective envelope (responsible for driving solar-like oscillators) in these stars on the main sequence. However, extending the emulator to model red giant solar-like oscillators would require expanding the grid up to \(\sim\SI{2.0}{\solarmass}\). We would need to compute thrice as many evolutionary tracks and evolve existing models further. Stars with \(M \gtrsim \SI{1.1}{\solarmass}\) have a convective core on the main sequence which introduces an additional model uncertainty from mixing at its boundary. Parametrising this process would further multiply the number of input tracks, increasing dimensionality and grid computation time. Therefore, we should research ways of selectively computing stellar models. For example, we could upsample the grid \citep[e.g.][]{Li.Davies.ea2022} where the neural network error is large. diff --git a/chapters/glitch-gp.tex b/chapters/glitch-gp.tex index 4d12e6b..b557321 100644 --- a/chapters/glitch-gp.tex +++ b/chapters/glitch-gp.tex @@ -63,7 +63,7 @@ \section{Data}\label{sec:glitch-data} \label{fig:glitch-test-obs} \end{figure} -\paragraph{Test Star} We created three sets of test data for worst-, better-, and best-case scenarios using stellar model S (defined in Section \ref{sec:model-s}). We recall that model S is similar to the Sun, with surface parameters of \(\teff = \SI{5682}{\kelvin}\), \(\log g = 4.426\) and \([\mathrm{Fe/H}] = 0.03\), and bulk parameters of \(M = \SI{1.00}{\solarmass}\), \(R = \SI{1.01}{\solarmass}\) and \(t_\star = \SI{4.07}{\giga\year}\). We calculated radial order modes (\(l=0\)) for the test star using the \textsc{GYRE} oscillation code \citep{Townsend.Teitler2013}. We then selected different numbers of modes (\(N\)) symmetrically about a reference frequency, \(\nu_\mathrm{ref} = \SI{2900}{\micro\hertz}\) (close to the expected frequency at maximum power of the star). For each test case, we added differing amounts of Gaussian noise (scaled by \(\sigma_\obs\)) to the frequencies. The parameters and mode frequencies for each test case are shown in Table \ref{tab:glitch-obs}. We also plotted the modes on an echelle diagram in Figure \ref{fig:glitch-test-obs}. We can see the effect of the helium glitch in the echelle diagrams, where there is a large `wiggle' at low frequency. The glitch signature is visible in the better case and clearest in the best-case scenario. +\paragraph{Test Star} We created three sets of test data for worst-, better-, and best-case scenarios using stellar model S (defined in Section \ref{sec:model-s}). Model S is similar to the Sun, with surface parameters of \(\teff = \SI{5682}{\kelvin}\), \(\log g = 4.426\) and \([\mathrm{Fe/H}] = 0.03\), and bulk parameters of \(M = \SI{1.00}{\solarmass}\), \(R = \SI{1.01}{\solarmass}\) and \(t_\star = \SI{4.07}{\giga\year}\). We calculated radial order modes (\(l=0\)) for the test star using the \textsc{GYRE} oscillation code \citep{Townsend.Teitler2013}. Then, we selected different numbers of modes (\(N\)) symmetrically about a reference frequency, \(\nu_\mathrm{ref} = \SI{2900}{\micro\hertz}\) (close to the expected frequency at maximum power of the star). For each test case, we added differing amounts of Gaussian noise (scaled by \(\sigma_\obs\)) to the frequencies. The parameters and mode frequencies for each test case are shown in Table \ref{tab:glitch-obs}. We also plotted the modes on an echelle diagram in Figure \ref{fig:glitch-test-obs}. We can see the effect of the helium glitch in the echelle diagrams, where there is a large `wiggle' at low frequency. The glitch signature is visible in the better case and clearest in the best-case scenario. \paragraph{16 Cyg A} We used the asteroseismic benchmark star 16 Cyg A as an example real star to test both methods. We adopted values for 16 radial mode frequencies identified by \citet{Lund.SilvaAguirre.ea2017} using observations from \emph{Kepler} \citep[][KIC 12069424]{Borucki.Koch.ea2010}. The mode frequencies and their associated uncertainties are given in Table \ref{tab:glitch-obs}. The glitch has been previously studied for 16 Cyg A with its binary companion 16 Cyg B in \citet{Verma.Faria.ea2014}, making it a useful subject for comparison. Similarly to model S, this target is a solar analogue. However, it is slightly hotter, more evolved and more metal-rich, with \(\teff = \SI{5825(50)}{\kelvin}\), \(\log g = \SI{4.33(7)}{\dex}\) and \([\mathrm{Fe/H}] = \SI{0.10(3)}{\dex}\) \citep{Ramirez.Melendez.ea2009}. Its bulk stellar parameters are \(M \approx \SI{1.1}{\solarmass}\), \(R \approx \SI{1.2}{\solarradius}\) and \(t_\star \approx \SI{7}{\giga\year}\) \citep{SilvaAguirre.Lund.ea2017}. @@ -79,12 +79,12 @@ \subsection{The V19 Method} f_A(n) = \tilde{f}_A(n) + \delta\nu_\helium + \delta\nu_\bcz; \quad \tilde{f}_A(n) = \sum_{k=0}^{4} b_k n^k, \end{equation} % -\sloppy where \(b_k \equiv a_{0k} \nu_0\) from Equation \ref{eq:poly}. The model parameters are given by \(\vect{\theta}_A = (b_0, \dots, b_4, a_\helium, \beta_\helium, \tau_\helium, \phi_\helium, a_\bcz, \tau_\bcz, \phi_\bcz)\), where the glitch amplitude parameters are modified to include \(\nu_0\) such that \(a_i \equiv \alpha_i\nu_0\). The \(\nu_0\) parameter is not explicitly included in the \citetalias{Verma.Raodeo.ea2019} model, but we find it useful to keep the scaling in mind and we include \(\nu_0\) in the GP model. +\sloppy where \(b_k \equiv a_{0k} \nu_0\) from Equation \ref{eq:poly}. The model parameters are given by \(\vect{\theta}_A = (b_0, \dots, b_4, a_\helium, \beta_\helium, \tau_\helium, \phi_\helium, a_\bcz, \tau_\bcz, \phi_\bcz)\), where the glitch amplitude parameters are modified to include \(\nu_0\) such that \(a_i \equiv \alpha_i\nu_0\). The \(\nu_0\) parameter is not explicitly included in the \citetalias{Verma.Raodeo.ea2019} model, but we find it useful to keep the scaling in mind, and we include \(\nu_0\) in the GP model. The model parameters are optimised by minimising a \(\chi^2\) cost function with a regularisation term, % \begin{equation} - \chi^2 = \sum_n \left[ \frac{\nu_n^\obs - f_{A}(n)}{\sigma_n^\obs} \right]^2 + \lambda^2 \sum_n \left[ \frac{\dd^3}{\dd n^3} \tilde{f}_A(n)\right]^2. + \chi^2 = \sum_n \left[ \frac{\nu_n^\obs - f_{A}(n)}{\sigma_n^\obs} \right]^2 + \lambda^2 \sum_n \left[ \frac{\dd^3}{\dd n^3} \tilde{f}_A(n)\right]^2, \end{equation} % where \(\nu_n^\obs\) and \(\sigma_n^\obs\) are the observed mode and its uncertainty at radial order \(n\), and \(\lambda\) is the regularisation parameter. The regularisation was introduced to avoid the polynomial over-fitting and absorbing the glitch terms. @@ -169,7 +169,7 @@ \subsection{The GP Method} % \phi_\helium, \phi_\bcz \sim \mathcal{U}\left(0, 2\pi\right), \end{gather*} % -centred on \(\overline{\nu}_0\) and \(\overline{\varepsilon}\) and scaled by \(s_{\nu_0}\) and \(s_\varepsilon\). For example, the location and scale parameters could come from global estimates of \(\langle\Delta\nu_n\rangle\) or a linear fit to the modes. For the test stars, we determined \(\overline{\nu}_0\) and \(\overline{\varepsilon}\) from a linear fit to the true mode frequencies and added representative uncertainties of 10 and 5 per cent respectively. For 16 Cyg A, we used measurements of \(\langle\Delta\nu_n\rangle\) and \(\nu_{\max}\) from \citet{Lund.SilvaAguirre.ea2017} to estimate \(\overline{\nu}_0\) and \(\overline{\varepsilon}\). +centred on \(\overline{\nu}_0\) and \(\overline{\varepsilon}\) and scaled by \(s_{\nu_0}\) and \(s_\varepsilon\). For example, the location and scale parameters could come from global estimates of \(\langle\Delta\nu_n\rangle\) or a linear fit to the modes. We determined \(\overline{\nu}_0\) and \(\overline{\varepsilon}\) for the test stars from a linear fit to the true mode frequencies, and added representative uncertainties of 10 and 5 per cent respectively. For 16 Cyg A, we used measurements of \(\langle\Delta\nu_n\rangle\) and \(\nu_{\max}\) from \citet{Lund.SilvaAguirre.ea2017} to estimate \(\overline{\nu}_0\) and \(\overline{\varepsilon}\). % Priors for the following parameters follow a log-normal distribution to ensure they are positive. We also exploit the property that the scale of a normal distribution in natural log-space is approximately the scale in real-space as a fraction of the distribution mean. @@ -250,7 +250,7 @@ \subsection{Test Star} In Figure \ref{fig:glitch-test-tau}, we plotted posterior distributions for the glitch acoustic depths, \(\tau_\helium\) and \(\tau_\bcz\), and compared them to the sound speed gradient of the test star from Figure \ref{fig:sound-speed-gradient}. We expect the acoustic depths to approximately line up with the sharp structural changes. For the worst case, both methods gave broad distributions for the acoustic depths, compatible with their respective initial guesses and priors. The \citetalias{Verma.Raodeo.ea2019} method initial guesses appeared to underestimate \(\tau_\bcz\), whereas the GP method prior was broad enough to encompass a wide range of possible \(\tau_\bcz\). In the better and best cases, we found that the \citetalias{Verma.Raodeo.ea2019} solutions were multimodal. For example, the better-case found solutions for \(\tau_\helium\) far deeper into the star than we would expect, at around \SI{1500}{\second} and \SI{2500}{\second}. % As predicted by \citet{Houdek.Gough2007} and shown in \citet{Verma.Faria.ea2014}... This is because \(\delta\nu_\helium\) does not include the smaller glitch component due to the first ionisation of helium, located at a smaller \(\tau\). -The the values for \(\tau_\helium\) obtained were under-predicted compared to the location of the trough due to the second ionisation of helium. We can see this for the best star fit with the \citetalias{Verma.Raodeo.ea2019} method which finds \(\tau_\helium = \SI{619(15)}{\second}\). The depression in \(\gamma\) due to He\,\textsc{ii} ionisation in the respective stellar model is located at \SI{733}{\second}. On the other hand, the GP method was closer with \(\tau_\helium = \SI{696(19)}{\second}\). +The values for \(\tau_\helium\) obtained were under-predicted compared to the location of the trough due to the second ionisation of helium. We can see this for the best star fit with the \citetalias{Verma.Raodeo.ea2019} method which finds \(\tau_\helium = \SI{619(15)}{\second}\). The depression in \(\gamma\) due to He\,\textsc{ii} ionisation in the respective stellar model is located at \SI{733}{\second}. On the other hand, the GP method was closer with \(\tau_\helium = \SI{696(19)}{\second}\). \begin{figure}[tb] \centering @@ -289,12 +289,12 @@ \section{Discussion}\label{sec:glitch-disc} \label{fig:best-smooth} \end{figure} -Throughout this work, we found a smaller \(\delta\nu\) amplitude at low frequency with the GP method than with the \citetalias{Verma.Raodeo.ea2019} method. This was particularly visible in the best case and in 16 Cyg A. We expected this was a result of the different smooth background models. In Figure \ref{fig:best-smooth} we plotted the smooth component of each model extended to lower order, unobserved modes. We found the GP background component had a turning point at \(\nu \approx \SI{1900}{\micro\hertz}\) which was higher than the \citetalias{Verma.Raodeo.ea2019} method at \(\nu \approx \SI{1500}{\micro\hertz}\). The smooth component of the \citetalias{Verma.Raodeo.ea2019} method was confidently incorrect outside of the observed frequencies. Conversely, the GP method predicted closer to the truth with increasing uncertainty further from the observations. It appeared that the GP provided a more accurate representation of the underlying function than the polynomial. +Throughout this work, we found a smaller \(\delta\nu\) amplitude at low frequency with the GP method than with the \citetalias{Verma.Raodeo.ea2019} method. This was particularly visible in the best case and in 16 Cyg A. We expected this was a result of the different smooth background models. In Figure \ref{fig:best-smooth} we plotted the smooth component of each model extended to lower order, unobserved modes. We found the GP background component had a turning point at \(\nu \approx \SI{1900}{\micro\hertz}\) which was higher than the \citetalias{Verma.Raodeo.ea2019} method at \(\nu \approx \SI{1500}{\micro\hertz}\). The smooth component of the \citetalias{Verma.Raodeo.ea2019} method was confidently incorrect outside the observed frequencies. Conversely, the GP method predicted closer to the truth with increasing uncertainty further from the observations. It appeared that the GP provided a more accurate representation of the underlying function than the polynomial. \begin{figure}[!tb] \centering \includegraphics[trim={0.4in 0.2in 0 0},clip]{figures/glitch-res.pdf} - \caption[The difference between the model without the glitch components predicted by theV19 method and the GP method.]{The difference between the model without the glitch components (\(f(n) - \delta\nu\)) predicted by the \citetalias{Verma.Raodeo.ea2019} method (A) and the GP method (B). The glitch components from the GP method are plot in gray for context. The 68 per cent confidence region is shaded for each line.} + \caption[The difference between the model without the glitch components predicted by theV19 method and the GP method.]{The difference between the model without the glitch components (\(f(n) - \delta\nu\)) predicted by the \citetalias{Verma.Raodeo.ea2019} method (A) and the GP method (B). The glitch components from the GP method are plot in grey for context. The 68 per cent confidence region is shaded for each line.} \label{fig:smooth-res} \end{figure} @@ -313,7 +313,7 @@ \section{Discussion}\label{sec:glitch-disc} \section{Conclusion} We introduced a new method for modelling acoustic glitches in solar-like oscillators using a Gaussian Process. Testing the method on a model star, we found that it more accurately characterised the underlying, smoothly-varying functional form of the radial modes than the -\citetalias{Verma.Raodeo.ea2019} method. Furthermore, our method appeared able to absorb the glitch component from He\,\textsc{i} ionisation, for which the polynomial was not flexible enough. However, this raises the question of whether He\,\textsc{i} ionisation glitch should be included in the model. +\citetalias{Verma.Raodeo.ea2019} method. Furthermore, our method appeared able to absorb the glitch component from He\,\textsc{i} ionisation, for which the polynomial was not flexible enough. However, this questions whether He\,\textsc{i} ionisation glitch should be explicitly included in the model. Additionally, the GP method provided more believable uncertainties on the glitch parameters, whereas the \citetalias{Verma.Raodeo.ea2019} method was over-confident with the best data and under-confident with the worst. Robust uncertainties are important when using the results to make further inference about helium enrichment. In this case, the GP marginalised over correlated noise in the model, not possible with the polynomial in the \citetalias{Verma.Raodeo.ea2019} method. From 3ebccb7952d3893e7871746d34d55eddbc3e5420 Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Thu, 27 Apr 2023 23:05:38 +0100 Subject: [PATCH 37/50] Finish intro --- chapters/introduction.tex | 78 +++++++++++++++++++-------------------- 1 file changed, 39 insertions(+), 39 deletions(-) diff --git a/chapters/introduction.tex b/chapters/introduction.tex index 3d990c5..be52313 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -12,7 +12,7 @@ % \chapter[Introduction]{Introduction} -\textit{In this chapter, we introduce the current state of modelling stars with asteroseismology and the types of stars being studied in this work. We start with a brief history of understanding the stars spanning the last century. In Section \ref{sec:seismo}, we introduce asteroseismology of stars which oscillate like the Sun. Then, we provide examples of asteroseismology being used to model large samples of dwarf and subgiant stars in Section \ref{sec:many-stars}. Finally, we introduce the concept of modelling stars the `Bayesian way' with some examples of current methods and their limitations.} +\textit{In this chapter, I introduce the current state of modelling stars with asteroseismology and the types of stars being studied in this thesis. I start with a brief history of understanding the stars spanning the last century. In Section \ref{sec:seismo}, I introduce asteroseismology of stars which oscillate like the Sun. Then, I provide examples of asteroseismology being used to model large samples of dwarf and subgiant stars in Section \ref{sec:many-stars}. Finally, I introduce the concept of modelling stars the `Bayesian way' with some examples of current methods and their limitations.} % Since the late 19th century, astronomers have been trying to understand the physical structure and evolution of the Sun and other stars. They wanted to determine whether other stars were like the Sun and whether they changed with time. Answers to these questions could tell us where we came from and what the future holds for the solar system. While little was known of this at the time, astronomers started by gathering data in an effort to map the night sky. The invention of the spectrograph by \citet{Draper1874} allowed researchers to systematically classify stars by their brightness in different wavelengths of light \citep{Maury.Pickering1897}. This was an pivotal early step towards the large-scale stellar surveys we are used to today. @@ -20,9 +20,9 @@ \section{Understanding the Stars}\label{sec:stars} -Early efforts to understand the stars began by finding relations between their spectral classification and magnitude in a given photometric band on what was later called a Hertzsprung-Russell (HR) diagram \citep[e.g.][]{Russell1914}. An HR diagram shows the absolute magnitude (or luminosity, \(L\)) of a star against its spectral class (or effective temperature, \(\teff\)). Astronomers found that stars were not uniformly distributed on the HR diagram, but were instead grouped in distinct sequences. For example, the region where most stars were found was called the \emph{main sequence}. +Early efforts to understand the stars began by finding relations between their spectral classification and magnitude in different photometric bands. On what was later called a Hertzsprung-Russell (HR) diagram \citep[e.g.][]{Russell1914}, astronomers plot the absolute magnitude (or luminosity, \(L\)) of a star against its spectral class (or effective temperature, \(\teff\)). They found that stars were not uniformly distributed on the HR diagram, but were instead grouped in distinct sequences. For example, the narrow diagonal band where most stars were found was called the \emph{main sequence}. Astronomers wondered why stars adhered to these sequences and to what extent they evolved with time. -Initial insight into stellar evolution came about when astronomers studied open clusters on the HR diagram \citep{Trumpler1930}. These are groups of stars found at a similar distance and close together on the sky. Assuming clusters formed at the same time with similar chemical abundances, the only expected difference between stars is their mass and multiplicity. Using stars of known mass (e.g. from orbital solutions to binary systems) and luminosity functions, astronomers could trace lines of similar mass and composition from younger to older clusters \citep[e.g.][]{Sandage1957}. Coupled with early arguments from stellar physics \citep{Chandrasekhar1939}, scientists formed stellar evolutionary tracks --- the path a star takes on the HR diagram during its evolution. Deriving the stellar radius (\(R\)) from the relation \(L \propto R^2 \teff^4\), they inferred that stars started on the left-hand edge of the main sequence and became brighter and larger throughout most of their lifetime. At some point, stars would leave the main sequence, rapidly cool and expand, and then ascend a region of the HR diagram known as the \emph{red giant branch}. +Initial insight into stellar evolution came about when astronomers studied open clusters on the HR diagram \citep{Trumpler1930}. These are groups of stars found at a similar distance and close together on the sky. Assuming clusters formed at the same time with similar chemical abundances, the only expected difference between stars is their mass and multiplicity. Using stars of known mass (e.g. from orbital solutions to binary systems) and luminosity distributions, astronomers could trace lines of similar mass and composition from younger to older clusters \citep[e.g.][]{Sandage1957}. Coupled with early arguments from stellar physics \citep[e.g.][]{Chandrasekhar1939}, scientists formed stellar evolutionary tracks --- the path a star takes on the HR diagram during its evolution. Deriving the stellar radius (\(R\)) from the relation \(L \propto R^2 \teff^4\), they inferred that stars started life on the dimmer edge of the main sequence and became brighter and larger throughout most of their lifetime. At some point, stars would leave the main sequence, rapidly cool, expand, and then ascend a region of the HR diagram known as the \emph{red giant branch}. % \citet{Kuiper1938} found an empirical mass-luminosity relation. @@ -32,15 +32,15 @@ \section{Understanding the Stars}\label{sec:stars} % The luminosity and effective temperature could be estimated from the magnitude and colour of the stars. From luminosity and temperature, we could derive the radius of stars. Early stellar mass estimates came from visual and spectroscopic binaries. Spectroscopy provides abundances of chemical species ionised in the stellar atmosphere. However, except for the Sun, stellar age and helium abundance has no model-independence. The latter ionisations at temperatures and densities higher than the surface of stars like the Sun. -The chemical composition, energy sources and evolutionary physics of stars were still uncertain. Spectroscopy revealed relative abundances of elements excited in stellar atmospheres, but determining their absolute abundances was difficult. \citet{Payne1925} proposed that stars comprised mostly hydrogen and helium, later reinforced by estimates of helium content in the Sun \citep[e.g.][]{Schwarzschild1946}. The idea was radical at the time because it did not match the abundance of elements found on Earth. Meanwhile, advancements in nuclear science spawned the theory of stellar nucleosynthesis \citep{Hoyle1946}. Stars produce elements heavier than hydrogen and helium through nuclear fusion reactions. This discovery explained the production of many elements in the universe, tying the formation and evolution of stars to planetary formation and the conditions for life in the universe. +The chemical composition, energy sources and evolutionary physics of stars were still uncertain. Spectroscopy revealed relative abundances of elements excited in stellar atmospheres, but it was difficult to determine their absolute abundances. \citet{Payne1925} proposed that stars comprised mostly hydrogen and helium, later reinforced by estimates of helium content in the Sun \citep[e.g.][]{Schwarzschild1946}. The idea was radical at the time because it did not match the abundance of elements found on Earth. Meanwhile, advancements in nuclear science spawned the theory of stellar nucleosynthesis \citep{Hoyle1946}. Stars produce elements heavier than hydrogen and helium through nuclear fusion reactions. This discovery explained the production of many elements in the universe, tying the formation and evolution of stars to planetary formation and the conditions for life in the universe. -In the second half of the 20th century, the theory of stellar evolution advanced. With this came computational methods for simulating stars \citep[e.g.][]{Kippenhahn.Weigert.ea1967}. Astronomers could start to compare observations of stars and clusters with simulated stars. We call this process \emph{modelling stars}. Early models of the Sun benefited from independent age estimates from geology and neutrino-production rates from observing cosmic rays. With these constraints, and the Sun as a calibrator, researchers could start to refine their models and test them on other stars. For example, \citet{Iben1967} tested stellar models on main sequence stars with similar masses to the Sun using observations of the clusters M67 and NGC 188. +In the second half of the 20th century, astronomers advanced the theory of stellar evolution. With this came computational methods for simulating stars \citep[e.g.][]{Kippenhahn.Weigert.ea1967}. Astronomers could start to compare observations of stars and clusters with numerical solutions to stellar structure equations. We call this process \emph{modelling stars}. Early models of the Sun benefited from independent geological age estimates and neutrino-production rates from observing cosmic rays. With these constraints, and the Sun as a calibrator, researchers could start to refine their models and test them on other stars. For example, \citet{Iben1967} computed stellar models for main sequence stars with similar masses to the Sun using observations of the clusters M67 and NGC 188. -On a similar timescale, a new field emerged which gave astronomers a model-independent way of studying the inside of stars. Identifying regular perturbations of the surface of the Sun, researchers realised that they pertained to high-order, stochastically-driven spherical harmonic oscillations. Measuring these oscillation modes, they could study the Sun similarly to how seismologists study the Earth. This new field, called asteroseismology, studied multiple non-radial oscillations, distinguishing itself from the well-know study of radially pulsating stars \citep[e.g. Cepheid variables;][]{Leavitt1908}. In Section \ref{sec:seismo} we give a brief history and theory of the asteroseismology for stars like the Sun. +On a similar timescale, a new field emerged which gave astronomers a model-independent way of studying the inside of stars. Identifying regular perturbations of the surface of the Sun, researchers realised that they pertained to high-order, stochastically-driven spherical harmonic oscillations. Measuring these oscillation modes, they could study the Sun similarly to how seismologists study the Earth. Astronomers studying multiple non-radial stellar oscillations distinguished this new field, called \emph{asteroseismology}, from the well-known study of radially pulsating stars \citep[e.g. Cepheid variables;][]{Leavitt1908}. In Section \ref{sec:seismo} we briefly elaborate on the history and theory of asteroseismology for stars like the Sun. % Approximations for the equation of state by \citet{Eggleton.Faulkner.ea1973}. -Today, astronomers have a huge abundance of data with which to determine stellar parameters and test theories of stellar evolution. In Section \ref{sec:many-stars}, we give some recent examples of modelling populations of solar-like oscillators and their context in the wider HR diagram. With big datasets and complex, multi-parameter models, there has been a recent shift towards Bayesian statistical methods. We introduce the concept of modelling stars `the Bayesian way' in Section \ref{sec:modelling-stars}. Here, we introduce some tools used to model these stars and then highlight some common assumptions made. +Today, astronomers have a huge abundance of data with which to determine stellar parameters and test theories of stellar evolution. In Section \ref{sec:many-stars}, we give some recent examples of Sun-like asteroseismic populations and their context in the wider HR diagram. With big datasets and complex, multi-parameter models, there has been a recent shift towards Bayesian statistical methods to model these stars. THerefore, we introduce the concept of modelling stars `the Bayesian way' in Section \ref{sec:modelling-stars}. There, we introduce some tools used to model these stars and then highlight some common assumptions made. % Large-scale surveys have lead to asteroseismology on a large scale. Combined with photometry, spectroscopy and astrometry (parallax, position and motion), asteroseismology has bePowerful tool in Section \ref{sec:many-stars}. @@ -52,26 +52,28 @@ \section{Understanding the Stars}\label{sec:stars} \section[Solar-Like Oscillators]{Asteroseismology of Solar-Like Oscillators}\label{sec:seismo} -In this section, we provide a brief history and theory of asteroseismology to provide a foundation from which to understand this thesis. However, we recommend the recent review by \citet{Aerts2021} or the lecture notes by \citet{Christensen-Dalsgaard2014} for a more detailed understanding of the field. +% In this section, we provide a brief history and theory of asteroseismology to provide a foundation from which to understand this thesis. However, we recommend the recent review by \citet{Aerts2021} or the lecture notes by \citet{Christensen-Dalsgaard2014} for a more detailed understanding of the field. \subsection{A Brief History of Asteroseismology} -Several decades ago, 5-minute oscillations in the radial velocity of the solar surface were observed by \citet{Leighton.Noyes.ea1962}, leading to the inference of acoustic waves trapped beneath the solar photosphere \citep{Ulrich1970}. A further decade of study culminated in the measurement of regular patterns of individual oscillation modes in the Doppler radial velocity \citep{Claverie.Isaak.ea1979} and total irradiance \citep{Woodard.Hudson1983a} of the Sun. Initially thought to be short-lived irregularities on the surface, these modes were found to be compatible with stochastically excited standing waves penetrating deep into the Sun. Later, \citet{Deubner.Gough1984} introduced the word \emph{helioseismology} (analogous to geo-seismology) to describe the study of the solar interior using observations of these modes. Helioseismology was soon responsible for breakthrough solar research, from measuring differential rotation \citep{Deubner.Ulrich.ea1979} to solving the mismatch between predicted and measured solar neutrino production \citep{Bahcall.Ulrich1988}. +Several decades ago, 5-minute oscillations in the radial velocity of the solar surface were observed by \citet{Leighton.Noyes.ea1962}, leading to the inference of acoustic waves trapped beneath the solar photosphere \citep{Ulrich1970}. A further decade of study culminated in the measurement of regular patterns of oscillation modes in the Sun; for example, from measurements of radial velocity \citep{Claverie.Isaak.ea1979} and total irradiance \citep{Woodard.Hudson1983a}. Initially thought to be short-lived irregularities on the surface, these modes were found to be compatible with stochastically excited standing waves penetrating deep into the Sun. Later, \citet{Deubner.Gough1984} introduced the word \emph{helioseismology} to describe the study of the solar interior using observations of these modes. Helioseismology was soon responsible for breakthrough solar research, from measuring differential rotation \citep{Deubner.Ulrich.ea1979} to solving the mismatch between predicted and measured solar neutrino production \citep{Bahcall.Ulrich1988}. -Astronomers initially debated the mechanism driving the modes of standing pressure waves (or \emph{p modes}) in the Sun. \citet{Goldreich.Keeley1977} suggested what became the prevailing theory, that the p modes were stochastically excited by near-surface convection. Hence, we might expect solar-like oscillations to be present in other stars which have a convective envelope similar to the Sun. Shortly thereafter, \citet{Christensen-Dalsgaard1984} was among those to introduce the term \emph{asteroseismology} --- the study of the internal structure of stars with many observable oscillation modes. Subsequently, solar-like oscillations were discovered in a few bright stars. Among the first were Procyon and \(\alpha\) Cen A \citep{Gelly.Grec.ea1986}, with individual modes later resolved by \citet{Martic.Schmitt.ea1999} and \citet{Bouchy.Carrier2001} respectively. +Astronomers initially debated the mechanism driving the modes of standing pressure waves (or \emph{p modes}) in the Sun. \citet{Goldreich.Keeley1977} suggested what became the prevailing theory, that the p modes were stochastically excited by near-surface convection. Therefore, astronomers expected solar-like oscillations to be present in other stars with a convective envelope like the Sun. Shortly thereafter, \citet{Christensen-Dalsgaard1984} was among those to introduce the term \emph{asteroseismology} --- the study of the internal structure of stars with many observable oscillation modes. Subsequently, solar-like oscillations were discovered in a few bright stars. Among the first were Procyon and \(\alpha\) Cen A \citep{Gelly.Grec.ea1986}, with individual modes later resolved by \citet{Martic.Schmitt.ea1999} and \citet{Bouchy.Carrier2001} respectively. -Instrumental and atmospheric noise limited the progress of asteroseismology with ground-based equipment to studies of small number of bright dwarf stars. Asteroseismology of solar-like oscillators requires high-cadence (\(\sim \SIrange{1}{10}{\minute}\)) brightness observations over long baselines (\(\sim \SI{1}{\year}\)) with noise of, for example \(\lesssim \SI{100}{ppm\per\hour}\) for \(I\)-band magnitude \(\lesssim 10\) \citep{Schofield.Chaplin.ea2019}. The first dedicated space-based missions which met these requirements arrived in the late 2000s, accelerating progress in the field. Initially, the \emph{CoRoT} mission \citep{Baglin.Auvergne.ea2006} detected solar-like oscillations in thousands of red giant stars \citep{DeRidder.Barban.ea2009,Mosser.Belkacem.ea2010}. Then, the \emph{Kepler} mission \citep{Borucki.Koch.ea2010} yielded oscillations in thousands more red giants \citep{Pinsonneault.Elsworth.ea2014} and hundreds of main sequence stars similar to the Sun \citep{Serenelli.Johnson.ea2017}. Most recently, \emph{TESS} \citep{Ricker.Winn.ea2015} added thousands more dwarf and giant stars to the roster of solar-like oscillators \citep{Hon.Huber.ea2021,SilvaAguirre.Stello.ea2020,Hatt.Nielsen.ea2023}. +Instrumental and atmospheric noise limited the progress of asteroseismology with ground-based equipment to studies of small number of bright dwarf stars. Asteroseismology of solar-like oscillators requires high-cadence (\(\sim \SIrange{1}{10}{\minute}\)) brightness observations over long baselines (\(\sim \SI{1}{\year}\)) with parts-per-million precision \citep[e.g.][]{Schofield.Chaplin.ea2019}. +% For example, noise of \(\lesssim \SI{100}{ppm\per\hour}\) for \(I\)-band magnitudes \(\lesssim 10\) \citep{Schofield.Chaplin.ea2019}. +The first dedicated space-based missions which met these requirements arrived in the late 2000s, accelerating progress in the field. Initially, the \emph{CoRoT} mission \citep{Baglin.Auvergne.ea2006} detected solar-like oscillations in thousands of red giant stars \citep{DeRidder.Barban.ea2009,Mosser.Belkacem.ea2010}. Then, the \emph{Kepler} mission \citep{Borucki.Koch.ea2010} yielded oscillations in thousands more red giants \citep{Pinsonneault.Elsworth.ea2014} and hundreds of main sequence stars similar to the Sun \citep{Serenelli.Johnson.ea2017}. Most recently, \emph{TESS} \citep{Ricker.Winn.ea2015} added thousands more dwarf and giant stars to the roster of solar-like oscillators \citep{Hon.Huber.ea2021,SilvaAguirre.Stello.ea2020,Hatt.Nielsen.ea2023}. For a more detailed review of modern asteroseismology, see \citet{Aerts2021}. \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \begin{figure}[tb] \centering \includegraphics[trim={0 0.4in 0 0},clip]{figures/spherical_harmonics.pdf} - \caption[Spherical harmonic oscillation modes for a few angular degrees ($l$) with azimuthal order \(m=0\).]{Spherical harmonic oscillation modes for a few angular degrees ($l$) with azimuthal order \(m=0\). The colour-map represents the radial displacement at the surface, with \emph{red} and \emph{blue} corresponding to displacement inward and outward respectively. These regions oscillate in and out, with the white regions representing stationary nodes on the surface.} + \caption[Surface spherical harmonic oscillation modes for a few angular degrees ($l$) with azimuthal order \(m=0\).]{Surface spherical harmonic oscillation modes for a few angular degrees ($l$) with azimuthal order \(m=0\). The colour-map represents radial displacement at the surface, with \emph{red} and \emph{blue} corresponding to displacement inward and outward respectively. The coloured regions pulsate in and out, and the white regions represent stationary nodes on the surface.} \label{fig:spherical-harmonics} \end{figure} -Oscillations on the surface of a star can be approximated by spherical harmonic functions with angular degree \(l\) and azimuthal order \(m\). The angular degree is the number of nodes on the surface of the star. We show a representation of the surface spherical harmonics for the first four angular degrees in Figure \ref{fig:spherical-harmonics}. For each \(l\), there exists \(2l+1\) solutions with different azimuthal order (\(m\)) corresponding to the different orientations of the nodes over the spherical surface. Additionally, the oscillation modes have unique frequency solutions for different radial orders (\(n\)), proportional to the number of nodes radially throughout the star. +We can approximate oscillations on the surface of a star by spherical harmonic functions with angular degree \(l\) and azimuthal order \(m\). The angular degree is the number of nodes on the surface of the star. We show a representation of the surface spherical harmonics for the first four angular degrees in Figure \ref{fig:spherical-harmonics}. For each \(l\), there exists \(2l+1\) solutions with different azimuthal order (\(m\)) corresponding to the different orientations of the nodes over the spherical surface. Additionally, the oscillation modes have unique frequency solutions for different radial orders (\(n\)) proportional to the number of nodes radially throughout the star. \begin{figure}[tb] \centering @@ -82,27 +84,27 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} In solar-like oscillators, p modes are stochastically excited by near-surface convection. Typically, the timescale of this process drives high-order modes in main sequence stars (\(n \sim 20\)). We can identify these modes in a frequency-power spectrum derived from photometric or radial velocity time series observations. For instance, both stars in the 16 Cygni planet hosting system are solar-like oscillators with similar properties to the Sun \citep{Metcalfe.Chaplin.ea2012,Davies.Chaplin.ea2015,Metcalfe.Creevey.ea2014}. Using 16 Cyg A as an example, we downloaded the power spectrum determined by the \emph{Kepler} Asteroseismic Science Operations Centre (KASOC) using \emph{Kepler} observations\footnote{\url{https://kasoc.phys.au.dk}}. Shown in Figure \ref{fig:seismo-psd}, the power spectrum of 16 Cyg A has a distinct power excess around \SI{2000}{\micro\hertz}. -The power excess has a Gaussian-like shape around frequencies compatible with the near-surface convective timescale responsible for mode excitation. We call the location of this Gaussian the \emph{frequency at maximum power}, \(\numax\). Dependent on near-surface conditions, \citet{Brown.Gilliland.ea1991} suggested \(\numax\) scales with the acoustic cut-off frequency --- the highest frequency at which acoustic waves can reflect near the stellar surface. Subsequently, \citet{Kjeldsen.Bedding1995} found that \(\numax \propto g\teff^{\,-1/2}\) where \(g\) and \(\teff\) are the near-surface gravitational field strength and temperature. +The power excess has a Gaussian-like shape around frequencies compatible with the near-surface convective timescale responsible for mode excitation. We call the centre of this Gaussian the \emph{frequency at maximum power} (\(\numax\)). Dependent on near-surface conditions, \citet{Brown.Gilliland.ea1991} suggested \(\numax\) scales with the acoustic cut-off frequency --- the highest frequency at which acoustic waves can reflect near the stellar surface. Subsequently, \citet{Kjeldsen.Bedding1995} found that \(\numax \propto g\teff^{\,-1/2}\) where \(g\) and \(\teff\) are the near-surface gravitational field strength and effective temperature. Hence, \(\numax\) for a solar-like oscillator decreases as it evolves, from \(\sim \SI{1000}{\micro\hertz}\) on the main sequence to \(\sim \SI{10}{\micro\hertz}\) near the tip of the red giant branch. -Looking closely at the power excess in Figure \ref{fig:seismo-psd}, we can see a comb of approximately equally spaced peaks. Each peak corresponds to one or more oscillation modes, with their central frequencies and frequency differences providing information about the internal stellar structure. Naturally, higher frequency modes correspond to higher \(n\). However, the angular degree and azimuthal order are harder to identify. We saw in Figure \ref{fig:spherical-harmonics} how modes of higher \(l\) have more anti-nodes on the surface. Therefore, the overall effect of the oscillations cancel out when measuring irradiance integrated over the stellar surface. Consequentially, observed mode amplitude decreases with \(l\), leaving only \(l \lesssim 3\) detectable. +Looking closely at the power excess in Figure \ref{fig:seismo-psd}, we can see a comb of approximately equally spaced peaks. Each peak corresponds to an oscillation mode, with their collective central frequencies and separations providing information about the internal stellar structure. Naturally, higher frequency modes correspond to higher radial order, \(n\). However, the angular degree and azimuthal order are harder to identify. We saw in Figure \ref{fig:spherical-harmonics} how modes of higher \(l\) have more anti-nodes on the surface. Therefore, the overall effect of the oscillations cancel out when measuring irradiance integrated over the stellar surface. Consequentially, observed mode amplitude decreases with \(l\), leaving only \(l \lesssim 3\) detectable. % Not Section 2.1 of JCD Lecture Notes (2014) derives this -We can assume the tallest peaks are \(l=0,1\), and the smaller peaks are \(l=2,3\), all modulated by the wider Gaussian-like envelope. +Therefore, we can assume the tallest peaks are \(l=0,1\), and the smaller peaks are \(l=2,3\), all modulated by the wider Gaussian-like envelope. Modes with different \(m\) cannot be distinguished for a spherically symmetric, non-rotating star. In the case of a rotating or distorted (asymmetric) star, the observed mode frequencies will split for different \(m\) via the Doppler effect. Measuring this splitting can constrain the rotation rate of the star \citep[e.g.][]{Davies.Chaplin.ea2015,Garcia.Ceillier.ea2014,Deheuvels.Garcia.ea2012}. Such studies have lead to breakthrough research on the rotational evolution of stars \citep[e.g.][]{Angus.Aigrain.ea2015,Hall.Davies.ea2021,vanSaders.Ceillier.ea2016}. However, we will hereafter consider only the case of a slowly rotating, spherically symmetric star. -If we consider an acoustic wave in a one-dimensional homogeneous medium, then we would expect each mode of oscillation to be an integer multiple of the fundamental mode. While the case for a star is more complicated, we can also approximate the frequencies for different modes as a multiple of a characteristic frequency. \citet{Tassoul1980} found that the modes could be approximated by assuming the asymptotic limit where \(l/n \rightarrow 0\), giving the following expression \citep[cf.][]{Gough1986}, +For an acoustic wave in a one-dimensional homogeneous medium, we would expect each mode of oscillation to be an integer multiple of the fundamental mode. While the case for a star is more complicated, we can also approximate the frequencies for different modes as a multiple of a characteristic frequency. \citet{Tassoul1980} found that the modes could be approximated by assuming the asymptotic limit, where \(l/n \rightarrow 0\). This lead to the following expression for the frequency of a mode for a given \(n\) and \(l\) \citep[cf.][]{Gough1986}, % \begin{equation} \nu_{nl} \simeq \left(n + \frac{l}{2} + \varepsilon\right) \nu_0 + O(\nu_{nl}^{-1}), \label{eq:asy} \end{equation} % -where \(\varepsilon\) is some constant offset and \(O(\nu_{nl}^{-1})\) represents higher order terms. The characteristic frequency, \(\nu_0\), is the inverse of the acoustic diameter, +where \(\varepsilon\) is some constant offset and \(O(\nu_{nl}^{-1})\) represents higher order terms. The characteristic frequency (\(\nu_0\)) is the inverse of the acoustic diameter, % \begin{equation} \nu_0 = \left(2 \int_{0}^{R} \frac{\dd r}{c(r)}\right)^{-1}, \end{equation} % -where \(c(r)\) is the sound speed as a function of radius (\(r\)) and \(R\) is the radius of the star. Similarly to other variable stars, \citet{Ulrich1986} found that this characteristic frequency relates to the mean density by \(\nu_0 \propto \overline{\rho}^{\,1/2}\). While \(\nu_0\) is not directly detectable in solar-like oscillators, we can approximate it by taking the difference between consecutive modes of the same angular degree, \(\Delta\nu_{nl} = \nu_{nl} - \nu_{n-1\,l}\). Thus, estimates of a global (or average) \emph{large frequency separation}, \(\Delta\nu \simeq \nu_0\), can provide information about the density of a star, leading to independent constraint on its mass and radius (e.g. from scaling relations to the Sun). +where \(c(r)\) is the sound speed as a function of radius (\(r\)), and \(R\) is the radius of the star. Similarly to other variable stars, \citet{Ulrich1986} found that this characteristic frequency relates to mean stellar density by \(\nu_0 \propto \overline{\rho}^{\,1/2}\). While \(\nu_0\) is not directly detectable in solar-like oscillators, we can approximate it by taking the difference between consecutive modes of the same angular degree, \(\Delta\nu_{nl} = \nu_{nl} - \nu_{n-1\,l}\). Thus, estimates of a global (or average) \emph{large frequency separation}, \(\Delta\nu\), can also provide information about the density of a star. This leads to independent constraint on stellar mass and radius (e.g. from scaling relations to the Sun). \begin{figure}[tb] \centering @@ -111,7 +113,7 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \label{fig:seismo-echelle} \end{figure} -The asymptotic expression helps us identify modes in a star. If the first term of Equation \ref{eq:asy} was exact, we would expect odd and even modes to be grouped together and separated by \(\dnu/2\). To show this, we revisit the power spectrum of 16 Cyg A and estimate a signal-to-noise ratio (SNR) by dividing out a moving median in steps of \SI{0.005}{\dex}. We can see the regular pattern predicted by Equation \ref{eq:asy} in the left panel of Figure \ref{fig:seismo-echelle}. Every other mode is approximately separated by \(\dnu\). To see this effect over the wider spectrum, we created an \emph{echelle} plot in the right panel. Folding the spectrum by an estimate of \(\dnu\) reveals a sequence of ridges corresponding to modes of different angular degree. Odd and even angular degree are grouped together, although do not lie on top of each other. The small difference between modes of different \(l\) is described by the higher order terms neglected from Equation \ref{eq:asy}. A faint ridge corresponding to \(l=3\) modes is also visible next to the \(l=1\) ridge. However, 16 Cyg A represents one of the highest SNR dwarf stars observed by \emph{Kepler}, and the \(l=3\) ridge is otherwise not usually visible. +The asymptotic expression helps us identify modes in a star. If the first term of Equation \ref{eq:asy} was exact, we would expect odd and even modes to be grouped together and separated by \(\dnu/2\). To show this, we revisit the power spectrum of 16 Cyg A and estimate a signal-to-noise ratio (SNR) by dividing out a moving median in log-frequency steps of \SI{0.005}{\dex}. We can see the regular pattern predicted by Equation \ref{eq:asy} in the left panel of Figure \ref{fig:seismo-echelle}. Every other mode is approximately separated by \(\dnu\). To see this effect over the wider spectrum, we created an \emph{echelle} plot in the right panel. Folding the spectrum by an estimate of \(\dnu\) reveals a sequence of ridges corresponding to modes of different angular degree. Odd and even angular degree are grouped together, although do not lie on top of each other. The small difference between modes of different \(l\) is described by the higher order terms neglected from Equation \ref{eq:asy}. A faint ridge corresponding to \(l=3\) modes is also visible next to the \(l=1\) ridge. However, 16 Cyg A represents one of the highest SNR dwarf stars observed by \emph{Kepler}, and the \(l=3\) ridge is otherwise not usually visible. % Once we have identified a solar-like oscillator, what information is there to gain from asteroseismology? We have discussed how parameters \(\numax\) and \(\dnu\) scale with global stellar properties. Scaling these parameters with respect to the Sun, we can obtain relations for the radius and mass of the star, % % @@ -124,19 +126,19 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} % We can also use individual modes and compare with models. Worth noting the surface term here. -Repeating this kind of analysis on a large-scale can provide a wealth of observables from which to model stars. In the next section, we give some examples where asteroseismology is used to model many solar-like oscillators. +Repeating this kind of analysis on a large-scale can provide a wealth of observables from which to model stars. In the next section, we give some examples where asteroseismology was used to model many solar-like oscillators. \section[Modelling Stars with Asteroseismology]{Modelling Many Stars with Asteroseismology}\label{sec:many-stars} -Recent space-based missions like \emph{Kepler} and \emph{TESS} enable asteroseismology with many stars. Like \emph{CoRoT}, these missions were primarily designed to detect exoplanets via the transit method. However, their instrumental requirements were also well-suited to asteroseismology. This allowed astronomers to study regions of the HR diagram in more detail, from main sequence dwarf stars red giants. We draw particular attention to \emph{Kepler}, the first of the two missions. \emph{Kepler}'s primary mission observed a single \SI{15}{\degree} diameter patch of sky for around 4 years \citep{Borucki.Koch.ea2010}, cut short by a mechanical failure which spawned the secondary \emph{K2} mission \citep{Howell.Sobeck.ea2014}. The mission took photometric measurements in short (\SI{60}{\second}) and long (\SI{30}{\minute}) cadences. During the primary mission, large asteroseismic datasets were gathered. The long baseline of 4 years allowed for high frequency precision not yet seen by \emph{K2} or \emph{TESS}. +Recent space-based missions like \emph{Kepler} and \emph{TESS} enable asteroseismology with many stars. Like \emph{CoRoT}, these missions were primarily designed to detect exoplanets using the transit method. However, their instrumental requirements were also well-suited to asteroseismology. This allowed astronomers to study regions of the HR diagram in more detail, from main sequence dwarf stars to red giants. We draw particular attention to \emph{Kepler}, the first of the two missions. \emph{Kepler}'s primary mission observed a single \(\sim\SI{100}{deg\squared}\) diameter patch of sky for around 4 years \citep{Borucki.Koch.ea2010}, cut short by a mechanical failure which spawned the secondary \emph{K2} mission \citep{Howell.Sobeck.ea2014}. \emph{Kepler} took photometric measurements in long (\SI{30}{\minute}) and short (\SI{60}{\second}) cadences, with the Nyquist frequency of the latter high enough to detect main sequence oscillators. During its primary mission, large asteroseismic datasets were gathered. The long baseline of 4 years allowed for high frequency precision not since seen by \emph{K2} or \emph{TESS}. % In this section, we introduce the sample of stars being studied in this thesis relative to the broader astronomical picture. -Of the asteroseismic targets found by \emph{Kepler}, we focus on dwarf and subgiant solar-like oscillators. These are stars with a similar mass to the Sun either on the main sequence or post-main sequence before ascending the red giant branch. From an asteroseismic perspective, the power spectra of dwarf stars are relatively simple. As solar-like oscillators approach the red giant branch, their modes begin to couple with buoyancy-driven modes in the core \citep{Bedding.Mosser.ea2011,Mosser.Barban.ea2011}. This leads to irregular patterns which can be used to separate evolutionary state and study properties of the core \citep[e.g.][]{Mosser.Vrard.ea2015}. On the other hand, the relative simplicity of main sequence oscillators make them a suitable place to start. Furthermore, the work in this thesis anticipates the upcoming \emph{PLATO} mission which aims to observe \(\sim 10^4\) dwarf and subgiant oscillators \citep{Rauer.Aerts.ea2016}. +Of the asteroseismic targets found by \emph{Kepler}, we focus on dwarf and subgiant solar-like oscillators. These are stars with a similar mass to the Sun on the main sequence or shortly after the turn-off before ascending the red giant branch. From an asteroseismic perspective, the power spectra of dwarf stars are relatively simple. As solar-like oscillators approach the red giant branch, their modes begin to couple with buoyancy-driven modes in the core \citep{Bedding.Mosser.ea2011,Mosser.Barban.ea2011}. This leads to irregular patterns which can be used to separate evolutionary state and study properties of the core \citep[e.g.][]{Mosser.Vrard.ea2015}. On the other hand, the relative simplicity of main sequence oscillators make them a suitable place to start our study. Furthermore, the work in this thesis anticipates the upcoming \emph{PLATO} mission which aims to observe \(\sim 10^4\) dwarf and subgiant oscillators \citep{Rauer.Aerts.ea2016}. % Modelling many stars with asteroseismology is complemented by other recent large-scale stellar surveys. High-precision astrometry from the \emph{Gaia} mission \citep{GaiaCollaboration.Prusti.ea2016} has provided improved distances and orbital solutions. The APOGEE large-scale spectroscopic survey \citep{Majewski.Schiavon.ea2017} has also yielded precise chemical abundances. These surveys have enabled studies of the assembly history of our galaxy. For example, \citet{Helmi.Babusiaux.ea2018} discovered a merger between the Milky Way and Gaia-Enceladus by analysing the motions and abundances from \emph{Gaia} and APOGEE. Asteroseismology has accompanied this work by helping determine the ages of stars tied to the merger \citep{Chaplin.Serenelli.ea2020,Montalban.Mackereth.ea2021}. -Modelling many stars with asteroseismology is complemented by other recent large-scale stellar surveys. High-precision astrometry from the space-based \emph{Gaia} mission provides precise distances and orbital solutions for over a billion sources \citep{GaiaCollaboration.Prusti.ea2016}. Additionally, the APOGEE large-scale spectroscopic survey provides precise chemical abundances for over half a million targets to date \citep{Majewski.Schiavon.ea2017,Jonsson.Holtzman.ea2020}. When combined, these surveys have enabled studies of the assembly history of our galaxy \citep[e.g.][]{Helmi.Babusiaux.ea2018} which have since been enhanced by asteroseismology \citep{Chaplin.Serenelli.ea2020,Montalban.Mackereth.ea2021}. +Modelling many stars with asteroseismology is complemented by other recent large-scale stellar surveys. High-precision astrometry from the space-based \emph{Gaia} mission provides precise distances and orbital solutions for over a billion sources \citep{GaiaCollaboration.Prusti.ea2016}. Additionally, the APOGEE large-scale spectroscopic survey provides precise chemical abundances for over half a million targets to date \citep{Majewski.Schiavon.ea2017,Jonsson.Holtzman.ea2020}. When combined, these surveys have enabled studies of the assembly history of our galaxy \citep[e.g.][]{Helmi.Babusiaux.ea2018} which have since been strengthened by asteroseismology \citep{Chaplin.Serenelli.ea2020,Montalban.Mackereth.ea2021}. \begin{figure} \centering @@ -144,13 +146,13 @@ \subsection{A Brief Theory of Solar-Like Oscillations}\label{sec:seismo-theory} \caption[Colour-magnitude diagram highlighting the region occupied by dwarf and subgiant solar-like oscillators.]{Colour-magnitude diagram highlighting the region occupied by solar-like oscillators. % The data are \emph{Gaia} G, G\textsubscript{BP}, and G\textsubscript{RP} band magnitudes respectively. The absolute magnitude (\(M_\mathrm{G}\)) is calculated from \emph{Gaia} parallaxes neglecting extinction. - \emph{Main plot:} The \emph{greyscale} 2D histogram shows the density of \emph{Gaia} DR3 targets with parallax \(>\SI{5}{\milli\aarcsec}\) (distance \(\lesssim\SI{200}{\parsec}\)). The \emph{coloured} 2D histogram shows the density of all \emph{Kepler} targets in \emph{Gaia} DR3. Scattered points are shown where the density is less than 10 points per bin. + \emph{Main plot:} The background 2D histogram (greyscale) shows the density of \emph{Gaia} DR3 targets with parallax \(>\SI{5}{\milli\aarcsec}\) (distance \(\lesssim\SI{200}{\parsec}\)). The foreground 2D histogram (blue-purple) shows the density of all \emph{Kepler} targets in \emph{Gaia} DR3. Scattered points are shown where the density is less than 10 points per bin. - \emph{Inset plot:} Stellar evolutionary tracks at solar metallicity are given by the \emph{black lines} for \SIrange{0.8}{1.4}{\solarmass} in steps of \SI{0.1}{\solarmass}. The points in the inset axes correspond to \emph{Kepler} solar-like oscillators with references given in the legend. The Sun is given by the `\(\odot\)' symbol.} + \emph{Inset plot:} Stellar evolutionary tracks with solar metallicity are given by the black lines for \SIrange{0.8}{1.4}{\solarmass} in steps of \SI{0.1}{\solarmass}. The circles in the inset axes correspond to \emph{Kepler} solar-like oscillators with references given in the legend. The Sun is given by the `\(\odot\)' symbol.} \label{fig:hr-diagram} \end{figure} -In Figure \ref{fig:hr-diagram}, we show a colour-magnitude diagram made using magnitudes and parallaxes from \emph{Gaia} Data Release 3 \citep[DR3;][]{GaiaCollaboration.Vallenari.ea2022}. For this illustrative plot, we have neglected the effect of extinction. The background distribution shows solar-neighbourhood \emph{Gaia} sources with a parallax greater than \SI{5}{\milli\aarcsec} for context. Over-plot is the distribution of \emph{Kepler} objects cross-matched with \emph{Gaia} DR3\footnote{The cross-matched dataset was obtained from \url{https://gaia-kepler.fun}}. The densest region lies in the low- to intermediate-mass main sequence (\SIrange{0.8}{1.2}{\solarmass}) but we can also see a clear red giant branch and red clump in the upper-right of the distribution. The inset plot draws attention to the region occupied by dwarf and subgiant solar-like oscillators, where we give some examples. +In Figure \ref{fig:hr-diagram}, we show a colour-magnitude diagram made using magnitudes and parallaxes from \emph{Gaia} Data Release 3 \citep[DR3;][]{GaiaCollaboration.Vallenari.ea2022}. For this illustrative plot, we have neglected the effect of extinction from dust in the interstellar medium. The background distribution shows solar-neighbourhood \emph{Gaia} sources with a parallax greater than \SI{5}{\milli\aarcsec} for context. Over-plot is the distribution of \emph{Kepler} objects cross-matched with \emph{Gaia} DR3\footnote{The cross-matched dataset was obtained from \url{https://gaia-kepler.fun}}. The densest region lies in the low- to intermediate-mass main sequence (\SIrange{0.8}{1.2}{\solarmass}) but we can also see a clear red giant branch and red clump in the upper-right of the distribution. The inset plot draws attention to the region occupied by dwarf and subgiant solar-like oscillators, where we give some examples. \citet{Chaplin.Kjeldsen.ea2011} identified the first large catalogue of \(\sim 500\) dwarf and subgiant solar-like oscillators (black circles in Figure \ref{fig:hr-diagram}) by measuring \(\dnu\) and \(\numax\) in \emph{Kepler} data. Later, \citet{Chaplin.Basu.ea2014} determined ages, masses and radii for these stars using \(\dnu\) and \(\numax\) complemented by photometry and ground-based spectroscopy where available. The subsequent arrival of APOGEE spectroscopy allowed \citet{Serenelli.Johnson.ea2017} to revisit this sample with a more consistent set of \(\teff\) and metallicity. By comparing observations to models of stellar evolution, they found radii, masses and ages with uncertainties of around 3, 5 and 20 per cent respectively. @@ -180,7 +182,7 @@ \section{Modelling Stars the Bayesian Way}\label{sec:modelling-stars} p(\vect{\theta} \mid \vect{y}) = \frac{p(\vect{y} \mid \vect{\theta})\,p(\vect{\theta})}{p(\vect{y})}, \end{equation} % -where \(p(\vect{y} \mid \vect{\theta})\) is the \emph{likelihood}, \(p(\vect{\theta})\) is the \emph{prior}, and \(p(\vect{y})\) is the \emph{evidence}. The prior encodes our expectation for \(\vect{\theta}\). The prior dominates if the observations are bad (i.e. have a small likelihood) and we return our current belief. However, if the observations are good enough, the likelihood has more influence on the posterior and our belief is updated. As a result, we can use the Bayesian methodology to systematically update our beliefs. This provides a statistical scheme from which to build our stellar models. +where \(p(\vect{y} \mid \vect{\theta})\) is the \emph{likelihood}, \(p(\vect{\theta})\) is the \emph{prior}, and \(p(\vect{y})\) is the \emph{evidence}. We should choose a prior to represent our expectation for \(\vect{\theta}\). The prior dominates if the observations are bad (i.e. have a small likelihood) and we return our current `belief' (or expectation). However, if the observations are good enough, the likelihood has more influence on the posterior and our belief is updated. As a result, we can use the Bayesian methodology to systematically update our beliefs. This provides a statistical scheme from which to build our stellar models. Let us consider a stellar model with three parameters, \(\vect{\theta} = t_\star, M, Z\). For instance, we might want the age of a star to help date a galactic merger, or its mass to characterise an exoplanetary system. To get the probability over age given our observations, we must \emph{marginalise} over the joint posterior with respect to all other parameters. The marginal posterior probability distribution for \(t_\star\) is, % @@ -190,24 +192,22 @@ \section{Modelling Stars the Bayesian Way}\label{sec:modelling-stars} % where the bounds of the integral are chosen across all space for \(M\) and \(Z\). In other words, the marginal distribution involves integrating over the uncertainty in all other model parameters. -The marginal posterior distribution depends on our choice of function which maps \(\vect{\theta}\) to predict observable parameters, \(\tilde{\vect{y}} = f(\vect{\theta})\). The likelihood then compares the predictions, \(\tilde{\vect{y}}\), with observations, \(\vect{y}\). When modelling stars, there is no analytical form to \(f\). Instead, we use numerical simulations which take in \(\vect{\theta}\) and approximate \(\vect{y}\). For example, the software package `Modules for Experiments in Stellar Astrophysics' (MESA), originally developed by \citet{Paxton.Bildsten.ea2011}, is a one-dimensional stellar evolution code. Assuming spherical symmetry, MESA simulates the structure and evolution of a star along a one-dimensional radial slice. In a nutshell, the code numerically solves a series of differential equations which govern the interior dynamics of the star \citep[see e.g.][]{Kippenhahn.Weigert.ea2013}. Often starting with a cloud of gas, MESA evolves the system over dynamical time steps recording physical properties at mesh points inside the star. +The marginal posterior distribution depends on our choice of function which maps \(\vect{\theta}\) to predict observable parameters, \(\tilde{\vect{y}} = f(\vect{\theta})\). The likelihood then compares the predictions (\(\tilde{\vect{y}}\)) with observations (\(\vect{y}\)). When modelling stars, there is no analytical form to \(f\). Instead, we use numerical simulations which take in \(\vect{\theta}\) and approximate \(\vect{y}\). For example, Modules for Experiments in Stellar Astrophysics \citep[MESA; originally developed by][]{Paxton.Bildsten.ea2011} is a one-dimensional stellar evolution code. Assuming spherical symmetry, MESA simulates the structure and evolution of a star along a one-dimensional radial slice. The code numerically solves a series of differential equations which govern the interior dynamics of the star \citep[see, e.g.][for an introduction to stellar structure and evolution]{Kippenhahn.Weigert.ea2013}. Often starting with a cloud of gas, MESA evolves the system over dynamical time steps recording physical properties at mesh points inside the star. -We can also predict oscillation mode frequencies (\(\nu_{nl}\)) with simulations. For example, the GYRE code developed by \citet{Townsend.Teitler2013} uses the output of MESA to compute oscillation modes for a given \(n\) and \(l\). While the physics of p mode propagation in the star is relatively well-known, our understanding of the atmospheric boundary conditions are not. As such, there is a known discrepancy between the simulated and observed p modes. The nature of this can lead to a systematic bias on modelled \(\dnu\) and \(\nu_{nl}\). Corrections for this effect exist \citep[e.g.][]{Ball.Gizon2014} but are still not fully understood. +We can also predict oscillation mode frequencies (\(\nu_{nl}\)) with simulations. For example, the GYRE code developed by \citet{Townsend.Teitler2013} uses the output of MESA to compute oscillation modes for a given \(n\) and \(l\). While the physics of p mode propagation in the star is relatively well-known, our understanding of the atmospheric boundary conditions are not. As such, there is a known discrepancy between the simulated and observed p modes. The nature of this can lead to a systematic bias when modelled \(\dnu\) and \(\nu_{nl}\). Corrections for this effect exist \citep[e.g.][]{Ball.Gizon2014} but are still not fully understood. -The complexity of stellar models means that the marginalised posterior distributions are not analytically derivable. Therefore, we use numerical methods like Markov Chain Monte Carlo (MCMC) to estimate the posterior. Typically, this involves exploring parameter space with multiple calls to \(f\) for different values of \(\vect{\theta}\). In the case of MCMC-based algorithms like Hamiltonian Monte Carlo (HMC) and the No U-Turn Sampler (NUTS), the gradient of \(f\) is also required. There are several open-source \textsc{Python} packages widely used to implement these algorithms including \texttt{pymc} \citep{Salvatier.Wiecki.ea2016} and \texttt{numpyro} \citep{Phan.Pradhan.ea2019}. +The complexity of stellar models means that the marginalised posterior distributions are not analytically derivable. One solution is to estimate the posterior with numerical methods like Markov Chain Monte Carlo (MCMC). Typically, this involves exploring parameter space with multiple calls to \(f\) for different values of \(\vect{\theta}\). In the case of MCMC-based algorithms like Hamiltonian Monte Carlo (HMC) and the No U-Turn Sampler (NUTS), the gradient of \(f\) is also required. There are several open-source \textsc{Python} packages widely used to implement these algorithms, for instance \texttt{pymc} \citep{Salvatier.Wiecki.ea2016} and \texttt{numpyro} \citep{Phan.Pradhan.ea2019}. -There are some existing methods for determining stellar parameters using this Bayesian approach. \citet{Bazot.Bourguignon.ea2008} used the MCMC algorithm to sample model parameters with on-the-fly stellar model calculation. While this method can be tailored to individual stars, it is very computationally expensive. Each proposed set of \(\vect{\theta}\) spawns a stellar simulation which evolves to a given age. Steps prior to this age may be discarded, and the simulation can take minutes to hours for each set of \(\vect{\theta}\). This is not a viable solution for modelling large numbers of stars. +There are some existing methods for determining stellar parameters using this Bayesian approach. As an early example, \citet{Bazot.Bourguignon.ea2008} used the MCMC algorithm to sample model parameters with on-the-fly stellar model calculation. While their method could be tailored to individual stars, it was very computationally expensive. Each proposed set of \(\vect{\theta}\) spawns a stellar simulation which evolves to a given age. Steps prior to this age may be discarded, and the simulation can take minutes to hours for each set of \(\vect{\theta}\). This was not a viable solution for modelling the large populations of stars being observed with high-precision today. -A more efficient solution is to sample a discrete grid of models \citep{Gruberbauer.Guenther.ea2012,Gruberbauer.Guenther.ea2013}. A recent tool which does this is the BAyesian STellar algorithm (BASTA) developed by \citet{AguirreBorsen-Koch.Rorsted.ea2022}. The authors pre-compute a grid of simulated stars corresponding to a set of relevant input parameters. They weight the grid points according to the prior and then compute the marginalised posterior for each model parameter. +A more efficient solution is to sample a discrete grid of models \citep{Gruberbauer.Guenther.ea2012,Gruberbauer.Guenther.ea2013}. For example, a recent tool which does this is the BAyesian STellar algorithm (BASTA) developed by \citet{AguirreBorsen-Koch.Rorsted.ea2022}. The authors pre-compute and interpolate a grid of simulated stars corresponding to a set of relevant input parameters. They weight the grid points according to the prior and then compute the marginalised posterior for each model parameter. We could extend this by interpolating a grid of stellar models to approximate \(f\) itself. This is done in the Asteroseismic Inference on a Massive Scale (AIMS) pipeline by \citet{Lund.Reese2018,Rendle.Buldgen.ea2019}. However, interpolation methods can be slow. They scale poorly with dimensionality and number of points on the grid. A small subset of the grid can be used to mitigate this issue, but we want a solution which can be extended to model many stars at once. -Alternatively, we could approximate \(f\) by interpolating a grid of stellar models. This is done in the Asteroseismic Inference on a Massive Scale (AIMS) pipeline by \citet{Lund.Reese2018,Rendle.Buldgen.ea2019}. Interpolation methods can be slow and scale poorly with dimensionality and number of points on the grid. A small subset of the grid can be used to mitigate this issue. However, we want a solution which can be extended to model many stars at once. +Asteroseismology has allowed us to model stars more precisely than ever before. This has exposed systematic uncertainties in common assumptions used when computing these models \citep[e.g.][]{Tayar.Claytor.ea2022}. For instance, fractional helium abundance (\(Y\)) cannot be determined spectroscopically in cool stars. Instead, we often assume it follows a linear enrichment law \citep{Chiosi.Matteucci1982,Ribas.Jordi.ea2000,Casagrande.Flynn.ea2007}. Such a law assumes that helium is enriched in the interstellar medium linearly with metallicity. However, \citet{Lebreton.Goupil.ea2014} found that varying \(Y\) by \(\pm\,0.03\) can have \(\gtrsim \mp\,20\) per cent effect on stellar age. Therefore, assuming \(Y\) instead of incorporating it in our model leads to underestimating our uncertainty on stellar age and other correlated parameters. In Chapters \ref{chap:glitch} and \ref{chap:glitch-gp}, we will explore one way to improve inference of \(Y\) using an asteroseismic signature of helium ionisation. In the meantime, we explore a better statistical treatment of our prior on such poorly characterised parameters. -Many of these methods do not inherently account for systematic uncertainty from approximations of stellar physics and other assumptions. Asteroseismology has allowed us to model stars more precisely. However, this has exposed systematic uncertainties in common assumptions used when computing these models \citep[e.g.][]{Tayar.Claytor.ea2022}. For example, fractional helium abundance (\(Y\)), which cannot be determined spectroscopically in cool stars, is often assumed to follow a linear enrichment law \citep{Chiosi.Matteucci1982,Ribas.Jordi.ea2000,Casagrande.Flynn.ea2007}. This law assumes that helium is enriched in the interstellar medium linearly with metallicity. \citet{Lebreton.Goupil.ea2014} found that varying \(Y\) by \(\pm\,0.03\) can have \(\gtrsim \mp\,20\) per cent effect on stellar age \citep{Lebreton.Goupil.ea2014}. - -Another assumption is the value of the mixing-length theory parameter (\(\mlt\)). This parametrises a common approximation of convective mixing used in stellar models \citep{Gough1977}. Many of the aforementioned methods assume a value of \(\mlt\) calibrated to the Sun. However, 3D hydrodynamical models have shown different values of \(\mlt\) do a better job of approximating convection for different stars \citep{Magic.Weiss.ea2015}. +% Another assumption is the value of the mixing-length theory parameter (\(\mlt\)). This parametrises a common approximation of convective mixing used in stellar models \citep{Gough1977}. Many of the aforementioned methods assume a value of \(\mlt\) calibrated to the Sun. However, 3D hydrodynamical models have shown different values of \(\mlt\) do a better job of approximating convection for different stars \citep{Magic.Weiss.ea2015}. % To properly model stars the Bayesian way, we need to accommodate \(Y\) and \(\mlt\) in our models. Unfortunately, the effect of these parameters on observables is degenerate. Observables are not precise enough to differentiate between the two on a star-by-star level. -Finally, there are many cases where we would expect stellar parameters to be correlated between stars. For example, we believe ages of stars in an open cluster belong to a distribution. With Bayesian methodology, we can encode this prior belief using a hierarchical model. In the next chapter, we introduce the concept of hierarchical Bayesian models. We explore a simple HBM in the context of astronomy and show how it can improve the estimates of stellar parameters. +Finally, it is important that our prior represents not only our belief for individual stars, but also the population. There are many cases where we would expect stellar parameters to be correlated between stars. Not accounting for such correlations cause us to overestimate uncertainty on individual stellar parameters. For example, we believe that \(Y\) belongs to a distribution governed by chemical enrichment of the interstellar medium. Stars formed in a similar place and time should share similar chemical abundances. We can incorporate prior beliefs like this using a hierarchical Bayesian model (HBM). In the next chapter, we introduce the concept of an HBM. We explore a simple HBM in the context of astronomy and show how it can improve our estimates of stellar parameters. Following that, we present a hierarchical model on a population of dwarf and subgiant stars observed by \emph{Kepler} in Chapter \ref{chap:hmd}. -Following that, we present a hierarchical model on a population of dwarf and subgiant stars observed by \emph{Kepler} in Chapter \ref{chap:hmd}. While such a method improves upon and tackles bias in our choice of \(Y\) and \(\mlt\), there is more helium abundance information to be gained from asteroseismology. Finally, in Chapter \ref{chap:glitch}, we explore signatures of helium abundance in the oscillation modes of stars. We present a new method for characterising these glitches. +% While such a method improves upon and tackles bias in our choice of \(Y\) and \(\mlt\), there is more helium abundance information to be gained from asteroseismology. Finally, in Chapter \ref{chap:glitch}, we explore signatures of helium abundance in the oscillation modes of stars. We present a new method for characterising these glitches. From d36f5528a4ae6bb33a6e43c1703ca5353b794817 Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Thu, 27 Apr 2023 23:05:47 +0100 Subject: [PATCH 38/50] Update title --- thesis.tex | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/thesis.tex b/thesis.tex index baf6e77..8d5cf43 100644 --- a/thesis.tex +++ b/thesis.tex @@ -54,7 +54,10 @@ % % Add your details here \author{Alexander J. Lyttle} -\title{Hierarchically Modelling Many Stars} +\title{% +Hierarchically Modelling Many Stars\\ +to Improve Inference with\\ +Asteroseismology} \group{Sun, Stars and Exoplanets Group} \school{Physics and Astronomy} \college{Engineering and Physical Sciences} From 2a583f03f9960ceb8ec39bc83f920384a534c89a Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Thu, 27 Apr 2023 23:42:42 +0100 Subject: [PATCH 39/50] Change pronoun to I from We in chapter summaries --- chapters/conclusion.tex | 2 +- chapters/glitch-gp.tex | 2 +- chapters/glitch.tex | 2 +- chapters/hbm.tex | 2 +- chapters/lyttle21.tex | 2 +- 5 files changed, 5 insertions(+), 5 deletions(-) diff --git a/chapters/conclusion.tex b/chapters/conclusion.tex index 66601e2..22c4da4 100644 --- a/chapters/conclusion.tex +++ b/chapters/conclusion.tex @@ -12,7 +12,7 @@ % \chapter{Conclusion \& Future Prospects} -\textit{We conclude this thesis by providing a summary of the work herein and reflecting upon key results. Then, we consider possible improvements to our hierarchical model and method for emulating stellar simulations. Finally, we discuss future prospects for applying our method to data from current and upcoming missions.} +\textit{I conclude this thesis by providing a summary of the work and reflecting upon key results. Then, I consider possible improvements to our hierarchical model (from Chapter \ref{chap:hmd}) and method for emulating stellar simulations. Finally, I discuss future prospects for applying our method to data from current and upcoming missions.} \section*{Summary} diff --git a/chapters/glitch-gp.tex b/chapters/glitch-gp.tex index b557321..d4abadd 100644 --- a/chapters/glitch-gp.tex +++ b/chapters/glitch-gp.tex @@ -12,7 +12,7 @@ % \chapter[Modelling Acoustic Glitches with a Gaussian Process]{Modelling Acoustic Glitch Signatures in Stellar Oscillations with a Gaussian Process}\label{chap:glitch-gp} -\textit{In this chapter, we apply a new method for modelling acoustic glitch signatures in the radial mode frequencies of solar-like oscillators. We compare our method with another using a model star with different levels of noise. Then, we apply both methods to the star 16 Cyg A to provide a real-world working example. We show that our method can be used to find the strength and location of glitches caused by the second ionisation of helium and the base of the convective zone. We also demonstrate that our method improves the treatment of the smoothly varying component of the mode frequencies.} +\textit{In this chapter, I apply a new method for modelling acoustic glitch signatures in the radial mode frequencies of solar-like oscillators. I compare this method with another using a model star with different levels of noise. Then, I apply both methods to the star 16 Cyg A to provide a real-world working example. I show that our method can be used to find the strength and location of glitches caused by the second ionisation of helium and the base of the convective zone. I also demonstrate that this method improves treatment of the smoothly varying deviation in the mode frequencies from their regular separation.} \section{Introduction} diff --git a/chapters/glitch.tex b/chapters/glitch.tex index d218d67..7d6f8d0 100644 --- a/chapters/glitch.tex +++ b/chapters/glitch.tex @@ -12,7 +12,7 @@ % \chapter[Acoustic Glitches in Solar-Like Oscillators]{Acoustic Glitches in Solar-Like Oscillators as a Signature of Helium Abundance}\label{chap:glitch} -\textit{Having demonstrated a hierarchical model over initial stellar helium abundance, we now explore an asteroseismic signature of helium which could provide more observational constraint. In this chapter, we introduce the concept of a glitch in the structure of a star producing a measurable signal in its observable oscillation modes. We start with a simple one-dimensional example in Section \ref{sec:1d-glitch}. Then, in Section \ref{sec:glitch-star} we introduce glitches due to helium ionisation and the base of the convective zone in solar-like oscillators. We review theoretical background on the glitches and their effect on stellar oscillation modes in advance of Chapter \ref{chap:glitch-gp}, where we apply a novel method for modelling these glitch signatures.} +\textit{In this chapter, I explore an asteroseismic signature of helium which could provide observables to add to the hierarchical model from Chapter \ref{chap:hmd}. I introduce the concept of an acoustic glitch in the structure of a star producing a measurable signal in its observable oscillation modes. In Section \ref{sec:1d-glitch}, I start with a simple one-dimensional example of a glitch. Then, I review the theory of glitch signatures due to helium ionisation and the base of the convective zone in Section \ref{sec:glitch-star}.} \section{Introduction} % \epigraph{\singlespacing``Ideals are like stars: you will not succeed in touching them with your hands, but like the seafaring man on the ocean desert of waters, you choose them as your guides, and following them, you reach your destiny.''}{\emph{Carl Schurz}} diff --git a/chapters/hbm.tex b/chapters/hbm.tex index 4c6237b..bfe8e08 100644 --- a/chapters/hbm.tex +++ b/chapters/hbm.tex @@ -12,7 +12,7 @@ % \chapter{Hierarchical Bayesian Models}\label{chap:hbm} -\textit{In this chapter, we introduce the concept of a hierarchical Bayesian model in the context of determining stellar parameters. We use a simplified analogy for measuring distances to stars in an open cluster to demonstrate the advantages and usage of a hierarchical model in Section \ref{sec:hbm-dist}. Finally, we identify parameters which could be treated hierarchically when modelling the physical properties of many stars, priming the reader for Chapter \ref{chap:hmd}.} +\textit{In this chapter, I introduce the concept of a hierarchical Bayesian model in the context of determining stellar parameters. Using a simplified analogy for measuring distances to stars in an open cluster, I demonstrate the advantages and usage of a hierarchical model in Section \ref{sec:hbm-dist}. Finally, I identify parameters which could be treated hierarchically when modelling the physical properties of stars, preparing the reader for Chapter \ref{chap:hmd}.} \section{Introduction} diff --git a/chapters/lyttle21.tex b/chapters/lyttle21.tex index 0035205..41dbef3 100644 --- a/chapters/lyttle21.tex +++ b/chapters/lyttle21.tex @@ -13,7 +13,7 @@ \chapter[Hierarchically Modelling Dwarf and Subgiant Stars]{Hierarchically Modelling \emph{Kepler} Dwarfs and Subgiants to Improve Inference of Stellar Properties with Asteroseismology}\label{chap:hmd} \textit{% - This chapter is taken verbatim from \citet{Lyttle.Davies.ea2021}. I confirm that all writing is my own except for the description of stellar models and input physics in Section \ref{sec:grid} which was written by Tanda Li\footnote{Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China}. In this work, we present a hierarchical Bayesian model which encodes information about the distribution of helium abundance (\(Y\)) and mixing-length-theory parameter (\(\mlt\)) in a population of dwarf and subgiant solar-like oscillators. + In this chapter, I present the entirety of \citet{Lyttle.Davies.ea2021} with minor modification. I confirm that all writing is my own except for the description of stellar models and input physics in Section \ref{sec:grid} which was written by Tanda Li\footnote{Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China}. In this work, we created a hierarchical Bayesian model to pool information about the distribution of helium abundance (\(Y\)) and mixing-length-theory parameter (\(\mlt\)) in a population of dwarf and subgiant solar-like oscillators. } \section{Introduction} From 4cfb411b26efc17308310f5fc5888d2011d30838 Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Fri, 28 Apr 2023 00:34:41 +0100 Subject: [PATCH 40/50] Fix bugs in references --- references.bib | 111 +++++++++++++++++++++++++++++-------------------- 1 file changed, 67 insertions(+), 44 deletions(-) diff --git a/references.bib b/references.bib index f63f216..623a7ba 100644 --- a/references.bib +++ b/references.bib @@ -103,15 +103,14 @@ @article{Ahumada.Prieto.ea2020 annotation = {ADS Bibcode: 2020ApJS..249....3A} } -@misc{Aigrain.Foreman-Mackey2022, +@article{Aigrain.Foreman-Mackey2022, title = {Gaussian {{Process}} Regression for Astronomical Time-Series}, author = {Aigrain, Suzanne and {Foreman-Mackey}, Daniel}, year = {2022}, month = sep, - number = {arXiv:2209.08940}, + journal = {arXiv e-prints}, eprint = {2209.08940}, primaryclass = {astro-ph}, - publisher = {{arXiv}}, doi = {10.48550/arXiv.2209.08940}, urldate = {2022-09-20}, abstract = {The last two decades have seen a major expansion in the availability, size, and precision of time-domain datasets in astronomy. Owing to their unique combination of flexibility, mathematical simplicity and comparative robustness, Gaussian Processes (GPs) have emerged recently as the solution of choice to model stochastic signals in such datasets. In this review we provide a brief introduction to the emergence of GPs in astronomy, present the underlying mathematical theory, and give practical advice considering the key modelling choices involved in GP regression. We then review applications of GPs to time-domain datasets in the astrophysical literature so far, from exoplanets to active galactic nuclei, showcasing the power and flexibility of the method. We provide worked examples using simulated data, with links to the source code, discuss the problem of computational cost and scalability, and give a snapshot of the current ecosystem of open source GP software packages. Driven by further algorithmic and conceptual advances, we expect that GPs will continue to be an important tool for robust and interpretable time domain astronomy for many years to come.}, @@ -176,7 +175,7 @@ @article{Anderson.Hogg.ea2018 doi = {10.3847/1538-3881/aad7bf}, urldate = {2020-05-27}, abstract = {Converting a noisy parallax measurement into a posterior belief over distance requires inference with a prior. Usually, this prior represents beliefs about the stellar density distribution of the Milky Way. However, multiband photometry exists for a large fraction of the Gaia-TGAS Catalog and is incredibly informative about stellar distances. Here, we use 2MASS colors for 1.4 million TGAS stars to build a noise-deconvolved empirical prior distribution for stars in color-magnitude space. This model contains no knowledge of stellar astrophysics or the Milky Way but is precise because it accurately generates a large number of noisy parallax measurements under an assumption of stationarity; that is, it is capable of combining the information from many stars. We use the Extreme Deconvolution (XD) algorithm\textemdash which is an empirical-Bayes approximation to a full-hierarchical model of the true parallax and photometry of every star\textemdash to construct this prior. The prior is combined with a TGAS likelihood to infer a precise photometric-parallax estimate and uncertainty (and full posterior) for every star. Our parallax estimates are more precise than the TGAS catalog entries by a median factor of 1.2 (14\% are more precise by a factor {$>$}2) and they are more precise than the previous Bayesian distance estimates that use spatial priors. We validate our parallax inferences using members of the Milky Way star cluster M67, which is not visible as a cluster in the TGAS parallax estimates but appears as a cluster in our posterior parallax estimates. Our results, including a parallax posterior probability distribution function for each of 1.4 million TGAS stars, are available in companion electronic tables.}, - keywords = {catalogs,Hertzsprung–Russell and C–M diagrams,methods: statistical,parallaxes} + keywords = {catalogs,Hertzsprung\textendash Russell and C\textendash M diagrams,methods: statistical,parallaxes} } @article{Ando.Osaki1975, @@ -659,7 +658,7 @@ @article{Bazot.Bourguignon.ea2012 doi = {10.1111/j.1365-2966.2012.21818.x}, urldate = {2023-04-17}, abstract = {Determining the physical characteristics of a star is an inverse problem consisting of estimating the parameters of models for the stellar structure and evolution, and knowing certain observable quantities. We use a Bayesian approach to solve this problem for {$\alpha$} Cen A, which allows us to incorporate prior information on the parameters to be estimated, in order to better constrain the problem. Our strategy is based on the use of a Markov chain Monte Carlo (MCMC) algorithm to estimate the posterior probability densities of the stellar parameters: mass, age, initial chemical composition, etc. We use the stellar evolutionary code ASTEC to model the star. To constrain this model both seismic and non-seismic observations were considered. Several different strategies were tested to fit these values, using either two free parameters or five free parameters in ASTEC. We are thus able to show evidence that MCMC methods become efficient with respect to more classical grid-based strategies when the number of parameters increases. The results of our MCMC algorithm allow us to derive estimates for the stellar parameters and robust uncertainties thanks to the statistical analysis of the posterior probability densities. We are also able to compute odds for the presence of a convective core in {$\alpha$} Cen A. When using core-sensitive seismic observational constraints, these can rise above {$\sim$}40 per cent. The comparison of results to previous studies also indicates that these seismic constraints are of critical importance for our knowledge of the structure of this star.}, - keywords = {Astrophysics - Solar and Stellar Astrophysics,methods: numerical,methods: statistical,stars: evolution,stars: fundamental parameters,stars: individual: α Cen A,stars: oscillations}, + keywords = {Astrophysics - Solar and Stellar Astrophysics,methods: numerical,methods: statistical,stars: evolution,stars: fundamental parameters,stars: individual: {$\alpha$} Cen A,stars: oscillations}, annotation = {ADS Bibcode: 2012MNRAS.427.1847B} } @@ -772,7 +771,7 @@ @article{Berger.Huber.ea2020 keywords = {205,484,555,Catalogs,Exoplanet systems,Fundamental parameters of stars} } -@misc{Betancourt.Girolami2013, +@article{Betancourt.Girolami2013, title = {Hamiltonian {{Monte Carlo}} for {{Hierarchical Models}}}, author = {Betancourt, M. J. and Girolami, Mark}, year = {2013}, @@ -1152,7 +1151,7 @@ @article{Casagrande.Ramirez.ea2010 } @misc{Caswell.Lee.ea2022, - title = {Matplotlib/Matplotlib: {{REL}}: V3.6.2}, + title = {{{matplotlib/matplotlib: REL: v3.6.2}}}, shorttitle = {Matplotlib/Matplotlib}, author = {Caswell, Thomas A. and Lee, Antony and Droettboom, Michael and de Andrade, Elliott Sales and Hoffmann, Tim and Klymak, Jody and Hunter, John and Firing, Eric and Stansby, David and Varoquaux, Nelle and Nielsen, Jens Hedegaard and Root, Benjamin and May, Ryan and Elson, Phil and Sepp{\"a}nen, Jouni K. and Dale, Darren and Lee, Jae-Joon and Gustafsson, Oscar and McDougall, Damon and {hannah} and Straw, Andrew and Hobson, Paul and Lucas, Greg and Gohlke, Christoph and Vincent, Adrien F. and Yu, Tony S. and Ma, Eric and Silvester, Steven and Moad, Charlie and Kniazev, Nikita}, year = {2022}, @@ -1968,7 +1967,7 @@ @article{Eggenberger.Charbonnel.ea2004 doi = {10.1051/0004-6361:20034203}, urldate = {2023-04-17}, abstract = {Detailed models of {$\alpha$} Cen A and B based on new seismological data for {$\alpha$} Cen B by Carrier \& Bourban (\textbackslash cite\{ca03\}) have been computed using the Geneva evolution code including atomic diffusion. Taking into account the numerous observational constraints now available for the {$\alpha$} Cen system, we find a stellar model which is in good agreement with the astrometric, photometric, spectroscopic and asteroseismic data. The global parameters of the {$\alpha$} Cen system are now firmly constrained to an age of t=6.52 {$\pm$} 0.30 Gyr, an initial helium mass fraction Yi=0.275 {$\pm$} 0.010 and an initial metallicity (Z/X)i=0.0434 {$\pm$} 0.0020. Thanks to these numerous observational constraints, we confirm that the mixing-length parameter {$\alpha$} of the B component is larger than the one of the A component, as already suggested by many authors (Noels et al. \textbackslash cite\{no91\}; Fernandes \& Neuforge \textbackslash cite\{fe95\}; Guenther \& Demarque \textbackslash cite\{gu00\}): {$\alpha$}B is about 8\% larger than {$\alpha$}A ({$\alpha$}A=1.83 {$\pm$} 0.10 and {$\alpha$}B=1.97 {$\pm$} 0.10). Moreover, we show that asteroseismic measurements enable to determine the radii of both stars with a very high precision (errors smaller than 0.3\%). The radii deduced from seismological data are compatible with the new interferometric results of Kervella et al. (\textbackslash cite\{ke03\}) even if they are slightly larger than the interferometric radii (differences smaller than 1\%).}, - keywords = {Astrophysics,stars: binaries: visual,stars: evolution,stars: individual: α Cen,stars: oscillations}, + keywords = {Astrophysics,stars: binaries: visual,stars: evolution,stars: individual: {$\alpha$} Cen,stars: oscillations}, annotation = {ADS Bibcode: 2004A\&A...417..235E} } @@ -2549,11 +2548,13 @@ @article{Gould.Huber.ea2015 annotation = {ADS Bibcode: 2015JKAS...48...93G} } -@article{Goupil2017, +@inproceedings{Goupil2017, title = {Expected Asteroseismic Performances with the Space Project {{PLATO}}}, + booktitle = {Eur. {{Phys}}. {{J}}. {{Web Conf}}.}, author = {Goupil, Mariejo}, year = {2017}, month = oct, + series = {European {{Physical Journal Web}} of {{Conferences}}}, volume = {160}, pages = {01003}, doi = {10.1051/epjconf/201716001003}, @@ -3111,15 +3112,19 @@ @article{Houdayer.Reese.ea2021 } @article{Houdayer.Reese.ea2022, - title = {Properties of the Ionisation Glitch {{II}}. {{Seismic}} Signature of the Structural Perturbation}, + title = {Properties of the Ionisation Glitch. {{II}}. {{Seismic}} Signature of the Structural Perturbation}, author = {Houdayer, Pierre S. and Reese, Daniel R. and Goupil, Marie-Jo}, year = {2022}, - month = feb, - journal = {arXiv e-prints}, - urldate = {2022-04-26}, - abstract = {In the present paper, we aim to constrain the properties of the ionisation region in a star from the oscillation frequency variation (a so-called glitch) caused by rapid structural variations in this very region. In particular, we seek to avoid the use of calibration based on stellar models thus providing a truly independent estimate of these properties. These include both the helium abundance and other physical quantities that can have a significant impact on the oscillation frequencies such as the electronic degeneracy parameter or the extent of the ionisation region. Taking as a starting point our first paper, we applied structural perturbations of the ionisation zone to the wave equation for radial oscillations in an isentropic region. The resulting glitch model is thus able to exploit the information contained in the fast frequency oscillation caused by the helium ionisation but also in the slow trend accompanying that of hydrogen. This information can directly be expressed in terms of parameters related respectively to the helium abundance, electronic degeneracy and extent of the ionisation region. Using a Bayesian inference, we show that a substantial recovery of the properties at the origin of the glitch is possible. A degeneracy between the helium abundance and the electronic degeneracy is found to exist, which particularly affects the helium estimate. Extending the method to cases where the glitch is subject to contamination (e.g. surface effects), we noted the importance of the slow glitch trend associated with hydrogen ionisation. We propose using a Gaussian process to disentangle the frequency glitch from surface effects.}, - keywords = {Astrophysics - Solar and Stellar Astrophysics}, - annotation = {ADS Bibcode: 2022arXiv220204638H} + month = jul, + journal = {\aap}, + volume = {663}, + pages = {A60}, + issn = {0004-6361}, + doi = {10.1051/0004-6361/202243298}, + urldate = {2023-04-27}, + abstract = {Aims: In the present paper, we aim to constrain the properties of the ionisation region of a star from the oscillation frequency variation (a so-called glitch) caused by rapid structural variations in this very region. In particular, we seek tof avoid the use of calibration based on stellar models, thus providing a truly independent estimate of these properties. These include both the helium abundance and other physical quantities that can have a significant impact on the oscillation frequencies, such as the electronic degeneracy parameter or the extent of the ionisation region. Methods: Building on previous findings, we applied structural perturbations of the ionisation zone to the wave equation for radial oscillations in an isentropic region. The resulting glitch model is thus able to exploit the information contained in the fast frequency oscillation caused by the helium ionisation but also that in the slow trend accompanying the ionisation of hydrogen. This information can be directly expressed in terms of parameters related to the helium abundance, electronic degeneracy, and the extent of the ionisation region, respectively. Results: Using Bayesian inference, we show that substantial recovery of the properties at the origin of the glitch is possible. We find a degeneracy between the helium abundance and the electronic degeneracy, which particularly affects the helium estimate. Extending the method to cases where the glitch is subject to contamination (e.g., surface effects), we note the importance of the slow glitch trend associated with hydrogen ionisation. We propose the use of a Gaussian process to disentangle the frequency glitch from surface effects.}, + keywords = {asteroseismology,Astrophysics - Solar and Stellar Astrophysics,stars: abundances,stars: interiors,stars: oscillations}, + annotation = {ADS Bibcode: 2022A\&A...663A..60H} } @article{Houdek.Gough2007, @@ -3482,15 +3487,19 @@ @article{Kashani.Ivry2021 } @article{Katz.Danielski.ea2022, - title = {Bayesian {{Characterisation}} of {{Circumbinary Exoplanets}} with {{LISA}}}, + title = {Bayesian Characterization of Circumbinary Sub-Stellar Objects with {{LISA}}}, author = {Katz, Michael L. and Danielski, Camilla and Karnesis, Nikolaos and Korol, Valeriya and Tamanini, Nicola and Cornish, Neil J. and Littenberg, Tyson B.}, year = {2022}, - month = may, - journal = {arXiv e-prints}, - urldate = {2022-05-17}, - abstract = {The Laser Interferometer Space Antenna (LISA) will detect and characterize \$\textbackslash sim10\^4\$ Galactic Binaries consisting predominantly of two White Dwarfs (WD). An interesting prospect within this population is a third object--another WD star, a Circumbinary Exoplanet (CBP), or a Brown Dwarf (BD)--in orbit about the inner WD pair. We present the first fully Bayesian detection and posterior analysis of sub-stellar objects with LISA, focusing on the characterization of CBPs. We used an optimistic astrophysically motivated catalogue of these CBP third-body sources, including their orbital eccentricity around the inner binary for the first time. We examined Bayesian Evidence computations for detectability, as well as the effects on the posterior distributions for both the inner binary parameters and the third-body parameters. We find that the posterior behavior bifurcates based on whether the third-body period is above or below half the observation time. Additionally, we find that undetectable third-body sources can bias the inner binary parameters whether or not the correct template is used. We used the information retrieved from the study of the CBP population to make an initial conservative prediction for the number of detectable BD systems in the original catalogue. We end with commentary on the predicted qualitative effects on LISA global fitting and Galactic Binary population analysis. The procedure used in this work is generic and can be directly applied to other astrophysical effects expected within the Galactic Binary population.}, - keywords = {Astrophysics - Earth and Planetary Astrophysics,Astrophysics - Instrumentation and Methods for Astrophysics,Astrophysics - Solar and Stellar Astrophysics,General Relativity and Quantum Cosmology}, - annotation = {ADS Bibcode: 2022arXiv220503461K} + month = nov, + journal = {\mnras}, + volume = {517}, + pages = {697--711}, + issn = {0035-8711}, + doi = {10.1093/mnras/stac2555}, + urldate = {2023-04-27}, + abstract = {The Laser Interferometer Space Antenna (LISA) will detect and characterize \textasciitilde 104 Galactic Binaries, consisting predominantly of two white dwarfs (WDs). An interesting prospect within this population is a third object - another WD star, a circumbinary exoplanet (CBP), or a brown dwarf (BD) - in orbit about the inner WD pair. We present the first fully Bayesian detection and posterior analysis of substellar objects with LISA, focusing on the characterization of CBPs. We used an optimistic astrophysically motivated catalogue of these CBP third-body sources, including their orbital eccentricity around the inner binary for the first time. We examined Bayesian evidence computations for detectability, as well as the effects on the posterior distributions for both the inner binary parameters and the third-body parameters. We find that the posterior behaviour bifurcates based on whether the third-body period is above or below half the observation time. Additionally, we find that undetectable third-body sources can bias the inner binary parameters whether or not the correct template is used. We used the information retrieved from the study of the CBP population to make an initial conservative prediction for the number of detectable BD systems in the original catalogue. We end with commentary on the predicted qualitative effects on LISA global fitting and Galactic Binary population analysis. The procedure used in this work is generic and can be directly applied to other astrophysical effects expected within the Galactic Binary population.}, + keywords = {Astrophysics - Earth and Planetary Astrophysics,Astrophysics - Instrumentation and Methods for Astrophysics,Astrophysics - Solar and Stellar Astrophysics,General Relativity and Quantum Cosmology,gravitational waves,planets and satellites: detection,white dwarfs}, + annotation = {ADS Bibcode: 2022MNRAS.517..697K} } @article{Khan.Hall.ea2018, @@ -3640,7 +3649,7 @@ @article{Kjeldsen.Bedding.ea2005 doi = {10.1086/497530}, urldate = {2020-09-21}, abstract = {We have made velocity observations of the star {$\alpha$} Centauri B from two sites, allowing us to identify 37 oscillation modes with l=0-3. Fitting to these modes gives the large and small frequency separations as a function of frequency. The mode lifetime, as measured from the scatter of the oscillation frequencies about a smooth trend, is similar to that in the Sun. Limited observations of the star {$\delta$} Pav show oscillations centered at 2.3 mHz, with peak amplitudes close to solar. We introduce a new method of measuring oscillation amplitudes from heavily smoothed power density spectra, from which we estimated amplitudes for {$\alpha$} Cen {$\alpha$} and B, {$\beta$} Hyi, {$\delta$} Pav, and the Sun. We point out that the oscillation amplitudes may depend on which spectral lines are used for the velocity measurements. Based on observations collected at the European Southern Observatory, Paranal, Chile (ESO Programme 71.D-0618).}, - keywords = {stars: individual (δ Pavonis),Stars: Individual: Constellation Name: α Centauri A,Stars: Individual: Constellation Name: α Centauri B,Stars: Individual: Constellation Name: β Hydri,Stars: Oscillations,Sun: Helioseismology} + keywords = {stars: individual ({$\delta$} Pavonis),Stars: Individual: Constellation Name: {$\alpha$} Centauri A,Stars: Individual: Constellation Name: {$\alpha$} Centauri B,Stars: Individual: Constellation Name: {$\beta$} Hydri,Stars: Oscillations,Sun: Helioseismology} } @article{Kjeldsen.Bedding.ea2008, @@ -3654,7 +3663,7 @@ @article{Kjeldsen.Bedding.ea2008 doi = {10.1086/591667}, urldate = {2020-09-18}, abstract = {In helioseismology, there is a well-known offset between observed and computed oscillation frequencies. This offset is known to arise from improper modeling of the near-surface layers of the Sun, and a similar effect must occur for models of other stars. Such an effect impedes progress in asteroseismology, which involves comparing observed oscillation frequencies with those calculated from theoretical models. Here, we use data for the Sun to derive an empirical correction for the near-surface offset, which we then apply to three other stars ({$\alpha$} Cen A, {$\alpha$} Cen B, and {$\beta$} Hyi). The method appears to give good results, in particular providing an accurate estimate of the mean density of each star.}, - keywords = {stars: individual: β Hyi α Cen A α Cen B,stars: oscillations,Sun: helioseismology} + keywords = {stars: individual: {$\beta$} Hyi {$\alpha$} Cen A {$\alpha$} Cen B,stars: oscillations,Sun: helioseismology} } @article{Kjeldsen.Bedding1995, @@ -3669,7 +3678,7 @@ @article{Kjeldsen.Bedding1995 issn = {0004-6361}, urldate = {2020-09-02}, abstract = {There are no good predictions for the amplitudes expected from solar-like oscillations in other stars. In the absence of a definitive model for convection, which is thought to be the mechanism that excites these oscillations, the amplitudes for both velocity and luminosity measurements must be estimated by scaling from the Sun. In the case of luminosity measurements, even this is difficult because of disagreement over the solar amplitude. This last point has lead us to investigate whether the luminosity amplitude of oscillations {$\delta$}L/L can be derived from the velocity amplitude (v\_osc\_). Using linear theory and observational data, we show that p-mode oscillations in a large sample of pulsating stars satisfy ({$\delta$}L/L)\_bol\_\{prop.to\} v\_osc\_/T\_eff\_. Using this relationship, together with the best estimate of v\_osc\_,sun\_=(23.4+/-1.4)cm/s, we estimate the luminosity amplitude of solar oscillations at 550nm to be {$\delta$}L/L=(4.7+/-0.3)ppm. Next we discuss how to scale the amplitude of solar-like (i.e., convectively-powered) oscillations from the Sun to other stars. The only predictions come from model calculations by Christensen-Dalsgaard \& Frandsen (1983, Sol. Phys. 82, 469). However, their grid of stellar models is not dense enough to allow amplitude predictions for an arbitrary star. Nevertheless, although convective theory is complicated, we might expect that the general properties of convection - including oscillation amplitudes - should change smoothly through the colour-magnitude diagram. Indeed, we find that the velocity amplitudes predicted by the model calculations are well fitted by the relation v\_osc\_\{prop.to\}L/M. These two relations allow us to predict both the velocity and luminosity amplitudes of solar-like oscillations in any given star. We compare these predictions with published observations and evaluate claims for detections that have appeared in the literature. We argue that there is not yet good evidence for solar-like oscillations in any star except the Sun. For solar-type stars (e.g., {$\alpha$} Cen A and {$\beta$} Hyi), observations have not yet reached sufficient sensitivity to detect the amplitudes we predict. For some F-type stars, namely Procyon and several members of M67, detection sensitivities 30-40\% below the predicted amplitudes have been achieved. We conclude that these stars must oscillate with amplitudes less than has generally been assumed.}, - keywords = {{DELTA} SCT,CEPHEIDS,STARS: INDIVIDUAL: {ALPHA} CEN,STARS: INDIVIDUAL: PROCYON,STARS: OSCILLATIONS,SUN: OSCILLATIONS} + keywords = {\{DELTA\} SCT,CEPHEIDS,STARS: INDIVIDUAL: \{ALPHA\} CEN,STARS: INDIVIDUAL: PROCYON,STARS: OSCILLATIONS,SUN: OSCILLATIONS} } @misc{Koposov.Speagle.ea2023, @@ -3968,12 +3977,14 @@ @article{Lund.Miesch.ea2014 annotation = {ADS Bibcode: 2014ApJ...790..121L} } -@article{Lund.Reese2018, +@inproceedings{Lund.Reese2018, title = {Tutorial: {{Asteroseismic Stellar Modelling}} with {{AIMS}}}, shorttitle = {Tutorial}, + booktitle = {Asteroseismol. {{Exopl}}. {{List}}. {{Stars Search}}. {{New Worlds}}}, author = {Lund, Mikkel N. and Reese, Daniel R.}, year = {2018}, month = jan, + series = {Astrophysics and {{Space Science Proceedings}}}, volume = {49}, pages = {149}, address = {{eprint: arXiv:1711.01896}}, @@ -4306,20 +4317,25 @@ @article{Miglio.Montalban2005 doi = {10.1051/0004-6361:20052988}, urldate = {2023-04-17}, abstract = {We apply the Levenberg-Marquardt minimization algorithm to seismic and classical observables of the {$\alpha$}Cen binary system in order to derive the fundamental parameters of {$\alpha$}CenA+B, and to analyze the dependence of these parameters on the chosen observables, on their uncertainty, and on the physics used in stellar modelling. We show that while the fundamental stellar parameters do not depend on the treatment of convection adopted (Mixing Length Theory - MLT - or ``Full Spectrum of Turbulence'' - FST), the age of the system depends on the inclusion of gravitational settling, and is deeply biased by the small frequency separation of component B. We try to answer the question of the universality of the mixing length parameter, and we find a statistically reliable dependence of the {$\alpha$}-parameter on the HR diagram location (with a trend similar to the predictions based on 2-D simulations). We propose the frequency separation ratios as better observables to determine the fundamental stellar parameters, and to use the large frequency separation and frequencies to extract information about the stellar structure. The effects of diffusion and equation of state on the oscillation frequencies are also studied, but present seismic data do not allow their determination.}, - keywords = {Astrophysics,stars: fundamental parameters,stars: individual: α Cen,stars: interiors,stars: oscillations}, + keywords = {Astrophysics,stars: fundamental parameters,stars: individual: {$\alpha$} Cen,stars: interiors,stars: oscillations}, annotation = {ADS Bibcode: 2005A\&A...441..615M} } -@article{Moedas.Nsamba.ea2020, - title = {Asteroseismic Stellar Modelling: Systematics from the Treatment of the Initial Helium Abundance}, - shorttitle = {Asteroseismic Stellar Modelling}, +@inproceedings{Moedas.Nsamba.ea2020, + title = {Asteroseismic {{Stellar Modelling}}: {{Systematics}} from the {{Treatment}} of the {{Initial Helium Abundance}}}, + shorttitle = {Asteroseismic {{Stellar Modelling}}}, + booktitle = {Dyn. {{Sun Stars}}}, author = {Moedas, Nuno and Nsamba, Benard and Clara, Miguel T.}, + editor = {Monteiro, M{\'a}rio J. P. F. G. and Garc{\'i}a, Rafael A. and {Christensen-Dalsgaard}, J{\o}rgen and McIntosh, Scott W.}, year = {2020}, - month = jul, - journal = {arXiv e-prints}, - urldate = {2021-02-11}, - abstract = {Despite the fact that the initial helium abundance is an essential ingredient in modelling solar-type stars, its abundance in these stars remains a poorly constrained observational property. This is because the effective temperature in these stars is not high enough to allow helium ionization, not allowing any conclusions on its abundance when spectroscopic techniques are employed. To this end, stellar modellers resort to estimating the initial helium abundance via a semi-empirical helium-to-heavy element ratio, anchored to the the standard Big Bang nucleosynthesis value. Depending on the choice of solar composition used in stellar model computations, the helium-to-heavy element ratio, (\$\textbackslash Delta Y/\textbackslash Delta Z\$) is found to vary between 1 and 3. In this study, we use the Kepler "LEGACY" stellar sample, for which precise seismic data is available, and explore the systematic uncertainties on the inferred stellar parameters (radius, mass, and age) arising from adopting different values of \$\textbackslash Delta Y/\textbackslash Delta Z\$, specifically, 1.4 and 2.0. The stellar grid constructed with a higher \$\textbackslash Delta Y / \textbackslash Delta Z\$ value yields lower radius and mass estimates. We found systematic uncertainties of 1.1 per cent, 2.6 per cent, and 13.1 per cent on radius, mass, and ages, respectively.}, - keywords = {Astrophysics - Solar and Stellar Astrophysics} + series = {Astrophysics and {{Space Science Proceedings}}}, + pages = {259--265}, + publisher = {{Springer International Publishing}}, + address = {{Cham}}, + doi = {10.1007/978-3-030-55336-4_34}, + abstract = {Despite the fact that the initial helium abundance is an essential ingredient in modelling solar-type stars, its abundance in these stars remains a poorly constrained observational property. This is because the effective temperature in these stars is not high enough to allow helium ionization, not allowing any conclusions on its abundance when spectroscopic techniques are employed. To this end, stellar modellers resort to estimating the initial helium abundance via a semi-empirical helium-to-heavy element ratio, anchored to the standard Big Bang nucleosynthesis value. Depending on the choice of solar composition used in stellar model computations, the helium-to-heavy element ratio, ({$\Delta$}Y/\,{$\Delta$}Z) is found to vary between 1 and 3. In this study, we use the Kepler LEGACY stellar sample, for which precise seismic data is available, and explore the systematic uncertainties on the inferred stellar parameters (radius, mass, and age) arising from adopting different values of {$\Delta$}Y/\,{$\Delta$}Z, specifically, 1.4 and 2.0. The stellar grid constructed with a higher {$\Delta$}Y/\,{$\Delta$}Z value yields lower radius and mass estimates. We found systematic uncertainties of 1.1\%, 2.6\%, and 13.1\% on radius, mass, and ages, respectively.}, + isbn = {978-3-030-55336-4}, + langid = {english} } @article{Mohammadi.Challenor.ea2022, @@ -4738,7 +4754,7 @@ @article{Nsamba.Monteiro.ea2018 doi = {10.1093/mnrasl/sly092}, urldate = {2023-04-17}, abstract = {Understanding the physical process responsible for the transport of energy in the core of {$\alpha$} Centauri A is of the utmost importance if this star is to be used in the calibration of stellar model physics. Adoption of different parallax measurements available in the literature results in differences in the interferometric radius constraints used in stellar modelling. Further, this is at the origin of the different dynamical mass measurements reported for this star. With the goal of reproducing the revised dynamical mass derived by Pourbaix \& Boffin, we modelled the star using two stellar grids varying in the adopted nuclear reaction rates. Asteroseismic and spectroscopic observables were complemented with different interferometric radius constraints during the optimization procedure. Our findings show that best-fitting models reproducing the revised dynamical mass favour the existence of a convective core ({$\greaterequivlnt$}70 per cent of best-fitting models), a result that is robust against changes to the model physics. If this mass is accurate, then {$\alpha$} Centauri A may be used to calibrate stellar model parameters in the presence of a convective core.}, - keywords = {Astrophysics - Solar and Stellar Astrophysics,method: asteroseismology,stars: convection and radiation,stars: fundamental parameters,stars: α Centauri A}, + keywords = {Astrophysics - Solar and Stellar Astrophysics,method: asteroseismology,stars: convection and radiation,stars: fundamental parameters,stars: {$\alpha$} Centauri A}, annotation = {ADS Bibcode: 2018MNRAS.479L..55N} } @@ -4882,8 +4898,9 @@ @phdthesis{Payne1925 author = {Payne, Cecilia Helena}, year = {1925}, month = jan, - journal = {Ph.D. Thesis}, + journal = {Annals of Harvard College Observatory}, urldate = {2023-04-05}, + school = {Harvard University}, keywords = {Astronomy}, annotation = {ADS Bibcode: 1925PhDT.........1P} } @@ -5707,11 +5724,13 @@ @article{SilvaAguirre.Stello.ea2020 langid = {english} } -@article{Skilling2004, +@inproceedings{Skilling2004, title = {Nested {{Sampling}}}, + booktitle = {24th {{Int}}. {{Workshop Bayesian Inference Maximum Entropy Methods Sci}}. {{Eng}}.}, author = {Skilling, John}, year = {2004}, month = nov, + series = {American {{Institute}} of {{Physics Conference Series}}}, volume = {735}, pages = {395--405}, doi = {10.1063/1.1835238}, @@ -5738,11 +5757,13 @@ @article{Skilling2006 keywords = {algorithm,annealing,Bayesian computation,evidence,marginal likelihood,Model selection,nest,phase change} } -@article{Skilling2009, +@inproceedings{Skilling2009, title = {Nested {{Sampling}}'s {{Convergence}}}, + booktitle = {29th {{Int}}. {{Workshop Bayesian Inference Maximum Entropy Methods Sci}}. {{Eng}}.}, author = {Skilling, John}, year = {2009}, month = dec, + series = {American {{Institute}} of {{Physics Conference Series}}}, volume = {1193}, pages = {277--291}, doi = {10.1063/1.3275625}, @@ -6047,16 +6068,15 @@ @incollection{Teh.Jordan2010 isbn = {978-0-521-51346-3} } -@misc{Tenachi.Ibata.ea2023, +@article{Tenachi.Ibata.ea2023, title = {Deep Symbolic Regression for Physics Guided by Units Constraints: Toward the Automated Discovery of Physical Laws}, shorttitle = {Deep Symbolic Regression for Physics Guided by Units Constraints}, author = {Tenachi, Wassim and Ibata, Rodrigo and Diakogiannis, Foivos I.}, year = {2023}, month = mar, - number = {arXiv:2303.03192}, + journal = {arXiv e-prints}, eprint = {2303.03192}, primaryclass = {astro-ph, physics:physics}, - publisher = {{arXiv}}, doi = {10.48550/arXiv.2303.03192}, urldate = {2023-03-15}, abstract = {Symbolic Regression is the study of algorithms that automate the search for analytic expressions that fit data. While recent advances in deep learning have generated renewed interest in such approaches, efforts have not been focused on physics, where we have important additional constraints due to the units associated with our data. Here we present \$\textbackslash Phi\$-SO, a Physical Symbolic Optimization framework for recovering analytical symbolic expressions from physics data using deep reinforcement learning techniques by learning units constraints. Our system is built, from the ground up, to propose solutions where the physical units are consistent by construction. This is useful not only in eliminating physically impossible solutions, but because it restricts enormously the freedom of the equation generator, thus vastly improving performance. The algorithm can be used to fit noiseless data, which can be useful for instance when attempting to derive an analytical property of a physical model, and it can also be used to obtain analytical approximations to noisy data. We showcase our machinery on a panel of examples from astrophysics.}, @@ -6064,14 +6084,17 @@ @misc{Tenachi.Ibata.ea2023 keywords = {Astrophysics - Instrumentation and Methods for Astrophysics,Computer Science - Machine Learning,Physics - Computational Physics} } -@article{Thompson.Christensen-Dalsgaard2002, +@inproceedings{Thompson.Christensen-Dalsgaard2002, title = {On Inverting Asteroseismic Data}, + booktitle = {First {{Eddington Workshop Stellar Struct}}. {{Habitable Planet Find}}.}, author = {Thompson, M. J. and {Christensen-Dalsgaard}, J.}, year = {2002}, month = jan, + series = {{{ESA Special Publication}}}, volume = {485}, pages = {95--101}, address = {{eprint: arXiv:astro-ph/0110447}}, + doi = {10.48550/arXiv.astro-ph/0110447}, urldate = {2023-01-20}, abstract = {Some issues of inverting asteroseismic frequency data are discussed, including the use of model calibration and linearized inversion. An illustrative inversion of artificial data for solar-type stars, using least-squares fitting of a small set of basis functions, is presented. A few details of kernel construction are also given.}, keywords = {Astrophysics,Stars: Oscillations,Stars: Structure}, From 87f005947b8ad9fe4e67a55b679208488ed8b971 Mon Sep 17 00:00:00 2001 From: Alexander Lyttle Date: Fri, 28 Apr 2023 00:42:36 +0100 Subject: [PATCH 41/50] Further update bib --- references.bib | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/references.bib b/references.bib index 623a7ba..5a5f917 100644 --- a/references.bib +++ b/references.bib @@ -1414,12 +1414,12 @@ @article{Christensen-Dalsgaard.Monteiro.ea1995 annotation = {ADS Bibcode: 1995MNRAS.276..283C} } -@article{Christensen-Dalsgaard.PerezHernandez1991, +@incollection{Christensen-Dalsgaard.PerezHernandez1991, title = {Influence of the {{Upper Layers}} of the {{Sun}} on the P-{{Mode Frequencies}}}, + booktitle = {Challenges to {{Theories}} of the {{Structure}} of {{Moderate-Mass Stars}}}, author = {{Christensen-Dalsgaard}, J. and P{\'e}rez Hern{\'a}ndez, F.}, year = {1991}, month = jan, - journal = {Chall. Theor. Struct. Moderate-Mass Stars}, volume = {388}, pages = {43}, doi = {10.1007/3-540-54420-8_49}, From 070d42a8415e4770e79f41873acbea0a7eac5a36 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Fri, 28 Apr 2023 10:44:42 +0100 Subject: [PATCH 42/50] Finish HBM chapter --- chapters/hbm.tex | 51 +++++++++++++++++++++++++++--------------------- 1 file changed, 29 insertions(+), 22 deletions(-) diff --git a/chapters/hbm.tex b/chapters/hbm.tex index bfe8e08..f37f097 100644 --- a/chapters/hbm.tex +++ b/chapters/hbm.tex @@ -16,20 +16,20 @@ \chapter{Hierarchical Bayesian Models}\label{chap:hbm} \section{Introduction} -When modelling a single star, we can use a variant of Bayes' theorem (Equation \ref{eq:bayes}) to estimate the posterior probability of its parameters. We choose priors to represent our current knowledge. For example, if we observe a star in the Milky Way at random, we expect its mass to belong to an Initial Mass Function \citep[IMF; e.g.][]{Chabrier2003} without other prior knowledge. In fact, an IMF prior is an input option for BASTA \citep{AguirreBorsen-Koch.Rorsted.ea2022}. Other prior assumptions include a helium enrichment law which determines the star's initial helium abundance from its metallicity, or that its age must not be older than the universe. +When modelling a single star, we can use Bayes' theorem (Equation \ref{eq:bayes}) to estimate the posterior probability of its parameters. We choose priors to represent our current knowledge. If we observe a star at random in the Milky Way, we expect its mass, age, and chemical composition to belong to some empirical or physical distribution. For example, the initial mass function \citep[IMF; e.g.][]{Chabrier2003} is used as a prior on mass by the BASTA stellar inference code \citep{AguirreBorsen-Koch.Rorsted.ea2022}. Another prior could restrict the age of the star to no older than the universe. -Let us consider modelling a large population of stars simultaneously. For example, we may want to create a catalogue of stellar parameters to use in exoplanet and galactic research. We could treat the parameters for each star independently, repeating the modelling procedure for each star in the sample. However, we know that stars belong to population distributions like the IMF. In some cases, these population distributions are not well understood and assuming one exactly can introduce bias into our model. Instead, it could be better to let the data inform such population priors. +Let us consider modelling a large population of stars simultaneously. For example, we may want to create a catalogue of stellar parameters to use in exoplanet and galactic research. We could treat the parameters for each star independently, repeating the modelling procedure for each star in the sample. However, we know that stars belong to population distributions like the IMF. In some cases, these population distributions are not well understood and assuming one exactly can introduce bias into our model. Instead, we could let the data inform such population-level priors. -Hierarchical (or multi-level) Bayesian models parametrise population-level prior distributions on individual stellar parameters. These population-parameters (\emph{hyperparameters}) which govern the population must themselves have prior distributions as per the Bayesian formalisation. The hierarchical aspect comes from the distinction between hyperparameters which take one value across a population and those which vary from star-to-star. In the next section, we go through a simple example hierarchical model. Then, we discuss a few parameters which could be given the hierarchical treatment in Section \ref{sec:hbm-phys}. +Hierarchical (or multi-level) Bayesian models parametrise population-level prior distributions on individual stellar parameters. The parameters of these population distributions (or \emph{hyperparameters}) should themselves have prior distributions as per the Bayesian formalisation. The hierarchical aspect comes from the distinction between hyperparameters, which take one value across a population, and parameters which vary from star-to-star. In the next section, we demonstrate a simple hierarchical model example. Then, we discuss a few parameters which could be given the hierarchical treatment in Section \ref{sec:hbm-phys}. \section[Stellar Distances]{Stellar Distances in an Open Cluster Analogue}\label{sec:hbm-dist} \newcommand{\appmag}{\ensuremath{\mathrm{v}}} \newcommand{\absmag}{\ensuremath{\mathrm{V}}} -In this section, we use the example of measuring distances to stars in an open cluster to demonstrate a hierarchical Bayesian model (HBM). This example is loosely based on the work of \citet{Leistedt.Hogg2017}, which presents a hierarchical model of the colour-magnitude diagram to improve distances from \emph{Gaia} \citep{GaiaCollaboration.Prusti.ea2016}. However, instead of considering the population distributions over magnitude and colour, we build a hierarchy over the stellar distances. +In this section, we demonstrate a hierarchical Bayesian model (HBM) with an analogy to measuring distances to stars in an open cluster. This example is similar to \citet{Leistedt.Hogg2017}, who present a hierarchical model of the colour-magnitude diagram to improve distances from \emph{Gaia} \citep{GaiaCollaboration.Prusti.ea2016}. However, instead of considering the population distributions over magnitude and colour, we build a hierarchy over the stellar distances. -We created a dataset analogous to an open cluster of \(N_\mathrm{stars}=20\) stars. We gave each \(i\)-th star a dimensionless distance (\(d_i\)) from the observer drawn randomly from a normal distribution with a mean of 10 and standard deviation of 0.1. Then, we converted these to dimensionless parallax using the relation \(\varpi = 1/d\). We also gave each star an absolute visual magnitude (\(\absmag_i\)) drawn randomly from a standard normal distribution. To get a quantity proportional to apparent magnitude (\(\appmag_i\)) for each star, we used the relation \(\appmag_i = \absmag_i + 5 \log_{10} d_i\). For the purpose of this example, we ignored additional real-world effects such as extinction and reddening. +We created a dataset analogous to an open cluster of \(N_\mathrm{stars}=20\) stars. We gave each \(i\)-th star a dimensionless distance from the observer (\(d_i\)) drawn randomly from a normal distribution with a mean of 10 and standard deviation of 0.1. Then, we converted these to dimensionless parallax using the relation \(\varpi = 1/d\). We also gave each star an absolute visual magnitude (\(\absmag_i\)) drawn randomly from a standard normal distribution. To get a quantity proportional to apparent magnitude (\(\appmag_i\)) for each star, we used the relation \(\appmag_i = \absmag_i + 5 \log_{10} d_i\). For the purpose of this example, we ignored additional real-world effects such as extinction and reddening. % \begin{table}[tb] % \centering @@ -40,7 +40,7 @@ \section{Introduction} We simulated noisy observations of \(\varpi_i\) and \(\appmag_i\) by randomly drawing from a normal distribution centred on their true values with standard deviations of \(\sigma_{\appmag,i} = 0.1\) and \(\sigma_{\varpi,i} = 0.01\) respectively. % We repeated this for \(N_\mathrm{stars}=20\) stars and present the true values and observables in Table \ref{tab:hbm-data}. -For real-world context, the uncertainties on \(\varpi\) from \emph{Gaia} \citep{GaiaCollaboration.Prusti.ea2016} Data Release 3 \citep[][]{GaiaCollaboration.Vallenari.ea2022} are approximately \SI{0.02}{\milli\aarcsec} for stars with \emph{Gaia} G-band magnitudes of less than 15. Therefore, if our choice of \(\sigma_{\varpi,i}\) was representative of \emph{Gaia} uncertainties, then the distance to our stellar cluster would be \(\sim \SI{5}{\kilo\parsec}\). An example open cluster at this distance is NGC 6791, at \SI{4}{\kilo\parsec} \citep{Brogaard.Bruntt.ea2011}. We note that by the same comparison, our chosen spread in dimensionless distances of 0.1 corresponds to an order of magnitude more than typical cluster sizes of a few parsecs. However, we chose to exaggerate the distance spread to make it easier to detect for the purpose of this example. +For real-world context, the uncertainties on \(\varpi\) from \emph{Gaia} Data Release 3 \citep[DR3;][]{GaiaCollaboration.Vallenari.ea2022} are approximately \SI{0.02}{\milli\aarcsec} for stars with \emph{Gaia} G-band magnitudes of less than 15. Therefore, if our choice of \(\sigma_{\varpi,i}\) was representative of \emph{Gaia} uncertainties, then the distance to our stellar cluster would be \(\sim \SI{5}{\kilo\parsec}\). An example open cluster at this distance is NGC 6791, at \SI{4}{\kilo\parsec} \citep{Brogaard.Bruntt.ea2011}. Typical cluster sizes are a few parsecs. On the same scale, the distance spread in our cluster analogue is an order of magnitude larger. However, we chose to exaggerate the distance spread to make it easier to measure and are not making a case for applying this example to the real-world. In Section \ref{sec:simple-model}, we describe a simple Bayesian model for determining distances and absolute magnitudes of stars in the cluster. Then, we define the HBM in Section \ref{sec:hbm-model} which incorporates a population-level distribution over distance. We outline our method for inferring model parameters in Section \ref{sec:hbm-inf} and then compare results from the models in Section \ref{sec:hbm-comp}. Finally, we explore how the HBM scales with the number of stars observed in Section \ref{sec:hbm-scale}. @@ -62,7 +62,7 @@ \subsection{Simple Model}\label{sec:simple-model} % where \(\mathcal{N}(x \,|\, \mu, \sigma^2)\) is a normal distribution over \(x\) with a mean of \(\mu\) and variance of \(\sigma^2\). -We assumed stars in the cluster were equally likely to be between a distance of 0 and 20. Moreover, we assumed the absolute magnitudes were likely to be normally distributed centred on 0 and scaled by 10. Therefore, the prior probability of the model parameters was, +We assumed stars in the cluster were equally likely to be at a distance from 0 and 20. Moreover, we assumed the absolute magnitudes were normally distributed centred on 0 and scaled by 10. Therefore, the prior probability of the model parameters was, % \begin{equation} p(d_i, \absmag_i) = \mathcal{U}(d_i \mid 0, 20) \, \mathcal{N}(\absmag_i \mid 0, 100), @@ -78,17 +78,16 @@ \subsection{Simple Model}\label{sec:simple-model} \label{fig:simple-pgm} \end{figure} -In Figure \ref{fig:simple-pgm}, we show a probabilistic graphical model of the simple model. This shows the connections between parameters in the model. There is no hierarchy in this model because each parameter exists within the box, meaning no parameters are shared between stars. However, we know that the stellar distances are correlated. For example, it is highly unlikely that one star in the cluster is at a distance of 5 and another is at a distance of 15. If we can exploit this expectation, we could improve our prior and thus improve the inference of \(d_i\) and \(\absmag_i\). +In Figure \ref{fig:simple-pgm}, we show a probabilistic graphical model of the simple model. This shows the connections between parameters in the model. There is no hierarchy in this model because each parameter exists within the box, meaning no parameters are shared between stars. However, we know that the stellar distances are correlated. For example, their distances should be tightly related; it is unlikely that one star in the cluster is at a distance of 5 and another is at a distance of 15. If we can exploit this expectation, we could improve our prior and thus reduce our uncertainty in \(d_i\) and \(\absmag_i\). \subsection{Hierarchical Model}\label{sec:hbm-model} -In this section, we present an HBM which includes the known correlation between distances to the stars in this open cluster analogue. We assumed the stars are all members of the same open cluster. Therefore, their distances can be modelled by some tight distribution. For this example, we assumed that each distance is drawn from a normal distribution characterised by new \emph{hyperparameters} \(\mu_d\) and \(\sigma_d\), +In this section, we present an HBM which includes expected correlation between distances to the stars in this open cluster analogue. We assumed the stars are all members of the same open cluster. Therefore, their distances can be modelled by some tight distribution. For this example, we assumed that each distance is drawn from a normal distribution characterised by hyperparameters \(\mu_d\) and \(\sigma_d\), % \begin{equation} d_i \sim \mathcal{N}(\mu_d, \sigma_d^2). \end{equation} % -The hyperparameters are so-called because they take a single value for the population of stars. Hence, the hierarchy of the model arises as some parameters represent how individual parameters are distributed in the population. Each stellar distance, \(d_i\), is no longer treated independently by the model. Hence, we modified the posterior probability distribution to account for this correlation, % @@ -104,7 +103,7 @@ \subsection{Hierarchical Model}\label{sec:hbm-model} % Our posterior now depends on all stars in the population, so we used bold symbols to represent the set of individual stellar parameters. For example, \(\vect{d} \equiv d_1, \dots, d_{N_\mathrm{stars}}\). -We modelled \(d_i\) as a random variable which depends on the hyperparameters. This is why our prior became the product of a conditional distribution, \(p(\vect{d} \,|\, \mu_d, \sigma_d)\) and a prior on the remaining independent parameters, \(p(\mu_d, \sigma_d, \vect{\absmag})\). We write the conditional distribution for distance as a product of normal distributions, +We modelled \(d_i\) as a random variable which depends on the hyperparameters. This is why our prior became the product of a conditional distribution, \(p(\vect{d} \,|\, \mu_d, \sigma_d)\) and a prior on the remaining independent parameters, \(p(\mu_d, \sigma_d, \vect{\absmag})\). We write the conditional distribution for distance as a product of independent draws from the population-prior, % \begin{equation} p(\vect{d} \mid \mu_d, \sigma_d) = \prod_{i=1}^{N_\mathrm{stars}} \mathcal{N}(d_i \mid \mu_d, \sigma_d^2), @@ -116,7 +115,12 @@ \subsection{Hierarchical Model}\label{sec:hbm-model} p(\mu_d, \sigma_d) = \mathcal{U}(\mu_d \mid 0, 20) \, \mathcal{N}(\ln\sigma_d \mid - \ln 10, 1). \end{equation} % -We chose to use a log-normal prior for \(\sigma_d\) to ensure that the parameter is positive. The standard deviation of the prior on \(\sigma_d\) approximately corresponds to a fractional uncertainty of 100 per cent. Finally, we inherited the prior on \(\absmag_i\) from the simple model, \(p(\vect{\absmag}) = \prod_{i=1}^{N_\mathrm{stars}} \mathcal{N}(\absmag_i \,|\, 0, 100)\). +We chose to use a log-normal prior for \(\sigma_d\) to ensure that the parameter is positive. The standard deviation of the prior on \(\sigma_d\) approximately corresponds to a fractional uncertainty of 1. Finally, we inherited the prior on \(\absmag_i\) from the simple model, +% +\begin{equation} + p(\vect{\absmag}) = \prod_{i=1}^{N_\mathrm{stars}} \mathcal{N}(\absmag_i \mid 0, 100). +\end{equation} +% \begin{figure}[tb] \centering @@ -129,9 +133,11 @@ \subsection{Hierarchical Model}\label{sec:hbm-model} \subsection{Inferring the Model Parameters}\label{sec:hbm-inf} -To infer the model parameters, we need to calculate the marginalised posterior distributions for each parameter. We could obtain these analytically by integrating the full posterior distribution over all model parameters except for the parameter of interest. Alternatively, we can approximate the marginalised posterior using a Markov Chain Monte Carlo (MCMC) sampling algorithm. We chose the latter approach because it is scalable to more complicated models where the marginalisation is not analytically solvable. +To infer the model parameters, we need to calculate the marginalised posterior distributions for each parameter. We could obtain these analytically by integrating the full posterior distribution over all model parameters except for the parameter of interest. Alternatively, we can approximate the marginalised posterior using a Markov Chain Monte Carlo (MCMC) sampling algorithm. We chose the latter approach because it is scalable to more complicated models where the marginalised posterior is not analytically solvable. -To sample from the approximate posterior distribution for both models, we used the No U-Turn Sampler \citep[NUTS;][]{Hoffman.Gelman2014} as implemented in the \textsc{Python} package \texttt{numpyro} \citep{Phan.Pradhan.ea2019,Bingham.Chen.ea2019}. We ran the sampler for 1000 steps following 500 warm-up steps (used to adapt the sampling procedure) and repeated for 10 MCMC chains. To reduce the number of divergences encountered during sampling, we increased the target accept probability from 0.8 to 0.98 for the HBM. The resulting marginalised posterior samples amounted to \num{10000} per parameter. +To sample from the posterior distribution for both models, we used the No U-Turn Sampler \citep[NUTS;][]{Hoffman.Gelman2014} as implemented in the \textsc{Python} package \texttt{numpyro} \citep{Phan.Pradhan.ea2019,Bingham.Chen.ea2019}. We ran the sampler for 1000 steps following 500 warm-up steps (used to adapt the sampling procedure) and repeated for 10 MCMC chains. +% To reduce the number of divergences encountered during sampling, we increased the target accept probability from 0.8 to 0.98 for the HBM. +The resulting marginalised posterior samples amounted to \num{10000} per parameter. \subsection{Comparing the Models}\label{sec:hbm-comp} @@ -153,7 +159,7 @@ \subsection{Comparing the Models}\label{sec:hbm-comp} \label{fig:hbm-corr} \end{figure} -In Figure \ref{fig:hbm-corr}, we compared the joint and marginalised posteriors of the distance to a few of the stars. The HBM found distances to higher precision, with typical standard deviations of \(s_{d,i} \approx 0.35\) compared to \(s_{d,i} \approx 1.2\) from the simple model. We also noticed a one-to-one correlation between \(d\) present in the HBM but not in the simple model. This was a result of the correlation introduction by the population-prior on \(d\). For each distance, we saw how a lower value in one corresponded to a lower value for the others. This represents our belief that cluster members should share a similar distance. On the other hand, the simple model suggested it was, for example, just as likely to find two stars at distances of 9 and 12 as it would for both to be at a distance of 11. +In Figure \ref{fig:hbm-corr}, we compared the joint and marginalised posteriors of the distance to a few of the stars. The HBM found distances to higher precision, with typical standard deviations of \(s_{d,i} \approx 0.35\) compared to \(s_{d,i} \approx 1.2\) from the simple model. We also noticed a one-to-one correlation between distances present in the HBM but not in the simple model. This was a result of the correlation introduction by the population-prior on \(d_i\). For each distance, we saw how a lower value in one corresponded to a lower value for the others. This represents our belief that cluster members should share a similar distance. On the other hand, the simple model suggested it was, just as likely to find two stars at distances of around 9 and 12 as it would for both to be at a distance of 11. \begin{figure}[tb] \centering @@ -162,11 +168,11 @@ \subsection{Comparing the Models}\label{sec:hbm-comp} \label{fig:hbm-global} \end{figure} -One consequence of the HBM was that it parametrised the population mean (\(\mu_d\)) and standard deviation (\(\sigma_d\)) of the distances to stars in the cluster separately from the observed noise. For the simple model, we estimated \(\mu_d\) and \(\sigma_d\) by taking the sample mean and standard deviation of distances in the cluster for each posterior sample. We compared the resulting posterior distributions for \(\mu_d\) and \(\sigma_d\) from the two models in Figure \ref{fig:hbm-global}. The mean distance from the simple model was \(\mu_d = 10.50 \pm 0.27\), whereas the HBM was more accurate with \(\mu_d = 10.01 \pm 0.24\). The simple model massively overestimated the standard deviation of cluster distances with \(\sigma_d = 1.815_{-0.293}^{+0.360}\), compared to the HBM's more accurate value of \(\sigma_d = 0.131_{-0.088}^{+0.280}\). The simple model could not distinguish between the uncertainty on individual distances and the spread of the population. +One consequence of the HBM was that it parametrised the population mean (\(\mu_d\)) and standard deviation (\(\sigma_d\)) of the distances to stars in the cluster separately from the observed noise. For the simple model, we estimated \(\mu_d\) and \(\sigma_d\) by taking the sample mean and standard deviation of distances in the cluster for each posterior sample. We compared the resulting posterior distributions for \(\mu_d\) and \(\sigma_d\) from the two models in Figure \ref{fig:hbm-global}. The mean distance from the simple model was \(\mu_d = 10.50 \pm 0.27\), whereas the HBM was more accurate with \(\mu_d = 10.01 \pm 0.24\). The simple model overestimated the standard deviation of cluster distances with \(\sigma_d = 1.815_{-0.293}^{+0.360}\), compared to the HBM's more accurate value of \(\sigma_d = 0.131_{-0.088}^{+0.280}\). The simple model could not distinguish between the uncertainty on individual distances and the spread of the population. \subsection{Scaling with the Number of Stars}\label{sec:hbm-scale} -To test how the model scales with number of stars, we repeated the HBM method for different \(N_\mathrm{stars}\). We randomly generated true and observed parameters in the same way as described at the beginning of \ref{sec:hbm-dist} for 320 stars. The observables were different to the last section, but drawn from the same distributions. We then sampled from the HBM for the first 20, 80 and 320 stars. These represented ratios in \(N_\mathrm{stars}^{1/2}\) of \(1\), \(2\) and \(4\) respectively. +To test how the model scales with number of stars, we repeated the HBM method for different \(N_\mathrm{stars}\). We randomly generated true and observed parameters for 320 stars in the same way as described at the beginning of the section. The observables were different to the last section, but drawn from the same true distributions. We then sampled from the HBM for the first 20, 80 and 320 stars. These corresponded to ratios in \(N_\mathrm{stars}^{1/2}\) of \(1:2:4\) respectively. \begin{figure}[tb] \centering @@ -175,16 +181,17 @@ \subsection{Scaling with the Number of Stars}\label{sec:hbm-scale} \label{fig:hbm-extended} \end{figure} -Figure \ref{fig:hbm-extended} shows how the distance estimates improved with increasing number of stars observed, \(N_\mathrm{stars}=(20,80,320)\). Unsurprisingly, the standard deviation on \(\mu_d\) decreased by a factor of \(N_\mathrm{stars}^{1/2}\) each time, with \(s_{\mu_d} \approx (0.24, 0.12, 0.06)\). We also expected a similar reduction in uncertainty for individual distances, albeit limited by the observational noise. Typical standard deviations for \(d_i\) were \(s_{d,i} \approx (0.30, 0.17, 0.15)\). As expected, the distance uncertainties also halved from \(20\) to \(80\) stars. However, this effect lessened with 320 stars. This demonstrated the method is still limited by the error budget across all observed parameters. Despite this, the HBM proved to be a useful way of getting more information from a limited number of noisy observations. +Figure \ref{fig:hbm-extended} shows how the distance estimates improved with increasing number of stars observed, \(N_\mathrm{stars}=(20,80,320)\). Unsurprisingly, the standard deviation of \(\mu_d\) decreased by a factor of \(N_\mathrm{stars}^{1/2}\) each time, with \(s_{\mu_d} \approx (0.24, 0.12, 0.06)\). We also expected a similar reduction in uncertainty for individual distances, albeit limited by the observational noise. Typical standard deviations for \(d_i\) were \(s_{d,i} \approx (0.30, 0.17, 0.15)\). As expected, the distance uncertainties also halved from \(20\) to \(80\) stars. However, this effect lessened with 320 stars. This demonstrated the method is still limited by the error budget across all observed parameters. Despite this, the HBM proved to be a useful way of getting more information from a limited number of noisy observations. \section[Stellar Properties]{Inferring Physical Properties of Stars}\label{sec:hbm-phys} We have seen how a simple HBM can be used to improve the distances and absolute magnitudes of stars in a cluster analogue. This kind of model only represents part of the puzzle for determining ages, masses, and radii in a population of stars. In these Bayesian stellar models (discussed previously in Section \ref{sec:modelling-stars}) we have more parameters which could be considered hierarchically. In this section, we discuss a few possible population distributions which could form a hierarchical model of stellar physics. -\paragraph{Mass} As mentioned in the introduction to this chapter, an IMF may be used as a prior on the mass of a star. If modelling stars with masses spanning a couple of orders of magnitude, it makes sense to draw their masses from an IMF. To make this hierarchical, we would parametrise the IMF as a conditional distribution which informs individual stellar masses. This way, the population of stars constrains the IMF while in-turn sharing information. In this thesis, we only consider stars in a narrow mass range (\SIrange{0.8}{1.2}{\solarmass}) where the IMF does not change much. However, a mass hierarchy like this may be useful in future work. +\paragraph{Mass} As mentioned in the introduction to this chapter, an IMF may be used as a prior on the mass of a star. If modelling stars with masses spanning a couple of orders of magnitude, it makes sense to draw their masses from an IMF. To make this hierarchical, we would parametrise the IMF as a conditional distribution which informs individual stellar masses. This way, the population of stars constrains the IMF while in-turn sharing information. In this thesis, we only consider stars in a narrow mass range (\SIrange{0.8}{1.2}{\solarmass}) where the IMF does not change much. -\paragraph{Age} We do not expect a star to be older than the universe. This makes the age of the universe a natural upper limit to a prior on stellar age. There are also populations of stars where we expect the age to be tightly related. For example, star systems like binaries and clusters are expected to have been formed at a similar time. Similarly to the distances in Section \ref{sec:hbm-dist}, a hierarchical model in age would parametrise the mean and variance of ages in these systems and in-turn improve other connected model parameters. This sort of analysis could be extended to find mixtures of age distributions in the galaxy which could indicate mergers \citep[e.g. \emph{Gaia}-Enceladus;][]{Helmi.Babusiaux.ea2018}. However, we do not further consider an HBM in age this in this work. +\paragraph{Age} We do not expect a star to be older than the universe. This makes the age of the universe a natural upper limit to a prior on stellar age. There are also populations of stars where we expect the age to be tightly related. For example, star systems like binaries and clusters are expected to have been formed at a similar time. Similarly to the distances in Section \ref{sec:hbm-dist}, a hierarchical model in age would parametrise the mean and variance of ages in these systems and in-turn improve other correlated model parameters. This sort of analysis could be extended to find mixtures of age distributions in the galaxy which could indicate mergers \citep[e.g. \emph{Gaia}-Enceladus;][]{Helmi.Babusiaux.ea2018}. +% While we do not consider HBM in age in this thesis, (Li et al., in preparation). -\paragraph{Chemical Abundances} Unlike mass and age, surface abundances of stars can be measured directly with spectroscopy. However, not all abundances are easily measured. For example, helium ionises below the surface of the cool stars being studied in this work. Also, diffusion and settling of elements during stellar evolution means surface abundances differ from formation to time of observation. Similarly to age, we can expect clusters of stars to share initial abundances. However, the distribution of abundances in the Milky Way may also be parametrised. As stars evolve they convert hydrogen into helium and heavier elements. Supernovae enrich the galaxy with these elements monotonically, which go on to constitute new stars. A hierarchical model could include this assumption to tie the abundance of helium to heavier elements. +\paragraph{Chemical Abundances} Unlike mass and age, surface abundances ratios can be measured directly in stellar atmospheres with spectroscopy. However, not all abundances are easily measured. For example, helium ionises below the surface of the cool stars being studied in this work. Also, diffusion and settling of elements during stellar evolution means surface abundances differ from formation to time of observation. Similarly to age, we can expect clusters of stars to share initial abundances. However, the distribution of abundances in the Milky Way may also be parametrised. As stars evolve they convert hydrogen into helium and heavier elements. Supernovae enrich the galaxy with these elements monotonically, which go on to constitute new stars. A hierarchical model could include this assumption to tie the abundance of helium to heavier elements. -\paragraph{Approximations of Stellar Physics} A hierarchical model need not be limited to real physical parameters. In stellar modelling, we often approximate stellar physics with \emph{pseudo}-physical parameters. For example, convective energy transfer is approximated by the mixing-length theory \citep[MLT;][]{Gough1977} parametrised by \(\mlt\). The value of \(\mlt\) which best models a star is likely to vary with other stellar parameters. It's effect on observables is small making \(\mlt\) difficult to constrain on a star-by-star basis. Hence, a hierarchical model would be able to condition individual values of \(\mlt\) on its population distribution. A distribution over pseudo-physical parameters is less intuitive, yet the HBM would allow us to build one empirically. In addition to the initial helium abundance, \(\mlt\) is modelled hierarchically in Chapter \ref{chap:hmd}. +\paragraph{Approximations of Stellar Physics} A hierarchical model need not be limited to real physical parameters. In stellar modelling, we often approximate stellar physics with \emph{pseudo}-physical parameters. For example, convective energy transfer is approximated by the mixing-length theory \citep[MLT;][]{Gough1977} parametrised by \(\mlt\). The value of \(\mlt\) which best models a star is likely to vary with other stellar parameters. The effect of \(\mlt\) on observables is small, making it difficult to constrain on a star-by-star basis. Hence, a hierarchical model would be able to condition individual values of \(\mlt\) on its distribution in the population. A distribution over pseudo-physical parameters is less intuitive, yet the HBM would allow us to build one empirically. In addition to the initial helium abundance, we model \(\mlt\) hierarchically in Chapter \ref{chap:hmd}. From 7208aa5343d4ca9ed8f0e17d52289574413ceb3d Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Fri, 28 Apr 2023 12:04:15 +0100 Subject: [PATCH 43/50] Finish glitch intro chapter --- chapters/glitch.tex | 71 +++++++++++++++++++++++---------------------- references.bib | 16 +++++----- 2 files changed, 44 insertions(+), 43 deletions(-) diff --git a/chapters/glitch.tex b/chapters/glitch.tex index 7d6f8d0..83405ae 100644 --- a/chapters/glitch.tex +++ b/chapters/glitch.tex @@ -12,20 +12,20 @@ % \chapter[Acoustic Glitches in Solar-Like Oscillators]{Acoustic Glitches in Solar-Like Oscillators as a Signature of Helium Abundance}\label{chap:glitch} -\textit{In this chapter, I explore an asteroseismic signature of helium which could provide observables to add to the hierarchical model from Chapter \ref{chap:hmd}. I introduce the concept of an acoustic glitch in the structure of a star producing a measurable signal in its observable oscillation modes. In Section \ref{sec:1d-glitch}, I start with a simple one-dimensional example of a glitch. Then, I review the theory of glitch signatures due to helium ionisation and the base of the convective zone in Section \ref{sec:glitch-star}.} +\textit{In this chapter, I introduce the concept of an acoustic glitch in the structure of a star producing a measurable signal in its observable oscillation modes. In Section \ref{sec:1d-glitch}, I start with a simple one-dimensional example of a glitch. Then, I review the theory of glitch signatures due to helium ionisation and the base of the convective zone in Section \ref{sec:glitch-star}.} \section{Introduction} % \epigraph{\singlespacing``Ideals are like stars: you will not succeed in touching them with your hands, but like the seafaring man on the ocean desert of waters, you choose them as your guides, and following them, you reach your destiny.''}{\emph{Carl Schurz}} % So far in this thesis, we have shown that a hierarchical Bayesian model can be used to infer the helium abundance distribution in a stellar population. We also found how this improves the inference of fundamental stellar parameters. However, there is limited information about helium abundance in the stellar observables used (e.g. \(L, \teff, \Delta\nu\)). -An acoustic glitch is a rapid variation of the sound speed inside a medium. The presence of a glitch induces a periodic signature in consecutive p mode frequencies. Acoustic glitches in Sun-like stars arise from sharp variations in their structure, such as the base of the convection zone (BCZ) and the first and second ionisation of helium (He\,\textsc{i} and He\,\textsc{ii}). There have been several attempts to measure this effect in the Sun and other stars over the past few decades. +An acoustic glitch is a rapid variation of the sound speed inside a medium. The presence of a glitch induces a periodic deviation from the regular separation of p mode frequencies. Acoustic glitches in Sun-like stars arise from sharp variations in their structure, such as the base of the convection zone (BCZ) and the first and second ionisation of helium (He\,\textsc{i} and He\,\textsc{ii}). There have been several attempts to measure this effect in the Sun and other stars over the past few decades. Early work identified glitches in solar p modes by analysing their second differences, \(\Delta_2\nu_{nl} \equiv \nu_{n-1\,l} - 2\nu_{nl} + \nu_{n+1\,l}\). Second and higher-order differences remove some slowly varying components of the mode frequencies and amplify faster varying components. Assuming a sharp localised discontinuity in sound speed at He\,\textsc{ii} ionisation and the BCZ, \citet{Basu.Antia.ea1994,Basu1997} modelled glitch signatures in the Sun to constrain the extent of convective overshoot at the BCZ. \citet{Monteiro.Thompson1998} further developed a model of the He\,\textsc{ii} ionisation glitch signature by accounting for its finite width in the star, later applying it to study the helium ionisation zone of the Sun \citep{Monteiro.Thompson2005}. Around the same time, \citet{Basu.Mazumdar.ea2004} showed that the amplitude of the He\,\textsc{ii} glitch signature correlated with the fractional helium abundance (\(Y\)) near the surface of Sun-like stars. A few years later, \citet{Houdek.Gough2007} proposed a closer physical approximation of the He\,\textsc{ii} glitch. They derived the glitch signature which was later used in several studies of solar-like oscillators. -Since helium ionises below the stellar atmosphere for cool stars (\(\teff \sim \SI{e5}{\kelvin}\)), asteroseismology was able to probe helium abundance where spectroscopy could not. Glitches could be further studied in stars other than the Sun with the advent of space-based missions. \citet{Miglio.Montalban.ea2010,Mazumdar.Michel.ea2012} were among the first to measure glitch signatures in other stars after \emph{CoRoT} provided evidence of solar-like oscillations in red giants. Then, studies of glitches in main sequence stars observed by \emph{Kepler} compared fitting to the modes frequencies directly with using second differences \citep{Mazumdar.Monteiro.ea2012,Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2017}. More recently, \citet{Verma.Raodeo.ea2019} used measurements of the glitch to estimate the helium abundance for the LEGACY sample of stars \citet{Lund.SilvaAguirre.ea2017}. +Since helium ionises below the stellar atmosphere for cool stars (\(\teff \sim \SI{e5}{\kelvin}\)), we can probe helium abundance with asteroseismology where we cannot with spectroscopy. Glitches could be further studied in stars other than the Sun with the advent of space-based missions. \citet{Miglio.Montalban.ea2010,Mazumdar.Michel.ea2012} were among the first to measure glitch signatures in other stars after \emph{CoRoT} \citep{Baglin.Auvergne.ea2006} provided evidence of solar-like oscillations in red giants. Then, studies of glitches in main sequence stars observed by \emph{Kepler} \citep{Borucki.Koch.ea2010} compared using modes frequencies directly with using second differences \citep{Mazumdar.Monteiro.ea2012,Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2017}. More recently, \citet{Verma.Raodeo.ea2019} used measurements of the glitch to estimate the helium abundance for the LEGACY sample of stars \citet{Lund.SilvaAguirre.ea2017}. -The reason that acoustic glitches cause a periodic signal in the mode frequencies is not obvious. Hence, we go through a simple, one-dimensional example in Section \ref{sec:1d-glitch}. Then, in Section \ref{sec:glitch-star}, we demonstrate how glitches in stellar structure due to helium ionisation and the BCZ affect the mode frequencies of solar-like oscillators. Starting with the variational principle \citep{Chandrasekhar1964}, we show how to get to the He\,\textsc{ii} glitch signature formula from \citet{Houdek.Gough2007}. +The reason that acoustic glitches cause a periodic signal in the mode frequencies is not obvious. Hence, we go through a simple, one-dimensional example in Section \ref{sec:1d-glitch}. Then, in Section \ref{sec:glitch-star}, we demonstrate how glitches in stellar structure due to helium ionisation and the BCZ affect the mode frequencies of solar-like oscillators. Starting with the variational principle \citep{Chandrasekhar1964}, we will show how to get to the He\,\textsc{ii} glitch signature formula from \citet{Houdek.Gough2007}. \section[1D Glitch Example]{A One-Dimensional Example of a Glitch}\label{sec:1d-glitch} @@ -49,7 +49,7 @@ \section{Introduction} \real\left[\xi(x, t)\right] = r \sin k x \sin(\omega t + \phi), \end{equation} % -where \(r\) and \(\phi\) are the amplitude and temporal phase respectively. Solutions for \(\omega\) which satisfy \(\xi(L, t)=0\) may then be found, +where \(r\) and \(\phi\) are the amplitude and phase respectively. Solutions for \(\omega\) which satisfy \(\xi(L, t)=0\) may then be found, % \begin{equation} \omega_n = c \frac{n \pi}{L}, \label{eq:omega-n} @@ -61,13 +61,13 @@ \section{Introduction} \centering % \includegraphics{figures/glitch-1d-example-diagram.pdf} \input{figures/glitch-1d-example.tex} - \caption[A diagram showing a one-dimensional medium with a small structural perturbation.]{A diagram showing a one-dimensional medium split into three regions. 1: Fixed at \(x=0\) with a constant speed of sound \(c\); 2: A small structural perturbation centred at \(x=x_\glitch\) with width \(2\delta x\) and constant speed of sound \(c + \delta c\); 3: Fixed at \(x=L\) with a constant speed of sound \(c\).} + \caption[A diagram showing a one-dimensional medium with a small structural perturbation.]{A diagram showing a one-dimensional medium split into three regions: (1) Fixed at \(x=0\) with a constant speed of sound \(c\); (2) A small structural perturbation centred at \(x=x_\glitch\) with width \(2\delta x\) and constant speed of sound \(c + \delta c\); (3) Fixed at \(x=L\) with a constant speed of sound \(c\).} \label{fig:1d-diagram} \end{figure} Now, let us suppose there is a small structural perturbation (or glitch) in the medium at position \(x_\glitch\) with half-width \(\delta x\). Figure \ref{fig:1d-diagram} shows this system divided into 3 regions, with region 2 containing the glitch. The speed of sound is \(c + \delta c\) in region 2, where we let the corresponding wave number be \(k + \delta k\). We want to find the frequencies which correspond to standing waves in this system and compare them to that of the homogeneous medium above. We will show that the resulting perturbation to the eigenfrequencies (\(\delta\omega\)) is periodic, with an amplitude and period that relates to the properties of the glitch. -Firstly, we propose solutions to the wave for each region by considering reflection and transmission at each boundary. Initially ignoring the wave superposed by a reflection at \(x=L\), +Firstly, we propose general solutions to the wave for each region by considering reflection and transmission at each boundary. Initially ignoring the wave superposed by the boundary at \(x=L\), % \begin{align} \xi_1(x, t) &= \ee^{i(\omega t - k x)} + A \ee^{i(\omega t + k x)}, \label{eq:xi1-r} \\ @@ -75,7 +75,7 @@ \section{Introduction} \xi_3(x, t) &= D \ee^{i(\omega t - k x)}, \label{eq:xi3-r} \end{align} % -where complex coefficients \(A\) and \(C\) represent reflections, and \(B\) and \(D\) represent transmissions at \(x_\glitch \pm \delta x\) respectively. Later, we will substitute the left-travelling wave (\(- \xi\{-k, -\delta k\}\)) after determining the values of the coefficients before solving for \(\omega\). +where complex coefficients \(A\) and \(C\) represent reflections, and \(B\) and \(D\) represent transmissions at \(x_\glitch \pm \delta x\) respectively. Later, we will substitute the left-travelling wave (\(- \xi\{-k, -\delta k\}\)) after determining the values of the coefficients. The internal boundary conditions for this system are given by enforcing spacial continuity at \(x_\glitch \pm \delta x\), % @@ -114,9 +114,9 @@ \section{Introduction} \real\left[\xi_1(x, t)\right] = \sin \omega t \left[2 \sin kx - r_A \sin(kx + \phi_A)\right]. \end{equation} % -However, this equation does not satisfy the outer boundary condition that the displacement is always zero at \(x=0\); in other words, \(\xi_1(0, t) = - r_A \sin \omega t \sin(\phi_A) \neq 0\), everywhere. To fix this, we introduce a small phase displacement \(\epsilon\) caused by the glitch and let \(x \rightarrow x + \epsilon\). +However, this equation does not satisfy the outer boundary condition that the displacement is always zero at \(x=0\); i.e. \(\xi_1(0, t) = - r_A \sin \omega t \sin(\phi_A) \neq 0\), everywhere. To fix this, we introduce a small phase displacement \(\epsilon\) caused by the glitch and let \(x \rightarrow x + \epsilon\). -Superposing the right-travelling wave and substituting Euler's formula into Equations \ref{eq:xi2-r} and \ref{eq:xi3-r}, we write the real components of the wave functions in trigonometric as, +Superposing the right-travelling wave and substituting Euler's formula into Equations \ref{eq:xi2-r} and \ref{eq:xi3-r}, we write the real components of the wave functions in trigonometric form as, % \begin{align} \real[\xi_1(x, t)] &= \sin \omega t \left\{2 \sin[k (x + \epsilon)] - r_A \sin[k(x + \epsilon) - \phi_A]\right\}, \label{eq:xi1-real} \\ @@ -132,7 +132,7 @@ \section{Introduction} \end{align} % -Finally, we use the boundary condition \(\xi_3(L, t) = 0\) to solve for \(\omega\). Setting Equation \ref{eq:xi3-real} to zero, we rewrite it in terms of the real and imaginary components of \(D\), +Finally, we use the boundary condition \(\xi_3(L, t) = 0\) to solve for \(\omega\). Letting Equation \ref{eq:xi3-real} equal zero at \(x=L\), we rewrite it in terms of the real and imaginary components of \(D\), % \begin{align} \sin \omega t \left\{r_D \sin[k(L + \epsilon) - \phi_D]\right\} &= 0, \quad (\div \sin \omega t) \notag \\ @@ -140,17 +140,17 @@ \section{Introduction} \real[D] \sin[k(L + \epsilon)] - \imag[D] \cos[k(L + \epsilon)] &=0. \label{eq:1d-glitch-sol} \end{align} % -The glitch affects the amplitude and phase of the wave, because if we set \(\epsilon = 0\) and \(D = 1\) we recover the homogeneous frequency solutions (Equation \ref{eq:omega-n}). Unfortunately, solving Equation \ref{eq:1d-glitch-sol} for \(\omega\) is not possible analytically. However, we can find individual roots (or oscillation modes) numerically. +The glitch affects the amplitude and phase of the wave, because if we set \(\epsilon = 0\) and \(D = 1\) we recover the homogeneous frequency solutions (see Equation \ref{eq:omega-n}). Unfortunately, solving Equation \ref{eq:1d-glitch-sol} for \(\omega\) is not possible analytically. However, we can find eigenfrequencies (or oscillation modes) numerically. \begin{figure} \centering \includegraphics{figures/glitch-1d-example-results.pdf} - \caption[The change in mode frequency induced by a rapid change in sound speed for the 1D example.]{The change in mode frequency induced by a change in sound speed of \(\delta c\) from \(x_\glitch - \delta x\) to \(x_\glitch + \delta x\) in a one-dimensional medium, bound such that \(x \in [0, 1]\) (see Figure \ref{fig:1d-diagram}). Outside the perturbation the speed of sound, \(c=1\). The frequency perturbations are offset by \(\omega_0\) given in the legend of the top panel. Points are joined by straight lines to guide the eye but do not represent real solutions. + \caption[The change in mode frequency induced by a rapid change in sound speed for the 1D example.]{The change in mode frequency induced by a change in sound speed of \(\delta c\) from \(x_\glitch - \delta x\) to \(x_\glitch + \delta x\) in a one-dimensional medium, bound such that \(x \in [0, 1]\) (see Figure \ref{fig:1d-diagram}). Outside the perturbation the speed of sound, \(c=1\). The frequency changes are offset by \(\omega_0\) given in the legend of the top panel. Points are joined by straight lines to guide the eye, but these lines do not represent real solutions. } \label{fig:1d-results} \end{figure} -Let us use \(\omega'_n\) to denote the solutions to Equation \ref{eq:1d-glitch-sol}, where \(n\) is a positive integer. We find \(\omega'_n\) by solving Equation \ref{eq:1d-glitch-sol} using Newton's method for \(n = 1,\dots,50\) modes. Using dimensionless units of length and time, we set \(c=1\), \(L=1\), and test several values of \(x_\glitch\), \(\delta x\), and \(\delta c\). Initial guesses for \(\omega'_n\) are obtained from the homogeneous medium solutions (\(\omega_n\)) in Equation \ref{eq:omega-n}. We show the difference between the glitch solutions and those from the homogeneous medium, \(\delta \omega_n = \omega'_n - \omega_n\), in Figure \ref{fig:1d-results}. We can see a periodic component to \(\delta\omega\) induced by the glitch. As the wave nodes pass in and out of the region with changing \(n\), the sensitivity of the wave to the glitch varies periodically. The overall sensitivity to the glitch depends on how much the wave changes inside the glitch, hence why low \(n\) modes have smaller \(\delta\omega\). +Let us use \(\omega'_n\) to denote the solutions to Equation \ref{eq:1d-glitch-sol}, where \(n\) is a positive integer. Using dimensionless units of length and time, we set \(c=1\), \(L=1\), and test several values of \(x_\glitch\), \(\delta x\), and \(\delta c\). We find \(\omega'_n\) by solving Equation \ref{eq:1d-glitch-sol} using Newton's method for \(n = 1,\dots,50\) modes. Initial guesses for \(\omega'_n\) are obtained from the homogeneous medium solutions (\(\omega_n\)) in Equation \ref{eq:omega-n}. We show the difference between the glitch solutions and those from the homogeneous medium, \(\delta \omega_n = \omega'_n - \omega_n\), in Figure \ref{fig:1d-results}. We can see a periodic component to \(\delta\omega\) induced by the glitch. As the wave nodes pass in and out of the region with changing \(n\), the sensitivity of the wave to the glitch varies periodically. The overall sensitivity to the glitch depends on how much the wave changes inside the glitch, hence why low \(n\) modes have smaller \(\delta\omega\). % Physically, this arises from the change in phase required to satisfy the boundary conditions of the glitch region. @@ -175,7 +175,7 @@ \section{Introduction} f(\omega) = a_1 \omega - a_2 \sin (\tau_1 \omega) \cos (\tau_2 \omega), \label{eq:1d-domega-func} \end{equation} % -where \(a_1\) and \(a_2\) are coefficients which are both functions of \(\delta x\) and \(\delta c\). Parameters \(\tau_1\) and \(\tau_2\) are the `frequencies' (with dimensionless units of time\footnote{An angular frequency in angular frequency space has units of time.}) of the periodic component to \(\delta\omega\). Given Figure \ref{fig:1d-results}, we expect \(\tau_1\) and \(\tau_2\) to be related to \(\delta x\) and \(x_\glitch\) respectively. +where \(a_1\) and \(a_2\) are coefficients which are both functions of \(\delta x\) and \(\delta c\). Parameters \(\tau_1\) and \(\tau_2\) are the `frequencies' of the periodic component to \(\delta\omega\). Given Figure \ref{fig:1d-results}, we expect \(\tau_1\) and \(\tau_2\) to be related to \(\delta x\) and \(x_\glitch\) respectively. \begin{figure}[tb] \centering @@ -208,9 +208,9 @@ \section{Introduction} \gamma = \left( \frac{\partial \ln P}{\partial \ln \rho} \right)_S, \end{equation} % -at constant entropy, \(S\). \citet{Chandrasekhar1939} introduced three adiabatic exponents (\(\Gamma_1,\Gamma_2,\Gamma_3\)) to describe the non-ideal gas inside a star. However, in this chapter we do not use the other two and hence refer the first as \(\gamma \equiv \Gamma_1\). +at constant entropy, \(S\). This is the first of three adiabatic exponents (\(\Gamma_1,\Gamma_2,\Gamma_3\)) introduced by \citet{Chandrasekhar1939} to describe the non-ideal gas inside a star. However, in this thesis we do not use the other two and hence refer the first as \(\gamma \equiv \Gamma_1\). -For the most part, \(\gamma\), \(P\), and \(\rho\) change smoothly with radius inside a star. However, a rapid structural glitch in these quantities would lead to a sudden change in sound speed. In the previous section, we showed how such a perturbation can lead to a periodicity in the eigenfrequencies for a homogeneous medium. Characterising this signal allowed us to measure the properties of the glitch. If similar glitches were present in a star, then we might be able to do the same. In this section, we explore the origins of glitches inside a solar-like star. Then, we see what effect these have on the eigenfrequencies, a quantity we can measure through asteroseismology. +For the most part, \(\gamma\), \(P\), and \(\rho\) change smoothly with radius inside a star. However, a rapid structural glitch in these quantities would lead to a sudden change in sound speed. In the previous section, we showed how such a perturbation can lead to a periodicity in the eigenfrequencies for an otherwise homogeneous medium. Characterising this signal allowed us to measure the properties of the glitch. If similar glitches were present in a star, then we might be able to do the same. In this section, we explore the origins of glitches inside a solar-like star. Then, we see what effect these have on the eigenfrequencies, a quantity we can measure through asteroseismology. Firstly, let us consider the sound speed profile of a Sun-like star. Particularly, we want to see how the sound speed changes on the timescale of a pressure wave travelling through the star. As discovered in Section \ref{sec:1d-glitch}, a convenient timescale to work with is the acoustic depth, \(\tau\). This is not to be confused with the symbol for stellar age in Chapter \ref{chap:hmd}. Here, we define \(\tau\) as the time taken for a pressure wave to travel from the surface (\(R\)) to some radius (\(r\)) in a star under the assumption of spherical symmetry, % @@ -218,7 +218,7 @@ \section{Introduction} \tau(r) = \tau_0 - \int_0^{r} \frac{\dd r'}{c(r')},\label{eq:tau} \end{equation} % -where \(\tau_0\) is the acoustic radius of the star. We recall from Section \ref{sec:seismo} that \(\nu_0 \equiv (2\tau_0)^{-1}\) may be approximated by the large frequency separation (\(\Delta\nu_{nl}\)) in the asymptotic limit that \(l/n \rightarrow 0\). +where \(\tau_0\) is the acoustic radius of the star. We recall from Section \ref{sec:seismo} that \(\nu_0 \equiv (2\tau_0)^{-1}\) may be approximated by the large frequency separation (\(\Delta\nu\)) in the asymptotic limit that \(l/n \rightarrow 0\). \begin{figure}[tb] \centering @@ -227,17 +227,17 @@ \section{Introduction} \label{fig:sound-speed-gradient} \end{figure} -In Figure \ref{fig:sound-speed-gradient}, we show the sound speed gradient with respect to \(\tau\) for a Sun-like stellar model (model S; see Section \ref{sec:model-s}). We see how the speed of sound changes smoothly throughout the star. In the convective zone, there is a noticeable wiggle around \SI{700}{\second} and a sharp change in direction at its base. The first is caused by the ionisation of helium, which we will explore further in Section \ref{sec:helium-glitch}. The second is due to a discontinuity in the temperature gradient as the stellar interior goes from unstable due to convection to radiative. This will be discussed in Section \ref{sec:bcz-glitch}. +In Figure \ref{fig:sound-speed-gradient}, we show the sound speed gradient with respect to \(\tau\) for a Sun-like stellar model (model S; see Section \ref{sec:model-s}). We see how the speed of sound changes smoothly throughout the star. In the convective zone, there is a noticeable wiggle around \SI{700}{\second} and a sharp change in direction at its base. The first is caused by the ionisation of helium, which we will explore further in Section \ref{sec:helium-glitch}. The second is due to a discontinuity in the temperature gradient as the dominant energy transport mechanism goes from convective to radiative. This will be discussed in Section \ref{sec:bcz-glitch}. \subsection{Sun-Like Model Star}\label{sec:model-s} -We computed a representative Sun-like model star (hereafter `model S') using MESA \citep[version 12115;][]{Paxton.Bildsten.ea2011,Paxton.Cantiello.ea2013,Paxton.Marchant.ea2015,Paxton.Schwab.ea2018,Paxton.Smolec.ea2019,Jermyn.Bauer.ea2023}. The model was computed with a mass of \SI{1.0}{\solarmass} to a central fractional hydrogen abundance of \(X_c=0.6\). We used initial fractional helium and heavy-element abundances of \(Y_\mathrm{init} = 0.28\) and \(Z_\mathrm{init}=0.02\) respectively. The mixing-length was parametrised by \(\alpha_\mlt=1.9\) and a turbulent pressure factor of 1. We evolved the star using element diffusion with MESA's default coefficients \citep{Stanton.Murillo2016}. In addition, we used MESA's default \citet{Grevesse.Sauval1998} solar chemical composition and opacity tables. We also output pulsation profile data in FGONG format to later use when computing oscillation modes in Chapter \ref{chap:glitch-gp}. Our resulting model S had an age of \SI{4.073}{\giga\year}, \(\teff \approx \SI{5682}{\kelvin}\), and \(R \approx \SI{1.014}{\solarradius}\). Since the model was evolved with element diffusion, the surface fractional helium and heavy-element were approximately \(Y_\mathrm{surf} \approx 0.2515\) and \(Z_\mathrm{surf} \approx 0.01810\) respectively. +We computed a representative Sun-like model star (hereafter `model S') using MESA \citep[version 12115;][]{Paxton.Bildsten.ea2011,Paxton.Cantiello.ea2013,Paxton.Marchant.ea2015,Paxton.Schwab.ea2018,Paxton.Smolec.ea2019,Jermyn.Bauer.ea2023}. The model was computed with a mass of \SI{1.0}{\solarmass} to a central fractional hydrogen abundance of \(X_c=0.6\). We used initial fractional helium and heavy-element abundances of \(Y_\mathrm{init} = 0.28\) and \(Z_\mathrm{init}=0.02\) respectively. The mixing-length was parametrised by \(\mlt=1.9\) and a turbulent pressure factor of 1. We evolved the star using element diffusion with MESA's default coefficients \citep{Stanton.Murillo2016}. In addition, we used MESA's default \citet{Grevesse.Sauval1998} solar chemical composition and opacity tables. We also output pulsation profile data in FGONG format to later use when computing oscillation modes in Chapter \ref{chap:glitch-gp}. Our resulting model S had an age of \SI{4.073}{\giga\year}, \(\teff \approx \SI{5682}{\kelvin}\), and \(R \approx \SI{1.014}{\solarradius}\). Since the model was evolved with element diffusion, the surface fractional helium and heavy-element were approximately \(Y_\mathrm{surf} \approx 0.2515\) and \(Z_\mathrm{surf} \approx 0.01810\) respectively. \subsection{Helium Ionisation Glitch}\label{sec:helium-glitch} -In this section, we will first show how the sound speed inside a solar-like star is affected by the ionisation of hydrogen and helium. Then, we will see that an increase in helium abundance (\(Y\)) increases the effect of helium ionisation on the speed of sound. Starting with the variational principle, we derive the frequently used equation for the glitch signature \(\delta\nu\) induced by the second ionisation of helium from \citet{Houdek.Gough2007}. +In this section, we will first show how the sound speed inside a solar-like star is affected by the ionisation of hydrogen and helium. Then, we will see that an increase in helium abundance (\(Y\)) increases the effect of helium ionisation on the speed of sound. Starting with the variational principle, we derive the commonly used equation for the glitch signature (\(\delta\nu\)) induced by the second ionisation of helium from \citet{Houdek.Gough2007}. -In the outer convective envelope of Sun-like stars, the temperature (\(\SIrange{e4}{e5}{\kelvin}\)) and density (\(\SIrange{e-5}{e-3}{\gram\per\centi\metre\cubed}\)) is suitable for ionising hydrogen and helium \citep{Eggleton.Faulkner.ea1973}. As a given chemical species in the star ionises, the number of particles and hence chemical potential of the species changes. This induces a gradient in the thermal free energy of the gas which relates to the pressure and entropy of the gas. Thus, we expect ionisation to cause a change in the pressure-density gradient at constant entropy, \(\gamma\). Since \(\gamma\) relates to the sound speed from Equation \ref{eq:sound}, any sudden change in \(\gamma\) will induce a change in \(c\). +In the outer convective envelope of Sun-like stars, the temperature (\(\SIrange{e4}{e5}{\kelvin}\)) and density (\(\SIrange{e-5}{e-3}{\gram\per\centi\metre\cubed}\)) is suitable for ionising hydrogen and helium \citep{Eggleton.Faulkner.ea1973}. As a given chemical species in the star ionises, the number of particles and hence chemical potential of the species changes. This induces a gradient in the thermal free energy of the gas which relates to the pressure and entropy of the gas. Thus, we expect ionisation to cause a change in the pressure-density gradient at constant entropy (\(\gamma\)). Since \(\gamma\) relates to the sound speed from Equation \ref{eq:sound}, any sudden change in \(\gamma\) will induce a change in \(c\). \begin{figure}[tb] \centering @@ -246,7 +246,7 @@ \subsection{Helium Ionisation Glitch}\label{sec:helium-glitch} \label{fig:gamma-zones} \end{figure} -In Figure \ref{fig:gamma-zones}, we show \(\gamma\) for model S against fractional acoustic depth, shaded by regions of ionisation. For an ideal monatomic gas, \(\gamma=5/3\), but we see that \(\gamma < 5/3\) in regions where helium and hydrogen ionise. Close to the surface of the star, hydrogen ionisation has the largest effect on \(\gamma\) because it makes up the majority of stellar matter. The first (He\,\textsc{i}) and second (He\,\textsc{ii}) ionisations of helium occur deeper in the star. We can see that the second ionisation of helium has a greater effect on \(\gamma\) than the first. The effect of the He\,\textsc{ii} ionisation causes a rapid change in \(\gamma\) over a few per cent in \(\tau\) (equivalent to \(\sim \SI{100}{\second}\) in model S). +In Figure \ref{fig:gamma-zones}, we show \(\gamma\) for model S against fractional acoustic depth, shaded by regions of ionisation. For an ideal monatomic gas, \(\gamma=5/3\), but we see that \(\gamma < 5/3\) in regions where helium and hydrogen ionise. Close to the surface of the star, hydrogen ionisation has the largest effect on \(\gamma\) because it makes up the majority of stellar matter. The first (He\,\textsc{i}) and second (He\,\textsc{ii}) ionisations of helium occur deeper in the star. We can see that the second ionisation of helium has a greater effect on \(\gamma\) than the first. He\,\textsc{ii} ionisation causes a rapid change in \(\gamma\) over a few per cent in \(\tau\) (equivalent to \(\sim \SI{100}{\second}\) in model S). \subsubsection{The Effect of Helium Abundance on \(\gamma\)} @@ -257,7 +257,7 @@ \subsubsection{The Effect of Helium Abundance on \(\gamma\)} \label{fig:gamma-temp-density} \end{figure} -We were able to isolate the contributions to \(\gamma\) from ionisation using a recent derivation by \citet{Houdayer.Reese.ea2021}. They obtained an approximate formula for \(\gamma\) as a function of temperature (\(T\)) and density valid for the outer convective zone of Sun-like stars. Using their formula, we plot \(\gamma\) for a range of \(T\) and \(\rho\) in Figure \ref{fig:gamma-temp-density}. The first panel illustrates the three ionisation regions of hydrogen and helium. The value of \(\gamma\) decreases when the ionisation reaction is occurring. The magnitude of this effect depends on the intersection with the temperature-density profile of the star (over-plot for model S). We can imagine a hotter star shifting this line such that ionisation occurs closer to the surface. Along the model S profile, we see that an increase in helium abundance by 0.1 decreases \(\gamma\) by up to \(\sim 0.03\) in the He\,\textsc{ii} ionisation region. This shows that helium abundance correlates with the depth of the glitch in \(\gamma\). However, there is clearly still some dependence on \(T\) and \(\rho\) which are governed by mass, chemical composition, and evolutionary phase. +We were able to isolate the contributions to \(\gamma\) from ionisation using a recent derivation by \citet{Houdayer.Reese.ea2021}. They obtained an approximate formula for \(\gamma\) as a function of temperature (\(T\)) and density, valid for the outer convective zone of Sun-like stars. Using their formula, we plot \(\gamma\) for a range of \(T\) and \(\rho\) in Figure \ref{fig:gamma-temp-density}. The first panel illustrates the three ionisation regions of hydrogen and helium. The value of \(\gamma\) decreases when the ionisation reaction is occurring. The magnitude of this effect depends on the intersection of the temperature-density profile of the star (over-plot for model S). We can imagine a hotter star shifting this line such that ionisation occurs closer to the surface. Along the model S profile, we see that an increase in helium abundance by 0.1 decreases \(\gamma\) by up to \(\sim 0.03\) in the He\,\textsc{ii} ionisation region. This shows that helium abundance correlates with the depth of the glitch in \(\gamma\). However, there is clearly still some dependence on \(T\) and \(\rho\) which are governed by mass, chemical composition, and evolutionary phase. % The exact effect of ionisation on \(\gamma\) is not known analytically. However, \citet{Houdayer.Reese.ea2021} recently approximated \(\gamma\) for the convective zone of a Sun-like star in their study of the helium ionisation glitch. In this subsection, we combine the equations derived in their work to help us understand the relation between ionisation and \(\gamma\). Let us consider an \(M\) mass star with fractional hydrogen and helium abundances of \(X\) and \(Y\) respectively. From \citet{Houdayer.Reese.ea2021}, we approximate the first adiabatic exponent as, % % @@ -302,14 +302,14 @@ \subsubsection{The Effect of Helium Abundance on \(\gamma\)} \label{fig:gamma-sound-speed} \end{figure} -To show the effect of small changes in \(Y\) on the sound speed, we plot the sound speed gradient in Figure \ref{fig:gamma-sound-speed}. The three models shown were evolved to the same \(X_c = 0.6\) as model S with initial helium abundances of 0.26, 0.28, and, 0.3. The dominant effect of helium abundance appears as a Gaussian-like depression in \(\gamma\) around the second ionisation of helium. We see how larger helium abundance increases the width and depression in \(\gamma\), which is reflected in the sound speed gradient. A larger \(Y\) also leads to a relative reduction in hydrogen abundance (\(X\)) which shrinks the width of the hydrogen ionisation region. +To show the effect of small changes in \(Y\) on the sound speed, we plot the sound speed gradient in Figure \ref{fig:gamma-sound-speed}. The three models shown were evolved to the same \(X_c = 0.6\) as model S with \(Y_\mathrm{init}\) of 0.26, 0.28, and, 0.3. The dominant effect of helium abundance appears as a Gaussian-like depression in \(\gamma\) around the second ionisation of helium. We see how larger helium abundance increases the width and depression in \(\gamma\), which is reflected in the sound speed gradient. A larger \(Y\) also leads to a relative reduction in hydrogen abundance (\(X\)) which shrinks the width of the hydrogen ionisation region. \subsubsection{The Effect of a Change in \(\gamma\) on p Mode Frequencies} To see how a change in \(\gamma\) affects the mode frequencies, we explore the sensitivity of a mode to rapid structural changes in the star. Starting with the variational principle, we can approximate the characteristic frequencies of a spherically symmetric, slowly rotating star by \citep{Chandrasekhar1964}, % \begin{equation} - \omega^2 = \frac{\mathcal{E}}{\mathcal{I}}\label{eq:var-prin} + \omega^2 \simeq \frac{\mathcal{E}}{\mathcal{I}}\label{eq:var-prin} \end{equation} % which is a ratio of \(\mathcal{E}\) (proportional to the mode's energy) @@ -346,7 +346,8 @@ \subsubsection{The Effect of a Change in \(\gamma\) on p Mode Frequencies} \frac{\delta\omega}{\omega} = \int_0^R \left(\mathcal{K}_{c^2,\rho} \frac{\delta c^2}{c^2} + \mathcal{K}_{\rho,c^2} \frac{\delta \rho}{\rho} \right) \dd r.\label{eq:kernels} \end{equation} % -where \(\mathcal{K}_{a, b}\) gives the relative effect on \(\omega\) at a given \(r\) due to a perturbation in a state variable \(a\) at fixed \(b\). The kernels, defined fully in \citet{Gough.Thompson1991}, can show the sensitivity of the wave to a change in state in the star. We plot \(\mathcal{K}_{c^2,\rho}\) and \(\mathcal{K}_{\rho,c^2}\) in Figure \ref{fig:kernels} for a few radial oscillation modes. Both decay through the star, meaning that the modes are less sensitive to structural changes deeper in the star. The kernels also oscillate at different frequencies corresponding to the radial order, meaning the amount they intersect with a glitch changes with \(\omega\). +where \(\mathcal{K}_{a, b}\) gives the relative effect on \(\omega\) at a given \(r\) due to a perturbation in a state variable \(a\) at fixed \(b\). The kernels, defined fully in \citet{Gough.Thompson1991}, can show the sensitivity of the wave to a change in state in the star. We plot \(\mathcal{K}_{c^2,\rho}\) and \(\mathcal{K}_{\rho,c^2}\) in Figure \ref{fig:kernels} for a few radial oscillation modes. Both decay through the star, meaning that the modes are less sensitive to structural changes deeper in the star. +% The kernels also vary faster radially with increasing mode frequency, meaning the amount they intersect with a glitch changes with \(\omega\). We can evaluate Equation \ref{eq:kernels} to find the effect on \(\omega\) due to a change in \(\gamma\) from helium ionisation. The first term can be easily rewritten in terms of a change in \(\gamma\) because \(c^2 \propto \gamma\), hence \(\mathcal{K}_{\gamma,\rho} \delta \gamma / \gamma \equiv \mathcal{K}_{c^2,\rho} \delta c^2 / c^2\). Helium ionisation has a negligible effect on density, so we can assume \(\delta\rho/\rho \approx 0\) in the ionisation region. Therefore, we may rewrite Equation \ref{eq:kernels} due to a change in \(\gamma\) as, % @@ -363,7 +364,7 @@ \subsubsection{The Effect of a Change in \(\gamma\) on p Mode Frequencies} \newcommand*{\propconst}{\ensuremath{{\mathcal{A}}}} -In order to solve Equation \ref{eq:delta-omega}, we first expand the mode inertia, \(\mathcal{I}\). The Lagrangian perturbation vector can be written in terms of a radial and horizontal component, \(\vect{\xi} = \xi_r \hat{r} + \vect{\xi}_h\). In the high-order limit where \(l/n \rightarrow 0\), the horizontal component \(\vect{\xi}_h\) is negligible. Therefore, we can rewrite the mode inertia as, +In order to solve Equation \ref{eq:delta-omega}, we first expand the mode inertia, \(\mathcal{I}\). The Lagrangian perturbation vector can be written in terms of a radial and horizontal component, \(\vect{\xi} = \xi_r \hat{r} + \vect{\xi}_h\). For low angular degree modes, in the limit where \(l/n \rightarrow 0\), the horizontal component \(\vect{\xi}_h\) is negligible. Therefore, we can rewrite the mode inertia as, % \begin{equation} \mathcal{I} \simeq \int_0^R \xi_r^2 \rho r^2 \, \dd r. @@ -377,14 +378,14 @@ \subsubsection{The Effect of a Change in \(\gamma\) on p Mode Frequencies} % where \(\propconst\) is a proportionality constant. The radial wave number \(K \simeq \omega / c\) for high-order acoustic modes. % Because the speed of sound changes slowly between nodes? -Assuming \(\tau\) is not close to the upper turning point (where the wave is reflected near the stellar surface), the phase term \(\psi \simeq \omega \tau + \epsilon\). The small offset \(\epsilon\) is a slowly varying function of \(\tau\). Substituting Equation \ref{eq:xi-r} into the mode inertia gives, +Assuming \(\tau\) is not close to the upper turning point (where the wave is reflected near the stellar surface), the phase term \(\psi \simeq \omega \tau + \epsilon\). Here, the small offset \(\epsilon\) is a slowly varying function of \(\tau\). Substituting Equation \ref{eq:xi-r} into the mode inertia gives, % \begin{align} \mathcal{I} &\simeq \propconst^2 \int_0^R K \cos^2\psi \, \dd r, \notag \\ &= \frac12 \omega \propconst^2 \int_0^R (1 + \cos 2 \psi) \frac{\dd r}{c}. \end{align} % -Finally, changing to an integral over the acoustic depth, \(\tau\) we can evaluate the integral for high-order modes where \(\omega_n \ll \tau_0^{\,-1}\), +Finally, changing to an integral over the acoustic depth (\(\tau\)) we can evaluate the integral for high-order modes where \(\omega_n \ll \tau_0^{\,-1}\), % \begin{equation} \mathcal{I} \simeq \frac12 \omega \propconst^2 \int_0^{\tau_0} [1 + \cos 2 (\omega\tau + \epsilon)] \, \dd \tau \simeq \frac12 \omega \propconst^2 \tau_0. \label{eq:inertia} @@ -397,7 +398,7 @@ \subsubsection{The Effect of a Change in \(\gamma\) on p Mode Frequencies} (\dive{\vect{\xi}})^2 \simeq \frac{\propconst^2 \omega^3}{\gamma P c r^2} \sin^2\psi, \label{eq:div-xi} \end{equation} % -into Equation \ref{eq:gamma-kernel} to get, +along with \(\mathcal{K}_{\gamma,\rho}\) from Equation \ref{eq:gamma-kernel} into Equation \ref{eq:delta-omega} to get, % \begin{align} \left.\frac{\delta\omega}{\omega}\right|_\gamma &\simeq \frac{1}{2\omega^2\mathcal{I}} \int_0^{R} \delta\gamma P (\dive{\vect{\xi}})^2 r^2 \, \dd r, \notag \\ @@ -422,7 +423,7 @@ \subsubsection{A Functional Form of the Helium Glitch Signature} % where \(\Gamma_\heII\) is the area, \(\Delta_\heII\) is the characteristic width, and \(\tau_\heII\) is the centre of the ionisation region. -Here, we verify the result for \(\delta\omega\) due to He\,\textsc{ii} ionisation from \citet{Houdek.Gough2007}. Substituting Equation \ref{eq:he-gamma} into Equation \ref{eq:omega-osc} with a change of variables to \(x = (\tau - \tau_\heII)/\Delta_\heII\), we get, +We will verify the result from \citet{Houdek.Gough2007} for \(\delta\omega\) due to He\,\textsc{ii} ionisation. Substituting Equation \ref{eq:he-gamma} into Equation \ref{eq:omega-osc} with a change of variables to \(x = (\tau - \tau_\heII)/\Delta_\heII\), gives, % \begin{equation} \left.\frac{\delta\omega}{\omega}\right|_{\heII, \mathrm{osc}} \simeq \frac{\Gamma_\heII}{2\sqrt{2\pi} \, \tau_0} \, \int_{-\infty}^\infty \ee^{- x^2/2} \cos 2 (\Delta_\heII \omega x + \widetilde{\epsilon_\heII}) \, \dd x \label{eq:omega-ii-osc} @@ -438,7 +439,7 @@ \subsubsection{A Functional Form of the Helium Glitch Signature} % \begin{align} I'(a) &= - 2 \Delta_\heII \omega \int_{-\infty}^\infty x \, \ee^{- x^2/2 } \sin 2 (\Delta_\heII \omega x a + \widetilde{\epsilon_\heII}) \, \dd x, \notag\\ - &= - 2 \Delta_\heII \omega \int_{-\infty}^\infty \sin 2 (\Delta_\heII \omega x a + \widetilde{\epsilon_\heII}) \, \dd \ee^{- x^2/2 }, \notag\\ + &= - 2 \Delta_\heII \omega \int_{-\infty}^\infty \sin 2 (\Delta_\heII \omega x a + \widetilde{\epsilon_\heII}) \, \dd\ee^{- x^2/2 }, \notag\\ &= 2 \Delta_\heII \omega \left\{ \left[ \ee^{- x^2/2 } \sin 2 (\Delta_\heII \omega x a + \widetilde{\epsilon_\heII}) \right]_{-\infty}^\infty - 2 \Delta_\heII \omega a \int_{-\infty}^\infty \ee^{- x^2/2 } \cos 2 (\Delta_\heII \omega x a + \widetilde{\epsilon_\heII}) \, \dd x \right\}, \notag\\ &= - 4 \Delta_\heII^2 \omega^2 a \, I(a), \end{align} @@ -475,7 +476,7 @@ \subsection{Base of the Convective Zone Glitch}\label{sec:bcz-glitch} \label{fig:bcz-density} \end{figure} -The sensitivity of p-modes to glitches gets smaller further into the star. However, we should not neglect the effect of the near-discontinuity at the base of the convective zone (BCZ). This arises from a jump in the second derivative of temperature at the BCZ which translated to discontinuities in the second derivative of density and sound speed. In Figure \ref{fig:bcz-density} we can see a fast change in direction of the density gradient around \(\tau/\tau_0 \approx 0.6\) (or at about \SI{2200}{\second} for model S). We also see that the relative location of the BCZ changes a little with \(Y\). +The sensitivity of p-modes to glitches gets smaller further into the star. However, we should not neglect the effect of the near-discontinuity at the base of the convective zone (BCZ). This arises from a jump in the second derivative of temperature at the BCZ which translated to discontinuities in the second derivative of density and sound speed. In Figure \ref{fig:bcz-density} we can see a fast change in direction of the density gradient around \(\tau/\tau_0 \approx 0.6\) (or at about \SI{2200}{\second} for model S). We also see that the relative location of the BCZ changes a little with initial \(Y\). % Overshoot actually causes discontinuity in first derivative, not second \citep{Zahn1991}. @@ -493,7 +494,7 @@ \subsection{Base of the Convective Zone Glitch}\label{sec:bcz-glitch} % is the change in second density derivative at the BCZ (\(r = r_\bcz\)). -For high-order modes, \(\tau_0 \omega \gg 1\) and thus \(\tan^{-1}(2\tau_0\omega) \simeq \pi/2\), and \((1 + {1}/{4\tau_0^2\omega^2})^{-1/2} \simeq 1\). Therefore, we can simplify this and write it in a more familiar form in terms of cyclic frequency, +For high-order modes, \(\tau_0 \omega \gg 1\). Hence, \(\tan^{-1}(2\tau_0\omega) \simeq \pi/2\), and \((1 + {1}/{4\tau_0^2\omega^2})^{-1/2} \simeq 1\). Therefore, we can simplify this and write it in a more familiar form in terms of cyclic frequency, % \begin{equation} \left.\frac{\delta\nu}{\nu}\right|_{\bcz,\mathrm{osc}} \simeq \frac{c_\bcz^2\Delta_\bcz}{32\pi^3}\frac{\nu_0}{\nu^3} \sin(4\pi\tau_\bcz\nu + \phi_\bcz) \label{eq:bcz-osc} diff --git a/references.bib b/references.bib index 5a5f917..cbf727f 100644 --- a/references.bib +++ b/references.bib @@ -175,7 +175,7 @@ @article{Anderson.Hogg.ea2018 doi = {10.3847/1538-3881/aad7bf}, urldate = {2020-05-27}, abstract = {Converting a noisy parallax measurement into a posterior belief over distance requires inference with a prior. Usually, this prior represents beliefs about the stellar density distribution of the Milky Way. However, multiband photometry exists for a large fraction of the Gaia-TGAS Catalog and is incredibly informative about stellar distances. Here, we use 2MASS colors for 1.4 million TGAS stars to build a noise-deconvolved empirical prior distribution for stars in color-magnitude space. This model contains no knowledge of stellar astrophysics or the Milky Way but is precise because it accurately generates a large number of noisy parallax measurements under an assumption of stationarity; that is, it is capable of combining the information from many stars. We use the Extreme Deconvolution (XD) algorithm\textemdash which is an empirical-Bayes approximation to a full-hierarchical model of the true parallax and photometry of every star\textemdash to construct this prior. The prior is combined with a TGAS likelihood to infer a precise photometric-parallax estimate and uncertainty (and full posterior) for every star. Our parallax estimates are more precise than the TGAS catalog entries by a median factor of 1.2 (14\% are more precise by a factor {$>$}2) and they are more precise than the previous Bayesian distance estimates that use spatial priors. We validate our parallax inferences using members of the Milky Way star cluster M67, which is not visible as a cluster in the TGAS parallax estimates but appears as a cluster in our posterior parallax estimates. Our results, including a parallax posterior probability distribution function for each of 1.4 million TGAS stars, are available in companion electronic tables.}, - keywords = {catalogs,Hertzsprung\textendash Russell and C\textendash M diagrams,methods: statistical,parallaxes} + keywords = {catalogs,Hertzsprung–Russell and C–M diagrams,methods: statistical,parallaxes} } @article{Ando.Osaki1975, @@ -658,7 +658,7 @@ @article{Bazot.Bourguignon.ea2012 doi = {10.1111/j.1365-2966.2012.21818.x}, urldate = {2023-04-17}, abstract = {Determining the physical characteristics of a star is an inverse problem consisting of estimating the parameters of models for the stellar structure and evolution, and knowing certain observable quantities. We use a Bayesian approach to solve this problem for {$\alpha$} Cen A, which allows us to incorporate prior information on the parameters to be estimated, in order to better constrain the problem. Our strategy is based on the use of a Markov chain Monte Carlo (MCMC) algorithm to estimate the posterior probability densities of the stellar parameters: mass, age, initial chemical composition, etc. We use the stellar evolutionary code ASTEC to model the star. To constrain this model both seismic and non-seismic observations were considered. Several different strategies were tested to fit these values, using either two free parameters or five free parameters in ASTEC. We are thus able to show evidence that MCMC methods become efficient with respect to more classical grid-based strategies when the number of parameters increases. The results of our MCMC algorithm allow us to derive estimates for the stellar parameters and robust uncertainties thanks to the statistical analysis of the posterior probability densities. We are also able to compute odds for the presence of a convective core in {$\alpha$} Cen A. When using core-sensitive seismic observational constraints, these can rise above {$\sim$}40 per cent. The comparison of results to previous studies also indicates that these seismic constraints are of critical importance for our knowledge of the structure of this star.}, - keywords = {Astrophysics - Solar and Stellar Astrophysics,methods: numerical,methods: statistical,stars: evolution,stars: fundamental parameters,stars: individual: {$\alpha$} Cen A,stars: oscillations}, + keywords = {Astrophysics - Solar and Stellar Astrophysics,methods: numerical,methods: statistical,stars: evolution,stars: fundamental parameters,stars: individual: α Cen A,stars: oscillations}, annotation = {ADS Bibcode: 2012MNRAS.427.1847B} } @@ -1967,7 +1967,7 @@ @article{Eggenberger.Charbonnel.ea2004 doi = {10.1051/0004-6361:20034203}, urldate = {2023-04-17}, abstract = {Detailed models of {$\alpha$} Cen A and B based on new seismological data for {$\alpha$} Cen B by Carrier \& Bourban (\textbackslash cite\{ca03\}) have been computed using the Geneva evolution code including atomic diffusion. Taking into account the numerous observational constraints now available for the {$\alpha$} Cen system, we find a stellar model which is in good agreement with the astrometric, photometric, spectroscopic and asteroseismic data. The global parameters of the {$\alpha$} Cen system are now firmly constrained to an age of t=6.52 {$\pm$} 0.30 Gyr, an initial helium mass fraction Yi=0.275 {$\pm$} 0.010 and an initial metallicity (Z/X)i=0.0434 {$\pm$} 0.0020. Thanks to these numerous observational constraints, we confirm that the mixing-length parameter {$\alpha$} of the B component is larger than the one of the A component, as already suggested by many authors (Noels et al. \textbackslash cite\{no91\}; Fernandes \& Neuforge \textbackslash cite\{fe95\}; Guenther \& Demarque \textbackslash cite\{gu00\}): {$\alpha$}B is about 8\% larger than {$\alpha$}A ({$\alpha$}A=1.83 {$\pm$} 0.10 and {$\alpha$}B=1.97 {$\pm$} 0.10). Moreover, we show that asteroseismic measurements enable to determine the radii of both stars with a very high precision (errors smaller than 0.3\%). The radii deduced from seismological data are compatible with the new interferometric results of Kervella et al. (\textbackslash cite\{ke03\}) even if they are slightly larger than the interferometric radii (differences smaller than 1\%).}, - keywords = {Astrophysics,stars: binaries: visual,stars: evolution,stars: individual: {$\alpha$} Cen,stars: oscillations}, + keywords = {Astrophysics,stars: binaries: visual,stars: evolution,stars: individual: α Cen,stars: oscillations}, annotation = {ADS Bibcode: 2004A\&A...417..235E} } @@ -3649,7 +3649,7 @@ @article{Kjeldsen.Bedding.ea2005 doi = {10.1086/497530}, urldate = {2020-09-21}, abstract = {We have made velocity observations of the star {$\alpha$} Centauri B from two sites, allowing us to identify 37 oscillation modes with l=0-3. Fitting to these modes gives the large and small frequency separations as a function of frequency. The mode lifetime, as measured from the scatter of the oscillation frequencies about a smooth trend, is similar to that in the Sun. Limited observations of the star {$\delta$} Pav show oscillations centered at 2.3 mHz, with peak amplitudes close to solar. We introduce a new method of measuring oscillation amplitudes from heavily smoothed power density spectra, from which we estimated amplitudes for {$\alpha$} Cen {$\alpha$} and B, {$\beta$} Hyi, {$\delta$} Pav, and the Sun. We point out that the oscillation amplitudes may depend on which spectral lines are used for the velocity measurements. Based on observations collected at the European Southern Observatory, Paranal, Chile (ESO Programme 71.D-0618).}, - keywords = {stars: individual ({$\delta$} Pavonis),Stars: Individual: Constellation Name: {$\alpha$} Centauri A,Stars: Individual: Constellation Name: {$\alpha$} Centauri B,Stars: Individual: Constellation Name: {$\beta$} Hydri,Stars: Oscillations,Sun: Helioseismology} + keywords = {stars: individual (δ Pavonis),Stars: Individual: Constellation Name: α Centauri A,Stars: Individual: Constellation Name: α Centauri B,Stars: Individual: Constellation Name: β Hydri,Stars: Oscillations,Sun: Helioseismology} } @article{Kjeldsen.Bedding.ea2008, @@ -3663,7 +3663,7 @@ @article{Kjeldsen.Bedding.ea2008 doi = {10.1086/591667}, urldate = {2020-09-18}, abstract = {In helioseismology, there is a well-known offset between observed and computed oscillation frequencies. This offset is known to arise from improper modeling of the near-surface layers of the Sun, and a similar effect must occur for models of other stars. Such an effect impedes progress in asteroseismology, which involves comparing observed oscillation frequencies with those calculated from theoretical models. Here, we use data for the Sun to derive an empirical correction for the near-surface offset, which we then apply to three other stars ({$\alpha$} Cen A, {$\alpha$} Cen B, and {$\beta$} Hyi). The method appears to give good results, in particular providing an accurate estimate of the mean density of each star.}, - keywords = {stars: individual: {$\beta$} Hyi {$\alpha$} Cen A {$\alpha$} Cen B,stars: oscillations,Sun: helioseismology} + keywords = {stars: individual: β Hyi α Cen A α Cen B,stars: oscillations,Sun: helioseismology} } @article{Kjeldsen.Bedding1995, @@ -3678,7 +3678,7 @@ @article{Kjeldsen.Bedding1995 issn = {0004-6361}, urldate = {2020-09-02}, abstract = {There are no good predictions for the amplitudes expected from solar-like oscillations in other stars. In the absence of a definitive model for convection, which is thought to be the mechanism that excites these oscillations, the amplitudes for both velocity and luminosity measurements must be estimated by scaling from the Sun. In the case of luminosity measurements, even this is difficult because of disagreement over the solar amplitude. This last point has lead us to investigate whether the luminosity amplitude of oscillations {$\delta$}L/L can be derived from the velocity amplitude (v\_osc\_). Using linear theory and observational data, we show that p-mode oscillations in a large sample of pulsating stars satisfy ({$\delta$}L/L)\_bol\_\{prop.to\} v\_osc\_/T\_eff\_. Using this relationship, together with the best estimate of v\_osc\_,sun\_=(23.4+/-1.4)cm/s, we estimate the luminosity amplitude of solar oscillations at 550nm to be {$\delta$}L/L=(4.7+/-0.3)ppm. Next we discuss how to scale the amplitude of solar-like (i.e., convectively-powered) oscillations from the Sun to other stars. The only predictions come from model calculations by Christensen-Dalsgaard \& Frandsen (1983, Sol. Phys. 82, 469). However, their grid of stellar models is not dense enough to allow amplitude predictions for an arbitrary star. Nevertheless, although convective theory is complicated, we might expect that the general properties of convection - including oscillation amplitudes - should change smoothly through the colour-magnitude diagram. Indeed, we find that the velocity amplitudes predicted by the model calculations are well fitted by the relation v\_osc\_\{prop.to\}L/M. These two relations allow us to predict both the velocity and luminosity amplitudes of solar-like oscillations in any given star. We compare these predictions with published observations and evaluate claims for detections that have appeared in the literature. We argue that there is not yet good evidence for solar-like oscillations in any star except the Sun. For solar-type stars (e.g., {$\alpha$} Cen A and {$\beta$} Hyi), observations have not yet reached sufficient sensitivity to detect the amplitudes we predict. For some F-type stars, namely Procyon and several members of M67, detection sensitivities 30-40\% below the predicted amplitudes have been achieved. We conclude that these stars must oscillate with amplitudes less than has generally been assumed.}, - keywords = {\{DELTA\} SCT,CEPHEIDS,STARS: INDIVIDUAL: \{ALPHA\} CEN,STARS: INDIVIDUAL: PROCYON,STARS: OSCILLATIONS,SUN: OSCILLATIONS} + keywords = {{DELTA} SCT,CEPHEIDS,STARS: INDIVIDUAL: {ALPHA} CEN,STARS: INDIVIDUAL: PROCYON,STARS: OSCILLATIONS,SUN: OSCILLATIONS} } @misc{Koposov.Speagle.ea2023, @@ -4317,7 +4317,7 @@ @article{Miglio.Montalban2005 doi = {10.1051/0004-6361:20052988}, urldate = {2023-04-17}, abstract = {We apply the Levenberg-Marquardt minimization algorithm to seismic and classical observables of the {$\alpha$}Cen binary system in order to derive the fundamental parameters of {$\alpha$}CenA+B, and to analyze the dependence of these parameters on the chosen observables, on their uncertainty, and on the physics used in stellar modelling. We show that while the fundamental stellar parameters do not depend on the treatment of convection adopted (Mixing Length Theory - MLT - or ``Full Spectrum of Turbulence'' - FST), the age of the system depends on the inclusion of gravitational settling, and is deeply biased by the small frequency separation of component B. We try to answer the question of the universality of the mixing length parameter, and we find a statistically reliable dependence of the {$\alpha$}-parameter on the HR diagram location (with a trend similar to the predictions based on 2-D simulations). We propose the frequency separation ratios as better observables to determine the fundamental stellar parameters, and to use the large frequency separation and frequencies to extract information about the stellar structure. The effects of diffusion and equation of state on the oscillation frequencies are also studied, but present seismic data do not allow their determination.}, - keywords = {Astrophysics,stars: fundamental parameters,stars: individual: {$\alpha$} Cen,stars: interiors,stars: oscillations}, + keywords = {Astrophysics,stars: fundamental parameters,stars: individual: α Cen,stars: interiors,stars: oscillations}, annotation = {ADS Bibcode: 2005A\&A...441..615M} } @@ -4754,7 +4754,7 @@ @article{Nsamba.Monteiro.ea2018 doi = {10.1093/mnrasl/sly092}, urldate = {2023-04-17}, abstract = {Understanding the physical process responsible for the transport of energy in the core of {$\alpha$} Centauri A is of the utmost importance if this star is to be used in the calibration of stellar model physics. Adoption of different parallax measurements available in the literature results in differences in the interferometric radius constraints used in stellar modelling. Further, this is at the origin of the different dynamical mass measurements reported for this star. With the goal of reproducing the revised dynamical mass derived by Pourbaix \& Boffin, we modelled the star using two stellar grids varying in the adopted nuclear reaction rates. Asteroseismic and spectroscopic observables were complemented with different interferometric radius constraints during the optimization procedure. Our findings show that best-fitting models reproducing the revised dynamical mass favour the existence of a convective core ({$\greaterequivlnt$}70 per cent of best-fitting models), a result that is robust against changes to the model physics. If this mass is accurate, then {$\alpha$} Centauri A may be used to calibrate stellar model parameters in the presence of a convective core.}, - keywords = {Astrophysics - Solar and Stellar Astrophysics,method: asteroseismology,stars: convection and radiation,stars: fundamental parameters,stars: {$\alpha$} Centauri A}, + keywords = {Astrophysics - Solar and Stellar Astrophysics,method: asteroseismology,stars: convection and radiation,stars: fundamental parameters,stars: α Centauri A}, annotation = {ADS Bibcode: 2018MNRAS.479L..55N} } From 05a5f59240acb06cdc764370a8cdff48fee0d85e Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Fri, 28 Apr 2023 14:10:07 +0100 Subject: [PATCH 44/50] Finish glitch GP chapter --- chapters/glitch-gp.tex | 84 +++++++++++++++++++++--------------------- 1 file changed, 42 insertions(+), 42 deletions(-) diff --git a/chapters/glitch-gp.tex b/chapters/glitch-gp.tex index d4abadd..486728d 100644 --- a/chapters/glitch-gp.tex +++ b/chapters/glitch-gp.tex @@ -12,7 +12,7 @@ % \chapter[Modelling Acoustic Glitches with a Gaussian Process]{Modelling Acoustic Glitch Signatures in Stellar Oscillations with a Gaussian Process}\label{chap:glitch-gp} -\textit{In this chapter, I apply a new method for modelling acoustic glitch signatures in the radial mode frequencies of solar-like oscillators. I compare this method with another using a model star with different levels of noise. Then, I apply both methods to the star 16 Cyg A to provide a real-world working example. I show that our method can be used to find the strength and location of glitches caused by the second ionisation of helium and the base of the convective zone. I also demonstrate that this method improves treatment of the smoothly varying deviation in the mode frequencies from their regular separation.} +\textit{In this chapter, I apply a new method for modelling acoustic glitch signatures in the radial mode frequencies of solar-like oscillators. I compare my method to another using a model star with different levels of noise. Then, I apply both methods to the star 16 Cyg A. I show that my method can be used to find the strength and location of glitches caused by the second ionisation of helium and the base of the convective zone. I also demonstrate that my method improves treatment of the smoothly varying component of the mode frequencies.} \section{Introduction} @@ -22,16 +22,16 @@ \section{Introduction} f(n, l) = \tilde{f}(n, l) + \delta\nu,\label{eq:general-glitch} \end{equation} % -where \(\delta\nu\) is some function of frequency which may, for example, be evaluated at \(\tilde{f}(n, l)\). +where \(\delta\nu\) is some function of frequency which may be evaluated at \(\tilde{f}(n, l)\). -In principle, \(\delta\nu\) could arise from any glitches expected in a particular star. For this work, we consider glitches in main sequence solar-like oscillators. In Chapter \ref{chap:glitch}, we derived approximations for \(\delta\nu\) arising from acoustic glitches caused by the second ionisation of helium and the BCZ. For this work, we choose to ignore contributions to \(\delta\nu\) from the first ionisation of helium. As a result, we let \(\delta\nu = \delta\nu_\helium + \delta\nu_\bcz\) where each component depends on parameters relating to properties of the glitches (cf. Equations \ref{eq:he-osc} and \ref{eq:bcz-osc}), +In principle, \(\delta\nu\) could arise from any glitches expected in a particular star. Here, we consider glitches in main sequence solar-like oscillators. In Chapter \ref{chap:glitch}, we derived approximations for \(\delta\nu\) arising from acoustic glitches caused by the second ionisation of helium and the BCZ. We choose to ignore contributions to \(\delta\nu\) from the first ionisation of helium. Consequently, we let \(\delta\nu = \delta\nu_\helium + \delta\nu_\bcz\) where each component depends on parameters relating to properties of the glitches (cf. Equations \ref{eq:he-osc} and \ref{eq:bcz-osc}), % \begin{align} \delta\nu_\helium &= \alpha_\helium \nu_0 \nu \, \ee^{-\beta_\helium \nu^2} \sin(4\pi\tau_\helium\nu + \phi_\helium), \label{eq:he-glitch}\\ \delta\nu_\bcz &= \alpha_\bcz \nu_0 \nu^{-2} \, \sin(4\pi\tau_\bcz\nu + \phi_\bcz). \label{eq:bcz-glitch} \end{align} % -The parameters \(\alpha_\helium \simeq \Gamma_\heII\) and \(\beta_\helium \propto \Delta_\heII^2\) relate to the area and variance of the Gaussian-like depression in \(\gamma\) caused by the second ionisation of helium. The amplitude parameter for the BCZ glitch, \(\alpha_\bcz \propto \Delta_{\bcz}\), is proportional to the difference in the second density derivative at the base of the convective zone and has units of frequency squared. The approximate acoustic depths of second helium ionisation and the BCZ are given by \(\tau_\helium\) and \(\tau_\bcz\) respectively, and \(\phi_\helium\) and \(\phi_\bcz\) are arbitrary phase constants. +The parameters \(\alpha_\helium \simeq \Gamma_\heII\) and \(\beta_\helium \propto \Delta_\heII^2\) relate to the area and variance of the Gaussian-like depression in \(\gamma\) caused by the second ionisation of helium (see Equation \ref{eq:he-gamma}). The amplitude parameter for the BCZ glitch is proportional to the difference in the second density derivative at the base of the convective zone (\(\alpha_\bcz \propto \Delta_{\bcz}\)) with units of frequency squared. The approximate acoustic depths of second helium ionisation and the BCZ are given by \(\tau_\helium\) and \(\tau_\bcz\) respectively, and \(\phi_\helium\) and \(\phi_\bcz\) are arbitrary phase constants. Providing \(\tilde{f}(n, l)\) is a good approximation of the mode frequencies, we can calculate \(\delta\nu\) at \(\nu = \tilde{f}(n, l)\) to predict \(f(n, l)\). For example, \(\tilde{f}(n,l)\) could be a \(K\)-th order polynomial in \(n\) with coefficients \(a_{lk}\) \citep[e.g.][]{Kjeldsen.Bedding.ea2005,Ulrich1986}, % @@ -39,11 +39,11 @@ \section{Introduction} \tilde{f}(n, l) = \nu_0 \sum_{k=0}^{K} a_{lk} n^k. \label{eq:poly} \end{equation} % -The linear component of this is equivalent to the asymptotic expression (see Equation \ref{eq:asy}) and the remaining terms describe curvature in the mode frequencies. However, there are drawbacks to using a polynomial for \(\tilde{f}(n, l)\). Whilst a polynomial with \(K = \infty\) can represent any function, this is impractical here. If \(K\) is too low, then it will not be flexible enough, biasing the inference of \(\delta\nu_\helium\) and \(\delta\nu_\bcz\). Yet, if \(K\) is too high, then the model will over-fit and the glitch component may be missed. Regularising the polynomial is one solution to over-fitting, but this adds extra parameters to tune. Additionally, a finite polynomial represents only a small fraction of function space, leading to our model to be systematically biased to a particular functional form of \(\tilde{f}\). +The linear component of this is equivalent to the asymptotic expression (see Equation \ref{eq:asy}) and the remaining terms describe curvature in the mode frequencies. However, there are drawbacks to using a polynomial for \(\tilde{f}(n, l)\). Whilst a polynomial with \(K = \infty\) can represent any function, this is impractical here. If \(K\) is too low, then it will not be flexible enough, biasing the inference of \(\delta\nu_\helium\) and \(\delta\nu_\bcz\). Yet, if \(K\) is too high, then the model will over-fit and the glitch component may be missed. Regularising the polynomial is one solution to over-fitting, but this adds extra parameters to tune. Additionally, a finite polynomial represents only a small fraction of function space, systematically biasing our model to a particular functional form of \(\tilde{f}\). \defcitealias{Verma.Raodeo.ea2019}{V19} -Directly fitting the glitch the above way has been done before \citep[e.g.][]{Verma.Faria.ea2014,Verma.Raodeo.ea2017,Mazumdar.Monteiro.ea2014}. In this work, we will compare the most recent version of this method \citep[][hereafter V19]{Verma.Raodeo.ea2019} with a new method. Our method will make use of a Gaussian Process (GP) to marginalise over our uncertainty in the functional form of \(f\). In the next section, we introduce the data used in comparing the two methods. Then, we define both modelling methods being compared in Section \ref{sec:glitch-methods}. Finally, we apply the methods to the data and present the results and discussion in Sections \ref{sec:glitch-results} and \ref{sec:glitch-disc}. +Directly fitting the glitch the above way has been done before \citep[e.g.][]{Verma.Faria.ea2014,Verma.Raodeo.ea2017,Mazumdar.Monteiro.ea2014}. In this work, we will compare the most recent version of this method \citep[][hereafter V19]{Verma.Raodeo.ea2019} with a new method. Our method will make use of a Gaussian Process (GP) to marginalise over the uncertainty in the functional form of \(f\). In the next section, we introduce the data used in comparing the two methods. Then, we define both modelling in Section \ref{sec:glitch-methods}. Finally, we apply the methods to the data and present the results and discussion in Sections \ref{sec:glitch-results} and \ref{sec:glitch-disc} respectively. \section{Data}\label{sec:glitch-data} @@ -59,17 +59,17 @@ \section{Data}\label{sec:glitch-data} \begin{figure}[tb] \centering \includegraphics{figures/glitch-test-obs.pdf} - \caption{Echelle plot of simulated true and `observed' radial mode frequency data from model star S for worst-, better-, and best-case scenarios.} + \caption{Echelle plot of simulated true and `observed' radial mode frequency data from model star S for worst, better, and best case scenarios.} \label{fig:glitch-test-obs} \end{figure} -\paragraph{Test Star} We created three sets of test data for worst-, better-, and best-case scenarios using stellar model S (defined in Section \ref{sec:model-s}). Model S is similar to the Sun, with surface parameters of \(\teff = \SI{5682}{\kelvin}\), \(\log g = 4.426\) and \([\mathrm{Fe/H}] = 0.03\), and bulk parameters of \(M = \SI{1.00}{\solarmass}\), \(R = \SI{1.01}{\solarmass}\) and \(t_\star = \SI{4.07}{\giga\year}\). We calculated radial order modes (\(l=0\)) for the test star using the \textsc{GYRE} oscillation code \citep{Townsend.Teitler2013}. Then, we selected different numbers of modes (\(N\)) symmetrically about a reference frequency, \(\nu_\mathrm{ref} = \SI{2900}{\micro\hertz}\) (close to the expected frequency at maximum power of the star). For each test case, we added differing amounts of Gaussian noise (scaled by \(\sigma_\obs\)) to the frequencies. The parameters and mode frequencies for each test case are shown in Table \ref{tab:glitch-obs}. We also plotted the modes on an echelle diagram in Figure \ref{fig:glitch-test-obs}. We can see the effect of the helium glitch in the echelle diagrams, where there is a large `wiggle' at low frequency. The glitch signature is visible in the better case and clearest in the best-case scenario. +\paragraph{Test Star} We created three sets of test data for worst, better, and best case scenarios using stellar model S (defined in Section \ref{sec:model-s}). Model S is similar to the Sun, with surface parameters of \(\teff = \SI{5682}{\kelvin}\), \(\log g = 4.426\) and \([\mathrm{Fe/H}] = 0.03\), and bulk parameters of \(M = \SI{1.00}{\solarmass}\), \(R = \SI{1.01}{\solarmass}\) and \(t_\star = \SI{4.07}{\giga\year}\). We calculated radial order modes (\(l=0\)) for the test star using the \textsc{GYRE} oscillation code \citep{Townsend.Teitler2013}. Then, we selected different numbers of modes (\(N\)) symmetrically about a reference frequency, \(\nu_\mathrm{ref} = \SI{2900}{\micro\hertz}\) (close to the expected frequency at maximum power of the star). For each test case, we added differing amounts of Gaussian noise (scaled by \(\sigma_\obs\)) to the frequencies. The parameters and mode frequencies for each test case are shown in Table \ref{tab:glitch-obs}. We also plotted the modes on an echelle diagram in Figure \ref{fig:glitch-test-obs}. We can see the effect of the helium glitch in the echelle diagrams, where there is a periodic feature which is strongest at low frequencies. The glitch signature is visible in the better case and clearest in the best case scenario. -\paragraph{16 Cyg A} We used the asteroseismic benchmark star 16 Cyg A as an example real star to test both methods. We adopted values for 16 radial mode frequencies identified by \citet{Lund.SilvaAguirre.ea2017} using observations from \emph{Kepler} \citep[][KIC 12069424]{Borucki.Koch.ea2010}. The mode frequencies and their associated uncertainties are given in Table \ref{tab:glitch-obs}. The glitch has been previously studied for 16 Cyg A with its binary companion 16 Cyg B in \citet{Verma.Faria.ea2014}, making it a useful subject for comparison. Similarly to model S, this target is a solar analogue. However, it is slightly hotter, more evolved and more metal-rich, with \(\teff = \SI{5825(50)}{\kelvin}\), \(\log g = \SI{4.33(7)}{\dex}\) and \([\mathrm{Fe/H}] = \SI{0.10(3)}{\dex}\) \citep{Ramirez.Melendez.ea2009}. Its bulk stellar parameters are \(M \approx \SI{1.1}{\solarmass}\), \(R \approx \SI{1.2}{\solarradius}\) and \(t_\star \approx \SI{7}{\giga\year}\) \citep{SilvaAguirre.Lund.ea2017}. +\paragraph{16 Cyg A} We used the asteroseismic benchmark star 16 Cyg A as an example real star to test both methods. We adopted values for 16 radial mode frequencies identified by \citet{Lund.SilvaAguirre.ea2017} using observations from \emph{Kepler} \citep[][KIC 12069424]{Borucki.Koch.ea2010}. The mode frequencies and their associated uncertainties are given in Table \ref{tab:glitch-obs}. The glitch has been previously studied for 16 Cyg A with its binary companion 16 Cyg B in \citet{Verma.Faria.ea2014}, making it a useful subject for comparison. Similarly to model S, this target is a solar analogue. However, it is slightly hotter, more evolved and more metal-rich, with \(\teff = \SI{5825(50)}{\kelvin}\), \(\log g = \SI{4.33(7)}{\dex}\) and \([\mathrm{Fe/H}] = \SI{0.10(3)}{\dex}\) \citep{Ramirez.Melendez.ea2009}. Its fundamental stellar parameters are \(M \approx \SI{1.1}{\solarmass}\), \(R \approx \SI{1.2}{\solarradius}\) and \(t_\star \approx \SI{7}{\giga\year}\) \citep{SilvaAguirre.Lund.ea2017}. \section{Methods}\label{sec:glitch-methods} -In this section, we describe the two models for radial mode frequency, \(\nu_n = f(n, l=0)\), and the fitting methods being compared in this work. The first is the \citetalias{Verma.Raodeo.ea2019} method, originally used in \citet{Verma.Faria.ea2014} to study the 16 Cyg binary star system. Secondly, we introduce our new `GP method'. Both methods use different statistical philosophy and formulation of the smoothly varying component, \(\tilde{f}(n)\). However, they both fit the same glitch component \(\delta\nu = \delta\nu_\helium + \delta\nu_\bcz\) as given in Equations \ref{eq:he-glitch} and \ref{eq:bcz-glitch}. +In this section, we describe the two models for radial mode frequency, \(\nu_n = f(n, 0)\), and the fitting methods being compared in this work. The first is the \citetalias{Verma.Raodeo.ea2019} method, originally used in \citet{Verma.Faria.ea2014} to study the 16 Cyg binary star system. Secondly, we introduce our new `GP method'. Both methods use different statistical philosophy and formulation of the smoothly varying component, \(\tilde{f}(n)\). However, they each fit the same glitch component \(\delta\nu = \delta\nu_\helium + \delta\nu_\bcz\) as given in Equations \ref{eq:he-glitch} and \ref{eq:bcz-glitch}. \subsection{The V19 Method} @@ -79,7 +79,7 @@ \subsection{The V19 Method} f_A(n) = \tilde{f}_A(n) + \delta\nu_\helium + \delta\nu_\bcz; \quad \tilde{f}_A(n) = \sum_{k=0}^{4} b_k n^k, \end{equation} % -\sloppy where \(b_k \equiv a_{0k} \nu_0\) from Equation \ref{eq:poly}. The model parameters are given by \(\vect{\theta}_A = (b_0, \dots, b_4, a_\helium, \beta_\helium, \tau_\helium, \phi_\helium, a_\bcz, \tau_\bcz, \phi_\bcz)\), where the glitch amplitude parameters are modified to include \(\nu_0\) such that \(a_i \equiv \alpha_i\nu_0\). The \(\nu_0\) parameter is not explicitly included in the \citetalias{Verma.Raodeo.ea2019} model, but we find it useful to keep the scaling in mind, and we include \(\nu_0\) in the GP model. +\sloppy where \(b_k \equiv a_{0k} \nu_0\) from Equation \ref{eq:poly}. The model parameters are given by \(\vect{\theta}_A = (b_0, \dots, b_4, a_\helium, \beta_\helium, \tau_\helium, \phi_\helium, a_\bcz, \tau_\bcz, \phi_\bcz)\), where the glitch amplitude parameters are modified to include \(\nu_0\) such that \(a_i \equiv \alpha_i\nu_0\). The \(\nu_0\) parameter is not explicitly included in the \citetalias{Verma.Raodeo.ea2019} model, but we find it useful to keep the scaling in mind, and we use \(\nu_0\) in the GP model. The model parameters are optimised by minimising a \(\chi^2\) cost function with a regularisation term, % @@ -87,9 +87,9 @@ \subsection{The V19 Method} \chi^2 = \sum_n \left[ \frac{\nu_n^\obs - f_{A}(n)}{\sigma_n^\obs} \right]^2 + \lambda^2 \sum_n \left[ \frac{\dd^3}{\dd n^3} \tilde{f}_A(n)\right]^2, \end{equation} % -where \(\nu_n^\obs\) and \(\sigma_n^\obs\) are the observed mode and its uncertainty at radial order \(n\), and \(\lambda\) is the regularisation parameter. The regularisation was introduced to avoid the polynomial over-fitting and absorbing the glitch terms. +where \(\nu_n^\obs\) and \(\sigma_n^\obs\) are the observed mode and its uncertainty at radial order \(n\), and \(\lambda\) is the regularisation parameter. The regularisation was used to avoid the polynomial over-fitting and absorbing the glitch terms. -We fitted the model using the \textsc{GlitchPy} code\footnote{\url{https://github.com/alexlyttle/GlitchPy}, adapted from \url{https://github.com/kuldeepv89/GlitchPy}.}. The fitting method is described in \citet{Verma.Raodeo.ea2019}, in which we adopted the same value for \(\lambda=7\) and bounds for the selection of initial parameters. We chose the initial parameters randomly within their bounds and optimised them using a BFGS minimisation of \(\chi^2\) \citep{Fletcher1987}, repeated 200 times until a global minimum was found. We further repeated this for 1000 realisations of the observed \(\nu_n\) with Gaussian noise scaled by \(\sigma_n^\obs\) to obtain a range of possible solutions. +We fitted the model using the \textsc{GlitchPy} code\footnote{\url{https://github.com/alexlyttle/GlitchPy}, adapted from \url{https://github.com/kuldeepv89/GlitchPy}.}. The fitting method is described in \citet{Verma.Raodeo.ea2019}, in which we adopted the same value of \(\lambda=7\) and bounds for the selection of initial parameters. We chose the initial parameters randomly within their bounds and optimised them using a BFGS minimisation of \(\chi^2\) \citep{Fletcher1987}, repeated 200 times until a global minimum was found. We further repeated this for 1000 realisations of the observed \(\nu_n\) with Gaussian noise scaled by \(\sigma_n^\obs\) to obtain a range of possible solutions. \subsection{The GP Method} @@ -106,9 +106,9 @@ \subsection{The GP Method} m(n) = \tilde{f}_B(n) + \delta\nu_\helium + \delta\nu_\bcz; \quad \tilde{f}_B(n) = (n + \varepsilon) \nu_0, \label{eq:asy-glitch} \end{equation} % -where \(\tilde{f}_B(n)\) is the asymptotic approximation of the mode frequency (cf. Equation \ref{eq:asy}). +where \(\tilde{f}_B(n)\) is the asymptotic approximation of the mode frequency (see Equation \ref{eq:asy}). -The kernel represents the expected covariance between the values of the function at different \(n\). We chose the squared exponential kernel function to be compatible with a smoothly-varying function of \(n\). Evaluating the kernel gives an \(N \times M\) matrix with element \((i,j)\) given by, +The kernel represents the expected covariance between the values of the function at different \(n\). We chose the squared exponential kernel function to be compatible with a smoothly-varying function of \(n\). Evaluating the kernel gives an \(N \times M\) matrix with an element \((i,j)\) given by, % \begin{equation} k(n_i, n'_j) = \alpha_k \nu_0 \, \ee^{- (n_i - n'_j)^2 / 2\lambda_k^2}, @@ -160,31 +160,31 @@ \subsection{The GP Method} \ln\mathcal{L}_B = - \frac12 \left[ {(\vect{\nu}_\obs - \vect{\mu})^\mathsf{T} \vect{\Sigma}^{\,-1} (\vect{\nu}_\obs - \vect{\mu})} + \ln(| \vect{\Sigma} |) + N\ln(2 \pi) \right], \end{equation} % -where \(\vect{\mu}\) and \(\vect{\Sigma}\) depend on \(\vect{\theta}_B\). The prior transform for \(\vect{\theta}_B\) is the inverse cumulative distribution function associated with the prior distribution \(p(\vect{\theta}_B)\). We defined the prior independently for each of \(\vect{\theta}_B\). The total prior distribution is the product of prior distributions for each parameter, \(p(\vect{\theta}_B) = \prod_j p(\theta_{j})\). We give the prior distributions and their shape parameters in Table \ref{tab:glitch-prior}. In the following paragraphs, we justify our choice of prior distributions based on approximate scaling relations between the parameters. % We also summarise the prior distributions and their parameters in Table \todo{Table summary} +where \(\vect{\mu}\) and \(\vect{\Sigma}\) depend on \(\vect{\theta}_B\). The prior transform for \(\vect{\theta}_B\) is the inverse cumulative distribution function associated with the prior distribution \(p(\vect{\theta}_B)\). We treated the prior for each of \(\vect{\theta}_B\) independently. The total prior distribution is the product of distributions for each parameter, \(p(\vect{\theta}_B) = \prod_j p(\theta_{j})\). We give the prior distributions and their shape parameters in Table \ref{tab:glitch-prior}. In the following paragraphs, we justify our choice of prior distributions based on approximate scaling relations between the parameters. % We also summarise the prior distributions and their parameters in Table \todo{Table summary} \paragraph{Smooth Component} Starting with the parameters for \(\tilde{f}_B(n)\), we chose to sample them from normal distributions, % -\begin{gather*} +\begin{gather} \nu_0 \sim \mathcal{N}(\overline{\nu}_0, s_{\nu_0}^2), \quad \varepsilon \sim \mathcal{N}(\overline{\varepsilon}, s_\varepsilon^2),%\\ % \phi_\helium, \phi_\bcz \sim \mathcal{U}\left(0, 2\pi\right), -\end{gather*} +\end{gather} % centred on \(\overline{\nu}_0\) and \(\overline{\varepsilon}\) and scaled by \(s_{\nu_0}\) and \(s_\varepsilon\). For example, the location and scale parameters could come from global estimates of \(\langle\Delta\nu_n\rangle\) or a linear fit to the modes. We determined \(\overline{\nu}_0\) and \(\overline{\varepsilon}\) for the test stars from a linear fit to the true mode frequencies, and added representative uncertainties of 10 and 5 per cent respectively. For 16 Cyg A, we used measurements of \(\langle\Delta\nu_n\rangle\) and \(\nu_{\max}\) from \citet{Lund.SilvaAguirre.ea2017} to estimate \(\overline{\nu}_0\) and \(\overline{\varepsilon}\). % Priors for the following parameters follow a log-normal distribution to ensure they are positive. We also exploit the property that the scale of a normal distribution in natural log-space is approximately the scale in real-space as a fraction of the distribution mean. -\paragraph{Acoustic Depths} We derived a prior on \(\tau_\helium\) and \(\tau_\bcz\) by observing how they scale with acoustic radius \(\tau_0\) in the grid of stellar models from \citet{Lyttle.Davies.ea2021} and in \citet{Verma.Rorsted.ea2022}. Typically, the fractional depth of He\,\textsc{ii} ionisation \(\tau_\heII/\tau_0 \approx 0.2\) and BCZ \(\tau_\bcz/\tau_0 \approx 0.6\) for main sequence solar-like oscillators (see, e.g. Figure \ref{fig:gamma-sound-speed}). We formed the prior distributions from these relations, estimating the acoustic radius from \({\tau}_0 = (2\nu_0)^{-1}\). To account for possible variance with stellar properties, we added a spread of 20 per cent. We defined their priors in natural log-space to ensure they remained positive, +\paragraph{Acoustic Depths} We derived a prior on \(\tau_\helium\) and \(\tau_\bcz\) by observing how they scale with acoustic radius (\(\tau_0\)) in the grid of stellar models from \citet{Lyttle.Davies.ea2021} and in \citet{Verma.Rorsted.ea2022}. Typically, the fractional depth of He\,\textsc{ii} ionisation and the BCZ are \(\tau_\heII/\tau_0 \approx 0.2\) and \(\tau_\bcz/\tau_0 \approx 0.6\), for main sequence solar-like oscillators (see, e.g. Figure \ref{fig:gamma-sound-speed}). We formed the prior distributions from these relations, estimating the acoustic radius from \({\tau}_0 = (2\nu_0)^{-1}\). To account for possible variance with stellar properties, we added a spread of 20 per cent. We defined their priors in natural log-space to ensure they remained positive, % -\begin{gather*} +\begin{gather} \ln\tau_\helium \sim \mathcal{N}\left[ \ln(\overline{\tau}_\helium), \, s_{\ln\tau}^2 \right], \quad \ln\tau_\bcz \sim \mathcal{N}\left[\ln(3\,\overline{\tau}_\helium), \, s_{\ln\tau}^2 \right],\\ - \overline{\tau}_\helium = (10\,\overline{\nu}_0)^{-1}, \quad s_{\ln\tau}^2 = (1/5)^2 + (s_{\nu_0}/\overline{\nu}_0)^2, -\end{gather*} + \overline{\tau}_\helium = (10\,\overline{\nu}_0)^{-1}, \quad s_{\ln\tau}^2 = (1/5)^2 + (s_{\nu_0}/\overline{\nu}_0)^2. +\end{gather} -\paragraph{Helium Glitch Amplitude} The prior on the glitch amplitude parameters was less trivial. By observing \(\gamma\) profiles in the grid of stellar models, we assumed that the width of the helium ionisation region was about 8 per cent of the acoustic depth of the region, \(\Delta_\heII/\tau_\heII \approx 0.08\). Thus, we centred the prior on \(\overline{\beta}_\helium\) obtained from this assumption and used the relation \(\beta_\helium = 8\pi^2\Delta_\heII^2\) from comparison to Equation \ref{eq:he-osc}. We then propagated the variance from the prior on \(\tau_\helium\), +\paragraph{Helium Glitch Amplitude} Deriving a prior on the glitch amplitude parameters was less trivial. By observing \(\gamma\) profiles in the grid of stellar models, we assumed that the width of the helium ionisation region was about 8 per cent of the acoustic depth of the region, \(\Delta_\heII/\tau_\heII \approx 0.08\). Thus, we centred the prior on \(\overline{\beta}_\helium\) obtained from this assumption and used the relation \(\beta_\helium = 8\pi^2\Delta_\heII^2\) from comparison to Equation \ref{eq:he-osc}. We then propagated the variance from the prior on \(\tau_\helium\), % -\begin{equation*} +\begin{equation} \ln\beta_\helium \sim \mathcal{N}\left[ \ln(\overline{\beta}_\helium), \, 4 s_{\ln\tau}^2 \right], \quad \overline{\beta}_\helium = \frac{32}{625} \, \pi^2 \overline{\tau}_\helium^2. -\end{equation*} +\end{equation} % Then, we centred the prior for \(\alpha_\helium\) to satisfy a depth of 0.1 in \(\delta\gamma/\gamma\) caused by helium ionisation using Equation \ref{eq:he-gamma}, % @@ -192,7 +192,7 @@ \subsection{The GP Method} \ln\alpha_\helium \sim \mathcal{N}\left[ 1/2 \, \ln({\overline{\beta}_\helium}/{400\pi}), \, s_{\ln\tau}^2 \right]. \end{gather*} % -We verified that the priors on \(\alpha_\helium\) and \(\beta_\helium\) were appropriate by checking that the prior amplitude at \(\nu_\mathrm{ref}\) peaks from \SIrange{0}{1}{\micro\hertz}, decaying thereafter. This was compatible with our prior belief for the helium glitch amplitude. +We verified that the priors on \(\alpha_\helium\) and \(\beta_\helium\) were compatible with our prior belief by checking that the amplitude at \(\nu_\mathrm{ref}\) peaked from \SIrange{0}{1}{\micro\hertz}, decaying thereafter. \paragraph{BCZ Glitch Amplitude} We constructed the prior on \(\alpha_\bcz\) such that the BCZ glitch amplitude was approximately \SI{0.1}{\micro\hertz} at \(\nu_\mathrm{ref}\) (an order of magnitude smaller than the helium glitch). This occurred when \(\alpha_\bcz/\nu_0 \approx \SI{30}{\micro\hertz}\). We also scaled the distribution by an additional 80 per cent, % @@ -210,9 +210,9 @@ \subsection{The GP Method} % \end{table} % The mean and variance for the aforementioned prior distributions are summarised in Table \ref{tab:glitch-prior} for the test star and 16 Cyg A. -Prior distributions for the remaining parameters were the same for both stars. For example, the prior on the phase parameters \(\phi_\helium\) and \(\phi_\bcz\) was uniformly distributed from 0 to \(2\pi\). We also used a weakly informative prior on the model uncertainty, \(\ln\sigma \sim \mathcal{N}( - \ln 100, 4)\), centred on an uncertainty of \SI{0.01}{\micro\hertz}. +Prior distributions for the remaining parameters were the same for both stars. For example, the prior on the phase parameters \(\phi_\helium\) and \(\phi_\bcz\) was uniformly distributed from 0 to \(2\pi\). We also used a weakly informative prior on the model uncertainty, \(\ln\sigma \sim \mathcal{N}( - \ln 100, 4)\), centred on \SI{0.01}{\micro\hertz}. -We sampled the posterior distribution using the nested sampling \textsc{Python} package \texttt{dynesty} \citep{Speagle2020,Koposov.Speagle.ea2023}. We applied the multi-ellipsoid bounding method \citep{Feroz.Hobson.ea2009} with 500 live points and the random walk sampling method \citep{Skilling2006} with a minimum of 50 steps before proposing a new live point. In addition, we enabled periodic boundary conditions for \(\phi_\helium\) and \(\phi_\bcz\) with the prior transform projecting to periodic space using \(\phi = \phi'\,\mathrm{mod}\,2\pi\), where \(\phi'\) is unconstrained. Otherwise, the nested sampler ran with its default parameters. We used the \textsc{Python} packages \texttt{jax} \citep{Bradbury.Frostig.ea2018} to make use of accelerated linear algebra (XLA) and just-in-time (JIT) compilation, and \texttt{tinygp} \citep{Foreman-Mackey.Yadav.ea2022} to build the GP. For analysis and comparison with the \citetalias{Verma.Raodeo.ea2019} method, we drew 1000 points randomly from the posterior samples according to their estimated weights. +We sampled the posterior distribution using the \textsc{Python} nested sampling package \texttt{dynesty} \citep{Speagle2020,Koposov.Speagle.ea2023}. We applied the multi-ellipsoid bounding method \citep{Feroz.Hobson.ea2009} with 500 live points and the random walk sampling method \citep{Skilling2006} with a minimum of 50 steps before proposing a new live point. In addition, we enabled periodic boundary conditions for \(\phi_\helium\) and \(\phi_\bcz\) with the prior transform projecting to periodic space using \(\phi = \phi'\,\mathrm{mod}\,2\pi\), where \(\phi'\) is unconstrained. Otherwise, the nested sampler ran with its default parameters. We used the \textsc{Python} package \texttt{jax} \citep{Bradbury.Frostig.ea2018} to make use of accelerated linear algebra (XLA) and just-in-time (JIT) compilation, and \texttt{tinygp} \citep{Foreman-Mackey.Yadav.ea2022} to build the GP. For analysis and comparison with the \citetalias{Verma.Raodeo.ea2019} method, we drew 1000 points randomly from the posterior samples according to their estimated weights. % Rearranged into dimensionless quantities, \(f = \nu/\nu_0\), \(t = \tau/\tau_0\), \(a_\helium = \nu_0\alpha_\helium\), \(b_\helium = \nu_0 \beta_\helium\), and \(a_\bcz = \alpha_\bcz/\nu_0^2\), % % @@ -234,11 +234,11 @@ \subsection{Test Star} \begin{figure}[!tb] \centering \includegraphics{figures/glitch-test-signal.pdf} - \caption[50 random draws from the V19 and GP methods showing the total glitch signal as a function of frequency.]{50 random draws from the V19 and GP methods showing the total glitch signal, \(\delta\nu = \delta_\helium(\nu) + \delta_\bcz(\nu)\), as a function of frequency, \(\nu\). The data points plot in blue are the median smooth component model at \(\vect{n}\) subtracted from the observed modes \(\vect{\nu}_\obs\) with their observed uncertainty \(\sigma_\obs\).} + \caption[50 random draws from the V19 and GP results showing the total glitch signal as a function of frequency.]{50 random draws from the V19 and GP methods showing the total glitch signal, \(\delta\nu = \delta_\helium(\nu) + \delta_\bcz(\nu)\), as a function of frequency, \(\nu\). The data points plot in blue are the median smooth component model at \(\vect{n}\), subtracted from the observed modes (\(\vect{\nu}_\obs\)) with their observed uncertainty (\(\sigma_\obs\)).} \label{fig:glitch-test-signal} \end{figure} -In Figure \ref{fig:glitch-test-signal}, we plot the predicted total glitch component (\(\delta\nu\)) using 50 draws from the posterior distribution for both methods. We subtracted the median smooth component (\(\tilde{f}(n)\)) from the observed \(\nu_n\) over-plot. For each scenario, both methods predicted similar glitch components. However, the \citetalias{Verma.Raodeo.ea2019} method showed more extreme multimodality in all cases. For example, the better-case showed a high-amplitude (\(\sim \SI{2}{\micro\hertz}\)) BCZ glitch solution which is not present with the GP method. Such a large BCZ signature would be unlikely. Additionally, the two methods differed by \(\sim \SI{1}{\micro\hertz}\) at low frequency in the best case. Despite this, the \citetalias{Verma.Raodeo.ea2019} method appeared to be more confident in its glitch solutions compared to the GP method. +In Figure \ref{fig:glitch-test-signal}, we plot the predicted total glitch component (\(\delta\nu\)) using 50 draws from the posterior distribution for both methods. We subtracted the median smooth component (\(\tilde{f}(n)\)) from the observed \(\nu_n\) over-plot. For each scenario, both methods predicted similar glitch components. However, the \citetalias{Verma.Raodeo.ea2019} method showed more extreme multimodality in all cases. For example, the better case showed a high-amplitude (\(\sim \SI{2}{\micro\hertz}\)) BCZ glitch solution which is not present with the GP method. Such a large BCZ signature would be unlikely. Additionally, the two methods differed by \(\sim \SI{1}{\micro\hertz}\) at low frequency in the best case. Despite this, the \citetalias{Verma.Raodeo.ea2019} method appeared to be more confident in its glitch solutions compared to the GP method. \begin{figure}[!tb] \centering @@ -247,7 +247,7 @@ \subsection{Test Star} \label{fig:glitch-test-tau} \end{figure} -In Figure \ref{fig:glitch-test-tau}, we plotted posterior distributions for the glitch acoustic depths, \(\tau_\helium\) and \(\tau_\bcz\), and compared them to the sound speed gradient of the test star from Figure \ref{fig:sound-speed-gradient}. We expect the acoustic depths to approximately line up with the sharp structural changes. For the worst case, both methods gave broad distributions for the acoustic depths, compatible with their respective initial guesses and priors. The \citetalias{Verma.Raodeo.ea2019} method initial guesses appeared to underestimate \(\tau_\bcz\), whereas the GP method prior was broad enough to encompass a wide range of possible \(\tau_\bcz\). In the better and best cases, we found that the \citetalias{Verma.Raodeo.ea2019} solutions were multimodal. For example, the better-case found solutions for \(\tau_\helium\) far deeper into the star than we would expect, at around \SI{1500}{\second} and \SI{2500}{\second}. +In Figure \ref{fig:glitch-test-tau}, we plotted posterior distributions for the glitch acoustic depths, \(\tau_\helium\) and \(\tau_\bcz\), and compared them to the sound speed gradient of the test star from Figure \ref{fig:sound-speed-gradient}. We expect the acoustic depths to approximately line up with the sharp structural changes. For the worst case, both methods gave broad distributions for the acoustic depths, compatible with their respective initial guesses and priors. The \citetalias{Verma.Raodeo.ea2019} method initial guesses appeared to underestimate \(\tau_\bcz\), whereas the GP method prior was broad enough to encompass a wide range of possible \(\tau_\bcz\). In the better and best cases, we found that the \citetalias{Verma.Raodeo.ea2019} solutions were multimodal. For example, the better case found solutions for \(\tau_\helium\) far deeper into the star than we would expect, at around \SI{1500}{\second} and \SI{2500}{\second}. % As predicted by \citet{Houdek.Gough2007} and shown in \citet{Verma.Faria.ea2014}... This is because \(\delta\nu_\helium\) does not include the smaller glitch component due to the first ionisation of helium, located at a smaller \(\tau\). The values for \(\tau_\helium\) obtained were under-predicted compared to the location of the trough due to the second ionisation of helium. We can see this for the best star fit with the \citetalias{Verma.Raodeo.ea2019} method which finds \(\tau_\helium = \SI{619(15)}{\second}\). The depression in \(\gamma\) due to He\,\textsc{ii} ionisation in the respective stellar model is located at \SI{733}{\second}. On the other hand, the GP method was closer with \(\tau_\helium = \SI{696(19)}{\second}\). @@ -255,21 +255,21 @@ \subsection{Test Star} \begin{figure}[tb] \centering \includegraphics{figures/glitch-test-amplitude.pdf} - \caption[Samples of the helium glitch amplitude at a reference frequency of \SI{3000}{\micro\hertz} fit with the V19 and GP methods for the worst-, better-, and best-case test data.]{Samples of the helium glitch amplitude at a reference frequency of \SI{3000}{\micro\hertz} fit with the V19 and GP methods for the worst-, better-, and best-case test data. The tallest bar in the V19 method panel is cropped with its value displayed in green text.} + \caption[Samples of the helium glitch amplitude at a reference frequency of \SI{3000}{\micro\hertz} fit with the V19 and GP methods for the worst, better, and best case test data.]{Samples of the helium glitch amplitude at a reference frequency of \SI{3000}{\micro\hertz} fit with the V19 and GP methods for the worst, better, and best case test data. The tallest bar in the V19 method panel is cropped with its value displayed in green text.} \label{fig:glitch-test-amplitude} \end{figure} -We compared helium glitch amplitudes at \(\nu_\mathrm{ref}\) from both methods in Figure \ref{fig:glitch-test-amplitude}. The \citetalias{Verma.Raodeo.ea2019} method preferred low-amplitude solutions for the worst and better cases than the best case compared to the GP method. On the other hand, the GP method reflects our prior in the worst case, with the width of the distribution shrinking as the data improves. The GP method does show some bi-modality in the better case, with higher solutions at \(A_\helium^\mathrm{ref} \approx 0.7\) which were not found by the \citetalias{Verma.Raodeo.ea2019} method. In the best case, the \citetalias{Verma.Raodeo.ea2019} method obtained \(A_\helium^\mathrm{ref} = 0.347_{-0.005}^{+0.006}\), whereas the GP method found \(A_\helium^\mathrm{ref} = 0.296_{-0.036}^{+0.042}\). +We compared helium glitch amplitudes at \(\nu_\mathrm{ref}\) from both methods in Figure \ref{fig:glitch-test-amplitude}. The \citetalias{Verma.Raodeo.ea2019} method preferred low-amplitude solutions for the worst and better cases compared to the GP method. On the other hand, the GP method reflected our prior in the worst case, with the width of the distribution shrinking as the data improved. The GP method did show some bi-modality in the better case, with higher solutions at \(A_\helium^\mathrm{ref} \approx 0.7\) which were not found by the \citetalias{Verma.Raodeo.ea2019} method. In the best case, the \citetalias{Verma.Raodeo.ea2019} method obtained \(A_\helium^\mathrm{ref} = 0.347_{-0.005}^{+0.006}\), whereas the GP method found \(A_\helium^\mathrm{ref} = 0.296_{-0.036}^{+0.042}\). % Why not use delta gamma / gamma as the probe of helium abundance? That is proportional to alpha / sqrt(beta). Then, an update to this method can use this and beta as a parameter and then work out alpha from that, since beta should scale with tau and the depth is a signature of helium abundance. Be careful as number of modes correlates with delta nu value of fit. \subsection{16 Cyg A} -We compared the results from both methods applied to the 16 Cyg A data in Figure \ref{fig:glitch-16cyga}. We found that the \citetalias{Verma.Raodeo.ea2019} method predicted extreme solutions for the glitch where the GP method did not. There was also a difference of about \SI{1}{\micro\hertz} at the low frequency end such that the \citetalias{Verma.Raodeo.ea2019} predicted a larger glitch amplitude. Otherwise, the two methods gave similar predictions for the glitch function. +We compared the results from both methods applied to the 16 Cyg A data in Figure \ref{fig:glitch-16cyga}. We found that the \citetalias{Verma.Raodeo.ea2019} method predicted extreme solutions for the glitch where the GP method did not. There was also a difference of about \SI{1}{\micro\hertz} at the low frequency end such that the \citetalias{Verma.Raodeo.ea2019} predicted a larger glitch amplitude. Otherwise, the two methods gave similar solutions for the glitch function. -Both methods found similar values for \(\tau_\helium\) but relatively different distributions for \(\tau_\bcz\). Regarding the helium glitch, the \citetalias{Verma.Raodeo.ea2019} method found \(\tau_\helium = 917_{-53}^{+50} \, \mathrm{s}\), and our GP method obtained \(\tau_\helium = 931_{-88}^{+59} \, \mathrm{s}\), within 1-\(\sigma\) of each other. Similarly to with the test star, the GP method found a slightly larger value of \(\tau_\helium\) than the \citetalias{Verma.Raodeo.ea2019} method, although not significantly in this case. \citet{Verma.Faria.ea2014} fit the glitch with \(l=0,1,2\) modes and found an acoustic depth of \SI{930(14)}{\second}, within 1-\(\sigma\) of both methods in this work. +Both methods found similar values for \(\tau_\helium\) but relatively different distributions for \(\tau_\bcz\). Regarding the helium glitch, the \citetalias{Verma.Raodeo.ea2019} method found \(\tau_\helium = 917_{-53}^{+50} \, \mathrm{s}\), and our GP method obtained \(\tau_\helium = 931_{-88}^{+59} \, \mathrm{s}\), within 1-\(\sigma\) of each other. Similarly to the test star, the GP method found a slightly larger value of \(\tau_\helium\) than the \citetalias{Verma.Raodeo.ea2019} method. \citet{Verma.Faria.ea2014} fit the glitch using \(l=0,1,2\) modes and found an acoustic depth of \SI{930(14)}{\second}, within 1-\(\sigma\) of both methods in this work. -We calculated the helium glitch amplitude at a reference frequency of \SI{2188.5}{\micro\hertz}, equivalent to the value of \(\nu_{\max}\) obtained by \citet{Lund.SilvaAguirre.ea2017}. Samples from both posteriors are shown in the bottom panel of Figure \ref{fig:glitch-16cyga}. We found \(A_\helium^\mathrm{ref} = 0.260_{-0.065}^{+0.050}\,\si{\micro\hertz}\) for the \citetalias{Verma.Raodeo.ea2019} method, and \(A_\helium^\mathrm{ref} = 0.333_{-0.073}^{+0.081}\,\si{\micro\hertz}\) for the GP method. Both values were about 1-\(\sigma\) apart. Additionally, both methods found \(A_\bcz^\mathrm{ref} \sim 0.1\,\si{\micro\hertz}\), with the GP method favouring a smaller value. The \citetalias{Verma.Raodeo.ea2019} method found some solutions with \(A_\bcz^\mathrm{ref}\) larger than \(A_\helium^\mathrm{ref}\), something we would not expect because the modes are less sensitive to structural changes deeper in the star. +We calculated the helium glitch amplitude at a reference frequency of \SI{2188.5}{\micro\hertz}, equivalent to \(\nu_{\max}\) obtained by \citet{Lund.SilvaAguirre.ea2017}. Samples from both posteriors are shown in the bottom panel of Figure \ref{fig:glitch-16cyga}. We found \(A_\helium^\mathrm{ref} = 0.260_{-0.065}^{+0.050}\,\si{\micro\hertz}\) for the \citetalias{Verma.Raodeo.ea2019} method, and \(A_\helium^\mathrm{ref} = 0.333_{-0.073}^{+0.081}\,\si{\micro\hertz}\) for the GP method. Both values were about 1-\(\sigma\) apart. Additionally, both methods found \(A_\bcz^\mathrm{ref} \sim 0.1\,\si{\micro\hertz}\), with the GP method favouring a smaller value. The \citetalias{Verma.Raodeo.ea2019} method found some solutions where \(A_\bcz^\mathrm{ref}\) was larger than \(A_\helium^\mathrm{ref}\), something we would not expect because the modes are less sensitive to structural changes deeper in the star. \begin{figure}[!tb] \centering @@ -289,7 +289,7 @@ \section{Discussion}\label{sec:glitch-disc} \label{fig:best-smooth} \end{figure} -Throughout this work, we found a smaller \(\delta\nu\) amplitude at low frequency with the GP method than with the \citetalias{Verma.Raodeo.ea2019} method. This was particularly visible in the best case and in 16 Cyg A. We expected this was a result of the different smooth background models. In Figure \ref{fig:best-smooth} we plotted the smooth component of each model extended to lower order, unobserved modes. We found the GP background component had a turning point at \(\nu \approx \SI{1900}{\micro\hertz}\) which was higher than the \citetalias{Verma.Raodeo.ea2019} method at \(\nu \approx \SI{1500}{\micro\hertz}\). The smooth component of the \citetalias{Verma.Raodeo.ea2019} method was confidently incorrect outside the observed frequencies. Conversely, the GP method predicted closer to the truth with increasing uncertainty further from the observations. It appeared that the GP provided a more accurate representation of the underlying function than the polynomial. +Throughout this work, we found a smaller \(\delta\nu\) amplitude at low frequency with the GP method than with the \citetalias{Verma.Raodeo.ea2019} method. This was particularly visible in the best case and in 16 Cyg A. We expected this was a result of the different smooth background models. In Figure \ref{fig:best-smooth} we plotted the smooth component of each model extended to lower order, unobserved modes. We found the GP background component had a turning point at \(\nu \approx \SI{1900}{\micro\hertz}\), higher than the \citetalias{Verma.Raodeo.ea2019} method at \(\nu \approx \SI{1500}{\micro\hertz}\). The smooth component of the \citetalias{Verma.Raodeo.ea2019} method was confidently incorrect outside the observed frequencies. Conversely, the GP method predicted closer to the truth with increasing uncertainty further from the observations. It appeared that the GP provided a more accurate representation of the underlying function than the polynomial. \begin{figure}[!tb] \centering @@ -298,13 +298,13 @@ \section{Discussion}\label{sec:glitch-disc} \label{fig:smooth-res} \end{figure} -In the test star's best case and 16 Cyg A, the GP method found a higher \(\tau_\helium\) than the \citetalias{Verma.Raodeo.ea2019} method. This could be because the GP was better able to distinguish between the He\,\textsc{ii} glitch and the smaller He\,\textsc{i} glitch. To explore this, we plotted the difference between each model's predictions for the smooth component in Figure \ref{fig:smooth-res}. We saw a clear periodic signal in the differences. The period of this signal in the test star corresponded to an acoustic depth of \(\sim \SI{500}{\second}\), matching the location of He\,\textsc{i} ionisation in model S. The periodic signal was less pronounced for 16 Cyg A, but corresponded to a plausible He\,\textsc{i} acoustic depth of \(\sim \SI{700}{\second}\). We expect that the polynomial was not flexible enough to pick up the He\,\textsc{i} ionisation signature, hence lowering its mean value for \(\tau_\helium\). +In the test star's best case and 16 Cyg A, the GP method found a higher \(\tau_\helium\) than the \citetalias{Verma.Raodeo.ea2019} method. This could be because the GP was better able to distinguish between the He\,\textsc{ii} glitch and the smaller He\,\textsc{i} glitch. To explore this, we plotted the difference between each model's predictions for the smooth component in Figure \ref{fig:smooth-res}. We saw a clear periodic signal in the differences. The period of this signal in the test star corresponded to an acoustic depth of \(\sim \SI{500}{\second}\), matching the location of He\,\textsc{i} ionisation in model S. The periodic signal was less pronounced for 16 Cyg A, but corresponded to a plausible He\,\textsc{i} acoustic depth of \(\sim \SI{700}{\second}\). We expected that the polynomial was not flexible enough to pick up the He\,\textsc{i} ionisation signature, hence lowering its mean value for \(\tau_\helium\). % Discuss the prior -The \citetalias{Verma.Raodeo.ea2019} method found several extreme solutions for \(\delta\nu\) whereas GP method did not. We expected this because the GP method used a prior over the model parameters. We tested relaxing the prior on \(\vect{\theta}_B\) and found similar multimodal posteriors to the \citetalias{Verma.Raodeo.ea2019} method. This showed that the prior helped eliminate unrealistic solutions. However, care should be taken over the choice of prior on \(\vect{\theta}_B\) to realistically reflect our expectation. If some of our prior assumptions are incorrect they may bias the results. For example, the outer convective region gets shallower as stars get hotter (approaching \(\teff \approx \SI{7000}{\kelvin}\)) making the assumption that \(\tau_\bcz/\tau_0 \approx 0.6\) an overestimate in these cases. We accommodate for this with a wide prior on \(\tau_\bcz\), but our prior could be more informed. There is a notable temperature dependence to \(\tau_\bcz/\tau_0\) observed in the grid of stellar models which could be exploited in the future when constructing the prior. +The \citetalias{Verma.Raodeo.ea2019} method found several extreme solutions for \(\delta\nu\) whereas GP method did not. We expected this because the GP method used a prior over the model parameters. We tested relaxing the prior on \(\vect{\theta}_B\) and found similar multimodal posteriors to the \citetalias{Verma.Raodeo.ea2019} method. This showed that the prior helped eliminate unrealistic solutions. However, care should be taken over the choice of prior on \(\vect{\theta}_B\) to realistically reflect our expectation. If some of our prior assumptions are incorrect, they may bias the results. For example, the outer convective region gets shallower as stars get hotter (approaching \(\teff \approx \SI{7000}{\kelvin}\)) making the assumption that \(\tau_\bcz/\tau_0 \approx 0.6\) an overestimate in these cases. We accommodate for this with a wide prior on \(\tau_\bcz\), but our prior could be more informed. There is a notable temperature dependence to \(\tau_\bcz/\tau_0\) observed in the grid of stellar models which could be exploited in the future when constructing the prior. -Additionally, the joint posteriors for \(\alpha_\helium\) and \(\beta_\helium\) were correlated in both methods. This was expected, because larger values of \(\beta_\helium\) can be compensated for with a larger amplitude factor \(\alpha_\helium\). In the GP method, we did not include this expected correlation in our prior. Hence, we found that having broader priors on \(\alpha_\helium\) and \(\beta_\helium\) lead to the prior predicting unrealistic glitches. We could improve upon this by using multivariate prior for the amplitude parameters to account for this. Despite this, our approach still improves on the \citetalias{Verma.Raodeo.ea2019} method by using a prior in the first place. +Additionally, the joint posteriors for \(\alpha_\helium\) and \(\beta_\helium\) were correlated in both methods. This was expected, because larger values of \(\beta_\helium\) can be compensated for with a larger amplitude factor (\(\alpha_\helium\)). In the GP method, we did not include this expected correlation in our prior. Hence, we found that having broader priors on \(\alpha_\helium\) and \(\beta_\helium\) lead to the prior predicting unrealistic glitches. We could account for this by using a multivariate prior for the amplitude parameters. Despite this, our approach still improves on the \citetalias{Verma.Raodeo.ea2019} method by using a prior in the first place. A potential limitation of the GP method is that we calculate the glitch at \(\tilde{f}_B(n)\) from the linear asymptotic equation (Equation \ref{eq:asy}). In regions where the gradient of \(\delta\nu\) is high, the difference between \(\tilde{f}_B(n)\) and the true frequency can be up to \(\sim \SI{0.1}{\micro\hertz}\). Our choice of kernel function cannot absorb this difference because is varies on a short length-scale. Instead, we accounted for this uncertainty by adding Gaussian noise to the model parametrised by \(\sigma\). However, for the best test star, we found \(\sigma \approx 0.05\), which was larger than the observational uncertainty, \(\sigma_\obs = 0.01\). This could limit our method's inference ability for the best asteroseismic targets. One solution is to replace \(\tilde{f}_B(n)\) with a quadratic \citep[e.g.][]{Nielsen.Davies.ea2021} which better approximates \(\nu_n\). However, the GP kernel would need to be adjusted to account for this. @@ -313,8 +313,8 @@ \section{Discussion}\label{sec:glitch-disc} \section{Conclusion} We introduced a new method for modelling acoustic glitches in solar-like oscillators using a Gaussian Process. Testing the method on a model star, we found that it more accurately characterised the underlying, smoothly-varying functional form of the radial modes than the -\citetalias{Verma.Raodeo.ea2019} method. Furthermore, our method appeared able to absorb the glitch component from He\,\textsc{i} ionisation, for which the polynomial was not flexible enough. However, this questions whether He\,\textsc{i} ionisation glitch should be explicitly included in the model. +\citetalias{Verma.Raodeo.ea2019} method. Furthermore, our method appeared able to absorb the glitch component from He\,\textsc{i} ionisation, for which the polynomial was not flexible enough. However, this questions whether the He\,\textsc{i} ionisation glitch should be explicitly included in the model. -Additionally, the GP method provided more believable uncertainties on the glitch parameters, whereas the \citetalias{Verma.Raodeo.ea2019} method was over-confident with the best data and under-confident with the worst. Robust uncertainties are important when using the results to make further inference about helium enrichment. In this case, the GP marginalised over correlated noise in the model, not possible with the polynomial in the \citetalias{Verma.Raodeo.ea2019} method. +Additionally, the GP method provided more believable uncertainties on the glitch parameters, whereas the \citetalias{Verma.Raodeo.ea2019} method was over-confident with the best data and under-confident with the worst. Robust uncertainties are important when using the results to make further inference about helium enrichment. In this case, the GP marginalised over correlated noise in the model, not possible with the \citetalias{Verma.Raodeo.ea2019} method. Future development of the method could involve building a prior for the glitch parameters. For example, we could start with fitting the model to simulated stars and using the results to build an empirical prior. Then, we could run the model on a larger asteroseismic sample of main sequence stars \citep[e.g.][]{Lund.SilvaAguirre.ea2017,Davies.SilvaAguirre.ea2016} and compare our results to those from \citet{Verma.Raodeo.ea2019}. Additionally, we could add parameters from the GP model to the hierarchical model introduced in \citet{Lyttle.Davies.ea2021}. Ultimately, our goal is to scale this method in anticipation of the \(\sim 10^4\) solar-like oscillators expected to be observed by \emph{PLATO} \citep{Rauer.Catala.ea2014}. From a5d112c9e336caf4d283da0a3782d1f5b8027bbf Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Fri, 28 Apr 2023 14:39:31 +0100 Subject: [PATCH 45/50] Finish conclusion --- chapters/conclusion.tex | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/chapters/conclusion.tex b/chapters/conclusion.tex index 22c4da4..422bd0b 100644 --- a/chapters/conclusion.tex +++ b/chapters/conclusion.tex @@ -18,13 +18,13 @@ \section*{Summary} In this thesis, we built a hierarchical Bayesian model (HBM) to improve the inference of stellar parameters with asteroseismology. Introducing the concept of an HBM in Chapter \ref{chap:hbm}, we showed how population-level distributions can be used as a prior over individual stellar parameters. We found that pooling parameters this way reduced their uncertainties by up to a factor of \(\sqrt{N}\) where \(N\) is the number of stars in the population. -In Chapter \ref{chap:hmd}, we built an HBM to estimate the masses, radii,and ages for a well-studied sample of \(\sim 60\) dwarf and subgiant solar-like oscillators. Limited by observational noise, existing modelling techniques typically struggle to account for the uncertainty in initial helium abundance (\(Y\)) and mixing-length theory parameter (\(\mlt\)) for these stars. We showed that applying a hierarchical prior over \(Y\) and \(\mlt\) allowed us to simultaneously marginalise over their uncertainty and model their distribution in the population. Pooling \(Y\) and \(\mlt\) in this way, we were still able to achieve statistical uncertainties of 1.2 per cent in radius, 2.5 per cent in mass, and 12 per cent in age. Notably, our HBM halved the uncertainty in stellar mass compared to the same model without parameter pooling. This provided a scalable and reproducible framework for modelling large populations of stars at the same time. +In Chapter \ref{chap:hmd}, we built an HBM to estimate the masses, radii, and ages for a well-studied sample of \(\sim 60\) dwarf and subgiant solar-like oscillators. Limited by observational noise, existing modelling techniques struggle to account for the uncertainty in initial helium abundance (\(Y\)) and mixing-length theory parameter (\(\mlt\)) for these stars. We showed that applying a hierarchical prior over \(Y\) and \(\mlt\) allowed us to simultaneously marginalise over their uncertainty and model their distribution in the population. Pooling \(Y\) and \(\mlt\) in this way, we were still able to achieve statistical uncertainties of 1.2 per cent in radius, 2.5 per cent in mass, and 12 per cent in age. Notably, our HBM halved the uncertainty in stellar mass compared to the same model without parameter pooling. This provided a scalable and reproducible framework for modelling large populations of stars at the same time. % In our HBM, we assumed a linear helium enrichment law as the mean of a population distribution over \(Y\). We marginalised over the uncertainty in the parameters of this law, improving upon other work which assume a fixed parametrisation of the law calibrated to the Sun \citep[e.g.][]{Serenelli.Johnson.ea2017}. We found the slope of this law (\(\Delta Y/\Delta Z\)) to be \(\approx 1\) and \(\approx 1.6\), with and without including the Sun-as-a-star in our population. Although these values of \(\Delta Y/\Delta Z\) were within 2-\(\sigma\) of each other and agreed with the literature, including the Sun had a clear effect on both \(Y\) and \(\mlt\). This offset may have been a result in our choice of \(\teff\) scale, suggesting an additional systematic we could add to the model. With some improvements to the HBM, we may be able to further break the degeneracy between \(\mlt\) and \(Y\). -The HBM required a function to map model parameters to observables. We built an emulator to approximate 1D numerical models of stellar evolution. Training a neural network on MESA stellar simulations, we were able to predict observable parameters (\(\teff, \dnu, L, [\mathrm{M/H}]\)) with typical precisions of less than \(\sim 0.1\) per cent (see Appendix \ref{apx:hmd}). This provided a simple, continuous and differentiable model well suited to modern, gradient-based MCMC algorithms. Another advantage to using a neural network emulator was its scalability. The simple matrix algebra involved is well suited to fast evaluations on a graphics processing unit (GPU) for large numbers of stars at the same time. Furthermore, the neural network could be scaled up to higher input and output dimensions with little performance impact, making the method transferable to other kinds of stars. For example, we recently trained a neural network to emulate the regularly spaced mode frequencies as a part of a Bayesian stellar model of \(\delta\) Scuti-type oscillators \citep{Scutt.Murphy.ea2023}. +The HBM required a function to map model parameters to observables. We built an emulator to approximate 1D numerical models of stellar evolution. Training a neural network on MESA stellar simulations, we were able to predict observable parameters (\(\teff, \dnu, L, [\mathrm{M/H}]\)) with typical precisions of less than \(\sim 0.1\) per cent (see Appendix \ref{apx:hmd}). This provided a simple, continuous, and differentiable model well suited to modern, gradient-based MCMC algorithms. Another advantage to using a neural network emulator was its scalability. The simple matrix algebra involved was well suited to fast evaluations on a graphics processing unit (GPU) for large numbers of stars at the same time. Furthermore, the neural network could be scaled up to higher input and output dimensions with little performance impact. This made our method transferable to other kinds of stars. For example, we recently trained a neural network to emulate the regularly spaced mode frequencies as a part of a Bayesian stellar model of \(\delta\) Scuti-type oscillators \citep{Scutt.Murphy.ea2023}. -In Chapter \ref{chap:glitch}, we recalled that glitches in stellar structure cause a periodic signal, \(\delta\nu\) in p mode frequencies. One such glitch arises from the second ionisation of helium, with the amplitude of \(\delta\nu_\helium\) correlating with helium abundance. Subsequently, we presented a new method for modelling the glitch signature using a Gaussian Process (GP) in Chapter \ref{chap:glitch-gp}. Past methods for measuring the glitch in the mode frequencies (\(\nu_{nl}\)) used a polynomial in \(n\) to approximate the smooth functional form of the frequencies, over which the periodic glitch signature could be modelled \citep[e.g.][]{Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2019}. We replaced the polynomial with a GP characterised by a kernel describing our prior belief of the function's smoothness and flexibility. We applied this method to model the glitch signature in radial mode frequencies for a fake, Sun-like star and 16 Cyg A. The GP allowed us to marginalise over our uncertainty in the functional form of \(\nu_{n\,0}\) with \(n\). In comparison, we found that the polynomial method was too restrictive and unable to marginalise over uncertainty in the model. +In Chapter \ref{chap:glitch}, we recalled that glitches in stellar structure cause a periodic signal (\(\delta\nu\)) in p mode frequencies. One such glitch arises from the second ionisation of helium, with the amplitude of \(\delta\nu_\helium\) correlating with helium abundance. Subsequently, we presented a new method for modelling the glitch signature using a Gaussian Process (GP) in Chapter \ref{chap:glitch-gp}. Past methods for measuring the glitch in the mode frequencies (\(\nu_{nl}\)) used a polynomial in \(n\) to approximate the smooth component, over which the periodic glitch signature could be modelled \citep[e.g.][]{Mazumdar.Monteiro.ea2014,Verma.Raodeo.ea2019}. We replaced the polynomial with a GP characterised by a kernel describing our prior belief of the function's smoothness and flexibility. We applied this method to model the glitch signature in radial mode frequencies for a simulated Sun-like star and 16 Cyg A. The GP allowed us to marginalise over our uncertainty in the functional form of \(\nu_{n\,l=0}\) with \(n\). In comparison, we found that the polynomial method was too restrictive and unable to marginalise over correlated noise in the model. % In Chapter \ref{chap:glitch}, we recalled that p mode frequencies carry information about acoustic glitches inside a star. However, the exact functional form of the modes with radial order is not known. We showed that a Gaussian Process (GP) could be employed to marginalise over the uncertainty in this functional form and improve detection of the helium glitch signature. Our method showed promise compared to those which have come before \citep[e.g.][]{Verma.Raodeo.ea2019}. We found the GP method was better able to find the true acoustic depth of He\,\textsc{ii} ionisation in our model star than the alternative, motivating a more quantitative comparison in the future. We hope to build a more informed prior on the model parameters and publish this method soon with more examples. @@ -32,7 +32,7 @@ \section*{Improving the Hierarchical Model} The helium glitch parameters for a given star correlate with its near-surface helium abundance. Therefore, a natural next step would be to include helium glitch parameters as an additional observable in our HBM. Our GP glitch model can be applied to both observed and modelled mode frequencies, providing extra parameters to include in our stellar model emulator. Adding these should improve inference of helium abundance for stars with individual modes identified \citep[e.g.][]{Davies.SilvaAguirre.ea2016,Lund.SilvaAguirre.ea2017}. Since our HBM simultaneously models the population distribution of helium, even a few stars with good helium constraint will in-turn improve helium estimates for the rest of the population. This introduces the possibility of testing more complex models of helium enrichment. -We also expect the HBM to scale to red giant solar-like oscillators for which observations are abundant. We trained the emulator on a grid of stellar models from the zero-age main sequence to the base of the red giant branch for masses from \SIrange{0.8}{1.2}{\solarmass}. The upper mass limit was motivated by the diminishing outer convective envelope (responsible for driving solar-like oscillators) in these stars on the main sequence. However, extending the emulator to model red giant solar-like oscillators would require expanding the grid up to \(\sim\SI{2.0}{\solarmass}\). We would need to compute thrice as many evolutionary tracks and evolve existing models further. Stars with \(M \gtrsim \SI{1.1}{\solarmass}\) have a convective core on the main sequence which introduces an additional model uncertainty from mixing at its boundary. Parametrising this process would further multiply the number of input tracks, increasing dimensionality and grid computation time. Therefore, we should research ways of selectively computing stellar models. For example, we could upsample the grid \citep[e.g.][]{Li.Davies.ea2022} where the neural network error is large. +We also expect the HBM to scale to red giant solar-like oscillators for which observations are abundant. We trained the emulator on a grid of stellar models from the zero-age main sequence to the base of the red giant branch for masses from \SIrange{0.8}{1.2}{\solarmass}. The upper mass limit was motivated by the diminishing outer convective envelope (responsible for driving solar-like oscillators) in these stars on the main sequence. However, extending the emulator to model red giant solar-like oscillators would require expanding the grid up to \(\sim\SI{2.0}{\solarmass}\). We would need to compute twice as many more evolutionary tracks and evolve existing models further. Stars with \(M \gtrsim \SI{1.1}{\solarmass}\) have a convective core on the main sequence which introduces an additional model uncertainty from mixing at its boundary. Parametrising this process would further multiply the number of input tracks, increasing dimensionality and grid computation time. Therefore, we should research ways of selectively computing stellar models. For example, we could augment the grid \citep[e.g.][]{Li.Davies.ea2022} by upsampling where the neural network error is large. There are a few additional systematic uncertainties we could also include in the HBM. For example, in Chapter \ref{chap:hmd} we did not consider the inaccuracies of near-surface physics which effect modelled mode frequencies. So-called `surface correction' methods exist \citep[e.g.][]{Ball.Gizon2014,Kjeldsen.Bedding.ea2008} but vary across the HR diagram when compared with 3D hydrodynamical simulations \citep{Sonoi.Samadi.ea2015}. \citet{Compton.Bedding.ea2018} found a range of surface corrections can shift modelled frequencies at \(\numax\) by up to \(\sim 0.5\) per cent. This would amount to a systematic effect on \(\dnu\) which we would expect to correlate with other stellar parameters. Therefore, a future iteration of the HBM should account for the surface term systematic. @@ -56,13 +56,13 @@ \section*{Current and Future Data} % \section{Current and Future Data}\label{sec:conc-future} -We tested the HBM on stars observed by \emph{Kepler}, but there are a few current and upcoming missions which we can utilise to increase our sample size. With larger sample sizes, we can further increase the precision of pooled parameters and better characterise their spread in the population distribution. Recently, \citet{Hatt.Nielsen.ea2023} identified a sample of \(\sim 4000\) solar-like oscillators in 120- and 20-second cadence \emph{TESS} data. Of these, around 50 are dwarf and subgiant stars which we could include in a future iteration of the HBM. However, we anticipate much bigger improvement with future missions expected to launch in a few years time. +We tested the HBM on stars observed by \emph{Kepler}, but there are a few current and upcoming missions which we can utilise to increase our sample size. With larger sample sizes, we can further increase the precision of pooled parameters and better characterise their spread in the population distribution. Recently, \citet{Hatt.Nielsen.ea2023} identified a sample of \(\sim 4000\) solar-like oscillators in 120- and 20-second cadence \emph{TESS} data. Of these, around 50 are dwarf and subgiant stars which we could include in a future iteration of the HBM. However, we anticipate greater improvements with future missions expected to launch in a few years time. % This has been exceptionally useful with galactic archaeology with the \(\sim 150,000\) oscillating red giants detected by \citet{Hon.Huber.ea2021}. -Towards the end of the 2020s, the \emph{PLATO} mission will observe tens of thousands of dwarf and subgiant solar-like oscillators \citep{Rauer.Catala.ea2014}. \emph{PLATO} aims to discover hundreds of exoplanets orbiting solar-type stars across a wider proportion of the sky than observed by \emph{Kepler}. Among its targets are around \num{20000} bright (V < 11) oscillating F-K dwarf stars to be observed over a baseline of around 2 years \citep{Goupil2017}. Using our HBM method on a sample this size could see a reduction in uncertainty (\(\sigma\)) on helium abundance from 0.01 to 0.0005. While this is the maximum expected uncertainty reduction (as discussed in Chapter \ref{chap:hbm}), it shows that we can start to consider more complex population distributions in helium and other stellar parameters. +Towards the end of the 2020s, the \emph{PLATO} mission will observe tens of thousands of dwarf and subgiant solar-like oscillators \citep{Rauer.Catala.ea2014}. \emph{PLATO} aims to discover hundreds of exoplanets orbiting solar-type stars across a wider proportion of the sky than observed by \emph{Kepler}. Among its targets are around \num{20000} bright (\(V\) < 11) oscillating F-K dwarf stars to be observed over a baseline of around 2 years \citep{Goupil2017}. Using our HBM method on a sample this size could see a reduction in uncertainty (\(\sigma\)) on helium abundance from 0.01 to 0.0005. While this is the maximum expected uncertainty reduction (as discussed in Chapter \ref{chap:hbm}), it shows that we can start to consider more complex population distributions in helium and other stellar parameters. -While \emph{PLATO} will offer unprecedented numbers of main sequence solar-like oscillators, we already have large samples of more evolved asteroseismic stars to include in a future HBM. Combined, \emph{Kepler}, \emph{K2}, and \emph{TESS} have yielded \(\sim 150,000\) red giant solar-like oscillators to date \citep{Hon.Huber.ea2021,Yu.Huber.ea2018}. Providing that we can extend our stellar model emulator to more these stars, expanding our dataset will allow us to test more complex population-distributions. For example, since \emph{TESS} is an all-sky survey, we could include kinematics and galactic positions in the helium enrichment law. Additionally, observations of open clusters and binary star systems introduce more population distributions over age, distance and chemical abundances. +While \emph{PLATO} will offer unprecedented numbers of main sequence solar-like oscillators, we already have large samples of more evolved asteroseismic stars to include in a future HBM. Combined, \emph{Kepler}, \emph{K2}, and \emph{TESS} have yielded \(\sim 150,000\) red giant solar-like oscillators to date \citep{Hon.Huber.ea2021,Yu.Huber.ea2018}. Providing that we can extend our stellar model emulator to more these stars, expanding our dataset will allow us to test more complex population-distributions. For example, since \emph{TESS} is an all-sky survey, we could include kinematics and galactic positions in the helium enrichment law. Additionally, observations of open clusters and binary star systems motivate population distributions over age, distance and chemical abundances. % The number of dwarf and subgiant solar-like oscillators expected from \emph{PLATO} will be comparable to the number of red giant oscillators already found with \emph{Kepler} and \emph{TESS} \needcite. In the meantime we could test extending our method to red giant stars to make use of the abundance of data. This comes with additional challenges. Oscillating red giants include masses \(\gtrsim \SI{1.2}{\solarmass}\) which would have had a convective core during their hydrogen-burning phase of evolution. In this case, we would have to consider overshooting at the convective core boundary. This is an approximation of the physics to simulate mixing at the boundary bringing fresh hydrogen fuel into the core and extending the main sequence lifetime. From 7f53532126d726e420204165337bb599d08d1a58 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Fri, 28 Apr 2023 14:39:40 +0100 Subject: [PATCH 46/50] Finish software --- chapters/software.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/chapters/software.tex b/chapters/software.tex index 3e8f773..fcd3fee 100644 --- a/chapters/software.tex +++ b/chapters/software.tex @@ -1,7 +1,7 @@ \chapter*{Data Availability \& Software} \addcontentsline{toc}{chapter}{Data Availability \& Software} -The data and code underlying this thesis are available in the online supplementary material of \citet{Lyttle.Davies.ea2021}, and in the Zenodo database at \url{https://dx.doi.org/10.5281/zenodo.4746353}. The code used to produce the remainder of this work will be made public at \url{https://github.com/alexlyttle/thesis} and in the Zenodo database upon publication. This thesis includes data collected by the \emph{Kepler} mission. Funding for the \emph{Kepler} mission is provided by the NASA Science Mission directorate This work has also used data from the European Space Agency (ESA) mission +The data and code underlying this thesis are available in the online supplementary material of \citet{Lyttle.Davies.ea2021}, and in the Zenodo database at \url{https://dx.doi.org/10.5281/zenodo.4746353}. The code used to produce the remainder of this work will be made public at \url{https://github.com/alexlyttle/thesis}, and in the Zenodo database upon publication. This thesis includes data collected by the \emph{Kepler} mission. Funding for the \emph{Kepler} mission is provided by the NASA Science Mission directorate This work has also used data from the European Space Agency (ESA) mission \emph{Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC From a76ae8e50c93c61f30310095aa12ebf8e3d9cb51 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Fri, 28 Apr 2023 14:39:57 +0100 Subject: [PATCH 47/50] Update introduction title --- chapters/introduction.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/chapters/introduction.tex b/chapters/introduction.tex index be52313..17eba6d 100644 --- a/chapters/introduction.tex +++ b/chapters/introduction.tex @@ -10,7 +10,7 @@ % all distributions of LaTeX version 2005/12/01 or later. % % -\chapter[Introduction]{Introduction} +\chapter[Inferring Stellar Parameters with Asteroseismology]{Inferring Stellar Parameters with Asteroseismology} \textit{In this chapter, I introduce the current state of modelling stars with asteroseismology and the types of stars being studied in this thesis. I start with a brief history of understanding the stars spanning the last century. In Section \ref{sec:seismo}, I introduce asteroseismology of stars which oscillate like the Sun. Then, I provide examples of asteroseismology being used to model large samples of dwarf and subgiant stars in Section \ref{sec:many-stars}. Finally, I introduce the concept of modelling stars the `Bayesian way' with some examples of current methods and their limitations.} From d0ba22ff49dce586d65e022a9382931e305137ea Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Fri, 28 Apr 2023 14:40:25 +0100 Subject: [PATCH 48/50] Update appendix intro --- appendices/lyttle21.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/appendices/lyttle21.tex b/appendices/lyttle21.tex index d43114f..13e8511 100644 --- a/appendices/lyttle21.tex +++ b/appendices/lyttle21.tex @@ -13,7 +13,7 @@ \chapter[Hierarchically Modelling Dwarf and Subgiant Stars]{Hierarchically Modelling \emph{Kepler} Dwarfs and Subgiants to Improve Inference of Stellar Properties with Asteroseismology}\label{apx:hmd} \textit{% - This chapter is taken with minor modification from the appendix of \citet{Lyttle.Davies.ea2021} and is my own original work. It follows on from Chapter \ref{chap:hmd}. In this chapter, we explain the methodology behind the neural network emulator in Section \ref{sec:ann}. We describe the training process and present results for the accuracy of the emulator. In Section \ref{sec:beta}, we briefly show the beta distribution which was used as a prior for some parameters in the main body of work. Finally, we test the hierarchical model on a synthetic population of stars in Section \ref{sec:test-stars}. + In this chapter, I present the appendix of \citet{Lyttle.Davies.ea2021} with minor modification. It follows from and is referenced to in Chapter \ref{chap:hmd}. Firstly, I explain the methodology behind the neural network emulator in Section \ref{sec:ann}. In Section \ref{sec:beta}, I briefly show the beta distribution which was used as a prior for some parameters in the main body of work. Finally, I test the hierarchical model on a synthetic population of stars in Section \ref{sec:test-stars}. } \section{Artificial Neural Network}\label{sec:ann} From b3de52aa143fa25ecb2a658982e340c86cee692b Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Fri, 28 Apr 2023 14:51:53 +0100 Subject: [PATCH 49/50] Final abstract and acknowledgements --- thesis.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/thesis.tex b/thesis.tex index 8d5cf43..83faa8e 100644 --- a/thesis.tex +++ b/thesis.tex @@ -93,9 +93,9 @@ %% ABSTRACT ------------------------------------------------------------------- % \abstract{% - Modelling stars to understand stellar physics and characterise their ages and masses is crucial for studying exoplanetary systems and the evolution of the Milky Way. As stellar modelling advances with the advent of high-precision asteroseismology, it is becoming increasingly important to account for the systematic effects that arise from our physical assumptions. In this thesis, I present a novel approach for improving the inference of fundamental stellar parameters using a hierarchical Bayesian model. I introduce a statistical treatment which `pools' helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)) to incorporate information about their distributions in the population. Specifically, I model \(Y\) as a distribution with a mean from a linear helium enrichment law parametrised by \(\Delta Y/\Delta Z\). I test our method on a sample of dwarfs and subgiants observed by \emph{Kepler} with a narrow mass range of \(0.8 < M/\mathrm{M}_{\odot} < 1.2\). Exploring various levels of pooling parameters, with and without the Sun as a calibrator, I report \(\Delta Y/\Delta Z = 1.05^{+0.28}_{-0.25}\) when the Sun is included in the sample. Despite marginalising over uncertainties in \(Y\) and \(\mlt\), I am able to report statistical uncertainties of 2.5 per cent in mass, 1.2 per cent in radius, and 12 per cent in age. Moreover, my approach can be extended to larger samples, enabling further uncertainty reduction in fundamental parameters and an improved characterisation of population-level distributions. + Modelling stars to understand stellar physics and characterise their ages and masses is crucial for studying exoplanetary systems and the evolution of the Milky Way. As stellar modelling advances with the advent of high-precision asteroseismology, it is becoming increasingly important to account for the systematic effects that arise from our physical assumptions. In this thesis, I present a novel approach for improving the inference of fundamental stellar parameters using a hierarchical Bayesian model. I introduce a statistical treatment which `pools' helium abundance (\(Y\)) and the mixing-length theory parameter (\(\mlt\)) to incorporate information about their distributions in the population. Specifically, I model \(Y\) as a distribution centred on a linear enrichment law parametrised by \(\Delta Y/\Delta Z\). I test our method on a sample of dwarfs and subgiants observed by \emph{Kepler} with a narrow mass range of \(0.8 < M/\mathrm{M}_{\odot} < 1.2\). Exploring various levels of pooling parameters, with and without the Sun as a calibrator, I report \(\Delta Y/\Delta Z = 1.05^{+0.28}_{-0.25}\) when the Sun is included in the sample. Despite marginalising over uncertainties in \(Y\) and \(\mlt\), I am able to report statistical uncertainties of 2.5 per cent in mass, 1.2 per cent in radius, and 12 per cent in age. Moreover, my approach can be extended to larger samples. This will enable further uncertainty reduction in fundamental parameters and data-driven insight into population-level distributions. - There is additional information on \(Y\) to be gained from detailed asteroseismology. Acoustic glitches, which arise from rapid changes in stellar structure (e.g. from helium ionisation), leave a periodic signature in the mode frequencies (\(\nu_{nl}\)) in solar-like oscillators. I present a new method for modelling acoustic glitch signatures in the radial mode frequencies using a Gaussian Process (GP). The GP provides a statistical treatment of uncertainty in the smooth component of our model for \(\nu_{nl}\) as a function of radial order, \(n\). Using a model star and 16 Cyg A, I compare this approach to another method which models the smooth component with a 4th-order polynomial. My results show that the GP method accurately determines the strength and location of acoustic glitches caused by He\,\textsc{ii} ionisation and the base of the convective zone. I find that using a prior to inform the glitch parameters in my method reduces the occurrence of extreme, unrealistic solutions in the posterior. Furthermore, I demonstrate that the GP approach outperforms the polynomial by absorbing the lesser signature of He\,\textsc{i} ionisation in the modes. However, the inclusion of the He\,\textsc{i} ionisation glitch in the model remains a question. Overall, my results suggest that the GP method should be further tested on more solar-like oscillators and then integrated into the hierarchical model presented in this work.} + There is additional information on \(Y\) to be gained from detailed asteroseismology. Acoustic glitches, which arise from rapid changes in stellar structure (e.g. from helium ionisation), leave a periodic signature in the mode frequencies (\(\nu_{nl}\)) in solar-like oscillators. I present a new method for modelling glitch signatures in the radial mode frequencies using a Gaussian Process (GP). The GP provides a statistical treatment of uncertainty in the functional form of our model for \(\nu_{nl}\). Using a model star and 16 Cyg A, I compare this approach to another method which models the smooth component of the function with a 4th-order polynomial. My results show that the GP method accurately determines the strength and location of glitches caused by He\,\textsc{ii} ionisation and the base of the convective zone. I find that using a prior to inform the glitch parameters in my method reduces the occurrence of extreme, unrealistic solutions in the posterior. Furthermore, I demonstrate that the GP approach outperforms the polynomial by marginalising over the lesser signature of He\,\textsc{i} ionisation. However, the inclusion of the He\,\textsc{i} ionisation glitch in the model remains a question. Overall, my results suggest that the GP method should be further tested on more solar-like oscillators and then integrated into the hierarchical model presented in this work.} %% ACKNOWLEDGEMENTS ----------------------------------------------------------- @@ -110,10 +110,10 @@ I also appreciate the support of the Head of School Bill Chaplin, and other postdoctoral researchers in our group, notably M. Nielsen, Tanda Li (李坦达), and W. Ball. I extend my thanks to the support staff in the School of Physics and Astronomy, particularly our office secretary Lou for her unwavering support over a challenging few years. I also thank my external collaborators, for example Nick Saunders at the University of Hawaii and Simon Murphy at the University of Southern Queensland, with whom I have worked during my graduate studies. \end{CJK*} -Last but not least, I am eternally grateful to the support of my loving family and friends. Particularly, I thank my fianc\'{e} Hannah, my parents Katy and Paul, and my brother Dom. - % FUNDING I acknowledge the support of the public who have indirectly funded the research in this work via various publicly funded agencies. Specifically, I thank the Science and Technology Facilities Council (STFC) who funded my PhD. I also acknowledge that this work is a part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (CartographY; grant agreement ID 804752). Finally, I am grateful for the financial and academic support from the Alan Turing Enrichment scheme. This scheme provided me the opportunity to learn from and collaborate with the wider world of machine learning in science. + +Last but not least, I am hugely grateful for the support of my loving family and friends. Particularly, I thank my fianc\'{e} Hannah, my parents Katy and Paul, and my brother Dom. } From c9f57aa9b1f6846f5084c291a8867d7d72bb6490 Mon Sep 17 00:00:00 2001 From: alexlyttle Date: Fri, 28 Apr 2023 14:54:34 +0100 Subject: [PATCH 50/50] Change n dash to m dash --- chapters/lyttle21.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/chapters/lyttle21.tex b/chapters/lyttle21.tex index 41dbef3..324cf8a 100644 --- a/chapters/lyttle21.tex +++ b/chapters/lyttle21.tex @@ -46,7 +46,7 @@ \section{Data}\label{sec:data} For this study, we selected the sample of 415 stars from the first APOKASC catalogue of dwarfs and subgiants \citep[][hereafter \serenelli]{Serenelli.Johnson.ea2017}. This sample provides an extensive set of dwarfs and subgiant stars with asteroseismic detections observed by the \emph{Kepler} mission. \citetalias{Serenelli.Johnson.ea2017} used grid-based modelling to determine the ages ($\tau$), masses ($M$), radii ($R$), and surface gravity ($\log g$) of stars in the sample, using global asteroseismic parameters, effective temperature ($\teff$), and metallicity ($\metallicity$) as inputs. -Using five independent pipelines, \citetalias{Serenelli.Johnson.ea2017} determined values for global asteroseismic parameters -- the large frequency separation $\dnu$ and the frequency at maximum power, $\numax$ with median uncertainties of 1.7 per cent and 4 per cent respectively. We chose to adopt the $\dnu$ determined in their work as inputs for our method. They also used $\metallicity$ published in Data Release 13 \citep[DR13;][]{Albareti.AllendePrieto.ea2017} of the APOGEE stellar abundances pipeline \citep[ASPCAP;][]{GarciaPerez.AllendePrieto.ea2016} with uncertainties of \SI{0.1}{\dex}. For their preferred set of results, they adopted $\teff$ from the Sloan Digital Sky Survey (SDSS) \emph{griz}-band photometry \citep{Pinsonneault.An.ea2012} with a median uncertainty of \SI{70}{\kelvin}. +Using five independent pipelines, \citetalias{Serenelli.Johnson.ea2017} determined values for global asteroseismic parameters --- the large frequency separation $\dnu$ and the frequency at maximum power, $\numax$ with median uncertainties of 1.7 per cent and 4 per cent respectively. We chose to adopt the $\dnu$ determined in their work as inputs for our method. They also used $\metallicity$ published in Data Release 13 \citep[DR13;][]{Albareti.AllendePrieto.ea2017} of the APOGEE stellar abundances pipeline \citep[ASPCAP;][]{GarciaPerez.AllendePrieto.ea2016} with uncertainties of \SI{0.1}{\dex}. For their preferred set of results, they adopted $\teff$ from the Sloan Digital Sky Survey (SDSS) \emph{griz}-band photometry \citep{Pinsonneault.An.ea2012} with a median uncertainty of \SI{70}{\kelvin}. We removed more evolved stars from the APOKASC sample by cutting those with $\log g < \SI{3.8}{\dex}$. We then kept stars within 1-$\sigma$ of $-0.5 < \metallicity < \SI{0.5}{\dex}$ to remove metal-poor and -rich stars. Main sequence stars with $M \gtrsim \SI{1.2}{\solarmass}$ are understood to have a convective, hydrogen-burning core, with some dependence on the chemical composition and choice of stellar physics \citep{Appourchaux.Antia.ea2015}. Stellar models with a convective core require the treatment of extra stellar physics such as overshooting, which is beyond the scope of this work. Therefore, we keep only stars with masses within 1-$\sigma$ of \SIrange{0.8}{1.2}{\solarmass} from the preferred set of results of \citetalias{Serenelli.Johnson.ea2017}. @@ -70,7 +70,7 @@ \section{Data}\label{sec:data} \label{fig:data} \end{figure} -The final sample comprised 81 stars for which we had complete data for $\teff$, $\metallicity$, $\dnu$, and $L$ to use as inputs for our stellar modelling method -- see Table \ref{tab:data}. In Fig. \ref{fig:data}, we show the HR diagram for the sample plot in context with a series of stellar evolutionary tracks at solar metallicity. +The final sample comprised 81 stars for which we had complete data for $\teff$, $\metallicity$, $\dnu$, and $L$ to use as inputs for our stellar modelling method --- see Table \ref{tab:data}. In Fig. \ref{fig:data}, we show the HR diagram for the sample plot in context with a series of stellar evolutionary tracks at solar metallicity. \section{Methods}\label{sec:meth} @@ -78,7 +78,7 @@ \section{Methods}\label{sec:meth} %%%%%%%%%%%%%%%%%%%% METHODS %%%%%%%%%%%%%%%%%%%%% -Firstly, we used a stellar evolutionary code to compute a grid of models to predict observable quantities (see Section \ref{sec:grid}). Subsequently, we trained an ANN on the grid of stellar models to map input parameters to output observables (see Appendix \ref{sec:ann} for further details). We then constructed three Bayesian models in Section \ref{sec:hbm} which each sampled the trained ANN to estimate stellar fundamental parameters as described in Section \ref{sec:sampling}. Evaluation of the ANN gradient is required during training. Consequently, estimating the gradient of the model likelihood is fast and simple when the observables are generated by an ANN. Hence, we open up the possibility of using a Hamiltonian Monte Carlo (HMC) algorithm -- for example, using the No-U-Turn Sampler \citep[NUTS;][]{Hoffman.Gelman2014} -- which requires the gradient to sample the model posterior. Once we had tested the accuracy of the model on a sample of synthetic stars, we evaluated each model on the subset of the APOKASC catalogue selected in Section \ref{sec:data}. +Firstly, we used a stellar evolutionary code to compute a grid of models to predict observable quantities (see Section \ref{sec:grid}). Subsequently, we trained an ANN on the grid of stellar models to map input parameters to output observables (see Appendix \ref{sec:ann} for further details). We then constructed three Bayesian models in Section \ref{sec:hbm} which each sampled the trained ANN to estimate stellar fundamental parameters as described in Section \ref{sec:sampling}. Evaluation of the ANN gradient is required during training. Consequently, estimating the gradient of the model likelihood is fast and simple when the observables are generated by an ANN. Hence, we open up the possibility of using a Hamiltonian Monte Carlo (HMC) algorithm --- for example, using the No-U-Turn Sampler \citep[NUTS;][]{Hoffman.Gelman2014} --- which requires the gradient to sample the model posterior. Once we had tested the accuracy of the model on a sample of synthetic stars, we evaluated each model on the subset of the APOKASC catalogue selected in Section \ref{sec:data}. \subsection{Grid of Stellar Models}\label{sec:grid}