Interior Modelling of Massive Stars in Multiple Systems (Springer Theses) 3030663094, 9783030663094

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Springer Theses Recognizing Outstanding Ph.D. Research

Cole Johnston

Interior Modelling of Massive Stars in Multiple Systems

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author's previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH.

More information about this series at http://www.springer.com/series/8790

Cole Johnston

Interior Modelling of Massive Stars in Multiple Systems Doctoral Thesis accepted by Institute of Astronomy, KU Leuven, Leuven, Belgium

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Author Dr. Cole Johnston Physics and Astronomy Institute of Astronomy, KU Leuven Leuven, Belgium

Supervisor Prof. Conny Aerts Institute of Astronomy, KU Leuven Leuven, Belgium

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-66309-4 ISBN 978-3-030-66310-0 (eBook) https://doi.org/10.1007/978-3-030-66310-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Supervisor’s Foreword

The theory of stellar structure lies on the basis of computer models mimicking the evolution of stars. Already for decades, simulated stellar models are being used throughout stellar, galactic, and extragalactic astrophysics. Yet, even in this modern era of computational astrophysics, these stellar evolution models still contain a variety of uncalibrated physical processes. Two important poorly understood phenomena are the transport of chemical elements and of angular momentum. The current Ph.D. thesis aimed at improving our knowledge of how the mixing of chemical elements takes place in the deep interiors of stars and how it connects to their internal rotation. Binary stars have since long been, and still are optimal laboratories to test the theory of stellar structure and evolution from their dynamical orbital motion. With properly assembled and calibrated photometric and spectroscopic data covering their orbit, eclipsing double-lined spectroscopic binaries allow us to deduce stellar masses in a model-independent way, reaching accuracies of  1%. These dynamical binary masses have led to inferences of element mixing taking place near the convective core of stars born with masses in the range from  1:2 to 25  M . The past decade, stars exhibiting numerous non-radial oscillations have joined the crowd of suitable calibrators for stellar evolution theory. This became possible thanks to the method of asteroseismology, requiring high-precision long-duration photometric light curves assembled from space. Gravity modes are a particularly interesting type of non-radial oscillations occurring in rotating stars with a convective core and a radiative envelope. Such modes have periods of the order of a day, which is of similar order as the rotation period of such stars. Gravity-mode asteroseismology offers the potential to infer element mixing in such rotating stars throughout the entire phase when they are fusing hydrogen into helium in their convective core. The development and application of gravity-mode asteroseismology to such stars was achieved only recently, in a data-driven way, by relying on their light curves with a duration of four years assembled with the Kepler space telescope.

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Supervisor’s Foreword

With his Ph.D. thesis, Cole Johnston brought new insights into the inner workings of stars born with a convective core, in particular into their internal mixing of chemical elements. He did so by bridging the fields of binarity and asteroseismology, with emphasis on mathematical modelling. Such a bridge is most powerful when built upon non-radial oscillation modes with periods in the range of the orbital period. Gravity modes fulfill this requirement and probe the interiors of such stars, all the way from their surface to their convective core. Cole Johnston developed new computational methods to perform asteroseismic modelling. This led him to introduce the novel concept of “isochrone cloud” fitting, with the aim to assess the level of deep mixing as a most important quantity to achieve proper stellar ages by measuring the mass of the convective core resulting from the internal mixing. The major result highlighted in this Ph.D. thesis is the need for more massive convective cores. This need for higher core masses was found and quantified for both single and binary stars. These higher than anticipated convective core masses have important implications for the chemical yield predictions resulting from stellar evolution models, used in studies of galaxy evolution. The new methods developed in this Ph.D. thesis offer both a fresh look upon stellar interiors and excellent potential for future interpretations of single stars, binaries and star clusters in our Milky Way. Leuven, Belgium December 2020

Prof. Conny Aerts

Preface

Positioning Astrophysics is a broadly interdisciplinary field concerned with a wide range of objects, such as asteroids, exoplanets, stars, galaxies, and interactions therein. Within the overall scope of astrophysics, stars represent a fundamental unit. Stars are the objects around which exoplanets orbit, they are the chemical engines that drive the chemical evolution of our Universe, they are the constituents of galaxies, and they are the progenitors of the most fascinating and energetic phenomena known to humankind. To this end, the theory of stellar structure and evolution is an essential component of astrophysics. This theory contains all of our knowledge on the constitution of stars, and how that constitution changes over time as the star evolves. The manifestation of the theory of stellar structure and evolution comes in the form of stellar evolution models. Numerous fields within astrophysics rely on the predictions made by stellar evolution models to be robust and accurate. However, modern computational resources require simplifications to be made to the processes that drive the time evolution of stellar models, which introduces the potential for models to deviate from reality. As such, calibrating theoretical models and the numerical implementation of the physics within them is greatly important. The comparison of predicted model parameters with observed stellar quantities lies at the heart of this calibration effort. The proper calibration of stellar models demands highly precise stellar parameters, which in and of itself is an immensely difficult task. Although there are numerous means and methods for determining precise stellar parameters, modelling stellar pulsations and stellar multiplicity provide some of the highest precision currently known to astronomers. Despite the solitary nature of our own Sun, most stars exist in pairs, triples, or groups, being formed at the same time and from the same material. Thus, even if we do not know the absolute age of a pair of stars before hand, we know that models must be able to reproduce the observed characteristics of the components of a binary while having the same age, which is an invaluable constraint in modelling. Stellar pulsations propagate through different vii

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layers of a star, probing the temperature, composition, and density of those layers. In fact, it is these very properties of the stellar interior that determine the frequency at which a star pulsates. Thus, much in the way that the size, shape, and composition of a bell determines how a bell rings, the size, shape, and composition of a star determine how it pulsates. By studying how a star pulsates, we can therefore infer the properties of the stellar interior. Individually, these two phenomena provide profound insight into the structure and evolution of stars. When combined, the analysis of pulsating stars in multiple systems promises a means of modelling the stellar interior and assuring that the model is consistent. Through various physical processes, stars transport angular momentum and chemicals throughout their interiors. These processes alter chemical gradients, change rotation rates, and can even directly extend the life of a star. For stars massive stars that have a convective core, any fresh hydrogen transported to the boundary of the core gets assimilated into the core, effectively enhancing the mass of the core and providing extra nuclear fuel for the star. Unfortunately, the efficiency with which chemicals are transported throughout a star is poorly known. However, through the precision that their analysis provides, stellar pulsations and stellar multiplicity enable a robust means for calibrating the effects that internal chemical mixing has on stellar structure and evolution.

Overview This monograph is concerned with efforts to improve the theory of stellar structure and evolution through modelling the observed properties of massive stars in multiple systems. In particular, this monograph focuses on the robust statistical calibration of the theoretical treatment of chemical transport within models of intermediate-to high-mass stars with fully mixed convective cores during their core-hydrogen burning phase. Instead of directly trying to model the efficiency of a particular chemical mixing process, we adopt a novel approach and model the properties of the stellar core, as they are directly altered by chemical mixing processes. We develop a modelling methodology that accounts for underlying model degeneracies, and incorporates both binary and asteroseismic information, when available, and apply it to binaries, pulsating binaries, and star clusters, consisting of slowly to moderately rotating stars. In this monograph, we develop a methodology consisting of binary and asteroseismic information to calibrate the implementation of physical mechanisms into stellar models. We address case studies, which serve as proof-of-concept work, as well as larger scale population studies. We apply this methodology to both pulsating and non-pulsating stars in binaries (multiple systems), and clusters to estimate the core masses of intermediate- to high-mass stars. We make use of both ground-based spectroscopy and photometry, as well as space-based photometry to characterise the stars that we study.

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In Chap. 2, we discuss the models and statistical methodologies employed in this thesis. We investigate the potential mixing mechanisms present in individual systems in Chaps. 3, 4, and 7. Furthermore, Chap. 4 serves as a proof-of-concept that demonstrates that we can uniquely constrain the core masses of stars in EBs and applies this interpretation to a sample of EBs. Chapter 5 expands the methodology discussed in Chap. 2 and applied to SB2 systems with at least one pulsating component in Chap. 4. Chapter 6 generalises the methodology to apply to clusters and investigates the extent to which enhanced core masses can explain the phenomenon of extended main-sequence turn-offs in young open clusters. Finally, Chap. 8 reflects on the analyses and discusses the future prospects of this work. Leuven, Belgium December 2020

Dr. Cole Johnston

Acknowledgements

Science, just as life, is not an individual venture. To that end, there are many people who have contributed to the work in this monograph and my development as a scientist. The two mentors who I am most thankful for are Prof. Conny Aerts and Prof. Andrej Prša. Beyond being a world-class scientist and mentor, Prof. Aerts is an expert in making sure people enjoy science, which is something I am extremely grateful for. Professor Prša has had a hand in my professional development since my Bachelor’s degree, and has never hesitated to help. Aside from these principal investigators, Dr. Andrew Tkachenko, Dr. Steven Bloemen, Dr. Paul Beck, and Dr. Dominic Bowman have proven themselves excellent mentors, wonderful collaborators, and great friends. Thank you to everyone for the journey thus far. I would also like to thank everyone at the Institute of Astronomy at KU Leuven. My years of coffee breaks, seminars, conferences, and receptions with all of you have made my time in Belgium something that I will never forget. I want to specifically thank Dr. Michael Abdul-Masih, who has been a friend and collaborator since our first undergraduate classes together in 2010. Obtaining my Ph.D. has brought me around the world, which is very helpful for gaining a perspective that I am sure I would not otherwise have. For that, I must thank everyone who has helped me to get here, both figuratively and literally. Finally, I would like to thank my family and friends for the support and patience you have all dedicated to me over the years.

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Contents

1 Scientific Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Stellar Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Energy Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Brunt–Vaïsälä Frequency . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Relevant Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Internal Chemical Mixing Processes . . . . . . . . . . . . . . . . . . . . 1.3.1 Convective Boundary Mixing . . . . . . . . . . . . . . . . . . . . 1.3.2 Radiative Envelope Mixing . . . . . . . . . . . . . . . . . . . . . 1.3.3 Overall Internal Mixing Profile . . . . . . . . . . . . . . . . . . . 1.4 Standard Stellar Evolution of Intermediate- to High-Mass Stars . 1.4.1 Stellar Evolution with Enhanced Internal Mixing . . . . . . 1.5 Asteroseismology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Stellar Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Types of Pulsating Stars . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Multiple Stellar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Binary Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Stellar Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 The Mass Discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Stellar Evolution Tracks, Isochrones, and Isochrone-Clouds 2.1 Stellar Structure and Evolution Tracks . . . . . . . . . . . . . . . 2.1.1 Initial Models and Numerical Controls . . . . . . . . . 2.1.2 Internal Mixing Processes . . . . . . . . . . . . . . . . . . 2.1.3 Input Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Fixed Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Convective Core Mass . . . . . . . . . . . . . . . . . . . . . 2.1.6 Computational Requirements . . . . . . . . . . . . . . . .

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2.2 Isochrone Construction . . . . . . . . . 2.2.1 Definition . . . . . . . . . . . . . 2.2.2 Construction . . . . . . . . . . . 2.2.3 Isochrone Fitting . . . . . . . . 2.3 Isochrone-Cloud Construction . . . . 2.3.1 Definition . . . . . . . . . . . . . 2.4 Forward Modelling Scheme . . . . . 2.4.1 Mahalanobis Distance . . . . 2.4.2 Error Estimation . . . . . . . . 2.4.3 Markov Chain Monte Carlo 2.5 Summary . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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3 The O+B Eclipsing Binary HD 165246 . . . . . . . . . . . 3.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . 3.2 Spectroscopic Analysis . . . . . . . . . . . . . . . . . . . . 3.2.1 Line Profile Variability . . . . . . . . . . . . . . 3.2.2 Interpretation as Variable Macroturbulence 3.2.3 Updated Atmospheric Solution . . . . . . . . . 3.3 Photometric Analysis . . . . . . . . . . . . . . . . . . . . . 3.3.1 K2 Photometry . . . . . . . . . . . . . . . . . . . . 3.3.2 Updated Binary Model . . . . . . . . . . . . . . . 3.3.3 Photometric Variability . . . . . . . . . . . . . . 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Estimating the Convective Core Mass for Stars in Eclipsing Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Target Overviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 CW Cephei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 U Ophiuchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 New Spectroscopy and Disentangling . . . . . . . . . . 4.3.2 Archival Photometry . . . . . . . . . . . . . . . . . . . . . . 4.4 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Spectroscopic Disentangling and Atmospheric Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 PHOEBE Models . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 CW Cep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 U Oph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Comparison of Results . . . . . . . . . . . . . . . . . . . . .

Contents

4.5 Evolutionary Modelling . . . . 4.5.1 Modelling Results and 4.5.2 Rotation and Tides . . 4.6 Interpretation . . . . . . . . . . . . 4.7 Expanded Sample . . . . . . . . . 4.8 Summary . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

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.......... Discussion . .......... .......... .......... .......... ..........

5 Binary Asteroseismology . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . 5.2 Stellar Models and Modelling Setup 5.2.1 Stellar Models . . . . . . . . . . . 5.2.2 Asteroseismic Diagnostic . . . 5.2.3 Parameter Estimation . . . . . . 5.2.4 Hare-and-Hound . . . . . . . . . 5.3 Kepler Sample . . . . . . . . . . . . . . . . 5.3.1 KIC 4930889 . . . . . . . . . . . 5.3.2 KIC 6352430 . . . . . . . . . . . 5.3.3 KIC 10080943 . . . . . . . . . . . 5.4 Discussion . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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6 The Effect of Enhanced Core Masses on the Observed Morphology of Young Clusters . . . . . . . . . . . . . . . . . . . 6.1 Introducing the Extended Main-Sequence Turn-off in Massive Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Modelling Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Application to NGC 1850 and NGC 884 . . . . . . . . . 6.3.1 NGC 1850 . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 NGC 884 . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Towards Constraining Tidal Mixing: U Gru 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 7.2 U Gru . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 TESS Light Curve . . . . . . . . . . . . 7.2.2 UVES Observations . . . . . . . . . . . 7.3 Pulsational Characteristics . . . . . . . . . . . . 7.4 Interpretation . . . . . . . . . . . . . . . . . . . . . 7.5 Summary and Conclusions . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Current and Future Prospects for the Precise Modelling of Pulsating Stars in Multiple Systems . . . . . . . . . . . . . . 8.2.1 Individual Targets . . . . . . . . . . . . . . . . . . . . . . . 8.3 Towards a Generalised Sample and Ensemble Modelling References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

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Appendix A: Base MESA Inlist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Appendix B: Moment Method Periodograms . . . . . . . . . . . . . . . . . . . . . . 181 Appendix C: Marginalised Posterior Distributions for Chap. 4. . . . . . . . 185 Appendix D: Secondary Component Parameter Correlation Plots From Chap. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Acronyms

AGB AM BJD CBM cc CCF CDM CEM CMD CNO DFT DR2 EEP EMDM EMEM eMSTO env EOS FCMD FT GA GOF GSSP HJD HPD HRD HST IGWs LMC LPV

Asymptotic giant branch Angular momentum Barycentric Julian Date Convective boundary mixing Convective core Cross-correlation function Classical dynamical mass Classical evolutionary mass Colour-magnitude Diagram Cycle carbon-nitrogen-oxygen cycle Discrete Fourier transform Data-release 2 Equivalent evolutionary phase Enhanced mixing dynamical mass Enhanced mixing evolutionary mass Extended main-sequence turn-off Envelope Equation of state Fractional core mass distribution Fourier transform Genetic algorithm Goodness-of-fit Grid Search in Stellar Parameters Helio-centric Julian Date Highest posterior density Hertzsprung-Russel Diagram Hubble Space Telescope Internal gravity waves Large Magellanic cloud Line profile variations

xvii

xviii

LSD MAMS MCMC MD MLE MLT MS Myr NLTE pp-chain REM RMS RV SB2 SMC SNR SPB SPD TAMS TAR YMC ZAMS

Acronyms

Least-squares deconvolution Middle-age main-sequence Markov chain Monte Carlo Mahalanobis Distance Maximum likelihood estimator Mixing length theory Main-sequence Million year(s) Non-local thermodynamic equilibrium Proton-proton chain Radiative envelope mixing Root-mean-square Radial velocity Double-lined spectroscopic binary Small Magellanic cloud Signal-to-noise ratio Slowly pulsating B-type Spectral disentangling Terminal-age Main-sequence Traditional approximation of rotation Young massive cluster Zero-age Main-sequence

Chapter 1

Scientific Context

1.1 Introduction Stars form an inter-sectional basis in astronomy and astrophysics. Stars are the fundamental units of galaxies, the progenitors of exotic objects such as neutron stars and black holes and the supernovae which produce them. They are the dynamical engines of clusters and galaxies, the engines which produce the observed elements in the Universe heavier than lithium, and the objects which host exoplanets. As such, a number of fields within astronomy rely on the theory of stellar structure and evolution to be robust in its ability to describe stars, and any inaccuracies in this theory propagate into these fields. To this end, the proper calibration of the theory of stellar structure and evolution and the accurate implementation of physical processes within this theory is of great importance. Whereas the theory of stellar structure and evolution is robust in a schematic sense, it has many shortcomings in its details, and in particular in the description of the processes at work in the stellar interior. Asteroseismology is the ideal tool for investigating such processes as it allows for the probing of stellar interiors which cannot be achieved through other means. Asteroseismology allows us to directly test the implementation of physical processes that influence stellar structure and evolution by comparing models which include these processes to asteroseismic observables, such as pulsation frequencies which are highly sensitive to the conditions of the stellar interior. Alternatively, the estimation of stellar parameters through binary modelling allows for the testing of the theory of stellar structure and evolution by providing parameter estimations which are so precise that they can discriminate between the predicted evolutionary trajectories of different stellar models. A major defining characteristic of the first stage of stellar evolution is the mass of the resulting helium core at the end of core-hydrogen burning. This quantity dictates the remainder of the star’s evolution as well as the properties of the stellar end product. To this end, calibrating any internal process which influences this quantity will thus © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0_1

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1 Scientific Context

reduce the error that propagates into later predictions of stellar evolution and into other fields of astronomy. In this monograph, we use combined asteroseismic and binary modelling to investigate the effects of internal mixing processes in stars. In particular, we investigate mixing in intermediate- to high-mass stars which have a fully mixed convective core on the main-sequence and produce white-dwarfs, neutron stars, and black holes. In the remaining pages of this chapter, we introduce the fundamentals of stellar structure, the processes at work in the stellar interior, stellar evolution and how it is affected by interior mixing processes, and the tools with which we can investigate these processes. The brief discussions in this chapter are based on the extensive monographs by Maeder (2009), Kippenhahn et al. (2012), and Aerts et al. (2010), and refer the reader to these monographs for a more in depth treatment of the concepts presented here.

1.2 Stellar Structure Under some simplifying assumptions, the theory of stellar structure and evolution can reasonably approximate real stars. These simplifying assumptions assert that a star is spherically symmetric, non-rotating, has no magnetic field, and experiences no externally applied body forces, such as the tidal effects of a close companion. Applying these, one can describe a star adequately along one dimension in terms of its pressure (P), density (ρ), temperature (T ), local radius (r ), luminosity (L), and chemical profiles (X j ), according to Eqs. (1.1)–(1.5). ∂P Gm =− , ∂m 4πr 4

(1.1)

∂r 1 = , ∂m 4πr 2 ρ

(1.2)

∂L = n +  g − ν , ∂m

(1.3)

∂T GmT ∂ ln T =− ∇; ∇≡ , ∂m 4πr 4 P ∂ ln P

(1.4)

1.2 Stellar Structure

3

mj ∂Xj = ∂t ρ

  i

ri j −



 r jk .

(1.5)

k

Equation (1.1) describes the state of mechanical (i.e. hydrostatic) equilibrium within a star as the balance of the radially inward gravity and outward pressure gradient, where G is the gravitational constant and m is the mass coordinate. This condition of hydrostatic equilibrium is maintained as long as there is no acceleration present in the gas, such as that due to the centrifugal force introduced by rotation. Equation (1.2) asserts the conservation of mass in the star and describes the radial distribution of mass. Equation (1.3) is the equation of energy conservation, which balances the energy generated by nuclear burning (n ) and gravitational contraction (g ) with the energy lost via neutrino emission (ν ). Equation (1.4) describes how energy is transported within the star, where ∇ is the temperature gradient. Calculation of ∇ requires special consideration in the cases where energy is transported via different mechanisms. We address these cases in Sect. 1.2.1. The system of Eq. (1.5) describes the chemical evolution of the star in time, where X j and m j are the mass fraction and mass of the j-th chemical species, respectively, ri j is the rate of change of species i to species j, and r jk is the rate of change of species j to species k. Modifications to this system of equations are required whenever chemical mixing processes occur, as will be discussed later. We note that Eqs. (1.1)–(1.5) are described in terms of the mass coordinate m. This approach is adopted as a convenience in consideration of the evolution of stars. Whereas the radius of a star can vary drastically over several orders of magnitude across the normal course of its evolution, the mass of the star stays more constant within the same order of magnitude, except for short phases of extreme mass loss or mass transfer. Generally speaking, as the star evolves, it will adjust to maintain hydrostatic equilibrium. In order to solve this set of equations, an equation-of-state (EOS) is required to relate the pressure P, temperature T , and density ρ. Considering an ideal gas with radiation, we have an EOS of the form: P=

1 R ρT + aT 4 , μ 3

where μ is the mean molecular weight, a is the radiation density constant, and R is the gas constant. The second term on the right accounts of the effects of radiation pressure on the gas. Solving this set of equations by adopting proper boundary conditions at the surface and centre of the star, as well as initial conditions at birth, we arrive at a stellar evolution model. The precise numerical considerations and description of included physics are discussed in detail in Chap. 2.

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1.2.1 Energy Transport The form of energy transport within a star depends on the temperature gradient, ∇, as defined in Eq. (1.4). In the case where radiation is solely responsible for energy transport, we can replace ∇ = ∇rad under the approximation that photons behave according to a diffusive description, and where ∇rad is the radiative temperature gradient given by: LP 3κ ∇rad = . 16πacG mT 4 Here, κ is the Rosseland mean opacity, and c is the speed of light. If the conditions in the stellar interior are such that the radiative temperature gradient, ∇rad , becomes large, energy transport via radiation may no longer be sufficiently efficient. In this case, energy transport is then achieved through convection. Convection is the large scale motions of mass elements, whose temperature, density, or molecular weight differ from their environment. Due to this, they move to neighbouring regions in the star where they dissipate and deposit their energy and chemicals. These macroscopic motions very efficiently transport energy and induce efficient chemical mixing. As such, a convective region can be considered fully chemically mixed and having an adiabatic temperature gradient, ∇ad , given as:   ∂ ln T , (1.6) ∇ad = ∂ ln P S where S is the entropy. In the case of non-rotating stars, a given region is determined to be unstable against convection according to either (i) the Schwarzschild criterion, given by: ∇ad < ∇rad ,

(1.7)

or (ii) the Ledoux criterion, given by: ∇ad +

ϕ ∇μ < ∇rad , δ

(1.8)

∂ ln μ ∂ ln P

(1.9)

where ∇μ = 

and δ=

∂ ln ρ ∂ ln T



 ; ϕ= P,μ

∂ ln ρ ∂ ln μ

 , P,T

where ∇μ is the chemical gradient. The Ledoux criterion accounts for the stabilising effect of a chemical gradient in a given region. The Schwarzschild criterion can be considered a special case of the Ledoux criterion which considers a homo-

1.2 Stellar Structure

5

geneous composition with ∇μ = 0. Any region which does not fulfil both criteria will be considered to undergo convection. In the case where a region is stable under the Ledoux criterion, but unstable under the Schwarzschild criterion (i.e., ∇ad < ∇rad < ∇ad + ϕδ ∇μ ), the region is said to be semiconvective. In such a region, a mass element will be considered vibrationally unstable and mix on the local thermal time scale (Langer et al. 1985). Conversely, in a region which is deemed stable against convection, any macroscopic mass element which experiences an acceleration will be returned to its equilibrium position via the buoyancy force. In 1-D stellar models, it is convenient to implement parameterised versions of convection. The most widely used description of convection is the time-independent mixing length theory (MLT ; Böhm-Vitense 1958). In short, this description considers the energy transported by convection in terms of the mean-free path m travelled by a given mass element before it dissolves and releases its heat and chemicals to its environment. This mean free path is proportional to the local pressure scale-height H p , where the proportionality factor αMLT is a free parameter. Due to the presence of large scale convective envelopes in low mass stars, including the Sun, the value of αMLT has been rigorously calibrated, and is generally accepted to be set at αMLT = 1.8 (Joyce and Chaboyer 2018; Viani et al. 2018). While the calibration of this value is important for stars with thick convective envelopes, as it effectively scales the efficiency of the energy transport via convection, it is of less consequence for those stars with strictly radiative envelopes. As with all simple parameterisations of complex processes, MLT has some well known shortcomings, which particularly affect the convective/radiative boundary region. By definition, a mass element transported via convection will no longer experience an acceleration at (or beyond) the convective boundary. However, the mass element has inertia and cannot abruptly stop moving once it reaches this boundary. Parameterisations of convection such as MLT do not account for this, and therefore underestimate any subsequent mixing at the convective boundary. Furthermore, numerical simulations predict efficient chemical and mass entrainment at the boundary caused by the turbulent eddies produced by departure from tangential laminar flow at the boundary (Meakin and Arnett 2007). Convective boundary mixing and its implications are key ingredients in this monograph and are more thoroughly discussed in Sects. 1.3 and 1.4.1.

1.2.2 Brunt–Vaïsälä Frequency One natural quantity of the stellar interior to consider is the Brunt–Vaïsälä frequency, or the buoyancy frequency. This local quantity governs the frequency with which a displaced mass element will oscillate around its equilibrium position. An approximate expression for the Brunt–Vaïsälä frequency is given by: N2 

 g  δ (∇ad − ∇) + ϕ∇μ , HP

(1.10)

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where g is the local gravitational acceleration and the remaining quantities have been previously defined. From this equation, we can see that the Brunt–Vaïsälä frequency is affected by the local pressure, chemical composition, gravitational acceleration, and temperature. The Brunt–Vaïsälä frequency is thus a function of location in the star, and has a spatially varying profile. An example of the spatial variation of the Brunt–Vaïsälä frequency can be seen in the right hand panel of Fig. 1.1. This frequency sets the limit for which a gravity wave can propagate in the stellar interior. Since this quantity is determined by the local conditions of the interior, measuring the frequency of such waves, as is done via asteroseismology (Sect. 1.5), provides information on the local conditions of the deep stellar interior.

1.2.3 Relevant Time Scales The processes involved in a star’s evolution span a large range of time scales. To contextualise stellar evolution and the processes which govern it, we consider three relevant time scales: (i) the dynamical, or free-fall, time scale, (ii) the Kelvin-Helmholtz, or thermal, time scale, and (iii) the nuclear time scale. For a star in hydrostatic equilibrium, the dynamical time scale represents the amount of time required for a star to return hydrostatic equilibrium after being perturbed. An order of magnitude estimate for this can be obtained using: tdyn ∼ (G ρ)−0.5 ,

(1.11)

where ρ is the average stellar density. This time scale ranges from several seconds in compact objects such as white dwarfs, to nearly 30 min for the Sun, and on the scale of several to tens of hours for larger stars. Whereas the dynamical time scale represents the time required for a star to adjust to a perturbation to the pressure (or gravity), the Kelvin-Helmholtz time scale represents the time required for a star to adjust to perturbations to the thermal equilibrium. The thermal equilibrium of a star is the balance between the rate of energy generation versus the rate of energy loss at the surface (i.e. the stellar luminosity, L). An estimate for this time scale is given by: G M2 . (1.12) tKH ∼ 2R L This time scale becomes important in those phases of evolution where the star loses its nuclear fuel source and must contract to provide energy to support the luminosity. For the Sun, this is of order 107 years, and decreases for increasing mass (luminosity). The longest time scale that we consider is the nuclear time scale, which is the amount of time that a star can support its luminosity with energy production via nuclear fusion. An estimate for this time scale is given by:

1.2 Stellar Structure

7

tnuc ∼

n , L

(1.13)

where n is the energy generation rate, whose exact value depends on the stage of nuclear burning (hydrogen, helium, etc.) as well as the type burning process ( pp−chain, CNO-Cycle, triple-α process). This is the longest time scale, on the order of 1010 years for the Sun, and 106 for a 25 M star. From these estimates, we can see that for any star: tdyn  tKH  tnuc .

1.3 Internal Chemical Mixing Processes The stellar interior is subject to many processes which can induce chemical (and angular momentum ; hereafter AM) transport throughout the star. The overall profile of the mass fraction of a chemical species, X j , within a star at a given time are then the result of the collective contribution of all nuclear reactions and mixing processes Dmix (r ) active at a given layer, r , working either individually or jointly. The system of equations describing the change in the mass fraction of chemical species X j then becomes:      mj  ∂Xj 1 ∂ 2∂Xj (1.14) = ρr ri j − r jk + 2 (Dmix (r )) . ∂t ρ ρr ∂r ∂r i k The first term on the right hand side of the equation is the net change due to nuclear reactions and the second term on the right hand side occurs due to various mixing mechanisms, all of which are treated as diffusive transport processes, described by diffusivity coefficients Dn (r ) (expressed in cm2 s−1 ). In the case where no enhanced mixing is active, only convective mixing needs to be considered in the mixing profile Dmix (r ). In the case where mixing processes are active near the boundary of the convective core, they will introduce fresh hydrogen (or helium, or other elements for later nuclear burning stages) to the core. This will, necessarily, divert the evolution of a star relative to the case where no near-core mixing is present. It follows that the total impact of near-core mixing is determined by its efficiency and the extent of the region where it is active. Derivation of this profile through 3D hydro-dynamical simulations is an active field of research. However, limitations of stellar evolution codes to 1-D require parameterised descriptions of the mechanisms which contribute to this overall mixing profile. As said previously, we adopt a diffusive approximation, allowing us to compute the diffusivity profile Dn (r ) from a parameterised model with free parameters. Given the impact that these processes have on stellar evolution, these free parameters must be calibrated through observational investigation (see Salaris and Cassisi 2017, for a recent review). The convective mixing in the core is based on the MLT formalism introduced in Sect. 1.2.1. The amount of convective mixing, denoted as Dconv , is calculated as:

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Dconv =

1 αMLT H p vconv , 3

(1.15)

where vconv is the convective velocity and αMLT is a free parameter. Over the normal course of stellar evolution, stars develop convective regions in their core and envelope. In this work, we are interested in the mixing profile as ranging from the convective core, over the core boundary, and up to the surface, i.e. rcc < r < R. In addition to the contribution from convection, we consider there to be two mixing regimes such that we arrive at the total chemical mixing profile of the form: Dmix (r ) = Dconv (r ) + DCBM (rcc < r < renv ) + DREM (renv < r < R) ,

(1.16)

Here, CBM stands for convective boundary mixing, and REM stands for radiative envelope mixing, where rcc is the radial location of the convective core boundary as determined by the Schwarzschild criterion (Eq. 1.7), renv is the radial coordinate of the base of the radiative envelope, and R is the total radius of the star, defined as where the optical depth reaches a value of τ = 2/3. It follows then, that Dconv , DCBM , and DREM are the mixing contributions from convection, CBM, and REM, respectively. When inserted into Eq. (1.14), we arrive at the overall change of the mass fraction X j described as:  



  2∂Xj ri j − r jk ρr (Dconv + DCBM + DREM ) . ∂r i k (1.17) This results in an overall mixing profile displayed in Fig. 1.1, where the grey area represents convective mixing, the blue area represents CBM, and the green area represents REM. The hatched and non-hatched profiles correspond to models with different combinations of free parameters to be introduced in the following section, which in turn result in different values of DCBM and DREM . Below we discuss the mechanisms which contribute to CBM and REM. mj ∂Xj = ∂t ρ



1 ∂ + 2 ρr ∂r

1.3.1 Convective Boundary Mixing Convective boundary mixing (CBM) collectively refers to all mixing processes active at the transition region between a fully chemically mixed convective region, and a stably stratified radiative region. Some of the mixing processes active in this region may include convective entrainment, convective penetration, diffusive exponential convective overshooting, internal gravity waves, as well as rotationally induced shear instabilities (see Viallet et al. 2015; Cristini et al. 2015, for detailed descriptions of these phenomena). The process of convective entrainment refers to the transport of mass (and therefore chemicals) via turbulent eddies formed between a convective

1.3 Internal Chemical Mixing Processes

9 −3.5

14

−4.0 12 −4.5 −5.0 −5.5

8

−6.0

6

log N 2/2π [s−2]

log Dmix [cm2 s−1]

10

−6.5 4 −7.0 2

0

−7.5

−1.0

−0.5

log(q)

0.0

−1.0

−0.5

0.0

−8.0

log(q)

Fig. 1.1 Comparative plots of stellar model interiors. Left: Mixing profiles for two stellar models of 3 M and X c = 0.3 for the convective core (grey), convective boundary region (CBM, blue) and radiative envelope region (REM, green) plotted against the logarithm of the mass coordinate q = m/M . The model with mixing profiles indicated as hatched regions has eight times higher CBM and 100 times higher REM than the other model, leading to an increase in core mass of 36.5%. Right: Logarithm of N 2 of the two models plotted against the logarithm of the mass coordinate q. The solid red line denotes N 2 of the model with a minimum amount of interior mixing, while the dashed black line denotes N 2 of the model with eight times higher CBM and 100 times higher REM. This figure was originally published by Johnston et al. (2019), their Fig. 1. Reproduced with c permission from Astronomy & Astrophysics, ESO

region and an adjacent radiative region (Meakin and Arnett 2007). While the inclusion of convective entrainment in 3D hydrodynamical simulation is common, it has only recently been implemented in 1-D stellar evolution codes (Staritsin 2013; Arnett and Moravveji 2017; Cristini et al. 2019). The convective boundary is set as the location where a mass element no longer experiences acceleration from convection. This, however, does not restrict convectively accelerated mass elements with inertia from passing beyond this boundary into the radiative region. Convective penetration and convective overshooting refer to the subsequent mixing generated by this process, and extend over a region whose length is determined by the local pressure scale height. The region covered by convective penetration is effectively an extension of the convective core, where the local temperature gradient of the region is assumed to be the adiabatic temperature gradient,

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making fully thermally and chemically mixed. In 1-D stellar evolution codes, the diffusion profile of convective penetration Dpen is approximated as a step function with the form: (1.18) Dpen (r ) = αpen H p , for rcc < r < renv , where αpen is a free parameter. The region mixed by convective overshooting, however, assumes the radiative temperature gradient, so that only the chemical structure of the region is altered. The diffusion profile produced by overshooting, Dov , is approximated as an exponentially decaying function of the form:   −2(r − rcc ) , for rcc < r < renv , (1.19) Dov (r ) = D0,cc exp f ov H p where D0,cc is the convective mixing coefficient at rcc , and f ov is the scaling factor which sets at what fraction of the local pressure scale height the exponential profile reaches half its starting value. These mixing phenomena described by Dpen and Dov produce similar effects on global stellar observables (Godart 2007; Claret and Torres 2017). However, thanks to their local probing power, asteroseismic observables can in principle distinguish between the two (Pedersen et al. 2018; Michielsen et al. 2019). A complete CBM profile would consider the joint effects of entrainment, penetration, and overshooting, acting over different distances from the Schwarzschild boundary (Hirschi et al. 2014). Based on 3D hydrodynamical simulations done by Augustson and Mathis (2019), Michielsen et al. (2019) have implemented a Gumble mixing profile, with a thermal structure consisting of a temperature gradient that transitions smoothly from the adiabatic gradient to the radiative gradient, as a potential CBM description. However, this and additional mechanisms that can contribute to CBM currently lack firm observational calibration. Furthermore, the CBM profile may be influenced by magnetic fields present in the stellar interior (Fuller et al. 2019) or by the tides generated by a close binary companion (Song et al. 2013). In addition to the near core mixing, convectively accelerated mass elements which travel beyond the convective boundary also excite a spectrum of travelling waves which propagate into the radiative region. Although this process is induced in the convective boundary region, its effects are strongest in the radiative envelope, and as such it is treated as an REM process below. For a non-zero chemical gradient in the near core region, it is possible that a layer is stable according to the Ledoux criterion, but unstable according to the Schwarzschild criterion. In this instance, the layer is said to be semiconvective, and will mix on a thermal time scale (Langer et al. 1985). This is significantly slower than convective mixing. Such semiconvective mixing is important to consider in the absence of additional CBM and has mainly been considered for stellar models in nuclear burning phases beyond the main-sequence (Wood et al. 2013; Moore and Garaud 2016). Since we are dealing with the case where other CBM is active in this work, we omit semiconvective mixing.

1.3 Internal Chemical Mixing Processes

11

1.3.2 Radiative Envelope Mixing Whereas CBM is a collective term to describe all of the mixing processes active in the near-core region, radiative envelope mixing (REM) is used as a collective term to describe all of the mixing processes active in the radiatively stratified regions of a star. Although convective zones can occur in the envelopes of stars, we consider the mixing there to be treated via MLT and any mixing at the boundaries of these convective zones with radiative regions to be a general case of the CBM processes described above. Thus, when we refer to REM processes, we mean to include any processes that do not fall under convection or CBM processes. These REM processes may include shear instabilities, meridional circulation, internal gravity waves, standing gravity mode pulsations, microscopic atomic diffusion, magnetism, and tides. In this monograph, we consider atomic diffusion to be the collective effect of gravitational settling, concentration diffusion, thermal diffusion, and radiative levitation (Thoul et al. 1994; Michaud et al. 2015; Dotter et al. 2017), which require time-scales on the order of the thermal or nuclear time scales to become significant contributors to the evolution of intermediate- to high-mass stars on the main-sequence. With the exception of microscopic atomic diffusion, these REM processes all act on time scales several orders of magnitude shorter than both the thermal and nuclear time scales. Therefore, in the event that any of these more rapid processes are active, the effects of atomic diffusion are expected to be effectively washed away. As such, we do not consider the effects of atomic diffusion in this monograph, although we acknowledge that these microscopic processes cannot be ignored when focusing on low-mass stars and their surface abundances (Dotter et al. 2017; Deal et al. 2017), as well as when considering the observed pulsations in compact subdwarf O and B stars (Hu et al. 2010; Bloemen et al. 2014). In the following Sections, we describe the major processes which contribute to REM.

1.3.2.1

Wave Induced Mixing

Wave induced mixing is the general term to describe the chemical mixing induced by both coherent standing pulsations as well as travelling internal gravity waves (IGWs) excited by convective plumes at the interface of the convective and radiative regions. While gravity modes (discussed in detail in Sect. 1.5) are standing waves which produce periodic variations in a star’s brightness, IGWs are travelling waves which are excited at one location, propagate, and eventually break at another location in the star. While IGWs also produce observable variability, it is stochastic in nature and not periodic. There is evidence to suggest, however, that IGWs can lead to the resonant excitation of coherent gravity modes through stochastic driving (Edelmann et al. 2019). Two-dimensional hydrodynamical simulations of a zero-age main-sequence star by Rogers and McElwaine (2017) demonstrated that IGWs also result in diffusive particle mixing. The averaged 1-D mixing profile induced by IGWs scales with

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increasing wave amplitude along a decreasing density gradient. Using a 1-D parameterised version of this profile, Pedersen et al. (2018) showed that the mixing induced by IGWs can both shift observed pulsation frequencies by altering the chemical gradient and produce surface abundance enhancements. Although there is no standard use of this mixing in stellar evolution computations yet, its profile can generally be expressed as:

DIGW (r ) = D0 ρ −ζ ; ζ ∈ [0.5, 1] , for r > rcc

(1.20)

where D0 is a free parameter, and ζ sets the rate at which the mixing scales with the density profile (Pedersen et al. 2018; Aerts 2021). An important characteristic of wave mixing by IGWs is that it is expected to occur in any star with a convective core, independent of metallicity. This makes IGWs an ideal candidate to explain enhanced mixing in slowly rotating, and/or metal poor stars.

1.3.2.2

Rotationally Induced Mixing

Rotation, mixing induced via rotation, and their impacts on stellar evolution are widely researched within astrophysics. In particular, the effects of rotation have been investigated on the properties and evolution of isolated, single field stars (Abt et al. 2002; Ekström et al. 2012; Zhang 2013), binary stars (Torres et al. 2010; Schneider et al. 2014; Brott et al. 2011a, b; de Mink et al. 2013), as well as of stellar clusters (Evans et al. 2006; Niederhofer et al. 2015a; Ahmed and Sigut 2017; Bastian et al. 2017). Rotation generates a multitude of instabilities and processes which induce chemical transport in stars (see, e.g. Heger et al. 2000, for extensive descriptions). For example, dynamical and secular shear instabilities, meridional circulation, horizontal turbulence, as well as barocline mixing are all produced by rotation (Maeder 2009). Although meridional circulation is naturally an advective process, it was shown by Chaboyer and Zahn (1992) that the chemical mixing induced by meridional circulation can be treated as a diffusive process in the same manner as the other mechanisms. However, the efficiency of each of these physical processes cannot be derived from theory or hydrodynamical simulations, given their complexity and (possibly nonlinear) interactions. Consequently, this requires at least one free parameter in the parameterised description of each mechanism. A comparison of stellar evolution models with different implementations of rotational mixing lead to differences of 0.1 dex in luminosity during the core-hydrogen burning and even up to 0.5 dex in later evolutionary phases (see Fig. 7 in Martins and Palacios 2013). It is therefore clear that one cannot rely on such models to fix the profile of Dmix (r ). Rather, asteroseismology offers a new way to calibrate this profile observationally (Aerts 2021). Rotation introduces structural deviations caused by the centrifugal force. When the rotation rate is below some 50% of the critical rate, these deviations are difficult to distinguish from those caused by other internal phenomena involved in CBM and REM using stellar surface quantities alone (Aerts et al. 2018). This however, is not

1.3 Internal Chemical Mixing Processes

13

the case when considering their effect on stellar pulsation frequencies, as will be discussed in more detail in Sect. 1.5.

1.3.2.3

Tidal Mixing

When a star is in a binary or higher order multiple system, tidal forces will cause equilibrium structure changes and induce numerous internal processes. The equilibrium tide generated by the presence of a close gravitating companion introduces an extra acceleration in Eq. (1.1). Similar to the case of sufficiently slow rotation, the structural changes induced by this acceleration can often be ignored in the deep stellar core. Hence, we can ignore the impact of these structural deviations on evolutionary calculations. There are, of course, cases in which the effects of the companion cannot be ignored. Most notably, these cases include the components experiencing spin up of the outer layers due to angular momentum exchange, over-contact systems, and mass transfer events. Over-contact systems are binary system which are so close that the mutual gravitation of the two stars deform each other into a peanut-shape. Mass transfer events in binary systems occur when one star loses some of its mass via winds or Roche-lobe overflow which is accreted onto the companion. Either of these cases obviously produce deviations from secular evolution and require proper theoretical treatment. In addition to the structural changes induced by the equilibrium tide, the dynamical tide will also induce chemical transport in stars. The dynamical tide will generate gravito-inertial waves in the stellar interior that break at a given layer. When these waves break, they will deposit both chemicals and angular momentum in the layer at which they break (Ogilvie 2014). This provides another form of mixing in stellar interiors which has no firm observational calibration to date and is lacking robust theoretical descriptions.

1.3.3 Overall Internal Mixing Profile For the remainder of this monograph, we consider CBM to be represented by convective overshooting (Eq. 1.19), REM to be represented by mixing induced by IGWs (Eq. 1.20), and we neglect the effects of atomic diffusion as described above. This leads to the functional form of the chemical mixing profile:  Dmi x = Dconv (r ) + D0,cc exp

−2(r − rcc ) f ov H p





+ D0,env ρ (r )−ζ .

(1.21)

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1.4 Standard Stellar Evolution of Intermediate- to High-Mass Stars Stellar evolution begins when localised fragments of a large molecular cloud experience a runaway isothermal collapse. At some point during this contraction, the material at the centre of this collapsing fragment becomes opaque, trapping the heat generated by gravitational collapse. At this point, a phase of adiabatic contraction occurs, resulting in an increasing temperature, density, and pressure until hydrostatic equilibrium is achieved. This gaseous body in hydrostatic equilibrium is known as a proto-star. The proto-star is still embedded in a cloud of material which will form an accretion disk that continues to donate mass to the proto-star until a later stage of evolution. Although the outer layers of this proto-star are in hydrostatic equilibrium, the core continues to contract, becoming increasingly hot and dense. At some point, the core temperature becomes sufficiently high to undergo the initial deuterium burning reactions of the proton-proton (pp-) chain. This reaction produces Helium-3, which is initially scarce in the stellar interior. This process continues until the core is sufficiently hot and abundant enough in 32 He to undergo the full pp-chain in equilibrium. Those stars which have accreted more than ∼1.2 M at this point will then also begin to burn hydrogen through the CNO-cycle, which is a catalytic reaction that fuses hydrogen into helium with a very high temperature dependence. This high temperature dependence then causes the core to become convective in order to efficiently transport the energy generated via the CNO-cycle. At the point of equilibrium hydrogen burning, the star is then said to have arrived on the zero-age main-sequence (ZAMS). For the remainder of this discussion, we only consider those stars massive enough to have a convective core on the ZAMS. Once on the ZAMS, the star is then in full thermal equilibrium, and will fuse hydrogen into helium on a nuclear time scale. The duration of this hydrogen burning main-sequence (MS) is determined by the amount of hydrogen available to participate in the fusion process. In secular evolutionary models with no internal mixing, this quantity is determined at the ZAMS. As a star slowly converts hydrogen into helium, the mean-molecular weight, μ, of the star changes, effectively driving the evolution of that star according to Eq. (1.5). This continues until core-hydrogen exhaustion (in this work defined numerically as when the central hydrogen content drops below X c < 10−12 ), where the star is said to arrive at the terminal-age main-sequence (TAMS). An example of evolutionary tracks of 1.7 and 5 M stars from ZAMS to the ascent of the red giant branch is shown in Fig. 1.2. Following our discussion of the time scales in Sect. 1.2.3, it is highly unlikely to catch a star between the TAMS and the onset of core-helium burning. As of the TAMS, the homogeneously mixed helium core begins to contract as it is not yet hot enough to fuse helium and thus no longer has an internally generated counter-balance to gravity. During this phase, the star evolves along the post-MS according to the thermal (Kelvin-Helmholtz) time scale. As the core contracts and becomes more dense, the temperature increases due to the energy released from gravitational contraction. While this occurs, the envelope expands and develops a large convective region. Furthermore, outside of the contracting core, a

1.4 Standard Stellar Evolution of Intermediate- to High-Mass Stars

15

1.5

logL/L

1.4 1.3 1.2 1.1 1.0

1.7 M 3.90

3.85

3.80

3.75

3.70

3.65

log (Teff /K) 3.4 3.3

logL/L

3.2 3.1 3.0 2.9 2.8

5M

2.7 4.2

4.1

4.0

3.9

3.8

3.7

3.6

log (Teff /K) Fig. 1.2 Top: Evolutionary tracks for a 1.7 M model with different amounts of CBM according to Eq. (1.19) ( f ov = 0.005—solid lines; 0.02—dashed line; 0.04—dotted line) and fixed REM according to Eq. (1.20) (D0,env = 100 cm2 s−1 ). Bottom: Same as top panel, but for a 5 M model. Core-hydrogen exhaustion (defined numerically as when the core-hydrogen mass fraction drops below 10−12 ) is denoted by black circles

shell of hydrogen burning occurs, temporarily providing the nuclear energy for the star. From this point forward, the remaining evolutionary trajectory is largely determined by the initial mass of the star (and hence the mass of the helium core at the TAMS). Those stars with initial masses below some ∼2.3 M will experience electron degeneracy in their contracting core, preventing a smooth transition to the temperatures necessary for helium burning via the triple-α process. Due to the nature of degenerate matter, the pressure does not increase with increasing temperature, and

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therefore, helium ignition occurs in what is known as a helium-flash, which instantly lifts the degeneracy and produces a convective helium-burning core. The cores of stars with initial masses higher than this ∼2.3 M are sufficiently massive that they can achieve the temperatures necessary for helium burning without a degenerate core. Once core helium burning starts, the extended envelope contracts as the star adjusts to the newfound energy source, and the star again evolves on the nuclear time scale on the helium-MS. For stars with initial masses below some ∼9 M , this is the last core burning phase that will occur, as their cores are not massive enough to achieve the temperatures required for subsequent nuclear burning cycles. These stars will then ascend the asymptotic giant branch (AGB) and end their lives as inert C/O or He white dwarfs, depending on their initial masses. Those stars with initial masses greater than ∼9 M can continue to experience subsequent core nuclear burning phases (and intermittent shell burning phases) until they develop iron/nickel cores. At this point, fusion to heavier elements becomes endothermic instead of exothermic, and the star no longer can produce energy in its now very dense core. The following contraction is the precursor to a supernova explosion, which will produce either a neutron star if the initial mass is lower than ∼15 M or a black hole if the initial mass is larger than this value.

1.4.1 Stellar Evolution with Enhanced Internal Mixing In the case of secular evolutionary calculations with no enhanced internal mixing, the chemical evolution of the star is dictated by Eq. (1.14) where the only mixing present is due to convection such that E n = Dconv according to Eq. (1.15). With the introduction of enhanced chemical mixing, however, we consider Eq. (1.17) to account for the increased transport of chemicals throughout the star. First, the enhanced mixing introduces new hydrogen into the fully mixed convective core, consequently increasing the overall mass of the convective core, as well as effectively replenishing the amount of hydrogen available for nuclear burning. As a direct result, a star which experiences CBM will have an extended MS life-time which is directly proportional to the amount of fresh Hydrogen introduced to the core. The extension of the MS as a function of increasing internal mixing is demonstrated by comparing the dashed and dotted tracks to the solid tracks in the panels of Fig. 1.2 (more detailed descriptions of evolutionary tracks than those in the current chapter follow in Chap. 2). Generalising this to models of all masses across the Hertzsprung-Russel Diagram (HRD), we notice that enhanced internal mixing effectively relocates the TAMS line, as seen in Fig. 1.3. The displaced TAMS line for the enhanced mixing tacks in Fig. 1.3 considers the numerical definition of the TAMS where the core-hydrogen content X c drops below X c < 10−12 , as this is the point past which the star no longer evolves on a nuclear time scale via hydrogen burning. Additionally, there is a complication in terms of the classical TAMS definition. Since the enhanced mixing continuously introduces hydrogen into the fully mixed convective core, the evolutionary track extends beyond the traditional location of the blue hook. This implies that the TAMS

1.4 Standard Stellar Evolution of Intermediate- to High-Mass Stars 4.0

3.5

7M

log (L/L )

Fig. 1.3 Evolutionary tracks for 3 to 7.5 M models in steps of 0.5 M for models with some minimum amount of internal mixing (solid grey) and some enhanced amount of internal mixing (dotted grey). ZAMS line denoted as solid black line, TAMS line for minimum mixing tracks and enhanced mixing tracks denoted by black dashed and dashed dotted lines, respectively

17

3.0

6M 5M

2.5

4M 2.0

3M 4.3

4.2

4.1

4.0

log (Teff /K)

is not a unique function in the parameter space, but instead spans a region depending on the level of internal mixing. The effect of enhanced CBM on the internal structure of a 3 M star can be seen in the left panel of Fig. 1.1, which shows the mixing profile as a function of the logarithm of the mass-coordinate, q. The grey regions correspond to the fully mixed convective core, the blue regions correspond to the CBM region, and the green corresponds to the REM regions. The hatched profile belongs to a stellar model with a factor eight larger CBM and a factor 100 larger REM, compared to the non-hatched profile. Both models were evolved to the point where they have a remaining fractional corehydrogen content of X c = 0.3. The model with enhanced mixing has a convective core some 36.5% more massive than its counterpart with less internal mixing. The panels of Fig. 1.4 display the location of the edge of the convective core (light grey) and the CBM region (dark grey), across the mass range 1.2 M ≤ M ≤ 25 M , for models with a small amount of CBM (hatched) and models with a large amount of CBM (non-hatched). The panels show the profiles at different stages of evolution, demonstrating the evolution of the core mass enhancement across a wide range of masses. This plot is equivalent to stacking the grey and blue regions for the profiles in the left panel of Fig. 1.1 across the mass range mentioned previously for different values of X c . This demonstrates the difference in core mass for stars with different CBM profiles across a wide stellar mass range. Crucially, CBM alters the chemical structure and (μ)-gradient in the layers surrounding the convective core. Alterations to the μ-gradient become important for the evolution of stars along the MS as the convective core shrinks. As a consequence of the enhanced core mass and altered μ-gradient, internal quantities such as the Brunt– Vaïsälä frequency are necessarily altered as well. This can be seen in the right panel

18

Mcc/M

Xc =0 .60 0.4

0.2

0.0

Mcc/M

Xc =0 .30 0.4

0.2

0.0

Xc =0 .10 Mcc/M

Fig. 1.4 Fractional mass of the convective core and CBM zones as a function of stellar mass, evaluated at a core hydrogen content of X c = 0.6 (top), X c = 0.3 (middle), and X c = 0.1 (bottom) for models with 1.2 M ≤ M ≤ 25 M . Light grey regions denote convective zones, dark grey regions denote CBM zones. Hatched regions denote models with minimum CBM (i.e., f ov = 0.005) and non-hatched regions denote models with maximum CBM (i.e., f ov = 0.040). Dashed horizontal line at Mcc /M plotted for reference

1 Scientific Context

0.4

0.2

0.0 0.5

1.0

log (M/M )

of Fig. 1.1, which shows the Brunt–Vaïsälä frequency as a function of mass coordinate for the two cases in the left panel. The panels of Fig. 1.5 show how some key stellar quantities, including the asymptotic period spacing of gravity modes (introduced in the following Section) change over time for models with different amounts of internal mixing. These changes in internal structure manifest as changes in the surface properties of the star, such as the temperature, radius, and hence luminosity and surface gravity. The left panels of Fig. 1.5 show the corresponding changing surface properties to the altered internal quantities in the right panels, for tracks with two amounts of enhanced mixing for 3 and 6 M models (solid and dashed tracks, respectively). The top panel of Fig. 1.5 demonstrates that enhanced internal mixing directly corresponds to an enhanced core mass relative to a model with less internal mixing, at any given point in the MS. These two modifications to stellar structure imply that stars of the same initial mass and initial chemical composition can have a wide range of properties as they transition off the MS, at the point of core-hydrogen exhaustion, to the post-MS. In general, the consequences of this have not been robustly and consistently accounted for in evolutionary calculations spanning the entirety of a stars evolutionary trajectory. There have been studies which investigated the impact on asteroseismic observables of individual types of objects such as red-giants (Montalbán et al. 2013; Constantino et al. 2015), as well as the impact of CBM on nucleosynthetic yields (Davis et al.

1.4 Standard Stellar Evolution of Intermediate- to High-Mass Stars 0.5

0.0

0.5

19 0.0

0.6 0.6

Δlog L/L

0.2 0.2

0.0

0.0

0.3

0.03

0.2

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Δπ0/d

Δlog R/R

0.4

ΔMcc/M

0.4

0.00

0.0 0.5

0.0

Xc

0.5

0.0

Xc

Fig. 1.5 Comparative plots of stellar quantities with respect to a base model with a minimum amount of internal mixing as a function of fractional core-Hydrogen content. Solid lines denote tracks with M = 3 M , dashed lines denote tracks with M = 6 M , grey lines denote tracks with a factor 4 enhanced CBM and 100 enhanced REM, and black lines denote tracks with a factor 8 enhanced CBM and 10 000 enhanced REM. All quantities are shown as differences with respect to the base model

2019). Importantly, however, there has not yet been attempts to consistently calibrate enhanced internal mixing from the MS through the post-MS, and onto the final stages of evolution. Thus, the consequences of enhanced mixing, in whatever form, are so far not yet robustly and consistently propagated into fields that rely on predictions of late stages of stellar evolution. Although other phenomena such as near critical rotation, wind mass-loss, strong magnetic fields, or binary interaction can greatly divert stellar evolution, we do not consider these phenomena in this monograph. This is supported by the observation that most stars with estimated rotational velocities have modest rotation rates compared to their critical rate. Furthermore, there is a very low magnetic field detection rate in intermediate- to high-mass stars (Wade et al. 2019). This does not, however, exclude the occurrence of an internal magnetic field in these stars.

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1.5 Asteroseismology Asteroseismology is the method by which one can investigate the stellar interior by analysing and modelling the observed oscillation frequencies of a star. As with all of other forms of astrophysical data (excluding gravitational waves), asteroseismology relies on the light emitted at the surface of a star. However, instead of considering a single observation, asteroseismology requires time-series in which we make repeated measurements of the emitted light. The frequencies at which this light varies are finely tuned by the internal structure of the star. Much in the same manner as how the size, shape, and composition of a bell determines the frequency with which it rings, the size, shape, and composition of a star determines the frequencies with which it oscillates. In this way, by measuring and modelling these frequencies, we can learn about the interiors of stars to a degree that is otherwise impossible. It is asteroseismology which provides us with the most stringent means of calibrating the internal mixing processes discussed in the previous Sections. To this end, we briefly discuss the characteristics of stellar pulsations and asteroseismology which are pertinent to this monograph.

1.5.1 Stellar Oscillations In the previous Sections, we have been describing the equations and physics that constitute an equilibrium stellar model. Generally speaking, stars will retain an equilibrium configuration. However, some situations will arise in which a star is perturbed from this equilibrium configuration. In this situation, the star will tend towards equilibrium again, unless the perturbation is constantly driven. In this case, the star will fluctuate or oscillate around its equilibrium position. The frequency with which these oscillations occur can be described in terms of spherical harmonics, which arise as the result of applying perturbation analysis in spherical coordinates to a model resulting from Eqs. (1.1)–(1.5), or any other included physics. Assuming a non-rotating, non-magnetic, spherically symmetric star, the perturbations caused by an oscillation mode produce a displacement ξ to a mass element. This displacement vector ξ has components in the r , θ , and φ coordinates given by: ξr = an (r ) Y m (θ, φ) exp (−2πi f t) , ξθ = bn (r ) ξφ =

∂ m Y (θ, φ) exp (−2πi f t) , ∂θ

bn (r ) ∂ m Y (θ, φ) exp (−2πi f t) , sin θ ∂φ

1.5 Asteroseismology

21

where n, , and m are the quantum numbers which describe the radial order, angular degree, and azimuthal order of the mode. an and bn are the radial and horizontal amplitudes, respectively, and f and t are the frequency of the mode and the time, respectively. Y m (θ, φ) are the spherical harmonics given by:  2 + 1 ( − m)! m m n P (cos θ ) exp (imφ) , Y (θ, φ) = (−1) 4π ( + 1)!

where P m (cos θ ) are Legendre polynomials. The number n corresponds to the number of radial nodal surfaces within the star, i.e. those shells at a given r which do not move during an oscillation. The numbers and m are related to the number of nodal lines on the surface. In the case where m = 0, all nodal lines are lines of latitude. When |m| < , m nodal lines are lines of longitude, and − |m| nodal lines are lines of latitude. When |m| = , all nodal lines are lines of longitude. Mode geometries are classified into two categories. The first category consists of radial modes, where

= m = 0. These modes are mainly sensitive to the density stratification of stars. The second category is known as non-radial modes, where > 0. Non-radial modes probe different depths of the star depending on their exact geometry, and as such carry information pertaining to the physical conditions of these layers. As (n, , m) dictate the layers which a mode probes, identification of these numbers is essential for the interpretation of asteroseismic data. Observational determination of these numbers remains the most difficult aspect of asteroseismic analysis. As discussed below, observed patterns can give strong constraints on these numbers.

1.5.1.1

Pressure Modes

In addition to its geometry, a pulsation is characterised according to the dominant force acting to restore the perturbation. When ignoring the effects of rotation and magnetism, the two dominant restoring forces are pressure and buoyancy (gravity). Those pulsation modes which are dominantly restored by pressure are termed pressure or p-modes. P-modes are generally higher in frequency and have primarily radial motions (i.e., an  bn ) at their outer turning point. The depth to which a p-mode propagates depends on the pulsation geometry, such that the low-frequency, low-

modes propagate through the deep interior of the star, whereas the high-frequency, high- modes are more trapped to the surface. As they are pressure waves, p-modes are able to freely propagate through convective regions, providing probing power where g-modes cannot. When considering the case where the pulsation geometry follows the condition that n  , p-modes with geometries of equal and consecutive n have frequencies set according by Tassoul (1980) as: νn

 

1 = ν n + + + α − D0 ( + 1) , 2 4

(1.22)

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where α ∼ 1 depends of the surface properties of the star and D0 is sensitive to the conditions of the near core regions, in particular the sound speed gradient. The term ν is known as the large frequency separation, and is given by:  ν = νn+1, − νn, = 2

R

0

dr c(r )

−1 ,

(1.23)

where c(r ) is the sound speed at radius r . Patterns according to these relations have been observed in thousands of solar-like pulsating MS stars and tens of thousands of solar-like pulsating red-giant stars (Chaplin et al. 2011; Yu et al. 2018). In addition to enabling easy mode geometry determination, these patterns have also enabled the determination of stellar properties via scaling relations with the large frequency separation, using the Sun as a calibrator.

1.5.1.2

Gravity Modes

Those modes dominantly restored by buoyancy are termed gravity or g-modes. Gmodes are lower in frequency and have primarily tangential motions (i.e., bn  an ). Furthermore, these modes propagate deep in the interior of the star. As buoyancy is the restoring force for g-modes, they can by nature only propagate in regions where the Brunt–Vaïsälä frequency is positive. Therefore, g-modes are trapped inside regions which are convectively stable. In low-mass stars with a convective envelope, g-modes are restricted to propagating in the radiative interior, which often makes them undetectable. For intermediate- and high-mass stars with a convective core and radiative envelope, g-modes can only propagate outside of the core. Thus, in intermediateand high-mass stars, g-modes cannot provide information on the conditions within the core itself, but are highly sensitive to the near-core regions. As with p-modes, if we assume the same conditions that n  , g-modes of constant and consecutive n are expected to be equally spaced, in period (as opposed to p-modes being equally spaced in frequency). The expression governing this spacing in a chemically homogeneous, non-rotating, non-magnetic, single star is given by Tassoul (1980) as: 0 (1.24) Pn = √ (n + α ) ,

( + 1) where α is a constant whose value is determined by the conditions of the g-mode cavity. For a star with a convective core, α becomes independent of . The asymptotic period spacing value 0 is given by:  0 = 2π

2

r2

r1

N (r ) dr r

−1

,

(1.25)

1.5 Asteroseismology

23

where N (r ) is the Brunt–Vaïsälä frequency, and r1 and r2 are the boundaries of the g-mode cavity. In the case where a μ-gradient is present, deviations from uniformity will be introduced into the pattern as certain modes will become trapped (Miglio et al. 2008). Furthermore, in the presence of rotation, the denominator of Eq. (1.24) is replaced by the eigen-values found by solving the Laplace tidal equation, which contains the Coriolis acceleration. These eigen-values depend on the rotation rate and the mode geometry. According to this, rotation will asymmetrically shift the period values of two consecutive modes (Bouabid et al. 2013), where prograde modes will have decreased spacings and retrograde modes will have increased spacings with increasing mode period. Assuming the detection of at least one g-mode period spacing pattern, one can simultaneously estimate the near-core rotation rate and the asymptotic period spacing value 0 . Since 0 is sensitive to any phenomenon that alters the spatial distribution of the Brunt–Väisälä frequency N (r ), 0 can serve as a diagnostic of the near core region. In particular, given Eq. (1.10), 0 allows for the probing of the temperature and chemical gradients deep inside the star. The bottom right panel of Fig. 1.5 shows how 0 varies across the MS for a 3 and 6 M model for different amounts of internal mixing. Figures 1 and 4 in Mombarg et al. (2019) demonstrate the variation of 0 across the MS for models of different mass, metallicity, as well as different amounts of internal mixing.

1.5.2 Types of Pulsating Stars In the years following the launch of space-based photometric missions, it has become increasingly clear that most, if not all, stars are variable at some level. Pulsations excited via various mechanisms are present all across the HRD, as seen in Fig. 1.6. This enables asteroseismology along the MS, in evolved giant stars, as well as in stellar remnants (pulsating white dwarfs) and stripped helium-core burning stars (subdwarf OB stars). While it is a general goal of stellar astrophysics to calibrate theory across all of these types of stars using asteroseismology, we restrict our discussion here to those pulsating stars studied in this monograph. As seen in Fig. 1.6, there are numerous types of pulsating stars along the MS. At the low mass end, we find solar-like oscillators. These are sun-like stars with an extended convective envelope. The turbulent motions of the convection in these stars stochastically drive oscillations across a range of frequencies. Some of these oscillations have frequencies which coincide with the natural eigen-frequencies of the star, causing the star to resonantly pulsate. This is known as stochastic driving. Moving up in mass (and temperature) we arrive at γ Doradus variables. These are gmode pulsating mid-F (1.4 M ) to late-A MS (1.9 M ) stars exhibiting multi-periodic behaviour with pulsation periods between several hours to a few days. With the advent of near continuous space-based photometry, thousands of new γ Dor variables were discovered. Furthermore, these data sets resulted in the detection of non-uniform g-mode period-spacing patterns (Van Reeth et al. 2015a, b), the detection of r-mode pulsations (Van Reeth et al. 2016; Saio et al. 2018), and the determination of the near

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Fig. 1.6 Theoretical Hertzsprung-Russel Diagram indicating the various classes of pulsating stars and their respective restoring forces. Pulsating stars exhibiting p-modes indicated by ‘\\’ hatching and stars exhibiting g-modes indicated by ‘//’ hatching. Colour scales with Teff as indicated across the top of the plot. For a full explanation of this figure, we refer to Chap. 3 of Aerts et al. (2010). Figure from Papics (2013) (his Fig. 1.4). Figure reproduced with permission from the author

core rotation rate in hundreds of γ Dor stars (Van Reeth et al. 2016; Li et al. 2019, 2020). Next, we arrive at δ Sct stars, which consist of early-F to early-A type p-mode pulsating stars, exhibiting either mono- or multi-periodic behaviour with pulsation periods ranging from minutes to hours. The pulsations in these stars are known to be driven by the κ-mechanism operating in the second partial ionisation zone of helium (Aerts et al. 2010). These variables have been observed to obey a period-luminosity relation (Breger 1990; Ziaali et al. 2019), and exhibit amplitude variability on time scales as short as several years (Bowman et al. 2016). Furthermore, these stars have

1.5 Asteroseismology

25

been observed to pulsate well outside of the theoretically predicted instability strip (Bowman and Kurtz 2018). Space-based photometry also revealed a large population of hybrid γ Dor/δ Sct pulsating stars. The study of such stars has led to the deduction of the full rotation profile from the surface to core in a handful of objects, revealing near rigid-body rotation (Kurtz et al. 2014; Saio et al. 2015; Schmid et al. 2015; Murphy et al. 2016; Van Reeth et al. 2018; Li et al. 2020). Slowly Pulsating B (SPB) stars are the high-mass analogue of γ Dor variables. These stars exhibit multiple g-mode pulsations with periods between hours to several days and cover a large mass range from some 2.5 to 8 M . SPBs are theoretically predicted to pulsate via the κ-mechanism operating on the iron-group opacity bump deep in the stellar envelope near T ∼ 105 K (Moskalik and Dziembowski 1992; Dziembowski et al. 1993; Gautschy and Saio 1993). Although only a handful have been observed from space (Pápics et al. 2017; Buysschaert et al. 2018; Pedersen et al. 2019), detailed modelling efforts have revealed near core mixing (Moravveji et al. 2015, 2016), derived the near core rotation rates of 7 SPB stars observed by Kepler (Pápics et al. 2017), and revealed counter envelope to core rotation (Triana et al. 2015). Recently, Pedersen (2021) analysed some 20 newly discovered apparently single SPB stars and provided estimates for their masses, core masses, ages, and interior mixing properties. Finally, we have β Cep variables , which are early B to O-type stars which pulsate in one or multiple low-order p- and g-modes, with periods from hours to days. Pulsations in β Cep stars are driven by the κ mechanism operating on the same iron-group opacity bump as in SPBs (Dziembowski and Pamiatnykh 1993), which now occurs further out in the envelope as β Cep stars are more massive. Space based photometry of β Cep variables has also revealed stochastically excited pulsations (Belkacem et al. 2008; Degroote et al. 2010), and a large increase in the number of known β Cep stars (Pedersen et al. 2019; Burssens et al. 2019; Handler et al. 2019). Furthermore, space photometry has revealed several new hybrid SPB/β Cep pulsators (Pedersen et al. 2019; Burssens et al. 2019). Simulations of IGWs predict a non-negligible surface perturbation resulting in stochastic photometric and line-profile variability (Aerts and Rogers 2015; Edelmann et al. 2019). This signal is predicted in any star that has a convective core. Blomme et al. (2011) and Aerts et al. (2017) detected such a signal in four O-stars observed by CoRoT and K epler . Using a sample of OB stars observed by CoRoT, K 2, and TESS, (Bowman et al. 2019a, b) detected and characterised a low-frequency stochastic signal in over 100 stars in different metallicity regimes. In both cases, the authors demonstrated that the signal is consistent with that predicted by simulations of IGWs (Edelmann et al. 2019), and is inconsistent with the signature of surface granuation as scaled from solar-like stars.

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1.6 Multiple Stellar Systems More often than not, stars do not evolve alone, but in binaries, triple systems, or in clusters of up to millions of stars. It has been shown for field stars that the rate of stellar multiplicity increases from some 30% for solar-type stars to nearly 100% for massive O stars (Raghavan et al. 2010; Sana et al. 2014; Moe and Di Stefano 2017). In this scenario, it is a reasonably good approximation that the components of a first-generation multiple system begin to collapse from the same giant molecular cloud at approximately the same time. This implies that the components of a multiple system have the same age and initial chemical composition. This leads to an extremely valuable constraint in stellar modelling. Here, we briefly discuss binary stars, multiple systems, clusters, and the information that can be derived from observations of each type of system.

1.6.1 Binary Stars It is generally assumed that the two components of a binary system are formed simultaneously, such that they have the same age and initial chemical composition. The observational determination of atmospheric abundances for the components of several wide binary systems supports this latter assumption of equal initial chemical composition, within observational uncertainties (Desidera et al. 2004, 2006). Furthermore, studies which compare young pre-MS binaries from the field with groups of stars observed to be in the same star forming region conclude that the components of the pre-MS binaries have smaller differences in apparent age than the stars from a given star forming region (Kraus and Hillenbrand 2009; Duchêne and Kraus 2013). These studies corroborate theoretical studies which predict binaries to form simultaneously from a collapsing mollecular cloud (Tohline 2002; Baraffe and Chabrier 2010). There are, however, notable cases where observations of young, low-mass pre main-sequence systems reveals an age discrepancy of up to a few hundred thousands of years (Stassun et al. 2008; Gómez Maqueo Chew et al. 2012). Alternatively, it is also possible that binaries can be formed through capture events, where two previously unbound stars become gravitationally bound after a close passing of the two. However, the frequency of events is low in all but the densest stellar environments, where it is still more likely to result in a scatter event than a capture (Tohline 2002). In the case of a single star, an observed spectrum can provide information on the stellar effective temperature, surface gravity, and projected rotational velocity. In the case of a double-lined spectroscopic binary (SB2), where light of both stars can be detected simultaneously in a spectrum, we can additionally derive the orbital 2 , where M1 and M2 are the masses motion of the system and the mass ratio, q = M M1 of the primary and secondary components, respectively. One can also determine the projected binary separation, a sin i, where a is the semi-major axis and i is the inclination of the orbital plane with respect to the line of sight. However, without

1.6 Multiple Stellar Systems

27

an estimate of the inclination, we cannot derive the individual masses, radii, or the un-projected binary separation. In the case where the orbital-plane of the system is inclined such that the two stars eclipse one another, we can further obtain model independent estimates for the absolute masses and radii of both components to a precision of 1% (Torres et al. 2010). Finally, combining the estimates on radius and effective temperature, eclipsing binaries (EBs) provide independent measures of the system luminosity, and hence, its distance. Determination of the absolute masses and radii of an eclipsing binary requires the detailed modelling of the eclipses and radial velocities of the system. While modelling of the radial velocities of each component (without any proximity effects, such as the Rossiter-McLaughlin effect) can be done using only Kepler’s equations, eclipse modelling is more involved. This generally involves accounting for the potential nonsphericity of the two components, the limb-darkening across a non-uniform disk, as well as detecting if an eclipse is occurring, and how much surface is covered as a function of time. Normally, this is achieved using the Roche model, which describes the gravitational potential surrounding two point masses that are orbiting around a common centre of mass. Under the Roche description, the surfaces of each component are described along equipotential surfaces. By modelling the eclipses present in a light curve, one obtains an estimate of the inclination, and thus the masses and radii. While there are a wide variety of codes which produce binary models using this description, we restrict ourselves to using two within this monograph: (i) PHOEBE (Prša and Zwitter 2005; Prsa et al. 2011), and (ii) ellc (Maxted 2016). The configuration of a binary system is not static. There is a complex interplay of angular momentum exchange between the orbit and the components. This exchange of angular momentum constantly works to circularise, synchronise, and align the system. In general, a binary system is born with two components which are rotating either sub- or super-synchronously with respect to the orbit, and these components may have a rotation axis which is misaligned with the normal of the orbital plane. Furthermore, the orbit of these stars may be eccentric. Over the course of time, the system will tend towards being synchronised, circularised, and aligned due to angular momentum exchange. In an eccentric system, there is no such thing as synchronous rotation, since the orbital velocity changes throughout one period. Instead, a star is said to be pseudo-synchronous when the rotational velocity of the star matches its orbital velocity at periastron in an eccentric orbit. The system is expected to first become (pseudo-)synchronous, and then circularise. This process is much more efficient in the case where the two stars in a binary have a large convective envelope, where tidal friction is the strongest dissipative factor. Deduction of the rotation rates and eccentricities of binaries is an active field of research to scrutinise tidal theory.

1.6.2 Stellar Clusters Whereas binaries consist of two gravitationally bound stars (typically) formed from the same molecular cloud at the same time, clusters can consist of up to millions of

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such coeval stars. There are, however, complications to this picture. In older globular clusters, Hubble Space Telescope photometry commonly reveals the presence of a split main-sequence (Piotto et al. 2015; Soto et al. 2017; Bastian and Lardo 2018). Furthermore, these multiple main-sequences have been linked to abundance differences (Milone 2015), suggesting multiple populations. Numerous mechanisms have been proposed to explain these phenomena, such as multiple episodes of starburst formation, ongoing star formation, binary interaction products, and periods of abundance enrichment from either asymptotic giant branch stars, massive stars, or binary interactions. However, no single mechanism is able explain the observations (Bastian and Lardo 2018). Despite the observation of an extended main-sequence in young massive clusters, there is no evidence to suggest an age spread caused by multiple stellar populations in these clusters (Portegies Zwart et al. 2010; Bastian and de Mink 2009; Niederhofer et al. 2015b). Unlike the case of globular clusters, the observations of young massive clusters can be reasonably well explained with single populations of stars and the consequences of stellar evolution (Bastian and Silva-Villa 2013). This phenomenon is the focus of Chap. 6 and will be discussed in more detail there. Analogous to the case of binary stars, modelling clusters can yield a wealth of information. While we model eclipses for EBs, we model the colour-magnitude diagram of clusters. The positioning of the stars within a colour-magnitude diagram of a cluster reveals the age, distance, and potentially the metallicity of the cluster. Assuming a single population, the age of the cluster determines the highest-mass star which can still exist on the MS. Thus, by modelling the point at which the main-sequence “turns-off” to the giant branch, we obtain the star of maximum mass still in the core-hydrogen burning. The nuclear time scale of this star is a good estimate of the age of that cluster. As will be discussed later, any deviations from a clear singular main-sequence can reveal the consequences of stellar evolution, allowing the opportunity to calibrate theories of stellar structure and evolution in large populations. This, of course is only possible in the case of clusters which contain a single population of stars, which is the case for young massive clusters. Since young massive clusters are of the order of a few millions of years to hundreds of millions of years old, the turn off stars of these clusters have spectral types ranging from late O- to late B-type stars, making them of interest for our work.

1.7 The Mass Discrepancy Binary stars have been exploited to investigate the consequences of enhanced internal mixing and hence and enhanced core mass for several decades (see e.g. Zahn 1977; Roxburgh 1978; Maeder and Meynet 1987; Andersen et al. 1990; Zahn 1991, for some early works). Herrero et al. (1992) first reported the systematic discrepancy between stellar masses determined through empirical spectral relations and those determined through fitting evolutionary tracks to spectroscopic parameters. In the years following, numerous studies have identified this discrepancy between

1.7 The Mass Discrepancy

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the highly precise dynamical masses determined through binary modelling and the masses of evolutionary models fit to spectroscopic parameters (Guinan et al. 2000; Ribas et al. 2000; Tkachenko et al. 2014; Claret and Torres 2019). Although present in single stars, this so-called mass discrepancy is revealed only when independent estimates of masses, radii, and/or ages become available, as is the case with eclipsing binary stars. Since its initial quantification, this mass discrepancy in binary stars has become the centrepiece for intense debate of the importance of different forms of internal mixing in stellar evolutionary theory (Ribas et al. 2000; Stancliffe et al. 2015; Constantino and Baraffe 2018; Claret and Torres 2018). The mixing induced by rapid rotation has been suggested as a means of addressing the mass discrepancy (Torres et al. 2010; Brott et al. 2011a, b; Schneider et al. 2014; Ekström et al. 2018). Using a sample of 18 binary systems (36 individual components) Schneider et al. (2014) demonstrated that mixing induced by rapid rotation can reasonably reproduce observations without recovering the mass discrepancy. However, this sample was heterogeneously collected from the literature, and requires stars with near-critical rotation to explain the observations of some of the stars in the sample. Furthermore, although rapid rotation has seen success in reproducing observed temperatures, surface gravities, and masses, it still cannot explain observed abundances of slowly rotating stars (Aerts et al. 2014; Pavlovski et al. 2018; Abdul-Masih et al. 2019). The use of enhanced internal mixing in the form of overshooting to resolve the discrepancy between dynamical and evolutionary masses has been reported over a large range of masses (Claret and Gimenez 1991; Schroder et al. 1997; Pols et al. 1997; Iwamoto and Saio 1999; Ribas et al. 2000; Torres et al. 2010; Tkachenko et al. 2014; Claret and Torres 2018). There is a lack of consensus, however, with different studies claiming that they can reliably reproduce observations of stars with a variety of masses and evolutionary progressions with and without overshooting (Andersen et al. 1990; Schroder et al. 1997; Pols et al. 1997; Claret 2007; Stancliffe et al. 2015; Claret and Torres 2016, 2017; Higl and Weiss 2017; Constantino and Baraffe 2018). It should be noted that not all of these studies use the same implementation of ‘overshooting’. Amongst these studies, convective penetration according to Eq. (1.18), exponential diffusive convective overshooting according to Eq. (1.19) as well as step convective overshooting (overshooting with the form of Eq. (1.18) but with the radiative temperature gradient) have all been used. In a series of studies, Claret and Torres (2016, 2017, 2018, 2019) (hereafter CT16, CT17, CT18, and CT19 respectively) have exploited a sample of well-detached, double-lined EBs in order to investigate the potential mass dependence of overshooting as theoretically predicted by Roxburgh (1992). In their original formulation, Roxburgh (1992) demonstrate that the overall extent of an overshooting region is proportional to and limited by the total energy of the stellar core, and hence the mass of the core and the overall mass of the star. The authors computed several tracks at the determined dynamical mass with varied overshooting (a step-overshooting prescription: αov in CT16, and a diffusive exponential description: f ov in CT17 and CT18) and αMLT values, then fit the tracks individually according to their respective observed quantities allowing for a 5% difference in age between the two components,

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as well as a difference in initial chemical composition between the two components. Their results revealed an apparent mass dependence of overshooting from 1.2 to 2 M with no significant mass dependence from 2 to 4 M . In contrast, given modern observational sensitivities and statistical methodologies, several studies have argued that the efficiency of a particular overshooting prescription cannot be uniquely constrained (Valle et al. 2016, 2017, 2018; Higl and Weiss 2017; Constantino and Baraffe 2018). Theoretically, Valle et al. (2016) found that models including some implementation of overshooting cannot uniquely be constrained given a set of observations with reasonable uncertainties. This result was corroborated by subsequent studies by Valle et al. (2018) and Constantino and Baraffe (2018) who demonstrated that global stellar quantities, accompanied by modern uncertainties, do not have sufficient discriminating power to constrain overshooting efficiencies when compared to grids of stellar evolutionary models. In response to Constantino and Baraffe (2018), CT19 then reported that their measured overshooting values agree with those predicted by Roxburgh (1992). Aside from the difficulties in comparing different parameterisations of overshooting, all of these studies make different assumptions on the binary nature of the studied systems and enforce different assumptions in their modelling. Whereas some studies (c.f. Schneider et al. 2014; Tkachenko et al. 2014) consider the components of a binary individually and evaluate the ages of the two components as an a-posteriori check, other studies, such as the ones by Claret and Torres (2016, 2017, 2018, 2019), allow for both different initial chemical compositions and up to a 5% age difference between the two components of a binary. Although they enforce the assumptions of common age and initial chemical composition, Constantino and Baraffe (2018) also enforce that both components of a binary have the same amount of overshooting. Furthermore, none of these studies correctly account for model degeneracies when determining the best fit parameters. It is well known from g-mode asteroseismology that the core mass, via CBM, is not only degenerate and correlated with the mass of the star, but also with metallicity and central hydrogen content (or more directly, the age) (Moravveji et al. 2015, 2016; Schmid and Aerts 2016; Buysschaert et al. 2018). When dealing with binaries, multiple systems, and clusters, however, aspects of these degeneracies are mitigated through the assumption of equal age and initial chemical composition, if applied. To this end, allowing for the components of a binary to have different ages and initial chemical compositions re-introduces this degeneracy, which will influence any conclusions drawn from results based on such a method. Moreover, estimation of f ov (DCBM ) also requires one to take into account the dependencies of the choice of nuclear network, chemical mixture, opacity tables, atomic diffusion (e.g. Aerts et al. 2018). Considering these factors, a comparative study of the literature cannot narrow down a possible unifying explanation for the mass discrepancy. To achieve this, we require a methodology which not only accounts for model degeneracies, but that enforces information to mitigate these degeneracies when possible, and incorporates independent information when available. In the following Chapters, we develop and apply such a methodology.

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Chapter 2

Stellar Evolution Tracks, Isochrones, and Isochrone-Clouds

This Chapter is based in part on: 1. Binary asteroseismic modelling: isochrone-cloud methodology and application to Kepler gravity mode pulsators6 JOHNSTON, C.; Tkachenko, A.; Aerts, C.; Molenberghs, G.; Bowman, D. M.; Pedersen, M. G.; Buysschaert, B.; Pápics, P. I. Monthly Notices of the Royal Astronomical Society, Vol. 482, 1231J, 15 pp., (2019) 2. Modelling of the B-type binaries CW Cephei and U Ophiuchi. A critical view on dynamical masses, core boundary mixing, and core mass JOHNSTON, C.; Pavlovski, K.; Tkachenko, A. Astronomy & Astrophysics, Vol. 628, A25, 16 pp. (2019) 3. Isochrone-cloud fitting of the extended main-sequence turn-off of young clusters JOHNSTON, C.; Aerts, C.; Pedersen, M. G.; Bastian, N. Astronomy & Astrophysics, Vol. 632, A74, 11 pp. (2019) 4. Forward Asteroseismic Modeling of Stars with a Convective Core from Gravity-mode Oscillations: Parameter Estimation and Stellar Model Selection Aerts, C.; Molenberghs, G.; Michielsen, M.; Pedersen, M. G.; Björklund, R.; JOHNSTON, C.; Mombarg, J. S. G.; Bowman, D. M.; Buysschaert, B.; Pápics, P. I.; Sekaran, S.; Sundqvist, J. O.; Tkachenko, A.; Truyaert, K.; Van Reeth, T.; Vermeyen, E. The Astrophysical Journal Supplement Series, Vol. 237, 15, 31 pp. (2018) Author Contribution: C. Johnston tested and benchmarked the model numerics (papers 1, 2, 3, 4), calculated the stellar model grid (1, 2, 3), calculated the isochrones (1, 2, 3), developed the concept for and calculated the isochrone-clouds (1, 2, 3), and with the help of G. Molenberghs, C. Aerts, and B. Buysschaert, developed the modelling scheme and error calculations (1, 2, 3). The modelling scheme based on the Mahalanobis distance and its application to gravity mode pulsating stars was introduced and developed by (4). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0_2

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As outlined in Chap. 1, the goal of this monograph is to investigate the impact of internal mixing mechanisms on stellar interiors and stellar evolution. This is achieved via matching observed quantities to those of grids of stellar models, or transformations thereof, such as isochrones and isochrone-clouds. This chapter discusses the stellar structure and evolution models, isochrones, and isochrone-clouds, the motivations for the choice of parameter ranges and input physics, as well as the forward modelling scheme used throughout this monograph.

2.1 Stellar Structure and Evolution Tracks While there exist several well established stellar structure and evolution codes, the work in this monograph is based on stellar models calculated using the MESA (Modules for Experiments in Stellar Astrophysics) code described in Paxton et al. (2011, 2013, 2015, 2018, 2019). As the mesa code is exhaustively detailed in this series of instrument papers, we only briefly describe the code and its physics and refer the reader to these papers for a detailed description. All of the models used in this monograph were calculated using mesa version r-10398, using a choice of numerical options and input physics optimised for g-mode asteroseismology, cf. the inlist in Appendix A.

2.1.1 Initial Models and Numerical Controls mesa provides both two options of starting points for their tracks, (i) a pre-computed library of ZAMS models from which a new ZAMS model can be computed by interpolation, and (ii) calculation of a full pre-MS track for the chosen input physics by the user. Stellar structure and evolution models numerically solve the set of equations introduced in Chap. 1 at a series of time points which are separated by some dynamically determined time-step. mesa simultaneously solves the structure and composition equations, starting from the surface and shooting inwards from the surface boundary condition to the boundary conditions at the core along a discretised chosen mesh grid. Although there are dozens of options which contribute to the overall number of mesh grid cells, we focus on three classes of control variables: (i) global resolution, (ii) local resolution, (iii) derivative based resolution. Globally, the resolution is set by a resolution coefficient, which increases the resolution for smaller values of this variable, and a maximum number of mesh points allowed globally, which we set to be relatively large (see mesh_delta_coeff in Appendix A) given the numerical precision demanded by asteroseismic analyses. Due to the sensitivity of g-modes to the inner regions of the star, and most importantly the near core-regions, we also impose an additional locally increased resolution around any nuclear burning or

2.1 Stellar Structure and Evolution Tracks

37

non-nuclear burning convective zone. Additionally, due to the increased mixing and impact on the local chemical gradient, we impose an increased resolution in any convective overshooting region. Finally, as the frequencies of g-modes are determined by the Brunt–Vaïsälä frequency, N (r ), we impose an increased mesh resolution at any location in the star where the derivative of the pressure, temperature, density (mass), temperature gradient, or chemical gradient exceeds a given value. Furthermore, we ensure that the controls are set such that the model achieves the condition of hydrostatic equilibrium to a relative numerical precision of 10−9 at every time-step in the track, including the sub- and red-giant phases of evolution. The choice of a time-step which is both adequately large to ensure efficient calculation and adequately small to ensure that there is not too large of a change in structure between two given times is a complex problem to address. mesa has two main options when considering time-steps. The first is a set of minimum and maximum time-steps allowed, and the second is a variable which considers the variation in internal structure from one step to the next. For our purposes, we choose to enforce a minimum time-step of 104 years, and unitless variation tolerance of 10−5 (see varcontrol_target in Appendix A). This ensures a rapid progression along phases of central nuclear burning, and sufficiently small time-steps at phases of rapid evolution (i.e. pre- and post-MS) such that the evolutionary track varies smoothly and does not contain discontinuities. In addition to these choices which actively dictate the evolution of the model and its numerical tolerances, we also make explicit choices to not superficially smooth the density or composition profiles, or re-interpolate the hydrogen and metal profiles for opacity calculations, as these would irretrievably alter the Brunt–Vaïsälä frequency.

2.1.2 Internal Mixing Processes As is standard in many stellar structure and evolution codes, mesa treats chemical mixing processes in the diffusive approximation, with each process having its own mixing coefficient. Thus, the overall chemical mixing process in a stellar model is then the sum of the individual mixing phenomena in the regions inside the star wherever that mechanism is active (see Eqs. 1.16 and 1.17). We point out that MESA’s diffusive approach is different from codes that adopt an advective numerical scheme for the treatment of rotation and its consequences for mixing and angular momentum transport (such as the Geneva/STAREVOL or CESTAM codes, cf. Ekström et al. 2012; Marques et al. 2013, respectively). The use of the diffusive approximation, however, leads to a degeneracy amongst the effects of different mixing mechanisms, such as rotationally induced mixing and convective core overshooting or any other radiative envelope mixing mechanisms.

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2.1.2.1

2 Stellar Evolution Tracks, Isochrones, and Isochrone-Clouds

Convection and Convective-Boundary Determination

As a default, mesa uses the mixing length theory (MLT, as described in Sect. 1.2.1) to describe regions which are unstable against convection. To this end, the mixing efficiency in convective regions within a stellar model are approximated by a large mixing coefficient. One shortcoming of the MLT description involves the determination of the convective boundary, which should occur where the radiative and adiabatic temperature gradients are equal (i.e. ∇rad = ∇ad ). Following revision r-10000, mesa enables a “predictive mixing” scheme which allows for the expansion of the convective region until it satisfies the condition ∇rad = ∇ad on both sides of the convective boundary. Building on the results of Gabriel et al. (2014), Paxton et al. (2018) show that when this method is not used, the convective core growth along the MS in intermediate mass stars is artificially hampered. As we are concerned with the mass of the convective core of stars on the MS, robust determination of the convective core boundary is required for our science goals, and we therefore use this predictive mixing scheme in our calculations. Additionally, we use the Ledoux criterion for convective stability, as given by Eq. (1.8).

2.1.2.2

Chemical Mixing Profiles

The ability to include a convective overshooting profile is built in to mesa by default. This profile comes in two forms, (i) step overshooting (according to Eq. 1.18), and (ii) exponential overshooting, (according to Eq. 1.19). Both of these profiles are treated as diffusive and adopt the radiative temperature gradient in the CBM region. One can integrate non-standard physics in stellar model calculations using mesa. We take advantage of this option to include two physical mechanisms not standardly included in mesa. The first is the chemical mixing profile induced by IGWs in the radiative zones of stellar models:   DIGW (r ) = Dext ρ (r )−1/2 , for r > renv ,

(2.1)

as implemented by (Pedersen et al. 2018), based on hydro-dynamical simulations by Rogers and McElwaine (2017). This profile has one free parameter, Dext , which is the value of the mixing profile at the convective core boundary. From this point, the efficiency of the mixing scales with the inverse of the square root of the density, ρ. The second non-standard physics included is convective penetration, as discussed in Chap. 1. We adopt the implementation discussed by Michielsen et al. (2019) (their ‘extended convective penetration profile’). In short, this profile enforces the adiabatic temperature gradient (as defined in Eq. 1.6) in the CBM zone, effectively enhancing the mass of the convective core over this fully mixed region. For this, we assume a step profile according to Eq. (1.18). In addition to convective overshooting, penetration, and rotational mixing, the near core region of stars with a shrinking convective core are expected to experience semiconvective mixing on a thermal time scale. As stated in Chap. 1, this occurs

2.1 Stellar Structure and Evolution Tracks

39

when a layer is stable according to the Ledoux criterion, but unstable according to the Schwarzschild criterion, due to the chemical gradient left by the receding core. Including this in our CBM profile would again introduce another free parameter. Furthermore, it has been shown that increased overshooting efficiency leads to weaker semiconvection, as overshooting reduces the chemical gradient required for semiconvection (Ding and Li 2014). To avoid unnecessary degeneracies, we do not consider semiconvection.

2.1.3 Input Physics The choice of input physics for the grid of stellar models used in this work is informed by the science goal, observational constraints, degeneracies between input parameters, and the ability to estimate a set of optimal parameters through forward modelling. This, in effect leads us to some minimum vector of free and fixed parameters  to be considered for model computations. This vector will consist of parameters that vary, θ , and parameters which are fixed ψ, such that  is those varied parameters, θ , given the fixed parameters, ψ:  =< θ | ψ > .

(2.2)

In order to establish a benchmark for comparative asteroseismic analysis, Aerts et al. (2018) investigated which input physics had the most consequence on asteroseismic analysis in the case of g-modes in stars with a convective core. The authors justify a hierarchy of which aspects of the input physics are most important to vary in stellar structure and evolution calculations, and which can be safely fixed. Following this study, we discuss and motivate our choices of input physics for .

2.1.3.1

Adjustable Free Parameters

Fundamentally, the initial mass and initial chemical composition lead to the four basic parameters of a stellar model which determine its evolution. The chemical composition of a stellar model is determined by the mass fractions X+Y+Z= 1, where X is the hydrogen mass fraction, Y the helium mass fraction, and Z the metal mass fraction. Since this relation is deterministic, we only need to vary two of the parameters to determine a unique initial chemical composition. As demonstrated in Fig. 2.1, there exists a well known degeneracy in stellar models between the initial mass Mini and the metal mass fraction Z, such that tracks with a lower value of Z mimic evolutionary tracks of higher Mini (with a corresponding higher Z) (Moravveji et al. 2015; Johnston et al. 2019). This degeneracy also occurs when Y is varied, as both amount to a change in the mean molecular weight of the star. To avoid this degeneracy and keeping in mind the types of stars under study, we fix Y = 0.276 to the cosmic B-star standard as found by Przybilla et al. (2008), and vary Z, such that we solve for X.

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Fig. 2.1 Evolutionary tracks for 6 M and 8 M for different values of initial metallicity (different colours) and amounts of internal mixing (dashed vs solid lines)

4.0

log L/L

3.8

3.6 Dmix 8M

3.4

3.2

Z 6M

3.0 4.45

4.40

4.35

4.30

4.25

4.20

4.15

log Teff

Since we are interested in the impact of internal mixing on evolutionary tracks, we allow the mixing profiles in our models to vary according to the free parameters in θ . As discussed in Chap. 1, the overall efficiency of the chemical mixing outside the convective core, Dmix (r > rcc ) is determined by numerous mixing mechanism acting together to produce a single chemical mixing profile, as is seen from examining Eq. (1.16). Furthermore, all mixing mechanisms in 1-D stellar structure and evolution codes are represented by parameterised functions, which introduces a large number of degenerate free parameters. Considering these degeneracies, we make the pragmatic assumption that internal mixing can be approximated by two parameterised mechanisms, one active in the CBM region and one active in the REM region. For the CBM region, we adopt either a convective penetration or convective overshooting description following Eqs. (1.18) or (1.19), respectively. For the REM region, we assume the mixing profile induced by IGWs, following Eq. (2.1). This approach allows us to remain agnostic to the actual causes of mixing, which have uncalibrated parameterised implementations, and instead consider the consequences of the mixing, such as enhanced core masses, increased luminosities, and altered μ-gradients. Furthermore, this allows us to parameterise internal mixing with only two free parameters, f ov (or αpen ) as f CBM and Dext as DREM . Although not explicitly set, the age along the stellar track τ , is required to be considered as a varied parameter in our scheme. The age implicitly follows from the calculations given a set of input parameters, and directly represents the time evolution of a stellar model. Including age, we arrive at the minimum vector of varied parameters: θ = Mini , Z , αMLT , f CBM , DREM , τ ,

2.1 Stellar Structure and Evolution Tracks

41

Table 2.1 The varied input parameters, their bounds, and step sizes for our grid of 100 224 stellar models. The initial mass of the stellar models was unevenly sampled depending on the mass range Unit Lower Upper Step Z ini Mini f CBM ( f ov /αpen ) log DREM /cm2 s−1 αMLT

– M – [dex] –

0.006 1.2 0.005 (0.05) 0 1.8

0.018 25 0.04 (0.4) 6 2.2

0.004 * 0.005 (0.05) 0.5 0.2

where f CBM is parameterised as f ov in the grid where diffusive exponential overshooting is used or as αpen in the grid where convective penetration is used. The ranges and steps used for the varied parameters in the grid of stellar models computed for our research are listed in Table 2.1. At the lower mass ranges of our grid, the parameter space in the HRD is densely populated, requiring a higher resolution in mass than at higher masses. As such, we use different step sizes in mass for different mass ranges. Between 1.2 and 2 M , we use M = 0.05 M , from 2–5 M , we use M = 0.1 M , from 5–10 M , we use M = 0.25 M , from 10–15 M , we use M = 0.5 M , and from 15–25 M , we use M = 1 M . The limits for DCBM (parameterised as f ov ) were taken from asteroseismic studies which estimated f ov in g-mode pulsating BAF-stars (Briquet et al. 2007; Moravveji et al. 2015, 2016; Schmid and Aerts 2016; Buysschaert et al. 2018). This range is compatible with those values obtained by studies which fit the components of well-detached eclipsing binaries to isochrones to estimate the amount of f ov or αpen required to obtain a fit. The range of DREM is limited to those values found to be appropriate for g-mode pulsating stars by Pedersen et al. (2018). A tremendous amount of work has gone into calibrating αMLT in stars with extended convective envelopes (Joyce and Chaboyer 2018), with a commonly accepted value for the Sun being αMLT = 1.8. Asteroseismic investigations of SPB stars have adopted higher values of αMLT = 2.0 (Moravveji et al. 2016). We vary αMLT to cover this range used in the literature, acknowledging that for stars without an appreciable convective layer in the envelope, the change across this range in αMLT is un-resolvable by current diagnostics for the mass range considered in this monograph.

2.1.4 Fixed Parameters As demonstrated by Aerts et al. (2018), choices of ‘fixed’ physics, such as the chemical mixture, reaction rate network, opacities, boundary conditions, EOS, etc., have a large impact on the outcome of seismic modelling. Our choice of input physics follows from the results of that study, which was specifically dedicated to intermediateand high-mass stars in the MS phase of evolution. We compute all mesa tracks

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from the beginning of pre-MS evolution, and include the full extension of the CNO nuclear reaction network (i.e., “pp_cno_extras_o18_ne22.net”). We make this choice as the other, less complete CNO network options do not include reactions involved in the branches of the CNO cycle that are only activated at higher temperatures. Furthermore, we set the surface boundary condition to adopt a simple photosphere and exclude microsopic atomic diffusion. As we include a global REM mechanism in the form Eq. (2.1) which acts on short time scales, we assume that the effects of microscopic atomic diffusion are washed away before having any consequence to the stars’ evolution. While we calculate models from the Hayashi track onward, we only include CBM and REM once the star has entered the MS, which we define as when the luminosity generated by nuclear burning makes up at least 99% of the total luminosity. Our models assume a uniform initial chemical composition where the helium fraction is fixed to the cosmic B-star standard (Y = 0.276 Przybilla et al. 2008; Nieva and Przybilla 2012). We fix the relative element fractions for the metals to those of the Sun, as determined by Asplund et al. (2009) but updated with reference to the cosmic B-star standard (Przybilla et al. 2008). We adopt the mesa defaults for the EOS and use OP opacity Table (Seaton 2005). Mass-loss is not included in models below 15 M . However, in models with Mini ≥ 15 M , we use the Vink wind scheme (Vink et al. 2000; Vinket al. 2001) with a scaling factor ξV ink = 0.3 (Vink et al. 2011; Puls et al. 2015). Our models model convection according the (Cox and Giuli 1968) MLT formulation. Additionally, convective stability is calculated following the Ledoux criterion. We do not consider semiconvective mixing (Langer et al. 1985), as this falls under the category of CBM. Aerts et al. (2018) demonstrate that in cases where a star is rotating below 50% its critical rate, non-rotating models are justified for asteroseismic modelling. As such, we calculate non-rotating spherically symmetric equilibrium models. These choices for fixed input physics are incorporated in the vector ψ from Eq. (2.2). Furthermore, the assumptions that our models do not include other physics such as a magnetic field or the effects of a close binary companion are included in this vector. Such assumptions can have differing degrees of impact, depending on the situation. In the case of binarity, for example, if mass transfer occurs, a model of mass M after mass transfer will have a systematically different age for a fixed combination of Teff and log g compared to a model of the same mass but not having undergone mass transfer. While this clearly impacts the estimated age, it is yet to be quantified how such events would impact estimated pulsation frequencies. To this end, ψ in effect not only includes our explicit and highly simplified assumptions, but must also be seen as encapsulating systematic uncertainty due to any missing or unknown physics. It is then clear that the dimension of parameters that enter into ψ as fixed values is tremendously high if the aim is to represent a real star by a stellar model relying on it. For the remainder of the manuscript, we consider ψ to be implicitly contained in .

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43

2.1.5 Convective Core Mass At several points in this monograph, we discuss the inference of the convective core mass from stellar evolutionary models. We adopt the definition used by MESA where the convective core mass is the mass contained within the Schwarzschild boundary. We choose to use the Schwarzschild boundary and not include the mass in the CBM region for several reasons. First, the core mass will already be enhanced by the presence of CBM as it transports material into the core. Second, including the mass in the CBM region would make our convective core mass estimates dependent on the CBM profile used. Although the inferred core masses are still ‘model dependent’, any difference in CBM profile simply reduces to a scaling issue, rather than introducing a dependence on the morphology of the profile as well.

2.1.6 Computational Requirements Calculating a grid of 100 224 stellar evolutionary model tracks as described above requires considerable computational resources. Considering our science goals, and in order to reduce the computational demands, we split the calculations into two categories. First, we calculate the pre-MS models without either CBM or REM, resulting in a two orders of magnitude decrease in the number of model tracks to be computed (N = 1044). For each of the 100 224 model tracks to be computed along the MS, we load the appropriate pre-MS model and switch on the mixing profiles according to the parameter values of . To compute these models, we applied for competitive time on the Flemish Tier-1 high-performance computing system BreNIAC (PI: Johnston). We were awarded the full 1092 node-days requested for our project.

2.2 Isochrone Construction 2.2.1 Definition An isochrone is a function in a grid of stellar models connecting all points in the grid which have the same age τ . Traditionally, an isochrone at the desired age τ is constructed via interpolation in a grid of evolutionary tracks which all have the same fixed input physics ψ, and vector of free parameters θ (see Eq. 2.2), to a desired age, τ . To define a unique function where a single effective temperature corresponds to single surface gravity, luminosity, mass, etc., we consider the vector: φi j = ϑi , τ j | ψ

(2.3)

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2 Stellar Evolution Tracks, Isochrones, and Isochrone-Clouds

to consist of the i-th permutation of the possible combinations of the vector ϑ = Z ini , f CBM , DREM , interpolated to the age τ j . We do not consider the mass in the vector ϑ because mass is used as the independent variable in the construction of the isochrone φi j .

2.2.2 Construction Constructing isochrones using simple linear interpolation in a grid of evolutionary tracks can result in under-sampled or altogether missed phases of rapid evolution. To avoid this, we follow the two-step isochrone calculation scheme described by Dotter et al. (2017), briefly detailed below. 1. We identify the main evolutionary phases in every evolutionary track. These phases include the start of the Pre-MS (PMS) to the zero-age MS (ZAMS), the ZAMS to the middle-age MS (MAMS), the MAMS to the TAMS, and the TAMS to the point of core helium ignition (RGBhb). Within any two main evolutionary phases, we identify n equidistantly spaced points, where the ‘distance’ between any two points is determined by a weighting function that accounts for the change in several surface and internal stellar quantities. We create interpolate equivalent evolutionary phase (EEP) tracks by interpolating the original evolutionary tracks onto these points via cubic piece-wise spline interpolation. 2. We create a monotonic mass-age relationship for each nth EEP point across all tracks. An isochrone is then constructed by using mass as an independent variable to interpolate in any stellar quantity we want to include.

2.2.3 Isochrone Fitting Isochrone fitting is standard practice in the fields of binary, or multiple star systems, and both open and globular clusters (Torres et al. 2010; Portegies Zwart et al. 2010; Bastian and Lardo 2018). Isochrone fitting is justified for these systems / populations under the assumption that binaries, multiple systems, and clusters are formed at the same time from the same molecular gas cloud. As discussed in Sect. 1.6, such an assumption is both theoretically and observationally supported in the case of binaries. Although this becomes more complicated for globular clusters, young massive clusters are still thought to consist of a single population of stars, thus meeting the requirement of equal age and initial chemical composition. Under this assumption, we can make two very powerful assertions that the stars are of the same age and have the same initial composition, thus greatly reducing the dimensionality of the grids used for modelling. In the case of binary stars, even stronger constraints are made possible through radial velocity and eclipse modelling, which can determine

2.2 Isochrone Construction

45

the fundamental parameters of the components to a precision of the order of 1% (Torres et al. 2010). The isochrones used in this classical type of modelling for stellar ages are commonly constructed from grids with a fixed amount of internal mixing, thus implicitly enforcing that all stars fit to the isochrone have the same amount of internal mixing. Numerous asteroseismic and binary modelling studies have shown that stars exhibit a wide range of internal mixing values. Figure 2.2 shows two isochrones of the same age calculated for different ϑi where f CBM = 0.005 and DREM = 1 cm2 s−1 (right-most isochrone) and f CBM = 0.04 and DREM = 10 000 cm2 s−1 (left-most isochrone). Comparing these isochrones illustrates how fitting along isochrones constructed for a single amount of internal mixing can artificially limit the allowed solution space.

2.3 Isochrone-Cloud Construction 2.3.1 Definition Johnston et al. (2019) introduced the concept of isochrone-clouds to synergistically combine Dmix (r ) profiles calibrated by asteroseismology with isochrone fitting of binary stars, multiple star systems, or clusters, all of which may or may not host pulsating components. Whereas an isochrone is the function interpolated in a grid of models with a given ϑi , given by the vector φi j , an isochrone-cloud is the region of parameter space covered at an age τ j bounded by all i-isochrones such that: φj =



φi j .

(2.4)

i

This effectively accounts for the parameter space otherwise excluded when only considering isochrones of a given ϑi . An example of a 50 Myr isochrone cloud is seen as the grey shaded region in Fig. 2.2. We note that an isochrone-cloud is always bounded by the two limiting isochrones which are constructed from models with the minimum and maximum amounts of internal mixing at that age. The grid of isochrone-clouds calculated here covers birth masses of M ∈ [1.2, 25.0] M , over three metallicity regimes Z = 0.006, 0.010, 0.014, and considers a wide range of CBM and REM efficiencies. The particular CBM (overshooting) efficiency is allowed to range between f CBM ∈ [0.005, 0.040], while we consider REM diffusivities with a base efficiency in DREM ∈ [0, 10 000] cm2 s−1 . These limits are derived from studies which performed asteroseismic modelling of single, isolated gravity mode pulsating stars observed by space-based photometric missions (see Briquet et al. 2007; Moravveji et al. 2015, 2016; Schmid and Aerts 2016; Buysschaert et al. 2018; Hendriks and Aerts 2019; Mombarg et al. 2019). See also Aerts (2021) for a summary of the properties of macroscopic mixing as deduced from asteroseismology.

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Fig. 2.2 Example evolutionary tracks for stellar models with 3–7 M for the minimum and maximum amounts of mixing in our grid, in solid and dashed grey, respectively. 50 Myr isochrones calculated from tracks with the most mixing (left) and least mixing (right) are shown in black. An example 50 Myr isochrone-cloud is denoted by the grey shaded region

By adopting this wide range of mixing properties, we allow for the case that stars of the same birth mass and metallicity can evolve differently due to a variety of mixing properties. This reflects, e.g., that some stars may have efficient wave mixing while others not, or some have effective rotational mixing and others less so, some may experience shear instabilities while others not, etc. Given that we have no understanding of any possible nonlinear interactions between these different individual physical phenomena, it makes sense to allow for very different levels of mixing inside stars of the same mass and metallicity because they can be born with very different rotational and magnetic properties. A given isochrone cloud is always constructed for a single metallicity, as to uphold the assumption that the components of a binary or cluster have the same initial chemical composition.

2.4 Forward Modelling Scheme Generally speaking, the forward problem in astrophysical modelling is defined in the following manner, adopting vector notations: Astronomical observations, Y∗ with uncertainties ε∗ are obtained and, if need be, transformed into form where they can more readily be compared to those produced by a model, M (). This stellar model is a parameterised version of some theory, which has an output vector Y corresponding to the observables Y∗ . This model may also have other outputs which are not directly comparable to observations, but which are important diagnostics for the theory of stellar evolution. These outputs are contained in the vector ζ .

2.4 Forward Modelling Scheme

47

We consider the example of a star with an observed luminosity L, effective temperature Teff , and surface gravity log g, all of which have some associated uncertainty in their measurement such that: Y∗ = L , Teff , log g and ε ∗ = σ L , σTeff , σlog g . We  whose values produce  want to obtain the set of parameters  Y which best match the observables Y∗ , given the uncertainties ε ∗ . To do this, we assume some cost function CF, which evaluates how well the model produced by the j-th permutation of parameter combinations  j reproduces the data. For convenience, we use the notation that the j-th permutation of combinations  j is implicitly contained in Mi j ,  parameter  such that Mi j = Mi  j , where Yi j is the model output to be compared to Yi∗ , with observational uncertainty εi∗ . Following the discussion in the previous sections, we  = Mini , Z , αMLT , f CBM , DREM , τ | ψ which optimise aim to find the values of    i = Mi   for the optimised set of paramethe cost function CF. We take M  , which produce the best matching outputs  ters  Y and  ζ , where in this example  ζ = Mcc , Rcc  contains the convective core mass and radius for the corresponding best model. However, it is possible that not only the observed measurements share some correlations, but also that the underlying grid of models against which we match our observations exhibit degeneracies as well. This is a crucial aspect of our motivation for choosing a cost function and means of error estimation, as discussed below.

2.4.1 Mahalanobis Distance  The choice of cost function is aimed at wanting to find the set of parameters,     , and the which minimises the difference between the  Y from the model, M  observations Y∗ . One of the most commonly used cost functions in astrophysics which accomplishes this is the χ 2 -metric of the form: χ 2j =

  Y∗ − Yi j 2 i

i

σi

,

  with σi2 being the variances resulting from the assumption that εi∗ ∼ N μ, σi2 .  . However, this Minimising this function, then provides the best set of parameters  cost function assumes that (1) the components of the vector  are independent and share no correlation, (2) that each component of  contributes an equal amount of information to the fit, and (3) that theoretical predictions Y are not subject to uncertainty and that their components are uncorrelated. However, in the case of correlated components of Y and , any correlated components would contribute less independent information to the resulting fit than in the case where they are truly independent. This is of particular importance in the case of matching observed quantities Yi∗ to those predicted from grids of stellar models which have well known and strong degeneracies in their free parameters, as demonstrated in Fig. 2.1. To address this situation, Aerts et al. (2018) used the Mahalanobis distance (hereafter

48

2 Stellar Evolution Tracks, Isochrones, and Isochrone-Clouds

MD) as the cost function. The MD is a standard cost function used in the statistical literature when dealing with models consisting of highly correlated free parameters, as well as correlated observables, each of which have their own variance (see e.g., Johnson and Wichern 2008, for a monograph). The MD in effect is a distance metric in n-dimensions which accounts for any parameter correlations. By adopting the MD as the cost function, we account for (1), (2) and (3) simultaneously. In the case where the theoretical predictions Y have larger uncertainties than Y∗ , which is the case for asteroseismic observables from space-based data, the optimised  as determined by the MD metric is of the form: vector  N

 min = arg min (Y − Y∗ ) V −1 (Y − Y∗ ) ,  i

(2.5)

where Y and Y∗ are the model outputs and observations respectively, and V is the variance-covariance matrix corresponding to Y. The vector V not only accounts for the correlation between parameters in the underlying  vector, but serves as a means to normalise the ranges covered by different parameters, which allows us to avoid  requires an estimation transforming the parameters themselves. Computation of  of V −1 . A pragmatic approach is to calculate V from the grid of models, assuming that the total variability within the grid is larger than the variability caused by the theoretical uncertainy of Y. Hence, we use: V =

N N  

1  1  Yj − Y Yj − Y ; Y = Yj, N −1 j N j

where N is the total number of grid-points and Y is the vector of mean values of Y. Additionally, to account for the observational uncertainty, we can define:  = V + λ, V where λ is the matrix whose values contain the observational variance-covariances. The theoretical uncertainties of Y are two orders of magnitude larger than those uncertainties of Y∗ considering the case of g-mode pulsation frequencies. This statistical framework allows for a flexible redefining of M () depending on the application, which is ideal for our purposes. We use the MD in Chaps. 4, 5, and 6, each with a different set of model choices  and ψ, as well as different observables Y∗ with uncertainties ε ∗ , all of which are discussed for their specific applications. The MD has been used by Aerts et al. (2018); Mombarg et al. (2019) and Pedersen (2021) for the asteroseismic modelling of g-mode pulsating B- and Fstars. Modelling applications concerning p-mode pulsations in stars have also used machine learning and considered theoretical model uncertainties for asteroseismic parameter estimations (Gruberbauer et al. 2013; Bellinger et al. 2016; Hendriks and Aerts 2019).

2.4 Forward Modelling Scheme

49

2.4.2 Error Estimation The MD functions as a maximum-likelihood point estimator, and as such inherently has no resulting uncertainty estimations associated with its calculation. While a point estimator provides a single best match to the data, it provides no information of the stability or robustness of the solution. This can lead to the instance where the first n best ranked model M j are located in well separated parts of the grid from which the models were evaluated. Also, uncertainty estimation is notoriously difficult for  min , because extremely elongated error highly correlated parameter components in  ellipses are anticipated. Here we discuss two methods for determining uncertainty estimates for solutions resulting from a grid evaluation using the MD.

2.4.2.1

Confidence Intervals

In the limiting case where the resulting parameter distributions are approximately normally distributed, the resulting MD values of grid evaluations follow a χ 2 distribution. This allows us to define the critical point c p = χ D2 which bounds the uncertainty region, where χ D2 is the critical χ 2 for D degrees-of-freedom (Aerts et al. 2018). Alternatively, we can define a limiting percentile to define the bounds of the uncertainty region. Assuming the limiting case of χ 2 distributed MD values, the 50th-percentile of the MD values corresponds to the 1σ region for normally distributed parameters. We note that in both of these estimation methods, the results depend on the construction of the grid. The possible width of the uncertainty region is defined by the range of parameters used to construct the grid, and also on the resolution of the grid. Aerts et al. (2018) suggested a possible solution to this grid-dependence by using a statistically modelled representation of the grid, which would then become resolution independent. In this case, the statistical model could consist of linear models, linear-mixed models, or fractional polynomials, with a given set of coefficients for the input vector , whose values determine the output vector Y. The best estimate and uncertainties are then determined through simple matrix operations, offering a fast and flexible means of parameter estimation.

2.4.2.2

Monte-Carlo Simulations

To circumvent the issue that the MD functions as a point estimator, we can apply Monte Carlo error estimation techniques to determine the uncertainty in the estimated parameters. To do this, we first calculate the MD for each grid point, and determine  min with its corresponding optimised output vector  Y according to Eq. (2.5). We  then define the vector    Y, ε ∗ ; n = 1...N , Yn = N 

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where N > 1000 is the total number of samples drawn for the Monte Carlo simulations. This vector samples the variability of our optimal solution within the observaY = Y∗ and re-calculate the tional uncertainty ε ∗ . For all N samples, we assert that  . For each MD for the entire grid, using the original variance-covariance matrix V  and append it to a list. The sample we only retain the single best point-estimate   n are then binned and the α% confidence interresulting distributions of estimated  vals are calculated as the 1 − α highest posterior density ranges. We make the explicit choice to only retain the single best point-estimate from each Monte Carlo iteration in order to estimate the robustness of our solution given the underlying grid. If we were to instead take the best M points from each iteration, we would be sampling the variance of our solution space given the underlying grid and observables. While this is an interesting feature to study, it is not within the scope of this monograph.

2.4.3 Markov Chain Monte Carlo Sections of this monograph make use of Markov Chain Monte Carlo (MCMC) techniques to determine an optimised model and error estimates. To do this, we use the emcee code (Foreman-Mackey et al. 2013). emcee uses an affine-invariant ensemble sampler to numerically approximate the posterior distribution p (θ |d) given by Bayes’ Theorem: p (θ |d) ∝ L (d|θ ) p (θ ) , (2.6) where θ is some vector of free parameters, d is the data, L (d|θ ) is the likelihood of the data, given the parameters θ , and p (θ ) is the prior probability of the sampled parameters. The vector p (θ ) incorporates any previously known information concerning the components of θ , and can take several forms. An uninformative, or flat, prior takes the form of a uniform distribution covering some range, whereas an informative prior can take the form of a normal distribution of a given width centred about some mean, or other forms such as logistic or Jeffery’s priors. The operative term is the likelihood function L = L (d|θ ). This function evaluates the likelihood that some generative model can reproduce the data given some set of parameters. The generative model and priors may change for different applications. A given run of the code consists of N iterations consisting of M chains, resulting in a total of N × M model evaluations. A given run continues until the chains are considered to be converged on a solution. While there are a number of methods to test for convergence, we consider a run to have converged when we have more than 50 times the number of iterations as given by the auto-correlation time. Once convergence has been achieved, we bin and histogram the posterior distributions in order to obtained a marginalised posterior distribution for each sampled parameter. Strictly speaking, these resulting distributions are the results, however, we want to quantify and simplify these distributions into a more compact form. To this end, we characterise the optimal parameter value and its uncertainty as the median and highest posterior density (HPD) confidence intervals, calculated using the marginalised

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posterior distributions. In the case of independent, normally distributed posteriors with some mean μ and standard deviation σ correspond to the median and 67% HPD confidence interval. In this monograph, we use MCMC techniques to optimise binary models and estimate model parameter uncertainties through characterisation of their posteriors as just described. Here, we have briefly laid out the theoretical background and leave the details to be described more thoroughly in the following chapters, where the particular application is adapted to the given science case.

2.5 Summary In this chapter we described the numerical setup for the stellar models used in this monograph. We discussed and justified our choice of input physics and outlined our implementation of internal mixing profiles in the grids used for this monograph. Furthermore, we discussed isochrones and introduced the concept of isochroneclouds which is used in this monograph. Finally, we discussed the forward modelling setup and error estimation scheme used throughout this monograph.

References Aerts, C. (2021). Probing the interior physics of stars through Asteroseismology. Reviews of Modern Physics, 93(1), 015001. arXiv:1912.12300. https://ui.adsabs.harvard.edu/abs/2021RvMP... 93a5001A. Provided by the SAO/NASA Astrophysics Data System. Aerts, C., Molenberghs, G., Michielsen, M., et al. (2018). ApJS, 237, 15. Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. (2009). ARA&A, 47, 481. Bastian, N., & Lardo, C. (2018). ARA&A, 56, 83. Bellinger, E. P., Angelou, G. C., Hekker, S., et al. (2016). ApJ, 830, 31. Briquet, M., Morel, T., Thoul, A., et al. (2007). MNRAS, 381, 1482. Buysschaert, B., Aerts, C., Bowman, D. M., et al. (2018). A&A, 616, A148. Cox, J. P. & Giuli, R. T. 1968, Principles of stellar structure Ding, C. Y., & Li, Y. (2014). MNRAS, 438, 1137. Dotter, A., Conroy, C., Cargile, P., & Asplund, M. (2017). ApJ, 840, 99. Ekström, S., Georgy, C., Eggenberger, P., et al. (2012). A&A, 537, A146. Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. (2013). PASP, 125, 306. Gabriel, M., Noels, A., Montalbán, J., & Miglio, A. (2014). A&A, 569, A63. Gruberbauer, M., Guenther, D. B., MacLeod, K., & Kallinger, T. (2013). MNRAS, 435, 242. Hendriks, L., & Aerts, C. (2019). PASP, 131, Johnson, R. A. & Wichern, D. W. 2008, Multivariate Analysis (American Cancer Society) Johnston, C., Tkachenko, A., Aerts, C., et al. (2019). MNRAS, 482, 1231. Joyce, M., & Chaboyer, B. (2018). ApJ, 856, 10. Langer, N., El Eid, M. F., & Fricke, K. J. (1985). A&A, 145, 179. Marques, J. P., Goupil, M. J., Lebreton, Y., et al. (2013). A&A, 549, A74. Michielsen, M., Pedersen, M. G., Augustson, K. C., Mathis, S., & Aerts, C. (2019). A&A, 628, A76. Mombarg, J. S. G., Van Reeth, T., Pedersen, M. G., et al. (2019). MNRAS, 485, 3248. Moravveji, E., Aerts, C., Pápics, P. I., Triana, S. A., & Vandoren, B. (2015). A&A, 580, A27. Moravveji, E., Townsend, R. H. D., Aerts, C., & Mathis, S. (2016). ApJ, 823, 130.

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Nieva, M.-F., & Przybilla, N. (2012). A&A, 539, A143. Paxton, B., Bildsten, L., Dotter, A., et al. (2011). ApJS, 192, 3. Paxton, B., Cantiello, M., Arras, P., et al. (2013). ApJS, 208, 4. Paxton, B., Marchant, P., Schwab, J., et al. (2015). ApJS, 220, 15. Paxton, B., Schwab, J., Bauer, E. B., et al. (2018). ApJS, 234, 34. Paxton, B., Smolec, R., Schwab, J., et al. (2019). ApJS, 243, 10. Pedersen, M. G., Aerts, C., Pápics, P. I., & Rogers, T. M. (2018). A&A, 614, A128. Pedersen, M. G., Aerts, C., Pàpics, P. I., Michielsen, M., Gebruers, S., Rogers, T. M., Molenberghs, G., Burssens, S., Garcia, S., and Bowman, D. M. (2021). Internal mixing of rotating stars inferred from dipole gravity models. Nat Astron. Portegies Zwart, S. F., McMillan, S. L. W., & Gieles, M. (2010). ARA&A, 48, 431. Przybilla, N., Nieva, M.-F., & Butler, K. (2008). ApJ, 688, L103. Puls, J., Sundqvist, J. O., & Markova, N. 2015, in IAU Symposium, Vol. 307, New Windows on Massive Stars, ed. G. Meynet, C. Georgy, J. Groh, & P. Stee, 25–36 Rogers, T. M., & McElwaine, J. N. (2017). ApJ, 848, L1. Schmid, V. S., & Aerts, C. (2016). A&A, 592, A116. Seaton, M. J. (2005). MNRAS, 362, L1. Torres, G., Andersen, J., & Giménez, A. (2010). ARA&A, 18, 67. Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. (2000). A&A, 362, 295. Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. (2001). A&A, 369, 574. Vink, J. S., Muijres, L. E., Anthonisse, B., et al. (2011). A&A, 531, A132.

Chapter 3

The O+B Eclipsing Binary HD 165246

This chapter is based in part on: 1. Detection of intrinsic variability in the eclipsing massive main-sequence O+B binary HD 165246 JOHNSTON, C.; Buysschaert, B.; Tkachenko, A.; Aerts, C.; Neiner, C. Monthly Notices of the Royal Astronomical Society: Letters, Vol. 469L, 118J, 5 pp. (2019) 2. Characterization of the variability in the O+B eclipsing binary HD 165246 JOHNSTON, C.; Aimar N.; Abdul-Masih M.; Bowman D. M.; White T.; Hawcroft C.; Sana H.; Sekaran S.; Dasilva K.; Tkachenko A.; Aerts C. Monthly Notices of the Royal Astronomical Society, Vol. 503, 1124J (2021) Author Contribution: (1) C. Johnston performed the binary modelling and performed the frequency analysis together with C. Aerts. B. Buysschaert performed a custom aperture light curve extraction. A. Tkachenko re-analysed the publicly available FEROS spectra to search for the contribution of the secondary. C. Aerts and C. Neiner were involved in the discussion and interpretation. (2) C. Johnston performed the binary and orbital modelling, calculated the LSD profiles and performed the line profile analysis. N. Aimar was a visiting master student working under the supervision of C. Johnston to normalise the spectra and search for periodicities in the line profiles. M. Abdul-Masih, C. Hawcroft, and H. Sana performed the FASTWIND atmospheric modelling. T. White provided the halo light curve. S. Sekaran and A. Tkachenko contributed to the line profile variability analysis. K. Dasilva contributed data-products related to the spectroscopic data set. C. Aerts was involved in the supervision of N. Aimar and discussion of results.

3.1 Introductory Remarks The evolution of massive stars is dictated by the complex interaction of numerous physical mechanisms, such as rapid rotation, mass loss, binary and tidal interaction, and stellar oscillations. Unfortunately, there is a general lack of predictive power from © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0_3

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the theoretical descriptions of these mechanisms which are expected to be active in massive stars. While theoretical evolutionary calculations can broadly predict the end products of massive star evolution, they cannot reproduce a number of observed phenomena in evolving massive stars. For example, massive OB-type stars have been observed to host low-frequency photometric and spectroscopic (line profile) variability in regions of the HRD where κ-mechanism driven g and p modes are not predicted (Simón-Díaz et al. 2017; Pedersen et al. 2019; Burssens et al. 2019). In recent literature, both turbulent subsurface convection (Cantiello et al. 2009; Lecoanet et al. 2019) and IGWs driven by the convective core (Aerts and Rogers 2015; Edelmann et al. 2019) have been proposed as mechanisms to explain this observed variability. Bowman et al. (2019a, b) identified and characterised the low-frequency power excess caused by IGWs in more than 100 OB stars observed by CoRoT , K epler , and K 2. Observational characterisation of sub-surface convection, however, is still small in sample size (Cantiello et al. 2009). This chapter deals with the characterisation and identification of the low-frequency variability observed in the O+B eclipsing binary HD 165246. HD 165246 was previously studied by Mayer et al. (2013) who used 13 FEROS spectra and 617 ASAS3 V-band observations to derive an initial orbital solution. The authors find a mass ratio of q = 0.17 from spectroscopic disentangling, and obtain Teff,1 = 33 300 ± 400 K and Teff,2 = 15 800 ± 700 K. Following the O8V spectral classification, the authors adopt a typical mass for such a star M1 = 21.5 M according to Martins et al. (2005). Furthermore, Sana et al. (2014) found HD 165246 to be the inner binary of a sextuple system, with the nearest resolvable companion being ∼ 3 arc-seconds away. To investigate and characterise the variability in HD 165246, we adopt an iterative modelling procedure. This involves first optimising the atmospheric and orbital solution in the spectroscopic dataset before using this information in the photometric analysis. The results of the photometric analysis are then used to inform the next iteration of the spectroscopic analysis, and so on. This is repeated until the solution no longer changes within the errors. For clarity, we discuss the analyses separately. We first discuss the spectroscopic analysis in Sect. 3.2 before discussing the photometric analysis in Sect. 3.3. In Sect. 3.4 we discuss the possible mechanisms and evolution of the system, and summarise the work in Sect. 3.5.

3.2 Spectroscopic Analysis This section was originally submitted as Johnston et al. (2021), their Sect. 4, titled: Spectroscopic Analysis. Reproduced with permission from author and OUP on behalf of MNRAS. We set out to orbitally and spectroscopically characterise HD 165246 and investigate the possible presence of line profile variability induced by stellar oscillations. To do

3.2 Spectroscopic Analysis

55

this, we obtained 160 observations between 3 May 2017 and 10 October 2019 with the hermes spectrograph (R = 85 000) (Raskin et al. 2011) attached to the 1.2 m Mercator telescope at El Observatario Roque de los Muchachos in Santa Cruz de La Palma. As many as 20 consecutive exposures were taken during eight nights in order to achieve a high temporal resolution. The observations have a mean of SNR = 85 at 550 nm (ranging from SNR = 63 to SNR = 112), with an average integration time of 120 s, and are well distributed across the orbit. These observations were subjected to background and bias subtraction, flat fielding, wavelength calibration (ThAr lamp spectrum), and order merging using the local hermes pipeline. The reduced spectra were subsequently normalised via spline fitting. To maximise the SNR for the spectra, we calculate a Least Squares Deconvolved (LSD) profile (Donati et al. 1997; Tkachenko et al. 2013). This method involves convolving a series of δ functions of given depths at a given set of wavelengths corresponding to a pre-determined mask to produce an average line profile √ from the entire spectrum. Thus, the expected increase in SNR is proportional to N , where N is the number of spectral lines used in the mask. Furthermore, by allowing for the simultaneous calculation of multiple average profiles, the LSD methodology enables the detection of multiple components in the spectrum. Whereas the spectra of O-stars feature strong He ii lines, the optical spectra of B-type or cooler stars feature strong He i or metal lines, depending on the effective temperature. To this end, the use of different masks allows for the detection of multiple components should they have a significant light contribution. Following these considerations, we subject all of the available HERMES spectra to this method considering two different masks: (i) helium lines between 4900 and 5900 Å, and (ii) metal lines, both of which were constructed from the VALD database (Kupka et al. 1999). Since hydrogen and helium lines are known to suffer from Stark broadening and the signature of radiation driven winds (should they be present), these lines are generally avoided in line profile variability studies. However, in the case of hot rapidly rotating stars, helium lines may be the only lines with sufficient SNR to be considered reliable (Balona et al. 1999; Rivinius et al. 2003). Figure 3.1 shows the average LSD profiles constructed using the helium-line mask (black) and the metal-line mask (grey). The average metal-line profile exhibits a trend in the red wing. Although it is unclear what introduces this ubiquitous trend in the LSD profiles produced with the metal-line mask, it is clear that these profiles are unsuitable for line profile analysis. The average helium-line profile is well behaved. As such, the remainder of the orbital and line profile analysis is carried out using the LSD profiles produced with the helium-line mask.

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1.000

0.975

< Flux >

Fig. 3.1 Top: Mean LSD profile for mask containing only He lines (black) and only metal lines (grey). Bottom: Standard deviation across LSD profiles. Figure adapted from Johnston et al. (2021), their Fig. 5. Reproduced with permission from author and OUP on behalf of MNRAS

0.950

0.925

0.900

σ

0.02

0.01 −200

0

200

RV [km s−1]

3.2.1 Line Profile Variability This section was originally submitted as Johnston et al. (2021), their Sect. 4.1, titled: Line profile variability. Reproduced with permission from author and OUP on behalf of MNRAS. The overall position and shape of a line profile is the combination of extrinsic and intrinsic broadening effects and perturbations, such as radial velocity (RV) shifts due to binarity, broadening due to rotation, micro- or macroturbulence, and stellar pulsations, some of which are variable in time. Thus, studying the line profile variations (LPVs) over time allows for the investigation of these signals. The velocity field produced by coherent/stochastic stellar pulsations induces strictly/quasi-periodic variations in the line forming regions near the stellar surface. These variations are detectable via time-resolved spectroscopic observations. Whereas pressure waves have a predominantly radial contribution to the line profile (Aerts and De Cat 2003), gravity waves produce predominantly tangential velocity variations in the line profile (De Cat and Aerts 2002), which only produce observable variations along the line of sight near the limb of the star. It is worth noting that line-driven winds of massive stars are thought to be inherently unstable, introducing yet an additional cause of variability into the line forming region (Puls et al. 2008; Sundqvist et al. 2011). However, the observational consequences of such winds are only important in cases where they are evident in the observations. Moreover, wind

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variability is stochastic and readily distinguishable from strictly periodic coherent oscillation modes. Spectroscopic asteroseismic frequency analysis of coherent pulsation modes employs one of two methods: (i) the moment method (Balona 1986; Aerts et al. 1992; Briquet and Aerts 2003), or (ii) the pixel-by-pixel method (Schrijvers et al. 1997; Mantegazza et al. 2000; Zima 2006; Zima et al. 2006). The moment method involves numerically integrating the moments of the extracted spectral line to describe the variability in terms of the equivalent width (0th moment), the centroid velocity (corresponding to RV; 1st moment), profile width (2nd moment), and profile skewness (3rd moment). The moment method is most robust for cases where the star is not rapidly rotating (v sin i < 50 km s−1 ). There are some notable exceptions, such as for studying rotational variability in rapidly rotating chemically peculiar stars (e.g. Lehmann et al. 2006). However, it can still be useful for identifying periodicities when combined with the pixel-by-pixel method for analysis of rapidly rotating stars. In contrast to the moment method which relies on the statistical properties of a line profile, the pixel-by-pixel method relies on the phase and amplitude caused by a stellar pulsation mode across the line profile. While the pixel-by-pixel method is more useful in cases where v sin i  50 km s−1 , it is limited by SNR and is not coupled to the theory of non-radial oscillations as is the case with the moment method. We use the FAMIAS software package (Zima 2008) to carry out the LPV analysis, using both the moment and pixel-by-pixel methods.

3.2.1.1

Orbital Variability

This section was originally submitted as Johnston et al. (2021), their Sect. 4.1.1, titled: Orbital variability. Reproduced with permission from author and OUP on behalf of MNRAS. The dominant source of variability among the spectra is the RV shift induced by the binary motion. In order to investigate any signal caused by stellar pulsations, we must first effectively model and remove this orbital signal. To do this, we calculate the first moment for all LSD profiles and fit a model to these RV shifts using the MCMC technique described in Sect. 2.4.3. In this application, we use the expression for RV curves produced by orbital motion: V1,2 = γ +

2πa1,2 sin i [e cos(ω0 ) + cos(θ + ω0 )] , √ Porb 1 − e2

where 1 and 2 refer to the primary and secondary component, a is the semi-major axis, i is the inclination, e is the eccentricity, Porb is the orbital period, ω0 is the argument of periastron, and θ is the true anomaly of the orbit.

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Table 3.1 Top: HPD estimates and uncertainties for orbital solution. Bottom: Derived values and uncertainties. Figure adapted from Johnston et al. (2021), their Table 3. Reproduced with permission from author and OUP on behalf of MNRAS Parameter Unit HPD estimate Porb tpp e ω0 K1 γ a1 sin i f(M)

BJD BJD – rad km s−1 km s−1 R M

4.592834 (fixed) 2457215.48 ± 0.06 0.030 ± 0.003 1.63+0.08 −0.09 53.0 ± 0.2 −7.9 ± 0.1 4.81+0.01 −0.01 0.071+0.001 −0.001

We fix the orbital period to Porb = 4.59270 d (as determined by Johnston et al. 2017) and sample the time of periastron passage tpp , the eccentricity e, the argument of periastron ω0 , the semi-amplitude of the primary K 1 , and the systemic velocity γ . We derive estimates and 1 − σ uncertainties as the median and 68.3 percentile HPD of the marginalised posteriors, which are listed in Table 3.1. The resulting best fit constructed from these values and the residuals are shown in the top and bottom panels of Fig. 3.2. The residuals show a peak-to-peak scatter of ∼ 20 km s−1 , indicating the presence of intrinsic variability. Our values for K 1 , γ , ω0 and e are different to those obtained by Mayer et al. (2013). However, given such a small eccentricity and an argument of periastron near 90◦ , differing solutions for small data sets, such as that used by Mayer et al. (2013), are not unexpected. We do not detect the presence of the secondary in any of the individual spectra, their LSD profiles, or in the first moment of these profiles. Additionally, we subject the original data set to spectroscopic disentangling using FDBinary (Ilijic et al. 2004; Pavlovski and Hensberge 2005), with a fixed orbital solution according to those values listed in Table 3.1 and only allow K 2 to vary, but are not able to reliably determine a solution.

3.2.1.2

Intrinsic Variability

This section was originally submitted as Johnston et al. (2021), their Sect. 4.1.2, titled: Intrinsic variability. Reproduced with permission from author and OUP on behalf of MNRAS. We remove the orbital motion of the primary from each of the normalised spectra according to the optimised parameters in Table 3.1. Following this, we calculate new LSD profiles using the helium line mask. These LSD profiles are then used in an

3.2 Spectroscopic Analysis

40 20

RV [km/s]

Fig. 3.2 Top: Observed RVs for HD 165246A in grey, best fit orbital solution in black. Bottom: Residuals after subtraction of best fit solution. The observed scatter is astrophysical in nature. Figure adapted from Johnston et al. (2021), their Fig. 6. Reproduced with permission from author and OUP on behalf of MNRAS

59

0 −20 −40

RV [km/s]

−60 10 0 −10 −0.4

−0.2

0.0

0.2

0.4

Φ

LPV analysis, where each LSD profile is assigned a weight according to its SNR. We calculate the zeroth, first, second, and third moments for the data set and subject them to iterative prewhitening until all periodicities with SNR > 4 are removed from the time-series of these four moments. We find 17 significant frequencies in the moments and in the variability across the LSD profiles as found by the so-called velocity pixel-by-pixel method (see Zima 2008). In this method one searches for frequencies that occur across the entire LSD profile in velocity space. For the latter application, we fix the frequency values found in the photometry or moments to get the highest-precision fit results for the amplitude and phase behaviour across the LSD profile, as is common practise in such applications (Zima et al. 2006). We show the periodograms of the moments in Appendix B and the amplitude and phase distributions across the LSDs in Fig. 3.3. This LPV frequency analysis result is indicative of low- to high-order p- and g-mode pulsational variability. Indeed, the frequencies found previously in rapidly rotating β Cep pulsators are markedly higher than those we find for HD 165246 (e.g. Schrijvers et al. 2004; Uytterhoeven et al. 2004, 2005), except for the frequencies f s,4 and f s,10 . Aside from these two frequencies, all the other ones are lower than those of the p modes found in the CoRoT space photometry of the slowly rotating O-type dwarf HD 46202 (Briquet et al. 2011), which is to date the β Cep star with the highest mass (24 M ). As expected we find common frequencies among the various spectroscopic diagnostics deduced from the LSD time series. Matches occur between f s,2 and f s,11 , f s,6 and f s,13 , and f s,8 and f s,16 . Additionally, we note that f s,9 = f p,2 . Of the five frequencies found in the pixel-by-pixel method, three match frequencies in the space

Amplitude[continuum]

−200

−200 ]

200

−200

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−200

−200

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fS,17 =0 .9920d −1

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0

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Fig. 3.3 Top: Amplitude across line profile. Bottom: Phase across line profile. Smoothed data displayed in blue, errors displayed in orange. Line profile overplotted in grey in both panels. Figure adapted from Johnston et al. (2021), their Fig. 7. Reproduced with permission from author and OUP on behalf of MNRAS

Φ[rad]

60 3 The O+B Eclipsing Binary HD 165246

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photometry. Of these, we note that f s,15 matches f p,3 , which we identified as the rotational frequency. Figure 3.3 shows the results of optimising the amplitude (top row) and phase (bottom row) and their errors, smoothed over a 15 km s−1 window in blue and orange, respectively. Additionally, we plot the average LSD profile in grey. With the exception of f s,13 , the amplitudes across the line profiles are constant within the uncertainties and do not allow us to interpret the results in terms of mode identification. Phase variability as expected for low-amplitude coherent modes can be seen in some of the bottom panels of Fig. 3.3, particularly for f s,13 . IGWs would result in more chaotic phase variability across the LSD profiles. The phase variability we detect for f s,14 , f s,16 , and f s,17 has a similar level to that found for some of the lowest-amplitude high-degree p modes found in ν Cen (Schrijvers et al. 2004), λ Sco (Uytterhoeven et al. 2004), and κ Sco (Uytterhoeven et al. 2005). In Sect. 3.3.3, we argued that f p,3 = f s,15 can be explained as the rotation frequency of the primary. The full 2π phase variation across the line profile is consistent with this interpretation. In conclusion, the complex interplay of frequencies, some of which found in both space photometry and high-resolution spectroscopy, is not exceptional (e.g., Cotton et al., submitted, treating the high-mass β Cep pulsator β Cru). This, along with the frequency regime found for the p modes of the O9V slowly rotating β Cep star HD 46202 (Briquet et al. 2011), makes us interpret the frequencies detected in the LSD and in the space photometry of HD 165246 as due to a mixture of coherent low-order p and g modes, along with IGWs, shifted into the gravito-inertial regime by the star’s fast rotation (Aerts et al. 2019).

3.2.2 Interpretation as Variable Macroturbulence This section was originally submitted as Johnston et al. (2021), their Sect. 4.2, titled: Interpretation as variable macroturbulence. Reproduced with permission from author and OUP on behalf of MNRAS. It is well known that rotational broadening alone is not sufficient to explain the shape of line profiles in massive stars (Gray 2005). A microturbulent velocity component is added during the atmosphere calculations, and can thus alter the line strength. Furthermore, a macroturbulent velocity component, ξmacro , being either anisotropic (with different radial and tangential contributions) or isotropic (with equal radial and tangential contributions), is required to better reproduce the line profile (Howarth et al. 1997; Gray 2005; Aerts et al. 2009; Simón-Díaz & Herrero 2014). Moreover, macroturbulence and rotational broadening are degenerate in atmospheric modelling codes, as both are included via convolution to an already computed spectrum. Additionally, those stars which undergo other phenomena such as sub-surface convection, spots, and/or stellar pulsations which impact the shape of the line profile and make

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Table 3.2 Significant frequencies, amplitudes, and SNRs extracted from 0th, 1st, 2nd, and 3rd moments and from the pixel-by-pixel method. Figure adapted from Johnston et al. (2021), their Table 4. Reproduced with permission from author and OUP on behalf of MNRAS d−1 – SNR Note Zeroeth moment f s,1 0.18 ± 0.02 f s,2 0.487 ± 0.004 f s,3 1.729 ± 0.003 f s,4 7.342 ± 0.002 First moment f s,5 1.476 ± 0.002 f s,6 1.821 ± 0.002 Second moment f s,7 2.225 ± 0.002 f s,8 2.638 ± 0.003 f s,9 0.501 ± 0.003 f s,10 4.873 ± 0.006 Third moment f s,11 0.489 ± 0.002 f s,12 0.800 ± 0.006 Pixel-by-pixel method f s,13 1.821 (fixed) f s,14 1.502 (fixed) f s,15 0.690 (fixed) f s,16 2.638 (fixed) f s,17 0.992 (fixed)

km s−1 2.3 ± 0.3 0.8 ± 0.1 0.9 ± 0.2 0.8 ± 0.2 km s−1 2.9 ± 0.4 3.3 ± 0.2 km2 s−2 500 ± 40 300 ± 40 400 ± 100 240 ± 40 km3 s−3 168000 ± 31000 150000 ± 14000 Continuum Units 3.1 ± 0.9 2.9 ± 1.0 2.7 ± 1.0 2.1 ± 1.0 3.9 ± 2.0

21.6 8.0 8.0 4.1

– – – –

7.5 8.2

– –

12.6 7.2 11.5 4.5

– – f p,2 –

9.6 8.2

– –

12.4 6.9 8.2 4.8.8 15.3

f s,6 f p,10 f p,3 f s,8 f p,4

it asymmetric require further consideration to accurately reproduce the line profile (Aerts et al. 2014). A non-radial pulsation produces asymmetric deviations from the static line profile that travel through the line profile over the pulsation phase. Thus, the presence of pulsations can directly influence the measurement of observed quantities, such as ξmacro . Aerts et al. (2009) demonstrated that the collective contribution of stellar pulsations can explain, at least in part, the observed macroturbulence deduced from the spectral line properties of massive stars. Aerts et al. (2009) also caution that determination of v sin i can be complicated by the presence of stellar pulsations which induce asymmetric time-dependent variations in the line profile, a problem made worse when spectra obtained at drastically different pulsation phases are stacked. Furthermore, Aerts et al. (2014) show that the macroturbulence needed to explain purely pulsational broadening can be on the order of, or larger than the value of the rotational velocity, and is variable over the pulsation cycle.

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As we observe pulsations and a large projected rotational velocity in the O-star primary of HD 165246, it is important that we obtain an independent estimate of v sin i to use as a constraint in the atmospheric modelling. We achieve this by using the iacob- broad tool developed by Simón-Díaz & Herrero (2014) to independently estimate v sin i for the cases considering no macroturbulence, using the first moment of the Fourier transform (FT) allowing for isotropic macroturbulence, and performing goodness-of-fit (GOF) calculations allowing for isotropic macroturbulence. We select 10 spectra from our data set, five of which span the range of the entire data set and the other five of which are taken during one night. This allows us to investigate the stability of the v sin i and ξmacro estimates over different timescales and at different points along any variability cycle. Given the SNR of our spectra, we use the He ii 4541 line for our calculations. The results of using iacob- broad on the He ii 4541 line are listed in Table 3.3. As expected, the estimates of v sin i are systematically higher when ξmacro is fixed at 0 km s−1 , yielding v sin i = 268 ± 25 km s−1 compared to v sin i FT = 238 ± 40 km s−1 and v sin i GOF = 230 ± 46 km s−1 . The presence of pulsations in HD 165246, however, complicates the interpretation of this. Aerts et al. (2014) demonstrated that even low amplitude pulsations can produce either over- or underestimations of v sin i, and hence ξmacro , by the FT method if a simple isotropic model of macroturbulence is used, depending on the pulsation phase that a spectrum is obtained. This is because a time-independent isotropic velocity is a wrong prior assumption when fitting profiles broadened by time-dependent pulsation modes. From this, Aerts et al. (2014) conclude, in agreement with Aerts et al. (2009), that the best means for estimating v sin i is via GOF with fixed ξmacro = 0 km s−1 . The estimate for v sin i(ξmacro = 0) = 268 ± 25 km s−1 is in agreement with the value for v sin i = 253 ± 7 km s−1 which is calculated assuming that f p,3 is the rotation frequency. Both the GOF and FT methods of determining ξmacro reveal that the estimates of ξmacro are variable on both inter- and intra-nightly timescales, with the mean estimates exceeding 100 km s−1 . This variability in the estimates of ξmacro can be understood in terms of the time-dependent asymmetries produced by pulsations in line profiles, which require different amounts of isotropic macroturbulence for a satisfactory fit. Thus, this is not indicative of actual variation in the macroturbulent velocity, but rather the consequence of measuring ξmacro at different phases of different pulsation cycles. Furthermore, we recall the 18 km s−1 peak-to-peak scatter observed in the residuals of the RV fit in Fig. 3.2. Both the variability in ξmacro and in the RVs are consistent with the effects of both non-radial coherent p and g modes, as well as IGWs propagating in the line-forming region of the primary of HD 165246 (Aerts et al. 2014). Such modes and waves occur in the frequency range covered by the values listed in Table 3.3.

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Table 3.3 Estimates for v sin i and ξmacro for 10 spectra. Reproduced with permission from author and OUP on behalf of MNRAS BJD v sin i (ξmacro = 0) v sin i FT v sin i GOF ξmacro,FT ξmacro,GOF d km s−1 km s−1 km s−1 km s−1 km s−1 2457896.7177510 2457962.4721318 2457964.5220527 2457965.4924402 2457965.4999635 2457965.5074870 2457965.5150105 2457965.5225342 2457967.4127213 2457971.5365183 Mean Range σ

256 264 282 247 274 265 244 254 333 258 268 89 25

235 239 253 225 235 248 217 226 257 241 238 40 12

234 237 252 232 213 223 207 224 246 237 230 46 13

118 119 136 113 154 120 132 137 222 108 136 113 31

117 118 136 113 189 163 153 137 246 108 148 137 41

3.2.3 Updated Atmospheric Solution This section was originally submitted as Johnston et al. (2021), their Sect. 4.3, titled: Updated atmospheric solution. Reproduced with permission from author and OUP on behalf of MNRAS. To determine an updated atmospheric solution, we co-add the normalised spectra (velocity corrected according to the orbital motion), as shown in grey in Fig. 3.4. We perform an atmospheric analysis using the numerical setup as described in AbdulMasih et al. (2019). In brief, we use a genetic algorithm (GA) wrapped around the non-local thermodynamic equilibrium (NLTE) radiative transfer code FASTWIND (Puls et al. 2005) to optimise the atmospheric parameters of the O-star primary (Charbonneau 1995; Mokiem et al. 2005). The GA allows for an efficient sampling of the expansive parameter space. Given that the GA is not setup to use the MD defined in Eq. 2.5, we instead choose a merit function which is proportional to the inverse of the chi-square of a given atmospheric model in comparison to a subset of lines from the co-added observed spectrum. The parameters for each generation of models are determined by combining parameters from the previous generation, where models with a lower chi-square have a higher chance of passing their parameters to the next generation. At each generation, parameter variations, or mutations, are introduced to effectively sample the parameter space. The GA analysis was carried out iteratively with the binary

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Fig. 3.4 Observed spectrum in grey and best fit model according to genetic algorithm optimisation in black. Location and identification of spectral lines indicated by vertical dashed lines. Figure adapted from Johnston et al. (2021), their Fig. 8. Reproduced with permission from author and OUP on behalf of MNRAS

modelling (discussed in Sect. 3.3.2), where the effective temperature was fixed in the binary modelling first, and then the surface gravity and light-dilution from the binary model was fixed in the next iteration of spectral fitting, until convergence was reached. The final optimised values for the FASTWIND model are listed in Table 3.4, and the best fit model is shown in black in Fig. 3.4. Our solution results in a primary effective temperature which is nearly 3000 K hotter than determined by Mayer et al. (2013). Furthermore, we note that we find a high microturbulent velocity in HD 165246, whereas previous studies typically fix this quantity. Finally, the macroturbulent velocity reported by the GA optimisation is significantly lower than that

Table 3.4 Estimated parameters returned from optimised FASTWIND model. Table adapted from Johnston et al. (2021), their Table 6. Reproduced with permission from author and OUP on behalf of MNRAS Parameter Unit Estimate Teff log g log M˙ β v∞ ξmicro ξmacro v sin i

K dex log(M/yr ) – km s−1 km s−1 km s−1 km s−1

36200+900 −600 4.05+0.07 −0.15 +0.2 −8.0−0.15 0.6787+0.15 −0.65 +60 2530−260 13+1.0 −1.3 20+5.2 −6.0 268 (fixed)

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obtained via the GOF and FT methods previously. This is due to differences in the way that macroturbulence is described, i.e. isotropic in the GOF and FT methods verses anisotropic in fastwind with the demand of equal radial and tangential components.

3.3 Photometric Analysis 3.3.1 K2 Photometry After the failure of a second reaction wheel, the K epler satellite was re-purposed under the K 2 mission, which scanned the ecliptic in 90 d long pointings (Howell et al. 2014). Due to the brightness of HD 165246, the K 2 pixels saturate during the 30 min cadence mode observations. Additionally, as HD 165246 lies near the ecliptic, it is situated in a densely crowded field and with several potentially contaminating sources nearby. In order to address the saturated pixels and potential contamination, we build two light curves with different methodologies. The first light curve is constructed by including the flux from a custom pixel mask which only contains saturated pixels (we refer to this light curve as lc-a). An image of the custom mask is displayed in blue in Fig. 3.5. Unfortunately, due to a combination of the saturated columns and the thruster firings required to stabilise the spacecraft, we can only recover ∼29.9 d of data with sufficient quality to analyse. The second light curve is constructed via the halo photometry method (hereafter lc-b). In the case of sufficiently bright targets, photons are scattered across the surrounding pixels. Halo photometry uses the scattered light from these surrounding pixels to build a light curve (White et al. 2017). However, since relative weights are applied to each pixel in order to scale their contribution, the resulting amplitudes of signals within the light curve are not to be taken at face-value. lc-b is more than twice the length of lc-a with a time span of ∼71.3 d. Unfortunately, the halo photometry method includes signal from the nearby objects within the pixel mask, with an artificially scaled contribution. As such, this potentially decreases the amplitude of the signal intrinsic to HD 165246 and makes it difficult to determine the individual contributions of all objects within the pixel mask. A comparison of the two light curves is seen in Fig. 3.6.

3.3.2 Updated Binary Model This section was originally submitted as Johnston et al. (2021), their Sect. 3.2, titled: Updated binary model. Reproduced with permission from author and OUP on behalf of MNRAS.

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Fig. 3.5 K 2 target pixel file for HD 165246 with original aperture mask shown in red and custom aperture mask shown in blue. Median flux per pixel represented in grey scale. Figure adapted from Johnston et al. (2017), their Fig. 1. Reproduced with permission from author and OUP on behalf of MNRAS

ΔKp [mag]

0.00

0.05

0.10

2670

2680

2690

2700

2710

2720

2730

2740

Time [BJD − 2454833]

Fig. 3.6 Custom extracted K 2 light curve of HD 165246 (lc-a; grey) and halo photometry light curve of HD 165246 (lc-b; black). Figure adapted from Johnston et al. (2021), their Fig. 1. Reproduced with permission from author and OUP on behalf of MNRAS

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We follow the Bayesian modelling methodology described in Sect. 2.4.3 and use the ellc code (Maxted 2016) to generate the binary model for the likelihood function. In order to incorporate the updated orbital and atmospheric analysis in Sects. 3.2.1.1 and 3.2.3, we impose Gaussian priors on the eccentricity e ∼ N (0.029, 0.003), the argument of periastron, ω0 ∼ N (1.63, 0.09) rad, the semi-amplitude of the primary K 1 ∼ N (53.0, 0.2) km s−1 , the effective temperature of the primary Teff,1 ∼ N (36150, 600) K, and the projected rotational velocity of the primary v1 sin i ∼ N (268, 25) km s−1 . The remaining parameters listed in Table 3.5 are given uniform priors. Since we did not recover a signal of the secondary, we allow the mass ratio, q = M2 /M1 = K 1 /K 2 , to vary freely as to not bias the fitting result. Similarly, we allow the third light, l3 , to vary freely as well. This, of course, will impact the mass ratio, temperature ratio, and inclination. Instead of directly fitting for the effective temperatures, ellc fits for the surface brightness ratio. To this end, the effective temperature and surface gravities of each component are only used to interpolate values for the limb-darkening and gravitydarkening coefficients in the Kepler passband from the tables published by Claret and Bloemen (2011). In order to address the presence of an asymmetry present in the out-of-eclipse photometric variability, we allow for the Doppler boosting factor for the primary, B1 , to vary as well. We fit both the light curve and RV curve of the primary simultaneously. We run 10 000 iterations with 128 walkers in our MCMC optimisation routine, discarding the first 5 000 iterations as burn-in. The parameter estimates and uncertainties listed in Table 3.5 are calculated as the median and 68.3-percentile HPD estimates of the marginalised posteriors for each parameter. The derived quantities, such as the masses and radii of each component are calculated at each iteration and saved along with the other sampled parameters, allowing us to calculate the estimates and uncertainties for these parameters in the same way. The best fit model, shown in black in the top panel of Fig. 3.7, is calculated from the values listed in Table 3.5. The residuals shown in the bottom panel of Fig. 3.7 have a root-mean-square (RMS) error of 0.89 mmag. To investigate whether or not the inclusion of the boosting factor, B1 , contributes to minimising the asymmetric out-of-eclipse signal mentioned previously, we build a second model with the same parameters as the first, but without Doppler boosting. This model produces residuals with a slightly higher RMS deviation of 0.91 mmag, and shows a brightening at ∼ −0.3, which is consistent with the phase when the O-star is moving along the line-of-sight. Furthermore, we note that a significant peak is present at the orbital frequency in the periodogram of the residuals for the model without beaming included. This peak is not detected in the residuals of the model with beaming included. Beyond the inclusion of the eccentricity, boosting factor, and updated Teff,1 in our improved binary model, we find a lower mass ratio compared to that of Mayer et al. (2013). We note that this may be caused by the inclusion of third light in our model. The estimated third light contribution of 26+2 −1 per cent is within the estimates of 7–43 per cent expected from the other members of the sextuple system as derived from the K-band magnitude contrasts published by Sana et al. (2014). As a combined

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Table 3.5 Estimated parameters returned from MCMC optimised ellc binary model. We refer to Maxted (2016) for the meaning of the symbols. Quantities marked with ∗ were only used to calculate limb- and gravity-darkening coefficients. Table adapted from Johnston et al. (2021), their Table 1. Reproduced with permission from author and OUP on behalf of MNRAS Parameter Unit Estimate R1 /a R2 /a a Sb q i Porb t0 fc fs F1 A1 B1 l3,a l3,b ∗ Teff,1 ∗ Teff,2 Derived parameters e ω0 k (r1 + r2 )/a L 2 /L 1 M1 R1 log g1 M2 R2 log g2

– – R – – deg d d – – – – – % % K K

0.208+0.001 −0.001 +0.001 0.069−0.002 35.3+0.6 −0.7 0.217+0.004 −0.006 0.16+0.01 −0.01 84.0+0.1 −0.1 4.59270+0.00001 −0.00001 2457215.4108+0.0004 −0.0005 0.0032+0.0008 −0.0008 0.1656+0.0009 −0.0007 3.1+0.2 −0.1 +0.1 0.30−0.2 +0.5 2.0−0.5 26+2 −1 67+1 −1 36100+200 −200 12100+600 −600

– rad – – – M R dex M R dex

+0.003 0.027−0.002 +0.01 1.55−0.01 0.333+0.007 −0.009 0.2766+0.002 −0.001 0.0241+0.002 −0.001 +1.1 23.7−1.4 +0.3 7.3−0.4 4.08+0.02 −0.04 3.8+0.4 −0.5 2.43+0.3 −0.1 4.24+0.02 −0.02

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3 The O+B Eclipsing Binary HD 165246 −0.02

ΔTp [mmag]

0.00 0.02 0.04 0.06 0.08 −0.003

ΔTp [mmag]

−0.002 −0.001 0.000 0.001 0.002 0.003 −0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

Φ

Fig. 3.7 Top: K 2 observations (grey) and best fit model (black) for HD 165246. Bottom: Residuals (grey) after removal of best fit model. Figure adapted from Johnston et al. (2021), their Fig. 2. Reproduced with permission from author and OUP on behalf of MNRAS

+1.1 +0.3 result, we calculate that the primary has M1 = 23.7−1.4 M with R1 = 7.3−0.4 R , +0.4 +0.3 and the secondary has M2 =3.8−0.5 M with R2 = 2.4−0.1 R . As mentioned previously, the modelling of lc-b is conducted differently to that of lc-a. Since we expect the physical binary model to be the same, but the third light contribution to be different, we fix the model and sample only the third light contribution for lc-b. From this, we find that lc-b has roughly 66 per cent composite contaminating light, which is significantly larger than the range expected from the other members of the sextuple system. This suggests that this light contains rescaled contributions from other stars in the K 2 pixel image.

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3.3.3 Photometric Variability This section was originally submitted as Johnston et al. (2021), their Sect. 3.3, titled: Photometric variability. Reproduced with permission from author and OUP on behalf of MNRAS. We use the residuals of lc-a and lc-b after removal of the optimised binary model (hereafter res-a and res-b, respectively) to calculate a Lomb-Scargle periodogram with the aim of searching for significant periodicities. The periodograms for resa and res-b are shown in the top and bottom panels of Fig. 3.8. We subject both res-a and res-b to an iterative prewhitening process to extract all variability with an amplitude signal-to-noise ratio (SNR) greater than four (i.e. SNR > 4; Breger et al. 1993), where the SNR is calculated over the full range ν ∈[0, 24.5] d−1 up to the

Amplitude [mmag]

0.2

0.1

0.0

Amplitude [mmag]

0.2

0.1

0.0

0

2

4

6

8

10

ν [d−1 ]

Fig. 3.8 Top: Lomb-Scargle Periodogram of res-a (black) and its residuals (red). Bottom: LombScargle Periodogram of res-b (black) and its residuals (red). Horizontal dashed dark grey and dashed-dotted light grey lines denote four times the white noise level in the residual periodograms after binary model removal and prewhitening, respectively. Figure adapted from Johnston et al. (2021), their Fig. 3. Reproduced with permission from author and OUP on behalf of MNRAS

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Table 3.6 Iterative prewhitening results for the residuals of res-a. Table adapted from Johnston et al. (2021), their Table 2. Reproduced with permission from author and OUP on behalf of MNRAS lc -a ID Frequency (d−1 ) Amplitude SNR Note (mmag) f p,1 f p,2 f p,3 f p,4 f p,5 f p,6 f p,7 f p,8 f p,9 f p,10 f p,11 f p,12 f p,13 f p,14 f p,15 f p,16 f p,17 f p,18 f p,19 f p,20

0.064 ± 0.003 0.502 ± 0.004 0.690 ± 0.003 0.992 ± 0.003 1.117 ± 0.003 1.160 ± 0.004 1.305 ± 0.004 1.382 ± 0.003 1.429 ± 0.003 1.502 ± 0.004 1.540 ± 0.003 2.004 ± 0.003 2.057 ± 0.004 2.165 ± 0.004 2.755 ± 0.004 3.448 ± 0.004 4.139 ± 0.004 5.529 ± 0.004 6.900 ± 0.004 7.601 ± 0.004

0.19 0.15 0.22 0.23 0.23 0.14 0.15 0.21 0.18 0.14 0.18 0.16 0.15 0.14 0.16 0.13 0.14 0.15 0.15 0.13

5.62 4.61 6.26 6.45 6.55 4.53 4.59 6.20 5.37 4.59 5.22 4.92 4.68 4.45 4.70 4.28 4.68 4.64 4.16 4.55

– f s,9 f s,15 f s,17 – – 6 f orb 2 f p,3 – f s,14 – – 3 f p,3 – 4 f p,3 5 f p,3 6 f p,3 8 f3 10 f p,3 11 f p,3

Nyquist frequency. Except for one frequency, f 17 , there is no overlap in the extracted frequency lists for res-a and res-b. Given the greater than factor two increase in contaminating light between the two light curves, and lack of similarity between the extracted frequency lists from res-a and res-b, we only consider those frequencies extracted from res-a in our subsequent analyses. The lack of overlap in extracted frequencies between the two lists does not imply that the signal is not present in res-b. Rather, given the increase in contaminating light, the signal is simply no longer significant according to our SNR > 4 criterion. We suggest that the frequencies extracted from res-b likely originate in one, or several, of the contaminating stars included in the halo-photometry mask, and not from the components of the HD 165246 system. The frequencies extracted from res-a are given in Table 3.6. The tabulated frequencies have been filtered for close frequencies occurring within 1.5 times the Rayleigh criterion (Degroote et al. 2009, 2010; Pápics et al. 2012; Bowman 2017). We identify only one harmonic of the orbital frequency in res-a ( f p,7 ). This is a consequence of the improved binary model. As indicated in Table 3.6, f p,2 , f p,3 ,

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f p,4 , and f p,10 are also extracted in the LPV analysis in Sect. 3.2.1.2. Furthermore, we find nine components (1, 2, 3, 4, 5, 6, 8, 10, 11) of a harmonic series with f p,3 = 0.690 ± 0.003 d−1 as the base frequency. Extended harmonic series are the result of non-sinusoidal signals in the light curve, such as binarity, rotational modulation, or high-amplitude pulsations. The latter option is excluded as all detected amplitudes are below 1 mmag. To investigate the former option, we phase res-a over f p,3 = 0.69 in Fig. 3.9 and find no obvious indication of a blended binary signal. Assuming f p,3 is the rotation frequency, one expects F1 = f p,3 / f orb = 3.17 ± 0.01, which is within 1 − σ of the value for F1 obtained from our binary modelling in Sect. 3.3.2, indicating that the rotational interpretation is feasible. Using f p,3 = f rot as well as R1 and i from Table 3.5 to compute v1 sin i yields v1 sin i = 253 ± 7 km s−1 . This matches well with the value estimated from spectroscopy in Sect. 3.2.3. This harmonic rotational signal could be caused by wind variability modulated by the stars rotation (i.e., a clumpy wind), although we do not detect strong wind signatures in the typical diagnostic lines for HD 165246. Nevertheless, the detection of f s,15 f p,3 in the LPVs also suggests an interpretation of this signal in terms of rotational modulation. The Lomb-Scargle periodogram of the prewhitened residuals of both res-a and res-b are shown in red in the respective panels of Fig. 3.8. These periodograms showcase stochastic low-frequency variability as found previously for a large sample −3

−2

ΔKp [mmag]

−1

0

1

2

3

−0.4

−0.2

0.0

0.2

0.4

Time mod 0.69 d

Fig. 3.9 res-a phase folded over f p,3 . Original data in grey, binned data in black. Figure adapted from Johnston et al. (2021), their Fig. 4. Reproduced with permission from author and OUP on behalf of MNRAS

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of CoRoT, K2, and TESS OB stars (Blomme et al. 2011; Aerts and Rogers 2015; Aerts et al. 2018; Simón-Díaz et al. 2018; Bowman et al. 2019a, b). Such a signal is predicted independently by 3D hydrodynamic simulations carried out by Edelmann et al. (2019) as well as by different 2D hydrodynamic simulations by Horst et al. (2020) and Ratnasingam et al. (2020). All of these simulations concerned single, young stars. However, given the complexity of this multiple system,other physical causes of this excess may be relevant as well. The stochastic variability occurs in both res-a and res-b and is significant according to the S/N > 4 level when the latter is computed from the residual periodogram computed from zero frequency up to the Nyquist frequency (see the dashed-dot horizontal line in Fig. 3.8). Our argument to consider such a broad range of frequencies to compute the noise level follows Blomme et al. (2011) and Bowman et al. (2020), who have shown that young massive O stars such as the primary component of HD 165246 have significant lowfrequency variability up to frequencies of order 100 d−1 .

3.4 Discussion This section was originally submitted as Johnston et al. (2021), their Sect. 5, titled: Discussion. Reproduced with permission from author and OUP on behalf of MNRAS. We detect multiple sources of variability in both the spectroscopic and photometric observations of HD 165246. We identify a series of harmonics in the K 2 photometry whose base frequency, f p,3 , is also present in the HERMES spectroscopy ( f p,15 ). The presence of such harmonics in the photometry indicates the presence of nonsinusoidal signal in the data, which in this context has two potential explanations: (i) rotational modulation, or (ii) background binary signal. Given the corroboration of the binary modelling, atmospheric modelling, as well as the identification of this signal in both the photometric and spectroscopic time series, we argue that this is the result of rotational modulation on the primary O8 V star. Assigning a single underlying mechanism to the remaining variance in the lowfrequency regime is challenging since it contains both coherent p and g modes selfexcited via the κ-mechanism as well as stochastically excited IGWs, which may also drive modes at resonant eigenfrequencies (Bowman et al. 2019a; Edelmann et al. 2019; Horst et al. 2020). Given the poor frequency resolution of both the spectroscopic and photometric data sets, we are currently unable to identify the modes/waves. Since we do, however, identify signal in both the photometry and spectroscopy independently, we are able to conclusively state that there is significant pulsational variability present at low frequencies which originates from the O-type primary. Moreover, the dominant frequencies detected in the moment variations listed in Table 3.2 all lead to a ratio of the tangential to radial velocity amplitude above

3.4 Discussion

75

unity. This ratio can be computed from the mass, radius and frequency of the mode following Eq. (3.162) in Aerts et al. (2010). This leads to ratios with a range covering roughly [3.3, 62.2], meaning that the pulsational variability is dominated by g modes or IGWs as these have dominant tangential motions, while p modes are dominated by radial motions. The signal corresponding to f s,4 and f s,10 leads to low K values and may correspond to either low-order p modes (Briquet et al. 2011) or high-order g modes shifted to high frequencies due to the Coriolis force (Buysschaert et al. 2018). The observed ξmacro variability may have various candidate sources: (i) coherent pulsations, (ii) IGWs, (iii) sub-surface convection, or iv) stochastic wind variability. As we have identified the presence of pulsations and/or IGWs, these invariably have at least some contribution to the variability in ξmacro on the basis of their contribution to the line profiles from which ξmacro is estimated. The parameters of the O8 V primary place it in a region on the HR diagram where the sub-surface convective velocity is theoretically estimated to be below 2.5 km s−1 (Cantiello et al. 2009, see their Fig. 9, top panel). Therefore, sub-surface convection cannot fully explain the large and variable tangential velocities that we observe. Furthermore, our estimates of ξmicro , log g, and v sin i place the O-star primary in an underpopulated region of the parameter space to compare with the predictions of Cantiello et al. (2009). This comparison, however, is complicated by the fact that HD 165246 is within the galaxy, whereas the majority of the sample analysed by Cantiello et al. (2009) consists of stars from the Large and Small Magellanic Clouds. In addition, stochastically variable wind signatures as computed by Krtiˇcka and Feldmeier (2018) stem from outflow and hence correspond dominantly to radial motions, while the observations point to dominant tangential velocity variations. Our observations of high v sin i, high ξmacro , and the presence of IGWs in the young O-star primary of HD 165246 are consistent with the results of both Simón-Díaz et al. (2017) and Bowman et al. (2019b, 2020). Simón-Díaz et al. (2017) observe a wide range of ξmacro from spectroscopy and Bowman et al. (2019b) observe a low-frequency excess in photometry (identified as IGWs) in both galactic and LMC O and B stars across the upper HRD. This suggests a common intrinsic mechanism and a relationship between macroturbulence as found in spectroscopy and stochastic low-frequency variability detected in space photometry (Burssens et al. 2020; Bowman et al. 2020). As demonstrated in Fig. 3.10, the O8 V primary is located close to the zero-age main-sequence in reference to both the non-rotating tracks with different amounts on internal chemical mixing (black tracks), calculated according to Chap. 2 using mesa (r-10398 Paxton et al. 2018) as well as tracks with a large initial rotational velocity (grey tracks), calculated by Brott et al. (2011). The non-rotating models are calculated such that the internal chemical element mixing is represented in two distinct regimes. The first regime corresponds to the convective boundary mixing (CBM) region. Here, the free parameter DCBM scales the slope with which the mixing profile decays beyond the core, as defined by the Schwarzschild criterion in terms of the local pressure scale height, H p . The second regime corresponds to radiative envelope mixing (REM), according to the IGW profile implemented by Pedersen et al. (2018). In this profile, the free parameter DREM sets the base efficiency of chemical mixing induced by this mechanism, and has units of cm2 s−1 . The

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dashed black (non-rotating) models plotted in Fig. 3.10 represent the case of minimal internal chemical mixing, with DCBM = 0.005 and DREM = 10 cm2 s−1 . The dashed-dotted black (non-rotating) models represent the case of maximum internal mixing with DCBM = 0.04 and DREM = 10 000 cm2 s−1 . These limiting values are those deduced by asteroseismology of intermediate-mass g-mode pulsating field stars (Briquet et al. 2007; Daszy´nska-Daszkiewicz and Walczak 2010; Daszy´nskaDaszkiewicz et al. 2013; Moravveji et al. 2016; Schmid and Aerts 2016; Buysschaert et al. 2018; Walczak et al. 2019; Wu and Li 2019). Aside from the distinct difference in accounting for rotation versus assuming non-rotating equilibrium models, the Brott et al. (2011) models assume Yini = 0.2638 and Zini = 0.008, whereas our models assume Yini = 0.276 and Zini = 0.014. Despite these differences, these two grids both indicate a massive primary having consumed less than 30% of its initial core hydrogen content, as seen in Fig. 3.10. Figure 3.11 compares the dynamical mass and surface gravity estimates for the O8V primary of HD 165246 with theoretical isochrones with different amounts of internal mixing using models calculated according to Chap. 2. The location of HD 165246A in Fig. 3.11 indicates it has an age between 2 and 3 Myr and a core hydrogen content of X c = 0.54, given the uncertainties on the dynamical surface gravity (black), or an age between 2 and 4.5 Myr given the uncertainties on the spectroscopic surface gravity (red). Our age estimates agree with those of Mayer et al. (2013) who used evolutionary tracks from Brott et al. (2011). The overlap between the dynamical and spectroscopic surface gravity estimates provides support for the mass estimate derived for the primary component. However, given the inability to detect the secondary star in the spectra, we have no means of independently verifying the solution for the secondary, making us wary of its absolute dimensions. With this in mind, we find that the secondary is not yet on the main-sequence when compared to

2.8 3.0 3.2

log g [dex]

Fig. 3.10 Evolutionary tracks for 25, 20, 15, and 10 M models. Solid black line denotes ZAMS line. Black tracks taken from Johnston et al. (2019) with DCBM = 0.005 + DREM = 10 cm2 s−1 (dashed lines) and DCBM = 0.040 + DREM = 10 000 cm2 s−1 (dashed-dotted lines). Dotted grey tracks taken from Brott et al. (2011) with vinit ∈[330,360] km s−1 , depending on the model. Figure adapted from Johnston et al. (2021), their Fig. 9. Reproduced with permission from author and OUP on behalf of MNRAS

3.4 3.6 3.8 4.0 4.2 4.4 4.60

4.55

4.50

4.45

4.40

log( Teff [K])

4.35

4.30

4.25

3.4 Discussion

77

3.8

τ =5 Myr

log g [dex]

3.9

τ =4 Myr

4.0

τ =3 Myr

4.1

τ =2 Myr

4.2

4.3

4.4

2

4

20

22

24

26

M[M ]

Fig. 3.11 Isochrones for 2,3,4, and 5 Myr shown in grey for tracks with DCBM = 0.005 + DREM = 10 cm2 s−1 (dashed lines) and DCBM = 0.040 + DREM = 10 000 cm2 s−1 (dashed-dotted lines). All models have Y = 0.276 and Z = 0.014. Dynamical (spectroscopic) estimates for mass and log g denoted by black (red) error-bars. Figure adapted from Johnston et al. (2021), their Fig. 10. Reproduced with permission from author and OUP on behalf of MNRAS

the evolutionary tracks in Fig. 3.11. We note that without smaller uncertainties on the parameters of the primary, we are not able to constrain the impact of internal chemical mixing on the evolution of this star. Additionally, without proper characterization of the secondary via detection of its RV variations, we are not able to perform isochronecloud fitting as introduced in Chap. 2.

3.5 Summary This chapter made use of an extensive set of time-series spectroscopy and K 2 photometry to study the massive eclipsing O+B binary HD 165246. Using both the photometry and spectroscopy, we identify variability consistent with gravity wave pulsations as well as rotational modulation. We observe a high and variable estimates of macro-turbulence in the spectroscopy, which can be explained in terms of time-dependent pulsational broadening. We determine a new atmospheric solution in which the O-star primary is estimated to be some 3 000 K hotter than the solution by

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Mayer et al. (2013). We used the RVs determined from this spectroscopy to optimise a new binary model with a small eccentricity, arriving at an updated mass ratio and fundamental parameter estimates for the O- and B-type components. The v sin i and ξmacro estimates for HD 165246A place it in a parameter space not explored by Cantiello et al. (2009) or Simón-Díaz et al. (2017) making it an ideal target for testing models of sub-surface convection and IGWs. We place an additional constraint on future models by providing mass, radius, and age estimates as well. Currently, the TESS mission is collecting up to a year of space-based photometry for several tens to hundreds of massive OB type pulsators. While this data set will be instrumental in establishing a population of and the instability strips for these pulsators, the asteroseismic modelling from space photometry will remain limited in the absence of frequency or period-spacing patterns. To this end, spectroscopic mode characterisation and the estimation of fundamental stellar quantities from binary modelling (when applicable) is crucial. Our study relied on an extensive ground-based spectroscopy, following the detection intrinsic variability in the K 2 light curve by Johnston et al. (2017). Asteroseismic inference for this particular target, however, remains limited. On average, only one in every ten OB stars detected to be intrinsically variable in space-photometry leads to mode identification suitable for detailed asteroseismic modelling (Burssens et al. 2019).

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Chapter 4

Estimating the Convective Core Mass for Stars in Eclipsing Binaries

This chapter is based in part on: 1. Modelling of the B-type binaries CW Cephei and U Ophiuchi. A critical view on dynamical masses, core boundary mixing, and core mass JOHNSTON, C.; Pavlovski, K.; Tkachenko, A. Astronomy & Astrophysics, Vol. 628, A25, 16 pp. (2019) 2. The mass discrepancy in intermediate- and high-mass eclipsing binaries Tkachenko A., Pavlovski K., JOHNSTON C., Pedersen M. G., Michielsen M., Bowman D. M., Southworth J., Tsymbal V., and Aerts C. Astronomy & Astrophysics, Volume 637, id.A60, 20 pp. (2020) Author contributions: (1) C. Johnston performed the binary modelling of both systems, developed the code for the evolutionary modelling, calculated the grid of models used and the resulting isochrone clouds, and interpreted the results in terms of core masses. C. Johnston led the competitive proposal for computing time on the Flemmish Tier-1 supercomputing system used to calculate the grid of stellar models. K. Pavlovski performed the spectroscopic disentangling of both systems and performed the atmospheric and abundance analysis for CW Cep. A. Tkachenko obtained and normalised the spectra and performed the atmospheric analysis for U Oph. (2) C. Johnston provided the grids used in this work and assisted in the interpretation of results.

4.1 Introductory Remarks Spectroscopic double-lined EB systems can provide stellar mass and radius estimatesto the order ∼1% precision (Torres et al. 2010). As was discussed in Sect. 1.7, the comparison of these fundamental quantities with stellar evolution tracks has iden© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0_4

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tified a discrepancy between the observed quantities and those predicted by theory (termed the mass-discrepancy). Several decades of literature have been dedicated to trying to use this mass-discrepancy to calibrate different types of physical mechanisms into stellar structure and evolution models. Amongst these studies, several have attempted to calibrate convective penetration (Ribas et al. 2000a; Tkachenko et al. 2014; Higl and Weiss 2017; Claret and Torres 2016), convective overshooting (Claret and Torres 2017; Constantino and Baraffe 2018), and rotation (Ekström et al. 2012; Schneider et al. 2014; Pavlovski and Southworth 2009; Abdul-Masih et al. 2019). However, different studies attempting to calibrate the same mechanism often do not agree on the numerical implementation of the mechanism, leading to discrepant calibrations and interpretations. In this chapter, we present the modelling of two well-studied, well-detached double-lined EBs using the isochrone-clouds as presented in Sect. 2.3. Following the discussion of internal mixing (Sect. 1.3) and its implementation in stellar structure and evolution codes (Sect. 2.1), we remain agnostic of the actual physical mechanism(s) active in the stars, and attempt to quantify the effects of internal mixing, whatever its cause, rather than attempting to calibrate a parameterised description of a given mechanism. As discussed in Sect. 2.1, this leads us to consider a single global mechanism to represent CBM and a single global mechanism to represent REM. The results of this Chapter serve as a proof-of-concept that we can infer the convective core masses of stars across a wide range of masses and interpret the combined effect of internal mixing processes in terms of enhanced convective core masses. In Sect. 4.2, we provide an overview of the two systems, including past modelling efforts. In Sect. 4.3 we discuss the data used in this work, and in Sect. 4.4.1 we discuss the determination of orbital and of spectroscopic quantities from the spectra. In Sect. 4.4.2 we detail the modelling procedure and results for the two systems with the mass ratio fixed as derived in the previous section. Section 4.5 covers our evolutionary modelling procedure. In Sects. 4.5.1 and 4.8 we discuss the newly determined mass and radii estimates for each system and the modelling results, and we place them in the context of the larger modelling efforts of the community. Finally, we discuss the expansion of the sample to include another 8 systems and span a larger mass range, as well as the results of modelling this updated sample.

4.2 Target Overviews 4.2.1 CW Cephei CW Cephei (CW Cep), is a well studied, bright, massive, double-line EB, with several reported mass and radius estimates in the literature. Although several estimates report percent level precision, there is a 13% uncertainty on masses for the primary (M1 = 11.82 − 13.49 M ) and secondary (M2 = 11.09 − 12.05 M ), as well as an 8% uncertainty on the radius estimates for the primary and secondary (Popper 1974,

4.2 Target Overviews

83

1980; Clausen and Gimenez 1991; Han et al. 2002; Erdem et al. 2004). This spread in reported fundamental parameters is largely due to differences in data sets and modelling methodologies, as well as a lack of a well determined light contribution from and metallicity of the components. This, combined with a lack of total eclipses, manifests as a degeneracy between light contribution, the ratio of the radii, and the system inclination that results in different solutions depending on how this degeneracy is addressed, if at all. Due to its brightness, CW Cep has enjoyed thorough spectroscopic characterisation in the past. Although the derived values of the primary semi-amplitude, K 1 are essentially equivalent throughout the literature, the derived values of K 2 change by 4σ between studies (Popper 1974; Popper and Hill 1991; Stickland et al. 1992). This necessarily results in different derived values of the mass ratio q = M2 /M1 = K 1 /K 2 . There are three likely sources of the discrepancy. First, different studies used data sets, each with different trends and observational issues. Second, the method for extracting radial velocities (RVs) is different between some studies, and furthermore, the use of a cross-correlation function to determine RVs (which was used by Popper and Hill (1991) and Stickland et al. (1992)) will produce different values depending on the type of mask used. Finally, due to the small number of RV measurements at their disposal, Stickland et al. (1992) decided to fix the eccentricity, whereas other studies let it vary in their solutions. Following the RV characterisation of CW Cep, there are several estimates of the components temperatures as well. As with the RV determination, there is a range of solutions for the primary effective temperature, spanning some 3 000 K. The highest temperature estimate comes from Clausen and Gimenez (1991) who derive Teff,1 = 28 000 ± 1 000 K using colour-colour photometric relations. The lowest temperature estimate comes from Terrell (1991) who find Teff,1 = 25 400 K by taking the typical temperature according to a B0.5 spectral classification. In both of these cases, the secondary effective temperature was estimated through lightcurve modelling, but have a smaller spread in solutions (only 1 300 K). Owing to the precise determination of its fundamental parameters, CW Cep has been the focus of several evolutionary modelling attempts. Current age estimates based on evolutionary modelling range from τ = 4.1 − 10.1 Myr, within uncertainties (Clausen and Gimenez 1991; Ribas et al. 2000b; Schneider et al. 2014). However, it is important to note that the youngest age estimate was based on models that allowed for variable metallicity and helium content (Ribas et al. 2000b), introducing the degeneracy discussed in Chap. 2 and displayed in Fig. 2.1. Using Bonn rotating stellar evolution models and matching the rotation rate to that of the observed v sin i values, Schneider et al. (2014) determined an age of approximately 6 Myr. However, Ribas et al. (2000b) used a more massive solution to model the system than Schneider et al. (2014) and Clausen and Gimenez (1991), necessitating a younger age determination. As it is a member of the Cep OB3 association (Blaauw et al. 1959), independent estimates of the age of CW Cep are possible. Exploiting this, Blaauw (1961) determined an average age of 10 Myr for the subgroup of stars containing CW Cep. Later work by Jordi et al. (1996) determined an age of 7.5 Myr

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for the subgroup containing CW Cep, placing the age inbewteen those estimates by Schneider et al. (2014) and Clausen and Gimenez (1991). Furthermore, CW Cep is known to undergo rapid apsidal motion, with the most recent solution having an apsidal period of 46.2 ± 0.5 yr (Nha 1975; Han et al. 2002; Erdem et al. 2004; Wolf et al. 2006). However, to date there is no commonly accepted explanation for the driving mechanism. Recently,

4.2.2 U Ophiuchi Similar to CW Cep, U Oph is a well studied, bright, albeit less massive, double-line EB, with several published lightcurve and orbital solutions. U Oph is comprised of two B5 dwarf stars in the middle of their MS evolution. AS with CW Cep, the range of reported masses and radii for the components of U Oph exceed the precision reported on individual solutions. For example, the primary has mass and radius estimates ranging from M1 = 4.93 − 5.27 M and R1 = 3.29 − 3.48 R , while the secondary has mass and radius estimates ranging from M2 = 4.56 − 4.78 M and R2 = 3.01 − 3.11 R (Holmgren et al. 1991; Vaz et al. 2007; Wolf et al. 2006; Budding et al. 2009). U Oph represents another case of a well studied object where several different studies report different mass ratios for the system, ranging from q = 0.89 − 0.94 (Popper and Hill 1991; Holmgren et al. 1991; Vaz et al. 2007; Budding et al. 2009). Again, the spectra from which these estimates were derived were taken with different instruments (Lick Observatory with plate spectra, Reticon slit spectra from the Dominion Astrophysical Observatory, ESO plate spectra and échelle spectra from the hercules spectrograph in New Zealand). Furthermore, the different number of spectra used in each study and the SNR of the spectra have consequences on the derived orbital solutions. Furthermore, analysis of these different spectra have lead to a large range of effective temperature estimates for both components, with differences bewteen solutions up 3 000 K (Clements and Neff 1979; Eaton and Ward 1973; Holmgren et al. 1991; Andersen et al. 1990; Budding et al. 2009). The existing light curve solutions to date are based on one of three data sets: (i) unfiltered photometric measurements (Huffer and Kopal 1951), (ii) single filter space photometry assembled by the OAO-2 space mission (Eaton and Ward 1973), or (iii) multi-colour Strömgren photometry taken with the ESO 1.5 m telescope (Vaz et al. 2007). As mentioned above, the solutions based on these data result in a range of reported fundamental parameters for this system. The extensive photometric and spectroscopic data sets of U Oph have enabled the detailed study of its apsidal motion, which is thought to be caused by a distant M3 ≈ 1 M third body on an P3 ≈ 38 yr orbit (Koch and Koegler 1977; Kaemper 1986; Wolf et al. 2002; Vaz et al. 2007; Budding et al. 2009). Despite the relatively well understood nature of the tertiary, estimates of the apsidal period, U , range from U ≈ 20 − 55 yr (Frieboes-Conde and Herczeg 1973; Panchatsaram 1981; Wolf et al. 2002; Vaz et al. 2007).

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85

Despite the extensive spectroscopic data sets, however, there remains uncertainty in the bulk metallicity of U Oph. This uncertainty has also propagated into evolutionary modelling attempts of the system through the age-mass-metallicity degeneracy discussed in Chap. 2. As a result, age estimates range from τ = 30 − 63 Myr, depending on the adopted metallicity and input physics for the underlying stellar models (Holmgren et al. 1991; Vaz et al. 2007; Schneider et al. 2014). The largest discrepancy in solutions come from using different metallicities, as demonstrated by Vaz et al. (2007) who find an age of τ = 62 Myr for isochrones with Z = 0.01 and τ = 30 Myr for isochrones with Z = 0.03.

4.3 Data In this Chapter, we make use of archival photometry and new modern high resolution spectroscopy to determine the fundamental parameters of the CW Cep and U Oph systems. In this section, we briefly discuss the data.

4.3.1 New Spectroscopy and Disentangling Both CW Cep and U Oph were observed using the high-resolution Échelle Spectrograph hermes (R∼85 000 Raskin et al. 2011), attached to the 1.2 m Mercator Telescope located at the Observatorio del Roque delos Muchachos in La Palma, Canary Islands, Spain. All data obtained with hermes is reduced and order merged by the local pipeline. Subsequent normalisation is performed manually by fitting a spline to the continuum. We obtained a total of 18 new observations of CW Cep with an average SNR of 110 and 11 new observations of U Oph with an average SNR of 145, over 1.5 years between January 2015 and August 2016 and 10 nights in April 2016, respectively. We do note, however, that the majority (15 of 18) spectra of CW Cep were obtained in the span of one month. Over the full range of their observations, we calculate that the argument of periastron of CW Cep advanced approximately 10◦ , whereas the argument of periastron of U Oph advanced approximately 4◦ , according to their latest apsidal motion calculations.

4.3.2 Archival Photometry For both CW Cep and U Oph, we utilise ground-based photometric data sets initially assembled and analysed by other studies. For CW Cep, we revisit the differential Strömgren ubvy data analysed by Clausen and Gimenez (1991). These data span three years and consist of 1396 photometric observations in the uby filters and 1318

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observations in the v filter, and have a stability of 0.004 mag in all filters, which we adopt as our uncertainty (Clausen and Gimenez 1991). All observations used HD 217035 and HD 218342 as comparison stars, and extinction corrections were applied nightly according to coefficients derived from both the comparison stars and other standard stars (Gimenez et al. 1990). In the case of U Oph, we utilise the 645 Strömgren bvy data taken by the 0.5 m European Souther Observatory (ESO) SAT telescope in La Silla, Chile. In order to determine precise differential magnitudes, three photometric comparison stars (HR 6367, HR 6353, and SAO 122251) were used. Furthermore, nightly extinction corrections were determined similarly to the case of CW Cep. As initially reported by Vaz et al. (2007), who initially analysed this data set, we adopt an uncertainty of 0.0037 mag in all filters.

4.4 Modelling Due to the similarities between the systems, much of the analyses methods are similar, if not identical. In order to avoid repetition, we briefly summarise the general methods we use. Notably, as the components of the systems are sufficient different in temperature, CW Cep and U Oph require different codes to perform atmospheric modelling. Aside from this, the methods are the same and vary only slightly in their direct application to each system. Finally, atmospheric and light curve modelling share overlapping stellar parameters and both suffer from degeneracies. In order to optimise the overlap of information and mitigate degeneracies, we adopt an iterative approach whereby we first model the atmospheres of the stars, then propagate that information into the light curve model. The results of the light curve modelling are then fed back into the atmospheric modelling, and so forth, until convergence is reached. For brevity, we describe each process individually.

4.4.1 Spectroscopic Disentangling and Atmospheric Modelling Since an observed spectrum of a binary star is composed of the light from both stars in the system, we must first disentangle the two individual component spectra before we can analyse them. To do this, we rely on the technique of Spectral Disentangling, hereafter SPD. Using a set of spectra that has good phase coverage, this technique solves for the radial velocity shifts of both stellar components simultaneously in order to optimise the orbital parameters and produce two disentangled spectra which are combined from all of the individual spectra used for disentangling. We use the FDBinary code, which utilises Fourier decomposition in order to efficiently perform SPD (Hadrava 1995; Ilijic et al. 2004). Furthermore, this technique avoids the use

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Table 4.1 Orbital parameters determined by method of spectral disentangling. The periods were fixed from photometry in these calculations. This table was originally published by Johnston et al. c (2019), their Table 1. Reproduced with permission from Astronomy & Astrophysics, ESO Param. Unit CW Cep U Oph P Tper e ω K1 K2

d d – deg km s−1 km s−1

2.72913159 57608.72 ± 0.05 0.0298 ± 0.0008 218.7 ± 5.7 211.1 ± 0.4 230.2 ± 0.4

1.67734590 – 0. 90. 181.1 ± 0.6 200.6 ± 0.8

q M1 sin3 i M2 sin3 i a sin i

– M M R

0.917 ± 0.002 12.66 ± 0.05 11.61 ± 0.04 23.78 ± 0.03

0.903 ± 0.005 5.08 ± 0.04 4.59 ± 0.04 12.65 ± 0.03

of artificially constructed masks for cross-correlation, which has been shown to introduce un-necessary uncertainties if the correct line mask is not used (Pavlovski and Hensberge 2010). The results of our disentangling for both systems are listed in Table 4.1. Once we have obtained the disentangled component spectra, we then subject them to atmospheric fitting. As mentioned above, the atmospheric modelling approach is different for each system. This is due to the fact that atmospheres well above 15 000 K require non-local thermodynamic equilibrium (NLTE) effects to be taken into account. As such, we employ different atmosphere modelling codes for the two systems.

4.4.1.1

CW Cep

Using the new high resolution heres échelle spectra, we derive an updated orbital solution for CW Cep. The results of the new orbital optimisation using spd are listed in Table 4.1. We find that our updated solution is within 1σ and 2σ of the solutions reported by Stickland et al. (1992) and Popper and Hill (1991), respectively. We subject the newly disentangled spectra to atmospheric fitting as well. Below, we briefly describe the atmospheric fitting process as follows from Pavlovski et al. (2018). The individual disentangled spectra have been re-normalised according to their respective light ratios as determined from successive iterations of the disentangling and light curve modelling processes. The atmospheric modelling is performed by optimising over a grid of pre-computed non-local thermodynamic equilibrium models with the starfit code (Tamajo et al. 2011; Kolbas et al. 2014). The pre-computed models are calculated using the spectral synthesis code detail/surface and atlas9 atmosphere models (Giddings 1981; Butler et al. 1984). The grid nominally covers

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Table 4.2 Atmospheric parameters derived from an optimal fitting of re-normalised disentangled spectra for the components of CW Cep and U Oph. For CW Cep a grid of NLTE synthetic spectra was used, whilst for U Oph a grid of LTE synthetic spectra was used. The quantities given without the uncertainties were fixed in the calculation. This table was originally published by Johnston et al. c (2019), their Table 2. Reproduced with permission from Astronomy & Astrophysics, ESO Component Teff (K) log g (dex) ξt (km s−1 ) v sin i (km s−1 ) CW Cep A CW Cep B U Oph A U Oph B

28 300 ± 460 27 550 ± 420 16 580 ± 180 15 250 ± 100

4.079 4.102 4.073 4.131

2.0 ± 0.5 1.5 ± 0.5 2.0 2.0

105.2 ± 2.1 96.2 ± 1.9 110 ± 6 108 ± 6

log g ∈ 3.5−4.5 dex and Teff ∈ 15 000−32 000 K. However, given the high precision and independent determination of log g from light curve modelling, we elect to fix log g in this optimisation. Furthermore we fix the metallicity to solar, i.e., [M/H]=0, and we fix the micro-turbulence ξmiro = 2 km s−1 in this atmospheric modelling. Thus, we are left with four free parameters per star: v sin i, light factor, doppler shift, and Teff . As an a-posteriori check, we ensure that the light factors sum to unity. We optimise these parameters over the spectral range from 4 000- 4 700 Å using the PIKAIA genetic algorithm, and determine parameter uncertainties via MCMC (Charbonneau 1995; Ivezi´c et al. 2014; Kolbas et al. 2014). We choose this spectral range to include H γ , H δ, as well as several helium lines. Including helium lines allows us to finely tune the effective temperature through optimising the ionisation balance. Furthermore, by fixing the log g, we are able to further constrain the effective temperature using the Balmer lines, which would otherwise suffer a degeneracy between Teff and log g. Our analysis reveals a Teff = 28 300 ± 460 K primary contributing 56.5 ± 0.5% of the light and a Teff = 27 550 ± 420 K primary contributing 42.5 ± 0.5%. The full atmospheric solution is listed in Table 4.2, and fits to select HeI and HeII lines of both components are displayed in Fig. 4.1. We note that our atmospheric solution is closest to that of Clausen and Gimenez (1991). Furthermore, we note that we detect H α emission in the new Hermes spectroscopy. The double-peaked H α emission can be seen in four separate spectra taken at phases quadrature, with a synthetic H α absorption profile at an orbital phase of 0.25 plotted in red for reference. If the emission were originating from one, or both, of the components, we would expect the intensity ratio between the two peaks and the velocity difference between the peaks to be variable over the course of the orbit. Instead, we observe that the intensity ratio is stable at roughly 0.95, and velocity difference is roughly constant with a value of ∼105 km s−1 . Normally, the presence of H α emission could indicate that we are dealing with Be stars or potentially recent mass-transfer. However, we find no evidence of phase variability of the emission. Therefore, we suggest that this observed emission is caused by either a circumbinary envelope or from the extensive HI nebula in which the Cep OB3 association is situated (Fig. 4.2).

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Fig. 4.1 Determination of the Teff for the components of CW Cep: the primary component (upper panels), the secondary component (bottom panels). The quality of the fits is presented for the Hei λ4388 and HeII λ4541 lines (left column), and the HeI λ4471 and HeII λ4686 lines (right column). This figure was originally published by Johnston et al. (2019), their Fig. 1. Reproduced c with permission from Astronomy & Astrophysics, ESO

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4 Estimating the Convective Core Mass for Stars in Eclipsing Binaries

Fig. 4.2 Selected spectra corresponding to quarter phases centred around H α showing constant emission throughout the orbit. A synthetic composite spectrum of CW Cep at quarter phase is shown in red for comparison. The RV is calculated in the rest frame of the system. This figure was originally published by Johnston et al. (2019), their Fig. 2. Reproduced with permission from Astronomy c & Astrophysics, ESO

Following the mean atmospheric determination, we also determine the photospheric composition for both components of CW Cep. We do this by calculating mean atmospheric models from the parameters listed in Table 4.2 and then varying the individual abundances of carbon, nitrogen, oxygen,magnesium, and silicon. The results of this optimisation are listed in Table 4.3 along side the abundances derived for a sample of OB binaries (Pavlovski et al. 2018) as well as from a sample of early B-type stars (Nieva and Przybilla 2012). Comparison of these values demonstrates general agreement. Finally, we are able to determine the overall metallicity of CW Cep by using MgII as a proxy in place of iron lines which are not present in early B-type stars such as CW Cep. Comparing the abundance of MgII, to values determined from the Sun log  (Mg)=7.55±0.02 (Asplund et al. 2009) and from a sample of 52 un-evolved B-type stars log (Mg) =7.59±0.15, we determine that CW Cep has a solar metallicity (Z = 0.014).

4.4.1.2

U Oph

As with CW Cep, we subjected the new hermes spectra of U Oph to spd. Despite the known small eccentricity, we decide to fix e = 0 given the relatively low number of spectra. The resulting optimised orbital parameters are listed in Table 4.1. Instead of using the same code to perform atmosphere modelling as with CW Cep, we instead use the LTE Grid Search in Stellar Parameters (GSSP, Tkachenko 2015) code since both stars are around 15 000 and are relatively un-evolved. As its name implies, GSSP optimises atmospheric parameters over a pre-computed grid of synthetic stellar spectra. The grid of synthetic spectra are calculated based on LLmodels model

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Table 4.3 Abundances determined for the components of binary system CW Cep. The atmospheric parameters used for the calculation of model atmospheres are given in Table 4.2. For comparison the mean abundances for a sample of OB binaries given in Pavlovski et al. (2018) and for the ‘presentday cosmic standard’ determined for a sample of a single sharp-lined B-type stars in Nieva and Przybilla (2012) are also presented. This table was originally published by Johnston et al. (2019), c their Table 3. Reproduced with permission from Astronomy & Astrophysics, ESO Star

C

N

O

(N/C)

CW Cep A

8.30 ± 0.07

7.79 ± 0.08

8.71 ± 0.07

−0.51 ± 0.11 −0.92 ± 0.11 7.55 ± 0.08

(N/O)

Mg

Si 7.49 ± 0.06

CW Cep B

8.24 ± 0.07

7.70 ± 0.08

8.70 ± 0.06

−0.54 ± 0.11 −1.00 ± 0.10 7.53 ± 0.09

7.45 ± 0.07

OB binaries

8.26 ± 0.05

7.70 ± 0.04

8.71 ± 0.04

−0.56 ± 0.06 −1.01 ± 0.06 7.59 ± 0.08

7.57 ± 0.10

B single stars

8.33 ± 0.04

7.79 ± 0.04

8.76 ± 0.05

−0.54 ± 0.06 −0.97 ± 0.06 7.56 ± 0.05

7.50 ± 0.05

atmospheres with the Synth radiative transfer code (Tsymbal 1996; Shulyak et al. 2004). GSSP employs a simple χ 2 metric for the goodness-of-fit of a given model and 1σ parameter uncertainties are estimated via critical χ 2 value based on the number of free parameters. GSSP has the functionality to fit the spectrum of a single star (GSSP_single), the disentangled spectra of a double-lined spectroscopic binary (GSSP_binary), or the composite spectra of a doubled-lined spectroscopic binary (GSSP_composite). In the case where the two components of a binary are sufficiently different in mass, their relative light contributions become strongly dependent on wavelength, which necessitates the simultaneous fitting of the disentangled component spectra. In the case where the stars have similar masses, and hence similar light contributions as a function of wavelength, the disentangled spectra can be fit separately with the a posteriori check that the derived light contributions sum to unity. For further discussion, we direct the reader to Tkachenko (2015). Given that the components of U Oph are of similar masses, we subject the disentangled spectra to independent fitting with the GSSP_single module. For U Oph, we also adopt an iterative approach, whereby we first determine light ratios from disentangling and GSSP, then use them as priors in the light curve modelling. We then take the dynamical surface gravitites derived from the light curve modelling and fix it in the next iteration of atmospheric modelling. The results of our atmospheric modelling of U Oph are displayed in Table 4.2. We find that the primary contributes 57.5±0.7% of the total light and the secondary contributes 42.5±0.8% of the light to the system. The best fitting synthetic spectra are overplotted against the disentangled component spectra in Fig. 4.3.

4.4.2 PHOEBE Models We make use of the PHOEBE binary modelling code (Prša and Zwitter 2005); Prsa et al. 2011) in order to model the eclipses and out of eclipse variability observed in both systems. PHOEBE is a Wilson-Devinney like code which has been updated to

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4 Estimating the Convective Core Mass for Stars in Eclipsing Binaries

Fig. 4.3 Quality of the fit of the disentangled spectra (black dots) of U Oph with the synthetic spectra computed from the best fit parameters listed in Table 4.2 (red and blue solid line for the primary and secondary component, respectively). The spectra of the secondary component were vertically shifted by a constant factor for clarity. This figure was originally published by Johnston c et al. (2019), their Fig. 3. Reproduced with permission from Astronomy & Astrophysics, ESO

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93

include newer physics. As discussed in Sect. 2.4.3, we make use of an MCMC routine to determine uncertainties on the parameters we optimise. In this application, the model produced by PHOEBE for a given set of input parameters serves as the generative model to compare against the data in the likelihood function L (d|θ ) in Eq. 2.6. As input, PHOEBE requires the light ratios for each component in all photometric passbands, the temperatures of both components, the potentials of both components, the gravity and limb-darkening coefficients of each components as well as the synchronicity parameter for each component. In addition to the parameters describing each component, PHOEBE also requires input describing the orbit, including the period, the time of primary conjunction, the mass ratio, the system inclination, the semi-major axis of the orbit, as well as the orbital eccentricity and the argument of periastron. Finally, both systems are known to experience contaminating third light, likely from a nearby companion. To this end, we also include a third light contribution in all photometric filters. The inclusion of third light introduces another degeneracy in our modelling. Third light effectively scales the depth of both eclipses, making the system appear to be less inclined. In effect, more third light implies a higher inclination, and effectively more massive stars. Thus, it is important that we include this in our models, lest we find under-massive solutions. As both systems undergo relatively rapid apsidal motion, we decide to phase bin the light curves of both objects into 300 phase bins that each span 0.0033 units of phase. We were motivated to set bin-widths of this size by the rate of apsidal motion observed in each system. In this way, we effectively mitigate any apparent phasesmearing induced by the apsidal motion of these systems. Thus, the data that we feed to our MCMC routine are the phase binned light curves in all filters. Using a Bayesian approach allows us to propagate any additional information from other sources in our modelling. As such, we make efficient use of the results of the spd and atmospheric modelling by applying Gaussian priors on several parameters. Namely, we apply Gaussian priors on the light ratios in the b and v filters, the mass ratio (q), as well as on the eccentricity (e) and argument of periastron (ω), as determined from spd. However, we do not sample e and ω directly, but rather sample e sin ω and e cos ω. We subsequently solve for e and ω and apply the priors. Similarly, we apply a Gaussian prior on v sin i, as determined from the atmospheric modelling. PHOEBE does not output this value, however, we calculate it for every model. This effectively constrains the values of f 1 = ωrot,1 /ωorb and f 2 = ωrot,2 /ωorb . In order to minimise potential degeneracies, we also fix several parameters. First, in both cases we fix the effective temperature of the primary to that determined from atmospheric modelling. We then sample for the secondary effective temperature since light curve modelling is not directly sensitive to the individual component temperatures, but rather to their ratio. Second, we fix the gravity darkening coefficients and albedos of all components to unity, as is theoretically expected for such massive stars (von Zeipel 1924). Thirdly, we use the Square-root limb-darkening law and interpolate in a grid of coefficients according to a models temperature and surface gravity. All other variables are sampled with uniform priors. We determine parameter estimates and 1σ uncertainties as median and the 67% Highest Posterior Density

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4 Estimating the Convective Core Mass for Stars in Eclipsing Binaries

(HPD) intervals of the resulting marginalised posterior distributions, as discussed in Sect. 2.4.3. Finally, throughout our MCMC routine, we keep track of several output parameters, such as the stellar masses, radii, surface gravitites and fractional radii. We determine estimates and uncertainties for these parameters in the same way as sampled parameters.

4.4.3 CW Cep Similar to the fitting of HD 165246 in Sect. 3.3.2, we use the Bayesian methodology described in Sect. 2.4.3 to fit the light curves of CW Cep. Following the results of the spd and atmospheric analysis, we impose Gaussian priors on the eccentricity e ∼ N (0.0298, 0.0008), on the argument of periastron ω ∼ N (218.7, 5.7), on the mass ratio q ∼ N (0.917, 0.002), on the projected semi-major axis a sin i ∼ N (23.78, 0.03), on the secondary effective temperature T,eff ∼ N (27550, 420), and on the light ratios in the v and b filters. Furthermore, we calculate the v sin i for every model using the synchronicity parameters and apply a Gaussian prior on the v sin i per component as v1 sin i ∼ N (105.2, 2.1) and v2 sin i ∼ N (96.2, 1.9). As mentioned previously, we also include third light in our model and apply uniform priors to all remaining sampled parameters listed in Table 4.4. The MCMC routine ran for 10 000 iterations with 128 independent walkers, where we evaluated convergence and cut all iterations before five auto correlation times. The resulting parameter estimates and their uncertainties determined from our binary modelling are listed in Table 4.4. Furthermore, we list all sampled parameters with their associated priors. We use the parameter estimates listed in Table 4.4 to build an average best fitting model for the v filter, as seen in the top panel of Fig. 4.4. The residuals in all filters after subtraction of the best fit models are shown in the remaining panels. The marginalised posterior distributions for the MCMC optimised light curve fit of CW Cep are located in Appendix C.

4.4.4 U Oph While we were able to reasonably fix the eccentricity to e = 0 for the RV orbit optimisation, the small eccentricity cannot be ignored in the light curve modelling. Thus, for U Oph, we adopt the same methodology as with CW Cep, however, we do not use the u band light curve due to poor data quality. Similarly, we apply Gaussian priors on the light ratios in the b and v filters, on the eccentricity e ∼ N (0.003, 0.0001) Vaz et al. (2007), on the mass ratio q ∼ N (0.903, 0.005), on the projected semi-major axis a sin i ∼ N (12.65, 0.03), on the secondary effective temperature T,eff ∼ N (15250, 100), and on the v sin i of both components as v1 sin i ∼ N (110, 6) and v2 sin i ∼ N (108, 6). All other sampled parameters receive a uniform prior and are listed in Table 4.4.

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Table 4.4 Binary model parameters for CW Cep and U Oph. This table was originally published by Johnston et al. (2019), their Table 4. Reproduced with permission from Astronomy & Astrophysics, c ESO CW Cep Parameter

Prior

U Oph HPD Estimate

Prior

HPD Estimate

+1.1 57.3−1.0

–

–

N (57.5, 0.7)

57.2+2.3 −5.2

Sampled parameters L1,u [%]

U (40, 70)

L1,v [%]

N (56.5, 0.5)

L1,b [%]

N (56.5, 0.5)

L1,y [%]

U (40, 70)

L3,u [%]

U (0, 15)

L3,v [%]

U (0, 15)

L3,b [%]

U (0, 15)

L3,y [%]

U (0, 15)

Teff,2 [K]

N (27550, 600)

Porb [d]

U (1, 5)

HJD0 [d]

U (−2, 2) + 2441669

i [deg]

U (70, 90)

e sin ω0

U (−0.0287, 0.0287)

e cos ω0

U (−0.0287, 0.0287)

a [R ]

U (5, 40)

q=

M2 M1

N (0.92, 0.002)

1

U (4.5, 9)

2

U (4.5, 9)

f1

U (0.5, 2)

f2

U (0.5, 2)

+1.1 56.7−1.0 +1.1 56.6−1.0 +1.1 56.5−1.0

0.6+0.6 −0.5 1.9+0.6 −0.5

2.8+0.5 −0.5 +0.6 3.6−0.4

27420+150 −120

2.7291316+4e−7 −3e−7 0.5831+0.0005 −0.0006

81.804+0.006 −0.004

−0.02544+2e−5 −2e−5 0.01329+4e−5 −3e−5 24.01+0.04 −0.04

0.919+0.005 −0.005 5.39+0.05 −0.03 5.43+0.04 −0.05 1.06+0.03 −0.03 1.03+0.03 −0.03

N (57.5, 0.7) U (40, 70)

57.1+2.2 −5.1

57.0+2.2 −5.2

–

–

U (0, 15)

0.8+0.2 −0.3

U (0, 15) U (0, 15) N (15620, 200) U (1, 4) U (−2, 2) + 2449161 U (70, 90) U (−0.003, 0.003) U (−0.003, 0.003) U (5, 40) N (0.90, 0.01) U (4, 9) U (4, 9) U (0.5, 2) U (0.5, 2)

1.1+0.2 −0.2 1.3+0.2 −0.2

15820+90 −90

1.67734590+2e−8 −2e−8 +0.00003 0.61101−0.00002

+0.1 87.86−0.08

0.00189+1e−5 −1e−5 0.00233+1e−5 −1e−5 12.66+0.03 −0.03 0.90+0.01 −0.01 4.64+0.02 −0.02 4.84+0.05 −0.05 1.07+0.07 −0.07 1.16+0.08 −0.07

Geometric parameters r1 r2

+0.001 0.227−0.002

0.212+0.002 −0.001

0.2715+0.0005 −0.0005 0.2408+0.0007 −0.0009

The MCMC routine ran for 10 000 iterations with 128 independent walkers, where we evaluated convergence and cut all iterations before five auto correlation times. The best estimated parameters and their associated uncertainties from our light curve modelling are listed in Table 4.4. The best model according to these parameters is displayed in Fig. 4.5. The marginalised posterior distributions for the MCMC optimised light curve fit of U Oph are located in Appendix C.

4.4.5 Comparison of Results Derived parameters for both CW Cep and U Oph are reported in Table 4.5 alongside other solutions from the literature. Our results for CW Cep are closest to those

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4 Estimating the Convective Core Mass for Stars in Eclipsing Binaries -0.5

-0.25

-0.5

-0.25

0.0

0.25

0.0

0.25

0.5

Δv

0.2 0.3 0.4 0.5

Δu

0.05 0.0

Δv

-0.05 0.05 0.0

Δb

-0.05 0.05 0.0

Δy

-0.05 0.05 0.0 -0.05 0.5

Orbital Phase

Fig. 4.4 Top panel: CW Cep PHOEBE model (solid black) for the Stromgren v light curve (grey circles) constructed from median values reported in Table 4.4. Bottom panels: Residual light curves in the uvby filters after the best model has been removed. The dashed black line denotes the zero point to guide the eye. This figure was originally published by (Johnston et al. 2019), their Fig. 4. c Reproduced with permission from Astronomy & Astrophysics, ESO

reported by Han et al. (2002) (their solution b). Our results for U Oph are closets to the solution reported by Budding et al. (2009), where our results are significantly more massive than the solution of Holmgren et al. (1991) and slightly less massive than that of Vaz et al. (2007). Additionally, we note that we report the highest precision on fundamental parameters from across other solutions in the literature. This is likely due to a combination of our updated RV solutions and the use of an MCMC routine. We do note, that methodological choices between the different studies mentioned in Table 4.5, such as the inclusion of third light, the fixing or varying of eccentricity, as well as the method to determine RVs from spectra, likely all contribute to the spread in reported solutions. Still, the difference between any two solutions is several times greater than the reported parameter uncertainties. In order to test the consistency of our results with other data, we also compare the summed luminosities from our models to luminosities derived from Gaia and Hipparcos parallaxes. To transform the Gaia and Hipparcos parallaxes into luminosities, we require estimates for the extinction, Av , and the Bolometric Corrections (BCs). For CW Cep, we adopt Av = 1.96, as derived from the E(b − y) value reported by Clausen and Gimenez (1991) and use an average bolometric correction of

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Table 4.5 Derived parameters CW Cep and U Oph, comparing derived fundamental parameters from this work to previous studies of CW Cep (top) and U Oph (bottom). This table was originally published by (Johnston et al. 2019), their Table 5. Reproduced with permission from Astronomy & c Astrophysics, ESO CW Cep Parameter

M1 [M ] M2 [M ] R1 [R ] R2 [R ] log g1 [dex] log g2 [dex] U Oph Parameter M1 [M ] M2 [M ] R1 [R ] R2 [R ] log g1 [dex] log g2 [dex]

Gimenez et al. Clausen and (1987) Gimenez (1991) 11.9 ± 0.1 11.82 ± 0.14 11.2 ± 0.1 11.09 ± 0.14 5.40 ± 0.1 5.48 ± 0.12 4.95 ± 0.1 4.99 ± 0.12 4.05 ± 0.02 4.03 ± 0.02 4.10 ± 0.02 4.09 ± 0.02

Han et al. (2002)a

Han et al. (2002)b

This work

13.49 12.05 6.03 4.60 4.01 4.19

12.93 11.84 5.97 4.56 3.99 4.19

13.00+0.07 −0.07 11.94+0.08 −0.07 5.45+0.03 −0.06 5.09+0.06 −0.03 4.079+0.010 −0.005 +0.005 4.102−0.010

Holmgren et al. (1991) 4.93 ± 0.05 4.56 ± 0.04 3.29 ± 0.06 3.01 ± 0.05 4.10 ± 0.01 4.14 ± 0.02

Budding et al. (2009) 5.13 ± 0.08 4.56 ± 0.07 3.41 ± 0.03 3.08 ± 0.03 4.08 ± 0.01 4.12 ± 0.01

This work

Vaz et al. (2007) 5.273 ± 0.091 4.783 ± 0.072 3.483 ± 0.020 3.109 ± 0.034 4.068 ± 0.010 4.128 ± 0.012

5.09+0.06 −0.05 4.58+0.05 −0.05 3.44+0.01 −0.01 3.05+0.01 −0.01 4.073+0.004 −0.004 4.131+0.004 −0.004

BCv = 2.95 ± 0.05 (Reed 1998). For U Oph, we adopt Av = 0.72 ± 0.2 according to Vaz et al. (2007) and BC = −1.5 ± 0.05 (Reed 1998). Our model of CW Cep results in a summed system luminosity of (log (L/L  ) = 4.48 ± 0.02, compared to Gaia and Hipparcos luminosities of log (L G /L  ) = 4.78 ± 0.41 (based on πG = 1.04 ± 0.49 mas, Luri et al. (2018); Lindegren et al. (2018)) and log (L G /L  ) = 4.42 ± 0.4 (based on π H = 1.57 ± 0.69 mas, van Leeuwen (2007)). Based on these, we find general agreement. Our model of  L U Oph results in a summed system luminosity of log L  = 3.12 ± 0.01, com  pared to a Gaia luminosity of log LL G = 3.27 ± 0.09 (based on πG = 3.74 ± 0.13 mas, Luri  et al. (2018); Lindegren et al. (2018)) and an Hipparcos luminosity of L log L  = 3.02 ± 0.1 (based on π H = 4.99 ± 0.41 mas, van Leeuwen (2007)). In this case, we find that our luminosity does not agree with the Gaia luminosity, but does agree with the Hipparcos luminosity to within 1σ . In both cases, it is likely that a distant third body has an impact on the estimated Gaia and Hipparcos parallaxes.

98

4 Estimating the Convective Core Mass for Stars in Eclipsing Binaries -0.5

-0.25

-0.5

-0.25

0.0

0.25

0.5

0.0

0.25

0.5

−0.2

Δv

0.0 0.2 0.4

Δv

0.05 0.0

Δb

-0.05 0.05 0.0

Δy

-0.05 0.05 0.0 -0.05

Orbital Phase

Fig. 4.5 Top panel: U Oph PHOEBE model (solid black) for the Stromgren v light curve (grey circles) constructed from median values reported in Table 4.4. Bottom panels: Residual light curves in the vby filters after the best model has been removed. The dashed black line denotes the zero point to guide the eye. This figure was originally published by (Johnston et al. 2019), their Fig. 5. c Reproduced with permission from Astronomy & Astrophysics, ESO

4.5 Evolutionary Modelling We now seek to use the updated fundamental stellar parameters of the components of CW Cep and U Oph to scrutinise stellar evolutionary models. We keep in mind, however, the results of Constantino and Baraffe (2018) and Valle et al. (2018) who demonstrate that, without independent, cross-constraining information, we cannot uniquely determine the efficiency of a given near-core mixing mechanism, even with such high precision. In light of this, as well as the discussions in Chaps. 1 and 2, we investigate our ability to uniquely determine the mass of the convective core, which will be influenced by the overall CBM and REM present in our models. As outlined in Chap. 2, we treat CBM, parameterised by a diffusive exponentially decaying overshooting prescription with a scaled extent f CBM = f ov (Eq. 1.19) as a nuisance parameter. In order to allow for the full range of CBM, we use the isochrone-clouds as described in Sect. 2.3. We employ a Monte Carlo approach using 10 000 iterations with the Mahalanobis distance (MD) as our merit function in order to carry out the evolutionary modelling using the isochrone-clouds. The Monte Carlo methodology is described in

4.5 Evolutionary Modelling

99

Table 4.6 Monte Carlo Isochrone-cloud modelling 95% confidence intervals for CW Cep and U Oph. This table was originally published by (Johnston et al. 2019), their Table 6. Reproduced c with permission from Astronomy & Astrophysics, ESO Parameter CW Cep U Oph Age [Myr] f ov,1 f ov,2 M1 [M ] M2 [M ] X c,1 X c,2 Mcc,1 [M ] Mcc,2 [M ]

7.0+1 −1 0.025+0.015 −0.02 +0.01 0.030−0.02 +0.1 13.00−0.16 +0.11 12.00−0.12 0.54+0.01 −0.03 0.57+0.01 −0.03 4.34+0.11 −0.29 3.86+0.12 −0.19

57.5+5.0 −2.5 0.025+0.015 −0.015 0.015+0.015 −0.01 5.08+0.07 −0.06 4.60+0.05 −0.05 0.48+0.02 −0.04 +0.03 0.51−0.02 +0.08 1.05−0.11 0.93+0.06 −0.05

Sect. 2.4.2.2, and is used here to derive uncertainties given that when used alone the MD is a maximum-likelihood point estimator with no associated uncertainties. We draw 1σ uncertainties as the 95% HPD confidence intervals of the resulting binned distributions. As discussed in Sect. 2.4.1 and Aerts et al. (2018), the MD accounts for underlying parameter correlations present in the grid of stellar models, or in our case in the isochrone-clouds. The resulting parameter estimates and their uncertainties are listed in Table 4.6.

4.5.1 Modelling Results and Discussion In agreement with the results of Constantino and Baraffe (2018) and Valle et al. (2018), we were not able to constrain the efficiency of CBM in either component of either system, even with such high precision. Instead, as can be seen in Table 4.6, we were able to constrain the values of the convective core mass, Mcc and remaining core hydrogen content X c for all components. Furthermore, we are able to determine precise ages for both systems using our isochrone-cloud methodology. We visualise the allowed values for the Mcc and X c of both components in isochrone-clouds within their estimated ages for CW Cep and U Oph in Figs. 4.6 and 4.6, respectively. The regions shown in blue are those allowed considering only models within the estimated dynamical masses, whereas the grey regions denotes the solution space allowed by considering the spectroscopic temperatures and dynamical surface gravities. When considering the solution space required by CW Cep B, we see that, at these ages, the secondary requires a more massive core to be co-eval with the primary. Additionally, the secondary and primary both have similar core hydrogen contents, despite the fact that the more massive primary should be more progressed along the MS. Together, these two observations concerning the Mcc and X c imply that both components

100

4 Estimating the Convective Core Mass for Stars in Eclipsing Binaries

require more massive cores, enhanced through internal chemical mixing, in order to have their observed properties at their age. When applying the same logic for the components of U Oph, we see that the primary requires a more massive Mcc and higher X c in order to match with the observed properties of the secondary at the considered ages. The solution for the secondary resides in the middle of the allowed solution space, implying that the required enhancement to the core is not as large as it is for the primary. The localisation of Mcc and X c for CW Cep and U Oph demonstrates that these inferred parameters can serve as an indicator of enhanced internal mixing, without invoking the myriad of issues concerning the implementation of a given CBM or REM mechanism. As mentioned above, the spread of reported dynamical parameters for CW Cep and U Oph is reason for concern, particularly in subsequent evolutionary modelling as is carried out here. This concern is largely directed towards any inferred age estimates or internal mixing efficiencies. Simply scanning the literature for different age estimates for the same system, as was briefly recounted in Sect. 4.2, demonstrates that our age estimates for both systems as listed in Table 4.6 largely agree with the values from literature. However, we do note that our age estimate of CW Cep is nearly a factor two that of Ribas et al. (2000a). Interestingly, our age estimate of CW Cep agrees best with the independent age estimate of the older sub-group within the Cep OB3 association by Jordi et al. (1996). Notably, our modelling of U Oph enforced a solar metallicity in light of the results of our atmospheric modelling. Our age estimate of 57.5 MYr for U Oph (assuming Z = 0.014) lies directly in between the two estimates of 62 and 52 Myr Vaz et al. (2007) which assumed Z = 0.017 and Z = 0.010, respectively. To this end, we find the best agreement with the solution of Vaz et al. (2007). We do note that our age estimate is nearly 20 Myr older than those of Budding et al. (2009) and Schneider et al. (2014), who used tracks with Z = 0.020 and Z = 0.014. In further comparison to the results of Schneider et al. (2014), who modelled both CW Cep and U Oph with rotating Bonn stellar evolution models with solar metallicity, we find much less evolved solutions for both systems. While Schneider et al. (2014) find the primary and secondary of CW Cep to be progressed ∼35−40% and ∼30−35% of the way through the MS, respectively, we find them to be less progressed at ∼27% and ∼21% of their total MS lifetime. Similarly, we find the components of U Oph to be ∼40% and ∼31% through the MS, while Schneider et al. (2014) find the components to be 50% and 40% through the MS. Although our study uses different evolutionary tracks to theirs, the cause of the discrepancy is likely the fact that Schneider et al. (2014) used different fundamental parameters for the systems from the literature. If one is to attempt to calibrate the efficiency of a given CBM or REM mechanism for a single system with different reported dynamical solutions, let alone a sample of heterogeneously analysed systems from across the literature, the results would be mired in different systematic offsets induced by different methodologies and assumptions. To this end, we argue that the only reliable means of scrutinising stellar evolution models is by using a sample of homogeneously analysed stars/systems. We return to this idea in Sect. 4.7.

4.5 Evolutionary Modelling

101

4.5.2 Rotation and Tides In our light curve analysis, we added a prior on v sin i and calculated the estimated v sin i for every model using the synchronicity parameter f k . This allowed us to constrain how rapidly the components are rotating with respect to the orbit. As can be seen by comparing the synchronicity parameters for all components in Table 4.4, the secondary component of CW Cep and the primary component of U Oph are rotating synchronously, while the primary of CW Cep and the secondary of U Oph are rotating super-synchronously. Using the estimated system inclinations, and assuming spinorbit alignment, we can estimate the un-projected equatorial rotational velocities for all components. We find that the components of CW Cep rotate with velocities v1 = 107 ± 3 km s−1 and v2 = 97 ± 3 km s−1 . For U Oph, we find that the components rotate with velocities v1 = 111 ± 7 km s−1 and v2 = 107 ± 6 km s−1 . We also compare these values to the (Keplerian) critical rotation rates (Maeder 2009; Aerts 2021) calculated according to the values listed in Table 4.5. According to our calculations, we find that the primary and secondary of CW Cep rotate at 19.6 ± 0.6% and 17.7 ± 0.6% of their critical values, respectively. Additionally, we find that the primary and secondary rotate at 25.6 ± 2% and 24.4 ± 1% of their critical values, respectively. With such precise determinations of the stellar parameters, we can test whether the objects agree with predictions for synchronisation and circularisation based on tidal theory (Zahn 1975, 1977). For CW Cep we find a synchronisation time of τsync = 0.606 ± 0.002 Myr and a circularistion time of τcirc = 62.9 ± 1 Myr. Interestingly, we find CW Cep to be a factor ten older than the time required to synchronise the system, and a factor ten younger than the time required to circularise the system. Given the current state of the synchronisation and the eccentricity of the system, we find it to be in agreement with theory. However, the system will evolve beyond the MS before the system reaches the circularisation time. For U Oph, we find a synchronisation time of τsync = 0.0879 ± 0.0005 Myr and a circularisation time of τcirc = 4.63 ± 0.02 Myr. Again, we find the system to be consistent with the estimates for synchronisation. However, the age we derive for U Oph is an order of magnitude older than the time required to circularise the system (Fig. 4.7). Despite this, we still observe a small, but significant eccentricity to the system. We postulate that this is likely caused by the presence the same distant third body which is causing the apsidal motion, and is likely causing Kozai-Lidov cycles (Naoz 2016).

4.6 Interpretation Despite the highly precise determination of fundamental stellar parameters, we were not able to statistically constrain the efficiency of CBM included in our stellar evolution models. This result is in line with the results of studies such as Viallet et al. (2015) which claim that modern observational precision on fundamental stellar parameters

102

4 Estimating the Convective Core Mass for Stars in Eclipsing Binaries 6.5

6.0 Myr 7.0 Myr 8.0 Myr

6.0

Mcc [M ]

5.5

5.0

4.5

4.0

3.5

3.0 0.60

0.55

0.50

0.45

Xc

Fig. 4.6 Convective core mass plotted against the core hydrogen content for the isochrone-clouds listed for CW Cep in Table 4.6. Those regions which are allowed by the spectroscopic uncertainties for CW Cep A and B are shown as grey cross and vertical hatched regions, respectively. Those regions corresponding to models that fall within the estimates of the dynamical masses for CW Cep A and B are shown as blue cross and vertical hatched regions, respectively

cannot constrain implementations of CBM processes. In lieu of this, however, we demonstrate that we are able to infer the age, convective core mass, and fractional core hydrogen content to a high precision. This results indicates a novel direction for the investigation of the ‘mass-discrepancy’ based on the convective core mass instead of the efficiency of a particular CBM process. Importantly, we highlight the spread in derived masses (and mass ratios) for CW Cep and U Oph from the literature and this work. This spread in solutions demonstrates that different methodologies can, and likely will, result in discrepant solution. To this end, any investigation into the ‘mass discrepancy’ and its cause requires a homogeneously analysed sample in order to mitigate any underlying systematic differences introduced by differing methodologies. Such a homogeneously analysed sample is the focus of the following section.

4.7 Expanded Sample

103 55.0 57.5 60.0 62.5

1.3

Myr Myr Myr Myr

1.2

Mcc [M ]

1.1

1.0

0.9

0.8

0.7

0.65

0.60

0.55

0.50

0.45

0.40

0.35

Xc

Fig. 4.7 Same as Fig. 4.6, but for U Oph

4.7 Expanded Sample Following the proof-of-concept that enhanced core masses can be inferred from EBs provided by the cases of CW Cep and U Oph, we consider the expansion of this to a larger sample involving eight more intermediate- to high-mass eclipsing systems. The sample is listed in Table 4.7 and is plotted in Fig. 4.8. These systems constitute a sample which was homogeneously analysed to remove any variance in the systematics due to different methodologies. The homogeneity of this sample enables us to investigate if convective core mass enhancement correlates with any observed, derived, or modelled quantities. Furthermore, this allows us to directly quantify the mass-discrepancy across a wide range of masses and evolutionary stages, as shown by the diversity of components in Fig. 4.8. As discussed in Sect. 1.7, the mass-discrepancy refers to the difference in dynamical mass (as obtained via binary modelling) and evolutionary mass (as determined via fitting spectroscopic quantities to a grid of stellar models, Guinan et al. 2000; Tkachenko et al. 2014). To quantify this mass-discrepancy, we consider the components of the binary systems to be single stars, and as such we do not require the full isochrone-cloud modelling as we no longer enforce the equal age assumption in the modelling. Furthermore, this solution considers a fixed amount of DREM = 100 cm2

104

4 Estimating the Convective Core Mass for Stars in Eclipsing Binaries

Table 4.7 Observed fundamental and atmospheric parameters of the sample targets. Error bars are given in parentheses in terms of the last digit(s). The superscript in the first column identifies the study the parameters have been taken from. For each object, the first/second line corresponds to the primary/secondary component. This table was originally published by (Tkachenko et al. 2020), c their Table 1. Reproduced with permission from Astronomy & Astrophysics, ESO Object/ M R log(Teff ) log g (M ) (R ) (dex) (dex) Parameter V578 Mon1 V453 Cyg2 V478 Cyg2 AH Cep2 V346 Cen3 V573 Car3 V1034 Sco3 V380 Cyg4 CW Cep5 U Oph5

14.54(8) 10.29(6) 13.90(23) 11.06(18) 15.40(38) 15.02(35) 16.14(26) 13.69(21) 11.78(13) 8.40(10) 15.14(39) 12.38(20) 17.07(12) 9.60(5) 11.43(19) 7.00(14) 13.00(7) 11.94(7) 5.09(5) 4.58(5)

5.41(4) 4.29(5) 8.62(9) 5.45(8) 7.26(9) 7.15(9) 6.51(10) 5.64(11) 8.26(16) 4.19(8) 5.41(5) 4.48(5) 7.49(7) 4.20(4) 15.71(13) 3.82(5) 5.45(5) 5.09(5) 3.44(1) 3.05(1)

4.477(7) 4.411(7) 4.459(8) 4.442(10) 4.507(7) 4.502(9) 4.487(8) 4.459(10) 4.417(5) 4.352(6) 4.504(5) 4.458(5) 4.508(7) 4.412(5) 4.336(6) 4.356(22) 4.452(7) 4.440(7) 4.220(4) 4.183(3)

4.133(18) 4.185(21) 3.710(9) 4.010(12) 3.904(9) 3.907(10) 4.019(12) 4.073(18) 3.675(17) 4.118(16) 4.151(7) 4.229(7) 3.921(8) 4.173(9) 3.104(6) 4.120(11) 4.079(10) 4.102(10) 4.073(4) 4.131(4)

1 Garcia et al. (2014); 2 Pavlovski et al. (2018); 3 Pavlovski et al. (in prep.); 4 Tkachenko et al. (2014); 5 This

chapter

s−1 , for all models and considers only one metallicity regime (Z = 0.014) to reduce the dimensionality and degeneracy of the solutions. We consider four cases: 1. Classical dynamical mass solution: (CDM solution) The stars are fit to a grid with only the lowest amount of CBM ( f CBM = 0.005), with the dynamical mass, surface gravity, and spectroscopic effective temperature in the cost-function. This solution represents the classical dynamical mass solution. 2. Classical evolutionary mass solution: (CEM solution) The stars are fit to a grid with only the lowest amount of CBM ( f CBM = 0.005), with the dynamical surface gravity, and spectroscopic effective temperature in the cost-function. In this solution, the stellar mass is an estimated parameter. This solution represents the classical evolutionary mass solution.

4.7 Expanded Sample

105

3.2

log (g/dex)

3.4

3.6

3.8

4.0

4.2

4.5

4.4

4.3

4.2

4.1

log (Teff /K)

Fig. 4.8 Sample of stars listed in Table 4.7 plotted over evolutionary tracks from 3 to 19 M , for models with minimum mixing (solid) and maximum mixing (dotted) in our grid. Primary and secondary components plotted as black circles and triangles respectively

3. Enhanced mixing dynamical mass solution: (EMDM solution) The stars are fit to a grid with varying CBM, including the dynamical mass, surface gravity, and spectroscopic effective temperature in the cost-function. 4. Enhanced mixing evolutionary mass solution: (EMEM solution) The stars are fit to a grid with varying CBM, including the dynamical surface gravity, and spectroscopic effective temperature in the cost-function. In order to quantify the mass discrepancy, we consider the CDM solution to be the reference case. We then compare the masses, convective-core masses and ages for all other solutions with reference to this in Table 4.8 in both absolute units and as a percentage relative to the CDM solution when applicable. Figure 4.9 shows the difference between the mass (upper left), convective core mass (upper right), and age (lower left) all as a function of surface gravity between the CDM and CEM solutions. This represents the mass-discrepancy as classically defined above. Examining these panels reveals that the classical mass-discrepancy can be interpreted in terms of a core mass discrepancy, where the mass discrepancy (upper left panel) increases proportionally to the discrepancy in core mass between the two solutions. Furthermore, the lower left and lower right plots, which show the age difference as a function of surface gravity and age difference as a function of mass difference, indicate that

12.19(15) B 13.66(30) 8.41(9) A 13.2(1.0) 15.95(40) B 2.77(32)

15.10(10) B 7.83(25) 16.28(15) B 4.90(20) 13.80(10) B 7.00(17)

14.10(40) B 10.9(3) 11.15(15) B 10.8(4) 15.60(20) B 7.40(20)

5.0(1.0)

3.80(50)

(Myr)

5.10(2) A 4.60(2) A

13.01(6) A 11.99(6) B

0.87(5)

57.5(7.5)

18.9(1.2)

18.2(1.1)

30.4(7)

29.5(8) 5.04(20) A 4.63(18) A

13.49(48) A 12.38(16) A

15.00(-5) B 8.55(75) A

19.90(1.0) A 10.64(24) A

16.88(39) A 12.89(30) A

14.90(20) A 8.42(22) A

17.30(40) A 15.20(30) A

17.60(60) A 17.20(60) A

16.80(70) A 12.38(50) A

(M ) 14.77(59) A 10.55(34) A

M

CEM solution

3.12(16)

7.00(60)

4.49(15)

6.41(25)

2.05(11)

3.42(2)

4.81(13)

6.09(20)

5.36(37)

5.49(40)

3.33(23)

4.43(30)

3.12(22)

(M ) 5.13(40)

Mcc

0.93(5)

3.69(14)

4.09(25)

57.6(10.0) 0.87(5)

51.3(5.8)

6.39(55)

6.50(56)

12.50(2.50) 2.12(30)

12.40(1.0) 2.41(1)

5.42(49)

5.00(50)

1.50(15)

2.50(20)

13.2(1.1)

10.50(15)

5.90(50)

4.46(29)

6.65(45)

6.32(45)

9.00(75)

8.65(35)

5.0(1.0)

3.74(55)

(Myr)

age

18.8(1.9)

18.5(1.7)

29.8(1.5)

30.3(3.1)

24.8(5.2)

16.1(1)

29.3(2.2)

35.2(5.0)

34.8(2.1)

38.0(2.4)

24.3(2.0)

23.0(4)

31.6(1.6)

35.2(2.0)

31.2(3.3)

31.2(3.5)

26.9(3.1)

26.4(3.0)

29.6(3.1)

34.7(4.3)

(%) 5.8(1.5)

4.13(77)

(Myr)

5.10(5) A 4.60(5) A

13.01(7) A 11.95(9) A

57.4(3.3)

52.9(3.0)

7.62(65)

7.09(65)

11.43(19) B,117.3(1.0) 7.06(13) B 22.6(2.7)

12.49(17) A 1.90(35) 17.15(13) B 7.00(15) 9.65(7) B 7.76(65)

12.02(14) B 15.96(18) 8.40(10) A 13.5(1.6) 15.80(30) B 3.06(28)

15.22(22) B 8.69(25) 16.26(18) A 5.41(30) 13.79(14) B 7.80(30)

14.00(20) B 12.38(40) 11.11(15) B 12.32(55) 15.58(22) B 8.21(25)

(M ) 14.55(9) A 10.30(6) A

age

EMDM solution M

0.93(7)

1.01(10)

3.86(45)

4.38(60)

1.81(30)

0.38(38)

2.88(5)

6.42(27)

4.37(25)

6.11(40)

2.04(30)

3.33(6)

4.72(35)

6.19(55)

5.25(37)

5.49(40)

3.32(34)

4.26(25)

3.21(35)

(M ) 5.12(40)

Mcc

20.2(1.8)

19.8(2.2)

32.3(4.0)

33.7(4.8)

25.6(4.8)

3.3(3.3)

29.8(8)

37.4(1.9)

35.0(2.5)

38.7(3.3)

24.3(4.1)

27.7(8)

34.2(2.9)

38.1(3.8)

34.5(3.0)

35.2(3.4)

29.9(3.5)

30.4(2.3)

31.2(3.6)

35.2(3.0)

(%)

(Myr)

2.26(30)

8.32(75) A

15.7(3.3)

3.86(5)

—–

—–

—–

—–

3.29(15)

7.44(50)

—–

6.58(30)

—–

4.14(20)

5.22(45)

—–

5.97(50)

6.05(50)

3.66(30)

4.82(20)

—–

(M )

Mcc

18.97(77) A 6.06(37) 10.47(24) A 6.51(58) 15.00(50) A,214.1(1)

16.62(45) A 2.89(23)

14.86(70) A 6.98(58) 13.76(40) A 13.11(55)

16.39(71) A 7.87(55)

14.95(35) A 11.00(40) 11.91(57) A 11.00(98) 16.74(73) A 7.47(52)

(M ) —–

age

EMEM solution M

error box; 1 Dynamical mass was enforced; 2 Maximum amount of the overshooting is assumed (see Sect.)

0.93(5)

3.65(6)

51.7(5.8)

3.84(8)

5.66(57)

22.8(1.2)

1.61(5)

28.3(6)

2.73(5)

3.3(3.3)

31.1(9)

5.33(12)

0.38(38)

34.2(1.3)

4.29(10)

35.9(3.1)

24.4(1.2)

5.72(35)

2.05(8)

29.7(6)

20.8(1.1)

4.10(5)

2.54(10)

33.8(9)

28.3(8)

5.51(9)

4.27(9)

29.1(8)

24.9(9)

2.78(6)

4.54(7)

22.8(1.3)

29.5(8)

3.04(7)

3.22(8)

33.7(1.0)

(%)

(M ) 4.90(13)

Mcc

6.70(56)

11.43(19) B,117.3(1.0) 7.07(14) B 19.6(1.3)

A/B Within/outside

U Oph

CW Cep

V380 Cyg

12.55(17) B 1.67(29) V1034 Sco 17.13(10) B 6.37(18) 9.66(7) B 6.42(60)

V573 Car

V346 Cen

AH Cep

V478 Cyg

V453 Cyg

Parameter

age

CDM solution

M

(M ) V578 Mon 14.55(9) A 10.30(6) A

Object/

27.2(6.6)

25.7(5)

31.4(2.2)

39.2(4.4)

40.0(2.5)

30.1(2.4)

35.1(4.9)

36.4(4.9)

36.1(4.8)

30.7(4.2)

32.2(2.2)

(%)

Table 4.8 Parameters for the different cases of the sample targets with error bars indicated in parentheses in terms of the last digit. The convective core mass is provided both in absolute (solar mass units) and relative (to the stellar mass) values. For each object, the first/second line corresponds to the primary/secondary component. Stars that do not have parameters for the EMEM solution could already be fitted in the EMDM solution and did not require variable initial mass on top of the overshooting parameter. This table was originally published by (Tkachenko et al. 2020), their Table 3. Reproduced with permission from Astronomy c & Astrophysics, ESO

106 4 Estimating the Convective Core Mass for Stars in Eclipsing Binaries

4.7 Expanded Sample

107

Fig. 4.9 Stellar mass (top left), convective core mass (top right), and stellar age (bottom left) difference between the CEM solution and the CDM solution as a function of stellar surface gravity. The age difference as a function of the stellar mass difference between these two solutions is shown in the bottom right panel. All differences are expressed in percent relative to the CDM solution values; the solid line depicts a linear fit in each of the four cases. The ρ and p parameters give Spearman’s rank correlation coefficient and the null hypothesis for linear correlation probability, respectively. This figure was originally published by (Tkachenko et al. 2020), their Fig. 3. Reproduced with c permission from Astronomy & Astrophysics, ESO

when treated as single stars, the mass-discrepancy also functions along the mass-core mass-age degeneracy. The age branch of this degeneracy is neutralised, however, in the case of isochrone or isochrone-cloud modelling, which localises the possible age range allowed in the solution space. The top row of Fig. 4.10 displays the difference in convective-core mass and age between the solution where we essentially fix the mass to the dynamical value and allow the core to be enhanced via internal mixing compared to the classical evolutionary mass solution, where the stellar mass was allowed to vary in the solution. This shows that the solutions converge to having the same core mass when enhanced internal mixing is accounted for, despite having different stellar masses. Finally, the bottom row of Fig. 4.10 shows the difference between the solutions which allowed for enhanced internal mixing. This panel serves as a quantification for the adjusted mass discrepancy, which is the remaining mass discrepancy after accounting for enhanced

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Fig. 4.10 Differences in convective core mass (left column) and age (right column) between the EMDM and CEM solutions (top row), and between the EMEM and EMDM solutions (bottom row). This figure was originally published as two figures by (Tkachenko et al. 2020). Their Figs. 4 and 5. c Reproduced with permission from Astronomy & Astrophysics, ESO

core masses via internal mixing. We note that a number of systems lie along zero difference in both core mass and age between the solutions, indicating that there is no longer a mass (or age) discrepancy in these cases. For these stars, the enhanced internal mixing provided enough of an enhancement to the core mass that the two solutions agree without need for further explanation. Those systems for which there is still a discrepancy, however, could require either still additional mixing, or could indicate that the spectroscopic solution is not correct. In the case of low-density atmospheres, which occur at low surface gravities, atmospheric solutions becomes increasingly degenerate. This is traditionally mitigated by fixing the surface gravity to that determined via binary modelling. However, the treatment of other physics is not agreed upon in literature. To this end, micro-turbulent broadening is generally fixed to a low value (i.e., ξmicro = 2 km s−1 ). However, a micro-turbulent velocity field produces the same effect as an increased effective temperature. Thus, those solutions for which micro-turbulence is artificially fixed to a lower value than is actually needed may have an overestimated effective temperature. This would make the stars appear more massive when fit to a grid of evolutionary models.

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4.8 Summary We have analysed archival photometry of two intermediate- to high-mass welldetached eclipsing binary systems using new high-resolution spectroscopic observations taken with the hermes spectrograph. We arrived at updated fundamental parameter estimates which were then used to model these systems using isochroneclouds as defined in Sect. 2.3. The precise fundamental parameters determined via binary modelling led to the inference of enhanced core masses for the four components of these two systems. Furthermore, by using generalised isochrone-cloud modelling, we did not encounter a mass discrepancy. The proof-of-concept inference of enhanced core masses in this Chapter meanwhile led to a follow-up study on an expanded sample of intermediate- to high-mass eclipsing binary systems to characterise the classical mass discrepancy. The analysis of these 10 systems in this sample led to the detection of enhanced core masses in the entire sample and the conclusion that the mass discrepancy can be interpreted in part as a core mass discrepancy. This finding implies that stellar structure and evolution models require some form of enhanced internal mixing, whatever the mechanism.

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Chapter 5

Binary Asteroseismology

This chapter is based in part on: 1. Binary asteroseismic modelling: isochrone-cloud methodology and application to Kepler gravity mode pulsators6 JOHNSTON, C.; Tkachenko, A.; Aerts, C.; Molenberghs, G.; Bowman, D. M.; Pedersen, M. G.; Buysschaert, B.; Pápics, P. I. Monthly Notices of the Royal Astronomical Society, Vol. 482, 1231J, 15 pp., (2019) Author Contributions: C. Johnston developed the isochrone-cloud methodology, calculated the grid of models and resulting isochrone-clouds used. C. Johnston, C. Aerts, and G. Molenberghs developed the forward modelling scheme and error estimation procedure. A. Tkachenko performed the atmospheric analysis of KIC 4930889. D. M. Bowman, M. G. Pedersen, B. Buysschaert, and P. I. Pápics contributed to the interpretation of results and the development of the modelling code.

5.1 Introduction Due to the complementary modelling constraints they apply, simultaneous modelling of binary and asteroseismic signal in a system is highly sought after. Early on, binary systems with at least one pulsating component were the focus of detailed photometric and spectroscopic campaigns aimed at total characterisation of the variability present in the system (Clausen 1996; De Cat et al. 2000, 2004; Aerts and Harmanec 2004; Miglio and Montalbán 2005). However, there are few studies which have simultaneously modelled a systems binary nature and pulsational signal. Instead, it is far more common to model each signal independently and subsequently check the results of the two modelling efforts to ensure (sometimes loose) agreement a posteriori. Such a methodology has successfully been applied to the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0_5

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case of p-mode pulsating solar-like oscillators in binaries (at various evolutionary stages) where both components are exhibit pulsations Miglio & Montalbán (see, e.g., Miglio and Montalbán 2005); Appourchaux et al. (see, e.g., Appourchaux et al. 2014, 2015); Metcalfe et al. (see, e.g., Metcalfe et al. 2015); White et al. (see, e.g., White et al. 2017); Bellinger et al. (see, e.g., Bellinger et al. 2017); Li et al. (see, e.g., Li et al. 2018); Beck et al. (see, e.g., Beck et al. 2018). Modelling efforts concerning binary systems in which the pulsations from only a single red giant component can be detected are more common (Beck et al. 2014; Themessl et al. 2018). Demonstrating the unique constraining power of simultaneous asteroseismic and binary modelling, Beck et al. (2014) was able to performed detailed modelling of the angular momentum and dynamical history of the evolved binary KIC 5006817. The simultaneous modelling of binarity and g-mode pulsations is somewhat more difficult, considering the signature of binarity often has time-scales similar to those of g-mode oscillation. Despite this, there have been several successful attempts to model both the g-modes and binarity of select systems. For example, Guo et al. (2017) investigated the internal rotation profile of the rejuvenated component of the oscillating Algol like system KIC 9592855. Crucially, Schmid and Aerts (2016) employed simultaneous binary and asteroseismic modelling of the twin hybrid pulsating binary system KIC 10080943. In their study, the authors were able to model multiple g-mode period spacing patterns in both components and subject their asteroseismic modelling to adhere to constraints placed by binary modelling. Although one of the most poignant examples to date, even Schmid and Aerts (2016) only found limited success in constraining the efficiency of convective core overshooting in their models, highlighting the need for new methodological considerations. In this chapter, we develop a simultaneous asteroseismic and binary evolutionary modelling methodology based on the isochrone-cloud modelling methodology discussed in Sect. 2.3 and applied to eclipsing binaries in Sect. 4.5. Our method properly accounts for parameter degeneracies in the underlying modelling grid. We apply this methodology to estimate the core masses of g-mode pulsating stars in three doublelined spectroscopic binary systems observed by the nominal K epler mission. We discuss the integration of the asteroseismic information and how the simultaneous application of binary and asteroseismic information changes and refines the resulting solution space.

5.2 Stellar Models and Modelling Setup 5.2.1 Stellar Models To evaluate the binary and asteroseismic properties of our target stars, we use a grid of stellar models calculated according to the numerical options and input physics described in Sect. 2.1. These assumptions constitute the vector ψ for our model. In order to asses the benefit (if any) to joint binary and asteroseismic modelling,

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we must consider the lowest possible number of free parameters. Thus, we only vary f CBM and impose a radially constant radiative envelope mixing of DREM = 10 cm2 s− 1. We parameterise CBM as diffusive exponential overshooting according to Eq. (1.19) and vary it as f CBM ∈ [0.005, 0.040] in steps of 0.005, such that the bounds agree with those values reported by asteroseismic analyses of single g-mode pulsating stars. Following the assumption that the components of a binary have the same initial chemical composition, we fix Z = 0.014 and Y = 0.276 (Przybilla et al. 2008; Nieva and Przybilla 2012). We calculate isochrone-clouds according to Sects. 2.2 and 2.3. We do not treat binary evolution in our grid of equilibrium models. Instead, given the detached nature of the components of the systems we are modelling, we assume that the two components of a binary have undergone secular, single star evolution without binary interaction.1

5.2.2 Asteroseismic Diagnostic As discussed in Sect. 1.5, gravity (g-)mode oscillations are excellent calibrators for near-core mixing processes because they propagate in the deep stellar interior near the core, and are sensitive to the processes at work there. Regular g-mode period spacing patterns, and deviations therefrom, are particularly sensitive to the near-core environment, such as the near-core rotation rate, convective core boundary mixing, and the convective core mass. The overall scale of a g-mode period-spacing pattern is set by the asymptotic period-spacing value 0 , defined in Eq. (1.25). This quantity is highly sensitive to any changes in the overall profile of the Brunt–Väisälä frequency N , as defined in Eq. (1.10). Thus, processes affecting the cavity where N is positive, as well as the density and the chemical gradient near the core, will change the value of 0 . As both a star’s evolutionary progress at a given and and 0 are sensitive to internal mixing, the combined modelling of these provides the unique opportunity to impose independent cross-constraints on estimates of the convective core mass. Since the value of 0 is influenced by any process which alters the Brunt–Väisälä frequency, we are interested in quantifying the deviations of 0 compared to a baseline value as caused by different values of f CBM , αMLT , and DREM in a stellar model. 0 is sensitive to the spatial distribution of the Brunt–Väisälä frequency throughout the stellar interior. The distribution of N is determined by the age, mass, and internal mixing profile of the a star. To this end, 0 is an excellent probe of the mass and radius of the core, as well as any process which might alter these properties. When considering the presence of diffusive exponential overshooting, which does not alter the thermal structure of a star, the location of the convective core according to the Schwarzschild boundary does not change. However, this process will enhance 1 The

base inlist and isochrone-clouds used in this chapter were made publicly available and are hosted in the MESA Marketplace: http://cococubed.asu.edu/mesa_market/inlists.html.

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the mass of the core due to the influx of chemicals. Alternatively, one can re-define the ‘core’ to extend up until the boundary of the CBM region (Claret and Torres 2016), but this alters the value of both the core mass and radius. As discussed in Chap. 2, we consider the core radius to be set at the Schwarzschild boundary, and the core mass only to account for the mass contained within that radius. The value of 0 is less dependent, however, on the value of αMLT . Table 5.1 compares the value of 0 for stellar models with different combinations of f CBM , αMLT , and DREM at fixed combinations of stellar mass and fraction core hydrogen content. Comparing the change in 0 for different parameters demonstrates that we will not be able to constrain αMLT and DREM through the modelling of 0 . As such, we only attempt to constrain f CBM in our modelling.

5.2.3 Parameter Estimation We use the forward modelling scheme introduced by Aerts et al. (2018) and detailed in Sect. 2.4 to estimate the parameters of the stars we model. In addition to the fixed input physics assumed in ψ, we also fix DREM = 10 cm2 s−1 , αMLT = 2.0, and Z = 0.014 to this vector for this application. This leaves us with the following  vector:  j = τ, f CBM , Mini | ψ , where j = 1 . . . N is the jth parameter combination and N is the total number of model combinations in the grid. As discussed in Sect. 2.4.1, each model  parameter  M j  j has two associated vectors of predicted values. The first associated vector, Y j is compared the the observations to be compared to the observations Y∗ with uncertainties ε∗ . The second vector, ζ j , contains predicted values that we will not compare to observations, but we will use to estimate the properties of the stellar core. In the single star case, we consider: Y j = (Teff , log g, 0 ) ,

(5.1)

and ζ j = Mcc , RCBM , X c  . Here, Mcc is the mass within the Schwarzschild radius, RCBM is the radial coordinate of the end of the CBM region, and X c is the remaining fractional core hydrogen  which best matches the observations is determined content of the model. The vector  using the Mahalanobis distance (Eq. 2.5), and has an accompanying outcome vector  ζ . We calculate 1σ uncertainties for both  and ζ according to the 50th-percentile of the resulting parameter distribution as described in Sect. 2.4. This same methodology was adopted by Mombarg et al. (2019) to estimate the age, X c and core mass of a sample of γ Dor pulsating variable stars.

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Table 5.1 Differences in 0 from one parameter combination to a baseline at fixed masses, and X C combinations. Baseline denoted by *. Observational uncertainties for 0 range from tens to hundreds of seconds. Table adapted from Johnston et al. (2019), their Table 2. Reproduced with permission from author and OUP on behalf of MNRAS M (M ) XC Zini αMLT f CBM log DREM (cm2 s−1 ) δ0 (s) 1.5 1.5 1.5 1.5 4.5 4.5 4.5 4.5 9.0 9.0 9.0 9.0 1.5 1.5 1.5 1.5 4.5 4.5 4.5 4.5 9.0 9.0 9.0 9.0 1.5 1.5 1.5 1.5 4.5 4.5 4.5 4.5 9.0 9.0 9.0 9.0

0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014

2.0 2.0 1.8 2.0 2.0 2.0 1.8 2.0 2.0 2.0 1.8 2.0 2.0 2.0 1.8 2.0 2.0 2.0 1.8 2.0 2.0 2.0 1.8 2.0 2.0 2.0 1.8 2.0 2.0 2.0 1.8 2.0 2.0 2.0 1.8 2.0

0.020 0.025 0.020 0.020 0.020 0.025 0.020 0.020 0.020 0.025 0.020 0.020 0.020 0.025 0.020 0.020 0.020 0.025 0.020 0.020 0.020 0.025 0.020 0.020 0.020 0.025 0.020 0.020 0.020 0.025 0.020 0.020 0.020 0.025 0.020 0.020

1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.5

0* 20 2 11 0* 2 1 2 0* 5 0 3 0* 51 30 15 0* 127 10 59 0* 202 4 6 0* 54 1 14 0* 166 6 70 0* 296 2 11

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In addition to this single star case, we extend this framework to include binary information in the form of the implicit assumptions of equal age and initial chemical 2 ), radii ratio (R = RR21 ), and effective composition, as well as the mass ratio (q = M M1 temperature and surface gravity of the secondary component. Importantly, we note that the models of the secondary also adhere to the same fixed parameter and numerical choices, meaning ψ remains the same. Thus, our predicted vector Y becomes:   Y = q, R, Teff,k , log gk , 0,k ; k = 1, 2, where k indicates either the primary (k = 1) or secondary (k = 2) component of the binary system in question. To impose the assumptions of equal age and initial chemical composition, we fit the stars to isochrone-clouds. This involves a change of basis from evolutionary tracks to isochrone-clouds and involves an expansion of the  vector to become:    = τ, Mk , f CBM,k ; k = 1, 2.

(5.2)

Whereas in Chap. 4 we directly used the mass and radius estimates per component, in this chapter we only consider non-eclipsing systems, which relegate us to using only mass and radius ratios as opposed to their absolute values. One target (KIC 10080943) is an eccentric binary system, for which Schmid et al. (2015) determine an inclination from the periodic periastron brightening. This enables the deduction of absolute mass and radius estimates for both components, and as such we use this case as a test to characterise how the resulting parameter estimation changes from the single star case to adding increasingly constraining information from the spectroscopic and eccentric binary cases. In order to enforce the equal age and initial chemical composition assumptions, as well as allow for both components of a binary to have different amounts of CBM, we construct sub-grids to be considered for MD calculations. A sub-grid consists of all of the combinations of grid points within a given isochrone-cloud that satisfy 3 − σ of the observed values of Teff and log g for either the primary and secondary, so long as at least one grid point satisfies the 3 − σ cut for both components. This combinatorial setup produces in excess of millions of combinations. We calculate the MD using the above Y for every grid point.

5.2.4 Hare-and-Hound This section was originally published by Johnston et al. (2019), their Sect. 3.3, titled: Hare-and-hound. Reproduced with permission from author and OUP on behalf of MNRAS.

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Table 5.2 Input and target parameters for the hare-and-hound exercise. Table adapted from Johnston et al. (2019), their Table 3. Reproduced with permission from author and OUP on behalf of MNRAS Parameter Primary Secondary Mini (M ) Zini αMLT f CBM Dmix (cm2 s−1 ) Teff (K) log g (dex) R (R ) rot (d−1 ) Age (Myr) Xc  (d) Mcc (M )

5.40 0.0135 2.10 0.025 20 16700 3.91 4.274 0.54 59.7 0.342 0.129 0.97

3.83 0.0135 1.95 0.010 15 14450 4.19 2.595 – 59.7 0.558 0.103 0.76

To test the methodology, we perform a hare-and-hound exercise, where the hare was computed from MESA using the input parameters listed in the top portion of Table 5.2. The system is evaluated at an age of 59.7 Myr, resulting in the parameters seen in the bottom half of Table 5.2. To simulate realistic observational errors for an SB2, we assume symmetric uncertainties of 200 K on Teff for both stars, 300 s on 0 , 0.01 on the mass ratio, and 0.05 on the radii ratio. The results of treating the hare as a single star and as the primary of an SB2 system can be seen in Figs. 5.1 and 5.2. Figure 5.1a shows the MD evaluations for each grid point. The vertical dotted line represents the 95th percentile and the dashed line represents the 50th percentile. Figure 5.1b shows the correlation structure for the components of the θ vector for all grid points with an MD below the 50th percentile cutoff. The binned distributions along the diagonal are projections of each component in θ onto one dimension. The colour represents the MD where all distances have been re-scaled between 1 and 1000 for ease of comparison across systems and cases. Since the MD is a maximum likelihood point (MLE) point estimator, we report the grid point with the lowest MD as the best model, where the MLE lower and upper bounds are taken as the minimum and maximum value of the parameter space within the 50th percentile (inter-quantile) cutoff, as listed in Table 5.3. We are interested in other astrophysical quantities such as Mcc , the radius R, core hydrogen content X c , and radial location of the overshoot region RCBM . However, since these quantities are not model input parameters in θ , but rather are output of a given model, we list their values corresponding to the best model without corresponding MLE bounds. In this example we can see that the application of binary information substantially reduces the parameter space compared to the single-star

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Fig. 5.1 Mahalanobis distance distribution (left) and 50th inter-quartile ranges (right) for the primary of the hare-and-hound exercise using a single-star evaluation. All points in the correlation plots are shaded according their rescaled Mahalanobis distance as indicated in the colour bar on the right. Diagonal plots are binned parameter distributions. Figure adapted from Johnston et al. (2019), their Fig. 2. Reproduced with permission from author and OUP on behalf of MNRAS

Fig. 5.2 Same as Fig. 5.1 but for the SB2 evaluation of the Hare-and-hound exercise. Figure adapted from Johnston et al. (2019), their Fig. 3. Reproduced with permission from author and OUP on behalf of MNRAS

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Table 5.3 MLE and 1 − σ errors of the parameters for the Hare-and-hound exercise. Table adapted from Johnston et al. (2019), their Table 4. Reproduced with permission from author and OUP on behalf of MNRAS Parameter Primary Secondary Single

SB2

Age (Myr) f CBM M (M ) R (R ) Mcc (M ) Rcc (R ) Xc Age (Myr) f CBM M (M ) R (R ) Mcc (M ) Rcc (R ) Xc

61 (0, 1593) 0.025 (0.005, 0.04) 5.4 (2.2, 9.9) 4.31 0.97 0.50 0.34 63 (38, 78) 0.025 (0.005, 0.04) 5.37 (3.86, 7.48) 4.33 0.96 0.49 0.34

– – – – – – 0.020 (0.005, 0.04) 3.80 (2.58, 5.91) 2.57 0.78 0.39 0.58

evaluation. With respect to the range of values recovered for the single-star evaluation, the application of binary information reduces the resulting uncertainty in age by 98% and reduces the uncertainty in inferred mass by over 53%. Furthermore, we can see that in both cases, the best models agree well with the input.

5.3 Kepler Sample This section was originally published by Johnston et al. (2019), their Sect. 4, titled: Kepler Sample. Reproduced with permission from author and OUP on behalf of MNRAS. We apply our methodology to three g-mode pulsating stars observed with Kepler. All targets are binary systems with at least one g-mode pulsating component with an estimate of 0 , as measured according to Van Reeth et al. (2016). These three binaries have been recently studied in the literature (Pápics et al. 2013; Schmid et al. 2015; Schmid and Aerts 2016; Pápics et al. 2017). The relevant spectroscopic and binary parameters are listed in Table 5.4.

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Table 5.4 Measured spectroscopic, binary, and asteroseismic parameters for Kepler targets. Table adapted from Johnston et al. (2019), their Table 5. Reproduced with permission from author and OUP on behalf of MNRAS Parameter KIC 4930889 KIC 6352430 KIC 10080943 Teff (K) log g (dex) 0 (s) 2 q (M M1 ) Porb (d) e

14 020 ± 12 820 ± 280 900 3.55 ± 0.24 4.38 ± 0.10 8712 ± 320 – 0.77 ± 0.09 18.296 ± 0.002 0.32 ± 0.02

12 810 ± 6805 ± 100 200 4.05 ± 0.05 4.26 ± 0.15 6944 ± 900 – 0.44 ± 0.03 26.551 ± 0.019 0.371 ± 0.003

7150 ± 250 7640 ± 240 3.81 ± 0.03 4.10 ± 0.10 3984 ± 38 4108 ± 51 0.9598 ± 0.0007 13.3364 ± 0.0003 0.449 ± 0.005

5.3.1 KIC 4930889 KIC 4930889 was found to be an SB2 system and was characterised by Pápics et al. (2017) as consisting of a B5 IV–V primary and B8 IV–V secondary. Their orbital and spectral analysis placed both components in the iron-bump theoretical instability strip for g modes in SPB stars. Their spectroscopic solution places the secondary as more evolved than the primary, which is an unphysical configuration unless binary evolution has altered the evolution of this system. We therefore revisit and re-normalised the original 26 spectra obtained by Pápics et al. (2017), and derive a new spectroscopic solution, which is presented in Table 5.4. This new solution reports a much lower surface gravity for the primary and a much higher surface gravity for the secondary compared to the original solution reported by Pápics et al. (2017). The newly determined radii ratio (R = 0.76) is consistent with evolutionary expectations. Additionally, the effective temperatures of both component are lower by ∼800 K compared to the solution of Pápics et al. (2017). Seismic modelling of this system has not been performed prior to this work. Pápics et al. (2017) report 297 significant frequencies in the 4-year Kepler light curve after filtering for close peaks and low-order combinations. From this list of 297 frequencies, the authors identify three separate period-spacing patterns (Figs. 15 and 16 from Pápics et al. (2017)). The first pattern consists of 20 consecutive radial orders and reveals a mean rotational frequency of f rot = 0.74 ± 0.01 d−1 , following the method of Van Reeth et al. (2016). The slope of this pattern reveals it to consist of dipole prograde modes, leading to 0 = 8712 ± 320 s. The second and third pattern are consistent with retrograde modes, but could not unambiguously be assigned a degree or component from which they originate. Figures 5.3 and 5.4 show the distributions of estimated parameters and their correlations. We find that the parameter estimates derived from the single star solution and the SB2 solution largely agree within their errors. While the best model returned by the binary evaluation is less massive by 0.28 M compared to the single-star case, the convective core mass and location of the overshooting zone are in very good agree-

5.3 Kepler Sample

123

Fig. 5.3 Same as Fig. 5.1 but for the single-star evaluation of KIC 4930889 A. Figure adapted from Johnston et al. (2019), their Fig. 5. Reproduced with permission from author and OUP on behalf of MNRAS

Fig. 5.4 Same as Fig. 5.1 but for the SB2 evaluation of KIC 4930889 A. Figure adapted from Johnston et al. (2019), their Fig. 6. Reproduced with permission from author and OUP on behalf of MNRAS

ment for the single and binary evaluations. This is due to the fact that the seismic diagnostic 0 is strongly sensitive to the convective core mass. As with the hareand-hound example, we note that the binary case greatly reduces both the mass-age and mass-overshoot degeneracies, which can be seen by comparing Figs. 5.3b and 5.4b. Compared to the single star evaluation, the binary evaluation reduced the uncertainty on the age and mass estimates by 94% and 44%, respectively. This is achieved because the binary constraints restrict the possible masses that can be considered

124

5 Binary Asteroseismology

valid at a given age, effectively lifting the degeneracy between the parameters. This propagates into the mass-overshoot degeneracy as the mass range is restricted. As the best solution, our analysis returns a system with an age of 103 Myr consisting of a +1.4 4.89+1.49 −1.09 M primary with a 0.54 M convective core and a 3.47−0.57 M secondary with a 0.60 M convective core near the ZAMS. Finally, as an a posteriori check, the mass ratio of the best model agrees with the value listed in Table 5.4 within 1σ .

5.3.2 KIC 6352430 KIC 6352430 is a close (Porb = 26.551 d) eccentric (e = 0.37) SB2 system consisting of a B7 V primary (KIC 6352430A) and an F2.5 V secondary (KIC 6352430B). It was observed by Kepler over 1459.5 d (Pápics et al. 2013). While some slight ellipsoidal variability was detected, the authors concluded that the remaining signal seen in the light curve could be explained by g modes excited via the κ-mechanism. Later analysis by Pápics et al. (2017) revealed 584 significant frequencies after cleaning for close and low-order combination frequencies. This led to the identification of a single tilted period-spacing pattern, consisting of modes spanning 24 radial orders. The mean asymptotic period spacing and slope of the pattern identify it as dipole prograde modes originating from the primary component, as much lower values are expected for the asymptotic dipole period spacing value in γ Dor stars (Pápics et al. 2017 vs. Van Reeth et al. 2015). Spectroscopic and orbital values taken from Pápics et al. (2013) are listed in Table 5.4. This system has not been the subject of asteroseismic modelling efforts prior to this work. Figures 5.5 and 5.6 and Table 5.5 show the results for the modelling of KIC 6352430. As in the case of KIC 4930889, the estimates derived for the binary case are significantly more precise than that of the single-star case, and agree within the errors of the single-star solution. For KIC 6352430, the binary evaluation reduces the uncertainty on the age estimate by 93% and on the mass estimate by 66%, with respect to the single star evaluation. While the stellar mass and convective core mass agree across both the single-star and binary evaluation cases, the MLE errors show that we do not have the capacity to estimate CBM. This suggests again that the core mass rather than the extent and shape of overshooting is the important astrophysical quantity (Constantino and Baraffe 2018). We can see that the correlation between mass and overshoot that is present in the single-star correlation plots (Fig. 5.5b) has been effectively lifted in the binary case. The binary solution reveals a system with +1.98 M primary with a 0.56 M convecan age of 205 Myr consisting of a 3.23−0.56 +0.68 tive core and a 1.34−0.14 M secondary with a 0.08 M convective core. The best estimated masses agree with the mass ratio reported in Table 5.4 to within 1σ .

5.3 Kepler Sample

125

Fig. 5.5 Same as Fig. 5.1 but for the single-star evaluation of KIC 6352430 A. Figure adapted from Johnston et al. (2019), their Fig. 7. Reproduced with permission from author and OUP on behalf of MNRAS

Fig. 5.6 Same as Fig. 5.1 but for the SB2 evaluation of KIC 6352430 A. Figure adapted from Johnston et al. (2019), their Fig. 8. Reproduced with permission from author and OUP on behalf of MNRAS

5.3.3 KIC 10080943 The KIC 10080943 system consists of two hybrid δ Sct/γ Dor pulsating components. Each of the components were observed to have multiple pulsation patterns, for which binary modelling could be performed to obtain independent estimates for the component masses and radii from modelling the periastron brightening as KIC 10080943 is an eccentric binary star system (Keen et al. 2015; Schmid et al. 2015; Schmid

126

5 Binary Asteroseismology

and Aerts 2016). In addition to identifying multiple patterns in each component, for both p- and g-modes, Schmid and Aerts (2016) were able to derive surface-to-core rotation rate estimates and an independent age estimate from binary modelling. For our purposes, we take the values of Teff , log g, M1,2 , and R1,2 from the binary modelling performed by Schmid et al. (2015) and the estimates of 0 from the analysis of Schmid and Aerts (2016), all of which are listed in Table 5.4. Given the characterisation of this system, we can test how the application of different information in the modelling impacts our results. We find that the single star and SB2 evaluation largely agree except for the overshoot and X c returned by the best model for each case. The application of mass and radius estimates to the modelling procedure results in yet again different mass, age, and overshoot estimates for the best model. Again, we see that we have no capacity to assess proper values for f CBM . We find approximate agreement between Mcc for all cases. The difference in solution between the SB2 and eccentric binary cases is caused by the use of the radii ratio in the SB2 case versus individual radii in the eccentric binary case. Since KIC 10080943 is comprised of two nearly identical stars, the application of the mass ratio and radii ratio does not provide any new information or strong constraints. Additionally, the covariance structure inherently changes between individual evolutionary tracks and the isochrone-clouds, which produces the differences between the single star and SB2 solutions. However, applying the absolute mass and radii estimates provides sufficient constraints to improve the solution. The best model from the eccentric binary evaluation reports a system with an age of +0.37 890+610 −190 Myr consisting of a 1.99−0.49 M primary with an 0.15 M convective core +0.41 and a 1.85−0.45 M secondary with an 0.17 M convective core. In both the binary evaluations, the uncertainty on the age was reduced by 68%. In the SB2 evaluation, the uncertainty on the mass was reduced by 87% with respect to the single star evaluation, and the full binary evaluation reduced the uncertainty an additional 2% with respect to the SB2 solution. Of the six models that Schmid and Aerts (2016) reported for KIC 10080943, we are interested in comparing Models 4 and 6 to our result. Model 4 corresponds to the solution from modelling the individual g-mode periods for both components accounting for the effect of rotation on pulsations using the Traditional Approximation of Rotation (TAR, Townsend 2003; Townsend and Teitler 2013). Model 6 corresponds to the solution from modelling the overall morphology of the g-mode period spacing pattern, while again accounting for rotation using the TAR. In both models, the diffusive exponential description of overshooting was used. Schmid and Aerts (2016) only enforced an equal age constraint in their modelling and employ a χ 2 evaluation using only the individual g modes (per star) in model 4 and the g-mode period spacing pattern (per star) in model 6. Our single star and SB2 solutions share approximate agreement with both model 4 and model 6 from Schmid and Aerts (2016). Our binary solution, which includes the mass ratio, absolute masses and radii, as well as the spectroscopic quantities, does not agree with either model 4 nor model 6 from Schmid and Aerts (2016). Due to the differences in the modelling methodology, we suggest caution at making a direct comparison between the two results.

5.4 Discussion

127

5.4 Discussion This section was originally published by Johnston et al. (2019), their Sect. 5, titled: Discussion. Reproduced with permission from author and OUP on behalf of MNRAS. Understanding the degeneracy between stellar mass, age, and extent of core overshooting is pivotal for asteroseismic modelling. The single star case evaluates 0 , Teff , and log g, all of which depend on a star’s mass, age, and convective core mass. In the binary cases, our methodology imposes a strict range of ages at which a solution is valid. Since 0 varies with mass and age, by constraining the valid age range, we also constrain the masses at which a given 0 can be considered a valid solution. In the SB2 case, the mass ratio (when sufficiently far from unity) drives the selection of stellar masses and core masses towards those combinations which satisfy the value of 0 , which is already constrained by the valid age range. In the eccentric binary case, the addition of absolute masses and radii fixes the stellar masses to be considered in the valid age range, leaving only the extent of core overshooting to influence the mass of the core, and thus 0 . While this is the most constrained case, the addition of the radii ratio in the SB2 case enables a cross-constraint on the evolution as well, constraining the mass-age-overshooting degeneracy to a manageable extent. This work shows that the application of additional information derived from binarity significantly improves the results of extracted parameters and their uncertainties due to the independent cross constrains that binary information provides. In particular, it is seen for each system that the single-star case has discrepant mass and age estimates compared to the SB2 case. Given the relationship between age and mass built into the isochrone-clouds, the restricted age range imposed by the binary constraints corresponds to a restricted mass range and thus shifts the best age and mass estimates. This comparison of single to binary star solutions reveals the hierarchy of what results to take as robust, and which to reference with caution. The binary constraints render some single-star configurations impossible, allowing the precision of the seismic diagnostic to take full effect. We note that the inclusion of absolute mass and radius estimates only becomes important in the case where both stars are similar, with a mass ratio near unity. In the case that the components of a binary are sufficiently different in mass, the inclusion of mass and radii ratios as well as the spectroscopic quantities of the secondary are sufficient to improve the model selection and parameter estimation. No clear trend emerges between mass and overshoot for this sample on only six individual stars, which span 1.34–4.89 M . We do not encounter the same mass dependence of overshooting as reported by Claret and Torres (2016, 2017, 2018, 2019). Even with the addition of the asteroseismic diagnostic, we recover the entire input range of overshooting as the uncertainty on its estimates. However, we do find consistent estimates of the convective core mass and location of the overshooting region (c.f. Miglio et al. 2008; Constantino and Baraffe 2018). Several overshoot

128

5 Binary Asteroseismology

Fig. 5.7 Same as Fig. 5.1 but for the single-star evaluation of KIC 10080943 A. Figure adapted from Johnston et al. (2019), their Fig. 9. Reproduced with permission from author and OUP on behalf of MNRAS

Fig. 5.8 Same as Fig. 5.1 but for the SB2 evaluation of KIC 10080943 A. Figure adapted from Johnston et al. (2019), their Fig. 10. Reproduced with permission from author and OUP on behalf of MNRAS

values can satisfy the observations of a given target in the single-star seismic case, as seen in the correlation plots in Figs. 5.3b, 5.5b, and 5.7b. We are able to remove this correlation structure by simultaneously applying asteroseismic and binary constraints, as seen in the correlation plots in Figs. 5.4b, 5.6b, 5.8b, and 5.9b. This leads to a unique determination of the stellar age and mass as seen by comparing the single and binary cases in Table 5.5.

5.5 Summary

129

Fig. 5.9 Same as Fig. 5.1 but for the eccentric binary evaluation of KIC 10080943 A. Figure adapted from Johnston et al. (2019), their Fig. 11. Reproduced with permission from author and OUP on behalf of MNRAS

5.5 Summary This section was originally published by Johnston et al. (2019), their Sect. 6, titled: Summary and Conclusions. Reproduced with permission from author and OUP on behalf of MNRAS. In this chapter we formulated a framework for the modelling of g-mode pulsating stars in binary systems and applied it to the three systems KIC 4930889, KIC 6352430, and KIC 10080943 (Pápics et al. 2013; Schmid et al. 2015; Schmid and Aerts 2016; Pápics et al. 2017). The addition of binary information proved useful for reducing the uncertainties on parameter estimates and reducing the correlation between model parameters. Comparison of the MLE estimates derived by the MD calculations did not reveal any obvious dependence of f CBM with mass or age. Most interestingly, we did not recover the traditional binary mass discrepancy in our results. This is likely due to the fact that even in our most constrained case (KIC 10080943) we do not have one per cent-level relative precision on the mass and radius estimates from binary modelling. In the future, binary systems with high precision mass and radius estimates and individually identified g modes need to be scrutinised to determine if the mass discrepancy persists with the inclusion of asteroseismic information in the modelling procedure. In this work, we do not include any possible effects of binary evolution or tidal effects on pulsations in our modelling. It has already been established for KIC 10080943 that rotation has a much larger impact than the tidal interaction (Schmid and Aerts 2016), and as both KIC 4930889 and KIC 6352430 have longer

6.39

0.58

0.37

0.06

R (R )

Mcc (M )

RCBM (R )

Xc

Eccentric binary

–

–

–

–

–

–

–

0.33 0.48

4.89 (3.80, 6.38)

6.72

0.54

0.35

0.05

R (R )

Mcc (M )

RCBM (R ) Xc

–

–

–

–

–

f CBM

M (M )

R (R )

Mcc (M )

RCBM (R )

Xc

–

–

–

–

–

– –

–

Age (Myr)

0.60

2.65

3.47 (2.9, 4.87)

0.025 (0.005, 0.04) 0.005 (–, 0.04)

f CBM

M (M )

103 (38, 150)

5.2 (2.4, 7.0)

M (M )

Age (Myr)

0.02 (0.005, 0.04)

f CBM

SB2

85 (0, 2000)

Age (Myr)

Single

KIC 4930889

Parameter

Case

–

–

–

–

–

–

0.42

0.34

0.56

3.01

–

–

–

–

–

–

–

–

–

–

–

–

–

–

0.55

0.11

0.08

1.42

1.34 (1.2, 2.02)

0.005 (–, 0.04)

205 (78, 215) 3.23 (2.67, 5.21)

0.04 (0.005, –)

0.38

0.31

0.51

2.91

3.4 (1.5, 8.9)

0.005 (–, 0.04)

140 (0, 2000)

KIC 6352430 –

0.06

0.14

0.15

3.11

1.99 (1.50, 2.36)

0.01 (0.005, 0.04)

0.28

0.17

0.19

–

–

–

0.46

0.17

0.18

1.79

1.56 (1.25, 2.21)

0.04 (0.005, –)

0.21

0.15

0.17

2.39

1.85 (1.40, 2.26)

0.005 (–, 0.04)

890 (700, 1500)

1.71 (1.26, 2.31) 2.56

– –

1440 (700, 1500) 0.04 (0.005, –)

0.006

0.103

0.096

2.66

1.7 (1.2, 9.7)

0.010 (0.005, 0.04) –

1446 (0, 2500)

KIC 10080943

Table 5.5 Maximum likelihood estimates (half inter-quartile range) of the model parameters for three Kepler binaries. Values reported with no lower and upper values are taken from the best model. Top panel is for single star solution. Middle panel is for SB2 solution. Bottom panel is for eccentric binary solution where applicable. Table adapted from Johnston et al. (2019), their Table 6. Reproduced with permission from author and OUP on behalf of MNRAS

130 5 Binary Asteroseismology

5.5 Summary

131

orbital periods and smaller mass ratios, the tide generating potential in these systems is even smaller than that of KIC 10080943. The addition of more or longer period-spacing patterns in the determination of 0 can aid in the improvement of the relative precision of this parameter and would thus improve the derived parameters compared to the results of this work. The methodology presented here can be extended to full modelling of g-mode period spacing patterns themselves, rather than just 0 . Our isochrone-cloud evaluation methodology provides a robust framework for investigating the overall internal mixing in binary stars—CBM + REM. Furthermore, the flexibility of this method allows for the easy inclusion of seismic information to impose an independent calibration of internal phenomena. Following the result that the inclusion of binary information modifies the solution space and effectively neutralises parameter degeneracies, future modelling of single stars will need to investigate whether or not the resulting solution is robust or if it is rather sliding along a parameter correlation. Such modelling is currently being undertaken in SPB stars by Pedersen (2021). The recently launched NASA TESS mission (Ricker et al. 2015) and the future ESA PLATO mission (Rauer et al. 2014) promise to deliver high-quality observations of thousands of new g-mode pulsators with sufficiently high precision on pulsation periods for stars in binaries and clusters (only in the continuous viewing zones for TESS), all of which will be suitable for analysis under the methodology put forth here. After their future release (2021+), the addition of Gaia astrometric binary solutions for non-eclipsing binary systems will enable the application of direct mass estimates in this methodology, instead of using only the mass ratios as was done for the SB2 systems here (Lindegren et al. 2018). Pedersen (2021) has recently shown that the inclusion of Gaia luminosities can greatly improve modelling of single pulsating stars that have an estimate of 0 . The addition of such information is readily possible within our methodology, and certainly merits future inclusion.

References Aerts, C. & Harmanec, P. (2004). Astronomical Society of the Pacific Conference Series. In Hilditch, R. W., Hensberge, H., & Pavlovski, K. (Eds.), Spectroscopically and Spatially Resolving the Components of the Close Binary Stars (Vol. 318, pp. 325–333). Aerts, C., Molenberghs, G., Michielsen, M., et al. (2018). ApJS, 237, 15. Appourchaux, T., Antia, H. M., Ball, W., et al. (2015). A & A, 582, A25. Appourchaux, T., Antia, H. M., Benomar, O., et al. (2014). A & A, 566, A20. Beck, P. G., Hambleton, K., Vos, J., et al. (2014). A & A, 564, A36. Beck, P. G., Kallinger, T., Pavlovski, K., et al. (2018). A & A, 612, A22. Bellinger, E. P., Basu, S., Hekker, S., & Ball, W. H. (2017). ApJ, 851, 80. Claret, A., & Torres, G. (2016). A & A, 592, A15. Claret, A., & Torres, G. (2017). ApJ, 849, 18. Claret, A., & Torres, G. (2018). ApJ, 859, 100. Claret, A., & Torres, G. (2019). ApJ, 876, 134. Clausen, J. V. (1996). A & A, 308, 151. Constantino, T., & Baraffe, I. (2018). A & A, 618, A177.

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Chapter 6

The Effect of Enhanced Core Masses on the Observed Morphology of Young Clusters

This Chapter is based in part on: 1. Isochrone-cloud fitting of the extended main-sequence turn-off of young clusters JOHNSTON, C.; Aerts, C.; Pedersen, M. G.; Bastian, N. Astronomy & Astrophysics, Vol. 632, A74, 11 pp. (2019) Author Contributions: C. Johnston calculated the isochrone-clouds and developed the fitting methodology used in the work. C. Johnston, C. Aerts, M. Pedersen, and N. Bastian contributed to the discussion and interpretation of results. Data for NGC 1850 was provided by N. Bastian. Data for NGC 884 was provided by C. Aerts and Chengyuan Li.

6.1 Introducing the Extended Main-Sequence Turn-off in Young Massive Clusters Colour-magnitude diagrams (CMDs) are an observational equivalent of the HRD, where the observed magnitude of an object, or of a population of objects, are plotted against the difference between the magnitudes of that (those) object(s) in two different photometric filters, or colours. This results in an observational diagnostic of the evolutionary progression of an object or population of objects. CMDs are particularly useful in the study of young massive clusters (YMCs). As YMCs are thought to consist of single, co-evolutionary population of stars born of the same material (Bastian and Lardo 2018), CMDs provide an excellent evolutionary diagnostic where one can determine the age of the cluster based on the morphology of the main-sequence turn-off point (MSTO). Since all stars in the cluster are assumed to be the same age, the most massive star in the cluster that is still on the MS is a direct

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0_6

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measure of the clusters age. However, observations of YMCs within the galaxy as well as in both the Large and Small Magellanic Clouds (LMC and SMC) reveal an extended MSTO (eMSTO) instead of a singular progression of stars (Mackey et al. 2008; Milone et al. 2018; Marino et al. 2018a; Cordoni et al. 2018). The eMSTO is observed as a broadened TAMS within the CMD of a cluster, effectively covering a wide range in both colour and magnitude. The presence of the eMSTO challenges the assumption that YMCs consist of a single co-eval population of stars. To date, there have been several proposed mechanisms to explain the eMSTO, e.g., formation delays, (near) critical stellar rotation, convective penetration, and binary evolution. However, none of these mechanisms can uniquely explain the full range of observed eMSTOs (Li et al. 2019; Gossage et al. 2019, and references therein). Crucially, the presence of eMSTOs has been linked to rapidly rotating stars as deduced from spectroscopic observations of stars associated with observed eMSTOs (e.g. Dupree et al. 2017; Kamann et al. 2018; Marino et al. 2018b; Bastian et al. 2018). These observations have motivated the fitting of eMSTOs with isochrones where stellar models are internally modified by the presence of rotationally induced mixing processes, while their surfaces and colours are modified according the the von Zeipel effect (e.g. Brott et al. 2011; Lagarde et al. 2012; Milone et al. 2018). While this improves fits for the youngest YMCs, the effects of rotation cannot fully explain observations of intermediate aged clusters (Niederhofer et al. 2015; Goudfrooij et al. 2017). Alternatively, clusters with ages greater than approximately one giga-year have eMSTOs consistent with a population of rapidly rotating stars that are slowed by the phenomenon of magnetic breaking (Georgy et al. 2019). To complicate this picture, observations of some clusters demonstrate that there is an apparent correlation between location in the CMD (the spread of the eMSTO) and the presence of H α emission (Bodensteiner et al. 2020a), whereas studies of other clusters such as the double cluster h and χ Persei demonstrate no such correlation (Li et al. 2019). Bodensteiner et al. (2020a) interpreted the presence of H α emission as being related to Be stars and their rapid rotation rates. While the Be phenomenon is related to rapid rotation, it has also been linked to binary evolution and the presence of stellar pulsations (Baade et al. 2017; Bodensteiner et al. 2020b). Further complications arise considering the mass range over which the eMSTO is observed. Generally, the eMSTO is observed in clusters with turn-off masses above ∼1.4 M , where the CMD narrows for stars below this mass (Goudfrooij et al. 2018). While this phenomenon coincides with the mass range below which magnetic breaking is thought to become efficient (Kraft 1967), it also coincides with the range above which stars develop a large convective core during the MS, and excite both non-radial g modes as well as stochastic internal gravity waves (IGWs) (Kippenhahn et al. 2012; Aerts et al. 2010). As discussed in Chap. 1, the spectrum of IGWs generated at the convective boundary with the radiative envelope also induce chemical transport within the star (Rogers and McElwaine 2017; Pedersen et al. 2018; Bowman et al. 2019a, b). In light of this, we postulate that CBM and pulsational REM plays a role in the formation of the eMSTO.

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The inclusion of convective penetration in model calculations has already been demonstrated to partially explain the eMSTO Yang and Tian (2017). Here, we look to more generally investigate the role of CBM and REM in the formation of the eMSTO. As discussed in Sect. 1.4.1, the inclusion of enhanced internal mixing naturally produces a range of potential TAMS locations for a given star. Figure 6.1 demonstrates how an isochrone-cloud evolves as a function of age in the CMD, naturally replicating the eMSTO phenomenon. Due to the presence of stars with different amounts of internal mixing, those stars with the least mixing in the isochrone-cloud evolve redwards towards the TAMS. Simultaneously, those stars increased mixing remain at earlier, bluer phases of the MS. This effectively results in a broadening region of the CMD as a function of increasing age. Furthermore, comparing the panels of Fig. 6.1 reveals that the maximum mass of an isochrone-cloud (on the MS) decreases with increasing age. Figure 6.2 demonstrates that the development on the extension of the isochrone-cloud is directly related to the evolution of the fractional core-mass distribution. This link allows us to investigate the eMSTO as a function of core-mass distribution. We note that, as discussed in Sect. 2.3, although we use parameterised versions of CBM and REM in our models, due to the degenerate implementations of mixing mechanisms, we do not make assumptions as to the cause of the enhanced

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Fig. 6.2 Distribution of fractional core masses for three isochrone-clouds shown in panels of Fig. 6.1. The dashed black distribution represents the fractional core masses for the 50 Myr isochrone-cloud, the dark grey dashed-dotted distribution is for the 150 Myr isochrone-cloud, and the light-grey dotted distribution is for the 300 Myr isochrone-cloud. This figure was originally published by (Johnston et al. 2019), their Fig. 4. Reproduced with permission from Astronomy & c Astrophysics, ESO

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mixing. To this end, this modelling can be interpreted as being consistent with the effects of rotation, as long as the latter justifies spherically symmetric equilibrium stellar models. In this chapter, we generalise the isochrone-cloud modelling method applied to binaries in Chaps. 4 and 5 to model the whole eMSTO of young massive clusters. This modelling enables the novel interpretation of the eMSTO phenomenon in terms of stars with enhanced core masses resulting from enhanced internal mixing. We fit isochrone-clouds to CMDs of two YMCs with eMSTOs to investigate to what extent isochrone-clouds can explain phenomenon. Furthermore, we investigate the correlation between modelled properties and pulsational characteristics, as well as with projected rotational velocity for stars with such measurements. We begin with a review of the eMSTO phenomenon in YMCs (Sect. 6.1) before discussing the generalisation of our isochrone-cloud modelling scheme (Sect. 6.2). We model the two clusters NGC 1850 and NGC 884 and investigate whether the modelled parameters scale with observed pulsational or rotational quantities (Sect. 6.3.2) and finally discuss the results of this modelling (Sect. 6.4).

6.2 Modelling Setup The isochrone-clouds we use to fit the clusters in this work have CBM and REM efficiencies in the range deduced from the results of binary and asteroseismic modelling of intermediate- to high-mass field stars, i.e., DCBM ∈ [0.005, 0.040] and

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log DREM ∈ [0, 4]. For the application to cluster fitting, we calculate isochroneclouds containing the absolute magnitudes and corresponding bolometric corrections for the Hubble Space Telescope (HST) F336W and F438W photometric filters, as well as for all photometric filters in the Johnson photometric system. In order to maintain the integrity of the observations, we convert the predicted absolute magnitudes of the isochrone-clouds to apparent magnitudes, requiring a distance estimate for the conversion. Furthermore, we use extinction coefficients calculated using the York Extinction Solver (YES, McCall 2004). This tool assumes the form Rλ = Aλ / E(B-V) = 3.1, where E(B-V) is the excess of the (B-V) colour index compared to models, RV = 3.07 is the scaling coefficient in the Vega system for the Fitzpatrick (1999) reddening law, and Aλ is the extinction at a given wavelength λ. Our fitting methodology is only focused on the eMSTO portion of the CMD. As such, we ignore the remainder of the cluster members by performing cuts in colour and magnitude, per cluster. To this end, we consider any agreement between the results of our fitting and the morphology of the lower main-sequence of each cluster to serve as supporting evidence for the fit. Our isochrone-clouds were calculated for ages between τ ∈ [5, 500] Myr, with a step size of 2.5 Myr below 50 Myr and a step of 10 Myr for isochrone-clouds older than 50 Myr. The goodness of fit of an isochrone-cloud of a given age is determined using the Mahalobis Distance (MD, Sect. 2.4.1). In order to calculate the total MD of an isochrone-cloud, we first calculate the MD for all N points in the isochrone-cloud, for all M cluster members being fit. We retain the best matching point for each cluster member and then sum the best MD for all cluster members, resulting in a total MD for that isochrone-cloud. The best matching isochrone-cloud is then selected as the one having the lowest total MD.

6.3 Application to NGC 1850 and NGC 884 6.3.1 NGC 1850 NGC 1850 is a well studied cluster located in the LMC. Given its association with the LMC, we assume a metallicity of Z = 0.006 and a distance of d = 42 658 pc using the distance modulus and extinction estimates from the literature (Bastian et al. 2016; Correnti et al. 2017; Yang and Tian 2017; Yang et al. 2018). Photometrically, NGC 1850 has been observed in several HST and Strömgren filters. These observations reveal the presence of an eMSTO, corresponding to an age of approximately 80 Myr. Furthermore, this age implies that the stars currently comprising the eMSTO have masses around 5 M . This places these stars firmly within the observed g mode instability strip of SPB stars (Salmon et al. 2012; Moravveji et al. 2016; Burssens et al. 2020). However, as there are no suitable time series of this cluster, we cannot determine if these low metallicity stars are in fact pulsating. While Yang et al. (2018) interpret the eMSTO as being the result of binary stars, D’Antona et al. (2017) claims that the presence of the observed eMSTO can be explained by a minor population of

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Fig. 6.3 Isochrone-cloud of co-eval population of single stars at ∼185 Myrs (blue) resulting from best eMSTO fitted for NGC 1850. This figure was originally published by Johnston et al. (2019), their Fig. 5. Reproduced with permission from Astronomy & c Astrophysics, ESO

stars rotating slowly with the remaining majority of stars rotating rapidly. This claim was supported by Bastian et al. (2017) who detected a large number of B stars with H α excess, linking the observed excess to rapidly rotating Be stars. However, without v sin i estimates to support this interpretation, other mechanisms for producing H α emission cannot be categorically ruled out. As previously stated, we perform a cut in colour and magnitude according to m F439W > 19 and m F336W − m F439W < 0, in following Bastian et al. (2016). We find a optimal match between observations and an isochrone-cloud with an age of ∼185 Myr, as shown in Fig. 6.3. The age we derive here is 32% older than the age derived by Bastian et al. (2017). However, given the difference in (i) methodology and (ii) stellar evolutionary models, this is expected. As a consequence of the increased age estimate, we also find a lower mass range for stars within the eMSTO. Whereas Bastian et al. (2017) found the eMSTO to be made up of stars with M ∈ [4.6] M , we find a mass range of M ∈ [1.9, 4.0] M . To investigate whether or not the presence of H α emission influences the interpretation of our results, we compare the position of those stars without H α emission in the CMD to those with H α emission in the left and middle panels of Fig. 6.4. There is no clear correlation between position in the CMD and the presence of H α emission. Furthermore, comparison of both populations against a single isochrone with an age of 185 Myr, calculated from models with the DCBM = 0.005 and log DREM = 0 (plotted in magenta in Fig. 6.4, demonstrates that both populations are poorly explained by a such an isochrone. The rightmost panel of Fig. 6.4 displays the distribution of the of core mass as a percentage of the total stellar mass for (i) all stars in the eMSTO according to their best matching model in the isochrone-cloud (black distribution), (ii) all stars without H α emission (blue dashed distribution), (iii) all stars with H α emis-

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Fig. 6.4 Fitting results for NGC 1850 using isochrone-clouds with Z = 0.006, leading to age of ∼ 185 Myrs. Left: Observed stars in NGC 1850 without H α emission, colour-coded according to the ratio of the convective core mass versus total stellar mass. Middle: Observed stars in NGC 1850 with H α emission colour-coded according to convective core mass. The isochrone with minimum amount of mixing at 185 Myr is plotted in magenta for reference in both panels. Right: Distribution of the fitted core mass as a fraction of total stellar mass, indicated in black for the full sample, in dashed blue for stars without H α emission and in dashed-dotted red for stars with H α emission. The distribution of fractional core mass for the 185 Myr isochrone with minimum mixing is shown in dotted magenta. This figure was originally published by Johnston et al. (2019), their Fig. 6. c Reproduced with permission from Astronomy & Astrophysics, ESO

sion (red dashed-dotted distribution), and (iv) the models in the 185 Myr isochrone in the left and middle panels. Comparison of these distributions demonstrates that there is no difference between the core mass distribution of stars with or without H α emission and that both populations of stars require a core mass distribution significantly different to that of the 185 Myr isochrone to reproduce the observations. The overall distribution of core masses ranges from 6 to 23%, with a clear skew and peak at approximately 20%. The distribution of stars with H α emission covers the same range, but does not have a clear peak. Finally, we note that the overall morphology of the best fitting isochrone-cloud closely matches the observed morphology of the lower MS portion of the CMD. It is necessarily implied through our results that location in the CMD, and thus in the eMSTO is directly related to the fractional core mass of a star. In this way, stars with larger fractional core masses are positioned bluewards, while stars with decreasing core masses are positioned redwards. The spread of the overall core mass distribution is then related to the width of the eMSTO. This interpretation does not forego the possibility that rapid rotation plays a role in the eMSTO. Since our methodology simply interprets the eMSTO in terms of core mass distribution, it is entirely possible that rotationally induced mixing could contribute to the spread of the

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core mass distribution. That being the case, these results demonstrate that isochroneclouds with mixing levels determined from asteroseismic and binary analysis of isolated field stars can at least partly explain the observed eMSTO of NGC 1850. This implies that an efficient chemical transport network is active in these stars in order to sufficiently increase the core mass. Determination as to which processes are active, such as waves, rotation, overshooting, tides, etc., requires detailed modelling of several stars from the cluster.

6.3.2 NGC 884 NGC 884 is an exceptionally well observed galactic cluster, otherwise known as χ Persei, member of the double cluster with h Persei . Due to its brightness, χ Persei has been the subject of much ground-based observational characterisation, namely several months long multi-colour photometry, as well as v sin i characterisation of several cluster members. Analysis of the multi-colour photometry revealed 65 pulsating B-type stars over a wide range of pulsation frequencies and amplitudes (Saesen et al. 2010, 2013). Ensemble asteroseismic analysis of eight β Cep pulsators using the CLÉS stellar evolution code (including convective penetration) resulted in an estimated age range of ∼13.2−19.1 Myr (Scuflaire et al. 2008). Recently, Li et al. (2019) arrived at an age estimate of 14 Myr based on Geneva stellar evolution models including the effects of rapid rotation. Given the location of NGC 884 within the galaxy, we adopt the galactic B-star metallicity of Z = 0.014 (Nieva and Przybilla 2012). This choice is supported by the presence of β Cep type pulsations in several cluster members, as the excitation mechanism requires sufficiently high metallicity, i.e. Z > 0.01, to operate (Miglio et al. 2007). Although numerous studies have suggested the correlation between the eMSTO and rotation, none have yet investigate whether or not the presence of pulsations is correlated with properties of the eMSTO in clusters. Given the detection of p and g modes combined with v sin i estimates in several stars, NGC 884 offers the first opportunity to comparatively investigate possible correlations between rotation, pulsations, and the eMSTO. We utilise the same Gaia DR2 observations of NGC 884 as used by Li et al. (2019) to carry out our modelling. However, as our isochrone-clouds were not calculated to output magnitudes in the Gaia filters, we first transform magnitudes from the Johnson to Gaia filters using transformations provided by the Gaia team. Following this transformation, we make make a cut to only include members of the eMSTO according to G r p < 14 and G bp − G r p < 0.75 (Li et al. 2019). Our fitting yields an optimal match with an isochrone-cloud with an age of 11 Myr, corresponding to stars in the eMSTO having masses between M ∈ [1.7, 17.8] M . The best matching isochrone-cloud is plotted over the Gaia data for NGC 884 in Fig. 6.5. Given the differences between methodology and stellar evolution models, we consider our

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Fig. 6.5 Isochrone-cloud of co-eval population of single stars at ∼11 Myrs (blue) resulting from best eMSTO fitted for NGC 884. This figure was originally published by Johnston et al. (2019c), their Fig. 7. Reproduced with permission from Astronomy & c Astrophysics, ESO

age estimate to agree with those of Scuflaire et al. (2008) and Li et al. (2019). Furthermore, the mass range in consistent with those stars expected to undergo p and g mode pulsations. The left panel of Fig. 6.6 displays the fit stars with their corresponding best matched fractional core mass. As with NGC 1850, we find that those stars with the highest fractional core mass are positioned the blue side of the eMSTO. Furthermore, we make use of the oscillation and v sin i data obtained by Saesen et al. (2010, 2013) in order to evaluate any correlation with the fractional core mass and (i) the v sin i, (ii) maximum oscillation amplitude, and (iii) the frequency of the oscillation mode with the highest amplitude. These values are plotted against one another in the middle panels of Fig. 6.6. To assess whether or not any statistically significant correlations are present, we subjected these combinations to a linear regression fit. Using the data available to us, we find no significant correlations and report the R2 and p values in Table 6.1. The right panel of Fig. 6.6 displays the inferred core mass distribution as a percentage of the total stellar mass for the total sample (solid black), those stars with observed pulsations (blue dashed), those stars with v sin i estimates (red dashed-dotted), and for an isochrone with an age of 11 Myr. In this case, the inferred distribution is shifted slightly to higher values, and does not have a sharp cut-off at the high end, compared to the distribution from the 11 Myr isochrone. It is important to note that, in order to be consistent, we restricted the range of REM efficiency based on the results of asteroseismic modelling of single, isolated, intermediate mass, field stars, with masses below 9 M . Despite this, there is reason to expect that more massive stars may have higher values of DREM , as induced by IGWs, than less massive stars

Fig. 6.6 Fitting results for NGC 884 using isochrone-clouds with Z = 0.014, leading to age of ∼ 11 Myrs. Left: Same as left panel of Fig. 6.4. The upper, middle and lower panel of the central plots show v sin i, the dominant oscillation frequency and its amplitude versus fractional core mass. Right: Same as right panel of Fig. 6.4, but blue denotes stars with measured oscillations and red denotes stars with measured v sin i. Pink dotted distribution is for those stars with the minimum amount of mixing. This figure was originally published by Johnston et al. (2019), their Fig. 8. Reproduced with permission from Astronomy & c Astrophysics, ESO

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Table 6.1 R 2 - and p-values of statistical regressions for various stellar parameters with the derived fractional core mass Mcc /M . This table was originally published by Johnston et al. (2019), their c Table 1. Reproduced with permission from Astronomy & Astrophysics, ESO y-variable R2 p v sin i f osc Aosc

0.007 0.003 0.034

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(Bowman et al. 2019b; Tkachenko et al. 2020). To this end, future work would benefit from including such increased DREM levels for the youngest clusters with stars with masses above 9 M .

6.4 Discussion and Conclusions In this chapter, we explored the novel application of isochrone-clouds to modelling the eMSTO of young clusters. Our applications involved both a galactic cluster and a cluster associated with the LMC, allowing us to probe different metallicity regimes. Through this application, we were able to demonstrate that levels of internal mixing, as independently calibrated by asteroseismic and binary studies, can explain a large portion of the observed eMSTO. Based on the results of Chaps. 4 and 5, we can interpret the eMSTO in terms of a widening distribution of fractional core masses. The width of the eMSTO is related to the width of the fractional core mass distribution where the evolution of the width of the core mass distribution is related to the amount of internal mixing present in the population of stars. Thus, we are able correlate position within the eMSTO to fractional core mass and amount of internal mixing. With this interpretation, two stars with the same age and similar birth mass, but different interior mixing levels would appear in different locations on the CMD, where the star with enhanced mixing would be located blueward of its less mixed analogue. Given the diversity of methodology and underlying stellar evolution models, combined with the typical uncertainties on age estimates found in the literature (Aerts et al. 2019), we find that our age estimates for NGC 1850 and NGC 884 largely agree with previous age estimates. Although our methodology can explain a large portion of the observed eMSTO, it still has shortcomings. First, this methodology currently cannot differentiate between single stars with different amounts of internal mixing and binaries whose apparent colours are shifted due to the contribution of a close luminous companion. In this case, the stars would simply appear as having a large amount of internal mixing. Similar limitations apply for the most rapidly rotating stars whose surfaces are deformed or for stars with significant surface magnetic fields. Second, all the points of an isochrone-clouds are subjected to the same extinction. Future work would benefit from the inclusion of differential extinction

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within the cluster members, as this would also improve distance estimates. This could also potentially address the possibility of ‘split main-sequences’, which may simply arise due to poor extinction determination as opposed to multiple populations with different rotation rates as suggested by D’Antona et al. (2015). This work demonstrates a novel explanation for the eMSTO, where we take the results of other fields and apply them to issues present in the field of stellar clusters. Currently, the two most widely accepted explanations rely on rapidly rotating stars and/or binary stars, which can both independently partly explain the phenomenon of the eMSTO (Bastian and Lardo 2018; Li et al. 2019; Beasor et al. 2019). The explanation we present here not only abides by the independent results of other fields, but also complies with the observation that the MS narrows for stars below ∼1.4 M (Goudfrooij et al. 2018), which is roughly the limit below which stars transition from having a convective to radiative core. This transition implies that neither CBM nor REM generated by IGWs would be active anymore. However, we do note that this is also entirely consistent with the mass regime for which magnetic braking becomes important (Kraft 1967; Georgy et al. 2019). Most notably, CBM and REM induced by IGWs alone cannot explain the apparent need for different amounts of internal mixing for two stars of the same mass. However, as discussed in Sect. 2.3, we consider the full range of mixing efficiencies as determined from asteroseismic analysis, such that we allow for not only pulsationally induced mixing, but different rotation rates, the occurrence of magnetic fields, and any possible nonlinear interactions between any of these mechanisms as well. Finally, we consider that our isochrone-cloud methodology needs to be calibrated on a larger sample of clusters with independent results. Furthermore, robust calibration requires clusters which have good spectroscopic characterisation to determine rotation rates, as well as good photometric time series to determine pulsational characteristics. An ideal cluster would also have at least one EB from which independent ages and distances can be determined. Thankfully, such a well observed data set is currently being assembled by the complementary Gaia and TESS missions. We have already made use of Gaia data in this Chapter, however, future Gaia data releases will also consist of spectroscopy. Combined with the ongoing TESS photometric mission which is covering some 85% of the sky, numerous clusters with sufficient data sets to extend this study will soon become available.

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Chapter 7

Towards Constraining Tidal Mixing: U Gru

This chapter uses UVES data obtained via application for ESO Directors Discretionary Time by the author. The spectroscopic analysis is currently ongoing by PhD student Sanjay Sekaran. The results of the spectroscopic and binary analysis will be submitted as follow-up paper(s) in the near future. Here, we present a status report. This chapter is based in part on: 1. Discovery of Tidally Perturbed Pulsations in the Eclipsing Binary U Gru: A Crucial System for Tidal Asteroseismology Bowman, D.M.; JOHNSTON, C.; Tkachenko, A.; Mkrtichian, D.E.; Gunsriwiwat, K.; Aerts, C. The Astrophysical Journal Letters, Vol. 883, L26, 6 pp. (2019) Author Contributions: C. Johnston and D.M. Bowman identified the target and signal. D.M. Bowman performed the frequency analysis. All authors participated in the discussion of potential mechanisms and interpretation of the frequency analysis.

7.1 Introduction In this monograph, we have made the explicit assumption that the binary nature of the studied systems has not impacted their evolution. Furthermore, by remaining agnostic as to the mechanisms contributing to the overall internal mixing profile, we effectively incorporated any tidally induced mixing (i.e. the collective mixing caused in the stellar interior due to either the equilibrium or dynamic tide) into Dmix (r ). Much like other forms of internal chemical mixing, tidally induced mixing is poorly constrained in stellar structure and evolution calculations. In addition to the chemical and angular momentum transport caused by tides, the tidal force is expected to induce changes to a star’s internal structure as a new force needs to

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0_7

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be included in the momentum equation. This will cause deviations from a secular evolution depending on the relative importance of the acting forces, amongst which are the Coriolis and centrifugal forces, the Lorentz force, and tidal forces. Pulsating stars in binaries offer a potential means to quantify any deviations in stellar structure, including increased mixing due to tides, via asteroseismology, should tidally induced or tidally perturbed pulsations be detected. Tidally induced pulsations occur due to the dynamical tidal response, which manifests itself as forced oscillations. The frequency of such oscillations (in the co-rotating frame) is determined by the orbital and rotational periods. In the case that these pulsations are driven at, or near a naturally occurring eigen-frequency in a star, the amplitude at this eigen-frequency will experience resonant excitation. When transformed to the inertial frame of the observer, these modes are observed at harmonics of the orbital frequency. Since the advent of high-quality nearly continuous photometric spacebased data-sets, tidally induced oscillations have been regularly observed in close eccentric binary systems, many of which exhibit a periastron-brightening event also known as a heart-beat event (Welsh et al. 2011; Thompson et al. 2012; Hambleton et al. 2016; Fuller 2017). This is caused by the mutual deformation of the two components when they are at their closest approach in the orbit. To date, there are some 170+ such systems known in the K epler and BRITE data (Hambleton et al. 2013; Kirk et al. 2016; Pablo et al. 2017). Analyses of these systems have revealed a dominant tidal response in terms of  = 2 modes, as expected from theory (Burkart et al. 2012; Weinberg et al. 2012; Fuller 2017). Phenomena such as resonant locking (Hambleton et al. 2016; Fuller 2017), rapid apsidal motion (Hambleton et al. 2018), and tidally driven p- and g-modes with amplitudes that are reasonably well predicted by theory have been observed in space photometry (Hambleton et al. 2013; Smullen and Kobulnicky 2015; Fuller 2017; Guo et al. 2017). In contrast to tidally induced oscillations, tidally perturbed oscillations are freely driven and damped stellar pulsations whose eigen-frequency is perturbed from the value it has under spherical symmetry due to the tidal response. Whereas these tidally perturbed oscillations have a well developed theoretical framework (Smeyers and Martens 1983; Smeyers 1997; Smeyers and Willems 1998; Reyniers and Smeyers 2003b, a), they have yet to be observationally confirmed. Reyniers (2002) applied the framework developed in Reyniers and Smeyers (2003a) to the δ Sct pulsating component of 14 Aurigae A, demonstrating that the pulsations in this system agree with tidal perturbations predicted by theory. However, their solution for this system was not unique. Different still to tidally induced and tidally perturbed pulsations are tidally trapped or tidally tilted pulsations. Whereas the former phenomena are the result of the dynamical tide and perturbations to the static stellar structure, tidally tilted pulsators are a class of pulsators where the pulsation axis, and hence axis of symmetry, is aligned along the line of apses connecting the two stars in a binary (Fuller et al. 2020). As a result, the visibility of a pulsation mode with respect to the observer can have a dependence on the orbital phase. This phenomenon has been observed in three stars observed by TESS to date (Handler et al. 2020; Kurtz et al. 2020; Fuller et al. 2020). Interestingly, one target is mono-periodic and fills its Roche-

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149

lobe, whereas the other is multi-periodic and significantly under-fills its Roche-lobe. A special case of tidally trapped oscillations was described in detail by Springer and Shaviv (2013), who investigated solar-like oscillations. Using the WKB wave approximation, these authors argue that the deformation caused by a close binary companion can create conditions such that most acoustic waves become evanescent in regions of low surface gravity before they can form standing modes. This creates an orbital phase visibility dependence for the pulsations which can form standing waves, such that, observationally it appears that the pulsations are exhibiting large scale amplitude variability. In this chapter, we present the case of the not yet understood tidally influenced pulsations observed in the post-mass transfer system U Gru. First, we discuss the phenomena present in the TESS light curves (Fig. 7.1) and the configuration of the U Gru system, before discussing the pulsational properties observed in this system. Finally, we discuss the potential mechanisms to explain the tidally influenced pulsations and outline future work to be done on the system.

7.2 U Gru U Gru was first identified as a semi-detached Algol-like EB with an A5V primary by Brancewicz and Dworak (1980). Using spectroscopy and sparse photometry, Brancewicz and Dworak (1980) was able to identify an orbital period of 1.88 days. Furthermore, using analytical calculations and assuming luminosity relations, Brancewicz and Dworak (1980) estimated the parameters of the primary as: Teff,1 = 8000 K, M1  2 M , and R1  2.5 R .

7.2.1 TESS Light Curve U Gru (TIC 147201138) was observed by the TESS satellite during both its nominal mission (sector 1) and the extended mission (sector 28) (Ricker et al. 2015). During sector 1 observations, U Gru was observed in both the fast two minute cadence mode and the longer 30 min full frame image (FFI) mode. During sector 28, U Gru was only observed in FFI mode, however, the cadence of the FFI mode increased from 30 min to 10 min at the start of the extended mission. As a result, we cannot combine the two data sets to improve the frequency precision due to the different observing cadence. We obtained the raw pixel data and FFIs from the Mikulski Archive for Space Telescopes (MAST) and extracted light curves using the lightkurve python package (Lightkurve Collaboration et al. 2018). We remove the background using the ‘regression corrector’ tool based on the flux determined from background pixels, remove outliers, and median normalise the light curve. The resulting light curves for sector 1 and 28 are displayed in the top and bottom panels of Fig. 7.1, respectively.

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7 Towards Constraining Tidal Mixing: U Gru

Fig. 7.1 TESS light curve of U Gru for Sectors 1 (top) and 28 (bottom)

The large disparity in eclipse depth (light ratio) combined with out of eclipse variability present in the system are consistent with classification of this system as an Algol-like system (Brancewicz and Dworak 1980). Algol systems are a class of binaries which have experienced a mass-ratio reversal as the consequence of binary interaction. The previously more massive component evolved and transferred a large fraction of its envelope to its companion, causing the companion to become the more massive component. In addition to the mass-ratio reversal, the more massive star now appears to be younger, making the system appear paradoxical. Close inspection of the light curve reveals flat-bottomed primary eclipses and triangular secondary eclipses, suggesting a high inclination and gravity darkening of the secondary. Additionally, there is clear out-of-eclipse variability consistent with Roche deformed stars and pulsational variability. When phase folded, the light curve displays an apparent broadening of flux approaching and receding from primary eclipse as seen in the bottom panel of Fig. 7.2. However, when shown against a reference binary model (red model in the upper panels of Fig. 7.2) for three different points of the TESS light curve, the flux at ingress and egress of primary eclipse changes as a function of time. We note that the flux levels flip from ingress and egress having higher and lower fluxes, respectively, to them having lower and higher fluxes, respectively, over the course of about 13 orbits. Unfortunately, this observed phenomenon has a period longer than the data set, restricting our ability to model it.

7.2.2 UVES Observations To characterise the components and the orbit of the system, we successfully applied for director’s discretionary time with the Ultraviolet and Visual Échelle Spectrograph (UVES on UT2@VLT; R ∼ 80 000/110 000 in blue/red; PI: C. Johnston). We obtained 12 epochs (listed in Table 7.1) over a three month period, which are

7.2 U Gru

151

Fig. 7.2 Bottom panel: binary model (red line) and the phase-folded TESS observations of U Gru (black points). Top panels: binary model for three sections of the TESS data demonstrating how the flux and shape of the light curve at ingress and egress in the primary eclipse changes throughout the observations. The flux offset is largest at the start (green panel) and end (blue panel) of the TESS light curve and reverses symmetry in the middle (yellow panel). The light curves from the three top panels are over plotted in the bottom panel to emphasise the changing flux at ingress and egress of the primary eclipse. This figure was originally published by Bowman et al. (2019), The Astrophysical Journal Letters, Volume 883, Issue 1, article id. L26, 6 pp. (DOI: 10.3847/2041-8213/ab3fb2). c AAS. Reproduced with permission Table 7.1 Barycentric Julian Date and orbital phase of UVES observations BJD (d) Phase RV1 (km s−1 ) RV2 (km s−1 ) 2458700.3238560315 2458672.2419380657 2458674.2601534864 2458661.240493133 2458678.321303489 2458633.3885257137 2458699.312911217 2458720.190228179 2458675.1921940376 2458707.3064664975 2458679.25040803 2458698.23880767

−0.47 −0.39 −0.32 −0.24 −0.16 −0.04 −0.01 0.09 0.17 0.24 0.33 0.42

−35.8 ± 0.6 −18.9 ± 0.8 8.0 ± 0.8 33.4 ± 0.7 59.5 ± 0.9 69.7 ± 0.7 68.3 ± 0.6 44.9 ± 0.6 – −15.3 ± 0.6 −38.8 ± 0.5 −48.7 ± 0.7

166 ± 3 – – – −148 ± 4 −159 ± 4 −150 ± 3 – 16 ± 1 – 150 ± 3 199 ± 3

RV3 (km s−1 ) 28.6 ± 0.2 33.7 ± 0.2 32.2 ± 0.2 34.4 ± 0.2 32.2 ± 0.2 35.8 ± 0.2 29.1 ± 0.2 18.0 ± 0.2 33.0 ± 0.2 25.2 ± 0.3 32.8 ± 0.2 29.0 ± 0.2

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equidistantly spaced in orbital phase. In order to increase the expected low signal of the faint secondary, we calculate an average profile for each spectrum using leastsquares deconvolution (hereafter LSD; Tkachenko et al. 2013), allowing for multiple components to be found. We calculated LSD profiles for spectra obtained with both the blue and red arms of UVES. The phase folded LSD profiles for the blue arm are shown in Fig. 7.3. Two components are clearly visible in the LSD profiles of the blue arm. One component is broadened with intrinsic line-profile variability throughout the motion along the orbit. The other component reveals a narrow profile which appears constant over the orbital period. We identify the broad-profile as the rejuvenated A-star primary, and identify the narrow profile as a potential tertiary component in the system. Although the cool faint secondary component was not detected in the blue arm, despite the increased SNR in the LSD profile, we were able to detect it at some phases in the LSD profiles of the red arm. The LSD profiles for both the blue and red arms of UVES are shown in black and grey, respectively, in Fig. 7.3. We note that an RV measurement could not be determined for the primary during primary eclipse, which is consistent with the flat-bottomed (total) eclipse observed for the primary. There were several phases for which we could not determine an RV measurement for the secondary, which can be explained in terms of a limited light contribution in phases near secondary eclipse, or overlap in RV space, where the presence of the primary, tertiary, or both otherwise dominate the profile. We extracted the RVs from the LSD profiles for all three components when possible. We fit an orbit to the inner binary using two setups: (A) assuming a circular orbit, (B) allowing the eccentricity to vary. Both solutions are listed in Table 7.2. Despite both the small number of spectra and small eccentricity, the standard deviation of

Fig. 7.3 LSD profiles from the UVES blue arm (black) and red arm (grey) phased over the binary orbit. Modelled orbits shown as dashed black lines

0.4

Φ

0.2

0.0

−0.2

−0.4

−300

−200

−100

0

RV[km s −1]

100

200

7.2 U Gru

153

Table 7.2 Orbital solution for the inner orbit of U Gru. Estimates taken as median and 95thpercentile HPD of marginalised posteriors Solution A Solution B 1.8812+0.0002 −0.0002 2458719.883+0.003 −0.003 60.5+0.3 −0.3 +1.5 181.6−1.4 0 (fixed) 0 (fixed) 2.4 0.333 ± 0.003

d d km s−1 km s−1 rad km s−1

Fig. 7.4 Top: Radial velocities for primary (circles) and secondary (triangles) with best models according to Table 7.2 for primary and secondary as solid and dashed grey lines, respectively. Bottom: Residuals after subtraction of best fit model

1.8819+0.0002 −0.0002 2458720.03+0.3 −0.3 59.8+0.3 −0.3 +1.5 181.5−1.4 0.041+0.004 −0.004 0.44+0.1 −0.1 1.5 0.330 ± 0.003

200

100

RV [km/s]

Porb Tpp K1 K2 e ω0 Residual scatter q

0

−100

RV [km/s]

20

0 −0.4

−0.2

0.0

0.2

0.4

Φ

the residuals of the solution B is smaller than solution A. Furthermore, the residuals of solution B have a peak-to-peak scatter of 5.4 km s−1 compared to a peak-to-peak scatter of 10.9 km s−1 for the residuals of solution A. We plot the LSD profiles according to orbital phase with the accompanying best fit solutions for the orbital motion of the primary and secondary (black dashed lines) in Fig. 7.3. Additionally, we plot the RV observations, the best model according to solution B in Table 7.2, and the residuals in Fig. 7.4. Both solutions result in a mass ratio of q = 0.33, which implies a secondary with M2 < 1 M assuming a typical mass M1 ∼ 2 M of an A-type star.

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Using solution B from Table 7.2, we performed spectroscopic disentangling (Hadrava 1995; Pavlovski et al. 2009) using the FDbinary code (Ilijic et al. 2004). We fixed the orbit of the tertiary to 1000 days since we could not determine its orbital period from the current data set. We were not able to disentangle the signal of the secondary component. We determined the best matching atmospheric parameters for the disentangled primary and tertiary using gssp (Grid Search in Stellar Parameters; Tkachenko 2015) having fixed the metallicity to solar. The resulting fit for the primary and tertiary are shown in the top and bottom panels of Fig. 7.5, respectively, with the estimated parameters listed in Table 7.3. From this solution, we can estimate that the tertiary is likely an evolved slowly rotating sub-giant or giant, making this system similar in configuration to the algol TW Draconis, which was identified to be a triple system with an Algol-like inner binary by Tkachenko et al. (2010). The detection of the tertiary from the UVES spectra is critical to the interpretation of the pulsations, as it reveals the presence of an additional tidal contribution. Furthermore,

Fig. 7.5 Top: Observed spectra of the primary (blue) and best fitting model according to the parameters in Table 7.3 (orange). Bottom: Observed spectra of the tertiary (blue) and best fitting model according to the parameters in Table 7.3 (orange). Figure used with permission of the author Sanjay Sekaran

7.2 U Gru

155

Table 7.3 Atmospheric solution from the disentangled primary and tertiary components of U Gru Primary Tertiary Teff log g v sin i f

K dex km s−1 %

8500 ± 100 3.48 ± 0.1 63 ± 4 87.7 ± 2

6300 ± 500 3.02 ± 1.4 10 ± 4 7±2

the determination of a mass ratio for the inner binary enables the determination of the dynamic masses and radii of its components. Without this information, our ability to discriminate between potential mechanisms is limited.

7.3 Pulsational Characteristics In many cases, the conditions governing the mass transfer in intermediate-mass close binaries result in the new primary residing in the classical instability strip where δ Sct or hybrid δ Sct/γ Dor pulsations are observed (Bowman and Kurtz 2018). Pulsations were first observed in an Algol system in RZ Cas (Mkrtichian et al. 2002, 2004), and have since been shown to be excellent candidates for probing the effects of mass transfer on stellar interiors (Guo et al. 2016). Furthermore, Algol-type systems have been observed to exhibit tidally induced oscillations by Guo et al. (2017). Pulsations are clearly visible in the TESS light curve of U Gru. However, the binary signal must first be treated in order to investigate the remaining pulsational signal. We determined the orbital frequency to be f orb = 0.531774 ± 0.000004 d−1 (i.e. Porb = 1.88050 ± 0.00001 d), from a multi-frequency nonlinear least-squares optimisation fitting the orbital frequency and 190 consecutive harmonics, up to 100 d−1 . The resulting amplitude spectrum (calculated via discrete Fourier transform; DFT; Deeming 1975; Kurtz 1985) is shown in Fig. 7.6. The DFT reveals both seemingly independent p-modes with frequencies as high as 66 d−1 , and a series of p-modes in a comb-like pattern with frequencies between 21 and 31 d−1 . Using an iterative pre-whitening scheme, we extract 22 frequencies from the residual light curve. The frequencies, amplitudes, and phases are iteratively optimised with a multi-frequency non-linear least-squares fit to the residual light curve. Of these 22 frequencies, 19 belong to the pattern of modes, 17 of which constitute an unbroken chain of consecutively split modes. These 17 modes are consistently separated by the orbital frequency, but are significantly separated from exact orbital harmonics (shown in Fig. 7.6 as grey vertical lines). Additionally, the frequencies that form this series also share a common separation from their nearest integer orbital harmonic of δ f = 0.074 d−1 . Although five of the frequencies in this pattern are below the traditional SNR>4 threshold criterion for significance, they follow the pattern and so are considered significant in our interpretation. Furthermore, two more frequencies have similar

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Fig. 7.6 Residual amplitude spectrum after pre-whitening the orbital harmonics (denoted as vertical grey lines in main panel and sub-panel for clarity) in the TESS light curve of U Gru revealing pulsation mode frequencies. This figure was originally published by Bowman et al. (2019), The Astrophysical Journal Letters, Volume 883, Issue 1, article id. L26, 6 pp. (DOI: 10.3847/2041c 8213/ab3fb2). AAS. Reproduced with permission

separations from their nearest orbital harmonics and are considered as part of this pattern. Three independent frequencies are also extracted which do not follow this pattern, but which fall within the expected range for freely self-excited p-modes in δ Sct stars (Breger 2000; Aerts et al. 2010; Bowman and Kurtz 2018). We note that the isolated frequency at 66.1853 d−1 (shown in the inset of Fig. 7.6) is still within the observed range for p-modes in δ Sct stars at the ZAMS, which is consistent with the A-star primary being rejuvenated by the recent mass-transfer.

7.4 Interpretation The tidal response in a binary or multiple system can manifest in a number of ways. First, we can decompose the tidal response into two components: (i) the equilibrium tide, and (ii) the dynamic tide. The equilibrium tide is the time-independent adjustment to the hydrostatic structure of a star due to the presence of other gravitating bodies. The dynamic tide results in perturbations to the star due to the time-dependent tidal forcing caused by orbital motion (e.g. Fuller 2017, for an extensive description). Expansion of these tidal responses shows that they are dominated by the quadrupole ( = 2) term. In the case of synchronous and circularised binaries, there is no variation in the tidal forcing across the orbit, and thus only the equilibrium tide remains. However, if either condition is broken, the dynamic tide forces oscillations in the components of the multiple system. Generally speaking, this is what is observed in ‘heartbeat’ systems, which display a periodic periastron brightening due to the combination of the equilibrium tide and the reflection effect. In a subset of these systems, tidally induced pulsations (the manifestation of the dynamic tide) are observed as well (Thompson et al. 2012; Fuller 2017). Furthermore, MacLeod et al. (2019)

7.4 Interpretation

157

showed that the dynamic tidal response in a close asynchronous binary, as could be induced by a recent mass transfer event, would generate resonances at decreasing combinations of  and m (angular degree and azimuthal order, respectively) as the orbit decays to attempt to synchronise. Alternatively, in the case of a triple or higher-order system, there is always a time-dependence to the tidal forcing. As outlined in Fuller et al. (2013), this causes a forcing at: (7.1) σm /2π = 2 (ωA − ωB ) − mωA , where ωA is the orbital frequency of the outer orbit, ωB is the frequency of the inner orbit, and m is the azimuthal angle of the forced oscillation being considered. Following this framework, Fuller et al. (2013) detected tidally induced oscillations in the compact hierarchical triple HD 181068. Furthermore, they note the lack of solar-like oscillations in the red-giant companion, suggesting that the tides play a role in mode damping. An absence of modes has been reported in various close red giant binaries as well (Gaulme et al. 2016). Fuller et al. (2013) comment that in the case of a triple system where either or both of the orbits are eccentric or where the orbits are inclined with respect to one another, more forcing frequencies would occur than dictated by Eq. (7.1). We are currently investigating seven possible scenarios for the presence of the tidally influenced pulsations in U Gru: (i): Tidally excited oscillations, driven at exact harmonics of the orbital frequency by the dynamic tide (Ogilvie 2014; Fuller et al. 2019). (ii): Tidally excited oscillations, driven at cascading combinations of  and m as a decaying orbit passing through natural resonances with stellar eigen-frequencies (MacLeod et al. 2019). (iii): Geometric mode visibility effects caused by the partial cancellation of pulsation modes as different parts of the surface are obscured during either primary or secondary eclipse. (iv): Tidally perturbed modes, which are naturally excited modes whose eigenfrequency is perturbed due to the hydrostatic adjustments induced by the equilibrium tide (Polfliet and Smeyers 1990; Reyniers 2002; Reyniers and Smeyers 2003b, a). (v): A chance resonance of the orbital frequency with the large frequency spacing, ν, in one of the components. (vi): Tidal forcing in a three body system (Fuller et al. 2013). (vii): Modes being trapped to one region/hemisphere of the star due to the tidal deformation caused by a close companion, as observed by Handler et al. (2019); Kurtz et al. (2020); Fuller et al. (2020). Since the extracted frequencies do not occur at integer multiples of the orbital frequency of the inner orbit, we can likely exclude scenario (i). Additionally, it is unlikely that we are observing such a rapid change in the orbital period to see the resonant excitation of modes described by MacLeod et al. (2019), so we exclude the evolutionary aspect of scenario (ii) as well. The UVES spectroscopy revealed a rapidly rotating primary and slowly rotating tertiary, as well as a potential eccentricity for the inner orbit, suggesting that pulsations could be driven via asynchronicity in

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either the primary or secondary (MacLeod et al. 2019). To investigate the possibility of scenario (iii), we break the light curve into three different sections: (a) out of eclipse, (b) in primary eclipse, and (c) in secondary eclipse. We calculate the DFT for each section as shown in the panels of Fig. 7.7. The vertical grey lines denote the locations of the extracted frequencies as listed in Table 7.4. Strikingly, we observe that the comb of frequencies split by the orbital frequency is not present in the DFT of the out of eclipse regions, whereas the comb seemingly increases in amplitude and extends in length in the primary eclipse regions. We note that this is also convolved with a complex window pattern due to the artificial selection of points in a given orbital phase. Additionally, signal still appears in the comb pattern in the secondary eclipse regions. This test reveals that the signal could be in part caused by partial mode cancellation as the disk is obscured during eclipse, causing a fluctuation in the amplitude. This does not explain the presence of the full series. Furthermore, the lack of the independent p-modes combined with the increase in amplitude of the comb-pattern in the DFT of the data in primary eclipse suggest that the independent p-mode pulsations are occurring in the A-type primary, and that the comb-like pattern is originating in either the secondary or tertiary. However, given the expected low light contribution of the secondary, we can conclude that this is more likely originating in the tertiary. In order to properly contemplate the possibility of scenarios (iv) and (v), we require the precise determination of the fundamental parameters of all components in the system, which is yet to be carried out. The likelihood of scenario (vi) is currently difficult to assess, as we do not have a clear characterisation of the orbit/separation of the third body. Finally, we comment that scenario (vii) cannot be excluded considering the current observations.

Fig. 7.7 Amplitude spectra of the residual TESS light curve (i.e. after subtracting the multiharmonic binary model) of U Gru using data from different binary phases (in primary: −0.09 ≤ φ ≤ 0.09 and in secondary: 0.41 ≤ φ ≤ 0.59; cf. Fig. 7.2). The vertical grey lines indicate the frequencies included in Table 7.4 identified as significant from the entire light curve. This figure was originally published by Bowman et al. (2019), The Astrophysical Journal Letters, Volume 883, Issue c 1, article id. L26, 6 pp. (DOI: 10.3847/2041-8213/ab3fb2). AAS. Reproduced with permission

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Table 7.4 Frequencies, amplitudes and phases of the significant pulsation modes in U Gru. 1σ uncertainties calculated from the multi-frequency non-linear least-squares fit and the SNR of each pulsation mode in the residual amplitude spectrum are given. The measured frequency difference between an independent pulsation mode frequency, ν, and the adjacent (lower frequency) harmonic, i, of the orbital frequency, i νorb , is also provided in the last column. This table was originally published by Bowman et al. (2019), The Astrophysical Journal Letters, Volume 883, Issue 1, article c id. L26, 6 pp. (DOI: 10.3847/2041-8213/ab3fb2). AAS. Reproduced with permission Frequency (d−1 )

Amplitude (mmag)

Phase (rad)

SNR

i

ν − i νorb (d−1 )

21.8802 ± 0.0029

0.217 ± 0.032

0.46 ± 0.15

4.27

41

0.078 ± 0.004

22.4077 ± 0.0021

0.296 ± 0.032

−2.88 ± 0.11

4.92

42

0.073 ± 0.003

22.9411 ± 0.0015

0.413 ± 0.032

0.15 ± 0.08

6.03

43

0.075 ± 0.002

23.4696 ± 0.0013

0.490 ± 0.032

−3.13 ± 0.07

6.26

44

0.072 ± 0.002

24.0022 ± 0.0012

0.520 ± 0.032

−0.03 ± 0.06

6.37

45

0.072 ± 0.002

24.5349 ± 0.0012

0.545 ± 0.032

2.95 ± 0.06

6.37

46

0.073 ± 0.001

25.0644 ± 0.0011

0.564 ± 0.032

−0.41 ± 0.06

6.75

47

0.071 ± 0.001

25.5979 ± 0.0015

0.417 ± 0.032

2.59 ± 0.08

5.72

48

0.073 ± 0.002

26.1303 ± 0.0016

0.381 ± 0.032

−0.69 ± 0.08

5.70

49

0.073 ± 0.002

26.6644 ± 0.0028

0.225 ± 0.032

2.26 ± 0.14

4.33

50

0.076 ± 0.003

27.1966 ± 0.0040

0.155 ± 0.032

−1.37 ± 0.21

3.51

51

0.076 ± 0.004

27.7289 ± 0.0057

0.109 ± 0.032

0.30 ± 0.30

2.72

52

0.077 ± 0.006

28.2758 ± 0.0061

0.103 ± 0.032

−2.95 ± 0.32

2.44

53

0.092 ± 0.006

28.7910 ± 0.0025

0.246 ± 0.032

−0.78 ± 0.13

3.97

54

0.075 ± 0.003

29.3180 ± 0.0025

0.247 ± 0.032

2.16 ± 0.13

4.12

55

0.071 ± 0.003

29.8512 ± 0.0035

0.181 ± 0.032

−1.34 ± 0.18

3.12

56

0.072 ± 0.004

30.3800 ± 0.0037

0.167 ± 0.032

1.69 ± 0.19

2.77

57

0.069 ± 0.004

31.4469 ± 0.0006

0.984 ± 0.032

1.01 ± 0.03

8.17

59

0.072 ± 0.001

33.0442 ± 0.0003

1.997 ± 0.032

−2.16 ± 0.02

8.22

62

0.074 ± 0.001

33.8598 ± 0.0004

1.571 ± 0.032

0.95 ± 0.02

8.78

63

0.358 ± 0.001

39.4689 ± 0.0020

0.323 ± 0.032

2.24 ± 0.10

6.72

74

0.112 ± 0.002

66.1853 ± 0.0023

0.276 ± 0.032

0.41 ± 0.12

5.97

124

0.245 ± 0.009

Determining the exact scenario(s) at work is the focus of the future work for this system. A full binary solution will provide the dynamic masses and radii of the inner binary required to obtain predictions from theory. Furthermore, it will provide the exact light contributions, and how they change over the orbit.

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7 Towards Constraining Tidal Mixing: U Gru

7.5 Summary and Conclusions In this chapter, we investigated the nature of the tidally influenced pulsations present in the light curve of U Gru. Using UVES spectroscopy obtained from an ESO DDT proposal, we discovered the presence of a tertiary component to the system, suggesting a complex dynamical configuration. We extracted 22 frequencies from the light curve after removal of a simple binary model consisting of 190 orbital harmonics. This revealed the presence of a series of 17 consecutive oscillation frequencies all split by the orbital frequency, with a near constant spacing from orbital harmonics. Additionally, we detect three pulsation modes which are apparently un-related to the series of modes, and are consistent with δ Sct p-mode pulsations self-excited via the κ-mechanism. We presented six possible mechanisms to explain the orbitally separated series of modes. We note that tidally excited oscillations may in fact be present in one of the components of U Gru. However, we cannot make a conclusive statement on this since the orbital harmonic model would forcibly remove any such signal artificially. Furthermore, the presence of the third component provides a possible explanation for the orbitally separated series of modes in terms according to Fuller et al. (2013). A full characterisation of the third body orbit is required to confirm or negate this hypothesis. We conclude that following positive confirmation of any scenario presented in this chapter would enable a unique constraining and characterisation of tidal interaction. Such analysis is required for future studies to be able to constrain the effects of tides as they contribute to internal mixing profiles of stars, as well as to the evolution of orbital dynamics, and their impact on the equilibrium structure of stars.

References Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. (2010). Asteroseismology. Astronomy and astrophysics library. Heidelberg: Springer. Bowman, D. M., Johnston, C., Tkachenko, A., et al. (2019). ApJ, 883, L26. Bowman, D. M., & Kurtz, D. W. (2018). MNRAS, 476, 3169. Brancewicz, H. K., & Dworak, T. Z. (1980). Acta Astronomica, 30, 501. Breger, M. (2000). δ Scuti stars (Review). In M. Breger & M. Montgomery (Eds.) (Vol. 210, p. 3). Astronomical society of the pacific conference series. Burkart, J., Quataert, E., Arras, P., & Weinberg, N. N. (2012). MNRAS, 421, 983. Deeming, T. J. (1975). Ap&SS, 36, 137. Fuller, J. (2017). MNRAS, 472, 1538. Fuller, J., Derekas, A., Borkovits, T., et al. (2013). MNRAS, 429, 2425. Fuller, J., Kurtz, D. W., Handler, G., & Rappaport, S. (2020). MNRAS, 498, 5730. Fuller, J., Piro, A. L., & Jermyn, A. S. (2019). MNRAS, 485, 3661. Gaulme, P., McKeever, J., Jackiewicz, J., et al. (2016). ApJ, 832, 121. Guo, Z., Gies, D. R., & Matson, R. A. (2017). ApJ, 851, 39. Guo, Z., Gies, D. R., Matson, R. A., & García Hernández, A. (2016). ApJ, 826, 69. Hadrava, P. (1995). A&AS, 114, 393. Hambleton, K., Fuller, J., Thompson, S., et al. (2018). MNRAS, 473, 5165.

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Hambleton, K., Kurtz, D. W., Prša, A., et al. (2016). MNRAS, 463, 1199. Hambleton, K. M., Kurtz, D. W., Prša, A., et al. (2013). MNRAS, 434, 925. Handler, G., Kurtz, D. W., Rappaport, S. A., et al. (2020). Nature Astronomy, 4, 684. Handler, G., Pigulski, A., Daszy´nska-Daszkiewicz, J., et al. (2019). ApJ, 873, L4. Ilijic, S., Hensberge, H., Pavlovski, K., & Freyhammer, L. M. (2004). In R. W. Hilditch, H. Hensberge, & K. Pavlovski (Eds.), Spectroscopically and spatially resolving the components of the close binary stars (Vol. 318, pp. 111–113). Astronomical society of the pacific conference series. Kirk, B., Conroy, K., Prša, A., et al. (2016). AJ, 151, 68. Kurtz, D. W. (1985). MNRAS, 213, 773. Kurtz, D. W., Handler, G., Rappaport, S. A., et al. (2020). MNRAS, 494, 5118. Lightkurve Collaboration, Cardoso, J. V. D. M., Hedges, C., et al. (2018). Lightkurve: Kepler and TESS time series analysis in Python. Astrophysics Source Code Library. MacLeod, M., Vick, M., Lai, D., & Stone, J. M. (2019). ApJ, 877, 28. Mkrtichian, D. E., Kusakin, A. V., Gamarova, A. Y., & Nazarenko, V. (2002). Pulsating components of eclipsing binaries: New asteroseismic methods of studies and prospects. In C. Aerts, T. R. Bedding, & J. Christensen-Dalsgaard (Eds.) (Vol. 259, p. 96). Astronomical society of the pacific conference series Mkrtichian, D. E., Kusakin, A. V., Rodriguez, E., et al. (2004). A&A, 419, 1015. Ogilvie, G. I. (2014). ARA&A, 52, 171. Pablo, H., Richardson, N. D., Fuller, J., et al. (2017). MNRAS, 467, 2494. Pavlovski, K., Tamajo, E., Koubský, P., et al. (2009). MNRAS, 400, 791. Polfliet, R., & Smeyers, P. (1990). A&A, 237, 110. Reyniers, K. (2002). PhD thesis, Instituut voor Sterrenkunde K.U.Leuven Celestijnenlaan 200B 3001 Leuven Belgium Reyniers, K., & Smeyers, P. (2003a). A&A, 409, 677. Reyniers, K., & Smeyers, P. (2003b). A&A, 404, 1051. Ricker, G. R., Winn, J. N., Vanderspek, R., et al. (2015). Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003. Smeyers, P. (1997). A&A, 318, 140. Smeyers, P., & Martens, L. (1983). A&A, 125, 193. Smeyers, P., & Willems, B. (1998). A&A, 336, 539. Smullen, R. A., & Kobulnicky, H. A. (2015). ApJ, 808, 166. Springer, O. M., & Shaviv, N. J. (2013). MNRAS, 434, 1869. Thompson, S. E., Everett, M., Mullally, F., et al. (2012). ApJ, 753, 86. Tkachenko, A. (2015). A&A, 581, A129. Tkachenko, A., Lehmann, H., & Mkrtichian, D. (2010). AJ, 139, 1327. Tkachenko, A., Van Reeth, T., Tsymbal, V., et al. (2013). A&A, 560, A37. Weinberg, N. N., Arras, P., Quataert, E., & Burkart, J. (2012). ApJ, 751, 136. Welsh, W. F., Orosz, J. A., Aerts, C., et al. (2011). The Astrophysical Journal Supplement Series, 197, 4.

Chapter 8

Final Remarks

8.1 Summary This monograph set out to investigate the impacts of internal mixing in intermediateand high-mass pulsating and binary stars. More specifically, we aimed to determine whether we could precisely deduce the enhancement in or alterations to internal quantities of stars experiencing internal mixing. This led us to the quantification of the convective core mass enhancement along the MS. This monograph consists of case studies and applications to large samples, in an attempt to characterise the consequences of enhanced convective core masses along the MS and calibrate stellar evolutionary models with this enhancement. Chapter 2 laid out the approach and assumptions used to calculate the stellar models. Additionally, we discussed the implementation of the internal mixing processes that we consider. This led us to the definition of isochrone-clouds, which generalise the concept of an isochrone to consider the consequences of enhanced near-core mixing. This chapter concluded with the discussion of our forward modelling methodology which attempts to best estimate stellar parameters while accounting for strong degeneracies and correlations present amongst the parameters of the underlying grid of stellar models. In Chap. 3, we characterised the variability present in the O+B eclipsing binary HD 165246. Using an extensive spectroscopic data set combined with space-based photometry assembled by K 2, we identified low-frequency spectral line-profile and photometric variability consistent with the signatures of IGWs, spot modulation, and κ-driven pulsations. We also found variability signal at harmonics of the orbital frequency, suggesting tidally induced pulsations in the variable star. We derived a mass estimate for the primary and concluded that the myriad of variability signals present will induce internal mixing originating from several mechanisms in at least one of the components. In Chap. 4, we used U Oph and CW Cep as case studies to demonstrate that we can determine the convective core masses of the components of eclipsing binaries using isochrone-clouds. This was then expanded to a larger, homogeneously analysed © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0_8

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sample consisting of 20 components (10 binaries) with precisely determined precise mass and radius estimates. Analysing this sample revealed that the mass discrepancy is in fact a core mass discrepancy. By enhancing the core masses of the stars via near-core mixing, we find a consistent reduction in the mass discrepancy in our sample. Chapter 5 involved the simultaneous binary and asteroseismic modelling of three g-mode pulsating SB2 systems observed by Kepler. The results of this modelling demonstrated that a given mixing profile cannot be constrained when considering the degeneracy in the underlying grid parameters. This is in full agreement with Constantino and Baraffe (2018) and Valle et al. (2018), who demonstrated this independently. Furthermore, this analysis demonstrated that the inclusion of asteroseismic information in part mitigates the effect of degeneracies when simultaneously applied with binary information, leading to appropriate estimation of the convective core masses of the components. Chapter 6 considers the novel application of isochrone clouds to the modelling of the eMSTO phenomenon in YMCs. This demonstrated that the apparently broadened TAMS can be, in part, understood as the consequence of a spread of enhanced core masses. The TAMS broadening (first demonstrated in Fig. 1.3), is naturally explained as a consequence of the fractional core mass distribution changing as a function of age, as seen in comparing Figs. 6.1 and 6.2. We re-iterate that this interpretation is consistent with internal mixing induced by rapid rotating, as this analysis quantified enhanced internal mixing, irrespective of its physical cause. Finally, Chap. 7 discusses the case of U Gru, which was discovered to be a triple system hosting tidally influenced pulsations in at least one of the components. U Gru offers the opportunity to characterise a number of tidal phenomena and investigate the effects of tidal forces throughout the host star. The major conclusion throughout the various chapters in this monograph is that intermediate- and high-mass stars experience internal mixing, leading them to have more massive convective cores than previously considered.

8.2 Current and Future Prospects for the Precise Modelling of Pulsating Stars in Multiple Systems Chapter 5 demonstrated that combining binary and seismic information is useful to constrain the modelling solution space. The natural next step to take for this methodology is to include the modelling of g-mode period spacing patterns of stars in spectroscopic or eclipsing binaries. The detection of period spacing patterns in EBs however, is proving a difficult task. Gaulme and Guzik (2019) systematically searched the Kepler Eclipsing Binary Catalogue (KEBC Kirk et al. 2016) for pulsating stars within EBs. They report 303 candidate systems, 115 of which are self exited heatdriven main-sequence pulsating A/F type stars where g mode period spacing patterns are expected. However, the authors do not attempt to search for such patterns. Li et al.

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(2020a) investigated a large sample of known γ Dor pulsating stars in eclipsing and spectroscopic binary systems from the literature, and detect g mode period spacing patterns in 35 systems. In a separate study, Sekaran et al. (2020) searched all of the well detached EBs from the KEBC and found only one system with a candidate g mode period spacing pattern. In their study, Li et al. (2020a) find that the observed distribution of near-core rotation rates in close binaries (Por b < 10 d) is quantifiably slower than the distribution of observed near core rotation rates in apparently single γ Dor pulsators (determined by Li et al. 2020b). This result is corroborated by the observation of ‘flat’ period spacing patterns in the components of KIC 10080943 and KIC 9850387. In these cases, the rotation is slow enough to not cause significant deviations in the g mode period spacing patterns. Only a handful of these systems have been subjected to detailed asteroseismic modelling to probe their evolution (Schmid and Aerts 2016; Guo et al. 2017; Zhang et al. 2018, 2020). The framework developed in Chap. 5 is poised to be extended to these cases to subject them to detailed evolutionary modelling. As was demonstrated in Chap. 6, the isochrone cloud fitting methodology can easily be applied to systems consisting of any number of components, from binaries, to triples, and even to clusters. In general, with our methodology, the more components to model simultaneously, the more we can mitigate model degeneracies. Considering the condition of equal age, each additional component provides an independent constraint to be met. In light of this, triply or multiply eclipsing systems which can be modelled photo-dynamically (Carter et al. 2011; Pál et al. 2011) are ideal for our modelling. Furthermore, young clusters with intermediate-mass g-mode pulsators are currently being observed by the TESS mission. Modelling these clusters will offer an order of magnitude increase in the number of stars for which we can determine near-core mixing and core masses. Additionally, modelling any cluster for which at least one eclipsing binary is known will provide independent cross-calibration of the results, much in the way that the addition of binary and seismic information calibrated the results of Chap. 5.

8.2.1 Individual Targets While there are numerous prospects for the modelling of lower mass γ Dor pulsating stars in binaries, there are fewer known SPBs in EBs and fewer still with detected period spacing patterns in binaries. To this end, the asteroseismic characterisation of massive pulsating stars in binaries still requires work and the identification of ideal systems for combined asteroseismic and binary modelling. Here we present several candidate systems we have identified for future modelling during the course of the PhD work. We highlight the various interesting objects that we discovered along the way, to be analysed and published in the near future.

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KIC 2696703

This section is based on ongoing work carried out by C. Johnston, T. Van Reeth, and A. Tkachenko. The spectra were normalised and analysed by A. Tkachenko. The frequency analysis and detection of period-spacing patterns was performed by T. Van Reeth. The binary and orbital modelling was carried out by C. Johnston.

The first candidate system is KIC 2696703, an eclipsing system consisting of two nearly identical F-type stars (q = 1.007). The system exhibits grazing eclipses, which are of much lower amplitude than the pulsational signal. This posed a challenge to the binary modelling, but was overcome by applying a non-equidistantly spaced binning to the phase folded Kepler lightcurve. We subjected the lightcurve to iterative pre-whitening to extract all frequencies with SNR>4. From these, three individual g-mode period spacing patterns were identified as shown in Fig. 8.1. We then combined the pre-whitened lightcurve and eight RV measurements obtained using hermes (Raskin et al. 2011) in an MCMC modelling approach according to the method described in Chap. 3. The results of this modelling are listed in Table 8.1. Considering the presence of multiple g-mode period spacing patterns as well as the precise determination of fundamental parameters from binary modelling, this system currently represents the best system for future application of our methodology for binary asteroseismology (Fig. 8.2).

Fig. 8.1 Top: Zoom in of the periodogram of KIC 2696703. Bottom: Period-spacing patterns derived from the upper panel. Figure courtesy of Timothy Van Reeth

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Table 8.1 Optimised parameters for KIC 2696703. Optimal values are taken as median and 95thpercentile HPD estimates of the marginalised posterior distributions Parameter Unit Primary Secondary Period T0 a q i e ω0 γ Teff L  Derived parameters M R log g

d d R deg rad km s−1 K %

M R dex

Fig. 8.2 Top: Phase folded radial velocities for the primary and secondary of KIC 2696703 with best fitting model according to Table 8.1. Bottom: Residuals after subtraction of optimal model

6.09458 (fixed) 2333.654 ± 0.003 20.7 ± 0.1 1.007 ± 0.008 79.5 ± 0.28 0.024 ± 0.002 0.59 ± 0.06 −20.1 ± 0.2 7160 ± 90 50.4 ± 0.5 11.8 ± 0.2

7130 ± 150 49.6 ± 0.6 11.9 ± 0.2

1.58 ± 0.3 1.93 ± 0.04 4.07 ± 0.02

1.60 ± 0.3 1.91 ± 0.05 4.08 ± 0.03

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KIC 8167938

This section is based on ongoing work by C. Johnston and. P. I. Pápics. The original frequency analysis and pre-whitening was performed by P. I. Pápics. The spectra used thus far were obtained by P. I. Pápics. C. Johnston led a successful proposal to obtain time-series spectroscopy of the system around primary eclipse to determine if the Rossiter–McLaughlin effect is present. These spectra have not yet been analysed.

KIC 8167938 (KOI-6052) is a B5III star (Frasca et al. 2016) which was first identified as an SPB by McNamara et al. (2012). In addition to displaying multiperiodic behaviour, KIC 8167938 displays large amplitude ellipsoidal variability and eclipses. The eclipse depth has much lower amplitude than the pulsations, and has a total duration of less than two hours. Given the 30 min cadence Kepler observations, we cannot distinguish between a grazing eclipse or a full transit (Fig. 8.3).

Amplitude [ppm]

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0

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Fig. 8.3 Top: Phase folded radial velocities for the KIC 8167939A and best fitting model according to Table 8.2. Bottom: Residuals after subtraction of optimal model

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Table 8.2 Optimised parameters for KIC 8167938. Optimal value and estimates taken as median and 95th-percentile HPD estimates of the marginalised posterior distributions Parameter Unit Primary Secondary d d rad km s−1 km s−1

2.5657 (fixed) 2456048.01+0.16 −0.16 0.013+0.005 −0.006 5.55+0.4 −0.4 8.1+0.2 −0.2 61.1+0.3 −0.3

M R

0.061+0.001 −0.001 3.10+0.02 −0.02

Fig. 8.4 Top: Periodogram of 4-year Kepler lightcurve of KIC 8167938. Bottom: Phase folded lightcurve of KIC 8167938 using pre-whitened lightcurve

–

60 40

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Period T0 e ω0 γ K Derived parameters f(M) a1 sin i

20 0

−20 −40

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−60 5 0 −5

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To investigate this system 24 spectra were obtained with the hermes spectrograph. Radial velocities were extracted from the LSD profile for each spectrum and modelled using an MCMC sampling technique. The solution listed in Table 8.2 reveals a small eccentricity. Furthermore, the secondary is not detectable in any spectrum to within a few percent of the continuum. As such, we have little information to constrain the nature of the secondary currently. No g-mode period-spacing patterns have yet been identified from this target. However, modelling of individual mode frequencies can still be achieved within our modelling framework, assuming particular mode degrees (Fig. 8.4).

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HD 46485

This section is based on ongoing work by C. Johnston and M. Abdul-Masih. HD 46485 is a rapidly rotating O7V((f)) star (Teff = 36000 K; Martins et al. 2015) recently revealed by TESS to be an eccentric eclipsing binary with an orbital period of Porb = 6.9443 d. In addition to being rapidly rotating (v sin i ≈ 320 km s−1 ), the TESS lightcurve of HD 46485 also reveals low-frequency variability (top panel, Fig. 8.5), similar to that observed in HD 165246 (Chap. 3), HD 188209 (Aerts et al. 2017), ρ Leo (Aerts et al. 2018), and the TESS lightcurves of CW Cep and U Oph. Furthermore, the RV measurements (lower panel Fig. 8.5) exhibit a large scatter around the orbital variation, similar to that observed in HD 165246 and ρ Leo. To date, we have collected 56 spectra of HD 46485 between 08 November, 2018 and 25 November, 2019. This dataset will allow us to characterise the inter- and

ΔTp

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Fig. 8.5 Top: Phased TESS lightcurve of HD 46485. Bottom: Phased RVs of HD 46485 collected from Mahy et al. (2009) and Hermes. Grey dots denote measurements obtained from individual lines (Mahy et al. 2009). Black x’s denote the mean RV across all lines and black triangles denote RVs extracted for the HeII 4686 line for data from Mahy et al. (2009). Grey stars denote RVs extracted from hermes data

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intra-nightly variability. The combined presence of these signals makes this system a crucial addition to the high-mass end of our growing sample of stars for which we can determine the core mass.

8.3 Towards a Generalised Sample and Ensemble Modelling This work demonstrates that multiple phenomena can consistently be interpreted under the premise of the consequences of enhanced core masses and the processes which produce this enhancement. We have taken advantage of the presence of binary, seismic, and cluster information to cross-calibrate our results and demonstrate that they are robust. This monograph culminates in the production of Fig. 8.6. This figure displays the inferred fractional convective core masses for all stars considered in this monograph (including the sample from Tkachenko et al. 2020), as well as for those stars modelled by Mombarg et al. (2019) and Pedersen (2021). The inferred measurements are compared against predictions by theoretical evolutionary tracks, where the solid black line represents the predicted fractional core mass at the ZAMS, and the dashed, dashed-dotted, and dotted lines represent the predicted fractional core mass for models with X c = 0.3 (red) and X c = 0.1 (black) with a minimum, intermediate, and maximum amount of internal mixing, respectively. The colour bar represents the inferred X c for stars with coloured points. This plot demonstrates that stars (with X c inferences) are shifted systematically to higher fractional core masses compared to where the models predict for the lowest amount of internal mixing. This further and more uniform population of this diagram with stars across a wider mass and evolutionary range is needed. Furthermore, a more diverse sample consisting of inferences from asteroseismology, binary star modelling, and combined binary and asteroseismic modelling is required to cross-calibrate this diagram. As it stands, this diagram does not rely on models implementing a particular form of either CBM or REM. Instead, this diagram requires an inference of the core mass and core hydrogen content, irrespective of what type of mixing was implemented in the stellar models used to derive that inference. Future work would benefit from the direct comparison of inferred Mcc and X c from models with different implementations of internal chemical mixing. Initial comparisons have been made, however, demonstrating that both models with convective penetration or convective overshooting result in the asteroseismic and binary fits (individually) for select systems (Moravveji et al. 2015; Claret and Torres 2017). The demise of Kepler gave rise to the K2 mission, which scanned the ecliptic for 90 days at a time, delivering exquisite observations of a number of different stellar populations before it finally ran out of fuel. Today, the TESS mission is currently observing millions of objects across the entire sky for at least 27 days in a search for Earth-like planets around bright, nearby stars (Ricker et al. 2015). This is producing

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Fig. 8.6 Inferred fractional convective core mass values from samples of eclipsing binaries (Chap. 4; blue with colour-coded vcrit estimates), pulsating SB2s (Chap. 5; orange), NGC 1850 and NGC 884 (Chap. 6; grey), single γ Dors from Mombarg et al. (2019) (green), and single SPBs from Pedersen (2021) (red). The solid line denotes the fractional core mass at the ZAMS. Dashed, dashed-dotted, and dotted lines denote the fractional core mass for all models with X c = 0.3 (red) and X c = 0.1 (black), for a minimum, intermediate, and maximum amount of internal mixing (respectively) according the grid input parameters in Table 2.1. All models considered have Z ini = 0.014 and Yini = 0.276

the most complete sample of stars with high-precision continuous space photometry to date and has already provided a large number of new g-mode pulsating A/F and O/B stars (Antoci et al. 2019; Pedersen et al. 2019). The nominal TESS mission observed in full-frame image (FFI) mode in 30 min cadence for the whole sky and observed some 10,000 selected targets in a short 2 min cadence mode for each pointing. The extended TESS mission is currently slated to continue observations for two additional years, but will include FFIs at a 10 min cadence and will allow for an ultra-fast 20 s cadence. In the future, the ESA PLATO mission (Rauer et al. 2014) is scheduled to launch (2026 at time of writing) and observe some hundreds of thousands of targets for at least two years in its nominal mission, across a 2250 square degree portion of the sky. These missions promise a bright future for the fields of binary stars, clusters, asteroseismology, and the combination of these efforts. With the incoming samples of pulsating stars in binaries, future versions of Fig. 8.6 will be used to more precisely calibrate the theories of stellar structure and evolution, but it is already clear now that

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stars in the same region of the HRD with the same chemical composition and rotation rate experience different levels of mixing. Having concluded this, the determination as to what specific mechanisms are at work and with what specific efficiency is now the task of future investigation.

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Appendix A

Base MESA Inlist

&star_job create_pre_main_sequence_model = .true. ! Set metal fractions consistently with the opacity kappa_file_prefix = ’OP_a09_p13’ kappa_lowT_prefix = ’lowT_fa05_a09p’ kappa_CO_prefix = ’a09_p13_co’ ! a09+Prz metal fractions initial_zfracs = 8 ! Nuclear Network ! We use the extended isotope network for surface abundances change_net = .true. new_net_name = ’pp_cno_extras_o18_ne22_extraiso.net’ change_initial_net = .true. ! Rotation flags ! We choose to calculate non-rotating equilibrium models new_rotation_flag = .false. change_rotation_flag = .false. change_initial_rotation_flag = .false. new_omega = 0 set_omega = .false. set_initial_omega = .false. / !end of star_job namelist &controls !! We use the other_D_mix profile according to Pedersen+2018 use_other_D_mix = .true. star_history_name = STAR_HISTORY_NAME initial_mass = INITIAL_MASS mixing_length_alpha = MIXING_LENGTH_ALPHA overshoot_f_above_burn_h_core = OVERSHOOT_F_ABOVE_BURN_H_CORE overshoot_f_above_burn_he_core = OVERSHOOT_F_ABOVE_BURN_H_CORE

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0

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min_D_mix = MIN_D_MIX initial_y = 0.276 initial_z = INITIAL_Z varcontrol_target = 1d-5 ! Saved all computed stellar parameters for each model in the history file history_interval = 1 ! Print output to be used with GYRE pulse_data_format = ’GYRE’ add_atmosphere_to_pulse_data = .true. add_center_point_to_pulse_data = .true. keep_surface_point_for_pulse_data = .true. add_double_points_to_pulse_data = .true. interpolate_rho_for_pulse_data = .true. threshold_grad_mu_for_double_point = 5d0 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! CORE BOUNDARY CONTROLS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! alpha_bdy_core_overshooting = 5 he_core_boundary_h1_fraction = 1d-2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! Mass loss !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! hot_wind_scheme = ’Vink’ Vink_scaling_factor = 0.3d0 ! Puls, J., (2015, IAU3017 proceeding) hot_wind_full_on_T = 1d4 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! STOPPING CONTROLS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! delta_lg_XH_cntr_max = -1 delta_lg_XH_cntr_limit = 0.05 ! stop when the center mass fraction of h1 drops below this limit xa_central_lower_limit_species(1) = ’h1’ xa_central_lower_limit(1) = 1d-12 ! target controls for exras_finish_step ! Used in setting IGW mixing profile x_logical_ctrl(1) = .true. x_ctrl(1) = D_EXT !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! MIXING CONTROLS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! remove_small_D_limit = 1d-6 use_Ledoux_criterion = .true. ! if > 0, N2 is smoothed, introducing noise in the period ! spacing patterns: we do not want this! num_cells_for_smooth_gradL_composition_term = 0

Appendix A: Base MESA Inlist alpha_semiconvection = 0d0 semiconvection_option = ’Langer_85 mixing; gradT = gradr’ thermohaline_coeff = 0d0 alt_scale_height_flag = .true. MLT_option = ’Cox’ mlt_gradT_fraction = -1 okay_to_reduce_gradT_excess = .false. ! Defining a minimum diffusive mixing (applicable in the ! radiative zones) set_min_D_mix = .true. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! OVERSHOOTING/CONVECTION CONTROLS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! D_mix_ov_limit = 5d-2 max_brunt_B_for_overshoot = 0 limit_overshoot_Hp_using_size_of_convection_zone = .true. overshoot_alpha = -1 predictive_mix(1) = .true. predictive_zone_type(1) = ’burn_H’ predictive_zone_loc(1) = ’core’ predictive_bdy_loc(1) = ’any’ predictive_mix(2) = .true. predictive_zone_type(2) = ’burn_He’ predictive_zone_loc(2) = ’core’ predictive_bdy_loc(2) = ’any’ predictive_mix(3) = .true. predictive_zone_type(3) = ’nonburn’ predictive_zone_loc(3) = ’shell’ predictive_bdy_loc(3) = ’any’ predictive_mix(4) = .true. predictive_zone_type(4) = ’burn_H’ predictive_zone_loc(4) = ’shell’ predictive_bdy_loc(4) = ’any’ conv_bdy_mix_softening_f0 = 0.002 conv_bdy_mix_softening_f = 0.001 conv_bdy_mix_softening_min_D_mix = 1d-1 ! Set overshooting overshoot_f0_above_burn_h_core = 0.002 overshoot_f0_above_burn_h_shell = 0.002 overshoot_f_above_burn_h_shell = 0.005 overshoot_f0_below_burn_h_shell = 0.002 overshoot_f_below_burn_h_shell = 0.005 overshoot_f0_above_burn_he_core = 0.002 overshoot_f0_above_nonburn_shell = 0.002 overshoot_f_above_nonburn_shell = 0.005 overshoot_f0_below_nonburn_shell = 0.002

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overshoot_f_below_nonburn_shell = 0.005 smooth_convective_bdy = .false. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! ELEMENTAL DIFFUSION CONTROLS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! do_element_diffusion = .false. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! ATMOSPHERE CONTROLS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! which_atm_option = ’simple_photosphere’ !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! OPACITY CONTROLS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! cubic_interpolation_in_X = .false. cubic_interpolation_in_Z = .false. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! ASTEROSEISMOLOGY CONTROLS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! num_cells_for_smooth_brunt_B = 0 interpolate_rho_for_pulsation_info = .true. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! MESH and RESOLUTION CONTROLS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! This is important to properly resolve the gravity modes ! near the convective core boundary max_allowed_nz = 40000 ! global mesh resolution factor mesh_delta_coeff = 0.2 mesh_adjust_use_quadratic = .true. mesh_adjust_get_T_from_E = .true. ! Additional resolution based on the pressure and temperature profiles P_function_weight = 40 T_function1_weight = 110 T_function2_weight = 0 T_function2_param = 2d4 gradT_function_weight = 0 xtra_coef_os_above_burn_h = 0.1d0 xtra_dist_os_above_burn_h = 2d0 ! resol coeff for chemical gradients mesh_dlogX_dlogP_extra = 0.15 ! additional resol on for gradient larger than this mesh_dlogX_dlogP_full_on = 1d-6

Appendix A: Base MESA Inlist ! additional resol off for gradient smaller than this mesh_dlogX_dlogP_full_off = 1d-12 ! taking into account abundance of He4 mesh_logX_species(1) = ’he4’ ! Additional resolution near the boundaries of the convective regions xtra_coef_czb_full_on = 1.0d0 xtra_coef_czb_full_off = 1.0d0 xtra_coef_a_l_hb_czb = 0.5d0 xtra_dist_a_l_hb_czb = 1d0 xtra_coef_b_l_hb_czb = 0.5d0 xtra_dist_b_l_hb_czb = 1d0 xtra_coef_a_l_hb_czb = 0.5d0 xtra_dist_a_l_hb_czb = 1d0 xtra_coef_b_l_hb_czb = 0.5d0 xtra_dist_b_l_hb_czb = 1d0 ! non-burning zone xtra_coef_a_l_nb_czb = 0.5d0 xtra_dist_a_l_nb_czb = 1d0 xtra_coef_b_l_nb_czb = 0.5d0 xtra_dist_b_l_nb_czb = 1d0 xtra_coef_a_l_nb_czb = 0.5d0 xtra_dist_a_l_nb_czb = 1d0 xtra_coef_b_l_nb_czb = 0.5d0 xtra_dist_b_l_nb_czb = 1d0 ! He burning zone xtra_coef_a_l_heb_czb = 0.5d0 xtra_dist_a_l_heb_czb = 1d0 xtra_coef_b_l_heb_czb = 0.5d0 xtra_dist_b_l_heb_czb = 1d0 xtra_coef_a_l_heb_czb = 0.5d0 xtra_dist_a_l_heb_czb = 1d0 xtra_coef_b_l_heb_czb = 0.5d0 xtra_dist_b_l_heb_czb = 1d0 ! Additional Resolution in overshooting region xtra_coef_os_full_on = 1.0d0 xtra_coef_os_full_off = 1.0d0 xtra_coef_os_above_burn_h = 0.5d0 xtra_dist_os_above_burn_h = 0.5d0 xtra_coef_os_below_burn_h = 0.5d0 xtra_dist_os_below_burn_h = 0.5d0 xtra_coef_os_above_nonburn = 0.5d0 xtra_dist_os_above_nonburn = 0.5d0 xtra_coef_os_below_nonburn = 0.5d0 xtra_dist_os_below_nonburn = 0.5d0

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180 xtra_coef_os_above_burn_he = 0.5d0 xtra_dist_os_above_burn_he = 0.5d0 xtra_coef_os_below_burn_he = 0.5d0 xtra_dist_os_below_burn_he = 0.5d0 / ! end of controls namelist

Appendix A: Base MESA Inlist

Appendix B

Moment Method Periodograms

See Figs. B.1, B.2, B.3, B.4. 0th−Moment 2

Amplitude [km s−1 ]

0 2 1 0 1.0 0.5 0.0

0.5 0.0

0

2

4

6

8

10

Frequency [d−1 ]

Fig. B.1 Periodograms of the 0th-moment of the LSD profiles. Extracted frequencies denoted by red vertical line. Dashed blue lines show noise level of pre-whitened data set, dotted blue lines denote four times the noise level. Figure adapted from Johnston et al. (2021), their Fig. A1. Reproduced with permission from author and OUP on behalf of MNRAS

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0

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1st−Moment 4 3

Amplitude[kms

−1

]

2 1 0

3 2 1 0

0

2

4

6

8

10

Frequency[d −1 ]

Fig. B.2 Same as Fig. B.1 but for 1st moment. Figure adapted from Johnston et al. (2021), their Fig. A2. Reproduced with permission from author and OUP on behalf of MNRAS

Appendix B: Moment Method Periodograms

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2nd−Moment 500 250

Amplitude[km 2 s−2 ]

0 400 200 0

200

0 200

0

0

2

4

6

8

10

Frequency [d −1 ]

Fig. B.3 Same as Fig. B.1 but for 2nd moment. Figure adapted from Johnston et al. (2021), their Fig. A3. Reproduced with permission from author and OUP on behalf of MNRAS

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Reference 3rd−Moment 200000 150000

Amplitude[km 3 s−3 ]

100000 50000 0

150000

100000

50000

0

0

2

4

6

8

10

Frequency [d −1 ]

Fig. B.4 Same as Fig. B.1 but for 3rd moment. Figure adapted from Johnston et al. (2021), their Fig. A4. Reproduced with permission from author and OUP on behalf of MNRAS

Reference Johnston, C., Aimar, N., Abdul-Maish, M., Bowman, D. M., White, T. R., Hawcroft, C., Sana, H., Sekaran, S., Dsilva, K., Tkachenko, A., & Aerts, C. (2021). Characterization of the variability in the O+B eclipsing binary HD 165246. Monthly Notices of the Royal Astronomical Society. 503(1): 1124–1137. arXiv:2102.08391. https://ui.adsabs.harvard.edu/abs/2021MNRAS.503.1124J. Provided by the SAO/NASA Astrophysics Data System.

Appendix C

Marginalised Posterior Distributions for Chap. 4

See Figs. C.1, C.2, C.3, C.4, C.5, C.6, C.7, C.8.

Fig. C.1 Marginalised posterior distributions for the primary and secondary passband luminosities for each observed filter. The median is shown by a solid vertical line, the upper and lower bounds for 68.27% CI by dashed vertical lines. This figure was originally published by Johnston et al. (2019), c their Fig. A1. Reproduced with permission from Astronomy & Astrophysics, ESO

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0

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Fig. C.2 Marginalised posterior distributions for the primary and secondary parameters. The median is shown by a solid vertical line, the upper and lower bounds for 68.27% CI by dashed vertical lines. This figure was originally published by Johnston et al. (2019), their Fig. A2. Reproc duced with permission from Astronomy & Astrophysics, ESO

Appendix C: Marginalised Posterior Distributions for Chap. 4

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Fig. C.3 Marginalised posterior distributions for the orbital parameters. The median is shown by a solid vertical line, the upper and lower bounds for 68.27% CI by dashed vertical lines. This figure was originally published by Johnston et al. (2019), their Fig. A3. Reproduced with permission from c Astronomy & Astrophysics, ESO

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Fig. C.4 Marginalised posterior distributions for the system parameters. The median is shown by a solid vertical line, the upper and lower bounds for 68.27% CI by dashed vertical lines. This figure was originally published by Johnston et al. (2019), their Fig. A4. Reproduced with permission from c Astronomy & Astrophysics, ESO

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Fig. C.5 Marginalised posterior distributions for the primary and secondary passband luminosities for each observed filter. The median is shown by a solid vertical line, the upper and lower bounds for 68.27% CI by dashed vertical lines. This figure was originally published by Johnston et al. (2019), c their Fig. B1. Reproduced with permission from Astronomy & Astrophysics, ESO

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Appendix C: Marginalised Posterior Distributions for Chap. 4

Fig. C.6 Marginalised posterior distributions for the primary and secondary parameters. The median is shown by a solid vertical line, the upper and lower bounds for 68.27% CI by dashed vertical lines. This figure was originally published by Johnston et al. (2019), their Fig. B2. Reproc duced with permission from Astronomy & Astrophysics, ESO

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Fig. C.7 Marginalised posterior distributions for the orbital parameters. The median is shown by a solid vertical line, the upper and lower bounds for 68.27% CI by dashed vertical lines. This figure was originally published by Johnston et al. (2019), their Fig. B3. Reproduced with permission from c Astronomy & Astrophysics, ESO

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Appendix C: Marginalised Posterior Distributions for Chap. 4

Fig. C.8 Marginalised posterior distributions for system parameters. The median is shown by a solid vertical line, the upper and lower bounds for 68.27% CI by dashed vertical lines. This figure was originally published by Johnston et al. (2019), their Fig. B4. Reproduced with permission from c Astronomy & Astrophysics, ESO

Appendix D

Secondary Component Parameter Correlation Plots From Chap. 5

See Figs. D.1, D.2, D.3, D.4. Fig. D.1 Same as Fig. 5.1b but for the SB2 evaluation of KIC 4930889 B. Figure adapted from Johnston et al. (2019), their Fig. A1. Reproduced with permission from author and OUP on behalf of MNRAS

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0

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Appendix D: Secondary Component Parameter Correlation Plots From Chap. 5

Fig. D.2 Same as Fig. 5.1b but for the SB2 evaluation of KIC 6352430 B. Figure adapted from Johnston et al. (2019), their Fig. A2. Reproduced with permission from author and OUP on behalf of MNRAS

Fig. D.3 Same as Fig. 5.1b but for the SB2 evaluation of KIC 10080943 B. Figure adapted from Johnston et al. (2019), their Fig. A3. Reproduced with permission from author and OUP on behalf of MNRAS

Appendix D: Secondary Component Parameter Correlation Plots From Chap. 5 Fig. D.4 Same as Fig. 5.1b but for the heartbeat star evaluation of KIC 10080943 B. Figure adapted from Johnston et al. (2019), their Fig. A4. Reproduced with permission from author and OUP on behalf of MNRAS

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Index

B β Cep stars, 25, 59 Binary stars, 26, 113 eclipsing, 81

C Carbon-Nitrogen-Oxygen (CNO), 7, 14 Convection, 4, 5, 8 convective boundary mixing, 43 Ledoux criterion, 4, 38 mixing length theory, 5, 38 Schwarzschild criterion, 8, 43, 115, 116 semiconvection, 5 Core mass, 43, 98, 116, 171 CW Cep, 82

D δ Sct stars, 23, 24 E Extended main-sequence turn-off, 28, 134

G Gaia, 144 γ Dor, 116 γ Dor stars, 23 G modes, 36, 39, 74, 114

H HD 165246, 54

Hydrostatic equilibrium, 6

I Internal gravity waves, 11, 61 Isochrone, 43, 45, 76 cloud, 45, 82, 98, 103, 118, 136

K Kepler, 148 KIC 10080943, 114, 118, 125 KIC 4930889, 122 KIC 6352430, 123, 124

M Mahalanobis distance, 47, 98, 116, 129, 137 Main-Sequence (MS), 14 Mass discrepancy, 29, 82, 103, 129 Mixing, 37 convective boundary mixing, 58, 114, 115 profile, 8 radiative envelope mixing, 8, 11 rotational, 12, 29, 134 wave induced, 11 Monte Carlo, 49, 98 Markov Chain, 50, 93

N NGC 1850, 137 NGC 884, 140 χ Per, 140

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Johnston, Interior Modelling of Massive Stars in Multiple Systems, Springer Theses, https://doi.org/10.1007/978-3-030-66310-0

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198 P P modes, 74 pp-chain, 7, 14 Pressure, 2 Pulsations g modes, 22 p modes, 21

S SPB stars, 25 Stellar models, 36

T Terminal-age main-sequence, 14, 134 TESS, 144 Tides, 13, 101, 148

Index Time scales, 6 dynamical, 6 Kelvin-Helmholtz, 5, 6, 10, 11 nuclear, 6, 11, 14, 16, 28

U U Gru, 149 U Oph, 84

Y Young massive clusters, 28, 133

Z Zero-Age Main-Sequence (ZAMS), 14, 36, 171