DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS CENTREFOR MATHEMATICAL SCIENCES UNIVERSITYOF CAMBRIDGE Magnetic Fields and Stellar Oscillations by Shyeh Tjing Loi CORPUS CHRISTI COLLEGE This dissertation is submitted for the degree of Doctor of Philosophy Submitted: December 2018 Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the Preface and specified in the text. It is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. I further state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. 1 Abstract Author: Shyeh Tjing Loi Title: Magnetic fields and stellar oscillations Stars are fluid bodies in which pressure, buoyancy, rotation and magnetism provide the restoring forces for wave propagation. Constructive interference of these waves generates global modes of oscillation. With sufficiently precise photometry, stellar oscillations can be detected through the periodic variations they induce in a star’s brightness. Understanding the connection between the internal structure of a star and the properties of its normal modes is the focus of stellar oscillation theory, which generally regards rotation and magnetism to be weak perturbations compared to pressure and buoyancy. The discovery several years ago that a sizable fraction of red giant stars exhibit abnormally low dipole mode amplitudes relative to the rest of the population (the “dipole dichotomy” problem), suggesting the existence of a damping process localised to the deep interior, has presented a puzzle for theoreticians as it is not predicted in the standard framework. The restriction of this phenomenon to those stars with masses high enough to previously have hosted core dynamos points to the possible role of a hidden, relic magnetic field. It is the goal of the work presented herein to explore the nature of the interactions between waves in deep stellar interiors (in particular, gravity waves) and magnetic fields of dynamically significant strengths, that it might enable a better understanding of global mode properties and the ability to reconcile with observations. The loss of spherical symmetry associated with the inclusion of magnetic fields renders it a mathe- matically complex problem to tackle, to which end we have invoked various complementary techniques: analytical treatment, numerical simulations and Hamiltonian ray tracing. We find that interactions be- tween gravity waves and a magnetic field rely on a resonance criterion, and are always possible in some part of phase space, regardless of the field strength and the plasma ¯. Under these circumstances, waves may either become trapped within the field region and thereafter dissipated, or reflected in a near-specular fashion, the outcome depending on the relative orientations of the magnetic field lines, the wavevector, and the direction of stratification. Estimates of the damping rates associated with the trapping process, for re- alistic field configurations and field strengths of megagauss order, are consistent with those inferred from red giant observations, providing a promising resolution to the dipole dichotomy problem. The results in this thesis pave the way for an extension of the theory of stellar oscillations that incorporates the effects of dynamically significant magnetic fields. 2 Preface This thesis has been organised into three parts, and comprises a total of twelve chapters, with between 3–5 chapters per part. The breakdown into such a large number of chapters stems from the use of a large num- ber of different techniques and approaches to tackling what is not a completely straightforward problem. However, it is hoped that the manner and order in which the content has been presented makes for logical reading. Part I, ‘Physics of Gaseous Spheres’, deals with stellar physics in general and comprises three chapters. Chapters 1 (‘Stellar Structure and Evolution’) and 2 (‘Magnetism in Stellar Interiors’) contain solely back- ground material on the subject and no new research. In Chapter 3 (‘Magnetohydrostatic Equilibria’), the first two sections (3.1 & 3.2) cover background material, while section 3.3 represents original work. Part II, ‘Stellar Oscillations’, comprises five chapters. Chapter 4 presents the theory of linear adiabatic stellar oscillations in the absence of rotation and magnetism, while Chapter 5 focuses on the specific back- ground to the problem addressed in this thesis, namely the “dipole dichotomy” problem. Both these chap- ters contain purely background material. The content of Chapter 6 is original and based on the publication ‘Torsional Alfvén resonances as an efficient damping mechanism for non-radial oscillations in red giant stars’ by Loi & Papaloizou (2017, MNRAS, 467, 3213). Chapters 7 and 8 also present original research, but in this case the work is preliminary and unpublished, and not related to magnetic fields. Part III, ‘Wave Propagation in a Magnetic Field’, presents further investigations into the problem of local wave interactions with a dynamically significant magnetic field. It comprises four chapters, the first three of which contain results published in ‘Effects of a strong magnetic field on internal gravity waves: trapping, phase mixing, reflection and dynamical chaos’ by Loi & Papaloizou (2018, MNRAS, 477, 5338). Some amount of background material is contained within Chapters 9 and 10, namely those found in Sections 9.1, 9.2 and 10.1. The fourth chapter, Chapter 12, contains original though preliminary work that is as yet unpublished. 3 Acknowledgments A large number of things go into making a PhD happen that aren’t just the research itself. In my particular case, I would like to acknowledge and thank my college, Corpus Christi, for giving me a roof over my head and being my home away from home; my scholarship benefactor, Charles Allen, for covering my mainte- nance and composition fees for the three years of my degree, thus making it possible for me to be here at all; and my department, DAMTP,for providing all the necessary logistical and administrative support for me as a student. Importantly, this includes the 24/7 card access required for me to sustain my slightly eyebrow- raising office hours, which have mostly fallen during the wrong 9-to-5 and where the seven days of the week only have as much practical meaning as an arbitrary permutation thereof. Much love to my friends and colleagues for the colour and personality they have painted into my exis- tence here in this little town, ensuring that, for better or worse, not a single week goes by without its own unique turn of events. Thanks to all of you, the limits of my own sanity continue to be chartered and pushed to new heights. Even more love goes to my family back in Sydney for their long-distance moral support and diligently weekly Skype calls which have meant that, despite their distance, we actually talk more nowadays than when I lived at home. They have promised to visit Cambridge if I graduate. The most significant character in this story to thank must be my PhD supervisor, John Papaloizou, for his time, encouragement, guidance and positivity, for putting up with my abysmal life management skills, for his patience with me in my darker and more dysfunctional moments, and in general, for showing far greater belief in me than I probably deserve. And of course, for being the clever one on this two-person team. Men- tion must also be made of my PhD advisor, Henrik Latter, for the eye he has kept out for me like a watchful older brother, for being there when my sanity has required shepherding, and for his (idiosyncratically dry, but always well intended) words of wisdom. During its incubation phases, this thesis visited some decidedly odd locations. There are possibly not many theses in existence which can lay claim to having been partially written in the backstage bowels of the famous Minack theatre, with its author huddled in pyjamas on the dim and drafty steps of a village hall, or on a laptop balanced precariously atop the ADC stage piano just minutes pre-performance. Photographic evidence exists in the lattermost case: see below. Given the whirlwind of adventures my poor thesis has been dragged through, for which I claim sole and shameful responsibility, it is a great relief for me (and it too, if it had feelings) that it is finally finished; hope is against hope that it won’t have been too much worse for the wear. Thank you in advance to the examiners for having to read through this right mess of a tome. I willingly accept any criticism for its rather diffuse and patchy structure, which is an amusingly appropriate metaphor for me in life: a jack of all trades, master of none. 4 Contents IPHYSICSOF GASEOUS SPHERES 9 1 Stellar Structure and Evolution 10 1.1 Introduction . 10 1.2 Solving for stellar structure . 11 1.2.1 Equations . 11 1.2.2 Polytropic models . 13 1.2.3 Stellar evolution codes . 14 1.2.4 Existence and uniqueness . 15 1.3 Evolutionary stages . 16 1.3.1 Pre-main sequence . 16 1.3.2 Main sequence . 17 1.3.3 Post-main sequence . 18 1.3.4 Degenerate stars . 19 2 Magnetism in Stellar Interiors 20 2.1 General concepts . 20 2.1.1 Magnetohydrodynamics . 20 2.1.2 Magnetic pressure and tension . 21 2.1.3 Magnetic flux . 22 2.1.4 Magnetic helicity .