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The Evolution and Pulsation of Crystallizing White Dwarf Stars THE EVOLUTION AND PULSATION OF CRYSTALLIZING WHITE DWARF STARS by MICHAEL HOUSTON MONTGOMERY, B.S., M.A. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Ful¯llment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN December 2002 Copyright by Michael Houston Montgomery 2002 THE EVOLUTION AND PULSATION OF CRYSTALLIZING WHITE DWARF STARS APPROVED BY DISSERTATION COMMITTEE: Supervisor: Acknowledgements I would like to thank Don and Ed for providing the environment and atmo- sphere of the WET lab. Their open approach to doing science is a model which others would do well to follow. From Ed I have learned the di®erence between models and reality, which I continually try not to forget. From Don I have learned how to have ideas and how to develop them, and that sometimes the craziest ideas are the best. The rest of my committee, Ethan, Craig, and Carl, have also been a valuable source of ideas and advice. The students in the WET lab, both past and present, are equally re- sponsible for the productivity of the group. Paul Bradley and Matt Wood have both made valuable contributions to my education, as well as to the calcula- tions in this thesis. They each deserve the title of \honorary Ph.D. committee member." Steve Kawaler also provided me with useful advice regarding the white dwarf evolution code, and evolution codes in general. The former students which I had signi¯cant overlap with while they were still at Texas are Judi, Chuck, Chris, Scot, and Ant^onio, all of whom have helped me and all of whom continue to be active in the ¯eld. Scot in particular has helped me understand the nature of the observations and of the stars themselves, even though we still have more questions than answers in that regard. He also taught me the basics of system administration on Unix machines, although my understanding of the inner workings of these systems is vii still fairly rudimentary. Ant^onio and I have had dozens of discussions ranging from the observations to the theory to the instrumentation (gasp). He is not content with the simple pat answers, but must actually understand. Conse- quently, it took hours to answer some of his questions, during which time I went from con¯dence to doubt and despair and ¯nally to a renewed and better understanding of the problem. Of course, sometimes we started not knowing the answer and matters only got worse. But in the end, we were certain that we did not know! Atsuko and I joined the lab at the same time and were two of the most junior graduate students for many years. She has made the lab a better and a brighter place not just for me, but for everyone else. Now that we are the \senior" graduate students, we are agreed upon one thing: it's no good being senior if there aren't junior students around to do the work! Eric did his second-year project as part of our group. In fact, the work on phase separation in Chapter 2 of this thesis is an extension of his Master's thesis. He has continued to be a collaborator and a good friend, and I hope that our friendship can continue to grow in the future. Travis is the most recent addition to the WET group. He has helped me with many computer-related problems and he promises to bring more modern computational techniques to the process of white dwarf model ¯tting. On a technical note, I would like to thank M. Salaris for generously providing the oxygen pro¯le shown in Figure 2.4. viii Finally, I would like to thank my friends and family for their support and encouragement over the years. My parents have been particularly under- standing of the various impractical choices I have made and of the large amount of time I have spent in school. My friends, being less practical themselves, have also been of great help. Thank you to all of you. While not being religious myself, I ¯nd it hard not to view the Universe itself as an `entity.' Consequently, I am grateful to the Universe for being understandable in a human sense, or at least being polite enough to give the appearance of being so. Hopefully, we will always be just smart enough to ¯gure out some of its secrets. ix THE EVOLUTION AND PULSATION OF CRYSTALLIZING WHITE DWARF STARS Publication No. Michael Houston Montgomery, Ph.D. The University of Texas at Austin, 2002 Supervisor: D. E. Winget This thesis addresses the physics relevant in crystallizing white dwarf stars. This problem is not merely of academic interest, since white dwarf stars provide us with one of the best methods for estimating for the age of the local Galactic disk. In addition, understanding these stars allows us to probe the physics of matter at temperatures and densities otherwise inaccessible in present-day laboratories. In the ¯rst part of my thesis, I explore the e®ect which phase separation of Carbon and Oxygen can have on the ages of white dwarf stars. I ¯nd that this additional energy source can lengthen white dwarf ages by at most 1.5 Gyr, » with more likely values being in the range 0.4{0.6 Gyr. The most important factors influencing the size of this delay are the total stellar mass, the initial composition pro¯le, and the phase diagram assumed for crystallization. These relatively small age delays are consistent with recent results that the oldest globular clusters may only be 2 Gyr older than the local Galactic disk. » xi In the second part of my thesis, I consider the e®ect which a crystalline core has on the pulsations of white dwarf stars. From global calculations of g-mode eigenfunctions which include the response of the crystalline core, I ¯nd that the amplitudes of the g-modes are greatly reduced in the solid region. As a result, the g-mode oscillations can be accurately modeled by a modi¯ed boundary condition in which the g-modes are excluded from the solid core. As the white dwarf models become more crystallized, the mean period spacing and the periods themselves are lengthened, and can increase by as much as 30% for a model which is 90% crystallized by mass. I also show how mode trapping information can be used to disentangle the e®ects due to the hydrogen layer mass from those due to crystallization. If we are able to obtain mode identi¯cations for enough modes in the DAV BPM 37093, then we may be able to \empirically" measure the degree of crystallization which is present in this object, and thereby test the theory of crystallization itself, now more than 30 years old. The simplicity of white dwarf stars makes them ideal targets of study, since we believe we can adequately model the physical processes occurring inside them. In addition, 98% of all stars are believed to end their lives as white dwarf stars. If we can understand these stars, then we can provide ¯nal boundary conditions for evolution of post-main sequence models. In this sense, they are fundamental objects. xii Table of Contents Acknowledgements vii Abstract xi Table of Tables xvi List of Figures xvii Chapter 1. Overview 1 1. Introduction . 1 2. Why We Study White Dwarf Stars . 2 2.1. Physics . 2 2.2. Astrophysical Relevance . 3 3. Overview of Thesis . 5 4. Coda . 5 Chapter 2. Phase Separation 9 1. Introduction . 9 2. Astrophysical Context . 9 3. The Physics of Phase separation . 12 3.1. Chemical Redistribution . 12 3.2. Energy Release . 16 4. Numerics . 19 4.1. The Melting Curve . 19 4.2. Implementation in WDEC . 20 4.3. Consistency Checks . 24 5. A Simple Test Problem . 28 6. Results . 30 6.1. 0.6 M¯ White Dwarf Models . 30 xiii 6.2. The Mass Dependence . 36 7. Conclusions . 40 Chapter 3. Pulsations and Crystallization 43 1. Astrophysical Context . 43 2. Review of Nonradial Oscillation Theory . 45 2.1. The Fluid Equations . 45 2.2. Asymptotic Theory . 49 3. The E®ect of a Crystalline Core . 56 3.1. The Torsional Modes . 56 3.2. The Spheroidal Modes . 59 4. Numerical Analysis . 62 4.1. The Global Solution . 62 4.2. The \Hard-Sphere" Boundary Condition . 64 5. The g-mode Periods as a Function of Mxtal=M? . 67 5.1. Asymptotic Relations . 67 5.2. Numerical Results . 76 6. ¢P as a Function of the Model Parameters . 85 h i 6.1. The Hydrogen Layer Mass, MH . 85 6.2. The Total Stellar Mass, M? . 89 6.3. The E®ective Temperature, Te® . 89 6.4. Scaling Relations . 92 7. Mode Trapping . 94 7.1. Physical Description . 94 7.2. Phenomenology . 96 8. Objective Fitting Procedures . 102 Chapter 4. Asteroseismological Signatures of Phase Separation105 1. The Question of P_ . 106 2. The E®ect of a Non-uniform C/O Pro¯le . 110 2.1. The 0.6 M¯ Models . 112 2.2. The 1.1 M¯ Models . 112 3. \Self-consistent" C/O Pro¯les . 114 xiv Chapter 5. Conclusion 121 1. Discussion of Results . 122 2. The Future . 124 3. Summing Up . 128 Appendices 129 Appendix A. Spheroidal Oscillation Equations in a Crystalline Medium 131 1. The Equations . 131 2. Central Boundary Conditions . 132 3. The Solid/Fluid Interface . 133 Appendix B. Mode Trapping in the Asymptotic Limit 135 1. A Particular Model . 136 2. A More General Result . 142 Appendix C.
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