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Quasi-Stars and the Schönberg–Chandrasekhar Limit

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Quasi-Stars and the Schönberg–Chandrasekhar Limit Quasi-stars and the Schönberg–Chandrasekhar limit Warrick Heinz Ball St Edmund’s College and Institute of Astronomy University of Cambridge arXiv:1207.5972v1 [astro-ph.SR] 25 Jul 2012 This dissertation is submitted for the degree of Doctor of Philosophy Declaration I hereby declare that my dissertation entitled Quasi-stars and the Schönberg–Chan- drasekhar limit is not substantially the same as any that I have submitted for a degree or diploma or other qualification at any other University. I further state that no part of my dissertation has already been or is being concurrently submitted for any such degree, diploma or other qualification. This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as specified in the text. Those parts of this dissertation that have been published or accepted for publication are as follows. • Parts of Chapters 1 and 2 and most of Chapter 3 have been published as Ball, W. H., Tout, C. A., Zytkow,˙ A. N., & Eldridge, J. J. 2011, MNRAS, 414, 2751 • Material from Chapters 5 and 6 and Appendix A has been published as Ball, W. H., Tout, C. A., Zytkow,˙ A. N., 2012, MNRAS, 421, 2713 This dissertation contains fewer than 60 000 words. Warrick Heinz Ball Cambridge, May 29, 2012 Abstract The mechanism by which the supermassive black holes that power bright quasars at high redshift form remains unknown. One possibility is that, if fragmentation is prevented, the monolithic collapse of a massive protogalactic disc proceeds via a cascade of triaxial instabilities and leads to the formation of a quasi-star: a growing black hole, initially of typical stellar-mass, embedded in a hydrostatic giant-like envelope. Quasi-stars are the main object of study in this dissertation. Their envelopes satisfy the equations of stellar structure so the Cambridge STARS code is modified to model them. Analysis of the models leads to an extension of the classical Schönberg– Chandrasekhar limit and an exploration of the implications of this extension for the evolution of main-sequence stars into giants. In Chapter1, I introduce the problem posed by the supermassive black holes that power high-redshift quasars. I discuss potential solutions and describe the conditions under which a quasi-star might form. In Chapter2, I outline the Cambridge STARS code and the modifications that are made to model quasi-star envelopes. In Chapter3, I present models of quasi-stars where the base of the envelope is located at the Bondi radius of the black hole. The black holes in these models are subject to a robust upper fractional mass limit of about one tenth. In addition, the final black hole mass is sensitive to the choice of the inner boundary radius of the envelope. In Chapter4, I construct alternative models of quasi-stars by drawing from work on convection- and advection-dominated accretion flows around black holes. To improve the accuracy of my models, I incorporate corrections owing to special and general relativity into a variant of the STARS code that includes rotation. The evolution of these quasi-stars is qualitatively different from those described in Chapter3. Most notably, the core black holes are no longer subject to a fractional mass limit and ultimately accrete all of the material in their envelopes. In Chapter5, I demonstrate that the fractional mass limit found in Chapter3, for the black holes in quasi-stars, is in essence the same as the Schönberg–Chandrasekhar limit. The analysis demonstrates how other similar limits are related and that limits exist under a wider range of circumstances than previously thought. A test is provided that determines whether a composite polytrope is at a fractional mass limit. In Chapter 6, I apply this test to realistic stellar models and find evidence that the existence of fractional mass limits is connected to the evolution of stars into the red giants. For dad, mom and Rudi. Acknowledgements First, I owe deep thanks to my supervisor, Chris Tout. His scientific insights are always invaluable. If I ever left our discussions without all the answers, I at least left with the right questions. I am also tremendously grateful to Anna Zytkow˙ for her keen interest and extensive input. Her insistence on precise language has not only helped me keep my writing clear but my thinking too. I have also appreciated the continuing support of my MSc supervisor, Kinwah Wu, during my PhD and I owe him substantial thanks for his prompt responses to my various needs. I would also like to thank the members of the Cambridge stellar evolution group who have come, been and gone during my time at the Institute. Particular thanks go to John Eldridge for being a sounding board during Chris’ sabbatical early in my studies, for helping me get to grips with the STARS code and for his support in my applications to academic positions. I’d also like to thank my comrade-in-arms, Adrian Potter, not least for lending me his code, ROSE, but, owing to our common supervisor, being a willing ear for discussions regarding italicized axis labels and their units, hanging participles, and the Oxford comma. Thanks also to Richard Stancliffe for taking the time to respond to my confused emails, which were usually about a problem that existed between the seat and the keyboard. On a personal note, I am tremendously thankful for my friendships, forged in Cambridge, in London or at home, that have endured my infrequent communication. I am also grateful for my fellow graduate students at the Institute who have come and gone during my degree. If anything, they have endured my frequent communication. Finally, to my family, thank you for providing inspiration, motivation and all manner of unconditional support to this marathon of mine. Warrick Heinz Ball Cambridge, May 29, 2012 Technical credits. The work presented in this dissertation was performed using a large amount of free and open-source software. Code and written work was edited with GNU EMACS and code compiled by the FORTRAN compiler in the GNU Compiler Collection. Day-to-day plotting was done using gnuplot. All the figures in this dissertation other than Figs 1.1, 1.2, 3.1 and 3.4 were made using VEUSZ by Jeremy Sanders. Production notes. This dissertation was compiled directly to PDF from LATEX source written in the book class with the times font family and the microtype package. Margins are specified with the geometry package and chapter headings are set using quotchap. Extra table layout features and caption formatting controls were included with the booktabs and caption packages, respectively. Contents Abstract v Acknowledgements ix Contents xi List of Figures xv List of Tables xvii 1 Supermassive black holes in the early Universe1 1.1 Supermassive black hole formation . .2 1.1.1 The early Universe . .2 1.1.2 Accretion and mergers of seed black holes . .4 1.1.3 Direct collapse . .6 1.2 The first luminous objects . .8 1.2.1 Population III stars . .8 1.2.2 Supermassive stars . 11 1.2.3 Quasi-stars . 12 1.3 Outline of this dissertation . 13 2 The Cambridge STARS code 15 2.1 Equations of stellar evolution . 16 2.1.1 Structure . 16 2.1.2 Matter . 18 2.1.3 Composition . 20 2.2 Boundary conditions . 21 2.2.1 The surface . 21 2.2.2 The centre . 22 2.3 Implementation . 22 2.3.1 Mesh-spacing . 23 2.3.2 Difference equations . 24 2.3.3 Solution and timestep . 25 xii 3 Bondi-type quasi-stars 27 3.1 Boundary conditions . 28 3.1.1 Radius . 28 3.1.2 Mass . 29 3.1.3 Luminosity . 31 3.2 Fiducial model . 32 3.2.1 Structure . 32 3.2.2 Evolution . 34 3.2.3 Inner mass limit . 36 3.2.4 Post-quasi-star evolution . 39 3.3 Parameter exploration . 39 3.3.1 Radiative and convective efficiencies . 39 3.3.2 Cavity properties . 41 3.3.3 Envelope mass loss and gain . 42 3.3.4 Initial envelope mass . 42 3.3.5 Inner boundary radius . 44 3.4 Comparison with Begelman et al. (2008) . 46 3.5 Conclusion . 49 4 CDAF-ADAF quasi-stars 51 4.1 The convective-advective boundary . 52 4.1.1 Black hole mass and accretion rate . 53 4.1.2 Luminosity conditions . 54 4.2 Additional physics . 55 4.2.1 Rotation . 56 4.2.2 Special and general relativity . 58 4.3 Fiducial model . 59 4.3.1 Evolution . 61 4.3.2 Structure . 62 4.3.3 Non-rotating and non-relativistic models . 63 4.4 Parameter exploration . 65 4.4.1 Rotation . 66 4.4.2 Radiative efficiency . 68 4.4.3 Advective efficiency . 70 4.4.4 Initial mass and total mass loss or gain . 70 4.5 Discussion . 73 4.6 Conclusion . 74 xiii 5 The nature of the Schönberg–Chandrasekhar limit 75 5.1 Fractional mass contours in the U–V plane . 76 5.1.1 The Schönberg–Chandrasekhar limit . 78 5.1.2 Quasi-stars . 79 5.1.3 Other polytropic limits . 81 5.2 Consequences for stellar evolution . 83 5.2.1 General limits . 83 5.2.2 Evolution beyond the limit . 85 5.2.3 Stability of polytropic cores . 85 5.2.4 Identifying mass-limited models . 87 6 Fractional mass limits and the structure of giants 89 6.1 A brief history of the red giant problem . 90 6.1.1 Convection, degeneracy and thermal stability of the envelope . 91 6.1.2 Molecular weight gradients, shell burning and polytropic indices .
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