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TRANSCRIPT
Multidimensional Simulations of Type Ia
Supernovae and Classical Novae
A Dissertation Presented
by
Brendan Kurt Krueger
to
The Graduate School
in Partial Fulfillment of the
Requirements
for the Degree of
Doctor of Philosophy
in
Physics
Stony Brook University
August 2012
Stony Brook University
The Graduate School
Brendan Kurt Krueger
We, the dissertation committee for the above candidate for the Doctor of Philosophydegree, hereby recommend acceptance of this dissertation.
Dr. Alan C. Calder – Dissertation AdvisorAssistant Professor, Department of Physics and Astronomy
Dr. Frederick M. Walter – Chairperson of DefenseProfessor, Department of Physics and Astronomy
Dr. Michael ZingaleAssociate Professor, Department of Physics and Astronomy
Dr. Peter StephensProfessor, Department of Physics and Astronomy
Dr. Marc-Andre PleierAssociate Physicist, Physics Department
Brookhaven National Laboratory
This dissertation is accepted by the Graduate School.
Charles TaberInterim Dean of the Graduate School
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Abstract of the Dissertation
Multidimensional Simulations of Type Ia Supernovaeand Classical Novae
by
Brendan Kurt Krueger
Doctor of Philosophy
in
Physics
Stony Brook University
2012
Explosive astrophysical phenomena have historically played a significant role inunderstanding the universe and our place within it. Stellar explosions are impor-tant distance indicators, allowing exploration of the structure and evolution of theuniverse. They also form and disperse heavy elements that are recycled into newastrophysical objects. Stellar explosions are not a uniform group; the progenitorsand mechanisms of stellar explosions vary tremendously. I used multidimensionalsimulations to study two distinct types of explosions that are believed to resultfrom similar progenitor systems: compact white dwarf stars that accrete mat-ter from stellar companions. The two types of explosions I studied are type Iasupernovae and classical novae.
Type Ia supernovae are thought to arise from a thermonuclear explosion originat-ing in the core of an accreting white dwarf and leave no remnant. These eventsare the premier distance indicators in cosmological studies, but questions remainabout systematic biases and intrinsic scatter. My investigation centered on thesystematic impact of the central density of the progenitor on the brightness ofthe supernova. Relating the progenitor’s central density to its age provided atheoretical explanation of the observed trend that type Ia supernovae from olderstars are dimmer. I also demonstrated the importance of a statistical study ofsuch problems, due to the strongly nonlinear evolution during the explosion.
Classical novae are important for the study of circumstellar dust formation andare significant contributors of specific isotopes found in our galaxy. They result
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from a thermonuclear runaway occurring in the accreted envelope on a whitedwarf. Only the envelope is consumed, so the white dwarf remains and the eventmay recur on time scales of 104 to 105 years. My study made use of a newsimulation code specialized for low-Mach number flows, such as convection justprior to the explosion. I developed hydrostatic initial models and physics modulesnecessary for simulations of classical novae. This problem provided unexpectedchallenges, but preliminary simulations of convection-driven mixing between theaccreted envelope and the underlying white dwarf are underway. Future resultswill explore the e!ects of convection, particularly the quantity and mechanismsof mixing.
My research into stellar explosions provided important insight into their mecha-nisms and required considerable development work, which improved our modelsand will allow more realistic simulations in the future.
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For Erin.
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Contents
List of Figures ix
List of Tables x
List of Abbreviations xi
Acknowledgements xiii
1 Introduction 11.1 Progenitor Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
I Type Ia Supernovae 3
2 Overview 42.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Current Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Width-Luminosity Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Goals of Research in SNeIa . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 The FLASH Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 E!ect of the Central Density - Paper I 93.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 E!ect of the Central Density - Paper II 154.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2.1 Initial White Dwarf Models . . . . . . . . . . . . . . . . . . . . . . . 184.2.2 Ensemble of Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 The Simulation Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.1 Burning Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.2 Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
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4.4.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4.3 Initial Morphology Correlations . . . . . . . . . . . . . . . . . . . . . 234.4.4 Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4.5 Distribution of 56Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4.6 Distinguishing Age Among SNeIa of Equal Brightness . . . . . . . . . 26
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.5.1 Recalibration of 56Ni Yield . . . . . . . . . . . . . . . . . . . . . . . . 274.5.2 Comparison to Observations . . . . . . . . . . . . . . . . . . . . . . . 284.5.3 Comparison to Other Theoretical E!orts . . . . . . . . . . . . . . . . 29
4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.7 Appendix: Randomized Initial Conditions . . . . . . . . . . . . . . . . . . . 334.8 Appendix: Recalibration of 56Ni Yields . . . . . . . . . . . . . . . . . . . . . 334.9 Appendix: Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Conclusions and Future Work 415.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
II Classical Novae 43
6 Overview 446.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.3 The State of Modern Simulations . . . . . . . . . . . . . . . . . . . . . . . . 466.4 The MAESTRO Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7 Initial Models 497.1 Outline of the Initial Model Builder . . . . . . . . . . . . . . . . . . . . . . . 49
7.1.1 Uniform Grid Interpolation . . . . . . . . . . . . . . . . . . . . . . . 507.1.2 Hydrostatic Equilibrium Integration — First Pass . . . . . . . . . . . 517.1.3 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.1.4 Hydrostatic Equilibrium Integration — Second Pass . . . . . . . . . . 567.1.5 Hydrostatic Equilibrium Verification . . . . . . . . . . . . . . . . . . 56
7.2 Correction of Entropy Discontinuity . . . . . . . . . . . . . . . . . . . . . . . 56
8 Modifications to the MAESTRO Code 598.1 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.2 Initial Convective Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9 Nuclear Reactions 629.1 Hydrogen Burning Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
9.1.1 The pp Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629.1.2 The CNO Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649.1.3 Rapid Proton Breakout . . . . . . . . . . . . . . . . . . . . . . . . . . 64
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9.2 Decoupling of Reactions and Hydrodynamics . . . . . . . . . . . . . . . . . . 659.3 Analytic CNO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669.4 Frank X. Timmes’s pphotcno Network . . . . . . . . . . . . . . . . . . . . . 679.5 Rebuilt Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9.5.1 Notes on Composition . . . . . . . . . . . . . . . . . . . . . . . . . . 689.5.2 Equation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689.5.3 A Simple Isobaric Network . . . . . . . . . . . . . . . . . . . . . . . . 699.5.4 A Faster Isobaric Network . . . . . . . . . . . . . . . . . . . . . . . . 709.5.5 The SDC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
10 Conclusions and Future Work 7510.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7510.2 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7510.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
– Bibliography 81
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List of Figures
3.1 Deflagration-Phase Neutronization . . . . . . . . . . . . . . . . . . . . . . . 113.2 56Ni Mass vs. Central Density . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Brightness vs. Cooling Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Stretch vs. Age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.1 Structure of the Progenitor WDs . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Phases of the DDT Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Evolution of Gravitational Binding Energy and Mass of IGEs . . . . . . . . . 234.4 Standard Deviation of 56Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Initial Morphology Correlations . . . . . . . . . . . . . . . . . . . . . . . . . 244.6 Final Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.7 Yields by Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.8 Average 56Ni Profiles by Central Density . . . . . . . . . . . . . . . . . . . . 264.9 Distribution of IGEs by Central Density . . . . . . . . . . . . . . . . . . . . 274.10 Central Density vs. Mass of Stable IGEs . . . . . . . . . . . . . . . . . . . . 274.11 Average 56Ni profiles by Mass of 56Ni . . . . . . . . . . . . . . . . . . . . . . 284.12 Distribution of IGEs by Mass of 56Ni . . . . . . . . . . . . . . . . . . . . . . 294.13 Recalibrated Final Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.14 Comparison to Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.15 Recalibration Proxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.16 Accuracy of Proxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.17 Independence of IGE Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.18 Recalibration Trend Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.1 Sample CN Initial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.2 Initial Model Grid Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.3 Aligning the Grid With the Peak Temperature . . . . . . . . . . . . . . . . . 527.4 Initial Model Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.5 Entropy Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.6 Relative Contributions of pp Chain Branches . . . . . . . . . . . . . . . . . . 58
9.1 The pp Chain Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639.2 The Cold CNO Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649.3 The Hot CNO Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
10.1 Maximum Mach Number Evolution . . . . . . . . . . . . . . . . . . . . . . . 7610.2 Preliminary CN Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
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List of Tables
3.1 Brightness-Age Fitting Parameters . . . . . . . . . . . . . . . . . . . . . . . 13
4.1 Properties of the Progenitor WDs . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Composition of the Progenitor WDs . . . . . . . . . . . . . . . . . . . . . . . 184.3 Spherical Harmonics of Initial Flame Surfaces . . . . . . . . . . . . . . . . . 344.4 Proxy Fitting Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.5 Trend Line Fitting Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 364.6 Recalibration Time Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.7 Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
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Abbreviations
This list is ordered alphabetically by abbreviation for ease of reference, not by order ofappearance in the text.
ADR advection-di!usion-reaction
AGB asymptotic giant branch
AMR adaptive mesh refinement
CFL Courant-Friedrichs-Lewy (Courant et al., 1928)
CN classical nova (plural: CNe, classical novae)
CNO carbon-nitrogen-oxygen
CV cataclysmic variable
DDT deflagration-to-detonation transition
DTD delay time distribution
EoS equation of state
GCD gravitationally-confined detonation
HSE hydrostatic equilibrium
IGE iron-group element
NSE nuclear statistical equilibrium
NSQE nuclear statistical quasi-equilibrium (also known in the literature as QSE, orquasi-statistical equilibrium)
pp proton-proton
rp rapid proton
SDC spectral deferred correction
SNIa type Ia supernova (plural: SNeIa, type Ia supernovae)
TFI turbulence-flame interaction
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V&V verification and validation
WD white dwarf
WLR width-luminosity relation
xii
Acknowledgements
Near the end of The Lord of the Rings1, when the One Ring finally exhausts Frodo,Samwise declares, “I can’t carry it for you, but I can carry you!” This dissertation may havemy name on the cover, but there were many who helped to carry me this far.
Firstly I would like to thank my Lord and Savior, who has been my constant companionand who gifted us with this beautiful universe to discover.
Without the encouragement and sacrifice of my wife, Erin, this dissertation would notexist. I could not hope to list all the ways she has been a blessing to me.
I never would have begun grad school without the support and encouragement of myparents, who taught me to work hard in pursuit of what I love.
Despite his protests to the contrary, Alan Calder has been an outstanding advisor. Hehas managed to balance when to step in and o!er direction and when to allow me to findmy own way.
The faculty of the Department of Physics and Astronomy have been wonderful. Inparticular, I would like to thank Mike Zingale and Doug Swesty, whose encouragement andguidance have been greatly appreciated.
Surviving grad school is a team sport. My thanks also go to my fellow grad students:Aaron, Chris, Josh, Rahul, Melissa, Adam, Yeunhwan, Kendra, Bryan, and Stephanie.
My collaborators, Dean Townsley, Ed Brown, and Frank Timmes, have endured an end-less string of revisions of my papers and have patiently helped me develop my research andrefine my writing into work worth publishing.
The text of this dissertation in part is a reprint of the materials as it appears in theAstrophysical Journal Letters and the Astrophysical Journal. The co-authors listed in thepublications directed and supervised the research that forms the basis for this dissertation.Several images included in this dissertation have been reprinted by permission of Frank X.Timmes.
The work described in this dissertation was supported by several sources, includingNASA through grant NNX09AD19G, the US Department of Energy through grant DE-FG02-87ER40317, and Lawrence Livermore National Laboratory through contracts B589924and B593287. Computational resources were provided by the New York Center for Compu-tational Sciences and the National Science Foundation through the TeraGrid and XSEDEprojects.
1If you didn’t expect me to put a Lord of the Rings quote in my dissertation, then you must not knowme very well.
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Chapter 1
Introduction
In this dissertation I explore two distinct, but related, topics: type Ia supernovae (SNeIa,discussed in Part I), and classical novae (CNe, discussed in Part II). Both are explosiveastrophysical phenomena involving thermonuclear reactions in or on the surface of a com-pact stellar object that is accreting matter from a companion star. Stellar explosions arescientifically interesting for many reasons; for example, they synthesize and distribute heavyelements, which strongly impacts galactic chemical evolution, and they can be used as dis-tance indicators, which has led to ground-breaking discoveries in cosmology. My researchinvolves the use of numerical simulations of these explosions, with the goal of understandingthe mechanisms which drive them and the processes involved.
1.1 Progenitor Systems
Despite significant di!erences in their mechanisms, SNeIa and CNe result from similarprogenitor systems: a binary star system with one compact star and one non-compact star1.The non-compact star, referred to as the companion star, is typically assumed to be eithera red giant or a star on the main sequence. In both cases, the compact star is a whitedwarf (WD). White dwarfs are the endpoint of stellar evolution of stars with zero-age main-sequence masses (MZAMS) less than approximately 8M! (Nomoto, 1984). Once such a starhas consumed its fuel, it contracts due to the loss of pressure support in the core. Thiscontraction continues until the pressure from electron degeneracy becomes su"cient to haltthe collapse2. The composition of the resulting WD will primarily be helium for the lowestMZAMS, a carbon-oxygen mixture for intermediate MZAMS, or an oxygen-neon mixture forthe highest MZAMS; both CNe and SNeIa are believed to result from carbon-oxygen WDs.Due to the properties of electron-degenerate material, WDs have an upper mass limit thatdepends on properties of the particular WD (e.g., the electron-to-baryon ratio and the spinperiod); this upper limit is known as the Chandrasekhar mass (MCh) and is typically around
1Other models exist for SNeIa; see Section 2.2.
2Electron degeneracy pressure is a quantum mechanical phenomenon that arises when fermions are com-pressed to the point where the Pauli exclusion principle forces a fermion to move to a higher energy state inorder to compress further, which manifests as a pressure (Fowler, 1926).
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1.4 M! (Chandrasekhar, 1931a,b, 1935). To give a sense of scale, a WD with a mass near1 M! will have a radius comparable to that of the Earth.
In binary systems, it is possible to transfer mass between stars. For the systems that Iam considering, several mechanisms exist to transfer mass from the non-compact companiononto the surface of the WD: primarily Roche-lobe overflow, in which the companion’s outerlayers are no longer gravitationally bound to the core of the companion due to the proximityof the WD, and stellar wind. A variety of phenomena may result from such mass transfer,including an assortment of cataclysmic variables (of which CNe are a subset) and SNeIa.
1.2 Numerical Simulations
My research is based on numerical simulations of these two phenomena. I do not simulateany stellar evolution or mass transfer, only the explosion itself. I use progenitor modelsinformed by stellar evolution research to capture the state of the star immediately prior tothe explosion. Therefore my simulations do not include the entire binary system, but onlythe WD (all or part of the WD, depending on the simulation).
I use two di!erent simulation tools for my simulations: FLASH and MAESTRO. Both codessolve modified versions of the Euler equations for inviscid flow (Euler, 1757a,b,c, 1761) on adiscretized grid of control volumes; that is, they are finite-volume Eulerian hydrodynamicscodes. Modifications to the underlying hydrodynamics equations include the ability to handlemulti-scale flows (e.g., through adaptive mesh refinement), gravity, nuclear reactions, andnon-ideal equations of state. Further discussion of these two codes will be deferred until later;for a discussion of FLASH see Section 2.5, and for a discussion of MAESTRO see Section 6.4.
These computational tools are modern research tools designed to run on the largest com-puting platforms available. The challenge of including all the significant physical processes,spanning the diverse scales that are important to the phenomena being studied, and scalingwell to get maximum use of the computational resources available makes these problemsinteresting from the perspective of computational science as well.
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Part I
Type Ia Supernovae
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Chapter 2
Overview
Type Ia supernovae have long provided a source of surprising discoveries about the struc-ture of our universe and our place within it. Tycho Brahe’s publication of De Nova Stella in1573 signalled the end of Aristotelian cosmology: the unchanging heavens had changed, andwith them our view of the universe we inhabit (Chaisson & McMillan, 2001). More recently,the Supernova Cosmology Project (Perlmutter et al., 1999) and the High-z Supernova SearchTeam (Riess et al., 1998) found evidence that the expansion of the universe is acceleratingin violation of prior expectations of the structure of the universe, suggesting the existenceof dark energy and revolutionizing our understanding of cosmology (Nobelprize.org, 2011).SNeIa continue to be the premier distance indicators for probing the structure and expansionhistory of the universe, but their intrinsic scatter in brightness and the inherent uncertaintyin measuring and calibrating them are becoming dominant sources of uncertainty in mea-surements of cosmological parameters. A better understanding of the mechanism of SNeIacould potentially lower these uncertainties and allow a deeper test of cosmology.
2.1 Historical Overview
The oldest known stellar explosion is recorded in the 5th century Chinese book TheBook of the Later Han, which discusses the appearance of a “guest star” in the year 185(Fan Ye, 5th century; see also Stothers, 1977). The modern study of stellar explosionsbegan with Tycho Brahe’s De Nova Stella (“On the New Star”; Brahe, 1573), describingthe appearance of a new star. From Tycho’s work we derive the modern name “novae” forthese phenomena. In the early twentieth century, studies of novae began to distinguish acategory that were significantly more luminous than other novae, which were termed “super-novae” by Baade and Zwicky (1934; see also Osterbrock, 2001). In 1941 Minkowski beganthe spectral system to distinguish between classes of supernovae, proposing two types: typeI, which lacks evidence of hydrogen in their spectrum, and type II, which have hydrogenpresent (Minkowski, 1941). Type I supernovae were further divided into types Ia, Ib, andeventually Ic (Bertola, 1964; Porter & Filippenko, 1987).
Hoyle & Fowler (1960) first proposed the idea of a thermonuclear runaway in the coreof a star supported by electron degeneracy as the mechanism for SNeIa. Whelan & Iben(1973) further refined this model by proposing what is now known as the single-degenerateparadigm: a carbon-oxygen WD, supported by electron degeneracy, approaches MCh through
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accretion from a binary companion and undergoes an explosion as an SNeIa. Truran et al.(1967) and Colgate & McKee (1969) developed the idea that the source of energy for thelight curve of SNeIa is the radioactive decay of 56Ni produced in the explosion.
Over the next few decades, theoretical models of SNeIa were refined. It was shown thatprompt detonations (in which the burning front propagates supersonically) do not reproducethe observational signatures of SNeIa (Nomoto et al., 1976; Woosley & Weaver, 1986), whilethe W7 model of Nomoto et al. (1984) showed that, in one dimension, a deflagration (inwhich the burning front propagates subsonically) is able to reproduce many observations.However, even the deflagration model su!ers from some shortcomings, and Khokhlov (1991a)introduced the delayed-detonation model, in which the explosion begins as a deflagration andtransitions to a detonation later in the supernovae outburst. This delayed-detonation modelis still the favored model in modern simulations of the single-degenerate paradigm of SNeIa.
2.2 Current Theory
The single-degenerate paradigm is not the only commonly-studied model for SNeIa. Thedouble-degenerate paradigm, introduced by Webbink (1984) and Iben & Tutukov (1984), isthe primary competing model to explain SNeIa. As with the single-degenerate paradigm,the double-degenerate paradigm assumes that a progenitor system is a binary star system.However, instead of one WD approaching MCh by accretion from a non-degenerate compan-ion, the double-degenerate paradigm assumes that both stars are WDs and the explosion isinitiated by a merger as the two WDs spiral together. My research focuses on the single-degenerate paradigm, so the double-degenerate paradigm will not be discussed further.
Even within the single-degenerate paradigm, there are a variety of mechanisms. SinceKhokhlov (1991a), most simulations of SNeIa have used the delayed detonation model, butthe details of how the detonation is initiated are as yet unknown. Various mechanismsexist, including the gravitationally-confined detonation first proposed by Plewa et al. (2004).My research focuses on the mechanism known as the deflagration-to-detonation transition(DDT), introduced in Khokhlov (1991a), in which the explosion begins as a deflagrationwithin the core of the WD, and eventually some critical set of conditions is reached thatcauses a transition to a detonation mode. The details of this transition are actively beingstudied in both terrestrial and astrophysical contexts. The method used in my researchis a simple model that assumes the transition occurs when the flame front reaches somecharacteristic density, the details of which will be explained below.
2.3 Width-Luminosity Relation
Phillips (1993) showed that the width and luminosity of the peak of SNIa light curves(brightness as a function of time) are correlated. In other words, knowing how fast thebrightness declines from maximum specifies the intrinsic maximum brightness of the SNIa.This relation, known as the Phillips relation or the width-luminosity relation (WLR), canbe calibrated against other methods in the astronomical distance ladder and used as a“standardizable candle”: knowing how bright a supernova actually is and how bright it
5
appears to be determines how far away it is. Using SNeIa as standardizable candles enablesstudies of cosmological structure and expansion history, and was instrumental in determiningthat the expansion of the universe is accelerating (Riess et al., 1998; Perlmutter et al., 1999).
The use of SNeIa to determine cosmological parameters has benefited from the rapidly-growing quantity of data available. The number of SNIa observations has decreased thestatistical uncertainty to the point where the systematics, such as variations of SNIa lightcurves with galaxy properties, will soon be the dominant source of uncertainty (Sullivanet al., 2011). In order to reduce the systematic uncertainty, it is necessary to gain a betterunderstanding of the mechanism that underlies SNeIa. Recent work has begun to accountfor correlations of the brightness and light curve of an SNIa with properties of the hostgalaxy (Conley et al., 2011). Improvements in our understanding of the mechanism of SNeIamay shed light on these host galaxy correlations, and lower the systematic uncertainty incosmological parameters.
2.4 Goals of Research in SNeIa
The vast majority of SNeIa obey this WLR, but an important question remains: Whatis the intrinsic scatter of these events? Or, to put it di!erently, how well does the WLRhold? Observations of SNeIa make use of a parameter known as the Hubble residual, whichis the di!erence between the distance inferred from a SNIa based on its light curve propertiesand the WLR and the distance inferred from its redshift and the expansion of the universe.This expansion was originally described by the Hubble Law (Lemaıtre, 1927; Hubble, 1929),hence the term Hubble residual. To quote from Kelly et al. (2010): “Correlations betweenHubble residuals and progenitor properties have the potential to introduce bias into SN Iacosmological measurements [...]. If correlations are identified, host measurements such as size,mass, or metallicity could be combined with light curve parameters [...] to improve luminositydistance estimates to SNe Ia and control systematics in cosmological measurements.”
As a prelude to addressing the Hubble residual, Townsley et al. (2009) presented a methodto investigate how certain properties of a progenitor WD impact the outcome of the associ-ated SNIa explosion, particularly the brightness of the SNIa. This collaborative study is anongoing research program, with several aspects presented in Townsley et al. (2009), Jacksonet al. (2010), Krueger et al. (2010), and Krueger et al. (2012). My research has focusedon the influence of the central density of the progenitor WD immediately prior to the for-mation of the deflagration front. Lesa!re et al. (2006) showed that two progenitors thatdi!er only in the time elapsed between the formation of the WD and the onset of accretionfrom the companion (known as the cooling time) will have di!erent central densities at thepoint where the conditions for the formation of a deflagration front are met, with longercooling times resulting in higher central densities. Thus the central density at the formationof the deflagration front is a proxy for the age of the progenitor star1. I investigated howthe brightness of an SNIa depends on this density, seeking a theoretical explanation of theobserved trend that older stars are systematically dimmer (Gallagher et al., 2008; Howell
1See Section 4.5.2 for clarification on the measurements of “age” for SNeIa
6
et al., 2009; Neill et al., 2009; Brandt et al., 2010).
2.5 The FLASH Code
This research was done by performing a suite of multidimensional full-star simulationsof SNeIa. My simulations were performed using a customized version of the FLASH code.FLASH is a hydrodynamics code designed to simulate nuclear-powered astrophysical flashes(Fryxell et al., 2000; Calder et al., 2002b). It solves a modified version of the Euler equationsfor inviscid flow that accounts for reactions. We may write these compressible reactivehydrodynamics equations as:
!"Xk
!t= !" · ("Xk u) + " #k (2.1a)
!"u
!t= !" · (u # "u) !"P ! "g (2.1b)
!" e
!t= !" · (u (" e + P )) + " enuc + "u · g, (2.1c)
where " is the mass density, Xk is the mass fraction for species k, u is the fluid velocity,#k is the rate of change of species k, P is the pressure, g is the acceleration of gravity, e isthe total energy density, and # is a dyadic product2. The mass fraction of a species is theratio of the partial mass density to the total mass density (Xk $ "k/"); for a given controlvolume, it is also the ratio of the mass of all nuclei of species k to the total mass in thecontrol volume. It follows from this definition that
!
k
"k = " and!
k
Xk = 1. (2.2)
Due to conservation of mass3,!
k
#k = 0. (2.3)
2The dyadic product is also known as an outer product or tensor product. The dyadic product oftwo vectors creates a tensor of order two; the statements P = v # w and Pij = viwj are equivalent.Equation (2.1b) may be clearer in component form:
!"ui
!t= !" · (ui"u) !
!P
!xi! "gi,
where this is a set of equations: one for each spatial dimension.
3In actual fact, mass is not conserved due to nuclear reactions converting mass to energy. The quantitiesXk and #k are not equivalent because Xk are fractions and by definition must sum to unity so that Xk mustsum to zero, while #k may account for this mass non-conservation and will therefore not generally sum tozero. However, even in an extreme case such as pure hydrogen converting to pure nickel, the change in massis less than 1%, so most hydrodynamic codes neglect this mass non-conservation and #k will sum to zero.
7
Given Equations (2.2) and (2.3), a summation over k of Equation (2.1a) yields
!"
!t= !" · ("u) . (2.4)
This summed form, along with Equations (2.1b) and (2.1c), can be seen as expressions ofthe conservation of mass, momentum, and energy respectively. The relationship between thespecific total energy, e, and the specific internal energy, $, is
e = $+1
2(u · u) . (2.5)
The term enuc represents a source term from the energy released by nuclear reactions. Thissystem of equations is not closed; it requires a relation between the pressure and the density.This is given by the equation of state (EoS), which relates the thermodynamic quantitiesof the system (e.g., density, pressure, energy, temperature, entropy); typically an EoS forastrophysical simulations will require two thermodynamic quantities and the composition.
FLASH makes use of adaptive mesh refinement (AMR), which allows grid cells to besubdivided or merged together in order to increase or decrease the resolution as needed.This is a powerful feature, which allows high resolution to capture important details in the“interesting” region of a simulation without wasting time computing unnecessary detailsin the “uninteresting” regions. What regions of a simulation are considered “interesting”depends on the goals of a particular simulation, and may include considerations such asenergy generation or steep gradients in fluid quantities.
The core of FLASH was developed at the Flash Center for Computational Science atthe University of Chicago, and is available directly from the Flash Center4. The detailsof the variant used to perform my simulations are described in Townsley et al. (2007),Townsley et al. (2009), and Jackson et al. (2010), with important support information oncertain components and/or implementations presented in Calder et al. (2007) and Seitenzahlet al. (2009b). The modifications made to this custom version of the FLASH code are alsosummarized in Krueger et al. (2010) and Krueger et al. (2012); see Chapters 3 and 4.
4http://flash.uchicago.edu/
8
Chapter 3
E!ect of the Central Density - Paper I
This chapter is a reproduction of the first paper I published on the problem of the centraldensity dependence of SNeIa. It was originally published in the Astrophysical Journal Letters(Krueger et al., 2010). In this paper I present the main findings of my research. Due to thelength restrictions of the Letters format, some details had to be omitted; those details, alongwith a more in-depth analysis, were presented in Krueger et al. (2012); see Chapter 4. Thispaper is reproduced by permission of the American Astronomical Society. The co-authorsdirected and supervised the research that forms the basis for this paper.
9
!" #$%&$'&!"( !) '*+ ,%&-*'"+(( !) './+ &0 (1/+%"!#$+ 2&'* '*+ $-+!) '*+ *!(' ('+33$% /!/13$'&!"
!"#$%&$ '( '")#*#"4+ ,&"-$ .( /&012-$4+ 3#&$ 4( 5-6$27#85+ ,7&$ 9( 9&7%#"4!6+ :%6&"% ;( !"-6$7+
&$% ;( <( 5=>>#284 9:;0<=>:?= @A /BCDEFD G $D=<@?@>CH 'B: (=0=: 1?EI:<DE=C @A ":J .@<KL(=@?C ,<@@KH (=@?C ,<@@KH ".H 1($
5 9:;0<=>:?= @A /BCDEFD 0?M $D=<@?@>CH 'B: 1?EI:<DE=C @A $N0O0>0H 'PDF0N@@D0H $3H 1($6 ":J .@<K Q:?=:< A@< Q@>;P=0=E@?0N (FE:?F:DH 'B: (=0=: 1?EI:<DE=C @A ":J .@<KL(=@?C ,<@@KH (=@?C ,<@@KH ".H 1($
7 9:;0<=>:?= @A /BCDEFD 0?M $D=<@?@>CH REFBES0? (=0=: 1?EI:<DE=CH +0D= 30?DE?SH R&H 1($8 (FB@@N @A +0<=B 0?M (;0F: +T;N@<0=E@?H $<EU@?0 (=0=: 1?EI:<DE=CH ':>;:H $VH 1($
!"#"$%"& '()( *+,#- )./ +##"01"& '()( 2345 '/ 0364$7-"& '()( 2345 ).
$,('%$Q'
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` 0N@?S:< F@@NE?S =E>: <0ED:D =B: F:?=<0N M:?DE=C 0= ES?E=E@? =B:<:OC ;<@MPFE?S N:DD 8\"E 0?M =BPD 0 ME>>:< :I:?=Y2BEN: @P< :?D:>ON: @A <:DPN=D M:>@?D=<0=:D 0 DES?E]F0?= =<:?MH J: ]?M F@?DEM:<0ON: I0<E0=E@? O:=J::? <:0NEU0=E@?DHE?MEF0=E?S =B: ?:F:DDE=C A@< 0I:<0SE?S @I:< 0? :?D:>ON: @A DE>PN0=E@?D =@ M:>@?D=<0=: 0 D=0=ED=EF0NNC DES?E]F0?= <:DPN=Y
8"5 9:,&7; BCM<@MC?0>EFD L ?PFN:0< <:0F=E@?DH ?PFN:@DC?=B:DEDH 0OP?M0?F:D L DP;:<?@I0:` S:?:<0N L JBE=: MJ0<AD
4Y &"'%!91Q'&!"
!OD:<I0=E@?D =0<S:=E?S =B: :?IE<@?>:?= @A =C;: &0 DP;:<?@I0:W(":&0X B0I: :T;@D:M @;:? _P:D=E@?D F@?F:<?E?S =B: M:;:?ZM:?F: @A O@=B =B:E< <0=:D 0?M 0I:<0S: O<ESB=?:DD @? :?IE<@?>:?=YR0??PFFE := 0NY W5aa\X DB@J =B0= =B: M:;:?M:?F: @A =B: ("&0<0=: @? M:N0C =E>: W:N0;D:M =E>: O:=J::? D=0< A@<>0=E@? 0?M =B:DP;:<?@I0 :I:?=X ED O:D= ]= OC 0 OE>@M0N M:N0C =E>: MED=<EOP=E@?W9'9X JE=B 0 ;<@>;= F@>;@?:?= N:DD =B0? 4 -C< 0A=:< D=0< A@<Z>0=E@? 0?M 0 =0<MC F@>;@?:?= D:I:<0N -C< N0=:<Y $N=B@PSB =B:FN0<E=C @A =BED :AA:F= ED FN@PM:M OC S0N0TC D0>;NE?S W)ENE;;:?K@5aabXH =B: O0DEF <:DPN= ED O@<?: @P= :I:? JE=BE? S0N0TE:D W%0DKE?:= 0NY 5aabXY -0NN0SB:< := 0NY W5aacX >:0DP<: 0 F@<<:N0=E@? O:Z=J::? =B: O<ESB=?:DD @A 0? ("&0 0?M E=D M:N0C =E>:H JBEFB =B:CD=0=: ED F@?DED=:?= JE=B :E=B:< 0 OE>@M0N @< 0 F@?=E?P@PD 9'9Y!=B:< <:F:?= D=PME:D OC *@J:NN := 0NY W5aabXH ":ENN := 0NY W5aabXH0?M ,<0?M= := 0NY W5a4aX 0ND@ ]?M 0 F@<<:N0=E@? O:=J::? =B: M:N0C=E>: 0?M O<ESB=?:DD @A 0? ("&0Y/BENNE;D W4bb6X EM:?=E]:M 0 NE?:0< <:N0=E@?DBE; O:=J::? =B:
>0TE>P> <ZO0?M >0S?E=PM: @A 0 NESB= FP<I: 0?M E=D <0=:@A M:FNE?:Y 'BED dO<ESB=:< :_P0ND O<@0M:<e <:N0=E@?DBE; B0DO::? :T=:?M:M =@ 0MME=E@?0N O0?MD JE=B =:>;N0=:D A<@> ?:0<OC:I:?=DH 0NN@JE?S (":&0 =@ O: F0NEO<0=:M 0D 0? :T=:?DE@? @A =B:0D=<@?@>EF0N MED=0?F: N0MM:< WD:: fB0 := 0NY 5aag A@< 0 M:DF<E;=E@?@A @?: >:=B@MXY 'B: O<ESB=?:DD @A 0? ("&0 ED M:=:<>E?:M;<E?FE;0NNC OC =B: <0ME@0F=EI: M:F0C @A 8\"E DC?=B:DEU:M MP<E?S=B: :T;N@DE@? W'<P<0? := 0NY 4b\gh Q@NS0=: G RFi:: 4b\bh$<?:== 4bc5h /E?=@ G +0D=>0? 5aaaXY
$ JEM:NC 0FF:;=:M ;<@;@D0N =@ :T;N0E? >0?CH EA ?@= >@D=H:I:?=D ED =B: =B:<>@?PFN:0< MED<P;=E@? @A 0 JBE=: MJ0<A W29X E?
0>0DDZ=<0?DA:<<E?S OE?0<C DCD=:> WA@< <:IE:JD A<@> I0<E@PD ;:<ZD;:F=EI:D D:: ,<0?FB := 0NY 4bb8h )ENE;;:?K@ 4bbgh *ENN:O<0?M=G "E:>:C:< 5aaah 3EIE@ 5aaah %j@;K: 5aa\XY &? =BED ;0<0MES>H0 N@?S:< M:N0C =E>: DPSS:D=D =B: ;@DDEOENE=C @A 0 N@?S:< :N0;D:M=E>: O:=J::? =B: A@<>0=E@? @A =B: 29 0?M =B: @?D:= @A 0FZF<:=E@?Y 9P<E?S =BED ;:<E@MH M:?@=:M B:<: 0D =B: 29 F@@NE?S=E>: W1F@@NXH =B: 29 ED ED@N0=:M A<@> 0?C DES?E]F0?= B:0= E?;P=0?M M:F<:0D:D E? =:>;:<0=P<:Y $ N@?S:< 1F@@N <:DPN=D E? 0 BESB:<F:?=<0N M:?DE=C JB:? =B: F@<: <:0FB:D =B: ES?E=E@? =:>;:<0=P<:W3:D0AA<: := 0NY 5aa\XH MP: =@ =B: N@J:< :?=<@;C 0= =B: @?D:= @A0FF<:=E@?Y 'BPDH 0 F@<<:N0=E@? O:=J::? F:?=<0N M:?DE=C 0?M =B:;:0K O<ESB=?:DD @A 0? :I:?= DPSS:D=D 0 F@<<:N0=E@? O:=J::? M:N0C=E>: 0?M =B: O<ESB=?:DD @A 0? :I:?=Y 2BEN: ;<:IE@PD J@<K E?MEZF0=:M 0 F@<<:N0=E@? O:=J::? F:?=<0N M:?DE=C 0?M ;:0K O<ESB=?:DDH?@?: B0D 0I:<0S:M @I:< 0 D=0=ED=EF0NNC DES?E]F0?= :?D:>ON: @A<:0NEU0=E@?D W,<0FBJE=U := 0NY 5aaah %j@;K: := 0NY 5aa\h *j@[EFB:= 0NY 5a4aXY 'B:<:A@<:H J: E?I:D=ES0=:H A@< =B: ]<D= =E>:H 0 D=0=EDZ=EF0NNC DES?E]F0?= F@<<:N0=E@? O:=J::? ;<@S:?E=@< F:?=<0N M:?DE=C0?M 0I:<0S: ;:0K O<ESB=?:DD @A (":&0Y'B: DP<<@P?ME?S D=:NN0< ;@;PN0=E@?H =B: >:=0NNEFE=C 0?M >0DD
@A =B: ;<@S:?E=@<H =B: =B:<>@MC?0>EF D=0=: @A =B: ;<@S:?E=@<H=B: F@@NE?S 0?M 0FF<:=E@? BED=@<C @A =B: ;<@S:?E=@<H 0?M @=B:<;0<0>:=:<D 0<: K?@J? =@ 0AA:F= =B: NESB= FP<I:D @A (":&0h =B:<@N:H 0?M :I:? ;<E>0FCH @A =B:D: I0<E@PD ;0<0>:=:<D ED =B:DPOk:F= @A @?S@E?S D=PMC W:YSYH %j@;K: := 0NY 5aa\h *j@[EFB:= 0NY 5a4ah f0FKD@? := 0NY 5a4aXY $MME=E@?0NNCH >0?C @A =B:D::AA:F=D >0C O: E?=:<F@??:F=:M E? F@>;N:T J0CD W3:D0AA<: := 0NY5aa\XY &? =BED D=PMCH J: ED@N0=: =B: ME<:F= :AA:F= @A I0<CE?S=B: ;<@S:?E=@< F:?=<0N M:?DE=C @? =B: ;<@MPF=E@? @A 8\"EY '@]<D= @<M:<H =BED CE:NM F@?=<@ND =B: O<ESB=?:DD @A 0? :I:?=h
10
D:F@?MZ@<M:< :AA:F=D @? =B: NESB= FP<I: 0<: N:A= A@< AP=P<:D=PMCY
5Y R+'*!9
!?F: 0 29 A@<>D E? 0 OE?0<CH E= ED E?E=E0NNC ED@N0=:M 0?MDN@JNC F@@ND E? 0 DE?SN:ZM:S:?:<0=: DF:?0<E@Y +I:?=P0NNCH >0DD=<0?DA:< O:SE?D =@ F0<<C NESB= :N:>:?=D A<@> =B: :?I:N@;: @A =B:F@>;0?E@? =@ =B: DP<A0F: @A =B:29Y &A =B: 0FF<:=E@? <0=: :TF::MD0 =B<:DB@NMH =B: E?A0NNE?S >0=:<E0N :T;:<E:?F:D D=:0MC OP<?E?SW"@>@=@ := 0NY 5aagXH 0?M :I:?=P0NNC =B: 29 S0E?D :?@PSB>0DD =@ F@>;<:DD 0?M B:0= =B: F@<:Y !?F: =B: =:>;:<0=P<: <ED:D:?@PSB =@ E?E=E0=: F0<O@? <:0F=E@?DH =B: F@<: O:SE?D =@ F@?I:F=W@< dDE>>:<eXY !P< ;<@S:?E=@< >@M:ND ;0<0>:=:<EU: =B: 29 0==B: :?M @A =BED DE>>:<E?S ;B0D:H kPD= ;<E@< =@ =B: OE<=B @A =B:[0>: =B0= :I:?=P0NNC JENN MED<P;= =B: :?=E<: 29 E? 0? ("&0Y2: F@?D=<PF=:M 0 D:<E:D @A ]I: ;0<0>:=:<EU:MH BCM<@D=0=EF
;<@S:?E=@< >@M:ND =B0= 0FF@P?= A@< DE>>:<E?S E? JBEFB J:I0<C =B: F:?=<0N M:?DE=C W"#XY 'B: @P=:< <:SE@?D 0<: ED@=B:<>0NH0N=B@PSB D@>: =:>;:<0=P<: D=<PF=P<: ED :T;:F=:M WiPBN:? := 0NY5aa\XH 0?M =B: F@<:D 0<: ED:?=<@;EF MP: =@ F@?I:F=E@? 0?M B0I:0 N@J:< Q$! <0=E@ W(=<0?E:<@ := 0NY 5aa6h /E<@ G ,ENMD=:? 5aachQB0>PN0K := 0NY 5aach /E<@GQB0?S 5aacXY f0FKD@? := 0NY W5a4aX:T;N@<:M =B:D: :AA:F=D 0?MJ: FB@D: =B: F@<: F@>;@DE=E@? 0D 7al45QH 8gl 4\!H 0?M 6l 55": 0?M =B: @P=:< N0C:< 0D 8al 45QH 7cl4\!H 0?M 5l 55":Y )@< @P< "#H J: FB@D: W4L8X ! 4ab S F>"6
E? D=:;D @A 4ab S F>"6Y 'B: F:?=<0N =:>;:<0=P<: >PD= O: E? =B:<0?S: @A F0<O@? ES?E=E@?H JBEFB ED 0;;<@TE>0=:NC WgLcX! 4ac iW:YSYH iPBN:? := 0NY 5aa\Xh J: D:N:F=:M g!4ac iY ,0D:M @? ;<E@<<:D:0<FBH J: FB@D: @=B:< >@M:N ;0<0>:=:<D =@ ;<@MPF: :T;:F=:M0>@P?=D @A ):ZS<@P; :N:>:?=D E? =B: :T;N@DE@? W'@J?DN:C := 0NY5aabh f0FKD@? := 0NY 5a4aXY 'B: I0NP:D J:<: K:;= F@?D=0?= E? 0NNDE>PN0=E@?D E? @<M:< =@ ED@N0=: =B: F:?=<0N M:?DE=C :AA:F=DY2E=B =B:D: ]I: ;<@S:?E=@< >@M:NDH J: P=ENEU: =B: D=0=ED=EF0N
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2: DE>PN0=:M 0 DPE=: @A 48a ("&0 >@M:ND JE=B 0 <0?S: @A"# =@ D=PMC =B: =<:?MD A@< 0 ;@;PN0=E@? @A (":&0Y 2: ]?M=B0= @? 0I:<0S: ;<@S:?E=@<D JE=B BESB:< "# ;<@MPF: N:DD 8\"EY*j@[EFB := 0NY W5a4aX 0<SP: =B0= 8\"E E? =B: F:?=<0N <:SE@?D @A=B: :T;N@ME?S 29 M@:D ?@= F@?=<EOP=: =@ =B: NESB= FP<I: 0=
>0TE>P>H 0?M =B:<:A@<: =B:C M@ ?@= D:: 0 DES?E]F0?= =<:?MJE=B F:?=<0N M:?DE=C E? =B: >0TE>P> =ZO0?M >0S?E=PM:H OP=<0=B:< E? N0=:Z=E>: O<ESB=?:DDY 'B: ;P<:ZM:[0S<0=E@? >@M:ND @A%j@;K: := 0NY W5aa\X :TBEOE= 0 DB0NN@J E?F<:0D: @A ;<@MPF:M 8\"E0D F:?=<0N M:?DE=C E?F<:0D:DH E? F@?=<0MEF=E@? @A @P< ]?ME?SDY&J0>@=@ := 0NY W4bbbX ]?M =B0= =B: =<:?M JE=B F:?=<0N M:?DE=CM:;:?MD @? =B: 99' =<0?DE=E@? M:?DE=Ch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
DP;;@<= =B: @OD:<I0=E@?0N ]?ME?S =B0= (":&0 A<@> @NM:< D=:NN0<;@;PN0=E@?D 0<: DCD=:>0=EF0NNC ME>>:<Y 2BEN: 0 M:S:?:<0FCO:=J::? 0S: 0?M >:=0NNEFE=C E? =B: E?=:S<0=:M NESB= @A D=:NN0<;@;PN0=E@?D :TED=DH =B: @OD:<I:M M:;:?M:?F: @A >:0? O<ESB=?:DD@A (":&0 @? >:0? D=:NN0< 0S: ED 0;;0<:?=NC =B: D=<@?S:< :AA:F=W-0NN0SB:< := 0NY 5aach *@J:NN := 0NY 5aabXY $FF@<ME?SNCH @P<FB@EF: =@ ?:SN:F= >:=0NNEFE=C :AA:F=D 0?M =@ F@?DEM:< @?NC =B::AA:F= @A F:?=<0N M:?DE=C @? 8\"E CE:NM 0NN@JD PD =@ @AA:< 0=B:@<:=EF0N :T;N0?0=E@? A@< =BED @OD:<I:M =<:?MY &AJ: 0MME=E@?0NNCF@?DEM:< =B: :AA:F= @A >:=0NNEFE=CH J: >0C D:: 0 DNESB=NC D=<@?S:<=<:?M @A M:F<:0DE?S O<ESB=?:DD JE=B E?F<:0DE?S 0S: 0D B0DO::? ;<:IE@PDNC DPSS:D=:M W'E>>:D := 0NY 5aa6XY !=B:< :AA:F=DO:DEM:D ;<@S:?E=@< F:?=<0N M:?DE=C 0?M >:=0NNEFE=CH DPFB 0D;<@S:?E=@< >0E? D:_P:?F: >0DDH >0C 0ND@ F@?=<EOP=: =@ =BED=<:?MY'B: E?D:?DE=EIE=C @A =B: @I:<0NN ):ZS<@P; CE:NM =@ F:?=<0N
M:?DE=CH 0?M =B:<:A@<: M:N0C =E>:H 0N@?S JE=B =B: M:;:?M:?F:@A =B: 8\"E CE:NM @? F:?=<0N M:?DE=CH E>;NE:D =B0= (":&0 @ADE>EN0< O<ESB=?:DD:D W0?M =B:<:A@<: DE>EN0< 8\"E CE:NMX A<@>;<@S:?E=@<D @A MEAA:<:?= 0S:D JENN ?@= B0I: =B: D0>: =@=0N ):ZS<@P; CE:NMY 'B@D: A<@> @NM:< ;@;PN0=E@?D JENNH @? 0I:<0S:HB0I: N0<S:< >0DD:D @A D=0ON: D;:FE:DY 'BED >0C 0<SP: A@< 0 DNESB=?@?ZP?EA@<>E=C E? =B: /BENNE;D <:N0=E@? O0D:M @? :?IE<@?>:?=W2@@DN:C := 0NY 5aagh *j@[EFB := 0NY 5a4aXY 'B: <:DPN=E?S FN@D:NC<:N0=:M A0>ENC @A O<ESB=?:DDLM:FNE?: =E>: <:N0=E@?D 0ND@ ;<@IEM:D0 ;BCDEF0N >@=EI0=E@? A@< E?=<E?DEF DF0==:< E? =B: /BENNE;D <:N0=E@?0D 0 <:DPN= @A F@>OE?E?S ;@;PN0=E@?D JE=B MEAA:<:?= >:0? D=:NN0<0S:DY &? =BED ;EF=P<:H =B: ;<E>0<C ;0<0>:=:< ED =B: M:S<:: @A:T;0?DE@? 0= 99'H M:=:<>E?:M OC =B: >@<;B@N@SC @A =B: :0<NC[0>: W0?M =B: 99' M:?DE=CH JBEFB J: B@NM F@?D=0?=XH 0?M=B: 0S: 0F=D 0 J:0K:< D:F@?M0<C ;0<0>:=:<Y &? 0?C F0D:H =B:;@DDEOENE=C @A DPFB 0? :AA:F= >@=EI0=:D AP<=B:< :T;N@<0=E@? @A =B:E>;0F= @A F:?=<0N M:?DE=C @? =B: NESB= FP<I: E=D:NAY
'BED J@<K J0D DP;;@<=:M OC =B: 9:;0<=>:?= @A +?:<SC=B<@PSB S<0?=D 9+Z)-a5Zag+%7484\H 9+Z)-a5Zac+%748gaH0?M 9+Z)-a5Zac+%748\8H 0?M OC "$($ =B<@PSB S<0?=""pab$94b-Y $YQYQY 0ND@ 0FK?@JN:MS:D DP;;@<= A<@> =B:9:;0<=>:?= @A +?:<SC P?M:< S<0?= 9+Z)-a5Zcg+%7a64gY9YRY'Y <:F:EI:M DP;;@<= A<@> =B: ,0<= fY ,@K A:NN@JDBE; 0==B: 1?EI:<DE=C @A $<EU@?0 A@< ;0<= @A =BED J@<KY 'B: 0P=B@<DS<0=:APNNC 0FK?@JN:MS: =B: S:?:<@PD 0DDED=0?F: @A /E:<<: 3:D0AZA<:H 0D J:NN 0D A<PE=APN MEDFPDDE@?D JE=B REK: VE?S0N:H 0?M =B:
13
PD: @A "(+ 0?M J:0K <:0F=E@? =0ON:D M:I:N@;:M OC &I@ (:E=:?ZU0BNY 'B: 0P=B@<D 0ND@ 0FK?@JN:MS: =B: B@D;E=0NE=C @A =B: i&'/HJBEFB ED DP;;@<=:M OC "() S<0?= /*.a8Z844\7H MP<E?S =B:;<@S<0>D d$FF<:=E@? 0?M +T;N@DE@?` =B: $D=<@;BCDEFD @A 9:ZS:?:<0=: (=0<De 0?M d(=:NN0< 9:0=B 0?M (P;:<?@I0:Ye 'B: D@A=ZJ0<: PD:M E? =BED J@<K J0D E? ;0<= M:I:N@;:M OC =B: 9!+ZDP;;@<=:M $(Q$$NNE0?F:D Q:?=:< A@< $D=<@;BCDEF0N 'B:<>@?PZFN:0< )N0DB:D 0= =B: 1?EI:<DE=C @A QBEF0S@Y 'BED <:D:0<FB P=ENEU:M<:D@P<F:D 0= =B: ":J .@<K Q:?=:< A@< Q@>;P=0=E@?0N (FE:?F:D0= (=@?C ,<@@K 1?EI:<DE=C$,<@@KB0I:? "0=E@?0N 30O@<0=@<CJBEFB ED DP;;@<=:M OC =B: 1Y(Y 9:;0<=>:?= @A +?:<SC P?M:<F@?=<0F= ?@Y 9+Z$Qa5ZbcQ*4acc\ 0?M OC =B: (=0=: @A ":J.@<KY
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4\aQ@NS0=:H (Y $YH G RFi::H QY 4b\bH $;fH 48gH \56)ENE;;:?K@H $Y #Y 4bbgH $%$G$H 68H 6ab)ENE;;:?K@H $Y #Y 5aabH E? (=:NN0< 9:0=B 0?M (P;:<?@I0: Q@?AY W(0?=0 ,0<O0<0H
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2YH -0?:DB0NE?S0>H RYH G )ENE;;:?K@H $Y #Y 5aacH $;fH \c8H g85-@NMB0O:<H -YH := 0NY 5aa4H $;fH 88cH 68b*ENN:O<0?M=H 2YH G "E:>:C:<H fY QY 5aaaH $%$G$H 6cH 4b4*j@[EFBH /YH := 0NY 5a4aH $;fH g4aH 777*@J:NNH 9Y $YH := 0NY 5aabH $;fH \b4H \\4&J0>@=@H iYH ,<0FBJE=UH )YH "@>@=@H iYH iEDBE>@=@H "YH 1>:M0H *YH *ETH2Y %YH
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14
Chapter 4
E!ect of the Central Density - PaperII
This chapter is a reproduction of the second paper I wrote on the problem of the centraldensity dependence of SNeIa, and is a follow-up to Krueger et al. (2010); see Chapter 3.This paper has been accepted by the Astrophysical Journal (Krueger et al., 2012). In thispaper I present additional details of the models and analysis that had to be omitted fromKrueger et al. (2010) due to length restrictions of the Letters format. Additionally, I presentnew analysis of those results and extend the conclusions based on the new analysis. Thispaper is reproduced by permission of the American Astronomical Society. The co-authorsdirected and supervised the research that forms the basis for this paper.
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36
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40
Chapter 5
Conclusions and Future Work
5.1 Summary and Conclusions
My research into SNeIa used a customized version of the FLASH code to perform a suiteof two-dimensional simulations. Applying the statistical method of Townsley et al. (2009), Igenerated five di!erent progenitor WDs, each with a di!erent initial central density, and per-formed thirty di!erent simulations of each progenitor with randomized initial flame surfaces.Based on analysis of these simulations, I provided a theoretical explanation of the observedtrend that SNeIa from older stellar populations are systematically dimmer. I showed that,due to the strongly non-linear processes involved, a statistical study of many simulationsis important to capturing the true trends in SNeIa. For the initial conditions that I used,15!20 di!erent initial flame surfaces are necessary to capture statistically significant trends.
Additionally, I find that SNeIa from my simulations match the qualitiative requirementsto fall along the observed WLR: the mass of 56Ni varies, correlating approximately to movingalong the WLR; the mass of IGEs is statistically constant, but has some small variation,which correlates approximately to the width of the WLR; the 56Ni is well-mixed out to ahigh enclosed mass. Despite being well-mixed, the 56Ni does show some clumpiness, whichcould potentially give rise to observable line-of-sight variations in observations of SNeIa.
Using the results of Lesa!re et al. (2006), I connected the central density of the progenitorimmediately prior to the ignition of the deflagration to the age of the progenitor, allowingmy research on the question of the central density to shed light on the question of variationsof SNeIa with the age of the host stellar population, a question which observers have beenstudying in order to control for systematic biases in cosmological studies. I presented apotentially-observable trend that could determine the relative ages of the progenitors of twoSNeIa with the same brightness. The mass of stable IGEs is tightly correlated with thecentral density, and therefore the age.
5.2 Future Work
The collaboration that I worked with on this research is currently investigating severalaspects of this problem, and is seeking to improve our models. Turbulence is inherently dif-ferent in three dimensions than it is in two dimensions, so a current goal of the collaboration
41
is to develop three-dimensional models. This includes modifying the method for randomiz-ing the initial conditions, and also includes a model for the interactions between turbulenceand the flame front. The collaboration is also pursuing post-processing of tracer particlesto verify the energetics and nucleosynthesis of our simulations, and present greater detail ofthe nucleosynthetic yield.
The next step for my own research within this collaboration is a study of the influenceof the carbon fraction in the core of the progenitor WD. At this stage, it will first require aliterature search to understand previous work on this topic, as well as to determine believableinitial conditions for progenitor WD models. This carbon fraction study will be performedanalagously to the central density study presented in Krueger et al. (2010) and Krueger et al.(2012), using a suite of two-dimensional simulations.
42
Part II
Classical Novae
43
Chapter 6
Overview
Classical novae are scientifically interesting for numerous reasons:
• They are potentially very important to the theory of SNeIa, as they may be the mech-anism by which the massive WDs that are needed as SNIa progenitors are made.However, this connection is still being debated.
• While CNe are not as bright as SNeIa, and are therefore not visible from such greatdistances, they can be calibrated through a relationship between their maximum mag-nitude and their rate of decline, and used as distance indicators (della Valle & Livio,1995).
• Novae have a strong impact on the interstellar medium: though they eject less matterper event than supernovae, novae are more numerous and likely contribute a compara-ble amount of matter to the interstellar medium. Due to the di!erence in nucleosyn-thesis between novae and supernovae, novae likely produce more intermediate-masselements, and may be the dominant sources of certain isotopes, possibly including 7Li,13C, 15N, 17O, 22Na, and 26Al.
• Because nova nucleosynthesis proceeds at temperatures and densities obtainable ina laboratory, they may be valuable tests for laboratory astrophysics (Jose & Iliadis,2011).
• Novae provide an opportunity to study circumstellar dust formation in real time.
• Most close binaries may go through a nova phase, so understanding novae may besignificant for binary stellar evolution.
• Novae pose many open questions related to light curves and spectra, and provide anexcellent opportunity to study non-steady-state radiation transport.
For recent compilations that review CNe observations and theory, see Hernanz & Jose (2002)and Bode & Evans (2008).
My study of CNe is still a work in progress. The next few chapters will detail the workI have done and my plans for future work to complete, and potentially extend, the project.
44
6.1 Historical Overview
Many phenomena can result from mass transfer onto a WD from a non-degenerate com-panion. Some are observed to brighten and dim (this can be periodic or aperiodic); suchsystems are referred to as cataclysmic variables (CVs). There are many CV subtypes, includ-ing classical novae, recurrent novae, dwarf novae, and a number of others that are typicallynamed after a prototype system representative of that class (Downes et al., 2001).
Theories about novae date back at least as far as Newton’s Principia Mathematica (New-ton, 1687; see also Jose & Iliadis, 2011): “So fixed stars, that have been gradually wasted bythe light and vapors emitted from them for a long time, may be recruited by comets that fallupon them; and from this fresh supply of new fuel those old stars, acquiring new splendor,may pass for new stars.” The study of novae also includes one of the most concise researchpapers ever written (Hartmann, 1925), which reads in its entirety, “Nova problem solved.Star inflates, bursts.” However, the modern theory of CVs could be argued to begin with Joy(1943), who first deduced the binary nature of a CV (specifically the AE Aquarii system).Walker (1954) showed that one of the two stars in the DQ Herculis system must be com-pact due to the short period, which implied very small separation of the two stars. Struve(1955) generalized these two findings and proposed that all CVs are close binary systems.Crawford & Kraft (1956) presented a model for AE Aquarii that serves as a prototype forall CV systems: one star overflowing its Roche lobe and transferring mass onto a compactcompanion; this model was shown to apply to ten nova systems by Kraft (1964).
Because of the high thermal conductivity of degenerate matter, it was initially believedthat heat would be transported away from the energy-generation region too rapidly for thetemperature to rise su"ciently to ignite a thermonuclear runaway. Giannone & Weigert(1967) show that the matter is not fully degenerate, therefore the thermal conductivity islow enough that the temperature is able to rise and ignite a thermonuclear runaway. Sparks(1969) performed simulations of shocks in the envelope of a star and showed that the resultinglight curves resemble novae.
6.2 Enrichment
Starrfield et al. (1972) determined that the fraction of carbon, nitrogen, and oxygen inthe WD’s accreted envelope prior to the explosion must be greater than the fraction presentin the sun in order to generate su"cent energy for a nova outburst; this is referred to asenrichment. Without enrichment, expansion quenches the burning before enough energy hasbeen released for an outburst. Carbon, nitrogen, and oxygen catalyze hydrogen burningthrough the CNO cycle (see Section 9.1), so the burning proceeds fast enough to generatesu"cient energy for an outburst before expansion can quench the burning. This theoreticalprediction was supported by observations that showed the abundances of nova remnantscould only result if the pre-outburst abundances were enriched (Gehrz et al., 1998).
The source of the enrichment is believed to be some sort of mixing between the accretedenvelope (primarily hydrogen and helium) and the underlying core (in the case of CNe,believed to be primarily carbon and oxygen). The mechanism(s) of this mixing are unknown,with suggestions including
45
• di!usion across the boundary, which builds a mixed layer that heats and induces con-vection (Prialnik & Kovetz, 1984; Kovetz & Prialnik, 1985; Iben & MacDonald, 1985;Iben et al., 1991, 1992; Fujimoto & Iben, 1992);
• shear instabilities along the boundary, particularly as caused by di!erential rotationduring the accretion phase (Durisen, 1977; Kippenhahn & Thomas, 1978; MacDonald,1983; Livio & Truran, 1987; Kutter & Sparks, 1987; Sparks & Kutter, 1987; Kutter &Sparks, 1989; Fujimoto, 1988);
• convection-induced shear instabilities, particularly breaking gravity waves on the core-envelope boundary (Rosner et al., 2001; Alexakis et al., 2002; Calder et al., 2002a;Alexakis et al., 2004);
• convective undershoot, where convective motions induced in the envelope begin topenetrate into the core and dredge up core material (Woosley, 1986; Glasner & Livne,1995; Glasner et al., 1997; Kercek et al., 1998, 1999; Glasner et al., 2011).
This question of enrichment and mixing is the focus of my research into CNe, with the goalbeing an understanding of how this mixing occurs, how fast it takes place, and how muchmaterial is mixed across the core-envelope boundary.
6.3 The State of Modern Simulations
The majority of simulation work on nova explosions has been one-dimensional, with con-vection described by mixing-length theory (see Starrfield, 2002, Glasner et al., 2011, and Jose& Iliadis, 2011 for reviews). However, most of the proposed mechanisms for enrichment areinherently multidimensional. Most multidimensional studies suggest that convective under-shoot may play a significant role in enrichment (Glasner et al., 1997, 2007; Casanova et al.,2010). However, some have seen qualitatively di!erent behavior (Kercek et al., 1998, 1999).Algorithmic di!erences and the dimensionality of the simulations may play a significantrole in the results (Kercek et al., 1999; Glasner et al., 2005; Casanova et al., 2011b). Thecurrent uncertainty in the field strongly suggests that multidimensional (especially three-dimensional) simulations may be necessary.
6.4 The MAESTRO Code
For this research I used MAESTRO, a new hydrodynamics tool designed for low-Mach-number astrophysical flows (Nonaka et al., 2010). Grid based codes are subject to theCourant-Friedrichs-Lewy (CFL) condition, which limits the time step such that informationcannot propagate more than a single cell in a single time step (Courant et al., 1928). Fora traditional compressible code (e.g., FLASH), the speed at which information propagates isequal to the fluid speed plus the sound speed. MAESTRO filters out acoustic waves as describedbelow, so that information propagates at the fluid speed. In the low-Mach regime in whichMAESTRO is valid (Mach number much less than unity, M % 1), this allows MAESTRO totake time steps longer than those of a traditional compressible code, typically by a factor
46
of approximately 1/M. Unlike other algorithms for low-Mach flow (e.g., incompressible oranelastic), MAESTRO captures important compressibility e!ects, including local heating (suchas from nuclear reactions) and movement through a stratified background. MAESTRO has beentested extensively against compressible algorithms in order to demonstrate the validity of theMAESTRO approximation for flows with Mach number less than approximately 0.3 (Almgrenet al., 2006a,b, 2008).
In order to achieve this low-Mach approximation, MAESTRO assumes a one-dimensionalbase state that is in hydrostatic equilibrium (HSE), and uses a variation of the Euler equa-tions1 to compute the multidimensional perturbations to this hydrostatic base state:
"P0 = !"0g (6.1a)
! ("Xk)
!t= !" · ("Xku) + "#k (6.1b)
! (u)
!t= !u · ("# u) !
1
""% !
"! "0"
g (6.1c)
! ("h)
!t= !" · ("hu) +
DP0
Dt+ " hnuc (6.1d)
" · (&0u) = &0
"
S !1
#1P0
!P0
!t!
fvd
#1P0
P0 ! PEoS
$t
#
, (6.1e)
where quantities marked by a bar (e.g., S) are averaged over a horizontal layer (i.e., an aver-age over all cells with the same radial coordinate), quantities marked with a subscript “EoS”are the result of a call to the EoS with the full-state quantities (e.g., PEoS = PEoS(", T, Xk)),and quantities marked with a subscript zero refer to the hydrostatic base state.
The first two equations are straightforward: Equation (6.1a) is the equation of hydrostaticequilibrium and defines the base state, while Equation (6.1b) is the same as Equation (2.4)from the usual Euler equations.
Equation (6.1c) corresponds to Equation (2.1b), with two changes. First, the equationfor HSE has been added in to change the pressure gradient to a gradient of the pressureperturbation (% = P ! P0, also known as the dynamic pressure). Second, the density hasbeen factored out of the ! ("u) /!t term to cast this as an equation for velocity instead ofmomentum density.
Equation (6.1d) corresponds to Equation (2.1c) except for two changes. First, the equa-tion has been reformulated to use specific enthalpy (h) instead of specific total energy (e) orspecific internal energy ($). This change (using some algebra, and substitution from Equa-tion (6.1c)) allows the pressure term to be recast using a notation referred to as a Lagrangian(or “material”) derivative:
DP0
Dt$!P0
!t+ u ·"P0. (6.2)
Accordingly, the energy source has been recast as an enthalpy source. Second, the pressurehas been changed from the full-state pressure (P ) to the base-state pressure (P0). This is
1See Equation (2.1) for the typical Eulerian equations for compressible reactive hydrodynamics.
47
the only explicit reference to the full-state pressure (other equations in this set use either thebase-state pressure or the pressure perturbation). This change decouples the pressure fromthe density, filtering out acoustic waves. However, because of this change, it is necessarythat the pressure perturbation be small; formally, the requirement is that |%| /P0 be of orderM2, leading to the requirement that the Mach number be small.
Equation (6.1e) is an implicit constraint on the velocity field, and is an expression ofthe EoS. The quantity &0 is a density-like variable derived in Almgren et al. (2006a), S isa generic source term, #1 is the logarithmic derivative of pressure with respect to densityat constant entropy (#1 = (d log P/d log ")s), $t is the time step, and fvd is known as the“volume discrepancy factor” (a dimensionless constant between zero and unity). Becausethis constraint equation is a linearization of a non-linear constraint, it is possible for thepressure to drift away from the pressure given by the EoS. The volume discrepancy termis included to counteract this drift and push the pressure into thermodynamic equilibrium(Pember et al., 1998).
48
Chapter 7
Initial Models
The initial models that I used in my CNe research were provided by Ami Glasner, andmatch some of the models used in Glasner et al. (2007). The models are a time sequenceof one-dimensional CN models from a Lagrangian stellar evolution code, and are named bythe peak temperature in the domain (in 107 K); the models supplied by Ami Glasner areT3, T3.5, T5, T7, and T9, with T3 being the earliest and T9 the latest as the base of theaccreted envelope heats up. Due to di!erences between a one-dimensional, Lagrangian stellarevolution code and a multidimensional, Eulerian hydrodynamics code, these initial modelsneeded to be modified through a series of steps to be appropriate for use in MAESTRO. Moredetail of these di!erences will be discusses in Section 7.1.1.
Figure 7.1 shows a sample profile based on Ami Glasner’s T5 model. The model extendsto a higher radius than is shown; beyond the radial limits of the plot the temperature andcomposition are constant, with the density determined as described in Section 7.1.2. Theupper panel shows the density profile, which clearly shows the sharp density gradient aroundthe core-envelope boundary. The central panel shows the temperature profile. The core is notfully isothermal, but approaches isothermal as the radius goes to zero. Immediately abovethe core-envelope boundary there is a nearly-isentropic region due to convection (additionaldiscussion of this region is in Section 7.2), and above the convective region the envelope isisothermal. The lower panel shows the composition: the core is an even mixture by mass of12C and 16O, while the envelope is approximately solar composition.
7.1 Outline of the Initial Model Builder
The goal of my simulations is to study mixing of carbon and oxygen from the coreinto the envelope. This mixing is expected to be related to convective velocities, whichare caused by nuclear energy generation. The peak energy-generation region is where thereis a large concentration of hydrogen (the fuel for the reactions) mixed with a significantquantity of carbon (the catalyst) at a high temperature (the reactions are extemely sensitiveto temperature). All of these processes are focused on the base of the accreted layer, wherethe mixing is occurring and the temperature is highest. Therefore, it is necessary to identifythe base of the accreted layer, rbase, as that location will be significant throughout the initialmodel building process.
49
Figure 7.1: This figure shows a sample profile based on Ami Glasner’s T5 model. The T5model has been the basis of all preliminary simulations to date. The model extends to ahigher radius than is shown here; the extension is at a constant composition and temperature.
7.1.1 Uniform Grid Interpolation
Hydrodynamics may be described in two frames of reference. The Lagrangian frame isbound to the material: a grid point tracks a parcel of material, and the grid points areallowed to move. The Eulerian frame is bound to space: the grid is stationary in space, andmaterial is allowed to move between grid points. Thus the input models from Ami Glasner’sLagrangian stellar evolution code have arbitrary grid point locations, while MAESTRO requiresfixed locations on a uniform grid1. Figure 7.2 shows a comparison between the Lagrangianinput model and a uniform grid model by plotting the density of grid points. Therefore, thefirst step of converting the initial model is to change the grid structure from the Lagrangianinput model by interpolating to a uniform grid. A simple linear interpolation is su"cient;the following steps will modify the profiles and ensure that the correct conditions are met,so it is not necessary to go beyond the simplest interpolation method.
Due to the temperature sensitivity of the nuclear reactions, it is important to capturethe peak temperature correctly. Depending on how the cell centers of the uniform grid alignwith the grid points of the input model, the peak temperature may be “clipped” o!: ifthe uniform grid cell center does not lie precisely at the peak, the interpolation scheme will
1Because MAESTRO uses adaptive mesh refinement, the grid points are not absolute, but may change in aprescribed way. However, MAESTRO initial models need to be defined on the finest grid, assuming that theentire domain will be at the finest resolution. MAESTRO will coarsen the initial model as appropriate basedon the criteria controlling the adaptive mesh.
50
Figure 7.2: This plot shows the density of grid points for the T5 model from Ami Glasner,as well as the same model after interpolation to a uniform grid. By definition, the uniformgrid will have a constant grid density, while the variable grid spacing of a Lagrangian codecan be seen in the high grid density near the core-envelope interface.
average the peak with the next point down, lowering the peak value. Certain alignmentsmay even cause the true peak to be lowered below the neighboring point, artificially movingthe peak and creating a di!erent profile in the vicinity of the new peak; this is shown inFigure 7.3. Therefore the initial model builder shifts the uniform grid slightly in order toplace the peak temperature in the center of a cell.
7.1.2 Hydrostatic Equilibrium Integration — First Pass
Because MAESTRO uses a hydrostatic base state, it is important to ensure that the initialmodel is in HSE. Due to di!erences in numerical methods and the included physics, HSEcan vary from one code to the next due to, e.g., gridding, the EoS, or gravity. Even if theinput model was in HSE, we must recompute the structure by integrating the HSE equation:
!P
!r= !"g. (7.1)
The acceleration of gravity, g, depends on assumptions made about the way that gravity iscalculated; for example, common gravity methods include constant gravity, enclosed-massgravity, and self-gravity. In the case of my simulations, I use a point-source inverse-squaredgravity to match the gravity derived for MAESTRO in Section 8.1. The entire initial domainis assumed to be in HSE, so that P = P0 and " = "0 at t = 0 s.
In practice, the integration becomes a pair of coupled equations: HSE and the EoS. If wetake the superscripts “c” and “p” to refer respectively to the current grid cell we are solving
51
Figure 7.3: A comparison of the temperature peak of the interpolated T5 model. Theonly di!erence between the two profiles is that the temperature peak is aligned to the cellcenter (red profile) or cell edge (blue profile). Notice how the blue profile has “clipped”o! the temperature peak, resulting in (a) a lower peak, (b) the peak moving to a higherradius, and (c) a shallower gradient immediately below the peak. The problem becomesmore exaggerated as the resolution decreases.
52
for and the previous cell already solved for,
P c = P p ! $r
"
"c + "p
2
#
g (7.2a)
P c = PEoS("c, T c, Xc
k), (7.2b)
where g is evaluated between the current and previous grid cells. A Newton-Raphson loopsolves this pair of equations by iterating on "c in order to bring these two expressions forthe pressure into agreement. The temperature and composition profiles given by the inter-polation from the input model are unchanged, so T c and Xc
k are known for all cells. Thisintegration also requires a boundary condition: due to the importance of the base of theaccreted layer, the density at rbase is taken as the boundary condition, with the pressureat rbase given by an EoS call. Integration proceeds in two sweeps: one integration sweepupwards from the base and one downward from the base. At the upper end of the domain,as the density falls o!, a tunable parameter determines a cuto! density below which the staris said to be in the “flu!”: a region of such low density as to be essentially irrelevant to thedynamics of the mixing. When the model falls into this region, the density is set to be equalto this cuto! density and the temperature is set to a constant value. The HSE integrationcore was adopted from previous initial model builders developed by the MAESTRO team.
7.1.3 Smoothing
Due to the conceptual di!erences between a Lagrangian fluid dynamics code and anEulerian fluid dynamics code, the two methods have di!erent capabilities. A Lagrangiancode is capable of having boundaries as thin as the spacing between two grid points, whileEulerian codes often have di"culties with sharp discontinuities and will tend to smooth themout. The input models supplied by Ami Glasner contain such a thin discontinuity: one gridpoint is a hot, low-density mixture dominated by hydrogen and helium, while the adjacentgrid point below it is a cool, high-density mixture of carbon and oxygen. This discontinuityis not expected to be so narrow in a real CN progenitor, so both for reasons of the physicsof the situation and the ability of the numerics, it is necessary to smooth out this interfaceacross a wider radial range. I implemented and tested a variety of smoothing methods.
Kernel Smoothing
Kernel smoothing takes all data points around the current grid point and performs aweighted average according to some smoothing kernel; essentially, this is a convolution ofthe original profile with the smoothing kernel function. The smoothing kernel function istypically symmetric and peaks in the center; for my implementation, I used a Gaussian.Unfortunately, the kernel smoothing method modifies the entire domain and does not pre-serve all of the important features of the initial model (e.g., the temperature peak), so it wasrejected.
53
Figure 7.4: Demonstration of the transition smoothing method on the temperature profile.The solid black profile shows the unsmoothed profile, while the dotted line shows where thetransition was cut out. The colored profiles show the new transition shape being addedin: green is the Gaussian, blue is the exponential, and red is the hyperbolic tangent. Thesmoothing length has been exaggerated to clearly show the transition shapes.
Transition Smoothing
Transition smoothing erases the transition region immediately below the base of theaccreted layer and replaces it with a specified transition shape. Centering the new transitionshape on rbase leads to changes in the thermodynamics of the base of the accreted envelope,hence only the region below the base is modified. The temperature and the composition aretypically the only two quantities smoothed by this method; the density and pressure will beupdated in the next step. This smoothing method requires the specification of a transitionshape and a smoothing length ('sm) which determines the width of the transition. Themethod of performing this smoothing is to mark the base of the accreted layer and the pointone smoothing length below the base (call this the cut point; rcut = rbase!'sm), then erase thethe region between the cut point and the base and replace it with a straight line at the samelevel as the cut point; i.e., for a quantity q, q(rcut < r < rbase) = q(rcut). Then the selectedtransition shape is added to the profile below the base, leaving the profile above the baseunchanged during the entire smoothing step. This is demonstrated in Figure 7.4: the solidblack profile shows the temperature profile prior to smoothing, then the dotted line showswhere the transition was removed. The colored profiles show various new transition shapesadded in to create a new transition. The three transition shapes intersect at both rcut andrbase, splitting the profile below the base into two regions: the upper core (rcut < r < rbase)and the inner core (r < rcut).
54
Gaussian: The first transition shape is a half-Gaussian, centered at the base and extendingto lower radii. The standard deviation of the Gaussian is set to 'sm, and the height is scaledto connect at rbase. This profile has the advantage of being smooth at rbase and rapidly fallsto the original profile away from the base. However, it changes the input model immediatelybelow the base (the upper core) more than is desired.
Exponential: The second transition shape is an exponential decay of the form exp(r/'sm),scaled to connect at rbase. This transition shape changes the upper core less than the Gaussianprofile, but still has a relatively sharp edge between the base of the accreted layer and thetop of the core. This shape also has more significant impact than the Gaussian in the innercore.
Hyperbolic tangent: The third transition shape is a hyperbolic tangent centered half asmoothing length below rbase and scaled to connect at rbase. This transition shape has asmoother connection at rbase than the exponential and falls away rapidly as the distanceincreases from rbase. This transition shape appears to balance the need for a smooth connec-tion at rbase (to avoid too sharp of a change between rbase and the next grid cell below) andnot changing the input model too much in the upper core. The hyperbolic tangent transitionshape is the smoothing technique that I am currently using, with some testing remaining todetermine if this is acceptible or if a better smoothing method is necessary.
Lowering the Composition Boundary
My research included brief testing of a new variation on the transition smoothing methodthat separates the composition transition and the hottest zone. One-dimensional stellar evo-lution codes cannot simulate convection, so convection must be approximated (often throughmixing-length theory). Depending on the choice and implementation of the boundary con-ditions for convection, the boundaries of the convective zone may or may not migrate. It ispossible that the convection has not migrated downward to abut the composition gradient.In addition, since Lagrangian codes are able to maintain arbitrarily thin discontinuities, it ispossible that a wider and smoother composition transition may result in a temperature peakfarther from the center of the transition. Therefore, the relative locations of the temperaturepeak, the composition transition, and the lower edge of the convective zone may not be cap-tured correctly. In particular, the temperature peak and lower edge of the convective zonemight be separated from the composition transition. If the composition transition were widerand farther-separated from the temperature peak, the catalysis of the nuclear reactions couldbe di!erent, which would change the heating, and therefore the convection, which would inturn feed back on the mixing across the composition transition. Based on these arguments,there may be a physical motivation to force a separation between the temperature peak andthe composition boundary. The simplest way to mock up this separation is to simply lowerthe composition boundary, which is what this new smoothing method was implemented todo. This method is only in preliminary testing stages, and requires some research on initialconditions in order to determine a more physically realistic implementation.
55
7.1.4 Hydrostatic Equilibrium Integration — Second Pass
The smoothing step changes the temperature and composition profiles, and both of thesefeed into the EoS and therefore impact the HSE through Equation (7.2b). This necessitatesa second pass of the HSE integrator in order to correct the density and pressure profiles.Running a single sweep of the integrator after both the interpolation and smoothing havebeen performed may lead to unexpected interactions between the interpolation and smooth-ing processes, so it is important to have two passes of HSE integration, one immediatelyafter the interpolation and one immediately after the smoothing.
7.1.5 Hydrostatic Equilibrium Verification
The final step is to verify the HSE conditions by comparing dP/dr to !"g and computingthe maximum error. This checks that the Newton-Raphson loop in the HSE integrationroutine converges and no unexpected errors have arisen. This also mimics the HSE checkthat MAESTRO itself will perform on importing the model, which tells you if MAESTRO willhave issues with the model.
7.2 Correction of Entropy Discontinuity
Early simulations with these initial models generated velocities in a region where there isno physical reason for such velocities to appear. I traced the problem back to a discontinuityin the initial model. A portion of the accreted envelope is nearly isentropic due to convection,except for a small discontinuity (in both entropy and temperature) in the middle of thisregion. See Figure 7.5: the upper panel shows the entropy profile, where the discontinuity ismore pronounced, but the lower panel shows the temperature profile, which also exhibits thisdiscontinuity. The discontinuity occurs when the temperature is approximately 2.5&107 K,which is the temperature when the pp II and pp III branches of the proton-proton reactionchain are equally important (see Figure 7.6). Above this temperature pp III is the dominantenergy source, and below it pp II dominates (with another critical temperature around1.7&107 K, below which the pp I branch takes over). While it is possible for some featureto appear here due to the change in reaction chains at this temperature, both the pp II andpp III reaction rates are continuous and smooth so a discontinuity is not expected. Whilethe discontinuity is somewhat small, it is still su"cient to cause two problems. The firstis that this convective region, which should mix thoroughly as a single region, is artificiallysplit into two disjoint convective zones that mix independently of each other. The second isthe appearance of the unphysical velocities which were problematic in early simulations.
In order to fix the problem, I had to correct for the discontinuity in entropy and tem-perature. The convection causes the region to be nearly, but not fully, isentropic. Changingit to a truly isentropic profile could change the dynamics by pushing part of the domainacross the boundary to being convective when it should not be or vice-versa. I found that acubic polynomial fit the profile very well when an arbitrary shift was permitted on one sideof the discontinuity. Using this cubic polynomial as a guide, I corrected the temperatureprofile between the discontinuity and the top of the convective region. Above the convective
56
Figure 7.5: Illustration of the discontinuity in the initial model. The top panel showsthe specific entropy profile, while the lower panel shows the temperature profile. The redcurve is the profile with the discontinuity, which can be seen clearly in the entropy (upperpanel) around 4700 km. The blue curve is the corrected version without the discontinuity.The shape of the entropy is unchanged, except for the vertical shift for radii greater than4700 km.
region the temperature profile is isothermal, which made it simple to connect the correctedprofile to the profile above the convective region. This correction was applied just prior tothe interpolation to a uniform grid.
57
# # # # # # #
Figure 7.6: Relative contributions of the three branches of the proton-proton chain of nuclearreactions. The critical temperature where the pp II branch gives way to the pp III branchcan be see to be around 2.5&107 K. This image is reprinted from http://cococubed.asu.edu/code_pages/burn_hydrogen.shtml by permission of Frank X. Timmes.
58
Chapter 8
Modifications to the MAESTRO Code
MAESTRO has been used to simulate multiple astrophysical phenomena (e.g., Malone et al.,2011; Zingale et al., 2011; Nonaka et al., 2012a), but my work is the first time that it has beenapplied to CNe. Therefore, it was necessary for me to implement a few new routines to includethe physics that is needed for CNe but was not needed for previous simulations performedwith MAESTRO. In particular, the plane-parallel geometry that I used for my simulationsneeded a point-source inverse-squared gravity routine, and I needed a new reaction networkdesigned to capture the nucleosynthesis and nuclear energetics involved in CNe.
8.1 Gravity
As CNe occur in a thin accreted layer on the surface of a WD and the core of the WDplays no part in the dynamics of the outburst, it is natural to perform simulations in a plane-parallel mode in order to eliminate the core of the WD from the simulation. In this domainthe gravity is dominated by the mass of the WD, with the contributions of the mass in theaccreted layer being negligible, so that a full self-gravity or an enclosed-mass gravity is notnecessary. However, a constant gravity is not su"cient: the radial extent is large enough thatthe di!erence in gravity between the top and bottom of the domain is non-negligible. Thusit was necessary to implement an inverse-squared gravity mode in MAESTRO, which modifiesthe constraint equation from Equation (6.1e). Instead of using Equation (6.1e) directly, Iwill use an alternate form that constrains the base-state expansion velocity (w0 = (u · er)),which is derived in the MAESTRO Users’ Guide (Lawrence Berkeley National Laboratory &Stony Brook University, 2012):
" · (w0er) = S !1
#1P0
DP0
Dt!
f
#1P0
P0 ! PEoS
$t. (8.1)
In the plane-parallel geometry, " · (w0er) = !w0/!r. This equation can be simplified bysplitting w0 into w0 and (w0 such that
! w0
!r= S !
f
#1P0
P0 ! PEoS
$t; (8.2)
59
this leaves behind! (w0
!r= !
1
#1P0
"
!P0
!t+ w0
!P0
!r
#
. (8.3)
The next step is to multiply through by #1P0, di!erentiate by r, and swap the order of amixed partial derivative:
!
!r
$
#1P0
!(w0
!r
%
= !!
!t
!P0
!r!!
!r
$
w0
!P0
!r
%
. (8.4)
Almgren et al. (2008) (Equation (30)) defines
)!(r) = (("! "0) w), (8.5)
where w = (u ! w0er) · er, and shows that (Equation (29))
!"0!t
= !" · ("0w0er) !" · ()!er) . (8.6)
Substitute the HSE equation for the base state (Equation (6.1a)) and Equation (8.6) intoEquation (8.4):
!
!r
$
#1P0
!(w0
!r
%
= !!
!t
!P0
!r!!
!r
$
w0
!P0
!r
%
=!
!t("0g) +
!
!r(w0"0g)
= "0
"
!g
!t+ w0
!g
!r
#
+ g
"
!
!t("0) +
!
!r(w0"0)
#
= "0Dg
Dt! g!)!!r
. (8.7)
Recall that the gravitational acceleration for a point source is g = Gmr"2, and m is constantas we are neglecting the gravitational contributions of mass within the domain. Therefore,
Dg
Dt=
D
Dt
"
Gm
r2
#
= Gm
"
!
!t
"
1
r2
#
+ w0
!
!r
"
1
r2
##
= 0 !2Gmw0
r3
= !2w0g
r. (8.8)
60
Combining Equations (8.7) and (8.8), then splitting w0 into w0 and (w0, yields the finalexpression for the constraint equation for point-source gravity in plane-parallel geometry:
!
!r
$
#1P0
!((w0)
!r
%
+2"0g (w0
r= !
2"0g w0
r! g!)!!r
. (8.9)
Although I derived this equation, the discretization and implementation of it was handledby the designers of MAESTRO, especially Michael Zingale. Details of the discretization (forboth uniform and adaptive grids) are given in Chapter 26 of the MAESTRO Users’ Guide.
To complete this constraint, it is necessary to specify a boundary condition. The expan-sion, which is constrained by the equation derived here, is driven by heating, which occursat the base of the accreted envelope. As the heat is predominantly carried to higher radii,the expansion should be contained in the region above the heating layer; therefore, thereshould be no expansion velocity at the base of the accreted layer. This gives the boundarycondition w0(rbase) = 0.
8.2 Initial Convective Field
A one-dimensional stellar evolution code, such as the one Ami Glasner used to generatethe input models that I used, can estimate the results of convection through mixing lengththeory. While mixing length theory can generate a mean radial velocity, it is unable togenerate a multidimensional convective velocity field that captures the turnover of convectiveeddies. Therefore, when I start my simulations in MAESTRO I have no initial velocities. Thisturns out to be a significant issue: without convective velocities, there is no way for energygenerated at the base of the accreted layer (the region of maximum energy generation) tobe transported away as it would be in a real CN. This leads to isolated zones going intothermonuclear runaway, causing spikes in the temperature and energy generation. Thesesmall runaways are completely unphysical. The convective velocity field would have longsince been established before temperatures reached the point where such runaways couldhappen, and would di!use the energy generated by temperature spikes into the surroundingmaterial. Thus it is necessary to mock up an initial convective velocity field.
The method developed by Michael Zingale was to start by inhibiting (but not forbidding)the runaway. This inhibition is done by passing the mean temperature of a layer (T ) to theEoS instead of the local temperature so that an entire layer must heat up in order to runaway, which slows the runaways. Thus there is still energy generation at the base of theconvective layer to drive the convective velocities, but the small regions do not run away asrapidly. Running with this modification allows a convective velocity field to develop thatis consistent with the initial model. Once the convective velocity field has stabilized, thethermodynamics and composition details are thrown out and the initial model is mappedback in, but now with a multidimensional velocity field. Then the simulation is restarted.
61
Chapter 9
Nuclear Reactions
Fundamentally important to the CN problem is the energy release from nuclear reactions.These reactions generate heat at the base of the envelope, creating a convective layer that isbelieved to cause mixing across the core-envelope boundary. This mixing in turn catalyzes thereactions, increasing the rate of heating and leading to a feedback loop that will culminate ina CN outburst. Over the course of this project, I have applied several methods to estimatethe heating from nuclear reactions. Each of these tools focuses on reactions that converthydrogen to helium.
9.1 Hydrogen Burning Chains
The important reaction processes are split into the pp chains, so named because theybegin by fusing a pair of protons, and the CNO cycles, so named because they rely on loopingthrough various isotopes of carbon, nitrogen, and oxygen. For a review of these processes,see the introduction to Wiescher et al. (2010) and references therein. Additionally, it ispossible for the reactions to escape the CNO cycle and begin forming heavier elements; thisis known as rp breakout, as it is based on the rapid proton process.
9.1.1 The pp Chains
The pp reaction chains begin by fusing a pair of protons, with a weak reaction convertingone into a neutron, leaving behind a deuterium nucleus. A third proton is added to this,yielding 3He. From this point, there are three branches. The pp I branch fuses two 3Henuclei to form a 4He nucleus and 2 protons, while the pp II and pp III branches merge a 3Heand a 4He, and eventually add an additional proton to create a pair of 4He nuclei. The ppchains are illustrated in Figure 9.1.
62
$
$
%
$
%
$
%
%
Figure 9.1: The three branches of the pp chain, which fuse protons to form nuclei of 4He. This image is reprinted fromhttp://cococubed.asu.edu/code_pages/burn_hydrogen.shtml by permission of Frank X. Timmes.
63
%
%
%
%
%
%
%
$
$
$
$
&
&
&
&
Figure 9.2: The four branches of the cold CNO cycle, which use carbon, oxygen, and nitrogenas catalysts for conversion of four protons to a 4He nucleus. This image is reprinted fromhttp://cococubed.asu.edu/code_pages/burn_hydrogen.shtml by permission of FrankX. Timmes.
9.1.2 The CNO Cycles
The CNO reaction cycles require the presence of carbon, nitrogen, and/or oxygen, whichserve as catalysts for the conversion of hydrogen to helium by enabling new reaction pathwaysthat are faster than those of the pp chains. There are a number of branches of the CNO cycle,but the common feature is adding four protons to a heavier nucleus (e.g., a 12C nucleus),interleaved with weak reactions that convert two of the protons to neutrons. The cycle isclosed by the emission of a 4He nucleus, returning to the original catalyst nucleus. Thefour branches of the cold CNO cycle are illustrated in Figure 9.2. Figure 9.3 adds severalbranches of the hot CNO cycle.
9.1.3 Rapid Proton Breakout
In the rapid proton (rp) process, a seed nucleus captures protons in quick succession,building to a large, proton-rich nucleus. A high temperatures is necessary for protons toovercome the Coulomb barrier of the seed nucleus, and a high concentration of protonsis necessary to capture multiple protons before weak processes begin to convert the excessprotons into neutrons and return the nucleus to stability. This process escapes the CNO cycleand forms heavier elements (Wallace & Woosley, 1981). Figure 9.3 shows some channels ofrp breakout from the CNO cycle.
64
%
%
%$
$
&
%
%
%
%
$
$
&
&&
%
%
$$
&
&
Figure 9.3: In addition to the four branches of the cold CNO cycle (shown in Figure 9.2), thisfigure adds branches of the hot CNO cycle, which are important at higher temperature, andrp breakout, which leaves the CNO cycle. This image is reprinted from http://cococubed.asu.edu/code_pages/burn_hydrogen.shtml by permission of Frank X. Timmes.
9.2 Decoupling of Reactions and Hydrodynamics
For a hydrodynamics code, fully coupling the reactions would involve adding the equa-tions for reactions to the hydrodynamics equation set (given by Equation (6.1) for MAESTRO).However, reactions have a very di!erent natural timescale from hydrodynamics; couplingthese equations would require much smaller time steps than the hydrodynamics allow, whichwould severely limit the types of phenomena that could be simulated with such a code. Inorder to avoid this problem, the hydrodynamics and reactions are decoupled. Such a decou-pling allows the hydrodynamics to proceed with a much longer time step than the reactions,which cycle through numerous smaller time steps to build up to a single hydrodynamics timestep. In MAESTRO, this is done by holding the hydrodynamics constant and reacting for half ofa hydrodynamics time step, then holding the reactions constant while performing a full timestep for the hydrodynamics, then finally reacting for the second half of the hydrodynamicstime step. Splitting the reactions into two half steps enables the method to be second-orderaccurate in time.
One issue that arises from this decoupling is the question of how the thermodynamicvariables (especially the density and temperature) evolve during the reaction step. Thereexist a number of approximations for how the thermodynamic variables should evolve dur-ing reactions. All of these methods are only approximations, and do not exactly match theevolution that would result from a fully couple set of hydrodynamics and reaction equations.Decoupling is common, as the di!erences between a decoupled method and a coupled method
65
are considered to be negligible, especially for small hydrodynamics time steps. Recent re-search has explored ways to decouple the equations, but eliminate the resulting changes (seeSection 9.5.5). I detail a few of these approximations for the evolution of the thermodynamicvariables.
Isochoric: For a grid-based code, a common approximation to the thermodynamic evo-lution is to assume that the reaction step is an isochoric (constant density) process. Thisassumption is made in order to conserve mass: because the hydrodynamics are not actingduring a reaction step, no matter is flowing in or out of a cell, so the mass in a cell cannotchange; combining this with the constant volume of the cell gives a constant density. Onecaveat is that mass is not truly conserved; the nuclear reactions convert some fraction of themass into energy, so density should actually change. However, even in an extreme case suchas pure hydrogen converting to pure nickel, the change in mass is less than 1%. Therefore,this mass loss is typically neglected.
Isobaric: For MAESTRO, it is important to carefully monitor the pressure due to the restric-tion that the ratio of the pressure perturbation to the base-state pressure (|%| /P0) be small.This suggests that MAESTRO should assume that reactions are an isobaric (constant pressure)process. Having an isobaric reaction network forces a choice between three di!erent incon-sistencies in the density. In the first case, the density would be evolved in a way that isthermodynamically consistent with the energy release. Doing so means that the density of agiven cell changes without any flow across the boundary, breaking conservation of mass. Thesecond case similarly evolves the density, but then neglects the density evolution from thereaction network in order to conserve mass, resulting in the hydrodynamics and the reactionshaving two di!erent densities. The third choice is to hold the density constant, which bringsthe density out of thermodynamic consistency with the energy release.
Hydrostatic: A simple method is to assume that the thermodynamic variables do notevolve. This approximation has a variety of names, including “hydrostatic” and “pure net-work”. This does not account for the feedback of temperature-sensitive reaction rates, e!ec-tively slowing the reactions from the speed at which they should proceed. However, this isthe simplest form of the network to implement.
I make use of the isobaric and hydrostatic approximations in the reaction network describedin Section 9.5. Details of my implementations of these approximations are presented below.
9.3 Analytic CNO
The first attempt at capturing the energy release from nuclear reactions, primarily ex-isting as a preliminary testing tool, was an analytic expression to approximate the energy
66
release of the CNO cycle (Kippenhahn & Weigert, 1996):
$ = 8.67&1027&
1 + 2.7&10"3 T 1/36 ! 7.78&10"3 T 2/3
6 ! 1.49&10"4 T6
'
& "XCNO X1H T"2/36 exp
&
!1.5228&102 T"1/36
'
, (9.1)
where T6 is the temperature divided by 106 K, and " and $ are assumed to be in cgs units.The symbol XCNO represents the combined mass fraction of all carbon, oxygen, and nitrogenisotopes. This only accounts for energy release without evolving the composition, density,or temperature, and acts as a source term to Equation (6.1d). When a full network wassubsequently implemented, this approximation was found to be a poor representation ofCNO energy generation for the conditions in a CN.
9.4 Frank X. Timmes’s pphotcno Network
The next step was to implement a nuclear reaction network. We chose to port FrankTimmes’s pphotcno network1 into MAESTRO. This network is designed to capture all of thenuclear processes described in Section 9.1. The core code of pphotcno was included essen-tially as it was provided by Frank Timmes (with minor modifications to integrate with theMAESTRO EoS), and I wrote new interface routines to connect MAESTRO and pphotcno.
The primary design decision for this process was which approximation to use. As dis-cussed in Section 9.2, MAESTRO typically uses isobaric reaction networks. However, whenporting pphotcno into MAESTRO, the isobaric mode presented some unexpected issues. Asa temporary solution, I instead chose to run pphotcno in a hydrostatic mode. This wassu"cient for preliminary simulations, but needed to be corrected for publishable results.
9.5 Rebuilt Network
The next implementation of nuclear reactions for my simulations involved rebuildingpphotcno. I did this for several reasons:
1. better integration with MAESTRO,
2. improved performance,
3. implementation of the isobaric assumption,
4. a code update to MAESTRO that implemented a technique known as spectral deferredcorrections (SDC; Nonaka et al., 2012b), which works to resolve the issues that arisefrom decoupling the reactions from the hydrodynamics.
1Available at http://cococubed.asu.edu/code_pages/burn_hydrogen.shtml.
67
The original pphotcno network included subroutines to compute the reaction rates andneutrino losses. I stripped these subroutines out of the code and reused them with minimalchanges (primarily for style consistency, tighter integration with MAESTRO, and eliminationof redundant code such as the EoS that was already included in MAESTRO). The rest of thecode was rebuilt, including the removal of the integration method built around the MA28linear system solver and replacing it with the VODE integrator.
9.5.1 Notes on Composition
Reaction networks in MAESTRO store the following information for each species: Ak, thenumber of nucleons in a nucleus of species k; Zk, the number of protons in a nucleus ofspecies k; and Bk, the binding energy of a nucleus of species k (given as a positive value).From these, the mass of a particle of species k is given by
mk = Zk mp + (Ak ! Zk) mn ! Bk/c2,
where mp and mn are the masses of protons and neutrons respectively. We can also computethe mass of a mole of particles of species k:
Wk = NA mk,
where NA is Avogadro’s number. An important caution: pphotcno makes a common ap-proximation: Wk = Ak. This assumes that mass is being measured in grams, and makes twofurther simplifying assumptions: (1) the mass of the proton and the mass of the neutron areboth approximately (1/NA) g, and (2) the binding energy is negligible. These assumptionsare not always appropriate, so at times it is necessary to use NA mk in place of Wk. I alsodefine Nsp to be the number of species in the reaction network.
Nuclear reaction networks often work in terms of the molar abundance (Yk), which is theratio of the number of moles of species k to the total mass, and therefore has units of oneover mass. This is given as follows:
Yk =(number of moles)k
(mass)tot
=(molar density)k
(mass density)tot
=nk/NA
"=
1
mk NA
"k"
=Xk
Wk,
where nk is the number density of nuclei of species k. The last form will be used frequentlyto convert betweeen molar abundance and mass fraction, which is the form used in thehydrodynamics and the EoS.
9.5.2 Equation Outline
Reaction networks are often framed as a set of coupled ordinary di!erential equations ofthe form
ui = fi
(
u1, . . . , uNeq
)
, i = 1..Neq,
where Neq is the number of equations. The first Nsp equations are the composition variables,and one or more thermodynamic quantities are also evolved. The fi function set is often
68
referred to as the “right-hand side” function. The choice of how many and which thermo-dynamic variables are evolved depends on the assumptions and design decisions that go intothe reaction network. I will demonstrate three di!erent choices: a simple isobaric network,an isobaric network that attempts to gain an improvement in run time, and MAESTRO’s SDCmethod.
9.5.3 A Simple Isobaric Network
The simplest method for an isobaric network is to evolve the composition and the energy,then use a call to the EoS to update the other quantities as necessary. The evolution of thethermodynamics is driven by energy release from nuclear reactions, so the energy is the mostobvious choice of which thermodynamic quantity to evolve. Coupling the energy with thequantity that does not evolve (in this case pressure) uniquely defines the thermodynamicstate through the EoS.
Evolution of the Composition
The fundamental job of the reaction network is to evolve the composition; networks thatuse the hydrostatic approximation merely evolve composition without allowing any feedbackto the thermodynamics, and non-hydrostatic reaction networks evolve the thermodynamicsbased on energy released by the changing composition. Evolution of the composition isdone by defining a set of reactions between the species present and constructing the ratesof these reactions. In the pphotcno network, the reaction rates are functions of the densityand temperature, and in some cases also the electron-to-baryon ratio (Ye, derivable fromthe composition). Using these reaction rates, we can compute the rate of change of thecomposition:
Yi =!
j
(
±Yj,1 [Yj,2 [Yj,3]] Rj(", T, Y1..Nsp))
, (9.2)
where the summation over j includes the reactions that consume (negative sign) or form(positive sign) species i, Rj is the rate for reaction j, and Yj," are the reactant species forreaction j. There may be 1, 2, or 3 reactants (hence why Yj,2 and Yj,3 are in brackets: theymay or may not be present for all reactions), and they may be the same species or di!erentspecies. For example, the triple-* reaction has three reactants (all of which are 4He), protoncapture has two reactants (1H and the capturing nucleus), and electron capture only has onereactant (the nucleus that captures the electron).
Evolution of the Energy
The root of the changing thermodynamics lies in the release of energy from the nuclearreactions. The rate of change of energy is simply a summation of the nuclear energy releasewith an additional term for neutrino cooling. The easiest form of energy to construct is the
69
specific internal energy:
$ = !!
k
*
Enuc,k
Mtot
+
! $#(", T, Y1..Nsp)
= !!
k
*
Mk c2
Mtot
+
! $#(", T, Y1..Nsp)
= !!
k
*
Xk Mtot c2
Mtot
+
! $#(", T, Y1..Nsp)
= !!
k
&
Yk Wk c2'
! $#(", T, Y1..Nsp)
= !NA c2!
k
&
Yk mk
'
! $#(", T, Y1..Nsp), (9.3)
where Enuc,k is the nuclear energy release from species k, Mtot is the total mass of all species,and $# encapsulates the energy losses due to the generation of neutrinos (this term is com-puted by subroutines included in Frank Timmes’s network). In the final step it is necessaryto use Wk = NAmk: the source of the energy release is the nuclear binding energy, and thetypical approximation (Wk = Ak) neglects this binding energy, so using NAmk in place ofWk ensures that this expression is correct, regardless of whether or not Wk is approximated.Both terms in this expression have negative signs. The nuclear binding energy summation isactually the sum of the mass lost from nuclei, which is the opposite of the energy gained fromthis mass loss. The neutrino term is actually the energy lost to neutrinos, so it is subtractedout of the energy source.
9.5.4 A Faster Isobaric Network
The previous method, although simple in concept and execution, requires a call to theEoS inside of each call to the right-hand side routine. For a simple EoS, such as the idealgas law, this may be acceptable. For a complex EoS, such as a realistic electron-ion plasma,this can result in a significant amount of time being spent in the EoS. Avoiding an EoS callcould result in a faster execution time for the reaction network.
This section attempts to derive the equations for an isobaric network in such a way asto avoid a call to the EoS. However, the derivation below is incomplete, in that it neglectsthe evolution of thermodynamic derivatives such as cP or !h/!Yk. It may be acceptable toconsider some of these as constant over the course of the integration, but which may be heldconstant would require testing. Preliminary testing has shown that assuming a constantvalue for cP gives noticeably incorrect results over long durations, but it is necessary toevaluate this approximation on the time scale of a hydrodynamic time step, which is theduration that a reaction network would run while coupled to MAESTRO. In order to make useof these equations, it would be necessary to evaluate which of the thermodynamic derivativesmust be evolved and then to augment the equation set to evolve the important quantities.
70
Coupled Evolution of Enthalpy, Temperature, and Density
Since the reaction rates are functions of the density and temperature, it makes the mostsense to evolve these two quantities. For faster execution, it is best to minimize the number ofvariables to be evolved. Also derived in this section is the evolution equation for the specificenthalpy; as will be discussed later, this may be an important quantity and its evolution iscoupled with that of the temperature and density.
The first expression comes from the relationship between specific enthalpy and specificinternal energy: h = $+ P/". Taking the time derivative of this expression and rearrangingslightly gives
h +P
"2" = $. (9.4)
This equation couples the evolution of density and specific enthalpy together, with the evo-lution of the energy (given by Equation (9.3) above) as a source term.
The second expression comes from the total di!erential of the specific enthalpy. InMAESTRO the EoS supplies the partial derivatives of specific enthalpy with respect to themass fraction, assuming that pressure and temperature are held constant. However, sincethe reaction rates available from Frank Timmes’s network are in terms of molar abundances,we will simply convert the derivatives according to
!h
!Yk
,
,
,
,
P,T
=!h
!Xk
,
,
,
,
P,T
Wk.
Now we can say that specific enthalpy should be a function of pressure, temperature, andcomposition and take the total di!erential:
dh =!h
!P
,
,
,
,
T,Y1..Nsp
dP +!h
!T
,
,
,
,
P,Y1..Nsp
dT +!
k
*
!h
!Yk
,
,
,
,
P,T
dYk
+
.
Since we are in the isobaric assumption, dP is zero. We can also identify the derivative withrespect to temperature as the isobaric specific heat capacity. If we divide this expression bydt and rearrange slightly, we get our second equation:
h ! cP T =!
k
"
!h
!YkYk
#
. (9.5)
This equation couples the evolution of the temperature and the specific enthalpy together,with the summation over the composition evolution expressions (given by Equation (9.2)above) as a source term.
The third expression comes from the total di!erential of the pressure. In MAESTRO theEoS supplies the partial derivatives of pressure with respect to mass fraction, assuming thatdensity and temperature are held constant. As with the previous expression, we convertderivatives with respect to mass fraction to derivatives with respect to molar abundances by
71
multiplying by Wi. The total di!erential of pressure therefore is
dP =!P
!"
,
,
,
,
T,Y1..Nsp
d"+!P
!T
,
,
,
,
!,Y1..Nsp
dT +!
k
*
!P
!Yk
,
,
,
,
!,T
dYk
+
.
Since we are in the isobaric assumption, dP is zero. Dividing by dt and rearranging slightlygives us our third equation:
!P
!""+!P
!TT = !
!
k
"
!P
!YkYk
#
. (9.6)
This equation couples the evolution of the density and the temperature together, with thesummation over the composition evolution expressions (given by Equation (9.2) above) as asource term.
The final expressions are simply the result of solving these coupled equations. In orderto simplify these expressions, let us define some shorthands:
%h $!
k
"
!h
!XkWk Yk
#
, (9.7a)
%P $!
k
"
!P
!XkWk Yk
#
, (9.7b)
and
$ $
"
cP!P
!"!
P
"2!P
!T
#"1
. (9.8)
With these we can summarize the equation set as
-
.
1 P!2 0
1 0 !cP
0 !P/!" !P/!T
/
0
-
.
h"T
/
0 =
-
.
$%h
!%P
/
0 .
Solving this system of equations yields our final, decoupled equations:
h = $
"
cPP
"2%P + cP
!P
!"$!
P
"2!P
!T%h
#
(9.9a)
" = $
"
!P
!T%h ! cP %P !
!P
!T$
#
(9.9b)
T = $
"
P
rho2%P +
!P
!"$+!P
!"%h
#
(9.9c)
Quasi-Isochoric Approximation
In some cases it may be useful to assume that the change in density is negligible, whichI refer to as the quasi-isochoric approximation. As discussed in Section 9.2, the isobaric
72
assumption forces a choice between di!erent inconsistencies. Given a choice between massnon-conservation, two di!erent densities, or thermodynamic inconsistency, it may be thatthermodynamic inconsistency is the “least-bad” option.
In this case, we can simplify the expressions derived in the previous section. Firstly,Equation (9.6) can be replaced by a statement of the quasi-isochoric approximation: " = 0.Equation (9.4) simplifies to the equivalence h = $, eliminating the need to consider theevolution of enthalpy separately from the evolution of energy. Using this equivalence, wecan simplify Equation (9.5) and solve for the temperature evolution2:
T =1
cP
*
$!!
k
"
!h
!XkWk Yk
#
+
. (9.10)
Using this approximation we eliminate the need to evolve density or enthalpy and arrive ata simpler expression for the evolution of the temperature.
Energy Release
One of the important outputs from a reaction network is the energy release, although inMAESTRO the enthalpy release is the desired quantity. There are several possible methods forcomputing the final energy release.
The simplest is to call the EoS using the initial and final states of the integration. Thisonly adds two EoS calls to the reaction network (one at the beginning of the integration andone at the end of the integration), instead of an EoS call to each iteration of the reactionnetwork’s sub-cycling (as the method outlined in Section (9.5.3) does).
The second method is to evolve the energy (or the enthalpy) directly. If the desiredquantity is not already being evolved to track the thermodynamic evolution, this adds anextra variable to the reaction network, which increases the execution time of each iteration ofthe network’s sub-cycling. Whether this is faster or slower than adding starting and endingEoS calls would have to be the subject of testing, and may depend on the duration overwhich the network needs to be evolved.
A third method is to use an integrated form of one of the evolution equations (Equa-tions (9.4) – (9.6) or Equations (9.9a) – (9.9c)) to deconvolve the total change in energyor enthalpy from the total change in the density, temperature, and composition. However,this method su!ers from the problem discussed above regarding the assumption that cP andother thermodynamic derivatives are constant, and therefore it would have to be modifiedconsistently with any other modifications implemented to address this issue.
2The MAESTRO Users’ Guide (Lawrence Berkeley National Laboratory & Stony Brook University, 2012)gives an alternate derivation that achieves this same result. See Chapter 19, “Notes on Prediction Types”,particularly Section 3, “Energy Evolution”.
73
9.5.5 The SDC Method
In the SDC version of the reaction network, there is no need to approximate the thermo-dynamic evolution (e.g., the isobaric approximation); the SDC method is intended to evolvethe reaction network consistently with the hydrodynamics by reducing the error that arisesfrom decoupling. The SDC method is an iterative process that applies a set of correctionsto reduce the decoupling error, which maintains the e"ciency of a decoupled method whileproviding the accuracy of a fully coupled method. For complete details of the SDC method,see Nonaka et al. (2012b). I did not implement the SDC method, but only implemented avariation of the reaction network that is consistent with the requirements of the SDC method.Here I detail how the equations for the reaction network are modified to integrate with theSDC method. Borrowing from the SDC documentation in the MAESTRO User’s Guide, thetwo key equations are
d("Xk)
dt= " #k + A!Xk
(9.11a)
d("h)
dt= " hnuc + A!h, (9.11b)
where A!Xkand A!h come from the hydrodynamics. They are defined as
A!Xk= !+" · ("Xk+u) (9.12a)
A!h = !+" · ("h+u) +DP0
Dt. (9.12b)
While the form of these equations comes from the Euler equations, we can make a loosecorrespondence with product rules for derivatives to give ourselves a mental aid in under-standing the relationships involved in these equations. For Equation (9.11a), the " #k termis akin to " Xk, and the A!Xk
term is akin to "Xk. For Equation (9.11b), we need to recallthe relationship between enthalpy density and internal energy density ("h = " $ + P ), andwe can see that the " hnuc term is akin to " $, and the A!h term is akin to " $+ P . This alsohelps identify the fact that hnuc is equivalent to $. The true derivation of these equationscomes from the Euler equations, so these product rules are merely a guide and not a trueequivalence. It should be noted, however, that the hnuc term appears in both the energy andthe enthalpy equations, so that it can be identified with $ formally.
The equations for Xk (except for a factor of Wk) and $ are given in Section (9.5.3).The SDC method evolves the partial densities instead of the mass fractions or molar abun-dances (which accounts for mass non-conservation as discussed previously when explainingthe isochoric approximation for reaction networks) and the enthalpy density instead of thespecific enthalpy. The density comes from a summation of the partial densities. All otherthermodynamics come from a call to the EoS in "-h mode.
74
Chapter 10
Conclusions and Future Work
10.1 Preliminary Results
This project is still a work in progress. Here I present results to date, which demonstratethe current status and suggest possible conclusions that may result from this study.
Figure 10.1 shows the evolution of the peak Mach number (the largest value of M inthe domain) over the course of a simulation of convection in the hydrogen-rich envelope ofa classical nova progenitor. For the first roughly 800 s, the value of the peak Mach numbervaries, but on average remains approximately constant around 0.04. After the 800 s mark,the scatter around the mean trend increases in frequency, and the mean trend begins torise. While I have not yet determined the reason(s) for these changes, this plot suggeststhat around 800 s the nuclear burning changes from some sort of steady-state burning to arunaway, leading towards a nova outburst.
Figure 10.2 shows a snapshot of another simulation just over 270 s after the start. Theregion shown is only a part of the simulation domain, focusing on the core-envelope interface(at approximately 4550 km as shown by the radial coordinate along the left side of theimage) and the convective region immediately above the interface. The central panel showsa region immediately above the interface (up to approximately 4600 ! 4700 km) that isenhanced in 12C, as evidenced by the greener color in this region in contrast to the morecyan color at a higher radius. By comparison with the right panel, this 12C-enhanced regionapproximately correlates with the region of nuclear energy release. The ripples on the core-envelope interface, while not fully understood as of yet, are reminiscent of breaking wavessuch as those discussed in Alexakis et al. (2004), although there is concern that these ripplesmay be an artifact due to boundary condition e!ects. While more analysis, and possiblyfurther simulations (perhaps at higher resolution), will be necessary for confirmation, theseresults suggest that breaking waves may play a significant role in the metallicity enrichmentof the envelope.
10.2 Summary and Conclusions
While I have done extensive background work to develop the tools necessary for simula-tions of CNe, this is an ongoing project. My work has primarily divided into two categories.
75
Figure 10.1: The evolution of the peak Mach number in a preliminary simulation of a classicalnovae in MAESTRO. Note the change in behavior around 800 s.
The first category is developing appropriate initial conditions for a multidimensional Eu-lerian code based on a one-dimensional Lagrangian code. As shown by Zingale et al. (2002),transferring between two such codes is not as trivial as copying the output of the Lagrangiancode into the Eulerian code. Many issues, both physical and numerical, must be addressedin transferring a model between two such codes: e.g., the sharpness of discontinuities, hy-drostatic equilibrium, the grid underlying the simulation, and convective velocities. Eachmuch be carefully considered, not only independently, but in conjunction with the other is-sues; for example, interpolation to a new grid and smoothing of discontinuities may interactin unexpected ways if care is not taken to ensure hydrostatic equilibrium after each suchtransformation of a model.
The second category is implementing an appropriate method for capturing the nucle-osynthesis and nuclear energetics of CNe. Prior research has shown that the appropriatereactions include the proton-proton chains and the carbon-nitrogen-oxygen cycles, alongwith rapid-proton breakout. I found that an analytic approximation to the CNO cycle is apoor representation of the energy release; in addition, it gives no information about the nu-cleosynthesis that occurs during a CNe. Due to time step considerations, reactions are oftendecoupled from hydrodynamics, which introduces errors from the approximations that areneeded to close the decoupled sets of equations. However, new methods are being developedto address these errors, including spectral-deferred corrections.
The study of CNe has proven more challenging than I initially expected. This is anongoing project, but promises to have the potential to explore new aspects of CNe andimprove our understanding of these complex astrophysical phenomena.
76
Figure 10.2: A snapshot of the core-envelope interface and convective region during a prelim-inary simulation of a classical nova in MAESTRO. Evidence of envelope enrichment by breakingwaves is suggested by this simulation, but further analysis will be necessary to confirm sucha tentative result.
77
10.3 Future Work
The current work on the CNe project is continued code development. This includesverification and validation (V&V) of both the initial model builder and the rebuilt nuclearreaction network. Preliminary V&V has been done, but is not yet extensive enough to ensurethat these codes are ready for use in publishable work.
I am preparing a suite of two-dimensional simulations designed to test the code compo-nents and begin clarifying the computational needs of the CN problem. The most physically-realistic simulations must be performed in three dimensions in order to accurately modelconvection, but three-dimensional simulations are very computationally expensive. Hence asuite of two-dimensional problems will help to map out the optimal use of computationalresources for three-dimensional simulations. I intend to investigate questions of:
• spatial resolution necessary to capture the processes in a CN outburst,
• boundary conditions (e.g., see Glasner et al. 2005),
• simulation domain size and its interaction with side boundary conditions,
• the possibility of a smaller reaction network that minimizes pp chain processes.
Once the code components under development have undergone rigorous V&V and the plannedtwo-dimensional preliminary suite has been performed, I can begin to move forward withexplorations of the physics of CNe. The intended goal of this project has been to studythe mixing processes across the core-envelope boundary in a CN outburst. With my com-putational tools, I can match the published literature on the subject for comparison andvalidation, then begin to expand on the current understanding of these mixing processes.
Classical novae present a challenging computational problem, particularly if results are tobe connected to observations. Although convection in the accreted envelope is not directlyobservable, the result of simulating convection-induced mixing in the burning region at thebase of the accreted envelope could potentially link to observations in several ways. First,any inhomogeneities following from the structure of the turbulent convective flow could seedinhomogeneities in the ejecta (see, e.g., Casanova et al., 2011b). Next, following the evolutionof a CN through the outburst would allow me to determine ejecta velocities. Simulating theoutburst is a significant computational challenge because the envelope will expand throughorders of magnitude in radius and a complete simulation would require interactions of theexpanding envelope with the accretion disk and possibly even the binary companion.
The nuclear burning and mixing processes that occur during my simulations will alsoprovide information about the quantity and distribution of certain nucleosynthesis products.The models and simulations presented in this dissertation assume simplified reactions forthe thermonuclear burning because of the expense (in computating power and memory) ofincluding variables for all participating nuclei. Greater fidelity and detail of the nucleosyn-thesis products would be achievable through post-processing of advected tracer particlesbased on their thermodynamic histories during a simulation. This information about nucle-osynthesis could be compared to studies of the ejecta (see, e.g., Gehrz et al., 1998). Theuse of radiation transport would allow even more observationally-oriented information, such
78
as calculating light curves and spectra from the simulations. For all of these issues, themost informative results would only be achievable by the di"cult process of continuing mysimulations beyond the mixing phase to the outburst phase, and perhaps even beyond tofollowing the evolution of ejecta. These di!erent phases will require di!erent computationaltools, which may require significant development.
With the computational tools I have been developing for my CN simulations, I will beable to pursue a variety of questions in addition to mixing across the core-envelope boundary.This includes:
• Studying the complete evolution of a CN outburst, including mixing and explosion.This could allow me to develop observable signatures for comparison with my simula-tions. This could also potentially allow an exploration of how much mass is ejected inan outburst, which is an important question in evaluating whether or not CNe couldbe a phase in the evolution of a SNIa progenitor.
• Studying the e!ect of the mass of the underlying WD, which sets the gravity andtherefore will be a significant factor in the pressure and density of the burning layeras well as the energy necessary to eject matter during an outburst. Through both ofthese e!ects, the mass of the WD may be important in determining whether or not aCNe accretes more matter between outbursts than it ejects during an outburst. Thisstudy would require the use of a stellar evolution code, such as the new open-sourcecode MESA, to generate realistic initial models with di!erent WD masses.
• Studying the e!ect of the composition of the accreted material, which will impactthe nucleosynthesis pathways favored during an outburst, which will in turn changethe rate of energy release and the total energy release before expansion quenches theburning.
• Performing a comparison of one-dimensional and multidimensional evolution. Due tothe time scales involved, it is not currently possible for a multidimensional hydrody-namics code to follow the evolution of a nova from the onset of accretion through thefirst outburst or between outbursts. Many studies of novae must therefore rely on one-dimensional, Lagrangian, stellar evolution codes. I could perform an important checkon some aspects of such codes by running an early-time model from the time-sequenceprovided by Ami Glasner until it matches a late-time model and comparing the evolu-tion between a one-dimensional stellar evolution code and a multidimensional reactivehydrodynamics code.
• Studying di!erences in novae on carbon-oxygen WDs and oxygen-neon WDs, exploringhow the resulting outbursts compare to classical vs. recurrent novae and searching forpotentially-observable signatures. The common assumption in nova theory is thatrecurrent novae occur on oxygen-neon WDs, while classical novae occur on carbon-oxygen novae, but this has not been conclusively confirmed. This could additionallyshed light on the question of whether these two di!erent classes of novae gain or losemass over long time scales.
79
• Evaluating the impact of varying uncertain reaction rates. This could shed light onpotential constraints on reaction rates that are currently not well known by comparingthe results of my simulations with observations.
Several of these points will require new computational tools, and some of them may not bepossible with current technology. However, as long-term goals, they serve to focus my e!orttowards exploring the important open questions in the field of CNe.
80
Bibliography
Albrecht, A. et al. 2006, arXiv Astrophysics e-prints
Alexakis, A. et al. 2004, ApJ, 602, 931
Alexakis, A., Young, Y., & Rosner, R. 2002, Phys. Rev. E, 65, 059904
Almgren, A. S., Bell, J. B., Nonaka, A., & Zingale, M. 2008, ApJ, 684, 449
Almgren, A. S., Bell, J. B., Rendleman, C. A., & Zingale, M. 2006a, ApJ, 637, 922
—. 2006b, ApJ, 649, 927
Arnett, W. D. 1982, ApJ, 253, 785
Aspden, A. J., Bell, J. B., Day, M. S., Woosley, S. E., & Zingale, M. 2008, ApJ, 689, 1173
Aspden, A. J., Bell, J. B., & Woosley, S. E. 2010, ApJ, 710, 1654
Baade, W., & Zwicky, F. 1934, Physical Review, 46, 76
Bertola, F. 1964, Annales d’Astrophysique, 27, 319
Blinnikov, S. I., & Khokhlov, A. M. 1986, Soviet Astronomy Letters, 12, 131
Bloom, J. S. et al. 2012, ApJ, 744, L17
Bode, F., & Evans, A. 2008, Classical Novae, Cambridge Astrophysics Series (CambridgeUniversity Press)
Brachwitz, F. et al. 2000, ApJ, 536, 934
Brahe, T. 1573, Copenhagen
Branch, D., Livio, M., Yungelson, L. R., Bo", F. R., & Baron, E. 1995, PASP, 107, 1019
Brandt, T. D., Tojeiro, R., Aubourg, E., Heavens, A., Jimenez, R., & Strauss, M. A. 2010,AJ, 140, 804
Bravo, E., & Garcıa-Senz, D. 2008, A&A, 478, 843
Bravo, E., Isern, J., Canal, R., & Labay, J. 1990, Ap&SS, 169, 19
81
Calder, A. C. et al. 2002a, in American Institute of Physics Conference Series, Vol. 637,Classical Nova Explosions, ed. M. Hernanz & J. Jose, 134–138
Calder, A. C. et al. 2002b, ApJS, 143, 201
—. 2007, ApJ, 656, 313
Casanova, J., Jose, J., Garcıa-Berro, E., Calder, A., & Shore, S. N. 2010, A&A, 513, L5
—. 2011a, A&A, 527, A5
Casanova, J., Jose, J., Garcıa-Berro, E., Shore, S. N., & Calder, A. C. 2011b, Nature, 478,490
Chaisson, E., & McMillan, S. 2001, Astronomy Today, 4th edn. (Prentice Hall)
Chamulak, D. A., Brown, E. F., Timmes, F. X., & Dupczak, K. 2008, ApJ, 677, 160
Chandrasekhar, S. 1931a, MNRAS, 91, 456
—. 1931b, ApJ, 74, 81
—. 1935, MNRAS, 95, 207
Colgate, S. A., & McKee, C. 1969, ApJ, 157, 623
Conley, A. et al. 2011, ApJS, 192, 1
Courant, R., Friedrichs, K., & Lewy, H. 1928, Mathematische Annalen, 100, 32
Crawford, J. A., & Kraft, R. P. 1956, ApJ, 123, 44
da Silva, L. A. L. 1993, Ap&SS, 202, 215
Davidson, P. A. 2004, Turbulence (Oxford: Oxford University Press)
della Valle, M., & Livio, M. 1995, ApJ, 452, 704
Domınguez, I., Hoflich, P., & Straniero, O. 2001, ApJ, 557, 279
Downes, R. A., Webbink, R. F., Shara, M. M., Ritter, H., Kolb, U., & Duerbeck, H. W.2001, PASP, 113, 764
Durisen, R. H. 1977, ApJ, 213, 145
Euler, L. 1757a, Memoires de l’academie des sciences de Berlin, 11, 217, Euler Archive E225
—. 1757b, Memoires de l’academie des sciences de Berlin, 11, 274, Euler Archive E226
—. 1757c, Memoires de l’academie des sciences de Berlin, 11, 316, Euler Archive E227
—. 1761, Novi Commentarii academiae scientiarum Petropolitanae, 6, 271, Euler ArchiveE258
82
Fan Ye. 5th century, o"cial Chinese historical text
Filippenko, A. V. 1997, ARA&A, 35, 309
Filippenko, A. V. 2009, in Stellar Death and Supernovae Conf. (Santa Barbara: UCSB),http://online.kitp.ucsb.edu/online/sdeath c09/filippenko/
Fisher, R., Falta, D., Jordan, G., & Lamb, D. 2010, in Gravitation and Astrophysics, ed.J. Luo, Z.-B. Zhou, H.-C. Yeh, & J.-P. Hsu, 335–344
Fowler, R. H. 1926, MNRAS, 87, 114
Fryxell, B. et al. 2000, ApJS, 131, 273
Fujimoto, M., & Iben, Jr., I. 1992, ApJ, 399, 646
Fujimoto, M. Y. 1988, A&A, 198, 163
Fuller, G. M., Fowler, W. A., & Newman, M. J. 1985, ApJ, 293, 1
Gallagher, J. S., Garnavich, P. M., Caldwell, N., Kirshner, R. P., Jha, S. W., Li, W.,Ganeshalingam, M., & Filippenko, A. V. 2008, ApJ, 685, 752
Gehrz, R. D., Truran, J. W., Williams, R. E., & Starrfield, S. 1998, PASP, 110, 3
Giannone, P., & Weigert, A. 1967, ZAp, 67, 41
Glasner, A. S., Livne, E., & Truran, J. W. 2011, ArXiv e-prints
Glasner, S. A., & Livne, E. 1995, ApJ, 445, L149
Glasner, S. A., Livne, E., & Truran, J. W. 1997, ApJ, 475, 754
—. 2005, ApJ, 625, 347
—. 2007, ApJ, 665, 1321
Goldhaber, G. et al. 2001, ApJ, 558, 359
Golombek, I., & Niemeyer, J. C. 2005, A&A, 438, 611
Hansen, C. J., Kawaler, S. D., & Trimble, V. 2004, Stellar interiors : physical principles,structure, and evolution, 2nd edn. (Springer-Verlag)
Hartmann, J. 1925, Astronomische Nachrichten, 226, 63
Hernanz, M., & Jose, J. 2002, Classical nova explosions: International Conference on Clas-sical Nova Explosions : Sitges, Spain 20-24 May 2002, Review of progress in quantitativenondestructive evaluation (American Institute of Physics)
Hillebrandt, W., & Niemeyer, J. C. 2000, ARA&A, 38, 191
83
Hoflich, P., & Khokhlov, A. 1996, ApJ, 457, 500
Hoflich, P., Khokhlov, A. M., & Wheeler, J. C. 1995, ApJ, 444, 831
Hoflich, P. et al. 2010, ApJ, 710, 444
Hoflich, P., Wheeler, J. C., & Thielemann, F. K. 1998, ApJ, 495, 617
Howell, D. A. et al. 2009, ApJ, 691, 661
Hoyle, F., & Fowler, W. A. 1960, ApJ, 132, 565
Hubble, E. 1929, Proceedings of the National Academy of Science, 15, 168
Iben, Jr., I., Fujimoto, M. Y., & MacDonald, J. 1991, ApJ, 375, L27
—. 1992, ApJ, 388, 521
Iben, Jr., I., & MacDonald, J. 1985, ApJ, 296, 540
Iben, Jr., I., & Tutukov, A. V. 1984, ApJS, 54, 335
Iwamoto, K., Brachwitz, F., Nomoto, K., Kishimoto, N., Umeda, H., Hix, W. R., & Thiele-mann, F.-K. 1999, ApJS, 125, 439
JabRef Development Team. 2010, JabRef, JabRef Development Team
Jackson, A. P., Calder, A. C., Townsley, D. M., Chamulak, D. A., Brown, E. F., & Timmes,F. X. 2010, ApJ, 720, 99
Jacoby, G. H. et al. 1992, PASP, 104, 599
Jha, S., Riess, A. G., & Kirshner, R. P. 2007, ApJ, 659, 122
Jose, J., & Hernanz, M. 2007, Journal of Physics G Nuclear Physics, 34, 431
Jose, J., & Iliadis, C. 2011, Reports on Progress in Physics, 74, 096901
Joy, A. H. 1943, PASP, 55, 283
Kelly, P. L., Hicken, M., Burke, D. L., Mandel, K. S., & Kirshner, R. P. 2010, ApJ, 715, 743
Kercek, A., Hillebrandt, W., & Truran, J. W. 1998, A&A, 337, 379
—. 1999, A&A, 345, 831
Khokhlov, A. M. 1989, MNRAS, 239, 785
—. 1991a, A&A, 245, 114
—. 1991b, A&A, 245, L25
—. 1995, ApJ, 449, 695
84
—. 2000, ArXiv Astrophysics e-prints
Khokhlov, A. M., Oran, E. S., & Wheeler, J. C. 1997, ApJ, 478, 678
Kippenhahn, R., & Thomas, H.-C. 1978, A&A, 63, 265
Kippenhahn, R., & Weigert, A. 1996, Stellar Structure and Evolution (Springer)
Kirshner, R. P. 2010, in Dark Energy: Observational and Theoretical Approaches, 1st edn.,ed. P. Ruiz-Lapuente (Cambridge University Press), 151
Kovetz, A., & Prialnik, D. 1985, ApJ, 291, 812
Kraft, R. P. 1964, ApJ, 139, 457
Krueger, B. K., Jackson, A. P., Calder, A. C., Townsley, D. M., Brown, E. F., & Timmes,F. X. 2012, accepted by the Astrophysical Journal
Krueger, B. K., Jackson, A. P., Townsley, D. M., Calder, A. C., Brown, E. F., & Timmes,F. X. 2010, ApJ, 719, L5
Kuhlen, M., Woosley, S. E., & Glatzmaier, G. A. 2006, ApJ, 640, 407
Kutter, G. S., & Sparks, W. M. 1987, ApJ, 321, 386
—. 1989, ApJ, 340, 985
Langanke, K., & Martınez-Pinedo, G. 2001, Atomic Data and Nuclear Data Tables, 79, 1
Lawrence Berkeley National Laboratory, C., & Stony Brook University. 2012, MAESTROUser’s Guide, distributed with the MAESTRO source code
Lemaıtre, G. 1927, Annales de la Societe Scietifique de Bruxelles, 47, 49
Lesa!re, P., Han, Z., Tout, C. A., Podsiadlowski, P., & Martin, R. G. 2006, MNRAS, 368,187
Li, W. et al. 2011, Nature, 480, 348
Lisewski, A. M., Hillebrandt, W., & Woosley, S. E. 2000, ApJ, 538, 831
Livio, M. 2000, in Type Ia Supernovae, Theory and Cosmology, ed. J. C. Niemeyer & J. W.Truran, 33
Livio, M., & Truran, J. W. 1987, ApJ, 318, 316
MacDonald, J. 1983, ApJ, 273, 289
Maeda, K., Ropke, F. K., Fink, M., Hillebrandt, W., Travaglio, C., & Thielemann, F.-K.2010, ApJ, 712, 624
Malone, C. M., Nonaka, A., Almgren, A. S., Bell, J. B., & Zingale, M. 2011, ApJ, 728, 118
85
Mannucci, F., Della Valle, M., & Panagia, N. 2006, MNRAS, 370, 773
Mazzali, P. A., & Podsiadlowski, P. 2006, MNRAS, 369, L19
Meakin, C. A., Seitenzahl, I., Townsley, D., Jordan, G. C., Truran, J., & Lamb, D. 2009,ApJ, 693, 1188
Minkowski, R. 1941, PASP, 53, 224
Neill, J. D. et al. 2009, ApJ, 707, 1449
Newton, I. 1687, Philosophi Naturalis Principia Mathematica (Jussu Societatis Regiae acTypis Josephi Streater. Prostat apud plures bibliopolas)
Niemeyer, J. C. 1999, ApJ, 523, L57
Niemeyer, J. C., & Hillebrandt, W. 1995, ApJ, 452, 769
Niemeyer, J. C., & Woosley, S. E. 1997, ApJ, 475, 740
Nobelprize.org. 2011, The 2011 Nobel Prize in Physics - Press Release
Nomoto, K. 1984, ApJ, 277, 791
Nomoto, K., Saio, H., Kato, M., & Hachisu, I. 2007, ApJ, 663, 1269
Nomoto, K., Sugimoto, D., & Neo, S. 1976, Ap&SS, 39, L37
Nomoto, K., Thielemann, F.-K., & Yokoi, K. 1984, ApJ, 286, 644
Nonaka, A., Almgren, A. S., Bell, J. B., Lijewski, M. J., Malone, C. M., & Zingale, M. 2010,ApJS, 188, 358
Nonaka, A., Aspden, A. J., Zingale, M., Almgren, A. S., Bell, J. B., & Woosley, S. E. 2012a,ApJ, 745, 73
Nonaka, A., Bell, J. B., Day, M. S., Gilet, C., Almgren, A. S., & Minion, M. L. 2012b,Combustion Theory and Modelling, accepted
Nugent, P. E. et al. 2011, Nature, 480, 344
Oda, T., Hino, M., Muto, K., Takahara, M., & Sato, K. 1994, Atomic Data and NuclearData Tables, 56, 231
Osterbrock, D. E. 2001, in Bulletin of the American Astronomical Society, Vol. 33, AmericanAstronomical Society Meeting Abstracts, 1330
Pember, R. P., Howell, L. H., Bell, J. B., Colella, P., Crutchfield, W. Y., Fiveland, W. A.,& Jessee, J. P. 1998, Combustion Science and Technology, 140, 123
Perlmutter, S. et al. 1999, ApJ, 517, 565
86
Phillips, M. M. 1993, ApJ, 413, L105
Phillips, M. M., Lira, P., Suntze!, N. B., Schommer, R. A., Hamuy, M., & Maza, J. 1999,AJ, 118, 1766
Pinto, P. A., & Eastman, R. G. 2000, ApJ, 530, 744
Piro, A. L., & Bildsten, L. 2008, ApJ, 673, 1009
Piro, A. L., & Chang, P. 2008, ApJ, 678, 1158
Plewa, T., Calder, A. C., & Lamb, D. Q. 2004, ApJ, 612, L37
Poludnenko, A. Y., Gardiner, T. A., & Oran, E. S. 2011, Physical Review Letters, 107,054501
Poludnenko, A. Y., & Oran, E. S. 2011a, Combustion and Flame, 157, 995
—. 2011b, Combustion and Flame, 158, 301
Pope, S. B. 1987, Annual Review of Fluid Mechanics, 19, 237
Porter, A. C., & Filippenko, A. V. 1987, AJ, 93, 1372
Prialnik, D., & Kovetz, A. 1984, ApJ, 281, 367
Raskin, C., Scannapieco, E., Rhoads, J., & Della Valle, M. 2009, ApJ, 707, 74
Riess, A. G. et al. 1998, AJ, 116, 1009
—. 2011, ApJ, 730, 119
Ropke, F. K. 2006, in Reviews in Modern Astronomy, Vol. 19, Reviews in Modern Astronomy,ed. S. Roeser, 127
Ropke, F. K. 2007, ApJ, 668, 1103
Ropke, F. K., Gieseler, M., Reinecke, M., Travaglio, C., & Hillebrandt, W. 2006, A&A, 453,203
Ropke, F. K., & Hillebrandt, W. 2005, A&A, 429, L29
Ropke, F. K., & Niemeyer, J. C. 2007, A&A, 464, 683
Ropke, F. K. et al. 2011, Progress in Particle and Nuclear Physics, 66, 309
Rosner, R., Alexakis, A., Young, Y.-N., Truran, J. W., & Hillebrandt, W. 2001, ApJ, 562,L177
Scalzo, R. A. et al. 2010, ApJ, 713, 1073
Scannapieco, E., & Bildsten, L. 2005, ApJ, 629, L85
87
Schmidt, W., Ciaraldi-Schoolmann, F., Niemeyer, J. C., Ropke, F. K., & Hillebrandt, W.2010, ApJ, 710, 1683
Schmidt, W., Niemeyer, J. C., & Hillebrandt, W. 2006a, A&A, 450, 265
Schmidt, W., Niemeyer, J. C., Hillebrandt, W., & Ropke, F. K. 2006b, A&A, 450, 283
Seitenzahl, I. R., Ciaraldi-Schoolmann, F., & Ropke, F. K. 2011, MNRAS, 414, 2709
Seitenzahl, I. R., Meakin, C. A., Townsley, D. M., Lamb, D. Q., & Truran, J. W. 2009a,ApJ, 696, 515
Seitenzahl, I. R., Ropke, F. K., Fink, M., & Pakmor, R. 2010, MNRAS, 407, 2297
Seitenzahl, I. R., Townsley, D. M., Peng, F., & Truran, J. W. 2009b, Atomic Data andNuclear Data Tables, 95, 96
Sparks, W. M. 1969, ApJ, 156, 569
Sparks, W. M., & Kutter, G. S. 1987, ApJ, 321, 394
Starrfield, S. 2002, in American Institute of Physics Conference Series, Vol. 637, ClassicalNova Explosions, ed. M. Hernanz & J. Jose, 89–98
Starrfield, S., Truran, J. W., Sparks, W. M., & Kutter, G. S. 1972, ApJ, 176, 169
Stothers, R. 1977, Isis. Journal of the History of Science Society, 68, 443
Straniero, O., Domınguez, I., Imbriani, G., & Piersanti, L. 2003, ApJ, 583, 878
Stritzinger, M., Mazzali, P. A., Sollerman, J., & Benetti, S. 2006, A&A, 460, 793
Struve, O. 1955, S&T, 14, 275
Sullivan, M. et al. 2011, ApJ, 737, 102
Timmes, F. X., Brown, E. F., & Truran, J. W. 2003, ApJ, 590, L83
Timmes, F. X., & Swesty, F. D. 2000, ApJS, 126, 501
Townsley, D. M., Calder, A. C., Asida, S. M., Seitenzahl, I. R., Peng, F., Vladimirova, N.,Lamb, D. Q., & Truran, J. W. 2007, ApJ, 668, 1118
Townsley, D. M., Jackson, A. P., Calder, A. C., Chamulak, D. A., Brown, E. F., & Timmes,F. X. 2009, ApJ, 701, 1582
Travaglio, C., Hillebrandt, W., Reinecke, M., & Thielemann, F.-K. 2004, A&A, 425, 1029
Truran, J. W., Arnett, W. D., & Cameron, A. G. W. 1967, Canadian Journal of Physics, 45,2315
Vladimirova, N., Weirs, G., & Ryzhik, L. 2006, Combust. Theory Modelling, 10, 727
88
Walker, M. F. 1954, PASP, 66, 230
Wallace, R. K., & Woosley, S. E. 1981, ApJS, 45, 389
Webbink, R. F. 1984, ApJ, 277, 355
Whelan, J., & Iben, Jr., I. 1973, ApJ, 186, 1007
Wiescher, M., Gorres, J., Uberseder, E., Imbriani, G., & Pignatari, M. 2010, Annual Reviewof Nuclear and Particle Science, 60, 381
Woosley, S. E. 1986, in Saas-Fee Advanced Course 16: Nucleosynthesis and Chemical Evo-lution, ed. J. Audouze, C. Chiosi, & S. E. Woosley, 1
Woosley, S. E. 1990, in Supernovae, ed. A. G. Petschek, 182–212
Woosley, S. E. 2007, ApJ, 668, 1109
Woosley, S. E., Kasen, D., Blinnikov, S., & Sorokina, E. 2007, ApJ, 662, 487
Woosley, S. E., Kerstein, A. R., Sankaran, V., Aspden, A. J., & Ropke, F. K. 2009, ApJ,704, 255
Woosley, S. E., & Weaver, T. A. 1986, ARA&A, 24, 205
Yuan, F. et al. 2010, ApJ, 715, 1338
Zingale, M. et al. 2002, ApJS, 143, 539
Zingale, M., Nonaka, A., Almgren, A. S., Bell, J. B., Malone, C. M., & Woosley, S. E. 2011,ApJ, 740, 8
89