Cataclysmic Variables and Double-Detonation Supernovae

A STUDY OF WHITE DWARFS: CATACLYSMIC VARIABLES AND DOUBLE-DETONATION SUPERNOVAE

by SPENCER CALDWELL DEAN M. TOWNSLEY, COMMITTEE CHAIR JEREMY BAILIN JULIA CARTWRIGHT

A THESIS

Submitted in partial fulﬁllment of the requirements for the degree of Master of Science in the Department of Physics in the Graduate School of The University of Alabama

TUSCALOOSA, ALABAMA

Novae, be it classical, dwarf, or supernovae, are some of the most powerful and luminous events observed in the Universe. Although they share the same root, they are produced by di↵erent physical processes. We research systems capable of experiencing novae with the intention of furthering our understanding of these astrophysical phenomena. A cataclysmic variable is a binary star system that contains a white dwarf with the potential of undergoing classical or dwarf novae. A recent observation of a white dwarf within one of these systems was found to have an unusually high surface temperature for its orbital period. The discovery contradicts current evolutionary models, motivating research to determine a theoretical justiﬁcation for this outlier. We simulated novae for a progenitor designed to represent a white dwarf in an interacting binary. We developed post-novae cooling timescales to constrain the temperature value. We found the rate at which classical

1 novae cool post-outburst (< 1Kyr )isingeneralagreementwiththefour year follow-up observation ( 2K).Theevolutionofwhitedwarfsduringdouble-detonation ⇠ type Ia supernovae was also studied. The progenitors capable of producing these events are not fully established, requiring a consistent model to be developed for parametric analysis. Three improvements were made to the simulation model used in (Townsley et al., 2019): the inclusion of a de-reﬁnement condition, a new particle distribution, and a burning limiter. The focus here was to enhance the computational eciency, o↵er better representation of particles in the supernova ejecta, and control the nuclear energy release. These developments were employed to test double-detonation scenarios capable of producing spectra analogous to type Ia supernovae, which will o↵er insight into their prevalence and strengthen their use in measuring cosmological distance.

ii ACKNOWLEDGMENTS

Foremost, I would like to express my profound gratitude to my advisor Dr. Dean Townsley. He has been integral to both my success and enjoyment as a graduate student at the University of Alabama. His support and guidance throughout all my research is and always will be appreciated. Thanks to Dr. Boris G¨ansicke and Dr. Anna Pala at the University of Warwick for providing the observational foundation for my work on cataclysmic variables. My special thanks to Dr. Broxton Miles for setting a lot of the groundwork for the double-detonation simulations performed and for composing the particle distribution implemented in this study. Thank you to my committee members Dr. Jeremy Bailin and Dr. Julia Cartwright for agreeing to participate and help me through the process. I also would like to thank Dr. Ken Shen at the University of California, Berkley who provided generous support with the double-detonation supernovae.

iii CONTENTS

ABSTRACT...... ii

ACKNOWLEDGMENTS...... iii

LISTOFTABLES...... vi

LISTOFFIGURES...... vii

1 CATACLYSMICVARIABLES ...... 1

1.1 INTRODUCTION ...... 1

1.2 METHODOLOGY ...... 5

1.3 RESULTS...... 7

1.3.1 OBSERVATIONAL INFLUENCE ...... 7

1.3.2 CLASSICAL NOVAE ...... 9

1.3.3 DWARFNOVAE...... 13

1.4 DISCUSSION...... 16

1.4.1 VARIABILITY OF IGNITION IN NOVAE EVOLUTION ...... 16

1.5 CONCLUSION ...... 21

2 SUPERNOVAE ...... 22

2.1 INTRODUCTION ...... 22

2.2 SIMULATIONSETUP...... 24

2.2.1 REFINEMENT...... 29

2.2.2 PARTICLEDISTRIBUTION ...... 31

2.2.3 BURNINGLIMITER ...... 34

iv 2.3 RESULTS...... 37

2.3.1 DE-REFINEMENTSTUDY...... 37

2.3.2 BURNING LIMITER STUDY ...... 38

2.3.3 RESOLUTIONSTUDIESONIGNITION ...... 41

2.4 DISCUSSION...... 46

2.5 CONCLUSION ...... 53

REFERENCES ...... 55

v LIST OF TABLES

2.1 ParametersforWhiteDwarfProgenitors ...... 27 2.2 NuclearReactionNetwork ...... 28

vi LIST OF FIGURES

1.1 Cumulative Distribution between Theoretical and Observed Te↵ ...... 8

1.2 Evolution of Te↵ and Tc ...... 10 1.3 Classical Nova Cooling Timescale for 0.95 M ...... 12 1.4 Binary Evolution Below the Period Gap ...... 14 1.5 Dwarf Nova Cooling Timescale for 0.819 M ...... 15

1.6 Initial Tc andClassicalNovaeEvolution ...... 17 1.7 AccretionRateandClassicalNovaeEvolution ...... 19 2.1 TimeSequenceofHeliumShellDetonation ...... 23 2.2 ShockAngleDetermination ...... 30 2.3 FinalParticlePositions...... 33 2.4 De-reﬁnement Test ...... 38 2.5 Burn Limiter Test ...... 39 2.6 New Burn Routine ...... 40 2.7 Ignition Reﬁnement Test ...... 42 2.8 Carbon Detonation Test ...... 45 2.9 Evolution of Helium Shell Detonation for 1.0 M ...... 47 2.10 Lead-up to Carbon Detonation ...... 48 2.11 FinalAbundancesofEjectafromDouble-Detonation ...... 49 2.12 Evolution of Reduced 16OHeliumShellDetonation ...... 51 2.13 Final Abundances of Ejecta for Reduced 16OProgenitor...... 52

vii 1

CATACLYSMIC VARIABLES

1.1 INTRODUCTION

Cataclysmic Variables (CVs) are close-interacting binary star systems that contain an accreting white dwarf accompanied by a low-mass main-sequence (MS) donor ( 0.6M ). . At the onset of CV formation, the more massive star in a binary star system matures to the giant phase and its matter encloses the entire binary, forming a common envelope. The less-massive donor falls to closer orbits around the degenerate core of the primary, which transfers energy outward and causes the envelope to be ejected. Once stripped of the common envelope, the binary is left detached, semi-detached, or close enough to merge. If detached, the stars still orbit one another, but evolve separately until angular momentum loss (AML) reduces the orbital separation or the donor expands on its nuclear timescale and overflows its Roche lobe, continuing mass transfer (Goliasch and Nelson, 2015). The Roche lobe is the equipotential surface surrounding each star that is at the same potential as the first Lagrange point, where material is equally bound to each star. At this point, material can shift from being bound to one star to the other by an arbitrarily small change in velocity. If the binary system is semi-detached after the common envelope is ejected, it is likely to continue its evolution as a CV. Semi-detached systems contain a donor star overflowing its Roche lobe that delivers matter to the accreting white dwarf, forming an accretion disk. Further evolution of the system is primarily dictated by AML, nuclear evolution of the donor (which will change its radius), and accretion rate. AML is caused by

1 two main sources, gravitational radiation and magnetic braking. Gravitational radiation is apersistentsourceofangularmomentumloss,drivingthesystemtosmallerorbitalperiods

(Porb). Gravitational radiation dominates in CVs when Porb < 3hr,whenthe

11 1 time-averaged accretion rate is M˙ =5 10 M yr (Patterson, 1984), where M is h i ⇥ solar mass = 1.989 1030 kg. AML for CVs with P > 3hrisdominatedbymagnetic ⇥ orb 9 8 1 braking (Spruit and Ritter, 1983) and have accretion rates 10 10 M yr .Magnetic braking is caused by interaction of the departing stellar wind with the magnetic field of the rotating star, reducing angular momentum of the rotating object. CVs with orbital periods between 2 and 3 hr are scarce, so that there is a dichotomy in the angular momentum loss mechanism for observed systems, separated by what is termed the “period gap.” The start of the period gap (P 3hr)signifieswhenthedonorstarhaslostenough orb ⇡ mass (Mdonor 0.23 M ) to become fully convective (Howell et al., 2001). Magnetic  braking is interrupted at this point, e↵ectively halting further accretion. The donor, which has become bloated, is then allowed to thermally adjust and shrink back to the equilibrium radius for a MS star with the relevant mass (Townsley and Gänsicke, 2009). During this stage, mass transfer ceases until AML caused by gravitational radiation drives the binary back to a semi-detached state (P 2 hr). Mass transfer onto the white dwarf orb ⇡ recommences once through the period gap. The orbital separation continues to shrink while the white dwarf accretes material from the donor, which continues to lose mass. Concurrently, while the mass-loss timescale increases, the Kelvin-Helmholtz timescale,

⌧thermal, increases faster. When ⌧thermal becomes greater than the mass-transfer timescale, the donor can no longer thermally adjust. This typically occurs when P 80 min and orb ⇡

Mdonor 0.06 M (Howell et al., 2001). Continued CV evolution leads to a gradual increase ⇡ in the orbital separation and decrease in accretion rate for the remainder of its lifetime. Observations performed by (Pala et al., 2017) were able to ascertain the surface temperatures for CV white dwarfs in their Hubble Space Telescope (HST) survey. Of those, one white dwarf exhibited an unusually high surface temperature (Te↵ =33855K)

2 for its orbital period (P 93.1 min). A CV white dwarf’s surface temperature is orb ⇡ determined by accretion heating and the T P distribution should reflect the e↵ orb accretion rate predicted by CV evolution (Goliasch and Nelson, 2015). From current evolutionary theory, we would expect to find a T 15,000 K based on the orbital period e↵ ⇡ of J1538. We were motivated by this observation to investigate whether the high Te↵ was due to the white dwarf experiencing a recent nova event. The CV in question, SDSS J153817.35+5123238.0 (hereafter J1538), is therefore the focus of our work. The underlying objective of our research is to provide a theoretical justification behind the nature of J1538 and by doing so acquire an understanding into the astrophysical phenomena CVs are capable of below the period gap. We focus on two classes of CV, dwarf and classical novae, that can produce temperatures relevant to our study. Dwarf novae CVs experience an outburst caused by a thermal instability in the accretion disk, which leads to rapid deposition of matter onto the surface of the white dwarf. Stability within the accretion disk returns after a sucient amount of mass is transferred and the accretion rate returns to its quiescent rate. These disk instabilities can reoccur periodically with the recurrence time a↵ecting the amount of energy released during outburst. The events can last for days and are separated by quiescent periods ranging from days to years (Piro et al., 2005). Classical novae outbursts, however, result from instabilities at the base of accreted material that leads to a thermonuclear runaway. Hydrogen rich material accreted at fairly

11 9 1 low rate of 10 10 M yr (Howell et al., 2001; Starrﬁeld et al., 2016; Townsley and Bildsten, 2004), accumulates in a thin shell around the white dwarf until the temperature and pressure at the base of the material reaches levels capable of hydrogen burning. When the accreted mass (Macc)equalstheignitionmass(Mign), the local timescale for nuclear burning becomes smaller than the local thermal timescale in the bottom layer of the accreted shell. The base of this layer will reach temperatures of 2 107 Kanddensities ⇠ ⇥ 3 3 10 gcm (Shen and Bildsten, 2009a). The layers surrounding the ignition point are ⇠ electron degenerate and unable to thermally adjust to the rising temperatures, causing a

3 runaway of proton-proton and carbon-nitrogen-oxygen (CNO) reactions (Denissenkov et al., 2013; Starrﬁeld et al., 2016). The result is a classical nova: the outer layer is ejected, the white dwarf relaxes beneath its Roche Lobe, and the orbital separation of the binary increases. The classical novae process can repeat itself with varying times of recurrence. The time-averaged accretion rate is low enough in both classical and dwarf novae CVs that the surface temperature, Te↵,ofthewhitedwarfcanbemeasuredduringperiodsof quiescence. We employed MESA (Modules for Experiments in Stellar Astrophysics) to simulate the stellar evolution of an accreting white dwarf through both classical and dwarf novae cycles. Three white dwarf progenitors were run through multiple outbursts after establishing an equilibrium core temperature, Tc,eq, to provide a model comparable to observed white dwarfs (Townsley and Bildsten, 2004). We chose progenitor models with masses of 0.6,

0.819, and 0.95 M to study, the latter being the estimated white dwarf mass, MWD,of J1538. This range covers the average mass of a CV white dwarf 0.83 0.23 M (Zorotovic ± et al., 2011). The cooling rate for each progenitor following the 50th outburst was determined, providing a constraint on Te↵ for follow-up observations to J1538. The theoretical temperature history was compared to the surveyed CV white dwarfs to measure the probability this temperature would be observed. Section 1.2 details the setup used to perform the MESA evolutionary simulations. Results follow in section 1.3, ﬁrst presenting a statistical analysis between the theoretical

Te↵ histories and those provided by observations detailed by (Pala et al., 2017) in section 1.3.1. The simulation results for classical and dwarf novae are in sections 1.3.2 and 1.3.3, respectively. Lastly, section 1.4 will discuss the current obstacles in developing a consistent model for future simulations and our conclusions obtained from the study in section 1.5.

4 1.2 METHODOLOGY

MESA1 is a powerful stellar astrophysics code that combines numerical and physical modules to simulate evolutionary scenarios (Paxton et al., 2011). MESA star is the one-dimensional evolutionary module that is capable of evolving a wide variety of stellar models by solving the fully coupled structure and composition equations simultaneously. It reads the input files and loads the physics modules, granting access to the initial setup parameters like reaction network, equations of state, opacity data, and more. Two types of Fortran namelist files are utilized, one specifying the evolutionary scenario to be conducted and the other details controls and procedures employed during the simulation. MESA provides a variety of progenitor models for use that allow modifications to structure and composition in order to meet a specific need. During a simulation, output files are generated to give data on the detailed evolutionary history. The output photo file contains the complete current state of the simulation, which can be used to restart the simulation. An output log records the evolutionary properties (age, mass, temperature, composition) over the course of the simulation. The output profile details the progenitor properties with various zones of the star at each timestep. One timestep in MESA star follows a four-step procedure. First, the model is re-meshed, if necessary, to prepare for a new timestep. Next, the model is adjusted to account for mass loss or gain by accretion. It adjusts abundances, determines convective di↵usion coecients, and solves for the current structure and composition. Then, the next timestep is estimated. And finally, the output files are produced (Paxton et al., 2011). We used MESA star to create three progenitors with properties resembling J1538, a white dwarf in an interacting-binary CV. We began by obtaining a 0.604, 0.819, and 1.025 M white dwarf from the collection of MESA models. The stellar structure and composition of an accreting white dwarf was reproduced by stripping each progenitor of its surface

7 7 layers, letting it cool to Tc =10 K, and then accreting 10 M of solar composition 1http://mesa.sourceforge.net

5 1 43 4 material [0.749 H, 0.29 10 He, and 0.237 He] onto the surface. The white dwarf is ⇥ cooled and models are saved for a range of core temperatures. We use models with core temperatures equal to 8 106 Kand9 106 Kfortherunspresentedhere.This ⇥ ⇥ progenitor represents a post-common envelope white dwarf that is allowed to cool before gravitational losses return the binary to a semi-detached state, consistent with previous work (Townsley and Bildsten, 2004). To simulate the outburst, the initial white dwarf model is run through three stages. First, we initiate the novae cycle by activating accretion onto the white dwarf until the

5 3 1 ignition mass is reached, Mign 10 10 M yr (Starrﬁeld et al., 2016; Townsley ⇡ and Bildsten, 2004). The accretion continues until the radius has grown to 0.3 R , ⇡ where R is solar radius = 6.955 108 meters. We denote this point as the beginning of ⇥ the outburst phase of the simulation. The radius expands > 0.8 R ,causingRochelobe overﬂow to prompt mass loss. The white dwarf shrinks back within its Roche lobe and surface hydrogen burning raises the temperature greater than 105 K. The end of the outburst is denoted when the radius has contracted under 0.2 R ,atwhichpointaccretion resumes and the process restarts. The progenitors are run through multiple outburst events until Tc,eq is reached. The use of Tc,eq provides results independent of the thermal state of the white dwarf, which is expected for the evolved state of observed white dwarfs (Townsley and Bildsten, 2005).

6 1.3 RESULTS

We ran simulations of classical and dwarf novae to determine whether the high surface temperature of J1538 could be caused by a thermonuclear event. We initiated accretion onto the progenitor white dwarfs which then accumulated enough mass to develop an instability at the base layer. This instability results in a thermonuclear outburst that produces temperatures of several 105 K. Following the outburst and ejecta of accreted material, the white dwarf enters a state of quiescence where it is allowed to cool. The results for the novae runs are presented in this section. We provide a low-sample statistical analysis of the measured white dwarf Te↵ from the HST survey in section 1.3.1. Cooling timescales were calculated for post-outburst classical and dwarf novae for comparison with follow-up observations of J1538. The initial core temperature and accretion rate were studied to determine their e↵ect on the evolution during classical novae.

1.3.1 OBSERVATIONAL INFLUENCE

The HST survey performed by (Pala et al., 2017) obtained Te↵ for white dwarfs in CVs. Thirty CV white dwarfs under the period gap (P 2 hr), including J1538, were orb  found to have an average surface temperature of 15,216 K. In order to accurately represent this observed sample, we began by determining the M˙ that will lead to this temperature h i during quiescence. The 0.819 M white dwarf progenitor was run through several classical 12 9 1 novae cycles, each with di↵erent accretion rates ranging from 10 10 M yr . Using the thermal history, we calculated the fraction of time spent during the evolution at each

Te↵.Consideringthewhitedwarfspendsthevastmajorityofitsevolutionbetween outbursts, the median quiescent temperature can be easily identiﬁed. We interpolated between the chosen range of M˙ and resulting quiescent T to ﬁnd the rate of accretion h i e↵ 12 that yielded a surface temperature of 15,216 K. The result was M˙ =1.989 10 M h i ⇥ 1 yr .Thisaccretionrateissmallerthanexpectedforbinariesinthisorbitalperiodrange.

11 As cited in section 1.1 and shown in section 1.3.3, the expected rate is around 5 10 ⇥

7 1 M yr . This di↵erence is due to the way ignition and heating is occurring in our models and will be discussed more in section 1.4.1. The 0.819 M model was put through multiple outbursts with this accretion rate to obtain its evolutionary history. We compared the results to the observed population of CVs with P 2hrtoprovideastatistical orb  framework for the 33,855 K observed on J1538.

Figure 1.1: Cumulative distribution between theoretical and observed Te↵.TheTe↵ values are based on fractional time during evolution. The theoretical temperature history was obtained from the 0.819 M progenitor having undergone 50 outburst events. Three random samples, each containing 30 values of Te↵ weighted with the theoretical history, were generated to compare with values obtained from survey (Pala et al., 2017).

Figure 1.1 displays the resulting cumulative distribution of the fractional time spent at each Te↵ during novae evolution. We included three random samples, containing thirty Te↵ measurements each, taken from and weighted with the theoretical distribution in order to

8 make a more accurate comparison with the observed population. The under-representation of high surface temperature observations ( 18,000 K) in comparison to the theoretical distribution may be attributed to observational biases that prevent a comprehensive population from being obtained. The cumulative distribution indicates there is less than a four percent chance of observing temperatures measured in J1538. Despite the small sample size and probability, the correlation is sucient to reason that the classical nova scenario remains a candidate for the mechanism producing the outlier Te↵ observed in J1538. A stronger conclusion cannot be inferred without an observed population that better represents the theoretical distribution, especially in the 18,000 23,000 K range.

1.3.2 CLASSICAL NOVAE

The current hypothesis for the high temperature observed on J1538 is that it either had recently undergone a period of high accretion or it was subject to a nova event (Pala et al., 2017). The progenitor models were run through 50 classical novae with a

12 1 time-averaged rate of M˙ =1.989 10 M yr to calculate the cooling rate following h i ⇥ an outburst. Figure 1.2 displays the evolution of both Te↵ and Tc for progenitors with masses 0.6 (red), 0.819 (green), and 0.95 (blue) M for the ﬁrst 2 Gyr of their evolution.

9 (a) Surface temperature history for 0.6 (red), 0.819 (green), and 0.95 (blue) M models.

(b) Core temperature history for 0.6 (red), 0.819 (green), and 0.95 (blue) M models.

Figure 1.2: Evolution of Te↵ and Tc.TheTc increases for all models during throughout the classical novae cycle, achieving Tc,eq sooner for the lower mass progenitor due to the higher accumulation of mass and more heating prior to ignition

10 The accreted layer grows thicker after each successive outburst, causing compressional heating and nuclear simmering to eventually heat the core to an equilibrium temperature.

The accretion rate, Tc, composition of accreted material, and MWD all factor in determining Mign (Shen and Bildsten, 2009a; Townsley and Bildsten, 2004). The results in

ﬁgure 1.2 here consider MWD as the only variable parameter, holding M,˙ composition, and

8 Tc the same in all three cases. The 0.6 M model evolved 1.48 10 yrs before the ﬁrst ⇥ 4 outburst ignited with Mign =2.94 10 M .Thecoretemperatureincreasesaftereach ⇥ subsequent outburst and roughly settles to an equilibrium after 2 Gyr. The 0.819 M 7 4 white dwarf evolved for 6.77 10 yrs before it ignited after accreting M =1.36 10 ⇥ ign ⇥ 7 M .The0.95M model took 4.55 10 yrs before it ignited at a lower Mign =9.05 ⇥ ⇥ 5 10 .Inagreementwithpastliterature(ShenandBildsten,2009a;TownsleyandBildsten,

2004, 2005), the ignition mass required to trigger a classical nova is less with higher MWD due to its higher surface gravity. The equilibrium temperature reached by the core also decreases with higher white dwarf mass, a consequence of accumulating less mass before igniting and the low Mign that will result in less compressional heating of the core. If J1538 experienced a classical nova, the high surface temperature would have been observed during a period of post-outburst cooling. The temperature of J1538 was used as a initial value to calculate the cooling rate over a four year period with the simulated theoretical thermal history, as shown in ﬁgure 1.3. This time constraint was chosen to match the estimated follow-up observation.

11 Figure 1.3: Classical nova cooling timescale for 0.95 M . Cooling curve after undergoing a classical 12 1 nova outburst induced with accretion of rate M˙ = 1.989 10 M yr . After 11,865 yrs the h i ⇥ 1 star cools to 33,855 K and then continues to cool at a rate of 0.774 K yr . The inset ﬁgure focuses on the four-year period used to determine the cooling rate.

The Te↵ calculated in the observation performed by (Pala et al., 2017) occurs 11,865

1 yrs after outburst and cools at a rate of 0.774 K yr . The inset ﬁgure enlarges the period in which the cooling rate was calculated, corresponding to the time of follow-up observation. The 0.6 and 0.8 M models possessed post-novae cooling rates of 0.359 and 1 0.614, K yr with time since outburst of 24,900 and 15,200 yrs, respectively. This produced a constraint on T for follow-up observations, expecting only a modest ( 3K) e↵ ⇠ decrease in temperature over a four year period from a post-outburst white dwarf. Furthermore, the estimated time since nova outburst will facilitate observations of a faint, previously ejected nova shell that would be expected from the classical nova scenario.

12 1.3.3 DWARF NOVAE

Dwarf novae are produced in CVs when the mass transfer from the accretion disk onto the white dwarf happens in short bursts of rapid accretion, separated by periods with little or no accretion onto the star. On long timescales, this corresponds to a time-averaged accretion rate, M˙ , as we have been using in previous sections. As previously mentioned, h i the 0.819 M progenitor was run through multiple novae to establish Tc,eq.Weallowedthe white dwarf to cool to a minimum surface temperature following the 50th outburst. Once cooled, an output photo ﬁle was saved and used as the initial model for the dwarf novae runs. The progenitor was formed this way because any observed object should have already gone through many nova outbursts. This procedure ensures an internal thermal state similar to that expected for such an object. The white dwarf was put through two month periods of high accretion followed by periods of rest where accretion is turned o↵. We ran four cases with recursion times 30, 100, 300, and 1,000 yrs. As mentioned in the previous section, the M˙ used for the h i 12 1 classical novae (1.989 10 M yr )islowerthanexpectedforthisorbitalperiod ⇥ range. Also, this low time-averaged accretion rate does not allow the white dwarf to reach surface temperatures relevant to J1538 ( 33,000 K). Therefore, we chose a rate by using ⇡ MESA to simulate the binary evolution of systems containing a white dwarf (0.6, 0.8, and 1.0 M )anda0.5M secondary star, as seen in ﬁgure 1.4.

13 Figure 1.4: Binary evolution below the period gap. Evolution of 0.6, 0.8, and 1.0 M white dwarfs with a 0.5 M donor. This displays the relation between accretion rate and orbital period below the period gap. The orbital period of J1538 (93.1 min) corresponds to an accretion rate 5 11 1 ⇡ ⇥ 10 M yr .

The binary system was given an initial period of 12 hrs and allowed to evolve for 1010 yrs. The bump at around P 100 min is a computational artifact due to MESA changing orb ' the equation of state table at this point. The transition is revealed due to the short timesteps used in the evolution of the binary. The binary evolves to a period minimum, as detailed in section 1.1, resulting in a gradual increase in period as the accretion rate steadily declines. The relationship shown in ﬁgure 1.4 between MandP˙ orb indicates the

11 average accretion rate for P 90 min, the period of J1538, is approximately 5 10 orb ⇡ ⇥ 1 M yr .Therateusedduringtheaccretioneventstosimulateadwarfnovawas

14 calculated by

M=˙ M˙ (t +t ) t (1.1) h i⇥ rest accr accr .

11 1 Here, M˙ =5 10 M yr ,taccr =twomonths,andtrest is the recurrence time. h i ⇥ This provides a constant time-averaged accretion rate during outbursts and assures the surface temperature reached between accretion events is the quiescent Te↵ (Townsley and G¨ansicke, 2009).

Figure 1.5: Dwarf nova cooling timescale for 0.819 M . Cooling curves for a 0.819 M white dwarf beginning at the time when Te↵ = 33,855 K. The recursion times of 30, 100, 300, and 1,000 yrs between two month periods of rapid accretion were used.

Figure 1.5 shows the cooling curve following Te↵ =33,855Kfora0.8M white dwarf run through two month periods of rapid accretion followed by periods of rest. The rate at

15 which the white dwarf cooled over a four year period was calculated to be 3,525, 3,700, 1 3,100, and 1,925 K yr for recurrence times of 30, 100, 300, and 1,000 years, respectively. Based on these rates, a follow-up observation of the temperature would need to be thousands of degrees cooler for this mechanism of heating to be considered a dwarf novae.

1.4 DISCUSSION

The composition and rate of the accreted material coupled with the white dwarf’s thermal properties provide complications in developing a consistent model for simulating

CVs. It has been demonstrated that the core temperature, Tc,ofthewhitedwarfcan reach equilibrium (where cooling equals heating during outbursts) in a classical novae cycle (Townsley and Bildsten, 2004). Late in the cycle, the accreted layer will grow deeper and compress the bottom layers to higher pressures and temperatures. This slow process, due to the local thermal time still being greater than the time to accrete, allows transfer of heat as the material burns on the surface (Townsley and Bildsten, 2007). We ﬁnd that the presence of 3He in the accreted envelope can change the amount of surface burning, impacting the evolution of novae and the equilibrium temperature of the core. The rate of accretion can also factor into the ignition during classical nova. We address these issues here, but a comprehensive study on their impact will have to be performed to acquire a sustainable model for the evolution of classical novae.

1.4.1 VARIABILITY OF IGNITION IN NOVAE EVOLUTION

A stark contrast in ignition behavior dependent on T and M˙ was discovered while c h i simulating classical novae. We discuss the e↵ect initial core temperature has on white dwarf evolution during classical novae to demonstrate how variation of the ignition can lead to di↵erent outcomes for the surface temperature. This represents an ongoing uncertainty in modeling these systems that will need to be explored more in future work. We compared the ﬁrst few outbursts between 0.819 M white dwarfs with initial core

16 temperatures of T =8 106 K(leftpanels)and9 106 K(rightpanels)inﬁgure1.6. c ⇥ ⇥ 12 1 Both have a time-averaged accretion rate M˙ =1.989 10 M yr .Themodel h i ⇥ possessing the lower core temperature does not transfer enough heat to the core, causing its temperature to drop with each consecutive outburst. However, accretion initiated on a white dwarf with an initial core temperature only 1 106 Kgreaterwillproducealayer ⇥ thick enough to heat the core prior to each nova event.

Figure 1.6: Initial Tc and classical novae evolution. Early evolution of 0.819 M white dwarfs with initial core temperatures of 8 106 K and 9 106 K accreting with a rate M˙ = 1.989 12 1 3 ⇥ ⇥ h i ⇥ 10 M yr .The He in the accreted material burns and depletes before triggering a runaway, resulting in a gradual decrease in core temperature. The higher Tc model stabilized the burning before early ignition, allowing the core to be heated by burning in the surface layer.

17 The dichotomy in the ignition is largely due to the creation of 3He on the white dwarf surface. Hydrogen fusion at low temperatures will produce 3He without consuming it. As the accreted layer increases in depth, the base temperature and density will increase enough that this 3He will begin to burn, increasing the energy release. This rise in the energy generation can, depending on the circumstances, lead to either a full nova outburst or just add additional heating before the outburst. The lower Tc progenitor experienced an early ignition that did not generate enough heat to raise the core temperature, causing it to steadily decrease with every outburst. The 9 106 Kwhitedwarf,however,underwent ⇥ 3 heating following the ﬁrst nova, gradually increasing Tc.Previousworkonthee↵ectof He on ignition has shown that even with accretion of solar abundance material, the presence of 3He can lead to an early ignition (Shen and Bildsten, 2009a) . This results in a shorter heating phase that leads to a lower Tc,eq (Townsley and Bildsten, 2004). Additionally, they established the ignition mass is larger at lower values of T . Although the 8 106 Kwhite c ⇥ 3 dwarf should have a higher Mign,the He burning triggers the nova outburst earlier, as can be seen comparing the left and right panels of ﬁgure 1.6. The absence of the slow increase in surface temperature before the nova, which is seen in the top left panel, suppresses the transfer of heat to the core, allowing it to cool.

18 Figure 1.7: Accretion rate and classical novae evolution. Temperature history for multiple outbursts through ﬁrst 108 years of evolution. A 0.819 M progenitor was used with accretion rates 9.46 12 1 12 1 ⇥ 10 M yr (top) and 9.5 10 M yr (bottom). The higher rate prevents early the ⇥ ignition of 3He that results in the initiation of an outburst as seen in the top ﬁgure.

19 Figure 1.7 o↵ers analysis to the e↵ect of 3He with varying accretion rates. We have

3 shown that at a low accretion rate and low Tc,thepresenceof He can lead to an early thermonuclear runaway resulting in a classical nova. Expanding on this, we ran the lower

12 11 1 Tc model through accretion rates from 10 10 M yr . Within this range, we discovered rates for which the initiation of 3He burning always triggers the full runaway

12 1 and outburst with mass ejection. Time-averaged rates between 2 10 M yr M˙ ⇥ h i 12 1 3 9.5 10 M yr undergo these pulses brought on by He, resulting in a decreasing  ⇥

Tc,eq. To show this, ﬁgure 1.7 compares temperature histories for 0.819 M with accretion 12 1 12 1 rates 9.46 10 M yr (top panel) and 9.5 10 M yr (bottom panel). The ⇥ ⇥ 3 higher rate experiences an early rise of Te↵ due to He burning, but accumulates enough material to stabilize and prevent early ignition. The resulting evolution of both classical and dwarf novae presented in sections 1.3.2 and 1.3.3, respectively, were impacted by this. The accretion rate obtained to produce the quiescent, median Te↵ found in the survey of white dwarfs (15,216 K) was much lower than expected. This result leads us to believe that the onset of 3He burning should lead to a thermonuclear runaway and nova outburst. The average accretion rate experienced by CVs

11 1 below the period gap (5 10 M yr ), would then be capable of producing the median ⇥ surface temperature observed, adhering to predicted evolutionary theory. A more thorough study will be required to address these concerns present in the current model for simulating novae.

20 1.5 CONCLUSION

Cataclysmic variables o↵er a wealth of information on how white dwarfs behave in interacting binaries. Studying CVs will establish a deeper understanding of how accretion and angular momentum losses both impact the evolution of white dwarfs and the system as awhole.TherestillremainsunansweredquestionsregardingtheimpacttheMWD, M,˙ and

Tc has on the Te↵ and Porb of CVs. This work o↵ers some insight into these broad questions through our research in providing a theoretical basis for the high surface temperature obtained from the white dwarf in J1538. Using MESA,wesimulatedbothclassicalanddwarfnovaefor0.6,0.819,and0.95M progenitors. These were evolved through multiple novae, allowing the core to reach an

1 equilibrium temperature, Tc,eq.Wecalculatedapost-novacoolingrateof0.774Kyr for the 0.95 M progenitor, the mass closest to the actual white dwarf in J1538. The cooling rate suggests a very small drop in temperature will be measured upon follow-up observation, which was confirmed with the four year temperature dropping only 2K. ⇡ We also performed a statistical analysis to ascertain the probability that a CV white dwarf would have this temperature during its evolution by comparing the theoretical thermal history obtained from the simulation with data from the HST survey by (Pala et al., 2017). Despite the small sample size, there remains a possibility the temperature observed from J1538 would be observed under favorable circumstances. The time since the nova outburst, as predicted from our simulation in section 1.3.2, provides observers a constraint to use in finding the remains of a classical nova shell. This discovery would all but verify a classical nova as the cause of the high Te↵,providingavaluablesystemtostudynovaetheoryand would be one of only a few where direct measurements of a post-nova white dwarf mass was obtained (Shara et al., 2017; Starrfield et al., 2016).

21 2

SUPERNOVAE

2.1 INTRODUCTION

Type Ia supernovae (SNeIa) are thermonuclear explosions that involve a carbon-oxygen (CO) white dwarf in an accreting single- or double-degenerate binary system. These highly luminous phenomena are powered by the radioactive beta decay of 56Ni 56Co 56Fe (Colgate and McKee, 1969). The half-life of 56Ni and 56Co are 6.1 and ! ! 77.7 days, respectively. Due to the nature of these explosions, the light curve they produce follows a general pattern that, when used with the luminosity-decline rate relation, provides very accurate distances out to z 1(Perlmutteretal.,1999;Riessetal.,1998). ⇠ The high luminosity and small dispersion in peak magnitude associated with SNeIa have been useful in determining the linearity of the Hubble parameter at low redshift (z 0.1)  and the expansion of space at high redshift (z 0.3 1) (Filippenko, 2005). The classic, single-degenerate model for type Ia supernovae consists of a white dwarf accreting from either a H or He donor until it reaches MCh 1.4M , the Chandrasekhar mass. When this ⇡ critical value is reached, electron degeneracy pressure in the core can no longer support the star against further gravitational collapse. Temperature and density will increase as the star contracts, resulting in runaway of nuclear fusion in the core. The double-degenerate model involves a merging CO white dwarf with either a He or CO white dwarf. These double-degenerate systems can either detonate when MCh is exceeded or by means of a double-detonation at sub-Chandrasekhar masses.

22 The sub-Chandrasekhar double-detonation scenario involves a CO white dwarf accreting He-rich matter, forming a helium envelope that surrounds the star. The white dwarf will accumulate matter until a critical mass is reached in the helium shell, causing a thermal instability at its base where 4He will burn to form 12Cviathetriple-alphaprocess (Townsley et al., 2016). Figure 2.1 illustrates a simulated double-detonation, displayed in temperature and density plots. Due to the high temperatures and pressures, once formed, 12Cwillimmediatelycapture4He and create heavier elements such as 44Ti and 48Cr, up to 56Ni. The detonation propagates along the surface, burning shell material until it converges at the opposite pole. A shock wave is generated at the detonation front that travels inwards, compressing carbon-oxygen material to a focused point that causes a second detonation. The shock front produced by the 12Cdetonationtravelsfromthepointof ignition outward through the entire star.

Figure 2.1: Time sequence of helium shell detonation. Temperature and density evolution of helium 5 3 shell detonation and propagation for progen2. Density contours above 10 gcm are shown in gray, emphasizing the inward traveling shock. Times displayed from top left are t = 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 s.

23 The double-detonation scenario is the focus of this work, specifically for thin helium shell white dwarfs that produce spectra resembling that of type Ia supernovae. This expands on the research previously performed (Townsley et al., 2019) by incorporating new strategies for the refinement, particle distribution, and limitations on nuclear burning. Double-detonation hydrodynamic simulations utilizing a sizable nuclear reaction network incur considerable computational cost. This requires developing techniques to increase eciency while maintaining the physics of the simulations. This work implements new procedures for double-detonation simulations while performing case studies within the parameter space to improve functionality and eciency. Section 2.2 describes setup procedures for the simulations performed using FLASH1.Weprovidedetailofthe improvements made; a de-refinement condition (section 2.2.1), particle distribution (section 2.2.2), and burning limiter (section 2.2.3). Section 2.3 gives the results from these additions and of two case studies performed to ascertain the proper procedure for a successful detonation. Section 2.4 presents the two successful double-detonation supernovae simulations with the procedural additions and details our final results. We o↵er our conclusions to this chapter in section 2.5.

2.2 SIMULATION SETUP

The details regarding the setup procedure, running the simulation, and new methods will be described in this section. The software and procedures utilized were nearly identical to those employed by (Townsley et al., 2019), but with the notable additions of a de-reﬁnement condition, new particle distribution, and burning limiter (Shen et al., 2018). We used FLASH to simulate the ﬂuid dynamics for a double-detonation supernova of a white dwarf progenitor. A logarithmically spaced grid along with the “new” multipole gravity solver is used. We establish a domain size large enough ( 8.8 1022 cm2)toprovidea ⇡ ⇥ full account of the ejected material. Outside the star, the grid is coarsened and a

1http://ﬂash.uchicago.edu, version 4.3

24 Lagrangian scalar is used to distinguish between stellar material and the surrounding

5 3 7 low-density (10 gcm )mediumpossessingauniformtemperatureof3 10 K. This ⇥ surrounding medium, or flu↵, was set to a lower density than in previous works (Townsley et al., 2019) in order to prevent the reverse shock from a↵ecting the ejecta. This would alter the final velocity distribution of material within the ejecta. The adaptive mesh grid is set to provide 19 levels of refinement (the 19th being 1 km resolution). The refinement structure used follows the strategy of (Townsley et al., 2009), which refines the mesh based on gradients in density, temperature, and abundance (4He and 12C). FLASH allows the refinement levels of specific regions to be designated in the initial parameter file of the

18 1 simulation. The reﬁnement threshold for nuclear energy generation was set to 10 erg g

1 s , permitting any cell containing values above this to refine at the maximum level allowed. We chose to refine energy-generating regions to 4 km resolution and non-energy-generating regions to 64 km in order to resolve regions of burning and allow those with no burning to coarsen. We designated an inner region, determined by the radius of the progenitor, that keeps the entire star refined to 32 km resolution. Any grid-space beyond this radius is allowed to de-refine to the non-energy-generating level. The core-shell interface is also fully resolved by refining a cell if the values of both 4He and 12C within the cell exceed 0.05 and 0.07, respectively. All 8 8cellblockswithmaximumabundances ⇥ less than two percent under these values will not fully refine. The refinement criteria was subject to change depending on a progenitor’s diculty to ignite (section 2.3.3), but generally follows a consistent pattern throughout all simulations. Once the helium shell detonation has propagated around the entire surface of the star, an interior focused region is refined to maximum resolution where the approaching shock front converges, ensuring the secondary detonation is well resolved. Upon carbon ignition, we relax the refinement of the focus region to that of the rest of the star. Once the shock propagates to the edge of the white dwarf, the maximum refinement is held to 1024 cells in radius, allowing the cell size to steadily increase as the star expands.

25 Nuclear burning in FLASH is calculated by integrating a set of di↵erential equations representing the abundances of the isotopes, which is then used to update the hydrodynamical variables (Timmes, 1999). The FLASH nuclear reaction capacity was ampliﬁed using a 55-isotope reaction network from MESA,shownintable2.2.Thenuclear reaction network is expanded to generate more realistic energy release via nuclear burning (Shen and Moore, 2014). The network yields only the energetics during the simulation, requiring the inclusion of tracer particles for accurate abundances. The simulation includes approximately 100,000 tracer particles that are used to post-process the abundances of the supernova ejecta. The temperature and density histories for each particle are logged and used to construct a mass-abundance proﬁle that is integrated to calculate the yields. The post-processed results are not presented in this work, but more information regarding the process can be found in (Miles et al., 2019). We used four white dwarf progenitors for the double-detonation simulations. The simulation begins by igniting the helium shell with a spherical hot spot on the symmetry axis (+y axis) at the base of the helium layer. The spot has a central temperature of 2 109 Kandedgetemperatureof8 108 Kwithasize200 105 cm, unless otherwise ⇥ ⇥ ⇥ stated. A band for tracking the shock front as it propagates along the surface is placed a few 100 km above the helium layer base. The mass, density, temperature, and composition of the helium shell and core for each progenitor are displayed in table 2.1.

26 Table 2.1: Parameters for white dwarf progenitors. progen1, progen2, and progen4 contain 0.009 14Nintheheliumshell,progen3 contains 0.0. The base temperature of the He-shell is 3 108 K ⇥ for progen1 and progen3, and 5 108 K for progen2 and progen4. ⇥

Parameters for White Dwarf Progenitors

region mass base density 4He 12C 16O 22Ne

7 3 core 0.817 M 1.35 10 gcm - 0.4 0.58665 0.01335 progen1 ⇥ 5 3 He-shell 0.033 M 2 10 gcm 0.891 0.05 0.05 - ⇥ 7 3 core 1.0 M 3.78 10 gcm - 0.4 0.59 0.01 progen2 ⇥ 5 3 He-shell 0.021 M 2 10 gcm 0.891 0.05 0.05 - ⇥ 7 3 core 1.0 M 4.03 10 gcm - 0.5 0.5 - progen3 ⇥ 5 3 He-shell 0.3 M 7 10 gcm 0.891 0.25 0.25 - ⇥ 7 3 core 1.0 M 3.8 10 gcm - 0.4 0.58665 0.01335 progen4 ⇥ 5 3 He-shell 0.021 M 2 10 gcm 0.926 0.05 0.015 - ⇥

27 Table 2.2: Nuclear reaction network. The reaction network was used for double-detonation simulations.

Nuclear Reaction Network

1 33 35 H Hydrogen-1 Cl Chlorine-33,34,35

3 4 36 39 He Helium-3,4 Ar Argon-36,37,38,39 11BBoron-1139Kr Krypton-39

12 13 40 CCarbon-12,13Ca Calcium-40

13 15 43 N Nitrogen-13,14,15 Sc Scandium-43

15 17 44 OOxygen-15,16,17Ti Titanium-44 18F Fluorine-18 47VVanadium-47

19 22 48 Ne Neon-19,20,21,22 Cr Chromium-48

22 23 51 Na Sodium-22,23 Mn Manganese-51

23 26 52 Mg Magnesium-23,24,25,26 Fe Iron-52

25 27 56 Al Aluminum-25,26,27 Fe Iron-56

28 30 55 Si Silicon-28,29,30 Co Cobalt-55

29 31 56 P Phosphorus-29,30,31 Ni Nickel-56

31 33 58 59 SSulfur-31,32,33 Ni Nickel-58,59

28 2.2.1 REFINEMENT

FLASH’s modular capabilities allow the user to enact custom controls to the refinement conditions, improving the accuracy and computational eciency of the simulation. Refinement control can allow for maximum refinement in high energy-generating regions, to ensure burning regions are fully resolved. Here, we implement a control to allow de-refinement in blocks along the core-shell interface that trail the shock front and reaction zone. The condition maintains the fully resolved shock front, but coarsens cells behind it that no longer contribute to its propagation. One of the goals of this improvement was to learn how to modify the FLASH code and gain the knowledge required to implement the particle distribution and burning limiter. We began by determining the maximum angle (theta) from the symmetry axis that the shock front has traveled. The position of the shock is a global quantity that is calculated in the FLASH subroutine writeIntegralQuantities, which also determines quantities including total energy, density, and velocity. A tracking band is placed to detect cells along the band radius that exceed 7 108 K, temperatures that are indicative of a ⇥ detonation front. The routine loops through all cells meeting this criteria and finds the maximum polar angle of shocked cells, the shock front. We define this angle sim theta and store its value in the simulation data where it’s accessible. We then added the new refinement condition to FLASH’s markRefineDerefine subroutine, which determines when and where the grid refines or de-refines a block. A block represents a sub-domain of 8 8 ⇥ computational cells that hold local physical information. First, the maximum angle for the grid blocks are calculated by

max theta = 90.0 180.0/⇡ arctan((Box(1,JAXIS)/Box(2,IAXIS)) (2.1) ⇥

max theta = 90.0 180.0/⇡ arctan((Box(1,JAXIS)/Box(1,IAXIS)) (2.2) ⇥

Where Box(1,2;I) and Box(1,2;J) are the bottom and top, left and right edges of a box,

29 respectively. This procedure determines the angle of the bottom right corner or bottom left corner of a block if located above (Eqn. 2.1) or below (Eqn. 2.2) the x-axis, respectively. Figure 2.2 is a visual representation of the grid blocks above and below the x-axis used to determine the maximum angle of the shock front.

Figure 2.2: Shock angle determination. Diagram of the maximum angle for grid blocks located at the shock front above (green) and below (blue) the x-axis. The orange semi-circle represents the star and the green and blue circles denote the box edges used in equations 2.1 and 2.2, indicating the location of a shock front above and below the x-axis.

A parameter, refine theta, is assigned to designate the angular position behind the shock which will be allowed to de-refine; shock theta = sim theta refine theta. Here, shock theta is the angle from the +y axis along the core-shell interface that will de-refine. With the maximum angle of each block and the maximum angle of the shock front known, we impose the boolean operator max theta < shock theta, which de-refines all blocks that satisfy the condition. This condition was tested using white dwarf progen2 and the results are detailed in section 2.3.1.

30 2.2.2 PARTICLE DISTRIBUTION

The motivation behind improving the particle distribution was to eliminate the need for binned wedges employed by (Townsley et al., 2019) and to account for under-sampled tracer particles in the outer edge of the ejecta. Although their method proved sucient, for further work the distribution should be divided into a 2-dimensional velocity grid without the need for wedges. In order to accomplish this, the particles need to be spaced such that the outer edge of the ejecta will have a finer particle distribution. The roughly 100,000 tracer particles included in the simulation track the temperature and density histories of the fluid. These particles are passive and have no e↵ect on the hydrodynamics of the simulation during the double-detonation, but are crucial to radiative transfer in post-processing. The previous particle distribution weighted each cell from the product of the cell’s volume and density. The total weight of each block is then just the sum of the weights of the cells. The global sum of block weights were normalized for use of a random number to place particles into blocks. This places the amount of particles determined by mass weighting into their respective blocks. The process is repeated to determine the number of particles in each cell, again depending on mass weighting. The new procedure was developed by Dr. Broxton Miles, a postdoctoral researcher at NC State University, for use in simulations of pair instability supernovae. It distributes particles into three zones: inner, outer, and polar. Inner mass is denoted by any mass located inside the outer edge of the core, designated by the parameter mdiv radius. Outer mass is any mass within a block that is located outside the mdiv radius. Polar mass is a subset of the outer mass and is located outside the mdiv radius within 10 degrees of the poles, which allows for a finer mass resolution in the region near helium ignition. The boundary between zones and fraction of particles in each zone are controllable parameters, allowing the distribution to be used on di↵erent progenitors. The routine loops through all blocks on each processor to calculate each block’s position and total mass. The amount of mass contained in each region (inner, outer, and polar) is also calculated and contingent on

31 the initial parameters chosen. If a block is fully contained inside a region, the number of particles for that block is just the total block mass divided by the mass weighting of particles in that region. The regional mass weighting is calculated by

total mass inner mdiv inner mass weight = ⇥ (2.3) inner part split num. particles wanted ⇥

total mass (1.0 inner mdiv) total polar mass outer mass weight = ⇥ (2.4) outer part split (1 frac polar) num. particles wanted ⇥ ⇥

total polar mass polar mass weight = (2.5) outer part split frac polar num. particles wanted ⇥ ⇥

Here, inner mdiv is the fraction of total mass within a chosen radius, inner part split is the fraction of particles inside the radius, outer part split is the fraction of particles outside the radius, and frac polar is the fraction of the outer particles at the poles. We chose values of 0.98, 0.80, 0.20, and 0.10 for each value, respectively, which provides suitable particle representation in all regions. A random number generator is used on each processor to distribute the correct number of particles to the grid according to their respective mass fractions and total mass with

total mass number of particles = (2.6) mass weight

Blocks that are located on the inner-outer boundary radius are handled by evaluating the location of each cell in the block with respect to the boundary. The center position of each cell is determined and its cell mass is added to the region (inner, outer, or polar) where the center is located. The total number of particles located within these boundary blocks is then determined by summing the number in each region based on their mass weighting.

32 The amount of mass each particle represents is then

total mass mass fraction mass per particle = ⇥ (2.7) particle fraction total number of particles ⇥

The implementation of the particles required augmenting FLASH particles initialization to distribute the particles by setting the new routine to override existing defaults. At the end of each hydrodynamic and nuclear burning timestep, the particle positions are integrated using second-order Runge-Kutta with the velocity ﬁeld, producing Lagrangian ﬂuid histories (Townsley et al., 2016). Abundance yields during the supernova are computed by integration of the tracer particles temperature and density histories acquired during the simulation.

Figure 2.3: Final particle positions. Position of particles in domain space at time t = 20 s for progen2. Inner region (red), outer region (blue), and polar region (dark blue). The ejecta outer edge is designated by the red line and gray velocity contours for 10, 20, and 30 103 km s 1 are ⇥ labeled.

33 Figure 2.3 displays the particles at 20 s following the double-detonation supernova for model progen2.Theinner,outer,andpolarregion’sparticlesaredesignatedbyred,blue, and dark blue, respectively. Gray velocity contours are labeled to designate 10,000, 20,000,

1 1 and 30,000 km s (black contours represent increments of 5,000 km s ). A much ﬁner

1 distribution of particles exists beyond 20,000 km s which encompasses the majority of heavier elements produced by the supernova (as seen in ﬁgures 2.11 and 2.13). The radiative transfer performed by (Townsley et al., 2019) su↵ered from under-sampled particles in the outer velocity bins of the ejecta as a result of equally massed tracer particles. This new particle distribution o↵ers better particle representation post-ejecta, allowing for more precise 2-dimensional radiative transfer calculations.

2.2.3 BURNING LIMITER

The intent of the burning limiter was to improve the treatment of nuclear energy release and its role in the detonation. Using a limiter allows the propagating shock front to be spatially resolved. Carbon detonations have length scales of 10 104 cm at the densities pertinent to the progenitors used, so we apply a burning limiter to artiﬁcially broaden the detonation front, allowing the interaction between the burning and shock to be resolved (Miles et al., 2019; Shen et al., 2018; Townsley et al., 2016). Reactions that occur in unresolved regions behind the shock must be accounted for to prohibit numerical di↵usion from over-inﬂuencing the reaction front (Fryxell et al., 1991). Any unresolved regions will create unphysical interactions between the shock and reaction zone (Townsley et al., 2016). Thickening the detonation front with the inclusion of the burning limiter increases the carbon-consumption length scale and allows maximum energy generation to be further behind the shock front. FLASH calculates hydrodynamics and reactions in an operator split approach. This method splits the original di↵erential equation in order to solve the hydrodynamic and nuclear reaction equations separately. The computations can then be performed using two di↵erent timesteps, thydro and tburn, the hydrodynamic and burning timesteps, respectively.

34 In order to artiﬁcially broaden the detonation front and keep it resolved, a limit is placed on the energy generation produced during the nuclear reaction timestep tburn.Fromthe second law of thermodynamics under constant volume (S = cv ln(T/T0)), the temperature change within a cell is related to the burning timestep by

lnT ✏¯ t c T(2.8) ⇠ · burn · v ·

where✏ ¯ is the average energy generation rate during tburn and cv is the speciﬁc heat at a constant volume (Kushnir and Katz, 2014). Introducing a limit to lnT in a cell allows the burning timestep to be reduced so that the burning is resolved in time without changing the overall hydrodynamic timestep. The limit (hereafter, Shen),

dlnT de =( ) c T(2.9) limit ⇢ max · v · determines the amount the temperature can change within a cell in one timestep during nuclear burning. Unless otherwise stated, a value of 0.1 was chosen for delimit for all double-detonation simulations. The procedure, originally implemented in (Miles et al., 2019; Shen et al., 2018), e↵ectively integrates the reactions for the full hydrodynamic timestep, thydro, but with reaction rates truncated by tburn/thydro.Thisbroadensthe detonation and resolves the shock front as desired. This work amends the burning routine by posing the limit as the energy release allowed during a sound crossing time, tsc, of a cell. While the previous form restricted the energy rate to tlimit/thydro,theuseoftsc makes the energy release rate independent of the choice of thydro.Thisisdesirablebecause,althoughthydro is typically similar to tsc,thydro is determined globally and therefore may not meet this expectation in some cases. Since the detonation is moving near the sonic speed, limiting the energy generation in this way will broaden the energy release in such a way that it is spatially resolved.

35 The new limit (hereafter, New), is calculated by

dlnT thydro delimit,new =( )max cv T (2.10) ⇢ · · · tsc

P where tsc =del/ c ⇢ , del is the global minimum block size, c =(@ ln P/@ ln ⇢)ad, P and ⇢ are pressure andq density, respectively. Here, the sound crossing time is computed using the local sound speed and velocity, but with the finest global grid scale which prevents coarser cells from having a smaller limit than finer cells. The reaction rate is obtained by utilizing a Newton-Raphson iteration that determines the dtburn for which the amount of energy generated in the local sound crossing time has rate delimit,new.TheShen burning limiter began with a guess of thydro for the root find (Shen et al., 2018). To reduce the time it takes to iterate, we started with a small guess time, dtguess,andlinearlyextrapolatedbased on times t = 0 and dtguess and energies de = 0 and the rate determined by dtguess.Blocks

9 with T < 10 Kareallowedtoguessthefullthydro as their burning is negligible. The idea is to reduce the original guess timestep to allow energy-generating regions to converge on the energy limit quicker, as dtburn will be smaller in these regions. The artiﬁcial nature of the limiter can present complications in the ignition and propagation of the detonation, which is discussed section 2.3.2, but ultimately provides a successful means of broadening the detonation front.

36 2.3 RESULTS

The de-refinement condition was first to be added as it only required minor alterations to the current grid refinement conditions in FLASH. As stated in section 2.2.1, the intention was to coarsen the grid on the core-shell interface in regions far enough behind the shock front where high levels of refinement are unnecessary. This required use of the position of the shock front, a quantity FLASH calculates. To apply this information, we chose to define an angular position behind the shock beyond which the grid will be allowed to de-refine. An angle of 30 degrees was chosen for the simulations performed in this work, allowing the reaction zone located behind the detonation front to remain fully refined. The results of the de-refinement condition are detailed in 2.3.1. The burning limiter was next to be added, requiring careful consideration of the limiting value to ensure successful propagation. We parameterize the limiter to explore the parameter space and control the energy generation rate similar to the study in (Shen et al., 2018). The results from the burning limiter test are in section 2.3.2 and resolution studies on ignition are detailed in section 2.3.3.

2.3.1 DE-REFINEMENT STUDY

Figure 2.4 displays the result due to the de-reﬁnement condition described in section 2.2.1. The condition is implemented at 0.535 s and assigned to de-reﬁne for blocks greater than 10 degrees behind the shock front that do not exceed the energy generation threshold

18 1 1 of 10 erg g s . This energy is not reached except in the detonation front and therefore de-reﬁnement is permitted in all blocks in this zone. The total blocks reduce from 5708 to 4758, a 17% reduction. Although not an overly signiﬁcant e↵ect on cost (a timestep change of 55 s to 49 s, 11% faster per step), it provides scalable functionality that should have a ⇡ greater e↵ect on thicker shells.

37 Figure 2.4: De-refinement test. Time series showing de-refinement behind helium shell detonation front for white dwarf progen2. Colors denote the log grid resolution near the shock front at 0.50 s and 0.65 s after ignition. In the figures, the darker blue is 128 km, lighter blue is 64 km, salmon is 32 km, and the shades of red lightest to darkest is 16, 8, and 4 km, respectively. The de-refinement condition was set to ten degrees at 0.535 s.

2.3.2 BURNING LIMITER STUDY

We tested implementing the burning routine on progen3,a50/50COwhitedwarf with a 1.0 M core and 0.03 M shell. The progenitor was chosen for the ease of initial ignition due to its highly enriched helium shell (see table 2.1). The white dwarf was ignited using a hotspot placed at the base of the helium layer and run, with the default FLASH burning routine, for 0.3 s. We then restarted the simulation with the New limiting

6 procedure detailed in section 2.2.3. We set our initial timestep guess dtguess =10 sand limiting parameter to 0.01.

38 Figure 2.5: Burn limiter test. Burn routine test with limiting parameter set to 0.01 for white dwarf progen3. The detonation was run to 0.3 s with the default FLASH burning routine, then restarted and run to 0.5 s with the New limiter. The inclusion of the burning limiter caused the detonation to fail due to the reaction zone uncoupling and lagging behind the shock front (denoted by the white arrow in right panel).

Figure 2.5 shows the temperature-density and nuclear energy plots of the helium shell while propagating along the surface of the white dwarf. The limiter has a small initial guess timestep and the inclusion of the sound crossing-time when determining energy generation rate. There are clear signs the detonation is failing to propagate along the helium shell. The reaction zone has uncoupled from the shock front, no longer fueling the detonation forward. This can be seen in the temperature-density plot that has cooled at the shock front and in the nuclear energy of ﬁgure 2.5 where the burning (darkest red) is disconnected from the front of the shock (made visible by the white arrow in the right panel). The thydro/tsc ratio was calculated to determine the New limiter’s e↵ect on the energy generation rate. A ratio of 0.1 was determined, indicating that by introducing ⇡ the tsc dependence, the burning is operating at a rate ten times smaller than is allowed with Shen. The burning rate is too slow to successfully propagate the detonation and to resolve this, we increased the limiting parameter by a factor of ten, from 0.01 to 0.1. Figure 2.6 is the result from adjusting the parameter and displays the e↵ect on the nuclear energy generation in the burning front compared to a detonation with no limiter. The increase to the burning rate preserves the structure of the detonation and successfully

39 propagates the shock front forward. Here, we compare with a case without the burning limiter and highlight its true intent, to broaden the detonation front.

Figure 2.6: New burn routine. Comparison of nuclear generation during helium shell detonation at 0.5 s for no burning limiter (left panel) and the New limiting routine (right panel). White dwarf model progen3 was used in both runs. The detonation front is broadened in the case with the burning limiter, resolving the shock front.

With the successful propagation of the detonation front using the New limiter, time tests were performed to measure any improvement to computation speed resulting from the smaller initial guess. As the detonation propagates around the shell, convective mixing between the core-shell interface changes the burning processes present for di↵erent times. 12Cand16O burning will become more prevalent during the core detonation phase, which has an impact on the reﬁnement level and energy generation. The burning timestep in these regions is small and an initial dtguess less than dthydro will converge on the correct limited energy rate quicker. This suggests the New limiter would prove more e↵ective during core burning. We compared the New limiter to the Shen limiter during this stage and found the time per step improved from 630 s to 378 s. The New limiter, using a smaller dtguess,provideda40%reductionintimebetweenstepscomparedtotheShen limiter.

40 2.3.3 RESOLUTION STUDIES ON IGNITION

HELIUM SHELL IGNITION

The refinement level chosen for cells undergoing energy generation is an important parameter that decides the success or failure of a detonation. In a double-detonation scenario, there are two stages in which careful consideration of resolution is necessary: the initial helium shell ignition and the shock-generated carbon detonation. Successful ignition of the helium layer requires initial conditions favorable to produce a resolved detonation front capable of propagating along the shell. Extensive research has been performed to determine the parameters needed for a successful detonation (Shen and Moore, 2014). Initially, it was assumed ignition occurs when the local helium burning timescale becomes less than the local dynamical timesecale (Bildsten et al., 2007; Shen and Bildsten, 2009b). However, this assumes the timescale for expansion is the sound crossing time for the scale height of the entire shell when in actuality the timescale is much shorter due to the detonation occurring in a very small region. As demonstrated in (Shen and Moore, 2014; Townsley et al., 2019), the addition of a full reaction network and a helium shell enriched with 12Cand16Oleadstoaboostintheheliumburningrateandcanproduceasuccessful detonation. The ignition of the helium shell for the 0.85 M white dwarf progen1 is shown in figure 2.7. We compare di↵erent levels of refinement for energy-generating regions during ignition with all other ignition parameters held constant.

41 Figure 2.7: Ignition refinement test. Ignition of progen1 with a refinement level for energy- generating regions of 4 km (top panel) and 1 km (bottom panel). Both are run with the New limiter and ignited with a 500 105 cm spherical hotspot. The lower level of refinement failed to ⇥ ignite due to the detonation front being unresolved.

42 Studies of white dwarf ignition performed by (Pakmor et al., 2012) and (Rosswog et al., 2009) asserted that resolution during ignition plays an important role in energy production. Hotspots that are under-resolved spread the energy generated over a larger area and prevent the critical temperature for ignition from being reached. Figure 2.7 shows both a failed (top panel) and successful (bottom panel) helium shell ignition for a 0.85 M white dwarf. Both runs included an inner region (out to 8 108 cm) that is reﬁned to 32 ⇥ km and non-energy-generating regions set to 64 km. A 500 105 cm spherical hotspot was ⇥ placed at the base of the helium layer with a central temperature of 2 109 Kandedge ⇥ temperature of 8 108 K. The top panel was run at 4 km resolution for energy-generating ⇥ regions while the bottom panel was run at 1 km resolution. The under-resolved ignition failed to trigger a burning front with a sustainable reaction zone. The energy generated in the under-resolved ignition is smoothed over larger grid cells that diminish the peak temperature and stop the detonation from propagating. Furthermore, the New burning limiter was used, which factors heavily in the energy generation rate and therefore success of an ignition. Lower resolution means larger cell size, which in-turn gives a longer sound crossing time. A larger tsc will yield a slower burning rate and cause less energy to be deposited into the burning front. The result is a failed ignition that will weaken without propagating as shown in ﬁgure 2.5.

CARBON IGNITION

The second detonation phase occurs o↵the axis of symmetry in the southern hemisphere, inside the core. It is triggered by the inward moving shock front spawned from the propagating helium shell detonation. The shock converges inside the star, raising temperatures to several 109 Kcausing12C ignition. As detailed in section 2.2, a focused region of 1 km resolution is placed at the location of convergence to facilitate detonation of the core material. Upon successful ignition, the focused region is relaxed back to the previous level of reﬁnement (4 km for energy-generating regions, 32 km in the interior) as

43 the detonation front will be fully resolved at 4 km (Townsley et al., 2009). However, the amount of de-refinement allowed in a timestep can have an e↵ect on the outward propagation of the carbon detonation. Figure 2.8 compares the propagating detonation following carbon ignition for progen2 when initiating a one-time decrease in maximum refinement. We compare the resulting carbon detonation when allowing the grid to reduce by 3 refinement levels (top panel) and 1level(bottompanel).Thecellsizechangesbyafactorof2n,wherenisthedi↵erence between levels (i.e. 3 levels of refinement is a factor of 8 in cell size). We initially allowed the grid to attempt a large change in refinement level, which resulted in a detonation that failed to propagate radially outward from the ignition point. The reason is similar to that of the failed helium shell detonation addressed in the previous section. When the grid attempts to coarsen too much at one time, the detonation structure becomes disrupted. Smoothing out the structure of the propagating detonation to a high degree causes the shock front to uncouple from the carbon burning. The ignited, shocked material (light blue) within the core has separated from the burning (red), halting its propagation as seen in the top panel of figure 2.8. The logical solution to this was to change the refinement pattern to try refining only one level at a time. This allows the detonation front to re-equilibrate after traveling across a cell, permitting further grid coarsening to occur without decoupling the burn front. With this change, the detonation can propagate through the stellar interior toward the surface unimpeded as seen in the bottom panel of figure 2.8.

44 Figure 2.8: Carbon detonation test. Resolution study of carbon detonation allowing changes in refinement at di↵erent levels for progen2. The top panel displays a change of 3 refinement levels at a time while the bottom panel tries 1 refinement level per step. The carbon detonation failed to propagate when refinement was changed too much in a timestep.

45 2.4 DISCUSSION

The advancements in the reﬁnement pattern, particle distribution, and burning limiter allow for a multi-dimensional double-detonation simulation similar to that performed in (Townsley et al., 2019), but with an improved particle representation, computational cost, and realistic burning. The base of the helium shell is ignited, forming a detonation front that burns through the shell, around the surface, towards the opposite pole. As the detonation front propagates, an inward-moving shock front develops that compresses the core material for an o↵-center detonation. Figure 2.9 is a time sequence of the helium shell detonation as it traverses the surface

3 of the star. The logarithmically spaced gray density contours (g cm )accentuatethe inward shock, which increases the temperature enough to detonate the core material. Each black contour represents a factor of ten density intervals per decade. The compression shock converges just after 2.5 s. The detonation continues outward through the white dwarf and helium shell, ejecting the burnt core material. The simulation is run to 20 s, allowing the asymmetric ejecta to continue to expand toward the edge of the grid domain. The asymmetry is due to the single hotspot ignition at the northern pole and the o↵-center ignition during core detonation. The successful carbon detonation occurs roughly 2.6 108 ⇥ cm south of center shown in ﬁgure 2.10. The focused region, 1.0 108 cm by 0.6 108 ⇥ ⇥ cm, is reﬁned to 1 km resolution and placed just ahead of the shock to ensure the convergence is fully resolved.

46 Figure 2.9: Evolution of helium shell detonation for 1.0 M . Temperature and density evolution 5 3 of helium shell detonation and propagation for progen2. Density contours above 10 gcm are shown in gray, emphasizing the inward traveling shock. Times displayed from top left are t = 0.25, 0.75, 1.0, 1.5, 1.75, and 2.25 s.

47 Figure 2.10: Lead-up to carbon detonation. The region is focused to maximum reﬁnement of 1 km prior to shock convergence, resulting in core ignition just after 2.5 s for progen2.

48 Figure 2.11: Final abundances of ejecta from double-detonation. Ejecta distribution of density and select species 20 s after helium shell ignition for progen2. Velocity contours in black with units 103 1 km s . The red line represents the boundary between the stellar material and low-density ﬂu↵. The majority of 44Ti and 48Cr was created during helium shell detonation near the ignition point due to favorable conditions for alpha capture.

49 Figure 2.11 shows the distribution of select species, 56Ni, 28Si, 40Ca, 44Ti, 48Cr, and density 20 s after ignition for progen2. As the ejecta propagates through the grid space, it strikes the ﬂu↵boundary (red line), which shocks and slows the ejecta’s descent further into the ﬂu↵. The asymmetry of the double-detonation is especially noticeable in the ejecta distribution. 56Ni is produced in high quantities during carbon detonation with very little

56 1 produced from initial helium shell ignition. Ni extends to 19,000 km s in the northern

1 28 40 hemisphere and roughly 15,000 km s in the southern. Both Si and Ca are produced in both the helium shell and carbon detonations, which is represented by the two levels of ejecta. The 28Si almost spreads to the edge of the ejecta in all directions and reaching over

1 40 30,000 km s ,while Ca reaches comparable velocities only near the helium shell ignition point. The majority of 44Ti and 48Cr was produced during the helium shell detonation near the ignition point, but in small amounts due to the fairly low mass (0.021 M )helium shell. The boundary between the core and helium shell ashes is evident in the density

1 1 contours. The core material is moving 15,000 km s and 22,000 km s in the south and north pole direction, respectively. Figure 2.12 shows the helium shell detonation for the model progen4 (see table 2.1 for details). The reduction of 16O in the helium shell produces a weaker ignition, likely due to less oxygen being readily available for alpha capture, resulting in a shallower detonation front when compared to figure 2.9. The final abundances for progen4 in figure 2.13 shows how the composition of the helium shell a↵ects the final abundances produced in the detonation. The higher amount of 4He in the shell compared to progen2 allows more alpha capture to take place overall, producing a greater abundance of isotopes higher in the reaction chain like 40Ca, 44Ti, and 48Cr.

50 Figure 2.12: Evolution of reduced 16O helium shell detonation. Temperature and density evolution 5 3 of helium shell detonation and propagation for progen4. Density contours above 10 gcm are shown in black. Times displayed from top left are t = 0.25, 0.75, 1.0, 1.5, 1.75, and 2.25 s.

51 Figure 2.13: Final abundances of ejecta for reduced 16O progenitor. Ejecta distribution of density and select species 20 s after helium shell ignition for progen4. Velocity contours in black with units 3 1 10 km s . The red line represents the boundary between the stellar material and low-density ﬂu↵.

52 2.5 CONCLUSION

This chapter continued the study of white dwarfs, but for a di↵erent class of thermonuclear explosion: type Ia supernova. SNeIa are of great importance in providing accurate values of cosmological distances, the Hubble constant, and both the matter (⌦M) and dark energy (⌦⇤)densityparameters.Thisworkisacontinuationoftheresearch performed by (Townsley et al., 2019) and focuses on the sub-Chandrasekhar double-detonation scenario. The intention here was to produce a consistent model that is proficient in testing the mechanisms of simulating double-detonation supernovae with spectra comparable to those in observed SNeIa. The simulations presented in this work were performed using FLASH in order to implement and test new procedures including a de-refinement condition, particle distribution, and burning limiter. The de-refinement technique allowed the adaptive mesh grid to coarsen behind the helium shell shock front, alleviating the computational cost involved with refining regions unimportant to the physics of the simulation. This condition provided moderate improvements to the computational eciency of the simulation with the test we ran granting a 17% reduction in total blocks and 11% faster timesteps. A new particle distribution was added to allow for a finer particle distribution at the outer edge of the ejecta. The goal was to improve the representation of particles in these regions, giving more accurate abundance measurements in post-processing. The new distribution

1 produced a signiﬁcant increase of particles in regions greater than 20,000 km s ,wherethe outer edge of the post-detonation ejecta reaches. Finally, the burning limiter was added to artiﬁcially broaden the detonation front so interaction between the reaction zone and shock front are fully resolved. This eliminates unphysical interactions and allows for maximum energy generation behind the shock front. We made improvements to the limiter used in (Shen et al., 2018) by adding a dependence on the sound crossing time of the cell, allowing it to be independent of the hydrodynamic timestep of the simulation. Additionally, we incorporated a smaller initial guess timestep in determining the limited rate of energy

53 generation to improve eciency in zones of high burning. This provided a decrease in the simulation time per step of 40% in these regions. After successful implementation of these changes, we incorporated them all into the two full double-detonation runs detailed in section 2.4. The research detailed in this chapter was intended to provide future work a physically consistent and ecient model to simulate double-detonations. This will establish constraints on the range of mass and CO pollution in the helium shell capable of producing these important phenomena, helping deﬁne the progenitors associated with double-detonation sub-Chandrasekhar systems.

54 REFERENCES

Bildsten, L., Shen, K. J., Weinberg, N. N., and Nelemans, G. (2007). Faint Thermonuclear Supernovae from AM Canum Venaticorum Binaries. The Astrophysical Journal, Letters, 662:L95–L98.

Colgate, S. A. and McKee, C. (1969). Early Supernova Luminosity. The Astrophysical Journal,157:623.

Denissenkov, P. A., Herwig, F., Bildsten, L., and Paxton, B. (2013). MESA Models of Classical Nova Outbursts: The Multicycle Evolution and E↵ects of Convective Boundary Mixing. The Astrophysical Journal,762(1):8.

Filippenko, A. V. (2005). Type Ia Supernovae and Cosmology. In Sion, E. M., Vennes, S., and Shipman, H. L., editors, White dwarfs: cosmological and galactic probes,volume332 of Astrophysics and Space Science Library,pages97–133.

Fryxell, B., Mueller, E., and Arnett, D. (1991). Instabilities and clumping in SN 1987A. I - Early evolution in two dimensions. The Astrophysical Journal,367:619–634.

Goliasch, J. and Nelson, L. (2015). Population Synthesis of Cataclysmic Variables. I. Inclusion of Detailed Nuclear Evolution. The Astrophysical Journal,809:80.

Howell, S. B., Nelson, L. A., and Rappaport, S. (2001). An Exploration of the Paradigm for the 2-3 Hour Period Gap in Cataclysmic Variables. The Astrophysical Journal, 550:897–918.

Kushnir, D. and Katz, B. (2014). Early Hydrodynamic Evolution Of a Stellar Collision. The Astrophysical Journal,785(2):124.

Miles, B. J., Townsley, D. M., Shen, K. J., Timmes, F. X., and Moore, K. (2019). Quantifying How Density Gradients and Front Curvature A↵ect Carbon Detonation Strength during SNe Ia. The Astrophysical Journal,871:154.

Pakmor, R., Edelmann, P., R¨opke, F. K., and Hillebrandt, W. (2012). Stellar GADGET: a smoothed particle hydrodynamics code for stellar astrophysics and its application to Type Ia supernovae from white dwarf mergers. Monthly Notices of the Royal Astronomical Society,424:2222–2231.

Pala, A. F., G¨ansicke, B. T., Townsley, D., Boyd, D., Cook, M. J., De Martino, D., Godon, P., Haislip, J. B., Henden, A. A., Hubeny, I., Ivarsen, K. M., Kafka, S., Knigge, C., LaCluyze, A. P., Long, K. S., Marsh, T. R., Monard, B., Moore, J. P., Myers, G., Nelson, P., Nogami, D., Oksanen, A., Pickard, R., Poyner, G., Reichart, D. E., Rodriguez Perez, D., Schreiber, M. R., Shears, J., Sion, E. M., Stubbings, R., Szkody, P., and Zorotovic,

55 M. (2017). E↵ective temperatures of cataclysmic-variable white dwarfs as a probe of their evolution. Monthly Notices of the Royal Astronomical Society,466:2855–2878.

Patterson, J. (1984). The evolution of cataclysmic and low-mass X-ray binaries. The Astrophysical Journal, Supplement,54:443–493.

Paxton, B., Bildsten, L., Dotter, A., Herwig, F., Lesa↵re, P., and Timmes, F. (2011). Modules for Experiments in Stellar Astrophysics (MESA). The Astrophysical Journal, Supplement,192:3.

Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R. A., Nugent, P., Castro, P. G., Deustua, S., Fabbro, S., Goobar, A., Groom, D. E., Hook, I. M., Kim, A. G., Kim, M. Y., Lee, J. C., Nunes, N. J., Pain, R., Pennypacker, C. R., Quimby, R., Lidman, C., Ellis, R. S., Irwin, M., McMahon, R. G., Ruiz-Lapuente, P., Walton, N., Schaefer, B., Boyle, B. J., Filippenko, A. V., Matheson, T., Fruchter, A. S., Panagia, N., Newberg, H. J. M., Couch, W. J., and Project, T. S. C. (1999). Measurements of ⌦and ⇤from 42 High-Redshift Supernovae. The Astrophysical Journal,517:565–586.

Piro, A. L., Arras, P., and Bildsten, L. (2005). White Dwarf Heating and Subsequent Cooling in Dwarf Nova Outbursts. The Astrophysical Journal,628:401–410.

Riess, A. G., Filippenko, A. V., Challis, P., Clocchiatti, A., Diercks, A., Garnavich, P. M., Gilliland, R. L., Hogan, C. J., Jha, S., Kirshner, R. P., Leibundgut, B., Phillips, M. M., Reiss, D., Schmidt, B. P., Schommer, R. A., Smith, R. C., Spyromilio, J., Stubbs, C., Suntze↵, N. B., and Tonry, J. (1998). Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. The Astronomical Journal, 116:1009–1038.

Rosswog, S., Kasen, D., Guillochon, J., and Ramirez-Ruiz, E. (2009). Collisions of White Dwarfs as a New Progenitor Channel for Type Ia Supernovae. The Astrophysical Journal, Letters,705:L128–L132.

Shara, M. M., Drissen, L., Martin, T., Alarie, A., and Stephenson, F. R. (2017). When does an old nova become a dwarf nova? Kinematics and age of the nova shell of the dwarf nova AT Cancri. Monthly Notices of the Royal Astronomical Society,465:739–745.

Shen, K. J. and Bildsten, L. (2009a). The E↵ect of Composition on Nova Ignitions. The Astrophysical Journal,692(1):324–334.

Shen, K. J. and Bildsten, L. (2009b). Unstable Helium Shell Burning on Accreting White Dwarfs. The Astrophysical Journal,699:1365–1373.

Shen, K. J., Kasen, D., Miles, B. J., and Townsley, D. M. (2018). Sub-chandrasekhar-mass white dwarf detonations revisited. The Astrophysical Journal,854(1):52.

Shen, K. J. and Moore, K. (2014). The Initiation and Propagation of Helium Detonations in White Dwarf Envelopes. The Astrophysical Journal,797:46.

56 Spruit, H. C. and Ritter, H. (1983). Stellar activity and the period gap in cataclysmic variables. Astronomy & Astrophysics,124:267–272.

Starrﬁeld, S., Iliadis, C., and Hix, W. R. (2016). The Thermonuclear Runaway and the Classical Nova Outburst. Publications of the Astronomical Society of the Paciﬁc, 128(5):051001.

Timmes, F. X. (1999). Integration of Nuclear Reaction Networks for Stellar Hydrodynamics. The Astrophysical Journal, Supplement,124:241–263.

Townsley, D. M. and Bildsten, L. (2004). Theoretical Modeling of the Thermal State of Accreting White Dwarfs Undergoing Classical Nova Cycles. The Astrophysical Journal, 600:390–403.

Townsley, D. M. and Bildsten, L. (2005). Classical Novae as a Probe of the Cataclysmic Variable Population. The Astrophysical Journal,628:395–400.

Townsley, D. M. and Bildsten, L. (2007). The Thermal Structure and Evolution of Accreting White Dwarfs. In Napiwotzki, R. and Burleigh, M. R., editors, 15th European Workshop on White Dwarfs,volume372ofAstronomical Society of the Paciﬁc Conference Series,page557.

Townsley, D. M. and G¨ansicke, B. T. (2009). Cataclysmic Variable Primary E↵ective Temperatures: Constraints on Binary Angular Momentum Loss. The Astrophysical Journal,693:1007–1021.

Townsley, D. M., Jackson, A. P., Calder, A. C., Chamulak, D. A., Brown, E. F., and Timmes, F. X. (2009). Evaluating Systematic Dependencies of Type Ia Supernovae: The Inﬂuence of Progenitor 22Ne Content on Dynamics. The Astrophysical Journal, 701:1582–1604.

Townsley, D. M., Miles, B. J., Shen, K. J., and Kasen, D. (2019). Double Detonations with Thin, Modestly Enriched Helium Layers Can Make Normal Type Ia Supernovae. arXiv e-prints, page arXiv:1903.10960.

Townsley, D. M., Miles, B. J., Timmes, F. X., Calder, A. C., and Brown, E. F. (2016). A Tracer Method for Computing Type Ia Supernova Yields: Burning Model Calibration, Reconstruction of Thickened Flames, and Veriﬁcation for Planar Detonations. The Astrophysical Journal, Supplement,225:3.

Zorotovic, M., Schreiber, M. R., and G¨ansicke, B. T. (2011). Post common envelope binaries from SDSS. XI. The white dwarf mass distributions of CVs and pre-CVs. Astronomy & Astrophysics, 536:A42.