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Evaluating Systematic Dependencies of Type Ia Supernovae: The Influence of Deflagration to Detonation Density

We explore the effects of the deflagration to detonation transition (DDT) density on the production of Ni-56 in thermonuclear supernova explosions (type Ia supernovae). Within the DDT paradigm, the transition density sets the amount of expansion during the deflagration phase of the explosion and therefore the amount of nuclear statistical equilibrium (NSE) material produced. We employ a theoretical framework for a well-controlled statistical study of two-dimensional simulations of thermonuclear supernovae with randomized initial conditions that can, with a particular choice of transition density, produce a similar average and range of Ni-56 masses to those inferred from observations. Within this framework, we utilize a more realistic"simmered"white dwarf progenitor model with a flame model and energetics scheme to calculate the amount of Ni-56 and NSE material synthesized for a suite of simulated explosions in which the transition density is varied in the range 1-3x10^7 g/cc. We find a quadratic dependence of the NSE yield on the log of the transition density, which is determined by the competition between plume rise and stellar expansion. By considering the effect of metallicity on the transition density, we find the NSE yield decreases by 0.055 +/- 0.004 solar masses for a 1 solar metallicity increase evaluated about solar metallicity. For the same change in metallicity, this result translates to a 0.067 +/- 0.004 solar mass decrease in the Ni-56 yield, slightly stronger than that due to the variation in electron fraction from the initial composition. Observations testing the dependence of the yield on metallicity remain somewhat ambiguous, but the dependence we find is comparable to that inferred from some studies.


INTRODUCTION
Type Ia supernovae are bright stellar explosions that are characterized by strong P Cygni features in Si and by a lack of hydrogen in their spectra (see Hillebrandt & Niemeyer 2000;Filippenko 1997 and references therein). Observations of type Ia supernovae (serving as distance indicators (Phillips 1993;Riess et al. 1996;Albrecht et al. 2006)) are at present the most powerful and best proved technique for studying dark energy (Riess et al. 1998;Perlmutter et al. 1999;Kolb et al. 2006;Hicken et al. 2009;Lampeitl et al. 2010), and, accordingly, there are many observational campaigns underway striving to gather information about the systematics of these events and to better measure the expansion history of the Universe (see Kirshner 2009 and references therein).
The most widely accepted model for these events is the "single-degenerate" scenario, which is the thermonuclear explosion of a white dwarf composed principally of 12 C and 16 O that has accreted mass from a companion (for a review of explosion models, see Hillebrandt & Niemeyer 2000;Livio 2000;Röpke 2006). The peak brightness of the supernova is set by the radioactive decay of 56 Ni produced in the explosion to 56 Co and then to 56 Fe. The empirical "brighter is broader" relation between the peak brightness of the light curve and the decay time from its maximum is understood to follow from the fact that both the luminosity and opacity are set by the mass of 56 Ni synthesized in the explosion (Arnett 1982;Pinto & Eastman 2000;Kasen & Woosley 2007). Because of the dependence of the light curve on the amount of radioactive 56 Ni synthesized during an explosion and the ability to infer the 56 Ni yield from observations (Mazzali et al. 2007), research into modeling thermonuclear supernovae typically focuses on the production and distribution of 56 Ni as well as other nuclides (such as 28 Si) as the measure with which to compare models to observations. One-dimensional simulations of the single-degenerate case showed that the most successful scenario is an initial deflagration (subsonic reaction front), born in the core of the white dwarf (WD) which at some point becomes a (supersonic) detonation, i.e. a deflagration-detonation transition (DDT, Khokhlov 1991;Höflich & Khokhlov 1996). A delayed detonation naturally accounts for the high-velocity Ca features (Kasen & Plewa 2005) and the chemical stratification of the ejecta. While these 1D models are able to reproduce observed features of the light curve and spectra, much of the physics is missing from these models. The presence of fluid instabilities during the deflagration warranted the development of multidimensional models, allowing a physicallymotivated calculation of the velocity of the burning front and thus removing a free parameter. By relaxing the symmetry constraints on the model, buoyancy instabilities are naturally captured leading to a strong dependence on the initial conditions of the deflagration that often result in too little expansion of the star by the time DDT conditions used in previous 1D studies are met (Niemeyer et al. 1996;Calder et al. 2003Calder et al. , 2004Livne et al. 2005). Multidimensional models may reach the expected amount of expansion prior to the DDT with the choice of particular ignition conditions and thus retain the desirable features from one-dimensional models (Gamezo et al. 2004;Plewa et al. 2004;Röpke et al. 2006;Jordan et al. 2008).
Due to the strong dependence on ignition conditions, multidimensional simulations of the DDT model are able to produce a wide range peak luminosities (via the production of a range of 56 Ni yields) consistent with a common explosion mode suggested from observations (Mazzali et al. 2007). Differences in the mass of synthesized 56 Ni can follow from properties such as metallicity and central density of the progenitor and/or differences in the details of the explosion mechanism such as the density at which the transition from deflagration to detonation occurs. Timmes et al. (2003) found that metallicity should affect the 56 Ni yield based on approximate lepton number conservation. The metallicity sets the fractional amount of material in nuclear statistical equilibrium (NSE) that is radioactive 56 Ni. Bravo et al. (2010) found a stronger dependence on metallicity due to a significant amount of 56 Ni that is synthesized during incomplete Si-burning.
Observational results to date are consistent with a shallow dependence of 56 Ni mass on metallicity but are unable to rule out a trend entirely (Gallagher et al. 2005(Gallagher et al. , 2008Neill et al. 2009;Howell et al. 2009). Determining the metallicity dependence is challenging not only because the effect appears to be small, but also due to the difficulty in measuring accurate metallicities for the parent stellar population and problems with strong systematic effects associated with the mass-metallicity relationship within galaxies (Gallagher et al. 2008). This effect is also difficult to decouple from the apparently stronger effect of the age of the parent stellar population on the mean brightness of SNe Ia (Gallagher et al. 2008;Howell et al. 2009;Krueger et al. 2010). Howell et al. (2009) note that the scatter in brightness of this observed relation is unlikely to be explained by the effect of metallicity. In general, the source of scatter can be explained by the development of fluid instabilities during the deflagration phase that contribute to differing rates of expansion between instances of supernovae. However, by considering the effect of metallicity on the DDT density, scatter in the metallicity relation may be enhanced beyond its intrinsic value inferred from fluid instabilities. Townsley et al. (2009) investigated the direct effect of metallicity via initial neutron excess and found it to have a negligible influence on the amount of material synthesized to NSE. However, the neutron excess sets the amount of material in NSE that favors stable Fe-group elements over radioactive 56 Ni. Therefore, the initial metallicity directly influences the yield of 56 Ni. In this work, we expand that study to include the potential indirect effect of metallicity in the form of the 22 Ne mass fraction (X 22 ) through its influence on the density at which the DDT takes place. To this end, we explore the effect of varying the transition density, a proxy for varying the microphysics that determine the conditions for a DDT. The conditions under which a DDT occurs are still a subject of debate. Niemeyer & Woosley (1997), Niemeyer & Kerstein (1997), and Khokhlov et al. (1997) proposed that a necessary condition is the transition to a distributed burning regime, in which turbulence disrupts the reaction zone of the flame. More recent numerical studies (Pan et al. 2008;Woosley et al. 2009;Schmidt et al. 2010) that describe the conditions for a DDT include dependencies on the turbulent cascade and the growth of a critical mass of fuel with sufficiently strong turbulence. For this study, we assume the DDT density to be the density at which thermonuclear burning is expected to enter the distributed regime. This choice links the explosion outcome to the dynamical evolution of the progenitor density structure during the deflagration phase. By analyzing the effect of transition density on the NSE yield, we can later analyze how the details of the microphysics affect the DDT density. For the purposes of this study, we will consider only the effect of X 22 on the DDT density. In reality, the 12 C mass fraction will also be important in determining the DDT density, but we choose to leave the exploration of the effect of 12 C to future work. Many other possible systematic effects exist that are outlined in Townsley et al. (2009), such as the central ignition density (Krueger et al. 2010), which are all held fixed in this study. The interdependence of all of these effects will be considered in the construction of the full theoretical picture in a future study.
We describe the details of our model in §2 and §3, and the properties of our statistical sample in §4. We present our findings on the dependence of transition density on NSE yield in §5.1. In §5.2 and §5.3, we assume a dependence of the transition density on 22 Ne content and construct the functional dependence of the 56 Ni yield on metallicity through the 22 Ne content. In §6, we discuss our conclusions and future work.

PARAMETERIZED REALISTIC WHITE DWARF PROGENITOR
In order to include relevant processes in explosion models, we first estimate the compositional profile of the progenitor white dwarf just prior to the birth of the flame. We begin by estimating the compositional profile resulting from the evolution of the post-main-sequence star that later becomes a white dwarf. Recall that the initial metallicity of the star is, by mass, almost entirely in the form of CNO. As a result of the C-N-O cycle, these all end up in helium layers as 14 N, the target of the slowest proton-capture in the cycle. Subsequent reactions during helium burning convert 14 N into 22 Ne; therefore, X 22 is proportional to the initial metallicity of the star (Timmes et al. 2003). The composition of the inner portion of the star (≈ 0.3 − 0.4 M ⊙ ) is set during core helium burning resulting in a reduced carbon mass fraction with respect to that of the outer layers, which is set by shell burning on the asymptotic giant branch (see Straniero et al. 2003, and references therein).
At some point after the white dwarf has formed, it begins to accrete material from its companion. As the mass of the accreting white dwarf approaches the Chandrasekhar mass limit, the core temperature and density increase such that carbon begins to fuse. The energy released by carbon burning drives convection in the core. The convective carbon burning ("simmering") phase lasts approximately 1000 years before the central temperature is high enough to spark a thermonuclear flame (Woosley et al. 2004). During the simmering phase, carbon is consumed from the convective core. Concurrently, though, the convective zone grows with increasing central temperature, pulling in relatively carbon-rich material from the outer layer. Thus, the net effect is to increase the carbon abundance in the convective region. We show the growth of the convective zone in Figure 1, where dashed lines show the compositional profile prior to the simmering phase and the corresponding solid lines show the compositional profile at the start of our simulations of the explosion.
For our WD models, we consider a parameterized 3-species compositional structure consisting of 12 C, 16 O, and 22 Ne, which is sufficient to capture the nuclear burning rates. Since 22 Ne is the only element in our parameterization that has more neutrons than protons, the neutron excess from simmering is accounted for by a parameterized 22 Ne mass fraction where X 22 is proportional to the initial metallicity coming from helium burning (Timmes et al. 2003) and ∆X 22 (∆Y e ) represents the effective enhancement of 22 Ne in the core following from the change in electron fraction during convective carbon burning. The electron fraction (Y e ) is related to X 22 by Y e = 10 22 such that a change in X 22 constitutes a change in Y e . The prime on X ′ 22 in Equation (1) indicates inclusion of effects of neutronization during carbon burning and this convention applies to expressions below. We choose the composition of the WD at the end of the simmering phase to consist of X 12 = 0.4 and X ′ 22 = 0.03 in the core and X 12 = 0.5 and X ′ 22 = 0.02 in the outer layer. Note that X 22 = X ′ 22 in the outer layer because neutronization due to carbon burning only occurs in the convective core.
For comparison, a compositional profile can be estimated for the WD prior to the onset of carbon burning. For simplicity, consider the production of one neutron for every 6 12 C burned (i.e., dY e /dY 12 ≈ 1/6, where Y 12 is the molar abundance of 12 C) (Chamulak et al. 2008;Piro & Bildsten 2008). For ∆X 22 = 0.01 in our progenitor model, this constitutes burning 0.04 M ⊙ of carbon during simmering. Assuming that prior to simmering, the core mass is ≈ 0.3 M ⊙ (Straniero et al. 2003) and the outer layer consists of X 12 = 0.5 material, we can account for all 12 C and conserve the total mass of the WD to estimate X 12 ≈ 0.2 in the carbon-reduced core of the WD prior to simmering as shown in Figure 1. Table 1 shows the composition of the progenitor white dwarf before and after the simmering phase using the parameterized 22 Ne mass fraction as well as the core mass. Note that throughout this study, we choose to neglect any variation of the amount of neutronization during simmering due to the initial Z. Accordingly, ∆X 22 (∆Y e ) is treated as a constant and, therefore, dX ′ 22 = dX 22 and derivatives involving the true 22 Ne mass fraction proportional to metallicity, X 22 , are equivalent to derivatives involving the parameterized 22 Ne mass fraction, X ′ 22 .
Because we consider the enhancement of the laminar flame speed by 22 Ne (Chamulak et al. 2007), we need to consider the effects of our parametrization of the 22 Ne content with care. In actuality, the neutronization during carbon simmering produces 13 C, 23 Na, and 20 Ne and not 22 Ne (Chamulak et al. 2008). A priori, this difference could alter the nuclear burning rates and hence the laminar flame speed and width. Table 2 shows the laminar flame speeds and widths for the same  Chamulak et al. (2007). The flame speeds and widths calculated using the parameterized 22 Ne mass fraction (3% by mass) are denoted with primes. For high densities ( 2.5 × 10 8 g cm −3 ) in which the flame speed is not dominated by the buoyancy-driven Rayleigh-Taylor instability, the difference is 5%; therefore, our parametrization of the neutronization via X 22 accurately captures the corresponding enhancement of the laminar flame speed in this regime. While the central temperature and central density of the progenitor just prior to the birth of the flame are primarily set by the accretion history of the white dwarf, which varies and is largely unknown, we choose a fiducial central density of ρ c = 2.2 × 10 9 g cm −3 and central temperature of T c = 7 × 10 8 K. We construct isentropic profiles of density and temperature in the (convective) core and isothermal profiles in the (thermally conductive) outer layer while maintaining hydrostatic equilibrium. We choose a fiducial isothermal temperature of the outer layer to be T iso = 10 8 K. The total mass of the WD progenitor model is 1.37 M ⊙ .
Due to the difference in composition between the core and outer layer, requiring a neutral buoyant condition at this interface no longer reduces to a continuous temperature as it does for a compositionally uniform white dwarf (Piro & Chang 2008). In reality, we might expect some convective overshoot and mixing between the core and the outer layer, but because the physics of convective overshoot are complex and not well understood, the width of the transition region is unknown. For simplicity, we assume no mixing region. The composition, density, and thermal profiles of the progenitor used for this study immediately prior to deflagration are the solid lines shown in Figure 1 We use a customized version of the FLASH Eulerian compressible adaptive-mesh hydrodynamics code (Fryxell et al. 2000;Calder et al. 2002). Modifications to the code include a nuclear burning model, composition-dependent laminar flame speeds, particular mesh refinement criteria, and instructions for determining the conditions for a DDT (for details, see Calder et al. 2007;Townsley et al. 2007Townsley et al. , 2009). In particular, the adaptive mesh refinement (AMR) capability of the code was utilized to achieve particular resolutions for burning fronts (4 km) and the initial hydrostatic star (16 km) for which the solution has converged ).
The burning model is used for both subsonic (deflagration) and supersonic (detonation) burning fronts. The laminar flame width for densities and compositions characteristic of a massive C-O WD is unresolved on the scale of our grid (4 km, Chamulak et al. 2007); therefore, we use an artificially thickened flame represented by the advection-reactiondiffusion equation (Khokhlov 1995;Vladimirova et al. 2006) to which our nuclear energetics scheme is coupled. Appropriate measures have been taken to ensure this coupling is acoustically quiet and stable such that the buoyant instability of the burning front is accurately captured ). The shock-capturing capabilities of FLASH naturally handle the propagation of detonation fronts given an accurate nuclear energetics scheme (Meakin et al. 2009). We note that the only common components between our code and that of Plewa (2007) are those publicly available as components of FLASH, which excludes all components treating the nuclear burning; differences are discussed in Townsley et al. (2007).
In this section we discuss an improvement to our burning model and an improvement to the method by which a transition from deflagration to detonation is made. The changes to the burning model reflect recent work to better match steady-state detonation structures that are partially resolved on the grid. This is important for obtaining accurate particle tracks for post-processing nucleosynthetic yields. Additionally, changes to the DDT method were necessary to ensure consistency between individual simulations using different DDT densities and/or different initial configurations of the flame.

Improved Burning Model
For the calculations presented here we utilize the latest revision of a parameterized three-stage model for the nuclear burning Townsley et al. 2007;Meakin et al. 2009;Seitenzahl et al. 2009b;Townsley et al. 2009). The details of this latest version will be published separately (Townsley et al., in prep.) along with extensive comparisons to nuclear network calculations of steadystate detonations, but we summarize the major changes here. This work represents the first time a burning model which correctly reproduces the nuclear statistical quasi-equilibrium (NSQE, Khokhlov 1989) to NSE transition time scales and length scales during incomplete silicon burning has ever been used in a multi-dimensional SN Ia calculation, as validated by comparison to steady-state detonation structures calculated out to the pathological point with the ZND model (see e.g. Fickett & Davis 1979). Accurate reproduction of this low-density burning regime is essential because a significant portion of the 56 Ni is produced in incomplete burning (Bravo et al. 2010) so that the overall 56 Ni yield is determined by the details of how this low-density cutoff of 56 Ni production occurs.
Obtaining a satisfactory reproduction of ZND detonation structures involved two main changes to the 3-stage burning model. First, we found that the progress variable representing the NSQE to NSE transition, φ qn , which also gives the mass fraction of Fe-group (NSE) material, did not match the time and space structure of this transition in steady state detonations calculated with a full nuclear network. The kinetics for this stage, first proposed by Khokhlov (1991) and adopted in Calder et al. (2007), are given by the simple form dφ qn /dt = (1 − φ qn )/τ NSE where τ NSE is a calibrated timescale that depends mainly on temperature. We have found, however that a far more appropriate match to steady-state detonations at densities important for incomplete silicon burning, ρ 10 7 g cm −3 , is obtained with the alternative form dφ qn /dt = (1 − φ qn ) 2 /τ NSE . This necessitates a recalibration of τ NSE , since it now plays a different role, but it is still sufficient for it to depend only on temperature. Although this change in derivative significantly improves the match between the parameterized burning and the ZND structure at the densities of interest, it is still not exact at all densities. It is therefore unclear if this form is indicative of some underlying physical process, and whether or not it is specific to detonations. There are several relaxation processes proceeding simultaneously, so that it is non-trivial to quantify separate contributions. This will be investigated in more detail in future work on post-processing nucleosynthesis.
The second major burning model change was motivated by needing to match the thermodynamic, i.e. T , ρ, profiles at densities at which the portion of the detonation structure representing the NSQE to NSE transition is resolved on the grid. Our previous treatment released all of the nuclear binding energy change to the NSE state by the end of the second stage, leaving the third stage NSQE to NSE transition energetically inert. This lack of an energy release on the NSE timescale leads to an incorrect progression of the density fall-off behind the shock front in the detonation. The abbreviated energy release leads to an under-prediction of the temperature immediately behind the unresolved earlier burning stages in a propagating detonation. A very good match of thermodynamic profiles to the full-network steady state detonation was obtained by tying the completion of the energy release to φ qn , so that energy release occurs in three distinct stages. Note that the previous change involving the kinetics used for φ qn is also an important contributor to the realism of the thermodynamic profiles obtained.
Finally, although detailed nucleosynthesis based on postprocessing tracer particle histories will be published in fu-ture work (Townsley et al., in prep.), we have performed two important tests for the burning model used in this work. The composition profiles obtained from post-processed histories for hydrodynamic simulations of steady-state detonations match the steady-state structure calculated via the ZND method with a large network with remarkable accuracy. Additionally, a preliminary version of the post-processing under development has been applied to the calculations presented here, and we have found good agreement between φ qn and the the fraction of material in the form of Fe-group nuclides found from post-processing on both an overall basis and in ejection velocity bins. Overall, the burning model changes led to a modest but significant (around 0.2M ⊙ ) increase in the overall Fe-group yields for the same explosion. The yields found here are not quite this much higher than similar cases from Townsley et al. (2009) because the progenitors used here have a lower central carbon fraction.

Improved Detonation Ignition Conditions
In order to study the systematic effects associated with changing the DDT density, we need to minimize any systematics in our method of starting a detonation. Previously, in Townsley et al. (2009), we visualized the simulation data from the deflagration phase and plotted a density contour at the DDT density. When the flame reached ≈ 64 km away from the contour, we picked a computational cell half-way between the flame front and the DDT density contour to place a detonation ignition point with a radius of 8 km. The placement and size of the detonation point was chosen to be as close to the flame front as possible while still allowing the detonation point to develop into a self-sustained, stable detonation front. If the detonation point is placed too close to the flame front, then the flame will interact with the detonation point before it develops into a self-sustained detonation. Comparisons of simulations from identical initial conditions with 8 and 12 km ignition radii finds the NSE yield differs by 0.5% throughout the evolution. This results indicates that the total yield is insensitive to the choice of detonation ignition radius for radii less than the characteristic size of a rising plume (as can be seen in Figure 2). We adopt 12 km for the detonation ignition radius in our study as it produces more robust detonations at low density.
To ensure that we do not introduce unintended systematic effects in this study, we improve our method of detonation ignition point placement over the previous "by hand" method by precisely defining the criteria for a DDT that is used in an algorithm. Parameters in this algorithm are chosen to be consistent with Townsley et al. (2009). Once the flame front reaches the specified DDT density in a simulation, a detonation is ignited 32 km radially outward away from the flame front. Here the reaction-diffusion (RD) front is defined by the variable representing progress of the subsonic burning wave, φ RD , with the leading edge defined as the region between the values 0.001 and 0.01 of this variable. During the deflagration phase of a simulation, φ RD is equivalent to the carbon-burning reaction progress variable, φ f a . At the leading edge of the RD front, very little carbon has burned and the local density is approximately equal to the unburned density. This provides a definition of the DDT density that is much more precise and accurate. We chose these criteria that ignite the detonation ahead of the RD front to avoid any issues with the detonation ignition point overlapping with the artificially thickened flame. If we were to choose criteria that would initiate a detonation inside a thickened flame, the detonation structure would need to be altered in some way to be consistent with the artificial nature of that region.
Our detonation ignition conditions also restrict detonation ignition points to be at least 200 km away from each other. This choice ensures that each rising plume starts 2-3 detonations, which is consistent with Townsley et al. (2009). In the case that multiple points within 200 km meet the detonation ignition conditions, the point furthest from the center of the star is preferred ensuring the ignition of detonations on plume "tops". In reality, the location of the spontaneous detonation points is not well known and is the subject of active research. For instance, Röpke et al. (2007) argue that a spontaneous detonation is triggered by the extreme turbulence found in the roiling fuel underneath the plume caps. Under the assumption that the DDT occurs when the flame reaches the low density for distributed burning, we place ignition points on the tops of the rising plumes. Future studies, however, will explore other physically motivated detonation methods in which the location of the detonation ignition is not necessarily specified relative to a plume, but rather determined by the local turbulence field, composition, density, and temperature Woosley et al. 2009). Figure 2 shows the placement of first one detonation point, and then another more than 200 km away from the first, both at the specified DDT density.
FIG. 2.-Snapshots of realization 2 just after one (left) and then another (right) detonation was ignited at the specified transition density of 1.26 × 10 7 g cm −3 (green contour). The reaction progress variable representing carbon burning is in color. The detonating regions are the rapidly expanding regions of completely burned carbon ahead of the plume.
Each detonation ignition point is defined by the profile of the reaction progress variable representing carbon burning, φ f a . Within the detonation ignition radius φ f a = 1, and, because we are igniting in fuel, φ f a = 0 outside this region. The subsequent energy release from this change in φ f a drives a shock strong enough to ignite the surrounding unburned material. This top-hat profile is a simple approach to igniting a detonation and has some drawbacks. At densities below ≈ 10 7 g cm −3 , this detonation ignition method does not produce a sufficiently strong shock to burn the fuel. The improvements to our DDT method place the detonation point radially outward from the specified DDT density; therefore, the density within the detonation point is actually somewhat less than that specified by the DDT density. This differs from the method described in Townsley et al. (2009) in that the detonation point was placed at a density that was somewhat higher than the specified DDT density. Igniting a detonation specifying a DDT density below 10 7.1 g cm −3 is impossible using a top-hat profile because the detonation ignition point is actually placed ≈ 10 7 g cm −3 . In future works, we will explore the use of a gradient in φ f a motivated by Seitenzahl et al. (2009a) to describe the detonation ignition point profile which should result in the formation of stronger shocks that will burn the surrounding low-density fuel.

PROPERTIES OF STATISTICAL SAMPLE AND METHOD
To study the systematic effect of transition density on the 56 Ni yield synthesized during a simulated explosion, we utilize the statistical framework developed in Townsley et al. (2009). In order to compare results between this study and Townsley et al. (2009), we use the same sample population using the same initial seed. The initial seed defines the starting point to a stream of random numbers used to characterize our sample population of thermonuclear supernovae as described in Townsley et al. (2009). In that study, we found that the sample dispersion in the estimated NSE yield does not asymptote until more than 10 realizations. Accordingly, our sample is made up of 30 realizations.
During the early part of the deflagration phase (≈ 0.1 s), the flame is most affected by convective motion in the core of the WD. Because the velocity field in the core and the number of ignition points are largely uncertain, we choose to characterize the initial flame surface using spherical harmonics, each with a random coefficient picked sequentially from the initial seed. Each realization is defined with a unique perturbation on the initial spherically symmetric flame surface using where r 0 is the radius of flame surface, a 0 is the amplitude of the perturbations, A l is a randomly-chosen coefficient corresponding to the spherical harmonic Y 0 l , and l min and l max set the range of spherical modes used to perturb the flame surface. This method serves to initialize Rayleigh-Taylor unstable plumes of random relative strengths, which we would expect from various distributions of ignition points and varying strengths of the convective velocity field found in the real population of progenitors. In this study, r 0 = 150 km, a 0 = 30 km, l min = 12, and l max = 16. The choices of these parameters are motivated by the resolution of our study and the desire to obtain reasonable 56 Ni yields inferred from observations. These choices are discussed in further detail in Townsley et al. (2009).
We choose to analyze the dependence on transition density by choosing 5 different transition densities equidistant in log space at Lρ DDT = {7.1, 7.2, 7.3, 7.4, 7.5} in cgs units, where Lρ DDT = log 10 ρ DDT . The range of DDT densities chosen for this study are motivated both by previous work and computational challenges. Below Lρ DDT = 7.1, a more realistic detonation structure is needed to successfully launch a detonation wave as described in the previous section and in Seitenzahl et al. (2009a). While recent studies have suggested a wider range of DDT densities are possible (Pan et al. 2008;Schmidt et al. 2010), we suggest that trends resulting from varying the DDT density are captured with a maximum Lρ DDT = 7.5.
A simulation is performed for each of the 30 realizations in our sample at each DDT density for a total of 150 simulations. We choose not to explore DDT densities below this specified range for computational reasons. Because of the approximate power-law density profile of the WD, DDT densities were chosen in log space because these densities correspond to relatively evenly spaced radial coordinates of the WD. The amount of 56 Ni synthesized in the explosion principally depends on the amount of expansion during the deflagration phase. Therefore, spatially separated detonation ignition conditions will allow the amount of expansion to vary a non-negligible amount and we can more easily analyze the dependence of the yield on DDT density. We find that extrapolating our results to a DDT density of Lρ DDT = 6.83 yields an estimated average 56 Ni yield of ≃ 0.60 M ⊙ with a standard deviation of 0.21 M ⊙ (from the fits described below and listed in Table 4) that is consistent with observations ). Because the actual DDT density is unknown and the subject of ongoing research (Pan et al. 2008;Aspden et al. 2008Aspden et al. , 2010Woosley et al. 2009;Schmidt et al. 2010), we are free to choose this value of the DDT density as the fiducial DDT density, Lρ DDT,0 = 6.83. This choice is relevant for analysis and comparison to other works as discussed in §5. The distribution of 56 Ni material synthesized during the explosion is shown in Figure 3 for different transition densities at Lρ DDT = 6.83, 7.10, and 7.30.
The NSE mass, M NSE , is defined as φ qn ρdV integrated over the star. We determine the NSE yield by running the simulation until M NSE is no longer increasing as a function of simulation time. This condition is defined as Because we estimate the 56 Ni yield as a fraction of the NSE yield, we consider the 56 Ni yield to have plateaued when the NSE yield has plateaued. Considering additional 56 Ni from incomplete Si-burning in NSQE material and the efficient capture of excess neutrons onto Fe-group elements changes the final 56 Ni estimate by 1%. Therefore, to good approximation, the 56 Ni yield is a fraction of the NSE yield. Figure 4 shows the evolution of the NSE yield as a function of simulation time showing the DDT time and the NSE yield plateau time for each DDT density for realization 2. Discernible from this figure, there is a clear dependence of the NSE yield on transition density.

RESULTS
Within the DDT paradigm for thermonuclear supernovae, the conditions under which a transition occurs are still not completely understood (Aspden et al. 2008(Aspden et al. , 2010. Generally, though, a hypothesized transition occurs when the flame enters the distributed burning regime, which occurs when the flame speed equals the turbulent velocity at the scale of the flame width assuming a Kolmogorov turbulent cascade (Niemeyer & Woosley 1997). Recent work has placed more stringent requirements on the DDT (Pan et al. 2008;Woosley et al. 2009;Schmidt et al. 2010) and fundamental questions concerning deflagration in the limit of disruptive turbulence remain (Poludnenko & Oran 2010). Regardless of the actual DDT mechanism, it is likely that the DDT conditions will depend on composition because both the width and burning rate of the flame depend on the abundances 12 C and 22 Ne. For the current study, we assume that a DDT will occur at a unique density given a particular composition, regardless of the microphysics involved. This assumption is reasonable given that the characteristics of the flame depend strongly on density. Using this assumption, we can delay the analysis of the particular microphysics that lead to a specific transition density and analyze the dependence of the amount of material synthesized to NSE during the explosion on the transition density. Therefore, we can think of each transition density as a proxy for changing the composition that determines the conditions for a DDT via the appropriate microphysics.
As discussed in detail in Townsley et al. (2009), many systematic effects exist that influence the outcome of a SN. In that study, the direct effect of 22 Ne was explored and found to have a negligible influence on the NSE yield. Therefore, we do not vary the initial 22 Ne mass fraction, but rather study the effect of varying the DDT density with the expectation that the NSE yield will be influenced indirectly by 22 Ne and 12 C abundances through the DDT density. We focus on the indirect effect of the 22 Ne abundance on the NSE yield, following up previous work in Townsley et al. (2009) and neglect the effect of varying the carbon abundance. Additionally, we neglect effects due to the central ignition density, compositional and thermodynamic WD structure, and the total WD mass. The effects due to these variables will be studied in turn in future works with the goal of addressing interdependencies once individual effects are better understood.

Dependence on Transition Density
The evolution of the amount of material above a density threshold (M ρ>ρthres = ρthres ρdV , where we take ρ thres = 2 × 10 7 g cm −3 ) principally determines the dependence of the NSE yield on DDT density. This is due largely to the linear relationship between the NSE yield and M ρ7>2 (t = t DDT ) shown in Figure 5, where t DDT is defined as the time the flame first reaches the specified DDT density. This definition is consistent with our assumption that DDT conditions are met on the -Mass burned to NSE as compared to the mass above a density of 2 × 10 7 g cm −3 at the first DDT time for each realization in our sample. The different shapes plotted (+, ×, * , 2, ) correspond to different transition densities (10 7.1 , 10 7.2 , 10 7.3 , 10 7.4 , 10 7.5 ), respectively. The solid line shows the linear fit to yields produced with a DDT density of 10 7.1 g cm −3 most closely matching the DDT density used in Townsley et al. (2009). The dashed line shows a 1:1 correlation between the NSE yield and mass above 2 × 10 7 g cm −3 . The lowest two DDT densities are less than the density threshold 2 × 10 7 g cm −3 and thus show more scatter about the linear relation due to increased dependence on the plume morphology. tops of rising plumes. As mentioned above, other DDT locations are possible, such as the highly turbulent region underneath plume "caps", which would result in a higher DDT density correlated to our present definition via the density structure local to the rising plume. As shown in Figure 4, the deflagration phase burns only a relatively small fraction of the white dwarf and the majority of material is burned during the detonation. Once a detonation has started, the propagation speed of the burning wave is much greater than the rate of expansion; therefore, the NSE yield is essentially independent of the evolution of M ρ7>2 for t > t DDT . The number of detonation points and their corresponding distribution in time and location contributes to the variance of the relation between the NSE yield and M ρ7>2 . The relation between the NSE yield and DDT density via M ρ7>2 also depends on the acceleration of the RT-unstable plumes not being too great near t DDT . For a constant plume rise-rate near t DDT , there is a linear relationship between the DDT density and t DDT . This relationship allows the evolution of M ρ7>2 to translate directly into a dependence of the NSE yield on DDT density. Figure 6 shows the evolution of M ρ7>2 and ρ min as a function of simulation time for 3 different realizations where ρ min is the minimum density at which the flame is burning material. The evolution of ρ min shows the linear relationship between the log of the DDT density and t DDT where the filled circles represent the t DDT times for each of 5 transition densities and the filled squares show the fiducial DDT density. For the times corresponding to the DDT conditions, the evolution of M ρ7>2 is in a region where the rate of expansion of the star becomes significant and M ρ7>2 begins to drop off relatively quickly.
We find, for each individual realization, the NSE yield depends quadratically on transition density. Table 3 lists the NSE yields for each realization at each of 5 transition densities equidistant in log-space. Table 4 shows the coefficients used to fit the quadratic dependence of NSE yield on transition density for M NSE (M ⊙ ) = a (Lρ DDT ) 2 + bLρ DDT + c.
(5) Figure 7 shows fits to the NSE yield for each realization and the average yields at each transition density. The NSE yields for individual realizations are plotted as blue crosses and the  (green) showing the expansion rate of the WD and the plume rise-time with DDT times (circles) associated with each DDT density emphasized. M ρ 7 >2 is defined as the total mass with density greater than 2 × 10 7 g cm −3 and ρ min is defined as the minimum density at which the flame is burning material. The translation into M ρ 7 >2 as a function of ρ min is shown in the right panel for comparison to M NSE as a function of ρ DDT shown in Figure 7. We also highlight the fiducial DDT density as black squares. These plots emphasize the importance of the rate of expansion of the WD and the plume rise-time as a function of density in determining the relation between the NSE yield and DDT density. Realization 2 produces NSE yields that are close to the average while realizations 10 and 18 produce the lowest and highest yields, respectively.
individual curves showing the dependence of NSE yield on DDT density are shown in grey. The average NSE yields at each DDT density are shown in red. At a transition density of 10 7.5 g cm −3 , notice that the dependence on DDT density has flattened out for almost all realizations and the variance of the yields among all realizations is small. This behavior is due to the fact that the progenitor white dwarf star has a finite mass of 1.37 M ⊙ . At the highest transition density, the expansion rate of the star is relatively small, and the evolution of M ρ7>2 is relatively slowly declining as compared to lower DDT densities (see Figure 6). -Quadratic fits (grey lines) to the NSE yields for each realization (blue crosses) and the average NSE yield at each transition density (red circles). The qualitative similarity between this figure and the right panel of Figure 6 is due to the near 1:1 correlation between M NSE and M ρ 7 >2 (t = t DDT ).
The curvature of the NSE yield dependence on DDT density, a, is well correlated with the NSE yield at a given DDT density. Figure 8 shows this correlation for Lρ DDT,0 = 6.83; however, a correlation exists for all DDT densities as may be discernible from Figure 7 noting that most black lines representing individual realizations do not cross. The lower the NSE yield for a given realization, the stronger the dependence on transition density. This result is likely due to realizations with lower yields having multiple competing plumes that release more energy allowing the star to expand more rapidly leading to a stronger dependence on the transition density.   After fitting the dependence on DDT density for each realization, we find an interesting correlation between the fitting parameters from Equation (5). Figure 9 shows the correlation between the fitting parameters a, b, and c with c on the horizontal axis. The tight correlation between these parameters indicates that a single parameter describes the dependence on DDT density for a given realization. The fitting parameters a and b can be expressed as a function of c: where α = 1.754 ± 0.003 × 10 −2 , β = −0.005 ± 0.002, δ = −0.2646 ± 0.0005, and γ = 0.21 ± 0.02. These parameters were calculated using a Least Squares method and the associated errors were calculated from the corresponding covariance matrix.  Table 4 for each realization. The fitting parameters are defined by Equation (5) and the relations between the fitting parameters are given by Equations (6) and (7). Each realization has a different random initial perturbation of the central ignition condition that sets the plume morphology. Dominant single plumes tend to allow the star to expand less prior to reaching the conditions for a detonation, while multiple competing plumes tend to release more energy during the deflagration phase, allowing more expansion prior to reaching the conditions for a DDT. Shown in Figure 10 are the Rayleigh-Taylor unstable plumes for the realizations with the highest and lowest yields a few tenths of seconds into the deflagration. Unfortunately, no strong correlation exists between the properties of the initial flame surface, such as mass enclosed, surface to volume ratio, or amount of power in the perturbation, for a given realization and the single fitting parameter c. This lack of a correlation implies there is no way to tell whether a particular initial condition will seed a single dominant plume or multiple competing plumes.
Physically, we might expect two competing effects that influence the NSE yield: plume morphology (rise time) and the rate of expansion. In our 2D simulations these two effects appear to be correlated as seen in Figure 6. Our results indicate that plumes near the symmetry axis tend to rise faster than plumes rising near the equator and we attribute this result to the fact that our simulations are two-dimensional. A plume that develops near the symmetry axis naturally becomes dominant in 2D and determines the DDT time and thus the NSE yield. In addition, a plume near the symmetry axis represents less volume than a plume of similar size near the equator. This indicates that energy is being deposited into a smaller volume allowing for a lesser rate of expansion. This explains why the overall rate of expansion of the star is correlated to the rise time of the first plume to reach DDT conditions.
In order to evaluate the dependence on DDT density without this unphysical correlation, we need to perform 3D simulations. A suite of 3D simulations will likely create a twoparameter family (plume rise and expansion rates) of solutions which describe the yield as a function DDT density. Additionally, we expect 3D plumes to behave similarly to 2D on-axis plumes implying a faster plume rise time. and Fe-group (NSE, black). Density in g cm −3 is indicated by contours (blue) logarithmically spaced at integer powers of 10 as well as the DDT density of 1.26 × 10 7 g cm −3 (red). As a result of the tight correlation for a and b and the determination of Equations (6,7), the average relation is characterized by just the average c-or equivalently, the average NSE yield at a specified ρ DDT . We evaluate the average and standard deviation of the fitting parameter c as well as the statistical properties of the NSE yield at the fiducial transition density of 10 6.83 g cm −3 shown in Table 5.

NSE Yield Dependence on 22 Ne
Recall that for the purposes of this study, we are neglecting the effects of varying the core carbon abundance, the central ignition density, the WD structure, etc. except for the indirect effect of X 22 , the neutron excess produced during simmering, and the DDT density. In order to derive the NSE yield dependence on X 22 , we need to expand the derivative of M NSE with respect to X 22 which involves only a couple of terms given our assumptions Townsley et al. (2009) showed that ∂MNSE ∂X22 = 0. In that study, they employed an estimate of the mass burned to NSE by measuring the amount of mass above a density of 2 × 10 7 g cm −3 . That correlation is confirmed by our current study as shown in Figure 5 where we plot the NSE yield of all realizations at each transition density. The correlation between NSE yield and mass above 2 × 10 7 g cm −3 at t DDT for a transition density of 1.26 × 10 7 g cm −3 , closest to the transition density used for that study, is highlighted. Given our assumptions and the result that the NSE yield does not depend directly on X ′ 22 ), the NSE yield only depends directly on the DDT density. The affect of X ′ 22 on the yield enters through the DDT density. Therefore, we will construct the functional dependence of Lρ DDT on X ′ 22 . The DDT density depends on X 22 via the microphysics involved in determining the conditions under which the flame transitions from a deflagration to a detonation, which, as noted above, is incompletely understood. We assume the transition occurs at a particular density at which the flame enters the distributed burning regime, but more stringent conditions for the DDT include dependences on the turbulent cascade and the growth of a critical mass of fuel with sufficiently strong turbulence (Pan et al. 2008;Woosley et al. 2009;Schmidt et al. 2010). The dependence on composition of these models may be explored and applied to the trends with DDT density presented in this study. Under our assumptions, the flame enters the distributed burning regime when the laminar flame width becomes of order the Gibson length (l G ) where Here, L is the length scale on which the strength of the turbulent velocity (u ′ ) is evaluated, and s l is the laminar flame speed (see e.g. Peters 2000). By using the laminar flame speeds and widths which depend on X 22 from Chamulak et al. (2007), we choose the conditions for a detonation are met at X 22 = 0.02 at a particular transition density (Lρ DDT ) when the Gibson length is equal to the laminar flame width. The strength of the turbulent velocity field is calculated assuming a Kolmogorov turbulence cascade: Using u ′ , we solve for the change in transition density, ∆Lρ DDT , by changing 22 Ne to X 22 = 0.06 and again setting the Gibson length equal to the laminar flame width where L cancels out of the equation. This evaluation yields a change in transition density over a change in X 22 , or dLρDDT dX22 . We use log-log fits to the flame speed and flame width as a function of density using the table generated by Chamulak et al. (2007). We use only densities above 10 8 g cm −3 for X 12 = 0.3 and above 5 × 10 8 for X 12 = 0.5. Within this parameter space, a power-law dependence of the flame speeds and widths on the log of the density is well-defined and the algorithm used to solve the flame characteristics is more stable at higher densities. Our resulting expressions for the density dependence on flame speed, s, and flame width, δ l , at the carbon mass fraction used in this study (X 12 = 0.4) for X 22 = 0.02, 0.06 are given by ln s = a ln ρ + b with coefficients given in Table 6.
Our derived expression for the derivative of transition density as a function of X 22 is given by where u = 0.4315 and v = −6.301. We solve this first-order differential equation and express the DDT density as a function of X ′ 22 .
where X ′ 22,0 is the parameterized 22 Ne mass fraction chosen to be associated with the fiducial transition density, Lρ DDT,0 . For all the transition densities considered in this study, the interface between the core composition and the composition of the outer layer is at a lower density. Therefore, the relevant 22 Ne content to consider is that of the core. Plugging in Equation (13) into Equation (5), we obtain the functional dependence of M NSE on X ′ 22 , such that Equation (14) is evaluated and plotted in Figure 11 for the fiducial transition density at the 22 Ne mass fraction used in this study, Lρ DDT (X ′ 22,0 = 0.03) = Lρ DDT,0 = 6.83.
We propagate the standard deviation of the mean evaluated at Lρ DDT,0 for a range of X ′ 22 . This is calculated by considering the relation between the standard deviation of the mean of the NSE mass and the standard deviation of the mean of the fitting parameter, c, given by where σ NSE is the standard deviation of the mean of the NSE mass and σ c is the standard deviation of the mean in the fitting parameter c. We evaluate σ c by inverting Equation (16) and solving for X ′ 22 = X ′ 22,0 . Plugging in this solution, the standard deviation of the mean of the NSE mass as a function of X ′ 5.3. 56 Ni Yield Dependence on Metallicity Now that we have constructed the functional dependence of NSE yield on 22 Ne we need to consider the dependence of the amount of 56 Ni synthesized in the explosion on metallicity through the 22 Ne content. A fractional amount of the NSE material is radioactive 56 Ni, which powers the supernova light curve. This fraction depends on Y e (and 22 Ne) as described in Timmes et al. (2003): In order to determine the dependence of M 56 on metallicity through the DDT density, we must construct f 56 and   (1) contains a term, ∆X 22 , which is a parameterization of the change in Y e due to neutronization during the carbon simmering phase. However, not all of the neutron excess evaluated at the M NSE plateau time comes from 22 Ne or the carbon simmering products. Some change in Y e is due to weak reactions occuring during the explosion that are included in our burning model Townsley et al. 2007).
First we consider the amount of non-56 Ni NSE material determined by X ′ 22 by equating the initial Y e to the Y e of material in NSE. Using baryon and lepton conservation for NSE material, we describe the electron fraction by contributions from 54 Fe, 58 Ni , and 56 Ni, where f non−56 is the mass fraction of non-56 Ni NSE material. For the following evaluation, we approximate the composition to be that of the core because most of the NSE material is within the core. We write the initial Y e as Y e,i = 10 22 We can then solve for f non−56 by equating (19) and (20) and add a term X n to represent the additional contribution due to neutronization from weak reactions occuring during the explosion Then the 56 Ni fraction of material in NSE is We note that the rate of weak reactions occuring during the explosion may depend on the initial composition as well. Accordingly, X n may have a dependence on X ′ 22 . We also expect X n to vary as a function of the transition density because a higher transition density will have less time for weak reactions to occur.
We construct a statistical sample of f 56 using the ratio of the 56 Ni and NSE yields produced in the simulations. We calculate the dependence of X n on transition density using a least-squares method and the result is shown in Figure 12. Evaluating the partial derivative of X n at the fiducial transition density, we find a shallow dependence: ∂X n ∂Lρ DDT LρDDT=LρDDT,0 = −0.096.  (22) and using the f 56 = M 56 /M NSE averaged over all realizations at each DDT density. The standard deviation of the sample (outer) and mean (inner) are shown as error bars at each DDT density. A quadratic best fit line is calculated using the standard deviation of the sample. The fiducial DDT density is shown as a black circle where Xn = 0.133.
We want to know the dependence of M 56 on X 22 . Since f 56 depends on the neutronization from weak reactions during the explosion, we must consider any dependencies of this neutronization on DDT density or X ′ 22 . We show that the dependence of f 56 on X ′ 22 via the effect of X ′ 22 on the in-situ neutronization is much weaker than the direct dependence due to lepton number conservation. Similar to Equation (18) Referring to Townsley et al. (2009), we can estimate ∂Xn ∂X22 by calculating the average ratio of M 56 to M NSE at X 22 = 0, 0.02 for the first 5 realizations whose detonation phases were simulated. The result is ∂Xn ∂X22 ∼ −0.2. The first term in Equation (27) can be evaluated from multiplying Equation (23) and Equation (12) for Lρ DDT in the range 7.0-7.5. For the fiducial transition density (Lρ DDT,0 ), we find Comparing to ∂ f56 ∂X22 ≃ −2.5, we find that the magnitude of dXn dX22 is much smaller and is unimportant for our evaluation of the dependence of the 56 Ni yield on X 22 . Therefore, we can ignore this term in the expansion of d f56 dX22 in Equation (26) and let dXn dX22 ∼ 0. The full derivative of f 56 with respect to X 22 can now be written as a partial derivative such that While we approximate X n as constant, we choose to evaluate it at Lρ DDT,0 using the best fit curve from Figure 12 obtaining X n = 0.133. Now we can relate the metallicity to 22 Ne since X 22 traces metallicity (Timmes et al. 2003). Substituting X 22 = 0.014(Z/Z ⊙ ) in Equation (1), we obtain where ∆X 22 = 0.01 for this study and X ′ 22 is our parametrization of the actual 22 Ne mass fraction, X 22 . The 56 Ni yield as a function of X ′ 22 is calculated by multiplying Equation (22) by Equation (14) and using X n = 0.133. The 56 Ni yield is plotted in Figure 13 using Equation (30)

CONCLUSIONS AND FUTURE WORK
We have analyzed the influence of the DDT density on the total 56 Ni synthesized during a thermonuclear supernovae. We determined that the dependence on DDT density is quadratic in nature, but this dependence can be described by a single parameter. We estimated the dependence of the supernova brightness ( 56 Ni yield) on metallicity by assuming a DDT occurs when the flame enters the distributed burning regime and extrapolating the laminar flame speeds and widths down to low densities. We find which is slightly steeper than Timmes et al. (2003) as seen in Figure 13. The uncertainty was calculated using the standard deviation of the mean of the fitting parameter c-or equivalently, the standard deviation of the mean of the NSE yield at a particular DDT density. The uncertainty in the assumptions about the normalization of the transition density as a function of X ′ 22 was not considered for the purpose of evaluating the uncertainty in the derivative. We find the effect of metallicity on DDT density influences the production of NSE material; however, the ratio of the 56 Ni yield to the overall NSE yield does not change as significantly and remains similar to the relation estimated from approximate lepton number conservation. We also find that the scatter in supernova brightness increases with decreasing transition density.
The very recent work of Bravo et al. (2010) on metallicity as a source of dispersion in the luminosity-width relationship of bolometric light curves addresses many of the same issues as our study and warrants discussion. In particular, Bravo et al. (2010) derive a metallicity dependence on the DDT density similarly to our study. However, the DDT density in their 1D simulations is very different from the DDT density in ours. The principal difference between the work described in this manuscript and that of Bravo et al. (2010) follows from the use of multi-dimensional simulations. While 3D simulations are required to correctly capture the effects of fluid dynamics and the Rayleigh-Taylor instability, 2D simulations incorporate these effects and relax the assumption of symmetry. Breaking the symmetry and assuming that a DDT initiates as a rising plume approaches ρ DDT produces an expansion history very different from what would be observed in 1D simulations. In fact, using 1D simulations implies that ρ DDT plays an unphysical role. The real physical description of the DDT in a type Ia supernova will depend on flame-turbulence interactions that will themselves depend on multi-dimensional effects in the flow. 1D models may be able to capture these effects, but such models must be motivated by more physical multi-dimensional studies.
In addition, our statistical framework with realizations from randomized initial conditions allows calculation of the scatter inherent in the multi-dimensional models. The standard deviations calculated for our models averaged over a set of realizations demonstrate that there can be considerable variation following from the randomized initial conditions. These variations follow from the different amounts of expansion occurring during the deflagration phase that follow from the different rising plume morphologies. In 1D models, even with the progenitor metallicity determining ρ DDT , there is a one-toone correspondence between expansion and ρ DDT for a given progenitor. Bravo et al. (2010) explored two scenarios, a linear and a non-linear dependence of the 56 Ni yield on Z, and report excellent agreement between the non-linear scenario and observations reported by Gallagher et al. (2008) with the caveat that they used 1D models to arrive at their conclusion. Without comparing results involving the metallicity dependence on ρ DDT , Bravo et al. (2010) find a stronger dependence of the 56 Ni synthesized from incomplete Si-burning on Z than 56 Ni coming from NSE material. We find a shallower dependence of the 56 Ni yield on Z than Bravo et al. (2010) in part due to our assumption that 56 Ni is synthesized as a fraction of NSE material; however, comparing their stratified models to the Timmes et al. (2003) relation results in only a ∼ 20% difference in the dependence of 56 Ni yield on Z. This effect is compounded in their "non-linear" scenario, but again, we must emphasize that ρ DDT plays a completely unphysical role in their simulations and the degree to which this effect is actually enhanced is unknown.
The light curve width calculations and subsequent popula-tion synthesis performed by Bravo et al. (2010) after finding a dependence of the 56 Ni yield on metallicity is useful in estimating a metallicity dependence in the hubble residual. For this purpose, we compare the results of the non-linear scenario of Bravo et al. (2010) to our study along with the expected dependence due to lepton number conservation (Timmes et al. 2003) in Figure 13. Bravo et al. (2010) find a ∼ 30% steeper dependence of the 56 Ni yield on Z/Z ⊙ evaluated at Z = 1Z ⊙ than our result in Equation (31). In addition, they find that for high Z the 56 Ni yield tends to flatten out becoming more similar to our results. For a particular subrange of metallicites, our results are very similar indicating that by performing the same analysis as Bravo et al. (2010) our results should also agree with the metallicity dependence found by Gallagher et al. (2008). This similarity depends on the choice of mean 56 Ni yield as both Timmes et al. (2003) and Bravo et al. (2010) relations are proportional to this quantity. This implies that the steepness of the relations are affected by the choice of mean 56 Ni yield. Our results are not sensitive to this choice. Choosing a higher fiducial DDT density of 10 7 g cm −3 that results in a higher mean 56 Ni yield of 0.77 M ⊙ increases Equation (31) by only 0.005 M ⊙ . We see less dependence on the mean 56 Ni yield because, although we have a similar dependence on X 22 due to lepton number conservation, the dependence of the yield on Lρ DDT is stronger at lower fiducial Lρ DDT,0 , as shown in previous sections. This effect is not captured in a simple proportionality relation like that quoted by Bravo et al. (2010), even if this effect is present in their calculations.
It has been discussed whether the source of scatter in peak brightness itself can be attributed to metallicity (Timmes et al. 2003;Howell et al. 2009;Bravo et al. 2010). We submit that multi-dimensional effects following from fluid instabilities during the deflagration phase leading to varying amounts of expansion provide scatter consistent with observations. The influence of metallicity on the DDT density affects the duration of the deflagration which will secondarily influence the magnitude of the scatter. Our results show that the primary parameter is the degree of (pre-)expansion before DDT-which determines the amount of mass at high density. This is, in turn, controlled by both the expansion rate and the plume rise time. These are also expected to be the basic ingredients in reality.
While our study is consistent with the expected theoretical brightness trend with metallicity (Timmes et al. 2003), observations to date have not been able to confirm this prediction (Gallagher et al. 2005(Gallagher et al. , 2008Howell et al. 2009). The progenitor age is difficult to decouple from metallicity given the mass-metallicity relationship within galaxy types (Gallagher et al. 2008). Additionally, the dependence of mean brightness of SNe Ia on the age of the parent stellar population appears to be much stronger than any dependence on metallicity (Gallagher et al. 2008;Howell et al. 2009). Seemingly, the only way to observe a dependence on metallicity is to constrain the mean stellar age by selecting galaxies of the same type. This approach was used by Gallagher et al. (2008), but the difficulties in accurately measuring metallicities for the parent stellar population have only constrained the magnitude of the effect on mean brightness, but thus far have not proved that a metallicity effect exists.
In order to determine the dependence of supernova brightness ( 56 Ni yield) on metallicity, a better understanding of how the transition density is affected by changes in the 22 Ne content is needed. Currently, we have extrapolated data from Chamulak et al. (2007) down to the range of expected transition densities using flame data from densities above 10 8 g cm −3 . The trend observed by Chamulak et al. (2007) that increasing X 22 increases the laminar flame speed is valid for densities above 10 8 g cm −3 ; however, it is unclear if this trend will hold for densities below 10 7 g cm −3 . Direct numerical simulations of the flame are needed for these lower densities to determine whether this trend still holds. However, in this parameter space, the laminar flame is extremely slow requiring a low-Mach-number treatment to model the flow. Regardless of the choice of model for the mechanism that produces a spontaneous detonation, a necessary condition is thought to be distributed burning. Properties of the laminar flame are necessary to estimate when burning becomes distributed. Future studies exploring other DDT mechanisms must also determine their compositional dependencies. So far, we have only explored the effect of varying the 22 Ne content directly ) and indirectly through the DDT density. Other effects exist that we have yet to study such as the carbon composition and flame ignition density (set by the average progenitor age). These properties may be influenced by metallicity such that the net affect on mean brightness is negligible. Additionally, if our assumptions about the dependence of DDT density on metallicity are incorrect and further studies indicate the opposite trend, the metallicity effect on DDT density could negate the effect due to lepton number conservation.
This study stresses the importance of multi-dimensional effects, but has provided evidence of the limitations of 2D simulations of supernovae. In the immediate future, we plan to perform 3D simulations for better realism. In reality, we might expect a two-parameter family in which the morphology of the dominant Rayleigh-Taylor plume is independent of the rate of expansion; however, for our 2D simulations, these two effects appear to be correlated by the choice of cylindrical geometry. A plume developing along the symmetry axis is like 3D RT compared to one near the equator, which is more like 2D RT, and the latter is weaker (slower to develop). In addition to developing faster, plumes near the symmetry axis represent less volume and, therefore, expand the star less. A fast rising plume combined with a slow rate of expansion indicates there will be a shallow dependence on DDT density (since the plume will move through the various DDT densities faster and with less expansion). This result may also depend on our choice that DDT conditions are met at the tops of rising plumes. For these reasons, 3D studies are necessary to ascertain whether there is a physically motivated correla-tion between the dominant plume morphology and the rate of expansion and whether this correlation depends on the choice of location of the DDT. In transitioning to 3D simulations, we expect the growth of many more plumes; however, the rate of expansion will likely not exceed that found in 2D. Because RT develops faster in 3D, the conditions for DDT will be met sooner leading to less overall expansion of the star and more mass at high density at the first DDT time. Therefore, we expect 3D results similar to the higher yielding 2D realizations with a shallower dependence on DDT density. In any case, this study indicates that it is possible to determine the susceptibility of a particular model to a change of DDT density (and, hence, a change in the composition which alters the microphysics that set the DDT density). Once the plume morphology has been established several tenths of seconds into the simulated explosion, it should be possible to estimate the total 56 Ni yield to fairly high accuracy given DDT conditions. However, the treatment of turbulent flame properties may be important.