TYPE Ia SUPERNOVA EXPLOSIONS FROM HYBRID CARBON–OXYGEN–NEON WHITE DWARF PROGENITORS

In the Chandrasekhar-mass scenario, the central temperature and density of the WD increase as it accretes mass from a binary companion and approaches the limiting Chandrasekhar mass. As the mass approaches the limit, central conditions become hot enough for carbon fusion to begin (via the ${}^{12}{\rm{C}}$–${}^{12}{\rm{C}}$ reaction), driving the development of convection throughout the interior of the WD (Baraffe et al. 2004; Woosley et al. 2004; Wunsch & Woosley 2004; Kuhlen et al. 2006; Nonaka et al. 2012). As the central temperature reaches ∼7 × 10⁸ K, the fuel in a convective plume burns to completion before it can cool via expansion (Nomoto et al. 1984; Woosley et al. 2004), and a flame is born.

The nature of this burning, be it a supersonic detonation or subsonic deflagration, largely determines the outcome of the explosion. It has been known for some time that a purely supersonic burning front cannot explain observations because the supersonic front very rapidly incinerates the star without it having time to react and expand (Arnett et al. 1971). The lack of expansion allows most of the star to burn at high densities, which produces an excessive ⁵⁶Ni yield and does not match the stratified composition of observed remnants (Mazzali et al. 2008). Instead, a subsonic deflagration must ignite, which allows the outer layers of the star to expand ahead of the burning front. In this case, the density of the expanding material decreases, which leads to incomplete burning of more mass and thus increased production of intermediate-mass elements. This deflagration must accelerate via instabilities and turbulent interaction, a topic that has been explored extensively in the past (Khokhlov 1993, 1995; Bychkov & Liberman 1995; Niemeyer & Hillebrandt 1995; Khokhlov et al. 1997; Cho et al. 2003, pp. 56–98; Röpke et al. 2003, 2004; Bell et al. 2004; Zingale et al. 2005; Schmidt et al. 2006a, 2006b; Zingale & Dursi 2007; Aspden et al. 2008; Ciaraldi-Schoolmann et al. 2009, 2013; Woosley et al. 2009; Hicks & Rosner 2013; Jackson et al. 2014; Hicks 2015; Poludnenko 2015).

A deflagration alone will not produce an event of normal brightness and expansion velocity (Röpke et al. 2007). Instead, the initial deflagration must transition to a detonation after the star has expanded some in order to produce abundances and a stratified ejecta in keeping with observations (Khokhlov 1991; Hoflich et al. 1995). The physics of this "deflagration-to-detonation transition" (DDT) are not completely understood, but there has been considerable study based on mechanisms involving flame fronts in highly turbulent conditions (Blinnikov & Khokhlov 1986; Woosley 1990; Khokhlov 1991; Hoflich et al. 1995; Höflich & Khokhlov 1996; Khokhlov et al. 1997; Niemeyer & Woosley 1997; Hoeflich et al. 1998; Niemeyer 1999; Gamezo et al. 2005; Röpke 2007; Poludnenko et al. 2011; Ciaraldi-Schoolmann et al. 2013; Poludnenko 2015). These models generally reproduce the observations under certain assumptions about the ignition (Townsley et al. 2009), but research has shown that the results are very sensitive to the details of the ignition (Plewa et al. 2004; Gamezo et al. 2005; García-Senz & Bravo 2005; Röpke et al. 2007; Jordan et al. 2008). In our simulations, we initialize a detonation once the deflagration front reaches a characteristic DDT fuel density, which controls the degree of expansion the star undergoes during the deflagration stage. The implementation details are described further in Section 3.4.

Contemporary observational campaigns typically investigate how the brightness and rates of SNe correlate to properties of the host galaxy such as mass and star formation rate (see Graur & Maoz 2013; Graur et al. 2015). Of particular interest is the delay-time distribution (DTD), the SN rate as a function of time elapsed from early, rapid star formation in the host galaxy, and how it may be used to constrain progenitor models (Hachisu et al. 2008; Bianco et al. 2011; Conley et al. 2011; Graur et al. 2011; Howell 2011; Maoz et al. 2012). See also the review by Maoz & Mannucci (2012). Very recent results indicate evolution of the UV spectrum with redshift, providing evidence for systematic effects with cosmological time (Milne et al. 2015).

Motivated by this interest in correlations between properties of the host galaxy and the brightness and rate of events, earlier incarnations of our group performed suites of simulations in the DDT scenario with a modified version the FLASH code (described below) to explore systematic effects on the brightness of an event measured by the yield of ⁵⁶Ni (Jackson et al. 2010; Krueger et al. 2010, 2012). The study we present here explores how explosions following from a new class of "hybrid" progenitors (Denissenkov et al. 2013b, 2015; Chen et al. 2014) compare with these previous results.

The combustion model in FLASH that we use for SN Ia simulations (Calder et al. 2007; Townsley et al. 2007, 2009, 2016; Seitenzahl et al. 2009; Jackson et al. 2014) separates the burning into four states: unburned fuel, C-fusion ash, a silicon-group-dominated nuclear statistical quasi-equilibrium (NSQE) state, and a full nuclear statistical equilibrium (NSE) state dominated by iron-group elements (IGEs). The progress of combustion from one of these states to the next is tracked by three scalar progress variables whose dynamics is calibrated to reproduce the timescales of reactions that convert material among these states. Previous work has focused on fuel mixtures composed principally of ¹²C, ¹⁶O, and ²²Ne. Simulation of the hybrid models required extension of this burning model to account for the presence of ${}^{20}\mathrm{Ne}$ as a large abundance in the fuel. Here we describe both how ²⁰Ne is processed during combustion and the modifications made to the burning model in order to incorporate 20Ne burning.

The burning stages above are determined by the hierarchy of timescales for the consecutive consumption of C and O via fusion and Si via photodisintegration and alpha capture. Investigation of the inclusion of ${}^{20}\mathrm{Ne}$ focused on whether an additional Ne-consumption stage would be required, and, if not, what stage should include Ne consumption. In order to characterize the physical burning sequence that we want to model, we performed a series of simulations of detonations propagating through WD material with the TORCH nuclear reaction network software (Timmes 1999, 2015).⁶ TORCH is a general reaction network package capable of solving networks with up to thousands of nuclides. A mode is implemented that computes the one-dimensional spatial thermodynamic and composition structure of a steady-state planar detonation using the the Zel'dovich, von Neumann, and Döring (ZND) model (Fickett & Davis 1979; Townsley et al. 2016). We use a reaction network composed of 225 nuclides consisting of the 200 nuclides in Woosley & Weaver (1995) in addition to the 25 neutron-rich nuclides added by Calder et al. (2007) to improve coverage of electron capture processes in the Fe group.

For the multi-species fuel and ash relevant to typical WD material, the ZND detonation exhibits the stages that motivate the combustion model. Figure 3 shows these stages as they appear in a ZND detonation calculation in fuel with the fractional composition (${}^{12}{\rm{C}}$ = 0.50, ${}^{16}{\rm{O}}$ = 0.48, ${}^{22}\mathrm{Ne}$ = 0.02) corresponding to the composition found in the interior of a C–O WD, and a fuel density of 10⁷ g cm⁻³. The evolution of the mass fractions in time following the passage of the shock through the zone of material is plotted below the density structure in Figure 3. The time ranges of the four states representing the burning stages are indicated by colors in the top panel of Figure 3, with the consecutive states separated by the ${}^{12}{\rm{C}}$–${}^{28}\mathrm{Si}$, ${}^{16}{\rm{O}}$–${}^{28}\mathrm{Si}$, and ${}^{28}\mathrm{Si}$–${}^{54}\mathrm{Fe}$ crossing times. It can be seen that the times of the density plateaus are directly comparable to the times at which the primary energy release transitions from one fuel source to another, as, e.g., when the ${}^{12}{\rm{C}}$ fraction has fallen to ≈1% of its initial value just before 10⁻⁸ s.

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In our combustion models for C–O progenitors, consumption of the initial fuel is modeled as a two-step process. The two stages represent the consumption first of ${}^{12}{\rm{C}}$, then of ${}^{16}{\rm{O}}$, mimicking the sequence seen in the detonation structure shown in Figure 3 for "Fuel" and "Ash" stages. At the end of this second stage the material is in a ${}^{28}\mathrm{Si}$-dominated NSQE state (Calder et al. 2007; Townsley et al. 2007). To determine how the burning stages change with the inclusion of ${}^{20}\mathrm{Ne}$, as is the case in the hybrid C–O–Ne progenitor, we perform ZND calculations with an admixture of ${}^{20}\mathrm{Ne}$ ranging from 0.01 to 0.45, at the expense of ${}^{12}{\rm{C}}$ content. For each composition we find the minimally overdriven solution, as was done in the C–O case. Out to the minimum in density, this solution is the same as the eigenvalue ZND solution, which corresponds to a self-supported detonation (Fickett & Davis 1979; Townsley et al. 2016). This computation gives the resolved detonation structure in C–O–Ne hybrid WD matter.

The eigenvalue detonation speeds from ZND calculations in material with ${}^{12}{\rm{C}}$ fraction varying from 0.05 to 0.5 are shown in Figure 4, demonstrating that self-supported detonations in this progenitor are feasible with only small variation in speed across this range of ${}^{12}{\rm{C}}$ fractions. Figure 5 shows the effect of simultaneously adding ${}^{20}\mathrm{Ne}$ and reducing ${}^{12}{\rm{C}}$ on the density profile. Lowering the ${}^{12}{\rm{C}}$ fraction weakens the shock and lengthens the timescales of the step features, corresponding to more slowly burning fuel and ash, as might be expected from the lower energy release afforded by the ${}^{20}\mathrm{Ne}$. However, it is noteworthy that no qualitatively new features arise from the change in fuel source that would suggest that more than four representative burning stages are needed.

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Qualitative similarity between C–O and C–O–Ne detonation structures is visible in the mass fractions as well. To demonstrate this, the abundance structures with initial ${}^{12}{\rm{C}}$ abundances of 0.5, 0.3, 0.15, and 0.05 are shown in Figure 6. We find that the ${}^{20}\mathrm{Ne}$ burns simultaneously with whatever ${}^{12}{\rm{C}}$ is present, producing primarily ${}^{28}\mathrm{Si}$. The Ne–C-burning stage is then followed by ${}^{16}{\rm{O}}$ burning to silicon-group NSQE elements and then on to NSE, just as in a model with no initial ${}^{20}\mathrm{Ne}$, except for the progressively later ${}^{16}{\rm{O}}$ burning time.

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A graphic representation of the most significant nuclides by mass fraction and the stage in which they are important is shown in Figure 7 for an initial ${}^{12}{\rm{C}}$ fraction of 0.15, representative of the majority of the hybrid stellar profile (see Figure 1). Nuclides are categorized based on the time at which they were maximally abundant in the network and shaded by their maximum abundance. Nuclides within the purple color palette were maximally abundant during the initial "fuel-burning" stage after the beginning of fusion and before the ${}^{28}\mathrm{Si}$ becomes equally abundant with ${}^{12}{\rm{C}}$, the ${}^{12}{\rm{C}}$–${}^{28}\mathrm{Si}$ crossing time t_fa. Blue nuclides were maximally abundant after t_fa but prior to the ${}^{16}{\rm{O}}$–${}^{28}\mathrm{Si}$ crossing time t_aq, and nuclides maximally abundant after t_aq but before the ${}^{28}\mathrm{Si}$–${}^{54}\mathrm{Fe}$ crossing time t_qn are plotted in green. Nuclides maximally abundant following t_qn are IGEs together with protons and alpha particles and are shaded in orange. Note that for the timescales we adopt the notation of Townsley et al. (2016), which describes the burning stages in detail.

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The burning model we are using computes the rates for progression through the burning stages from the local temperature and the energy release from a set of major fuel abundances, previously including ${}^{12}{\rm{C}}$, ${}^{16}{\rm{O}}$, and ${}^{22}\mathrm{Ne}$. We have found here that any ${}^{20}\mathrm{Ne}$ is consumed along with ${}^{12}{\rm{C}}$ and that otherwise the burning is quite similar to that with just ${}^{12}{\rm{C}}$ and ${}^{16}{\rm{O}}$ as principal constituents. In consequence, the only necessary modification to our burning model is to include ${}^{20}\mathrm{Ne}$ in the abundances of the initial state in the burning model. This accounts for the difference in binding energy of ${}^{20}\mathrm{Ne}$ compared to ${}^{12}{\rm{C}}$ and gives lower burning temperatures. Additionally, throughout the majority of the progenitor outside the core ignition region, due to prior convective mixing, the ${}^{12}{\rm{C}}$ content is high enough that we will extrapolate the laminar flame speeds of Timmes & Woosley (1992) and Chamulak et al. (2008). We consider this a reasonable approximation since much of the flame propagation is dominated by Rayleigh–Taylor overturn and turbulence.

The temperature and composition at the base of the convective zone in the hybrid C–O–Ne WD provide a natural flame initialization region for simulation of thermonuclear runaway in the DDT scenario, so we map the MESA profile into the FLASH domain, preserving its features at 4 km spatial resolution. We do this by first converting the MESA model to a uniform grid by mass-weighted averaging of quantities in MESA zones with spacing less than 4 km and using quadratic interpolation to estimate quantities where MESA zones have spacing greater than 4 km. Although our combustion model in FLASH does not evolve nuclide abundances, it uses the initial abundances of ${}^{12}{\rm{C}}$, ${}^{20}\mathrm{Ne}$, and ${}^{22}\mathrm{Ne}$ (assuming that the rest is ${}^{16}{\rm{O}}$) to compute the initial mean nuclear binding energy and electron fraction. Therefore, we also represent the full set of nuclides in the MESA profile by this reduced set of four nuclides in the uniformly gridded profile, requiring the carbon mass fractions to be identical because there is still sufficient ${}^{12}{\rm{C}}$ in the star to sustain a detonation front. In addition, we use ${}^{22}\mathrm{Ne}$ in the reduced set to account for the Y_e of the full set of nuclides, and we constrain ${}^{20}\mathrm{Ne}$ and ${}^{16}{\rm{O}}$ to be in the same ratio R in both sets of abundances. These constraints provide the following definitions for the reduced abundances used for FLASH:

$$\begin{eqnarray}&&{X}_{{}^{12}{\rm{C}}}^{\mathrm{FLASH}}={X}_{{}^{12}{\rm{C}}}^{\mathrm{MESA}},\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}&&{X}_{{}^{22}\mathrm{Ne}}^{\mathrm{FLASH}}=22\times \left(\displaystyle \frac{1}{2}-{Y}_{e}^{\mathrm{MESA}}\right),\end{eqnarray} \tag{ 1b }$$

$$\begin{eqnarray}&&R={X}_{{}^{20}\mathrm{Ne}}^{\mathrm{MESA}}/{X}_{{}^{16}{\rm{O}}}^{\mathrm{MESA}},\end{eqnarray} \tag{ 1c }$$

$$\begin{eqnarray}&&{X}_{{}^{16}{\rm{O}}}^{\mathrm{FLASH}}=\displaystyle \frac{1-{X}_{{}^{12}{\rm{C}}}^{\mathrm{FLASH}}-{X}_{{}^{22}\mathrm{Ne}}^{\mathrm{FLASH}}}{R+1},\end{eqnarray} \tag{ 1d }$$

$$\begin{eqnarray}&&{X}_{{}^{20}\mathrm{Ne}}^{\mathrm{FLASH}}=R\times {X}_{{}^{16}{\rm{O}}}^{\mathrm{FLASH}}.\end{eqnarray} \tag{ 1e }$$

Nothing constrains the resulting uniformly gridded profile to be in HSE, however, so we next construct an equilibrium profile by applying the HSE pressure constraint

$$\begin{eqnarray}&&{P}_{\mathrm{EOS}}({\rho }_{i},{T}_{i},{X}_{i})=\displaystyle \frac{g{\rm{\Delta }}r}{2}({\rho }_{i}+{\rho }_{i-1}).\end{eqnarray} \tag{ 2 }$$

Starting at the uniformly gridded profile's central density and using its temperature and composition in each zone together with the Helmholtz equation of state (EOS) of Almgren et al. (2010), we solve Equation (2) for the density in each zone. In Equation (2), Δr indicates the zone width, T_i is the temperature in zone i, X_i is the composition in zone i, and g is the gravitational acceleration at the boundary of zone i and $i-1$ due to the mass enclosed by zone $i-1$ and below. The resulting uniformly gridded equilibrium profile (FLASH) and the original profile (MESA) are shown for comparison in Figures 1 and 2. The values of the total mass before and after this procedure differ by 6 × 10⁻³ M_⊙. This procedure produced a structure that was stable in FLASH, with fluctuations in central density less than 3%, for at least 5 s with no energy deposition. Finally, we replaced the EOS routine in the public FLASH distribution with that from CASTRO  (Timmes & Swesty 2000; Almgren et al. 2010) to obtain a more consistent tabulation.⁷

To simulate the explosion from either hybrid or traditional progenitor models, we use a modified version of FLASH,⁸ an Eulerian adaptive mesh compressible hydrodynamics code developed by the ASC/Alliances Center for Astrophysical Thermonuclear Flashes at the University of Chicago (Fryxell et al. 2000; Calder et al. 2002). While FLASH is capable of evolving thermonuclear reaction networks coupled to the hydrodynamics, in order to treat a ≤1 ${\rm{cm}}$ flame front in full-star simulations of SNe Ia, we use a coarsened flame model that uses an advection-diffusion-reaction scheme to evolve a scalar variable representing the progression of burning from fuel to ash compositions as detailed in Calder et al. (2007) and Townsley et al. (2007, 2009). An additional scalar represents the burning progress from ash to intermediate-mass silicon-group elements in NSQE. A final scalar represents the burning of silicon-group elements to IGEs in NSE. The timescales for evolving these scalars are density and temperature dependent and are determined from self-heating and steady-state detonation calculations with a 200+-species nuclear reaction network (Calder et al. 2007; Townsley et al. 2016). For our evaluation of the suitability of this burning scheme for fuel with an admixture of ${}^{20}\mathrm{Ne}$, see Section 3.1.

The two-dimensional (2D) models we present do not utilize a subgrid-scale model for the turbulence–flame interaction (TFI). Earlier work produced subgrid-scale TFI models by extending terrestrial combustion models to consider astrophysical flames (Schmidt et al. 2006a, 2006b; Jackson et al. 2014); however, the resulting subgrid-scale models only capture TFI in three-dimensional (3D) simulations. 2D models inherently lack true turbulence, so instead we use the mimimal enhancement based on the Rayleigh–Taylor strength introduced by Townsley et al. (2007). This assumes that the TFI will self-regulate on resolved scales so that results are insensitive to the detailed treatment of the TFI. We believe this to be sufficient for 2D comparative studies like that undertaken here.

This study presents simulations of the thermonuclear explosions of both a hybrid C–O–Ne WD progenitor (Denissenkov et al. 2015) and a C–O WD progenitor similar to that used in previously published suites of SN Ia simulations (Krueger et al. 2012). In both cases, we initialize the simulations with a "matchhead" consisting of a region near or at the WD's center that is fully burned to NSE. The energy release from this initial burn ignites a subsonic thermonuclear flame front that buoyantly rises and partially consumes the star while the star expands in response. As detailed in Townsley et al. (2009) and Jackson et al. (2010), in order to effect a DDT, we suppose the DDT point to be parameterizable by a fuel density ρ_DDT at which the subsonic flame reaches the distributed burning regime where the flame region has become sufficiently turbulent that a supersonic detonation front may arise, self-supported by the energy release from the nuclear burning proceeding behind the detonation shock front.

We thus use a similar DDT parameterization for our C–O and hybrid C–O–Ne simulations as in the C–O WD SN Ia simulations of Krueger et al. (2010, 2012) for consistency. Our choice is to ignite the detonation when a rising plume reaches the threshold density, but reality may be quite different. For example, the turbulent intensity may be greatest at the underside of a rising plume, which would imply igniting a detonation deeper in the core and possibly later, both of which would affect the yield. For purposes of this paper, which was to compare the yield from a hybrid model to a traditional C–O model, we prefer to use a simply consistent condition across cases. Jackson et al. (2010) address the dependence of the DDT density on composition separately.

Consequently, when the deflagration reaches a point where it is at ρ_DDT = 10^7.2 g ${\rm{cm}}$⁻³, we place a region with its fuel stage (representing ${}^{12}{\rm{C}}$ and ${}^{20}\mathrm{Ne}$, if present) fully burned 32 km radially outward from this point that is of size 12 km in radius. Multiple DDT points may arise, but they are constrained to be at least 200 km apart. Our choice of DDT density is slightly high in order to ensure robust ignition of a detonation shock in all cases. We note that this choice strongly influences the ${}^{56}\mathrm{Ni}$ yield because it determines the amount of expansion of the star prior to the detonation, which determines the overall density profile (Jackson et al. 2010).

We performed our calculations in 2D z − r cylindrical coordinates, extending radially from 0 to 65,536 km and along the axis of symmetry from −65,536 to 65,536 km. We selected a maximum refinement level corresponding to 4 km resolution using the PARAMESH adaptive mesh refinement scheme described in Fryxell et al. (2000). This resolution permitted efficiency in performing many repeated simulations with different initial conditions. The 4 km resolution was also informed by previous resolution studies in Townsley et al. (2007, 2009), which found that in 2D DDT simulations of C–O WD explosions the trends with resolution of total mass above the DDT density threshold at the DDT time are fairly robust. The amount of high-density mass at the DDT is important because it reflects the extent of neutronization during the deflagration and thus correlates with the IGE yield of the explosion. In addition, we wish to compare the hybrid IGE yields and other explosion characteristics with those of explosions from C–O WD progenitors previously explored in Krueger et al. (2012), which used 4 km resolution in FLASH with the same z − r geometry we describe above. We can thus make our comparison robust by controlling for resolution and geometry factors.

To compare the resulting features of SNe Ia produced by the hybrid model to those of previous studies of centrally ignited C–O WDs in the DDT paradigm (Krueger et al. 2010, 2012), we generate a reference C–O model. The reference model has the same central density as the hybrid model with a central temperature of 7 × 10⁸ K, a core composition of (${}^{12}{\rm{C}}$ = 0.4, ${}^{20}\mathrm{Ne}$ = 0.03, ${}^{16}{\rm{O}}$ = 0.57), and an envelope composition of (${}^{12}{\rm{C}}$ = 0.5, ${}^{20}\mathrm{Ne}$ = 0.02, ${}^{16}{\rm{O}}$ = 0.48). For comparison, Figures 8 and 9 show the density and temperature profiles of the C–O and C–O–Ne hybrid WD.

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We use the deflagration initialization of Krueger et al. (2012), which followed the method of Townsley et al. (2009). Townsley et al. (2009) reported that the scatter in ${}^{56}\mathrm{Ni}$ yield from SNe Ia could be produced in a theoretical sample of SN Ia simulations from C–O WD progenitors in the DDT scenario if the evolution of the flame surface shortly following the start of deflagration is described by a deformed spherical burned region within the progenitor core. This deformation consists of a perturbation of the sphere's surface by a set of spherical harmonic functions with randomly chosen amplitudes. The theoretical SN Ia sample is formed by thus constructing a set of randomized burned geometries and then using each as the initial condition for a DDT simulation of the explosion.

We adopt this procedure to burn a spherical ignition region extending to a nominal radius of 150 km before applying a set of 35 randomly seeded amplitude perturbations, including those of Krueger et al. (2012), to ignite a suite of 35 C–O WD explosions. Finally, previous work has made the point that the explosion yield from simulations such as these may be sensitive to the initial conditions, motivating us to perform suites of 2D simulations instead of a modest number of 3D simulations (Calder et al. 2011). Although this kind of "vigorous" ignition is disfavored by convection zone ignition simulations (Nonaka et al. 2012), it appears necessary for DDT simulations to reproduce observed SNe (e.g., Seitenzahl et al. 2013; Sim et al. 2013).

Given the temperature and composition profile of the hybrid model, if it is to undergo thermonuclear runaway, ${}^{12}{\rm{C}}$ ignition will begin at the base of the convective zone, where temperatures are highest. This coincides with a ${}^{12}{\rm{C}}$ abundance of ≈0.14 at a stellar radius of 350 km. Because the hybrid progenitor model was produced by the one-dimensional MESA code and has only radial dependence, a shell-like ignition region naturally results when mapping the model to a multidimensional grid. Because this scheme cannot adequately capture the effects of any low-mode, large-scale convection present in such a star, it is not clear what is the correct choice of ignition geometry for this hybrid WD model.

Lacking constraints on the exact geometry of the ignition region, we modify the ignition scheme described in Section 4.1 minimally for the hybrid progenitor by replacing the burned sphere with a thin burned shell modulated by a spherical harmonic function with variable amplitude and harmonic number, as shown in Figure 10 for different choices of the harmonic number. With this parameterization, the harmonic number controls the spatial localization and the amplitude controls the amount of initially burned mass. We therefore use our ignition parameterization to generate a suite of 35 hybrid realizations corresponding to a range of initially burned masses from 3.0 × 10⁻³ ${M}_{\odot }$ to 1.3 × 10⁻² ${M}_{\odot }$ and a number of initial burned regions from 1 to 10. We note that our implementation of the initial conditions produced a slight asymmetry in the initially burned regions as a function of angle, as may be observed in Figure 10.

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We demonstrate the influence of the number-and-amplitude parameters using the final ${}^{56}\mathrm{Ni}$ yield as a proxy for the explosion results in Figures 11 and 12. Because we use the same ignition geometry and conditions for the C–O realizations as were used in Krueger et al. (2012), the general size of the initially burned region for the C–O realizations remains very near 0.0084 ${M}_{\odot }$. However, because the nature of the ignition in the C–O–Ne hybrid progenitor is unknown, we chose to sample the number-and-amplitude parameter space to provide a range of initially burned masses for comparison. In spite of the scatter in Figure 11, we performed a linear fit between the ${}^{56}\mathrm{Ni}$ yield and initially burned mass for the C–O–Ne realizations, obtaining a slope indistinguishable from zero within uncertainties and an intercept that matches the average ${}^{56}\mathrm{Ni}$ yield for the C–O–Ne realizations (Table 1). We also show in Figure 12 that most of the variation in ${}^{56}\mathrm{Ni}$ yield from the C–O–Ne realizations originates from the interplay between the number of ignition regions and their size. The more ignition regions that are used, the greater effect the variation on their size has on the spread in ${}^{56}\mathrm{Ni}$ yields.

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Table 1.  Average Yields and Kinetic Energy

Progenitor	${}^{56}\mathrm{Ni}$	IGE	Kinetic Energy
Type	(${M}_{\odot }$)	(${M}_{\odot }$)	(×1051 $\mathrm{ergs}$)
C–O	0.97 ± 0.06	1.12 ± 0.07	1.39 ± 0.05
C–O–Ne	0.86 ± 0.10	0.98 ± 0.11	1.06 ± 0.10

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Using these parameters to vary the initial burned mass and its distribution within the progenitor, we evaluate the effect of this parameterization on the estimated ${}^{56}\mathrm{Ni}$ yield, IGE yield, binding energy, and other explosion properties in the following section.

We simulate the explosions of the hybrid and C–O realizations through the end of the detonation phase and compare their features in Figures 13, 14, and 16–20 below. We compare the ${}^{56}\mathrm{Ni}$ yields in the C–O and hybrid models, estimated from Y_e and the NSE progress variable, by assuming that the composition in NSE is ${}^{56}\mathrm{Ni}$ plus equal parts ${}^{54}\mathrm{Fe}$ and ${}^{58}\mathrm{Ni}$, as described in Townsley et al. (2009) and Meakin et al. (2009).

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Production of ${}^{56}\mathrm{Ni}$ is comparable between the C–O and hybrid cases (Figure 13), with the full range of values from each suite of simulations shown in the shaded regions and the mean values shown by solid curves. The DDT event can be distinguished in the ${}^{56}\mathrm{Ni}$ evolution by the sharp increase in the rate of ${}^{56}\mathrm{Ni}$ production around 1.5 ${\rm{s}}$ that rapidly yields over 0.5 ${M}_{\odot }$ of ${}^{56}\mathrm{Ni}$. While the C–O cases show a wider variation in the time at which the DDT occurs, these also have a narrower spread in final ${}^{56}\mathrm{Ni}$ mass relative to the hybrid models. The hybrid models also tend to produce more ${}^{56}\mathrm{Ni}$ in the deflagration phase, and some of them show a temporary plateau in ${}^{56}\mathrm{Ni}$ production between 1.5 and 2 s. The same feature is also evident in the binding energy curves of Figure 14, computed by summing the realization's gravitational potential, internal, and kinetic energies.

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This feature is a peculiarity of the off-center ignition in the hybrid models that is absent in the C–O cases and results from the relatively ${}^{12}{\rm{C}}$-poor, cooler core region burning about 0.25 s after the detonation front has swept through the rest of the star. This delayed burning is shown in Figure 15, which demonstrates the progression of the detonation front into the core. Although a feature evolving over so short a time this early in the explosion will likely not be visible in the SN light curves, the delayed contribution of the core to ${}^{56}\mathrm{Ni}$ production may modify the ${}^{56}\mathrm{Ni}$ distribution in space and velocity, potentially yielding spectral differences compared to nondelayed hybrid as well as C–O WD explosions.

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The dynamical qualities of the explosion shown in the binding energy curves of Figure 14 indicate that the time distribution of unbinding is narrower for the hybrid models than for the C–O models, though the hybrid models have a wider distribution of final binding energies in all cases lower than the binding energies of the C–O models within 1 s of becoming unbound. This should correlate to a lower expansion velocity of the ejecta and thus slower cooling and delayed transparency relative to ejecta from C–O models. The binding energy curves also explain the differences in expansion the models undergo during deflagration and detonation, shown by the mass above the density threshold 2 × 10⁷ g ${\rm{cm}}$⁻³ in Figure 16. For times prior to ≈1.2 ${\rm{s}}$, the C–O mass curves lie slightly lower than the average hybrid mass curve, indicative of a greater degree of expansion on average for the C–O models. However, the hybrid mass curve range encompasses that of the C–O mass curves until ≈1.4 ${\rm{s}}$, reflective of the fact that until then, some hybrid realizations are more tightly bound than all the C–O realizations due to burning less mass and thus expanding less. During the detonation phase, however, the C–O models show a much wider variation in expansion than do the hybrid models in spite of having a smaller range of kinetic energies and mass burned to IGEs (Figures 14 and 20) once unbound. This is due to the C–O models demonstrating a much wider range of DDT times than the hybrid models.

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Figure 17 compares the final IGE yield of the C–O and C–O–Ne models with the degree by which the models expand during the deflagration phase. The latter is characterized by the mass above 2 × 10⁷ g ${\rm{cm}}$⁻³ at the DDT time, with more high-density mass indicating less expansion during deflagration. The averages of both the C–O and C–O–Ne suites along both axes are indicated by the shaded regions with ±1σ widths. The trend for both C–O and C–O–Ne models is that less expansion during the deflagration phase results in greater IGE yields, expected because low expansion results in there being more high-density fuel for the detonation to consume. In addition, both the C–O and C–O–Ne models expand over similar ranges during deflagration on average, showing that they are dynamically comparable in spite of having qualitatively different deflagration ignition geometries. Furthermore, for similar deflagration expansion, the C–O models tend to yield consistently greater IGE mass, suggesting that the lower IGE yields from C–O–Ne models are not a result of these models expanding differently than the C–O models. Rather, we interpret this disparity as indicating that the lower IGE yield in C–O–Ne models results from their lower ${}^{12}{\rm{C}}$ abundance and the fact that given similar fuel density, their ${}^{20}\mathrm{Ne}$-rich fuel will burn to cooler temperatures than fuel in the C–O models. This in turn will result in slower burning to IGEs and thus a lower IGE yield.

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The estimated ${}^{56}\mathrm{Ni}$ yields are shown in Figure 18 across the range of masses burned to IGEs for all C–O and C–O–Ne realizations at 4.0 ${\rm{s}}$ simulation time, at which point the total mass burned to ${}^{56}\mathrm{Ni}$ had become constant (see Figure 13). For comparable masses burned to IGEs, the hybrid models tend to consistently produce slightly more ${}^{56}\mathrm{Ni}$ than the C–O models, although the ratio of IGE mass producing ${}^{56}\mathrm{Ni}$ given by the slope is the same in both cases, within the fit error. The reason for this trend is evident from Figure 19, which shows the fraction by mass of IGE material producing ${}^{56}\mathrm{Ni}$ evolving in time, and Figure 20, which shows the concurrent evolution of mass burned to IGE. During the deflagration phase, the C–O–Ne models on average burn more material to IGEs and also had a significantly higher fraction of IGE material producing ${}^{56}\mathrm{Ni}$, yielding more ${}^{56}\mathrm{Ni}$ than the C–O models. This may be due to greater neutronization in the early deflagration of the C–O models, which are ignited closer to the center and thus at slightly higher density than the initial deflagration of the C–O–Ne models. However, during the subsequent detonation phase, the C–O models on average burn more mass to IGE while maintaining a ${}^{56}\mathrm{Ni}$/IGE fraction very similar to that of the C–O–Ne models, yielding significantly more ${}^{56}\mathrm{Ni}$ by the end of the detonation phase. For reference, Table 1 summarizes the average values of ${}^{56}\mathrm{Ni}$ yield, IGE yield, and final kinetic energy for the 35 C–O and 35 C–O–Ne realizations with one standard deviation uncertainties.

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