EARLY EMISSION FROM TYPE Ia SUPERNOVAE

For the analysis of this section, we assume that the shock accelerates down the density gradient following the self-similar solution of Gandel’Man & Frank-Kamenetskii (1956) and Sakurai (1960), v∝ρ^−β₀ with β = 0.19. This description of the shock velocity is valid for explosions in which the energy is released at small radii, r ≪ R_* (see Matzner & McKee 1999), i.e., ignoring the fact that part of the energy is released by nuclear burning at large radii. The distributed energy release is accurately described in our numerical analysis (Sections 3 and 4). As we show there, the simple model described in this section provides an acceptable description of the velocity profile of the ejecta, and a rough description of the pressure profile in regions where radiation dominates the pressure (see Figure 1).

The early, t < 1 day, emission from SNe is dominated by the outer ≲ 10⁻³ M_☉ shells of the progenitor, which are heated by the SN shock and are emitting radiation while expanding and cooling. We discuss the opacity and EOS effects in Sections 2.1.1 and 2.1.2, respectively, using the analytic model of Rabinak & Waxman (2011). This simple model's assumptions are that the initial density profile of the progenitor is given by a power law of the form ρ₀∝δⁿ where δ = (R_*/r_i − 1) and r_i is the initial radius of the shell, that the post-shock energy density is dominated by radiation and that the post-shock expansion is adiabatic. Since the results are not sensitive to the value of n, we adopt below n = 3 (the value obtained for non-degenerate radiative envelopes).

The model assumptions are similar to those used in other analytic studies of the early emission (e.g., Chevalier 1992; Waxman et al. 2007; Chevalier & Fransson 2008; Rabinak & Waxman 2011; Nakar & Sari 2010). We use the Rabinak & Waxman (2011) analysis since it includes a realistic description of the opacity (beyond the Thomson opacity of fully ionized plasma used in other analyses). Another comment is in place here regarding the various analyses. For the calculation of the luminosity, the diffusion of radiation below the photosphere is ignored in Rabinak & Waxman (2011) while it is taken into account in Chevalier & Fransson (2008) and Nakar & Sari (2010). As pointed out in Rabinak & Waxman (2011), the effects of the diffusion on the luminosity are indeed negligible, and the results obtained with and without diffusion are nearly identical (e.g., compare Rabinak & Waxman 2011; Nakar & Sari 2010). The effects of diffusion are, on the other hand, important for determining the spectrum or color temperature. The color temperatures obtained in Rabinak & Waxman (2011) differ from those obtained in Nakar & Sari (2010) due to the more accurate description of the opacity.

2.1.1. Opacity

As long as photospheric temperature is well above the recombination temperature, the ejecta is nearly fully ionized and the opacity is time independent and dominated by electron scattering. At this stage the bolometric luminosity and the effective temperature are given by (see Equations (13) and (15) in Rabinak & Waxman 2011)

$$\begin{equation} L(t) = 3.2 \times 10^{39} \frac{E^{0.85}_{51} R_{8.5} }{f^{0.16}_{\rho } (M/1.4\, M_{\odot })^{0.69}\kappa ^{0.85}_{0.2} }t_2^{-0.31} \mbox{ erg}\, {\rm s}^{-1}, \end{equation} \tag{ 1 }$$

and

$$\begin{equation} T_{{\rm eff}}(t) = 3.5 \, f_{\rho }^{-0.022} \frac{E_{51}^{0.016} R_{8.5}^{1/4} }{(M/1.4M_{\odot })^{0.03} \kappa ^{0.27}_{0.2}} t_2^{-0.47} \mbox{ eV}. \end{equation} \tag{ 2 }$$

Here t = 10²t₂ s is the time after breakout, κ = 0.2κ_0.2 cm² g⁻¹, M is the ejecta mass, E = 10⁵¹E₅₁ erg is the explosion energy, and f_ρ is a dimensionless factor of order unity that depends on the envelope density profile (see Rabinak & Waxman 2011). Fitting an n = 1.5 profile to the outer 10⁻² M_☉ pre-detonation density profiles of the simulations described in Section 3 gives 0.5 < f_ρ < 2. The opacity is κ_0.2 ≈ 1 + X, where X is the H mass fraction and R_* = 10^8.5R_8.5 cm. For DDT explosions of Chandrasekhar mass progenitors, R_8.5 ∼ 1. These simple results for the luminosity and temperature are compared to numerical simulations of DDT explosions in Section 4.

The color temperature T_col is set by photons that can diffuse and reach the photosphere after thermalizing in an inner layer. As a result, T_col is higher than the effective temperature and is approximately given by T_col ≈ 1.2T_eff (Rabinak & Waxman 2011). Although this result was obtained for explosions of RSG, BSG, and WR stars, we expect it to hold also for explosions of Type Ia SNe, since the pressure profile of the Type Ia ejecta is similar to that obtained in the analytic model presented here (see Figure 1), and since the composition of the ejecta is similar to that considered for WR progenitors.

When the photospheric temperature approaches the recombination temperature of the ejecta, the approximation of constant opacity is no longer valid and the decline of the opacity, which is composition dependent, has to be taken into consideration. For H poor envelopes, the decline in the opacity becomes important for T_eff ≲ 3 eV. The bolometric luminosity for He and C/O envelopes is given in this regime by (see Equations (25) and (29) in Rabinak & Waxman 2011)

$$\begin{equation} L^{[{\rm He}]}(t) = 3.1\times 10^{39} \frac{E^{0.84}_{51} R_{8.5}^{0.85} }{f^{0.15}_{\rho } (M/1.4\, M_{\odot })^{0.67} }t^{-0.02}_3 \mbox{ erg}\, {\rm s}^{-1}, \end{equation} \tag{ 3 }$$

and

$$\begin{equation} L^{[{\rm C/O}]}(t) = 4.4\times 10^{39} \frac{E^{0.83}_{51} R^{0.8}_{8.5} }{f^{0.14}_{\rho } (M/1.4\, M_{\odot })^{0.67}}t_3^{0.07} \mbox{ erg}\, {\rm s}^{-1}, \end{equation} \tag{ 4 }$$

respectively (note that these equations correct for typos that appeared in Equations (25) and (29) in Rabinak & Waxman (2011), where the subscript of R should be “12” and not “13” and the exponent of t should be −0.02 and 0.07 and not 0.03 and −0.07, respectively). Here, t = 10³t₃ s. The effective temperature is given by (see Equations (23) and (27) in Rabinak & Waxman 2011)

$$\begin{equation} T_{{\rm eff}}^{[{\rm He}]}(t) = 1.5 \mbox{ eV}f_{\rho }^{-0.02} R_{8.5}^{0.2} t_3^{-0.38} \end{equation} \tag{ 5 }$$

and

$$\begin{equation} T_{{\rm eff}}^{[{\rm C/O}]}(t) = 1.6 \mbox{ eV}f_{\rho }^{-0.017} R_{8.5}^{0.19} t_3^{-0.35}, \end{equation} \tag{ 6 }$$

both with very weak dependence on E and M.

The composition of the WD's outer mass shells at the onset of the explosion is uncertain. The estimated maximum mass of H and He shells are ≲ 10⁻⁵ M_☉ (Shen & Bildsten 2009a) and ≲ 10⁻³ M_☉ (Iben & Tutukov 1989; Shen & Bildsten 2009b), respectively. As the diffusion sphere propagates inward to larger mass shells it may pass through shells with different compositions. For t₂ ≲ 1, the plasma in the photosphere is fully ionized for all types of compositions, and the luminosity and temperature are given by Equations (1) and (2), respectively. For t₂ > 3 the photosphere propagates beyond the outer ∼10⁻⁵ M_☉ and the effective temperature drops below 3 eV. At t₂ > 3 we therefore expect the radiation to be emitted from shells dominated by He or C/O, with luminosity and temperature given by Equations (3) and (5) or (4) and (6), depending on the composition. For 1 ≲ t₂ ≲ 3, the detailed properties of the emission depend on the amount of H (X as function of mass). A comparison of Equations (1) and (3) or (4) implies that the modification of the opacity with time leads to a nearly time-independent luminosity at t larger than few hundred seconds.

Piro et al. (2010) found $L\sim 3\times 10^{\textbf {}40}t_3^{-0.16} \mbox{ erg}\mbox{ s}^{-1}$. The time dependence they obtain is different than ours due to the fact that opacity evolution is not included in their calculation. The difference in normalization (their predicted luminosity is ≈10 times larger than ours) is mainly due to an inaccuracy of their Equation (17) (see comment preceding Equation (13) below). The luminosity normalization (but not its time dependence) obtained by Nakar & Sari (2010) is closer to the one obtained here.

2.1.2. EOS

As long as the pressure, p, is dominated by radiation and the evolution is adiabatic, the radiation pressure of a given fluid element declines as p = p_rad∝ρ^4/3∝r⁻⁴. For shells at which the pressure is dominated by the plasma, the pressure declines as p = p_gas∝ρ^5/3∝r⁻⁵, which implies, assuming thermal equilibrium, p_rad∝ρ^8/3∝r⁻⁸. The faster decline of the radiation energy density in shells dominated by plasma pressure implies that once the diffusion front reaches shells dominated by plasma pressure, a strong suppression of the luminosity is expected, by a factor ∼(ρ/ρ₀)^4/3 ∼ (r/R_*)⁻⁴ compared to the luminosity given in Section 2.1.1, which is calculated assuming radiation pressure domination. In what follows we estimate the time t_drop, defined as the time at which the diffusion front reaches layers in which the plasma pressure is equal to the radiation pressure.

In the discussion of the preceding paragraph, we have ignored the possible contribution of recombination to the energy density. This contribution is small for shells in which radiation pressure dominates. In these shells the photon number density n_γ is much larger than the electron number density n_e (n_γ/n_e does not change significantly during adiabatic expansion), so that although the recombination temperature T_rec is a few times smaller than the ionization energy (due to the low density, ≲ 1 g cm⁻³), the energy released by recombination is negligible.

For shells in which the post-shock energy density of radiation and plasma is similar, ρ₀ ≃ ρ_{0, e} (see Equation (14)), n_γ ∼ n_e, the contribution of recombination to the energy density (∼T_recn_e) is significant. In order to estimate the effect of recombination we have calculated the adiabatic cooling of a gas in thermal equilibrium, using an EOS calculation as outlined in Cox & Giuli (1968), Timmes & Arnett (1999), and Timmes & Swesty (2000). We calculated the temperature T after shell expansion, for shells with initial density 10⁵ < ρ₀/(1 g cm⁻³) < 10⁷ and initial post-shock temperature T₀ ∼ 3m_pv²/16 ∼ 2 × 10⁵ eV. At t = 3 × 10³ the ratio of the shell's density ρ to its original density is ρ/ρ₀ ∼ (R_*/vt)³ ≈ 10⁻¹²(R_*/v₉)³. For 10⁻¹⁴ < ρ/ρ₀ < 10⁻¹⁰, we find (T/T₀)⁴ < 10⁻³(ρ/ρ₀)^−4/3, indicating that a strong suppression in the radiation energy density and luminosity are expected at t > t_drop compared to those obtained assuming radiation pressure domination (which yields (T/T₀)⁴ = (ρ/ρ₀)^−4/3). Finally, we note that the assumption of thermal equilibrium is valid, since for the range of final densities and temperatures obtained the free–free absorption time is much smaller than t (note also that the photon and electron energies are kept similar by Compton scattering, as the number of collisions required to modify the photon energy is ∼c/v while the number of collisions a photon undergoes over a dynamical time is ∼τc/v).

Let us proceed with the calculation of t_drop. Assuming that the post-shock pressure is dominated by radiation (p_rad = aT⁴/3, where T is the temperature and a is the radiation constant) and that the plasma pressure is given by p_gas = ρ_shT/μm_p (where ρ_sh = [(γ + 1)/(γ − 1)]ρ₀ is the post-shock density, m_p is the proton mass, and μ is the molecular weight), and using the self-similar shock description (assuming an adiabatic index of γ = 4/3, e.g., Rabinak & Waxman 2011, for details), the post-shock ratio of radiation to gas pressure is given by

$$\begin{equation} \eta _{{\rm sh}}\equiv \frac{p_{{\rm rad}}}{p_{{\rm gas}}} = \frac{1.2\times 10^3 E_{51}^{3/4} (\mu / 2)}{(\rho _0/\mbox{g}\mbox{ cm}^{-3})^{0.53} R^{0.84}_{8.5} (M/1.4 \,M_{\odot })^{0.47}} \end{equation} \tag{ 7 }$$

(electron degeneracy pressure and pair production, which were neglected in this analysis, do not change this result substantially). The post-shock pressure is not dominated by radiation for

$$\begin{equation} \rho _0 \gtrsim \rho _{0,e} \equiv 6.6\times 10^5 \frac{ E_{51}^{1.4} (\mu /2)^{1.9}}{R^{1.6}_{8.5} (M/1.4\, M_{\odot })^{0.89}} \mbox{ g}\mbox{ cm}^{-3}. \end{equation} \tag{ 8 }$$

Noting that ρ₀ < 3M/4πR³, we find that a region in which the post-shock pressure is not dominated by radiation exists only for compact progenitors,

$$\begin{equation} R_{8.5} \lesssim 11 (M/1.4\, M_{\odot })^{4/3} E^{-1}_{51}(\mu /2)^{-4/3}. \end{equation} \tag{ 9 }$$

Let us next estimate the evolution of the radiation to gas pressure ratio, η, with time. Assuming adiabatic evolution of an ideal gas in equilibrium with radiation, the density ρ for which η = 1 for a shell with post-shock density ρ_sh and initial pressure ratio η = η_sh, is

$$\begin{equation} \rho /\rho _{{\rm sh}}= \eta _{{\rm sh}}^{-1}e^{-8(\eta _{{\rm sh}}-1)}. \end{equation} \tag{ 10 }$$

To estimate the time when the diffusion sphere reaches a layer with η = 1, we use the (time-dependent) density and pressure profiles of the ejecta given in Rabinak & Waxman (2011), and use the method described there for determining the (time-dependent) mass of the shell reached by the diffusion front. For time-independent opacity we find

$$\begin{equation} t_{\rm drop}\approx 5.3 \times 10^3 \frac{E^{0.83}_{51} R^{1.1}_{8.5} \kappa ^{0.5}_{0.2} (\mu /2)^{1.44}}{f^{0.17}_{\rho } (M/1.4\, M_{\odot })^{0.69}} \rm s, \end{equation} \tag{ 11 }$$

while for C/O opacity we find (using the power-law approximation for the opacity suggested in Rabinak & Waxman 2011)

$$\begin{equation} t_{\rm drop}\approx 3 \times 10^3 \frac{E^{0.66}_{51} R_{8.5} (\mu /2)^{1.2}}{f^{0.15}_{\rho } (M/1.4\, M_{\odot })^{0.56}} \rm s \end{equation} \tag{ 12 }$$

(in both cases we neglected logarithmic corrections).

It is illustrative to compare t_drop obtained above to that obtained using the more detailed description of the ejecta profiles suggested by Piro et al. (2010). These profiles are obtained under the following assumptions: the pre-detonation envelope is in hydrostatic equilibrium and the pressure is dominated by degeneracy pressure (see Equation (A2)), the velocity of the shock is given by Equation (A5), the terminal velocity of a shell is v_f = 2v_s (instead of v_f = (6/7)v_s taken by Piro et al. 2010, which is more appropriate for the planar phase), and the diffusion front is located at an optical depth τ = c/v_f. The pressure in the ejecta is calculated using the relation p = p_sh(ρ/ρ_sh)^γ, where p_sh is the initial post-shock pressure (correcting Equation (17) of Piro et al. 2010, which uses ρ₀ instead of ρ_sh in the denominator). For these profiles we find

$$\begin{equation} \eta _{{\rm sh}}\equiv \frac{p_{{\rm rad}}}{p_{{\rm gas}}} = 910 \frac{(\mu / 2) v_9^{3/2} \rho _6^{0.27}}{(\rho _0/\mbox{g}\mbox{ cm}^{-3})^{0.52} }, \end{equation} \tag{ 13 }$$

where V_run = 10⁹v₉ cm s⁻¹ and ρ_run = 10⁶ρ₆ g cm⁻³ are the velocity and density where the detonation wave transforms to a shock wave, respectively. Thus, the assumption of radiation pressure dominance breaks for

$$\begin{equation} \rho _0 \gtrsim 5\times 10^5 v_9^{2.9} \rho _6^{0.52} (\mu /2)^{1.9}\mbox{ g}\mbox{ cm}^{-3}. \end{equation} \tag{ 14 }$$

For constant opacity we find

$$\begin{equation} t_{\rm drop}\approx 3.6 \times 10^3 \frac{R_{8.5}^2 v_9^{2.16} \kappa _{0.2}^{0.5} \rho _6^{0.39} (\mu /2)^{1.8}}{\, (M/1.4 \, M_{\odot })^{0.5}}\ {\rm s}, \end{equation} \tag{ 15 }$$

while for C/O opacity we find

$$\begin{equation} t_{\rm drop}\approx 2.2 \times 10^3 \frac{R_{8.5}^{1.74} v_9^{1.76} \rho _6^{0.31} (\mu /2)^{1.5}}{(M/1.4 \, M_{\odot })^{0.44}}\ {\rm s}. \end{equation} \tag{ 16 }$$

When the shock reaches layers with optical depth τ = δM_BOκ/(4πR²_*) comparable to c/v_s, where δM_BO is the mass exterior to the shell, photons outrun the shock and escape, producing a breakout flash. Our simulations’ pre-detonation profiles (described in Appendix A.1) are different from those used by Piro et al. (2010). However, as expected, the breakout energy is not sensitive to the details of the profiles. Using the simulation based (extrapolated) profiles derived in Appendix B, and the shock description given in Appendix A.2, the shock velocity at τ = c/v_s ≡ β⁻¹_s is mildly relativistic (for κ_0.2 = 1). For mild-relativistic shocks, v_s in Equation (A5) is replaced with Γ_sv_s, where Γ_s = [1 − (v_s/c)²]^−1/2 (Tan et al. 2001; Piro et al. 2010; Nakar & Sari 2012). The resulting breakout velocity is

$$\begin{equation} \Gamma _s v_s \approx 1.8 c \,\frac{R^{0.31}_{8.5} v_9^{1.16} f_\beta ^{0.05} \kappa ^{0.84}_{0.16} (2 \rho _6/\mu)^{0.21} }{(M/1.4 \, M_{\odot })^{0.16} }, \end{equation} \tag{ 17 }$$

and the rest-frame breakout energy is

$$\begin{equation} E_{\rm BO} \approx 10^{40} \frac{R^{2.3}_{8.5} v_9^{1.16} f_\beta ^{0.05} (2 \rho _6/\mu)^{0.21} }{(M/1.4 \, M_{\odot })^{0.16} (\Gamma _s/2.1)^{1.16} \kappa ^{0.84}_{0.2} }\, \rm erg. \end{equation} \tag{ 18 }$$

E_BO depends weakly on f_β ≡ β⁻⁴(1 − β), where 1 − β = L/L_edd, the ratio of the luminosity escaping the pre-detonation progenitor to the Eddington luminosity, is not accurately determined by our simulations. Our results are in agreement with previous estimates (Imshennik et al. 1981; Piro et al. 2010; Nakar & Sari 2010), suggesting that the higher breakout energy obtained by Höflich & Schaefer (2009) is due to the coarse numerical resolution near the stellar edge. In the case that an optically thick detached shell is present Equations (17) and (18) still hold, replacing the stellar radius with the detached shell radius.

Since the shock velocity is mildly relativistic at breakout, relativistic effects must be taken into consideration in estimating the properties of the observed breakout emission. As shown by Budnik et al. (2010) and Katz et al. (2010), for mildly relativistic, v/c ≳ 0.1, shocks the radiation within the shock transition region is far from thermal equilibrium. The photon spectrum extends to ≈30 keV for v/c = 0.2 and to ≈500 keV as v approaches c. These results apply to the structure of steady shocks. Unfortunately, a solution for the shock structure at the non-steady conditions at breakout is not available for relativistic velocities (unlike the non-relativistic case, for which the solution is given in Sapir et al. 2011; Katz et al. 2012). A very crude estimate of the properties of the emitted radiation may be obtained by assuming that an energy ≈Γ_escE_BO escapes over an observed timescale of R_*/Γ²_escc with a spectrum given by boosting the steady state shock transition region spectrum by Γ_esc, where Γ_esc is the characteristic Lorentz factor of the plasma at the time when photons escape, and Γ_esc > Γ_s is expected since the plasma accelerates during breakout. Nakar & Sari (2012), for example, assume Γ_esc = Γ_f, where Γ_f is the terminal Lorentz factor that the plasma reaches, and use the estimate of Tan et al. (2001), Γ_fβ_f ≃ Γ_sβ_s(2.03 + (Γ_sβ_s)^1.732).

We derive the emission of radiation using the density and temperature profiles at the end of the simulations, t_s = 100 s (note that t_s is measured from the onset of deflagration, while we use t to denote the time measured from breakout). At this stage the ejecta has expanded significantly and the radius r of a relevant mass shell is ≫R_* and is approximately given by tv_f, where the shell's velocity v_f is approximately its terminal velocity. The simulations’ mass resolution at the outer cells, ∼3 × 10⁻⁷ M_☉, sets a lower limit to the time from which the emission can be reliably calculated based on the simulations, t ≳ 3 × 10² s.

We neglect the reheating by the radioactive decay of elements synthesized in the explosion process (valid for the outer layers of the ejecta, in which only a small fraction of the elements are fused), and extrapolate the profiles to later times assuming that the ejecta expand adiabatically and that the fluid elements reached their final velocity. We use an EOS of radiation in equilibrium with an ideal gas (μ = 1.7455 for fully ionized equal mass fractions of C and O). Note that although we assume full ionization for the EOS, we do not make this assumption for the opacity. This approximation for the EOS is accurate as long as the thermal energy density is dominated by radiation, and becomes less accurate when the plasma pressure becomes significant. This implies that our light curves are not accurate at t ≳ t_drop.

In Figure 1, we compare the pressure and velocity profiles at the end of the simulations to the profiles suggested by Rabinak & Waxman (2011; for n = 3) and Piro et al. (2010; corrected as explained in the text preceding Equation (13)), denoted by RW and PCW subscripts, respectively. For the model of Rabinak & Waxman (2011) we used E₅₁ = 1, R_8.5 = 1, and M = 1.4 M_☉, similar to the values obtained from the simulations (see Table 1). The discontinuity in the PCW profiles results from a transition between regions with different pre-detonation density profiles. Both the RW and PCW models do not properly describe the pressure behavior at large optical depth, where the pressure is no longer purely dominated by radiation and thus drops faster with time, as explained in Section 2.1.2.

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For the lower deflagration velocity the pressure is higher by ∼70% at 10² ≲ τ_es ≲ 10⁴, where τ_es is the optical depth for electron scattering (κ_0.2 = 1). This difference is due to the fact that for the low velocity explosions the envelope is heated to higher temperatures during the deflagration phase. Thus, for a given pre-detonation density ρ₀ the mass fraction δM (integrated mass to the stellar surface) is larger compared with that in high velocity explosions. For γ = 4/3, as τ_es∝δM(tv_s)⁻² and the density in the ejecta ρ∝δM(tv_s)⁻³, the pressure in the ejecta p∝v²_s(ρ/ρ₀)^γ∝(δM/ρ₀)^1/3τ_est⁻², implying that for lower ρ₀ at given δM the pressure is larger at a given τ_es.

We use the density and pressure profiles obtained as described in Section 4.1 to derive the properties of the emitted radiation. Due to the deviation from pure radiation pressure domination, the pressure profiles deviate significantly from p∝τ (which is valid for the core collapse progenitors). This implies that photon diffusion may have a non-negligible effect on the predicted luminosity (Rabinak & Waxman 2011). We therefore estimate the luminosity at time t as dE/dt, where E(t) is the radiation energy that escapes by diffusion up to time t. The diffusion depth is determined as the depth at which τ = c/v_f, and τ is calculated with the opacity taken from the OP project tables (Seaton 2005). The effective temperature is calculated as T_eff = (L/4πr²_phσ)^1/4, where r_ph is the radius of the photosphere (τ = 1). The temperature at the photosphere (τ = 1) is typically larger than T_eff by 10%–20%, which implies that neglecting diffusion leads to an overestimate of L by a factor of 1.5–2. The simulations’ resolution was not fine enough to allow an accurate determination of the color temperature.

The resulting luminosities and effective temperatures are given in Figure 2. The luminosity depends weakly on ρ_crit and is lower for larger deflagration velocities, reflecting the lower pressure at given τ_es (see Figure 1). Comparing the results presented in the figure with Equations (4), (6), and (12), we find that the simple model of Section 2.1 provides a good description of the properties of the emitted radiation. For the slow deflagration velocity the luminosity is similar to that predicted by the simple analytic analysis, while for the larger velocity it is 2–3 times lower.

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An estimate of the luminosity expected for a pure He composition of the outer shells may be obtained by extrapolating the simulation profiles assuming an EOS with μ = 2, and calculating the emission using the opacity of He. The luminosity obtained in this case is ∼50% lower and the effective temperature is ∼15% lower than the results obtained for C/O, consistent with the simple model of Section 2.1.

We consider the pressure and density profiles of the outer layers of the progenitor, which are modified by weak shocks generated during the deflagration phase, focusing on the profiles at t = t_det. During the deflagration phase the progenitor WD expands. In regions, where the expansion velocity is similar to or higher than the sound speed, weak shock waves form, which heat the envelope and modify its pressure profile. During this phase, two physical processes affect the pressure profile: shock wave heating and diffusion of energy. Diffusion significantly affects regions in which the photon diffusion time (to the stellar surface) is similar to the deflagration time. The outer part of the progenitor's envelope may be divided into three regions distinguished by the physical processes that affect them: (1) the inner region, which is negligibly affected by shock heating and diffusion; (2) the intermediate region, which is affected by shock heating but is negligibly affected by diffusion; (3) the outer region, which is affected by both shock heating and photon diffusion. The mass resolution in our simulations does not allow us to fully resolve region (2). We denote by (2A) and (2B) the parts of this region that are resolved and unresolved by the simulations, respectively. Below we describe the simulations’ results for regions (1) and (2A). The unresolved regions, (2B) and (3), are discussed in Appendix B.

Let us first discuss the pressure profile expected at the outer shells of the progenitor based on simple theoretical arguments. For simplicity, we neglect the shells’ self-gravity and thickness (relative to the progenitor radius R_* at the onset of detonation). We denote the density of a shell at the onset of detonation by ρ₀. Assuming that the star is in hydrostatic equilibrium, we have for the outer layers

$$\begin{equation} \delta M(\rho _0) \approx \frac{4\pi R_*^4 P_0(\rho _0)}{G M}, \end{equation} \tag{ A1 }$$

where M is the total mass of the progenitor, P₀ is the pressure, and δM is the integrated mass from the point where the density equals ρ₀ out to the stellar surface. For a polytropic EOS, P₀∝ρ^{1 + 1/n}₀, the density profile is ∝δⁿ, where δ = (R_*/r − 1).

We consider first the inner region (1). In this region, the pressure is negligibly affected by the deflagration process and is therefore dominated by the electron degeneracy pressure. In this case P₀(ρ₀) is polytropic with index n = 3/2 and may be approximated by (following Piro et al. 2010)

$$\begin{equation} P_{\rm deg} \equiv 9.91 \times 10^{12} \left(\frac{\rho _0}{\mu _e\mbox{ g}\mbox{ cm}^{-3}}\right)^{5/3} \mbox{ erg}\mbox{ cm}^{-3}, \end{equation} \tag{ A2 }$$

where μ_e is the molecular weight per electron. The sound speed in these layers is c_s = 4.1 × 10⁶(ρ₀/g cm⁻³)^1/3μ^5/6_e cm s⁻¹.

Let us consider next region (2A). The simulations show that when the deflagration front reaches a density ρ_crit ∼ 10⁷ g cm⁻³, at time t_det, the outer layers of the progenitor expand at a speed 3–6 × 10⁸ cm s⁻¹. The pressure profile deviates from that of region (1) at densities ∼3 × 10⁴ g cm⁻³, beyond which it becomes flatter and nearly linear in ρ₀. This change in the density profile is the result of shock heating during expansion. We use the simulations to obtain an approximation for P₀(ρ₀). In Table 1, we give the best-fit parameters for a pressure profile of the form

$$\begin{equation} P_0(\rho _0)= K_{{\rm ch}}\times 10^{15} \left(\frac{\rho _0}{\mbox{g}\mbox{ cm}^{-3}}\right)^{\gamma _p} \mbox{ erg}\mbox{ cm}^{-3}, \end{equation} \tag{ A3 }$$

for t = t_det and ρ₀ < 10⁴ g cm⁻³. In all of the fits the index γ_p is in the range 0.9–1, implying a nearly isothermal profile, in agreement with the simulations of Höflich & Schaefer (2009, see their Figure 1), which have a similar resolution to that of our simulations, <10⁻⁶ M_☉.

The pressure in region (2A) is different than that assumed by Imshennik et al. (1981), Piro et al. (2010), and Nakar & Sari (2010; for the same ρ₀ range). In particular, it corresponds to a negative polytropic index (n < 0) and does not correspond to a density profile declining as a power of δ (distance from the progenitor's edge). The density profile we find falls approximately exponentially with r, similar to the profile produced by an isothermal gas. The shells of region (2A) determine the emission on timescales of minutes to hours.

In the left panel of Figure 3, we compare the results of the simulations to the approximations of Equations (A2) and (A3). Since the profile is nearly isothermal at the outer regions, we normalize P₀ to

$$\begin{equation} P_{\rm thrm}(\rho _0) \equiv 4.79\times 10^{15} \left(\frac{\rho _0}{\mbox{g}\mbox{ cm}^{-3}}\right) \frac{ T_{10\, {\rm keV}}}{\mu /2}\, \mbox{ erg}\mbox{ cm}^{-3}, \end{equation} \tag{ A4 }$$

where T = 10T_10 keV keV and μ is the molecular weight. The approximation of Equation (A2) is good to a few percent in the density range 3 × 10⁴ g cm⁻³ < ρ₀ < 10⁶ g cm⁻³ and the approximation of Equation (A3) is good to tens of percent for ρ₀ < 10⁴ g cm⁻³. In the latter density range, P₀ is given by P_thrm to within a factor of two.

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In the right panel of Figure 3, we compare δM(ρ₀) obtained in the simulations to that given by Equation (A1) using the pressure approximations of Equations (A2) and (A3) (with the values given in Table 1). Equation (A1) with P₀ given by Equation (A2) is accurate to a factor ∼2 in the density range 3 × 10⁴ g cm⁻³ < ρ₀ < 10⁶ g cm⁻³. The discrepancy is partially due to deviations from hydrostatic equilibrium and partially due to the fact that R_* overestimates r by up to 15% in this density range. Equation (A1), with P₀ given by Equation (A3), is accurate to a few tens of a percent in the density range ρ₀ < 10⁴ g cm⁻³. In this density range, substituting P_thrm for P₀ gives an approximation for δM, which is good to up to a factor of two.

As the detonation wave propagates in the declining density profile of the star, the entropy produced by the burning of elements declines, until eventually this entropy is smaller than the entropy that is produced by the shock compression. At that point the detonation wave transforms to a shock wave. The shock wave continues to accelerate in the declining density profile. In regions where the density falls as δⁿ, shock acceleration is described by the Gandel’Man–Frank-Kamenetskii–Sakurai self-similar solutions (Gandel’Man & Frank-Kamenetskii 1956; Sakurai 1960). In regions where the density falls exponentially with δ, the acceleration is described by the self-similar solution of Grover & Hardy (1966). In both types of solutions, the shock velocity is given by

$$\begin{equation} v_s = V_{{\rm run}}\left(\frac{\rho _0}{\rho _{{\rm run}}}\right)^{-\beta _1}. \end{equation} \tag{ A5 }$$

For a radiation-dominated shock, with post-shock adiabatic index of γ_s = 4/3, β₁ ≈ 0.19 for density profiles declining as a power law with n = 3, 3/2 (Grassberg 1981; Matzner & McKee 1999) and β₁ ≈ 0.176 for exponential density profiles (Grover & Hardy 1966). The self-similar solutions also show that the terminal velocity of the shells, v_f, is ∝v_s. For power-law density profiles v_f≅2v_s, while for exponential density profiles v_f≅1.5v_s.

The transition from a detonation wave to a shock wave was analyzed by Piro et al. (2010). The transition density ρ_run was defined as the density at which the “induction length” λ = (q/)ρ_det, where q, are the energy and energy generation rate of the detonation process, respectively, is smaller than the scale over which the progenitor density changes, assuming ρ_det is given by the Chapman–Jouguet detonation velocity at the corresponding density. For a WD equally composed of ¹²C and ¹⁶O they found ρ_run ≈ 2 × 10⁶ g cm⁻³, in agreement with our simulations (although the equation for was take in Piro et al. 2010 at the wrong electron screening regime). It should be noted here that the values of V_run, ρ_run may be affected by instabilities that are not considered here (e.g., Domínguez & Khokhlov 2011).

In the left panel of Figure 4, we compare the shock velocity and the terminal velocity obtained in a simulation to the approximation of Equation (A5). This figure shows that although the density profile changes from a power of δ to an exponential, the approximation Equation (A5) holds. The figure also shows that the values estimated analytically for ρ_run and V_run are accurate to ∼10%. The ratio v_f/v_s is not independent of ρ₀ and declines from a ratio of ≈2.4 at ρ₀ = 10⁵ g cm⁻³ to ≈1.4 at ρ₀ = 10 g cm⁻³. The decline in the ratio v_f/v_s with ρ₀ can be attributed to the transition from a power law to an exponential density profile and to spherical affects (see Matzner & McKee 1999). In the right panel of Figure 4, we compare the velocity at t_s = 100 s (terminal velocity) to the approximation

$$\begin{equation} v_f = 2.5 f_v \, v_s (\rho _0/\rho _{{\rm run}})^{0.05}. \end{equation} \tag{ A6 }$$

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