WAIT FOR IT: POST-SUPERNOVA WINDS DRIVEN BY DELAYED RADIOACTIVE DECAYS

The identity of the progenitors of SNe Ia remains an unsolved mystery (Maoz et al. 2014 and references therein). One of the most promising scenarios at present invokes helium-rich accretion from a helium WD or low-mass C/O WD. If mass transfer is dynamically stable, the resulting convective burning episodes may eventually yield a helium shell detonation (Bildsten et al. 2007; Shen & Bildsten 2009) and a subsequent core explosion in a variant of the double detonation scenario (Woosley et al. 1980; Fink et al. 2007; Shen & Bildsten 2014). For systems that merge unstably, which may in fact constitute most double WD binaries (Shen 2015), dynamical instabilities caused by the interaction of the direct impact accretion stream with freshly accreted material may trigger a helium detonation during the merger (Guillochon et al. 2010; Dan et al. 2012; Raskin et al. 2012; Pakmor et al. 2013), also yielding a double detonation SN Ia.

In both of these cases, the WD donor may survive the explosion of the accretor. In the case of dynamically stable mass transfer, the donor is only mildly tidally distorted as it overflows its Roche lobe and will remain self-bound following the SN Ia. For dynamically unstable double WD mergers, the ease with which helium polluted with CNO nuclei can be detonated (Shen & Moore 2014) allows for the possibility that a helium detonation is triggered soon after unstable mass transfer begins in helium-rich double WD binaries. If this is the case, the primary WD may explode and produce a SN Ia before the donor is tidally disrupted.⁴

The companion WD may survive the SN Ia explosion, but it will not emerge unchanged. The focus of this work is the ${}^{56}\mathrm{Ni}$ that is synthesized by the primary WD's explosion at velocities low enough such that the ${}^{56}\mathrm{Ni}$ is gravitationally bound to and captured by the WD companion. We estimate the bound ${}^{56}\mathrm{Ni}$ mass using the approximation of Dwarkadas & Chevalier (1998) to the density profile of the homologously expanding SN Ia ejecta:

$$\begin{eqnarray}&&\rho (v,t)=\displaystyle \frac{{M}_{\mathrm{ej}}}{8\pi {v}_{\mathrm{ej}}^{3}{t}^{3}}\exp (-v/{v}_{\mathrm{ej}}),\end{eqnarray} \tag{ 1 }$$

where M_ej is the ejecta mass, t is the time from explosion, v is the magnitude of the velocity in the SN's center-of-mass frame, and ${v}_{\mathrm{ej}}^{2}={E}_{\mathrm{kin}}/6\,{M}_{\mathrm{ej}}$, where E_kin is the ejecta's total kinetic energy. Ignoring the work that is done on the ejecta by the shocked material interacting with the companion, the mass bound to the companion is

$$\begin{eqnarray}&&{M}_{\mathrm{bound}}=\int \rho ({\boldsymbol{v}},t){dV}\end{eqnarray} \tag{ 2 }$$

such that ${({\boldsymbol{v}}-{{\boldsymbol{v}}}_{2})}^{2}\lt 2{{GM}}_{2}/d$, where ${v}_{2}=\sqrt{G({M}_{1}+{M}_{2})/a}$ is the magnitude of the companion's velocity in the SN ejecta's rest frame, a is the binary separation at the time of explosion, d is the distance from the material to the companion's surface, and M₂ is the companion's mass. Using the explosion values from Sim et al. (2010) for an M_ej = 0.97 ${M}_{\odot }$  pure detonation explosion with 1.04 × 10⁵¹ erg of kinetic energy, a mass–radius relation for 10⁷ K WDs from MESA (Paxton et al. 2011, 2013, 2015), and Paczyński's (1971) fit to the Roche radius, and evaluating at $t=1\,{\rm{s}}$, we find bound ${}^{56}\mathrm{Ni}$ masses of 0.006, 0.03, and 0.08 ${M}_{\odot }$  for 0.3, 0.6, and 0.9 ${M}_{\odot }$  WD companions, respectively.

We note that these are upper limits to the actual bound masses, because we have neglected the work that is done on the ejecta by previously shocked material surrounding the companion. Furthermore, the surface structure of the surviving companion WD will be affected by the passage of the shock and by the Roche lobe overflow that takes place prior to the SN Ia explosion, particularly for merger-induced detonations. These effects will influence the amount of ${}^{56}\mathrm{Ni}$ that is captured by the surviving WD donor, as well as contributing to the light curve when the thermalized shock energy is radiated outwards. We parameterize these uncertainties in Section 5 by varying the bound envelope masses, but we emphasize that future work with more realistic initial conditions is necessary for more accurate results.

SNe Iax are a recently designated class of peculiar SNe with characteristics akin to the prototype SNe 2002cx (Li et al. 2003) and 2005hk (Phillips et al. 2007). While some debate remains (Valenti et al. 2009; Moriya et al. 2010), the current observational evidence suggests that they are thermonuclear explosions of WDs with non-degenerate donor stars (Foley et al. 2013 and references therein). In one case, SN 2012Z, the progenitor system was detected and is consistent with a helium-burning star donor (McCully et al. 2014a).

The prevailing SN Iax progenitor scenario (e.g., Foley et al. 2013) is very similar to the formerly standard SN Ia progenitor scenario (e.g., Nomoto et al. 1984): a WD accretes matter from a non-degenerate donor, grows toward the Chandrasekhar mass, and ignites nuclear burning in its core due to the increasing density. The energy release from carbon-burning leads to a convective simmering phase of ~1000 yr (Woosley et al. 2004; Piro & Chang 2008) that ends when turbulent fluctuations give rise to hotspots that heat faster than convective eddies can dissipate their excess entropy, resulting in the birth of a deflagration. Whether or not this deflagration transitions to a detonation determines whether the outcome is a SN Ia or a SN Iax. Without this transition, the buoyant deflagration plume rises and breaks out of the star, burning the WD incompletely and not releasing enough energy to unbind the entire WD. The ejecta mass and the ejected ${}^{56}\mathrm{Ni}$ mass are thus smaller than for a regular SN Ia, leading to fainter and more rapidly evolving light curves.

A very important consequence of this model is the surviving WD remnant and the deflagration ashes that remain bound to it. Simulations show that much of the deflagration ash is ejected (Jordan et al. 2012; Kromer et al. 2013; Fink et al. 2014; Long et al. 2014). However, a significant fraction of the WD remains unburned and gravitationally bound following the ejection of the burned material, and this surviving WD has a large enough potential well to retain 0.03–0.2 M_⊙ of burned material as a gravitationally bound envelope. Radioactive decays in this burned material will drive a wind that neatly explains the persistent photosphere, low velocities, and lack of nebular emission that are some of the hallmarks of SNe Iax (Jha et al. 2006; Foley et al. 2013). As in the case of the surviving SN Ia WD donors discussed in Section 2.1, the dense, hot ${}^{56}\mathrm{Ni}$-rich matter at the surface of the bound WD accretor remnant of the SN Iax will be fully ionized, leading to delayed radioactive decays and a prolonged WD wind.

We use the results from Fink et al. (2014) to motivate our models of SN Iax WD winds. Fink et al. performed a large suite of simulations with a varying number of deflagration ignition kernels. Their best fits to the prototype SN 2005hk were for models with 5–20 ignition kernels, resulting in unburned masses of 0.91–0.49$\,{M}_{\odot }$ and bound, burned masses of 0.12–0.06 ${M}_{\odot }$ containing ${}^{56}\mathrm{Ni}$ masses of 0.02–0.004 ${M}_{\odot }$. These numbers are in rough agreement with the other simulations in the literature (Jordan et al. 2012; Kromer et al. 2013; Long et al. 2014). We bracket the uncertainties by considering two models with core + envelope masses of 0.9 + 0.1 and 0.5 + 0.05 ${M}_{\odot }$  and ${}^{56}\mathrm{Ni}$ mass fractions within the envelopes of 0.2 and 0.1, respectively. The non-${}^{56}\mathrm{Ni}$ material in the envelope is assumed to be ${}^{56}\mathrm{Fe}$, because it is radioactively inert.

One complication is the possibility that, following the ejection of material, the bound WD remnant continues to convectively burn carbon in its core, which would eventually lift the degeneracy of the core and significantly impact the subsequent evolution. However, the ejection of a large amount of material may yield enough adiabatic expansion to quench further convective core burning. We test this by constructing a near-Chandrasekhar mass 1.35 ${M}_{\odot }$ C/O WD in MESA and heating the center until carbon-burning causes convection. We find that a localized burning region begins to run away in the center when the convective region's entropy is $8\,\times \,{10}^{7}\,\mathrm{erg}\,{{\rm{g}}}^{-1}\,{{\rm{K}}}^{-1}$. We then construct less massive C/O stars with the same entropy, approximating the SN Iax WD remnants after the ejection of the burned material, and follow their evolution to ascertain whether or not they continue to burn convectively. We find that even for a remnant mass of 1.2 ${M}_{\odot }$, which is larger than we consider in this paper, the removal of the inner 0.15 ${M}_{\odot }$ from the simmering near-Chandrasekhar mass WD decreases the gravitational potential sufficiently that the remnant's center cools without reigniting convective carbon-burning. We are thus justified in assuming that the bound SN Iax remnants we consider are simply degenerate WD cores with hot ${}^{56}\mathrm{Ni}$-rich envelopes.

In order to determine the ionization state of the material, we solve the two Saha equations for the occupation fractions of the fully ionized (f₀), hydrogen-like (f₁), and helium-like $({f}_{2})$ states.⁵ We do not need to track lower-ionization states since the electron density at the nucleus is primarily determined by the K-shell. Therefore we have

$$\begin{eqnarray}&&\displaystyle \frac{{f}_{0}}{{f}_{1}}=\displaystyle \frac{{g}_{{\rm{e}}}{g}_{0}}{{g}_{1}}\left(\displaystyle \frac{{n}_{{\rm{Q}}}}{{n}_{{\rm{e}}}}\right)\exp \left(-\displaystyle \frac{{\chi }_{1}}{{k}_{{\rm{B}}}T}\right)\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{f}_{1}}{{f}_{2}}=\displaystyle \frac{{g}_{{\rm{e}}}{g}_{1}}{{g}_{2}}\left(\displaystyle \frac{{n}_{{\rm{Q}}}}{{n}_{{\rm{e}}}}\right)\exp \left(-\displaystyle \frac{{\chi }_{2}}{{k}_{{\rm{B}}}T}\right)\end{eqnarray} \tag{ 4 }$$

along with the constraint f₀ + f₁ + f₂ = 1. The degeneracies are ${g}_{{\rm{e}}}={g}_{1}=2$ and g₀ = g₂ = 1, and the quantum concentration is ${n}_{{\rm{Q}}}={({m}_{{\rm{e}}}{k}_{{\rm{B}}}T/2\pi {{\hslash }}^{2})}^{3/2}$. The values χ₁ and χ₂ are the ionization energies of these states, which are drawn from Kramida et al. (2015) and are listed in Table 1.

Table 1.  Information about ${}^{56}\mathrm{Ni}$ and ${}^{56}\mathrm{Co}$

Reaction	"Lab" Rate	Qν	χ1	χ2
	(s−1)	(MeV)	(keV)	(keV)
${}^{56}\mathrm{Ni}{\to }^{56}\mathrm{Co}$	1.32 × 10−6	0.41	10.775	10.289
${}^{56}\mathrm{Co}{\to }^{56}\mathrm{Fe}$	1.04 × 10−7	0.84	10.012	9.544

Note. The values of the "lab" rate and Q_ν (the average energy of the emitted neutrino) are from Nadyozhin (1994). The values χ₁ and χ₂ are the ionization energies of the hydrogen-like and helium-like states, respectively, and are drawn from Kramida et al. (2015).

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Furthermore, we make the assumption that the electron density is independent of the f values. If the ${}^{56}\mathrm{Ni}$ is a trace isotope, this is essentially exact. In the opposite limit of pure ${}^{56}\mathrm{Ni}$, this is justified since we are considering only the highest ionization states: if all ${}^{56}\mathrm{Ni}$ goes from fully ionized to having two bound electrons, the electron density changes by a fraction $\approx 2/Z$, which is only ≈7%. We use the electron density reported by MESA, which is for a fully ionized composition, i.e., ${n}_{{\rm{e}}}=\rho {Y}_{{\rm{e}}}/{m}_{{\rm{p}}}$. With these occupation fractions, the average number of bound electrons is equal to 1 when

$$\begin{eqnarray}&&{n}_{{\rm{e}}}=2{n}_{{\rm{Q}}}\,\exp \left(-\displaystyle \frac{{\chi }_{1}+{\chi }_{2}}{2{k}_{{\rm{B}}}T}\right).\end{eqnarray} \tag{ 5 }$$

This is the dashed line shown in Figure 1, where the χ values used are those of ${}^{56}\mathrm{Ni}$.

As shown in Figure 1, the tabulated weaklib rates do not extend to densities lower than $\rho {Y}_{{\rm{e}}}=10\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. Therefore, in order to construct smooth decay rates for the fully ionized atoms, we make the following simple extensions of these tables.

In the case of ${}^{56}\mathrm{Ni}$, since the positron decay rate is so slow, we only need to consider electron capture from the continuum. The electron capture rate corresponding to a single transition can be written as

$$\begin{eqnarray}&&{\lambda }_{\mathrm{ec}}\propto {\int }_{1}^{\infty }\displaystyle \frac{{\epsilon }^{2}{(\epsilon +q)}^{2}}{1+{z}^{-1}\exp (\theta \epsilon )}G(Z,\epsilon )d\epsilon \end{eqnarray} \tag{ 6 }$$

with the usual thermodynamic definitions $\beta ={({k}_{{\rm{B}}}T)}^{-1}$, $\theta =\beta {m}_{{\rm{e}}}{c}^{2}$, and $z=\exp (\beta {\mu }_{{\rm{e}}})$, where T is the temperature and μ_e is the electron chemical potential (Bahcall 1964). We have also defined a dimensionless energy difference $q=Q/({m}_{{\rm{e}}}{c}^{2})$ and a dimensionless energy for the electron, $\epsilon =E/({m}_{{\rm{e}}}{c}^{2})$. In the non-relativistic limit, $G(Z,\epsilon )\approx 2\pi \alpha Z$ (Equation 5(c) in Fuller et al. 1980) where α is the fine-structure constant. In the non-degenerate limit, we can replace the Fermi–Dirac distribution with a Boltzmann distribution where $z=({n}_{{\rm{e}}}/{n}_{{\rm{Q}}})\exp (\theta )$. Thus, to leading order in θ,

$$\begin{eqnarray}&&{\lambda }_{\mathrm{ec}}\propto \displaystyle \frac{1}{\theta }\displaystyle \frac{{n}_{{\rm{e}}}}{{n}_{{\rm{Q}}}}\propto \rho {T}^{-1/2}.\end{eqnarray} \tag{ 7 }$$

Therefore, when we are off the table, we use the rate

$$\begin{eqnarray}&&{\lambda }_{\mathrm{off},\mathrm{Ni}}(\rho {Y}_{{\rm{e}}},T)=\left(\displaystyle \frac{\rho {Y}_{{\rm{e}}}}{10\,{\rm{g}}\,{\mathrm{cm}}^{-3}}\right){\left(\displaystyle \frac{T}{{10}^{7}{\rm{K}}}\right)}^{-1/2}{\lambda }_{\mathrm{ref},\mathrm{Ni}}\end{eqnarray} \tag{ 8 }$$

where ${\lambda }_{\mathrm{ref},\mathrm{Ni}}$ is the weaklib rate at $\rho {Y}_{{\rm{e}}}=10\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and $T={10}^{7}\,{\rm{K}}$.

In the case of ${}^{56}\mathrm{Co}$, at densities below those included in weaklib tables, the dominant decay for fully ionized material is positron emission. The positron emission rate corresponding to a single transition can be written as

$$\begin{eqnarray}&&{\lambda }_{\mathrm{pe}}\propto {\int }_{1}^{q}{\epsilon }^{2}{(\epsilon -q)}^{2}G(-Z,\epsilon )d\epsilon \end{eqnarray} \tag{ 9 }$$

with the symbols having the same definitions as in Equation (6). We have assumed that positron phase space is empty. In the non-relativistic limit, $G(-Z,\epsilon )\approx 2\pi \alpha Z\exp (-2\pi \alpha Z\epsilon /\sqrt{{\epsilon }^{2}-1})$ (Equation 5(d) in Fuller et al. 1980). Therefore Equation (9) is simply a constant, and thus the rate has no density or temperature dependence. This is only true for a single transition. A significant temperature dependence can still exist, and does in this case, due to the thermal occupation of the first nuclear excited state at $E=0.158\,\mathrm{MeV}$, allowing additional positron-decay transitions to become important. However, this does not introduce a density dependence. Therefore, when we are off the table, we use the rate

$$\begin{eqnarray}&&{\lambda }_{\mathrm{off},\mathrm{Co}}(\rho {Y}_{{\rm{e}}},T)={\lambda }_{\mathrm{ref},\mathrm{Co}}(T)\end{eqnarray} \tag{ 10 }$$

where ${\lambda }_{\mathrm{ref},\mathrm{Co}}(T)$ is the weaklib rate at $\rho {Y}_{{\rm{e}}}=10\,{\rm{g}}\,{\mathrm{cm}}^{-3}$.

We use the ionization fractions (Section 3.1), "lab" rates (Table 1), and the extended weaklib rate tables (Section 3.2) to construct rates that are applicable over the regime of interest. We assume that atoms with two (or more) bound electrons decay at the "lab" rate and that atoms with exactly one bound electron decay at one-half of the lab rate. Atoms that are fully ionized decay at the rate given by the extended weaklib tables. Therefore, the total decay rate is

$$\begin{eqnarray}&&{\lambda }_{\mathrm{total}}=({f}_{2}+0.5{f}_{1}){\lambda }_{\mathrm{lab}}+{f}_{0}{\lambda }_{\mathrm{weaklib}}.\end{eqnarray} \tag{ 11 }$$

The rates that we use in our MESA calculations are shown in Figure 2.

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Figure 3 shows profiles at 500 days of our main fiducial model of the surviving companion of a SN Ia, a $0.6\,{M}_{\odot }$ WD that has captured a $0.03\,{M}_{\odot }$ envelope that is initially pure ${}^{56}\mathrm{Ni}$. The radii of various enclosed masses, separated by ${10}^{-4}\,{M}_{\odot }$, are shown as bullets, some of which are labeled with vertical numbers. The third panel shows the mass fractions of ${}^{56}\mathrm{Ni}$ (green), ${}^{56}\mathrm{Co}$ (red), and ${}^{56}\mathrm{Fe}$ (black); a significant amount of undecayed ${}^{56}\mathrm{Ni}$ remains even after 82 "lab" half-lives have elapsed. Due to the steady wind (as demonstrated in the bottom panel by the constant value of $\dot{M}=3.5\times {10}^{-4}\,{M}_{\odot }$ yr⁻¹ within the acceleration region), the part of the envelope near $0.1\,{R}_{\odot }$ has expanded to a density and temperature where ${}^{56}\mathrm{Ni}$ is no longer fully ionized and electron captures can occur. At this location, the energy generation rate reaches a maximum (third panel, dashed line), which is also seen in the density–temperature profiles in Figure 2. As expected from steady-state super-Eddington wind solutions (Nugis & Lamers 2002; Ro & Matzner 2016), the location where the luminosity surpasses the local Eddington rate (second panel) is nearly coincident with the sonic point with respect to the isothermal sound speed of the gas, ${k}_{{\rm{B}}}T/\mu {m}_{p}$, where μ is the mean molecular weight (top panel).

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Due to the continued energy release from ${}^{56}\mathrm{Ni}$ and ${}^{56}\mathrm{Co}$ decays, the luminosity and velocity increase above the wind-launching region. The increase in luminosity ends at the $\tau =2/3$ photosphere (dotted line in all panels), because the energy from radioactive decays is assumed to escape this optically thin region. To be more precise, the kinetic energy from the positrons is likely thermalized even in this optically thin region. We account for this small effect in the light curves shown later in this section, but we neglect this energy when evolving the models with MESA. The velocity and mass loss rate, however, continue to increase above the photosphere. This is due not to continual acceleration, but instead to previously higher velocities and higher mass loss rates earlier in the model's evolution.

In Figure 4, we show the linear (left-hand plot) and logarithmic (right-hand plot) temporal evolution of our main fiducial model as well as fiducial models with different WD and envelope masses. Black, red, and green lines represent core + envelope masses of $0.6+0.03$, $0.3+0.006$, and $0.9+0.08\,{M}_{\odot }$, respectively. In each plot, the top panel shows the evolution of the total luminosity, which is the sum of the luminosity at an optical depth of $\tau =2/3$ and the power from positron kinetic energy released by ${}^{56}\mathrm{Co}$ decays above the photosphere. We assume the positrons are able to thermalize and deposit their energy locally while the gamma-rays emitted during the ${}^{56}\mathrm{Ni}$ and ${}^{56}\mathrm{Co}$ decays escape the wind and the surrounding SN ejecta. In practice, however, the positron kinetic energy in the optically thin region is smaller than the luminosity at the photosphere, so this contribution is not significant. Note that the luminosity from the SN Ia ejecta is not included. For comparison, the dotted line in the top panels represents the total power from radioactive decays of $0.001\,{M}_{\odot }$ of initially pure ${}^{56}\mathrm{Ni}$ under "lab" conditions.

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The effective temperature, derived from the luminosity and radius at the $\tau =2/3$ surface, is shown in the middle panels. The velocity at the $\tau =2/3$ location is shown as a solid line in the bottom panels, and the mass-averaged velocity in the optically thin region is shown as a dashed line.

The observed quasi-bolometric late-time light curve of the best observed SN Ia to date, SN 2011fe, is shown as bullets (Nugent et al. 2011; Tsvetkov et al. 2013; Kerzendorf et al. 2014; Shappee et al. 2016). The late-time flattening of SN 2011fe and other SNe Ia above simple predictions for ${}^{56}\mathrm{Co}$ decay within the SN ejecta has been attributed to light echoes and the decay of long-lived radioactive isotopes such as ⁵⁷Co and ⁵⁵Fe (Seitenzahl et al. 2009; Röpke et al. 2012; Kerzendorf et al. 2014; Graur et al. 2016; Shappee et al. 2016). Our models represent an additional contribution to the total luminosity, and thus the observed light curve represents an upper limit to the luminosity for our wind models.

Because the photospheres recede and the models become very hot, peaking at shorter wavelengths than the quasi-bolometric light curves include, it is conceivable that the radiation from the surviving WD companion goes undetected by current observations. However, the SN ejecta is likely optically thick to these UV photons due to line-blocking even during the nebular phase (Li & McCray 1996). Thus, the WD wind's effective temperature while the SN Ia is younger than ∼10 yr is not representative of the expected emission. Instead, the surviving WD wind's luminosity will be absorbed and reradiated in the optical by the SN ejecta. A more quantitative analysis of the expected emission awaits future radiative transfer calculations, but it is likely that the observed late-time light curve of SN 2011fe is indeed an applicable upper limit to the allowed contribution to the quasi-bolometric luminosity from a surviving WD companion.

It is clear that our models do not fade at the simple ${}^{56}\mathrm{Co}$ decay rate at late times. The delayed decays of ${}^{56}\mathrm{Ni}$ cause the slope of the evolution of luminosity to be shallower and, for most models, to flatten to the Eddington-limited value where it remains for years to decades. This is because the WD envelope regulates itself in such a way that the power released by the delayed radioactive decays balances gravity at the Eddington limit. This self-regulation is very similar to the evolution of classical novae, for which the post-nova WD burns hydrogen at the Eddington limit for an extended period of weeks to decades.

Due to this flattening of luminosity, it is obvious that the upper limits from Section 2.1 for the captured ${}^{56}\mathrm{Ni}$ envelopes on the 0.6 and $0.9\,{M}_{\odot }$ WDs lead to late-time light curves that are too luminous. Meanwhile, the $0.3\,{M}_{\odot }$ WD ejects its much smaller $0.006\,{M}_{\odot }$ envelope more rapidly and continues to fade such that its late-time light curve remains less luminous than the latest observations of SN 2011fe. However, such a low-mass helium WD cannot be a typical donor for the bulk of SNe Ia due to its long previous main-sequence lifetime.

Note that the very sharp drop in luminosity and rise in temperature for the $0.9+0.08\,{M}_{\odot }$ model ∼30 yr after the explosion (green lines in the lower plot of Figure 4) is somewhat misleading. The actual evolution of this model as the WD reaches its maximum temperature and then falls onto the WD cooling track occurs at a similar rate to the other models but appears different due to the logarithmic time axis. This can also be seen in the figures to follow for some of the other models that reach the WD cooling track decades after the explosion.

Several important assumptions have gone into our modeling of these post-SN Ia winds, and we now demonstrate the effects of relaxing some of these assumptions. In Figure 5, we show the influence of the choice of envelope mass. Black, red, and green lines represent core + envelope masses of $0.6+0.03$, $0.6+0.003$, and $0.6+0.0003\,{M}_{\odot }$, respectively. In order to eject the envelope rapidly enough that the WD can cool and remain less luminous than SN 2011fe at late times, the envelope mass needs to be much smaller than the upper limit of $0.03\,{M}_{\odot }$ calculated in Section 2.1. This remains a reasonable possibility, because the envelope masses of our fiducial models derived in Section 2.1 are a strict upper limit. Work done on the ejecta by previously shocked material will reduce the actual mass that remains bound to the surviving WD companion. Furthermore, although we are assuming that the companion WD is not fully disrupted by the onset of the merging process, it will still be significantly expanded when the SN Ia explosion occurs. Its shallower potential well will also result in a less massive captured envelope. Future work will use the output of merger simulations to account for these effects and provide more quantitative initial conditions that may yield late-time luminosities within the constraints of SN 2011fe.

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A related uncertainty is the entropy structure of the captured envelope. It is plausible that our fiducial model is more bound than a realistic simulation would find, and so the fiducial envelope is ejected later than for a realistic initial configuration. In Figure 6, we show the effect of changing the entropy of the envelope. All three models have the same core + envelope mass of $0.6+0.03\,{M}_{\odot }$. The black line, as for all the figures in this subsection showing temporal evolution, is the main fiducial model with an entropy twice that for which the envelope's integrated internal energy is half its integrated gravitational binding energy. The red and green lines have 0.5 and 1.5 times the entropy of the fiducial model, respectively. As expected, less bound envelopes are ejected sooner, allowing the WD to dim more rapidly.

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Our final test quantifies the effect of the opacity, as shown in Figure 7. The black line is our fiducial model, with a constant opacity $\kappa =0.2\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$ in the region with iron-group mass fraction ${X}_{56}\gt 0.1$ and density $\rho \lt 1\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. The red line represents a model with a constant opacity $\kappa =2.0\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$. A higher opacity causes the envelope to be ejected somewhat later but also reduces the Eddington limit, so the luminosity flattens at a lower value. The physical opacity is likely to be higher than our fiducial choice of $0.2\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$ because of the large opacity of iron-group elements and effects such as line broadening in the accelerated wind. Radiative transfer calculations of post-SN winds will inform a more physical treatment of opacity in future work.

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In summary, the main fiducial model shown in Figure 4 is overluminous by several orders of magnitude at the latest observed phases of SN 2011fe, which suggests that if SNe Ia arise from double WD systems, both WDs are destroyed during the explosion. However, there are several effects that can reduce the discrepancy between late-time observations and the luminosity predicted from a surviving WD donor's wind. The largest effect comes from ejecting the ${}^{56}\mathrm{Ni}$-rich envelope more rapidly, which can in turn be caused by a less massive and less bound envelope. This is a reasonable expectation since the envelope mass of the fiducial model is an upper limit. More realistic initial configurations for our wind models from future merger simulations will allow us to use late-time observations to place meaningful constraints on SN Ia progenitor scenarios.

We now turn to models of winds from bound remnants of SN Iax explosions. Similar to the post-SN Ia surviving WD donors of the previous subsection, the post-SN Iax surviving WD accretors that we model in this subsection will be degenerate cores surrounded by hot ${}^{56}\mathrm{Ni}$-rich envelopes. However, as described in Section 2.2, the envelopes will not be pure ${}^{56}\mathrm{Ni}$, but instead will be a mixture of carbon deflagration ashes with a ${}^{56}\mathrm{Ni}$ mass fraction of 0.1–0.2.

The black and red lines in Figure 8 represent core + envelope masses of $0.5+0.05\,{M}_{\odot }$ with initial ${}^{56}\mathrm{Ni}$ mass fractions in the envelope of 0.1. The green and blue lines represent masses of $0.9+0.1\,{M}_{\odot }$ with initial ${}^{56}\mathrm{Ni}$ mass fractions in the envelope of 0.2. The black and green models begin with entropies twice that for which the envelope's integrated internal energy is half its gravitational binding energy; the red and blue models have entropies 50% higher than the black and green models, respectively. Models with initial envelope entropies half that of the black and green models were also run. However, the envelopes were too gravitationally bound for the delayed and diluted radioactive decays to launch winds. Since such a wind is necessary to explain observations of SNe Iax, we do not discuss these models further.

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With the exception of the model with higher mass and lower entropy (green), our simulations do not significantly exceed the best-observed late-time SN Iax bolometric light curves in the literature: SNe 2005hk (squares; Phillips et al. 2007; Sahu et al. 2008; McCully et al. 2014b), 2008A (bullets; Ganeshalingam et al. 2010; Hicken et al. 2012; McCully et al. 2014b), and 2008ha (diamond; Foley et al. 2014). The model with higher mass and higher entropy (blue) is somewhat marginal, although it is not ruled out given the uncertainties inherent in our simulations. The photospheric velocities of our models are also in good agreement with the late-time SN Iax velocities reported in the literature (Jha et al. 2006; McCully et al. 2014b).

The large optical depths of the relatively massive SN Iax bound envelopes result in very extended photospheres and low effective temperatures that cool to 1000–2000 K by ~400 days. The IR detection of SN 2014dt at 300–400 days (Fox et al. 2016) may be a common feature of SNe Iax and may be due to the low effective temperature of the photosphere, as opposed to, or in addition to, dust formation (Foley et al. 2016).

The more massive post-SN Iax bound envelopes take longer to be ejected than the very low-mass envelopes of the SN Ia models favored by the late-time observations of SN 2011fe in the previous subsection. This is further compounded by the lower initial ${}^{56}\mathrm{Ni}$ mass fractions and subsequently lower specific energy release in the SN Iax remnant envelopes. Thus, it seems to be a robust prediction that the late-time luminosities of these bound remnants will approach the Eddington limit and remain there for years to decades. In this framework, the $1.5\times \,{10}^{38}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, $2100\,{\rm{K}}$ source detected at the location of SN 2008ha four years after the explosion (Foley et al. 2014) is consistent with being emission from the bound remnant's wind and not the donor, which should be much hotter if it is a helium star. We predict a constant bolometric luminosity for SN 2008ha and other old SNe Iax for at least a decade, but the sources will likely become bluer on a shorter timescale of several years.

While the envelope masses in our SN Iax simulations are taken from previous studies of explosions and are thus somewhat better motivated than for our SN Ia models, the entropy structures are still a large uncertainty. More quantitative constraints on SN Iax progenitor models await future studies using hydrodynamic simulations of explosions that spatially resolve the bound remnant for more accurate initial conditions. These future studies should also account for the influence of the surviving donor within the post-SN Iax wind, both as an illuminating source and as a mechanism for shaping and accelerating the wind.