TYPE Ia SINGLE DEGENERATE SURVIVORS MUST BE OVERLUMINOUS

Type Ia supernovae (SNe Ia) are among the most luminous explosive events known, and are observable from halfway across the universe (e.g., Rodney et al. 2012). The peak luminosities of SNe Ia are well correlated with the decline rate of their light curves, thereby permitting their use as standardizable candles (Phillips 1993; Hamuy et al. 1995; Riess et al. 1995). The combination of these two properties has made SNe Ia very useful for measuring cosmological parameters (Riess et al. 1998; Perlmutter et al. 1999). As surveys have grown in scope, systematic errors are becoming the dominant source of uncertainty in Type-Ia-SNe-based cosmological constraints (Wood-Vasey et al. 2007; Kessler et al. 2009; Guy et al. 2010; Conley et al. 2011).

Even as SNe Ia have been crucial in shaping our understanding of the universe, the nature of their progenitor systems remains theoretically ambiguous and observationally elusive (for a review see Wang & Han 2012). While it is commonly accepted that SNe Ia result from the thermonuclear explosion of a carbon-oxygen white dwarf (WD) in a close binary system, the nature of the binary companion and the chain(s) of events leading up to the SN explosion are still uncertain. Broadly speaking, there are two dominant progenitor models. The first is the double degenerate (DD) scenario where the companion is also a WD (Tutukov & Yungelson 1979; Iben & Tutukov 1984; Webbink 1984). The second is the single degenerate (SD) scenario where the companion is a non-degenerate object: a main sequence (MS) star, a red giant (RG), a sub-giant, or a He star (Whelan & Iben 1973; Nomoto 1982). While the SD scenario is widely considered to be the more plausible channel for SNe Ia, there is mounting evidence suggesting that this view is incorrect. First, SNe Ia progenitor models predict outflows during the presupernova evolution creating windblown cavities, however, Badenes et al. (2007) find that the X-ray observations of seven young SN Ia remnants are inconsistent with this picture.² Second, it has recently been shown that rate of WD mergers in the Galactic disk is comparable to that of the SNe Ia rate in Milky Way like galaxies, seemingly in support of the DD scenario (Badenes & Maoz 2012). Additionally, giant donors seem to be ruled out observationally as the dominant channel³ (e.g., Mattila et al. 2005; Leonard 2007; Hayden et al. 2010; Bianco et al. 2011; Brown et al. 2012a; Li et al. 2011; Bloom et al. 2012; Brown et al. 2012b; Edwards et al. 2012; Schaefer & Pagnotta 2012), and in this study we do not consider them as possible progenitor systems. An He star donor was recently ruled out for the nearby Type Ia SN 2011fe (Li et al. 2011), while main sequence and sub-giant companions have mounting evidence against them (e.g., Leonard 2007; Brown et al. 2012a; Bloom et al. 2012; Schaefer & Pagnotta 2012), but might still remain a significant channel for the production of SNe Ia.

One approach to constraining the SD channel is to examine the effects of the explosion on the donor star (Colgate 1970; Cheng 1974; Wheeler et al. 1975; Fryxell & Arnett 1981; Taam & Fryxell 1984; Livne et al. 1992; Marietta et al. 2000; Meng et al. 2007; Pakmor et al. 2008; Pan et al. 2012). While many aspects of the interaction of a Type Ia explosion with the donor star, such as its kick velocity (e.g., Ruiz-Lapuente et al. 2004), early-time shock emission (e.g., Kasen 2010; Hayden et al. 2010; Bianco et al. 2011; Brown et al. 2012b), the statistical properties of the surviving companion population (Han 2008; Wang & Han 2010), and the amount of material removed by the explosion (Mattila et al. 2005; Leonard 2007), have received close attention, little has been made of the dramatic changes in the properties of the donor following the explosion. Pan et al. (2012) recently emphasized that fully 65% of the mass removed from the companion is due to ablation (shock-mediated heat transfer), not stripping (momentum transfer),⁴ so mass cannot be removed without heating the star because ablation and heating are not independent variables. Thus, as mass is ablated from the companion, the entire stellar interior is also shock heated and the remaining envelope is significantly puffed up (Marietta et al. 2000). Podsiadlowski (2003) modeled the future evolution of SN Ia companions by considering a constant amount of mass stripped from the donor star while treating the energy injected into the envelope of the companion as a free parameter, and varied it almost by an order of magnitude. While not favored by Podsiadlowski (2003), models with small amounts of energy injected into the remaining star compared to the binding energy of the stripped material could actually become less luminous than the star's luminosity prior to the explosion. These low luminosity models are not, however, consistent with the simulations.

In this paper we focus on the post-explosion state of the SD donor star. They are required to be more luminous than before the explosion by the physics observed in simulations of shock interactions. As a result, the recent observations of nearby SN remnants (SNRs) to detect or constrain the surviving star are far more stringent than generally assumed. In Section 2, we model the evolution of the donor star following the interaction with the SN ejecta and compare our results to previous discussions. In Section 3, we investigate nearby systems individually, first looking at SNRs in the Galaxy and the Large Magellanic Cloud (LMC) and then discussing nearby extragalactic SNe Ia. Section 4 briefly discusses the effect of shock interactions on the binary companions of core-collapse SNe. In Section 5 we review our conclusions.

There have been several numerical studies of the impact of the SN ejecta on a companion star of a SN Ia. Marietta et al. (2000) performed two-dimensional simulations of the impact of the SN ejecta on MS stars, sub-giants, and RGs using an Eulerian hydrodynamics code and found that MS and sub-giant donors lost ∼15% of their mass as a result of the impact. Meng et al. (2007) then pointed out that the pre-supernova mass transfer required by the SD model would change the stellar structure of the companion, causing it to become more compact than an isolated MS star. From this they concluded that less mass should be removed from the donor envelope than predicted by the models of Marietta et al. (2000). Pakmor et al. (2008) confirmed this using a three-dimensional smoothed particle hydrodynamics (SPH) code to model the impact of SN ejecta on a companion whose structure was specified by the binary evolution study of Ivanova & Taam (2004). Finally, Pan et al. (2012) performed the most detailed study to date, with the inclusion of symmetry-breaking effects (i.e., orbital motion, rotation of the companion, and Roche-lobe overflow) in a simulation using the multi-dimensional adaptive mesh refinement code FLASH. They found that these effects again raise the mass unbound from the secondary to 16% for an MS donor.

These studies show that there is a generic sequence of events during the impact of the SN ejecta on the companion. The initial impact sends a shock into the donor's stellar envelope, which also sends a reverse shock back into the SN shell and creates a contact discontinuity between the two. The reverse shock develops into a bow shock around the companion which deflects much of the SN ejecta around the donor. Meanwhile, the forward shock penetrates the companion and the center of the shock slows as it encounters the steep density gradient of the stellar interior, accelerating once more after passing through the core. The wings of the shock, which propagate through the outer envelope, move faster than they do through the center because they are propagating through regions of lower density, causing the shock to become highly curved. The curved shock meets itself at the backside of the companion, depositing much of its energy in the donor's envelope and ablating significant amounts of mass from the surface. This creates a back pressure on the rest of the companion, decreasing the kick imparted by the SN blast wave. After most of the SN ejecta has passed the donor, the shock-heated outer envelope of the companion begins to expand and some of it becomes unbound. The majority of the unbound material is at low velocities and is mixed with the inner iron-group layers of the SN ejecta. Even though some material is unbound from the companion, the majority of its mass remains bound to the star. The net heating of the star is not reported, but in the standard 1 M_☉ MS model of Marietta et al. (2000) the star is left with a hot, extended, asymmetric envelope containing ∼10% of the mass, while the central pressure is halved. There is almost no discussion of the subsequent appearance of the stars other than in Marietta et al. (2000). They note that its core nuclear reaction rates will be diminished for a time-scale of 10³–10⁴ yr but that the luminosity will be driven by the Kelvin-Helmholtz collapse of the extended envelope. Their expectation was for the star to become significantly more luminous (500–5000 L_☉ for a MS 1 M_☉ star).

Ideally, to study the long-term evolution of the impacted companion, one would want to model the three-dimensional interaction of the SN ejecta with a suite of possible donor stars until the companion is relaxed and roughly spherical. Then one would use the mean density, temperature, and composition profiles as the initial conditions for a one-dimensional stellar evolution code and follow the companion until it settles back on the MS at the location of its current mass. Ivanova & Taam (2004) show that the mass of the donor star can range from ∼0.6–1.5 M_☉. We only explore the 1 M_☉ MS model of Marietta et al. (2000) because that paper contains the information needed to calibrate our model. Hydrodynamic simulations to calibrate our simple models at other masses or evolutionary states are not available, but the results presented in this paper should generally hold.

We study this problem using the approach of Podsiadlowski (2003), who followed the evolution of a star which loses mass and is heated over a short period of time in order to mimic the effects of the explosion. We used the stellar evolution code MESA (Paxton et al. 2011), allowing for fluid velocities. We added a module to first have a rapid phase of mass loss followed by a rapid phase of extra heating. In the heating phase we also included an outer boundary R_out, adjusting the mass loss rate so that any material at R > R_out is lost in a wind on the escape time scale from that radius. This allowed us to treat mass loss as in Podsiadlowski (2003) or to allow the heating phase to be responsible for some or all of the mass loss when the amount of energy added becomes large. Thus, the models had four parameters: the amount of mass lost in the first phase (ΔM), the amount of energy added in the second phase (E_heat), the truncation radius (R_out), and the time scale used for both the mass loss and heating phases (Δt). In each interval, the mass loss and heating rates were linearly increased and then symmetrically decreased. The internal energy added per unit mass was distributed as the ratio of the enclosed mass at any radius to the total mass. Thus more energy is deposited at the edge of the star than near the center. This leads to results more consistent with the Marietta et al. (2000) simulations than, for example, making the energy added per unit mass independent of radius. It also makes sense physically, as the shock will deposit more energy per unit mass in the lower density regions of the star.

We first considered models where ΔM = 0.15 M_☉ to match the Marietta et al. (2000) hydrogen cataclysmic variable (HCV) model⁵ and E_heat = 0, 0.5, 1, 2, 3, 4, 5, or 6 × 10⁴⁷ erg. We used time scales of Δt = 1, 3, and 10 yr, and R_out = 10, 30, and 100 R_☉. The E_heat = 5 and 6 × 10⁴⁷ erg models did not converge for Δt = 1 yr and R_out = 100 R_☉ due to the development of an oscillation as the transient ends and the outer envelope collapses. Figure 1 shows the evolution of the companion in luminosity and radius as a function of E_heat and the duration Δt assuming R_out = 100 R_☉ after the transient phase ended. The expected luminosity and radius evolution depends little on Δt once a time period Δt has elapsed after the transient phase. The main effect appears to be an underestimate of the luminosity in that period, presumably because of the additional energy radiated during the transient phase. Starting with the 4 × 10⁴⁷ erg model, the star expands to R_out = 100 R_☉, leading to a small amount of additional mass loss. The inclusion of the outer boundary appears to drive the high E_heat models to a similar final structure so that there is little difference in their subsequent evolution. The luminosity of the higher E_heat models can be restricted by reducing R_out because it is the rapid collapse of a greatly over-expanded envelope that produces the highest luminosities. For example, using R_out = 10 R_☉ caps the luminosity of the high E_heat models at roughly 80 L_☉, but they sustain that luminosity for a long time and then roughly merge onto the evolution of the models with R_out = 100 R_☉.

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Figure 2 presents the fractional change in the central temperature, density, and pressure for the Δt = 3 yr, R_out = 100 R_☉ models as a function of the energy added, E_heat. This figure shows that E_heat ≃ 4 × 10⁴⁷ erg is needed to reproduce the fractional change in the HCV model of Marietta et al. (2000). For the remainder of the paper we will take the E_heat = 4 × 10⁴⁷ erg, Δt = 3 yr, and R_out = 100 R_☉ model as our fiducial companion model. Depositing the energy per unit mass uniformly or significantly changing E_heat from this value, leads to models that do not reproduce the Marietta et al. (2000) simulations well. Figure 3 shows the evolution of the effective temperature (T_eff) for the fiducial model and models with higher and lower E_heat. In general, we find that if E_heat is large enough to initiate mass loss (the fiducial and higher E_heat models) then the temperature decreases, and models with insufficient heating to initiate mass loss (the lower E_heat models) become hotter. Since the ultimate mechanism of the mass loss is ablation due to heating, only the models with enough energy input to initiate mass loss are realistic. The total energy in SNe Ia ejecta is ∼1.2 × 10⁵¹ erg, of which ∼3.3 × 10⁴⁹ erg is incident on the star since, assuming the companion must overflow its Roche lobe, a/R ≃ 3. The binding energy of the 0.15 M_☉ of stripped material is E_strip ≃ 6 × 10⁴⁷ erg, so the total energy transferred to the star is approximately E_heat + E_strip ≃ 10⁴⁸ erg and ≃ 60% of that energy is used to strip material from the surface, and ≃ 40% heats the remaining star. Only a small fraction (f ≃ 0.03) of the incident shock energy is transferred to the star. Simply assuming that 3% of the shock energy is transferred to the star with 60% of that energy going into stripping mass does not perfectly reproduce the scaling of mass loss with a/R in Marietta et al. (2000), but it comes considerably closer to doing so than the analytic estimates of Wheeler et al. (1975). More generally, the transfer of energy into heating the surviving star is a crucial parameter that needs to be reported for any future simulations.

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In the final analysis, the basic physics is relatively easy to understand. The shocks heat the star, depositing more energy per unit mass near the surface than near the center. However, the shocks have no physical knowledge of which material will eventually escape and the star has no sharp density jumps to create any features in the energy deposition. The heated material then expands, but with a continuous density distribution between material which ultimately escapes and material which remains bound. As a result, the energy deposited in material which eventually escapes cannot be significantly larger than the energy deposited in the material which does not escape. Mass is stripped beyond some effective radius R_out, leaving a puffed-up envelope that then begins to cool and collapse. The more extended this envelope, the higher the peak luminosity since a lower density envelope has a shorter thermal time scale. Ultimately, however, for the same envelope extent the star produces similar luminosities. Because there are no means in this process of distinguishing between escaping and bound material until the escaping material is stripped, the low E_heat models following Podsiadlowski (2003) are unphysical because they effectively make the heating of the unbound material independent of the heating of the bound material. When the mass loss is driven by ablation, the energy used to strip mass must be tightly correlated with the energy used to heat the surviving star and is not an independent variable.

In this sense, alternate models where we drive the mass loss with only heating are far more physical. For small values of E_heat the star expands but does not reach R_out and has no mass loss. Larger values of E_heat (>4 × 10⁴⁷ erg) expand the star to R_out, and the stars begin to lose mass. The post-transient evolution of all stars that have had mass loss is very similar because the primary driver of their luminosity is the collapse of the envelope from R_out to its eventual equilibrium radius. This is illustrated in Figure 4, which shows the evolution as a function of E_heat labeled by whether or not the stars lost any mass assuming R_out = 100 R_☉. Ignoring the luminosity during the transient, much of which is radiating energy injected into the expanding envelope, the subsequent evolution of the mass losing transients is essentially independent of the deposited energy. The amount of mass lost rises steadily with E_heat, and for the maximum energy E_heat = 10⁴⁸ erg, the mass loss is approximately the same as in the HCV model. The central pressure of this model is, however, too low, which means that we should concentrate more of the energy deposition toward the surface.

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Although challenging, searching for the donor star after an SN Ia explosion has significantly faded is a popular method for testing the SD model. This can be done either by examining sources near the center of historical SNRs or by observing the sites of recent SNe Ia in external galaxies. What has not been clearly factored into these discussions is that the surviving star must be cooler and more luminous than it was before the explosion. This is an unavoidable consequence of mass loss by ablation. Our fiducial model with E_heat ≃ 4 × 10⁴⁷ erg and R_out = 100 R_☉, the best match to simulations (shown in Figures 1 and 3), indicates that initially after the SN the companion has L ∼ 300–1000 L_☉ and T_eff ∼ 5000–6000 K, depending slightly on Δt. Then, after ∼10 yr, we find L ∼ 100–200 L_☉ and T_eff ∼ 5000–5200 K. After ∼100 yr, we see L ∼ 50–60 L_☉ and T_eff ∼ 5300–5400 K. Finally, after ∼1000 yr L ∼ 15–20 L_☉ and T_eff ∼ 5500 K.

3.1. Supernova Remnants

3.2. Recent Supernovae

Besides observing the explosion sites of historical SNRs, one could also look at the location of more recent SNe Ia once the SN has faded. This method benefits both from having very accurate search positions and from the greater luminosity of the surviving stars at earlier times (L ∼ 50–300 L_☉). The main problem is that these directly observed SN are far more distant, with the closest being at a distance of >3 Mpc. We briefly discuss four of the closest SNe Ia observed in the last century.

3.2.1. SN 1937C, SN 1972E, and SN 1986G

Three historic SNe Ia, SN 1937C, SN 1972E, and SN 1986G, are particularly good candidates for a study of their post-explosion systems because they are nearby and were well observed. SN 1937C in IC 4182 at ∼4.4 Mpc (Saha et al. 2006) was discovered by Zwicky in the fall of 1937 at Palomar Observatory (Baade & Zwicky 1938). SN 1972E was discovered in NGC 5253 at ∼4.2 Mpc (Saha et al. 2006). Finally, the closest SN Ia discovered in the digital imaging era, SN 1986G, was discovered in NGC 5128 (Centaurus A) at a distance of ∼3.4 Mpc (Evans et al. 1986; Ferrarese et al. 2007). Given the historical SN explosion images, it should be possible to centroid the SN explosion to a fraction of an arcsecond and with HST observations one could plausibly find or rule out a SD companion for these SNe Ia.

3.2.2. SN 2011fe

The recent Type Ia SN 2011fe in M 101 is the nearest Ia in the last 25 yr. At a mere 6.4 Mpc (Shappee & Stanek 2011), this SN Ia provides an exceptional opportunity to put the strongest constraints on the progenitor system of any SNe Ia. Patat et al. (2013) obtained 12 epochs of high resolution spectra of SN 2011fe, finding that it is only slightly reddened and lies in a "clean" environment. Horesh et al. (2012) used upper limits from both radio and X-ray observations of SN 2011fe to rule out circumstellar material from a giant companion. Chomiuk et al. (2012) then strengthened these radio constraints with EVLA observations, ruling out most popular SD progenitor models. The proximity and the low extinction of SN 2011fe allowed Li et al. (2011) to place 10–100 times stronger constraints on the visible light from the progenitor system than previous studies using deep archival pre-explosion HST/ACS observations. These observations rule out luminous red giants (>3.5 M_☉) and most helium stars as plausible donors.

SN 2011fe was discovered very quickly by the Palomar Transient Factory (Law et al. 2009) on 2011 August 24, less than one day after explosion, allowing multi-wavelength follow-up observations to be quickly carried out. With the early time UV, optical, and X-ray observations Nugent et al. (2011) and Bloom et al. (2012) concluded that the radius of the primary was smaller than 0.1 R_☉ and 0.02 R_☉, respectively. Bloom et al. (2012) then went onto show that their constraints leave compact objects as the only viable primary. Brown et al. (2012a) and Bloom et al. (2012) also used the very early-time UV and optical observations to constrain the shock heating which would occur as the SN ejecta collides with the companion. They rule out giants and main sequence secondaries more massive than the sun. These constraints seem to imply that the DD model is the only viable progenitor model. However, these studies both rely on the calculations of Kasen (2010), which assumes a Roche-lobe-overflowing secondary with a typical stellar structure. As previously discussed, Meng et al. (2007) argued that the secondary should be smaller and more compact than a typical main sequence star. Such a smaller, more tightly bound companion would produce a weaker signal at even earlier times which could have been missed. Additionally, slightly less massive companions can also be too small for the SN ejecta shock to have been observed.

Very late time observations of SN 2011fe would place further constraints on the progenitor system by detecting, or putting upper limits on, the existence of a shocked donor. The luminosity of the surviving donor star is expected to be L ∼ 300 L_☉, which corresponds to V ∼ 27 mag. In Figure 5, we present the BVRI light curves of SN 2011fe of Richmond & Smith (2012), Munari et al. (2013), and Shappee et al. (2013) along with a linear fit for the late-time decline as compared to the expected evolution of our fiducial model. If the current decline rate continues, the shocked companion should begin to dominate the SN light in the R ∼ 900 days after maximum light. At that point, HST observations monitoring the very late-time light curve of SN 2011fe should see it level off as most of the observed flux will then be coming from the shock heated companion and not the expanding, cooling shell of ejecta.

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However, there are other effects which might also cause the late-time SN light curve to flatten. First, light echoes from the SN explosion off nearby dust can add additional late-time flux as was seen from SN 1987A (Crotts 1988). Fortunately, the environment surround SN 2011fe shows no evidence for gas or dust (Patat et al. 2013). Second, Auger and internal conversion electrons emitted in the decay of ⁵⁷Co are expected to become a significant source of heating ∼750 days after the explosion and dominate the positron decay of ⁵⁶Co after ∼1000 days (Seitenzahl et al. 2009; Seitenzahl 2011). The late-time bolometric light curve is then expected to flatten (Figure 4 of Röpke et al. 2012). Fortuitously, the evolution of a nearby companion will occur on time scales much longer than the half-life of ⁵⁷Co (271.79 days). Thus, a single degenerate companion would still be observable at even later times.

Interestingly, a similar process may also be possible during stripped core-collapse supernova (SN IIb, Ib, and Ic) explosions. One of the theoretical models for producing stripped core-collapse supernovae progenitors is using binary mass transfer to remove the outer envelope of the primary (e.g., Yoon et al. 2010), a common envelope phase (e.g., Nomoto et al. 1995), or both. In such scenarios, the companion is again forced to lie relatively close to the primary, with similar issues about mass stripping and shock heating to the SD SNe Ia case.

The maximum shock energy is essentially the same as for the Ia case because it is determined by the geometry of Roche-lobe overflow. This means that the effects on main sequence binary companions are likely greatly reduced. First, the escape velocity of a higher mass star is somewhat larger, so less mass is stripped for a given amount of energy. Second, the internal energy of the higher mass star is much larger, so the fractional change in the energy created by the shock is much smaller. Third, the luminosity of the star is much larger, so any adjustment leading to a roughly constant source of additional luminosity is a much smaller fractional change in the luminosity. As a result, we would expect that the effects on an 8 M_☉ main sequence companion with the same shock energy and geometry will not very dramatic, and this is confirmed in our numerical experiments. Additionally, the typical ccSN binary companion (without mass transfer) can be at much larger distances than in a SNe Ia, so the effects will likely be unobservable for MS companions.

However, if the companion has evolved into a giant star, the effects can be much greater due to the significantly reduced binding energy of the companion's envelope. It may even be possible, if the separation between the SN and a giant companion is small, for the SN ejecta to completely strip the envelope of the companion, leaving only the dense core, as seen in hydrodynamic simulations of SNe Ia explosions with a giant companion (e.g., Marietta et al. 2000). The problem is that most of the time the companion star to a ccSN will still be on the main sequence (see Kochanek 2009) unless the binary members were nearly exact twins (Pinsonneault & Stanek 2006).

Main-sequence (MS) donor stars surviving the explosion of a SD SNe Ia must be strongly heated and puffed up in a manner tightly coupled to their mass loss. In stellar evolution models that roughly match the final state of a 1 M_☉ star losing 0.15 M_☉ in the hydrodynamic simulations of Marietta et al. (2000), the star must be heated by E_heat ≃ 4 × 10⁴⁷ erg, and it commences its post explosion evolution with the radius and luminosity dramatically increased and the effective temperature modestly decreased. When mass is lost by ablation, it is not physically possible to avoid these effects. This means that the "leftover" stars of Tycho's SN and in the LMC supernova remnants SNR 0609 − 67.5 and SNR 0519 − 69.0 should still be ∼20 times more luminous than before the explosion. This greatly strengthens the arguments of Schaefer & Pagnotta (2012) against a SD scenario for the LMC SNR 0609 − 67.5 and essentially rules out Tycho Star G as the survivor of SN 1572. Finally, such luminous donors should be observable with HST within ∼10 Mpc. We point out that there are a number of recent nearby Type Ia SNe, including SN 1937C, SN 1972E, SN 1986G, and SN 2011fe ∼2 yr after the explosion that should be studied in detail. For the latter, difference imaging between late time and pre-explosion HST images should be a particularly powerful probe. Thus, the SD channel has been ruled out for at least two nearby SNe Ia far more strongly than assumed and can be easily tested for several more.

Recent works have suggested the "spin-down model" as a way to avoid the existing limits by using rotational support of the WD to delay the explosion long enough to render the companion far fainter than at the time of accretion (Justham 2011; Di Stefano et al. 2011; Di Stefano & Kilic 2012). In addition to several problems of fine tuning, there are no known physical systems where accretion leads to rotational velocities sufficiently close to breakup for rotation to be of much relevance to the hydrostatic support of the system. The fastest spinning WD known (RX J0648.0–4418, P_spin ≃ 13.2 s, Mereghetti et al. 2011) is still spinning at less than half the rate (P_spin ≲ 6 s) needed for rotation to be important (Yoon & Langer 2005). Moreover, most of the scenarios considered by Di Stefano & Kilic (2012) lead to lower mass companions which will be even more drastically affected by shock heating, so our conclusions largely hold for their scenarios as well.

Existing hydrodynamic simulations of the effects of explosions on SNe Ia companions have largely ignored the consequences of the explosions for the future evolution of the stars, although Marietta et al. (2000) comment on the problem and provide enough information to calibrate our model. At a minimum, simulations should report the final outer radius of the bound material and the final energies of the companion star (as compared to the initial energies). Better still would be to report asymmetrically averaged density and energy profiles that could then be used as the basis for more self-consistent models of their subsequent evolution.

We thank Marc Pinsonneault, Jennifer van Saders, Bill Paxton, Philipp Podsiadlowski, Todd Thompson, Dale Mudd, and Joe Antognini for discussions and encouragement. We also thank the anonymous referee for their helpful comments and suggestions improving this manuscript. We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research. B.J.S. was supported by a Graduate Research Fellowship from the National Science Foundation. B.J.S., C.S.K., and K.Z.S. are supported in part by NSF grant AST-0908816. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has made use of NASA's Astrophysics Data System Bibliographic Services.