PRE-EXPLOSION COMPANION STARS IN TYPE Iax SUPERNOVAE

Type Iax supernovae (SNe Iax) were recently proposed as one of the largest classes of peculiar SNe (Foley et al. 2013). Members of this subclass, have several observational similarities to normal SNe Ia (Li et al. 2003; Branch et al. 2004; Jha et al. 2006; Phillips et al. 2007; Foley et al. 2009, 2010, 2013), but also present sufficiently distinct observational properties. SNe Iax are significantly fainter than normal SNe Ia (Foley et al. 2013, 2014). They have a wide range of explosion energies (10⁴⁹–${10}^{51}\;\mathrm{erg}$), ejecta masses (0.15–$0.5\;{M}_{\odot }$), and ${}^{56}\mathrm{Ni}$ masses (0.003–$0.3\;{M}_{\odot }$). The spectra of SNe Iax are characterized by lower expansion velocities (2000–$8000\;\mathrm{km}\;{{\rm{s}}}^{-1}$) than those of normal SNe Ia (15,000 km s⁻¹) at similar epochs. Moreover, two SNe Iax (SN 2004cs and SN 2007J) were identified with strong He lines in their spectra (Foley et al. 2009, 2013).

SNe Iax are the most common of all types of peculiar SNe by both number and rate, with an estimated rate of occurrence of about 5%–30% of the total SN Ia rate (Li et al. 2011; Foley et al. 2013). Most SNe Iax are observed in late-type, star-forming galaxies (Foley et al. 2013; Lyman et al. 2013; White et al. 2015). The observational constraints suggest relatively short delay times for the progenitor systems of SNe Iax ($\lt 500\;\mathrm{Myr}$, see Foley et al. 2014). However, there is at least one Iax event, SN 2008ge, that was observed in an S0 galaxy with no signs of star formation (which suggests a long delay time, see Foley et al. 2010, 2013).

It is accepted that SNe Iax are either thermonuclear explosions of accreting WDs or the core-collapse of massive stars. However, the nature of the progenitor systems that give rise to SNe Iax has not been yet elucidated (Foley et al. 2013, 2014; Liu et al. 2013a, 2015; McCully et al. 2014; Fisher & Jumperx 2015). Given the diversity of SNe Iax, multiple different progenitor channels are likely to contribute to the observed population. The most likely progenitor scenarios currently proposed for SNe Iax are summarized as follows.

1. The Chandrasekhar-mass (Ch-mass) scenario. A WD accretes hydrogen (H)- or helium (He)-rich material from a non-degenerate companion until its mass approaches the Ch-mass, ${M}_{\mathrm{Ch}}\approx 1.4\;{M}_{\odot }$, at which point a thermonuclear explosion ensues (Whelan & Iben 1973; Nomoto 1982; Nomoto et al. 1984). Here, the companion star could be a H-rich donor (either a main-sequence; MS star, a subgiant, or a red giant; RG) or a He-rich donor (He star). It has been suggested that a Ch-mass WD can undergo either a deflagration, or a detonation, or a delayed detonation to lead to an SN Ia explosion (Arnett 1969; Woosley et al. 1986; Khokhlov 1989). In this work, we only focus on SNe Iax that were recently thought to likely be produced from the deflagration explosion of a WD. Turbulent deflagrations in WDs can cause strong mixing of the SN ejecta (e.g., Röpke 2005), and the small amount of kinetic energy released from deflagration explosions is in good agreement with the low expansion velocities of SNe Iax (Foley et al. 2013). Furthermore, the (weak) pure deflagration explosion of a Ch-mass WD has recently been proposed as a promising scenario for SNe Iax (Kromer et al. 2013, 2015; Fink et al. 2014). In this specific deflagration explosion scenario, a weak deflagration explosion fails to explode the entire WD, leaving behind a polluted WD remnant star (Jordan et al. 2012; Kromer et al. 2013; Fink et al. 2014). Hydrodynamical simulations have shown that the off-center-ignited weak deflagrations of Ch-mass C/O or hybrid C/O/Ne WDs (García-Berro et al. 1997; Chen et al. 2014; Denissenkov et al. 2015) are able to reproduce the characteristic observational features of SNe Iax quite well (Jordan et al. 2012; Kromer et al. 2013, 2015; Fink et al. 2014), including the faint 2008ha-like SNe Iax (Kromer et al. 2015). However, it is impossible to determine the exact ignition condition of a Ch-mass WD in this work because the WD is treated as a point mass in our binary calculations (see Section 2). We therefore assume that all Ch-mass WDs in our binary calculations will lead to (weak) pure deflagration explosions for SNe Iax.

In addition, the pulsational delayed detonation explosion of a near Ch-mass WD has been proposed to be a possible scenario for the SN Iax SN 2012Z (Stritzinger et al. 2014) which has been suggested to come from a progenitor system consisting of a He-star donor and a C/O WD (McCully et al. 2014). However, further and more detailed hydrodynamic and radiative-transfer simulations for this pulsational delayed detonation explosion model are still needed.

2. The sub-Ch-mass scenario (e.g., Iben et al. 1987; Livne 1990; Woosley & Weaver 1994; Woosley & Kasen 2011). The WD accretes He-rich material from a companion star (a non-degenerate He star or a He WD) to accumulate a He-layer on its surface. If a He-shell accumulation reaches a critical value ($\simeq 0.02$–$0.2\;{M}_{\odot }$), a single-detonation (i.e., ".Ia model," see Bildsten et al. 2007; Shen et al. 2010) or double-detonation is triggered to cause the thermonuclear explosion of the sub-Ch-mass WD. However, current works show that theoretical spectra and light-curves predicted from the simulations with a thick He shell (0.1–$0.2\;{M}_{\odot }$) do not match the observations of SNe Ia/Iax (Kromer et al. 2010; Woosley & Kasen 2011; Sim et al. 2012). Therefore, we will assume a thin He shell ($\approx 0.05\;{M}_{\odot }$) in our subsequent calculations for the sub-Ch-mass scenario. Recent population synthesis calculations for the sub-Ch-mass double-detonation scenario suggest that the rates and delay-time distribution from this scenario seem to be consistent with the observations of SNe Iax (Ruiter et al. 2011; Wang et al. 2013). However, current explosion models for the double-detonation scenario (Bildsten et al. 2007; Fink et al. 2010; Kromer et al. 2010; Sim et al. 2010; Shen et al. 2010; Woosley & Kasen 2011) show that this kind of explosion struggles to reproduce the characteristic features of SNe Iax, for example, the low ejecta velocity of SNe Iax, low ${}^{56}\mathrm{Ni}$ mass, and strong mixing of their explosion ejecta. Also, the sub-Ch-mass explosions are not going to undergo a weak pure deflagration as in the case for Ch-mass WDs. Nevertheless, further investigations are still required and numerous complications remain to be solved in such a model (Piersanti et al. 2014; Shen & Moore 2014; Piro 2015). For instance, Piro (2015) suggests that including turbulent mixing in He-accreting WDs can lead to more than 50% C/O in the accreted layer at the time of ignition, which probably causes the nucleosynthetic products to be different, or similar. It is unclear if any extended explosion model within this scenario can successfully reproduce most observational features of the least luminous SNe Iax. We therefore also include the sub-Ch-mass explosion as a potential scenario for producing SNe Iax in our presented calculations.

3. The massive-star core-collapse scenario (e.g., Umeda & Nomoto 2005; Valenti et al. 2008; Moriya et al. 2010). A core-collapse SN is triggered by the gravitational collapse of the Fe core of a massive star (with an initial MS mass above about $10\;{M}_{\odot }$. that had its H and a significant amount of He stripped from its outer layers. In particular, the fallback core-collapse explosions of massive stars have been proposed to explain the peculiar SN Iax SN 2008ha because these specific SN explosions could produce a low explosion energy and ${}^{56}\mathrm{Ni}$ mass to account for the properties of SN 2008ha (Moriya et al. 2010). However, this core-collapse scenario seems difficult to explain the full diversity of SNe Iax. Also, a TP-AGB-like source at the position of SN 2008ha has been detected by a recent Hubble Space Telescope (HST) analysis (Foley et al. 2014). This source has been further suggested to be a candidate of either the bound remnant of the WD, or the companion star of the WD, or the companion of the massive-star progenitor (although it is very unlikely, see Foley et al. 2014). If future observations confirm that this source is indeed a bound remnant of the WD, a massive star as the progenitor of SN 2008ha would be ruled out. Also, pre-explosion imaging for another SN Iax, SN 2008ge, has ruled out particularly massive stars as potential progenitors (Foley et al. 2010, 2015).

Analyzing pre-explosion images at the SN position provides a direct way to constrain SN progenitor systems (e.g., see Foley et al. 2014, 2015; McCully et al. 2014). To date, no progenitors of normal SNe Ia have yet been directly observed, even for detections of relatively nearby SNe Ia, SN 2011fe, and SN 2014J. However, the probable progenitor system of an SN Iax SN 2012Z (i.e., SN 2012Z-S1) has been recently discovered from pre-explosion HST images (McCully et al. 2014). It is further suggested that SN 2012Z had a progenitor system consisting of a He star donor and a C/O WD (McCully et al. 2014).

In the accreting-WD explosion scenario, the WD can only be observed directly in our own Milky Way and several very nearby galaxies because the WD would be faint. Consequently, the companion stars generally play a major role in determining the pre-explosion signatures of progenitor systems. It may be possible to determine the nature of the companion star of SNe Iax, and thus their progenitors, by analyzing pre-explosion images at the SN positions. In this work, we predict pre-explosion signatures of companion stars for different progenitor systems that have been recently proposed for SNe Iax. This will be very helpful for analyzing future pre-explosion observations of SNe Iax. In Section 2, we describe the numerical method we used in this work. The pre-explosion signatures of companion stars are presented in Section 3. A comparison between our results and the observation of SN 2012Z-S1 are presented in Section 4. The conclusions are summarized in Section 5.

We use the Cambridge stellar evolution code STARS, which was originally written by Eggleton (1971, 1972, 1973) and then updated several times (Han et al. 2000). The detailed description of the version we used can be found in Han et al. (2000) and Han & Podsiadlowski (2004). Generally, Roche-lobe overflow (RLOF) is treated by the prescription of Han et al. (2000). All of our stars have metallicity $Z=0.02$ and start in circular orbits. For H-rich stars (MS, subgiant, and RG), we set the ratio of the typical mixing length (l) to the local pressure scale height (H_P), $\alpha =l/{H}_{{\rm{P}}}=2.0$, and the convective overshooting parameter, ${\delta }_{\mathrm{ov}}=0.12$, which roughly corresponds to an overshooting length of $\approx 0.25\;{H}_{{\rm{P}}}$. The He-rich star is evolved without enhanced mixing, i.e., the convective overshooting parameter, ${\delta }_{\mathrm{ov}}=0$.

In our binary evolution calculations, the accreting WDs (i.e., C/O WDs or hybrid C/O/Ne WDs) are treated as a point mass, only detailed structures of the mass donor stars are solved in the code. We trace detailed mass-transfer of the binary systems until the WDs increase their mass to the critical limit (assuming SN Iax explosions occur). For different companion star models, the mass-growth rate of the WD, ${\dot{M}}_{\mathrm{WD}}$, is defined as follows.

For the H-rich donor star channel, the mass growth rate of the WD, ${\dot{M}}_{\mathrm{WD}}={\eta }_{\mathrm{He}}{\dot{M}}_{\mathrm{He}}={\eta }_{\mathrm{He}}{\eta }_{{\rm{H}}}| {\dot{M}}_{\mathrm{tr}}|$, where ${\dot{M}}_{\mathrm{He}}$ is the mass-growth rate of the He layer under the H-shell burning, and $| {\dot{M}}_{\mathrm{tr}}|$ is the mass-transfer rate from the non-degenerate H-rich companion star. Also, ${\eta }_{\mathrm{He}}$ is the mass-accumulation efficiency for He-shell flashes, which is shown below in detail (see also Kato & Hachisu 2004), and ${\eta }_{{\rm{H}}}$ is the mass-accumulation efficiency for H burning, which is defined as follows (see also Liu et al. 2015)⁴

$$\begin{eqnarray}{\eta }_{{\rm{H}}}=\left\{\begin{array}{ll}{\dot{M}}_{\mathrm{cr},{\rm{H}}}/| {\dot{M}}_{\mathrm{tr}}| , & | {\dot{M}}_{\mathrm{tr}}| \gt {\dot{M}}_{\mathrm{cr},{\rm{H}}},\\ 1, & {\dot{M}}_{\mathrm{cr},{\rm{H}}}\geqslant | {\dot{M}}_{\mathrm{tr}}| \geqslant \displaystyle \frac{1}{8}{\dot{M}}_{\mathrm{cr},{\rm{H}}},\\ 0, & | {\dot{M}}_{\mathrm{tr}}| \lt \displaystyle \frac{1}{8}{\dot{M}}_{\mathrm{cr},{\rm{H}}},\end{array}\right.\end{eqnarray} \tag{ 1 }$$

where ${\dot{M}}_{\mathrm{cr},{\rm{H}}}=5.0\times {10}^{-7}(1.7/X-1)({M}_{\mathrm{WD}}/{M}_{\odot }-0.4){M}_{\odot }{\mathrm{yr}}^{-1}$ is the critical accretion rate for stable H burning, X is the H mass fraction, ${M}_{\mathrm{WD}}$ the mass of the accreting WD. Here, the optically thick wind (Hachisu et al. 1996) is assumed to blow off all unprocessed material if $| {\dot{M}}_{\mathrm{tr}}|$ is greater than ${\dot{M}}_{\mathrm{cr},{\rm{H}}}$, and the lost material is assumed to take away the specific orbital angular momentum of the accreting WD. The effect of magnetic braking is also included by adopting the description of angular-momentum loss from Sills et al. (2000). If the C/O WD increases its mass to near the Ch-mass limit ($\approx 1.4\;{M}_{\odot }$), we assume the SN Iax explosion ensues.

Very recently, considering the uncertainties of the C-burning rate, Chen et al. (2014) have suggested that hybrid C/O/Ne WDs as massive as $1.3\;{M}_{\odot }$ can be formed if the convective undershooting of the C-burning layer is large enough and the carbon fraction below the flame is largely reduced. These hybrid C/O/Ne WDs in close binary systems can eventually accrete H- or He-rich material from their companion stars to reach the Ch-mass limit and explode as faint SNe (see García-Berro et al. 1997; Chen et al. 2014; Denissenkov et al. 2015). In this work, this hybrid WD scenario has also been included in our detailed binary evolution calculations for the Ch-mass scenario. Because all WDs are treated as a point mass in our calculations, the C/O WD and the hybrid WD channel are distinguished just based on their initial WD mass (see Section 4). The initial WD masses of ${M}_{{WD}}^{i}=0.65,0.7,0.8,0.9,1.0,1.1,1.2$, and $1.3{M}_{\odot }$ are considered for the H-accreting Ch-mass scenario (see also Liu et al. 2015). With a low initial WD mass of $0.65\;{M}_{\odot }$, only two systems (out of about 120 binary systems that are calculated) can lead to SN Iax explosions, $0.65\;{M}_{\odot }$ is thus set to be the minimum WD mass for producing SNe Iax from this channel. A WD mass of $1.3\;{M}_{\odot }$ is the maximum initial mass we assumed for a hybrid C/O/Ne WD (see Chen et al. 2014; Denissenkov et al. 2015).

For the He-rich donor star channel, the mass growth rate of the WD, ${\dot{M}}_{\mathrm{WD}}={\eta }_{\mathrm{He}}| {\dot{M}}_{\mathrm{tr}}^{\prime }|$ (see also Wang et al. 2010; Liu et al. 2015), where $| {\dot{M}}_{\mathrm{tr}}^{\prime }|$ is the mass-transfer rate from the He-rich companion star, the ${\eta }_{\mathrm{He}}$ is the mass-accumulation efficiency for He burning, which is set to be

$$\begin{eqnarray}{\eta }_{\mathrm{He}}=\left\{\begin{array}{ll}{\dot{M}}_{\mathrm{cr},\mathrm{He}}/| {\dot{M}}_{\mathrm{tr}}^{\prime }| , & | {\dot{M}}_{\mathrm{tr}}^{\prime }| \gt {\dot{M}}_{\mathrm{cr},\mathrm{He}},\\ 1, & {\dot{M}}_{\mathrm{cr},{\rm{H}}}\geqslant | {\dot{M}}_{\mathrm{tr}}^{\prime }| \geqslant {\dot{M}}_{\mathrm{st}},\\ {\eta }_{\mathrm{He}}^{\prime }, & {\dot{M}}_{\mathrm{st}}\geqslant | {\dot{M}}_{\mathrm{tr}}^{\prime }| \geqslant {\dot{M}}_{\mathrm{low}},\\ 1(\mathrm{no}\ \mathrm{burning}), & | {\dot{M}}_{\mathrm{tr}}^{\prime }| \lt {\dot{M}}_{\mathrm{low}},\end{array}\right.\end{eqnarray} \tag{ 2 }$$

where ${\dot{M}}_{\mathrm{cr},\mathrm{He}}=7.2\times {10}^{-6}({M}_{\mathrm{WD}}/{M}_{\odot }-0.6){M}_{\odot }\;{\mathrm{yr}}^{-1}$ is the critical accretion rate for stable He burning; ${\dot{M}}_{\mathrm{st}}$ is the minimum accretion rate for stable He-shell burning (Kato & Hachisu 2004; Piersanti et al. 2014); ${\dot{M}}_{\mathrm{low}}=4\times {10}^{-8}\;{M}_{\odot }\;{\mathrm{yr}}^{-1}$ is the minimum accretion rate for weak He-shell flashes (Woosley et al. 1986); ${\eta }_{\mathrm{He}}^{\prime }$ is obtained from linearly interpolated from a grid computed by Kato & Hachisu (2004). As mentioned above, if the WD increases its mass to the near Ch-mass limit ($\approx 1.4\;{M}_{\odot }$), we assume the SN explosion ensues. Here, the initial WD masses of ${M}_{\mathrm{WD}}^{i}=0.865,0.9,1.0,1.1,1.2$, and $1.3\;{M}_{\odot }$ are considered for the He-accreting Ch-mass scenario (see also Wang et al. 2010; Liu et al. 2015). The lower mass limit of $0.865\;{M}_{\odot }$ is corresponding to the minimum WD mass for producing SNe Iax from this channel, and an upper-limit of $1.3\;{M}_{\odot }$ is the maximum initial mass we assumed for the hybrid C/O/Ne WD (see Chen et al. 2014; Denissenkov et al. 2015).

For the low mass-transfer-rate He-accreting case, $| {\dot{M}}_{\mathrm{tr}}^{\prime }| \;\lesssim 4\times {10}^{-8}\;{M}_{\odot }\;{\mathrm{yr}}^{-1}$, a thick layer of helium is believed to grow on the surface of the WD. As a result, once a He-shell accumulation reaches a critical value ($0.05\;{M}_{\odot }$ is adopted in this work, see Woosley & Kasen 2011), we assume that a single detonation or double detonation is triggered, the sub-Ch-mass WD explodes as an SN Iax. When the mass-transfer rate is low, $| {\dot{M}}_{\mathrm{tr}}^{\prime }| \lesssim 1\times {10}^{-9}\;{M}_{\odot }\;{\mathrm{yr}}^{-1}$, it is suggested that the flash when the He layer ignites is too weak to initiate a carbon detonation, which results in only a single He detonation wave propagating outward (Nomoto 1982). We do not especially distinguish this low-mass-transfer case in our binary evolution calculations because the condition of He ignition is still uncertain (Bildsten et al. 2007; Shen et al. 2010; Woosley & Kasen 2011; Shen & Moore 2014). This is different from the setup in Wang et al. (2013) who did not include the case of $| {\dot{M}}_{\mathrm{tr}}^{\prime }| \lesssim 1\times {10}^{-9}\;{M}_{\odot }\;{\mathrm{yr}}^{-1}$ because they only focused on the double-detonation explosions. In the present work, the initial WD masses of ${M}_{\mathrm{WD}}^{i}=0.8,0.9,1.0$, and $1.1\;{M}_{\odot }$ are adopted for the He-accreting sub-Ch-mass scenario. Here, $0.8\;{M}_{\odot }$ corresponds to the prediction for the minimum WD mass for carbon burning from recent hydrodynamic simulations. The detonation of a C/O WD may not be triggered for lower mass (Sim et al. 2012). We also expect that the initial C/O WD masses are below $\simeq 1.1\;{M}_{\odot }$ as more massive WDs are usually formed consisting of O and Ne, i.e., ONe WDs (Siess 2006; Doherty et al. 2015, but see Ruiter et al. 2013).

In the He-accreting sub-Ch-mass scenario, the companion star could also be a He WD. However, the long delay times of the He-WD donor channel ($\gtrsim 800\;\mathrm{Myr}$, see Ruiter et al. 2011) are inconsistent with the observational constraints on the ages of SN Iax progenitors ($\lt 500\;\mathrm{Myr}$; Foley et al. 2013; Lyman et al. 2013; White et al. 2015), and this channel is generally proposed to produce calcium-rich SNe such as SN 2005E (Foley 2015). Therefore, the He-WD donor scenario is not addressed in our calculations.

Using the method described in Section 2, we performed detailed calculations for a set of binary systems with various initial WD masses (${M}_{\mathrm{WD}}^{i}$), companion masses (M₂ⁱ), and orbital periods (Pⁱ) in different progenitor scenarios. Consequently, the range of companion stars of progenitor properties at the moment of SN Iax explosions, e.g., the bolometric luminosity (L), effective temperature (${T}_{\mathrm{eff}}$), photospheric radius (R), orbital velocity (${V}_{\mathrm{orb}}$), and surface gravity (g) are directly determined in our binary calculations with the STARS code. In addition, some properties of binary progenitor systems when the SN explosion occurs can also be calculated, e.g., the orbital periods of binary systems, the amount of mass lost during the optically thick stellar wind phase, mass-transfer rate at the moment of SN explosion, and even equatorial rotational velocities of companion stars (if we assume that the companion stars co-rotate with their orbits). All of these pre-explosion signatures of companion stars and binary progenitors can be compared with pre-explosion observations of SNe Iax to help understand the nature of the SN Iax progenitor (e.g., McCully et al. 2014).

Furthermore, to facilitate direct comparison with optical observations, the bolometric luminosity is converted to broadband magnitudes by using the same method as Pan et al. (2013) under the assumption that the companion photosphere emits a blackbody radiation spectrum:

$$\begin{eqnarray}&&{m}_{{S}_{\lambda }}=-2.5\;{\mathrm{log}}_{10}\left[\displaystyle \frac{\int {S}_{\lambda }(\pi {B}_{\lambda })d\lambda }{\int ({f}_{\nu }^{\;0}\;c/{\lambda }^{2}){S}_{\lambda }d\lambda }\ {\left(\displaystyle \frac{R}{d}\right)}^{2}\right],\end{eqnarray} \tag{ 3 }$$

where ${S}_{\lambda }$ is the sensitivity function of a given filter at wavelength λ, ${B}_{\lambda }$ is the Planck function, d is the distance of the star, and ${f}_{\nu }^{0}=3.631\times {10}^{-20}\;\mathrm{erg}\;{\mathrm{cm}}^{-2}\;{{\rm{s}}}^{-1}\;{\mathrm{Hz}}^{-1}$ is the zero-point value in the AB magnitude system. Here, the effective temperature (${T}_{\mathrm{eff}}$) and photospheric radius (R) of a pre-explosion companion star obtained from our binary evolution calculations are needed. Specifically for this work, the different filters of the HST/WFC3 system and their corresponding sensitivity functions are considered in this study. The absolute magnitudes are calculated in the AB magnitude system.

Figure 1 illustrates the Hertzsprung–Russell (H–R) diagram in different representations using the HST wavebands. As it is shown, companion stars in the H-rich donor and He-rich donor channel are independently located in the H–R diagram at the moment of SN explosion. We can also clearly distinguish the location of companion stars between the He-rich donor Ch-mass channel (i.e., C/O WD+He star and hybrid WD+He star channel) and the He-rich donor sub-Ch-mass channel. This indicates that we can probably rule out (or confirm) the proposed progenitor scenarios for SNe Iax with pre-explosion observations at the SN Iax position. But, unfortunately, we cannot fully distinguish the hybrid C/O/Ne WDs' companions and C/O WDs' companions in the Ch-mass explosion scenarios because the accreting WDs are treated as a point in our detailed binary evolution calculations (but see the discussion in Section 4).

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Recently, a luminous blue source (SN 2012Z-S1) was detected in pre-explosion HST images that was coincident with the location of SN 2012Z McCully et al. (2014), and the non-degenerate He companion star to a C/O WD (originally proposed for SNe Iax by Foley et al. 2013) was suggested to be the likely explanation of the observations (McCully et al. 2014). However, they also suggested that the possibility of a massive star and an accretion disk around the exploding WD cannot still be excluded (see Figure 2 of McCully et al. 2014). Here, we compare our model results with the observations of SN 2012Z-S1.

4.1. Comparison with Detailed Binary Calculations

A comparison between the results of our binary evolution calculations for different potential progenitor channels and the observations of SN 2012Z-S1 are displayed in Figure 1. As shown in this figure, the predictions from the Ch-mass scenario with a He star donor are consistent with the observations, which agrees with the previous suggestion for the progenitor system of SN 2012Z (see McCully et al. 2014).

However, as previously mentioned, it is impossible to distinguish the C/O WDs and the hybrid WDs in our binary calculations because the accreting WD is set up as a point mass. To further discuss the differences of companion stars in the C/O WD+He star and hybrid WD+He star channel, we simply assume the range of initial WD mass for the C/O WD+He star channel of 0.865–$1.2\;{M}_{\odot }$, the range of initial WD mass for the hybrid WD+He star channel of 1.1–$1.3\;{M}_{\odot }$ (left column of Figure 2, see also Chen et al. 2014; Denissenkov et al. 2015). In Section 4.2, our binary population synthesis (BPS) calculations for the hybrid WD+He star channel will obtain that most binary systems in this channel have an initial WD mass larger than $1.1\;{M}_{\odot }$, the binary systems with a lower initial WD mass are quite rare. In addition, only a few C/O WDs are formed with a mass exceeding $1.1\;{M}_{\odot }$. This implies that our assumptions on the initial masses of C/O WDs and hybrid WDs are appropriate.

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The left column in Figure 2 shows the locations of companion stars at the moment of SN explosions for the C/O WD+He star and hybrid WD+He star Ch-mass scenario, respectively. It is shown that the observation of SN 2012Z-S1 is only consistent with the predicted companion locations from the hybrid WD+He star scenario and the C/O WD+He star scenario with an initial C/O WD mass of $1.2\;{M}_{\odot }$. Taking the problem of the origin of very massive C/O WDs into account, our detailed binary evolution calculations seem to disfavor that SN 2012Z-S1 is a non-degenerate companion star to a C/O WD, it is more likely to be a He star with a hybrid C/O/Ne WD. Because the hybrid WDs have much lower C to O abundance ratios at the moment of the explosive C ignition than their pure C/O counterparts (Denissenkov et al. 2015), which probably lead to different observational characteristics from those of the Ch-mass C/O WDs after the SN explosions and thus being distinguished by spectroscopy observations. Recently, an off-center deflagration in a near Ch-mass hybrid C/O/Ne WD has been simulated by Kromer et al. (2015). This showed that deflagrations in near Ch-mass hybrid C/O/Ne WDs can explain the faint SN Iax SN 2008ha (Kromer et al. 2015). However, only a simple near Ch-mass hybrid C/O/Ne WD model was used, more future works with considering different initial conditions are needed to explore the deflagration explosions of hybrid WDs.

4.2. Comparison with BPS Calculations

4.3. Uncertainties

Some potential progenitor scenarios have been proposed to explain the observational properties of SNe Iax. Although weak deflagrations of Ch-mass WDs seem to be the most promising scenario for SNe Iax (Jordan et al. 2012; Foley et al. 2013; Kromer et al. 2013; Fink et al. 2014; McCully et al. 2014), including the faint SNe Iax such as 2008ha-like events (Kromer et al. 2015). However, the nature of the progenitor systems of SNe Iax is still unclear. In this work, we predict observational features of companion stars of binary progenitor systems at the moment of the SN Iax explosion by performing detailed binary evolution calculations with the 1D stellar evolution code STARS, which will be very helpful for constraining the progenitor systems of SNe Iax through their pre-explosion imaging (e.g., McCully et al. 2014; Foley et al. 2015).

Comparing our results with the observations of SN 2012Z-S1, it is found that our detailed binary evolution calculations for the C/O WD+He star and hybrid WD+He star Ch-mass channel can contribute significantly to the number of systems in the vicinity of SN 2012Z-S1, but it seems the hybrid WD+He star Ch-mass channel is more favorable for the observations of SN 2012Z-S1. However, our BPS calculations for both the C/O WD+He star and the hybrid WD+He star Ch-mass channel produces companion stars as cool as the SN 2012Z-S1 only at very low probability. We thus conclude that the possibility of the He donor star as a companion of SN 2012Z is low based on our BPS predictions, but the possibility cannot be excluded. A further confirmation needs future observations after SN 2012Z will have faded below the brightness of SN 2012Z-S1.

We thank the anonymous referee for valuable comments and suggestions that helped us to improve the paper. We thank Elvijs Matrozis for his helpful discussions. This work is supported by the Alexander von Humboldt Foundation. R.J.S. is the recipient of a Sofja Kovalevskaja Award from the Alexander von Humboldt Foundation. B.W. acknowledges support from the 973 programme of China (No. 2014CB845700), the NSFC (Nos. 11322327, 11103072, 11033008, and 11390374), the Foundation of Yunnan province (No. 2013FB083).