A Physical Basis for the H-band Blue-edge Velocity and Light-Curve Shape Correlation in Context of Type Ia Supernova Explosion Physics

TL;DR

This paper links the velocity of a specific spectral feature in Type Ia supernovae to their light-curve shape, providing insights into explosion physics and distinguishing between different explosion models.

Contribution

It introduces a physical interpretation of the $v_{edge}$ velocity and its correlation with light-curve parameters, enhancing understanding of supernova explosion mechanisms.

Findings

$v_{edge}$ correlates with light-curve shape across supernova types.

Explosion models near the Chandrasekhar limit align with observed $v_{edge}$ values.

Sub-luminous SN 2015bo suggests diverse explosion scenarios.

Abstract

Our recent work demonstrates a correlation between the high-velocity blue edge, $v_{edge}$, of the iron-peak Fe/Co/Ni $H$-band emission feature and the optical light curve shape of normal, transitional and sub-luminous type Ia Supernovae (SNe Ia). We explain this correlation in terms of SN Ia physics. $v_{edge}$ corresponds to the sharp transition between the complete and incomplete silicon burning regions in the ejecta. It measures the point in velocity space where the outer $^{56}$Ni mass fraction, $X_{\rm{Ni}}$, falls to the order of 0.03-0.10. For a given $^{56}$Ni mass, $M(^{56}Ni)$, $v_{edge}$ is sensitive to the specific kinetic energy $E_{\rm kin}$($M(^{56}Ni)/M_{WD}$) of the corresponding region. Combining $v_{edge}$ with light curve parameters (i.e., s$_{BV}$, $\Delta m_{15,s}$ in $B$ and $V$) allows us to distinguish between explosion scenarios. The correlation between $v_{edge}$ and light-curve shape is consistent with explosion models near the Chandrasekhar limit. However, the available sub-$M_{Ch}$ WD explosion model based on SN 1999by exhibits velocities which are too large to explain the observations. Finally, the sub-luminous SN 2015bo exhibits signatures of a dynamical merger of two WDs demonstrating diversity among explosion scenarios at the faint end of the SNe Ia population.

Full text

supernova

††journal: ApJ

1 Introduction

Significant evidence supports the idea that type Ia supernovae (SNe Ia) result from the thermonuclear explosions of at least one carbon-oxygen (C-O) white dwarf (WD) in a binary system. There are two main progenitor channels which have been hypothesised to produce these cosmic explosions. These are the single degenerate scenario (SDS), and the double degenerate scenario (DDS). In the SDS a WD accretes material from a non-degenerate companion star such as a H/He or red giant star (Whelan & Iben, 1973; Livne, 1990; Woosley & Weaver, 1994; Nomoto et al., 1997), whereas in the DDS the system consists of two WDs.

Within each progenitor scenario, multiple explosion mechanisms have been explored. One method for exploding a SN Ia is with heat released during the dynamical merging of two WDs (e.g.,  Dan et al., 2014, 2015).

Alternatively, when a WD approaches the Chandrasekhar mass ($M_{Ch}$) the explosion can be triggered by compressional heating in the center. This possibility may occur in both the SDS (Hoeflich & Khokhlov, 1996; Nomoto et al., 1997), as well as in the DDS scenario where a tidally disrupted WD accretes onto the primary WD on secular time scales, which are much longer than the hydro-dynamical time scales (Piersanti et al., 2003). In a $M_{Ch}$ explosion the nuclear burning flame front begins as a sub-sonic deflagration wave and then at a particular transition density, $\rho_{tr}$, it evolves into a supersonic detonation wave. This is known as a delayed detonation (DDT) model and has been shown to provide a good match to spectra and light curves of SNe Ia (Khokhlov, 1991; Yamaoka et al., 1992; Hoeflich & Khokhlov, 1996; Gamezo et al., 2005; Poludnenko et al., 2011; Blondin et al., 2011, 2015; Hoeflich et al., 2017; Ashall et al., 2018). For DDT explosions the luminosity of the SN is correlated with the amount of burning during the deflagration phase (Gamezo et al., 2005; Röpke et al., 2007; Jordan et al., 2008), which in spherical models is parameterized by $\rho_{tr}$, where less luminous objects have a lower $\rho_{tr}$ value (Hoeflich & Khokhlov, 1996; Hoeflich et al., 2017). Effectively, a low $\rho_{tr}$ produces more intermediate mass elements (IME) at the expense of 56Ni, and for less luminous objects the remaining 56Ni is located at lower velocities.

Another possibility is the explosion of a sub-$M_{Ch}$ WD in the so-called edge-lit, or double detonation scenario. Here, He accreted from a companion star onto the surface of the WD detonates, which drives a shock wave inward igniting the centre of the WD, while the outer layer is consumed by the initial detonation (Woosley & Weaver, 1994; Livne & Arnett, 1995; Hoeflich & Khokhlov, 1996; Nugent et al., 1997; Fink et al., 2007; Pakmor et al., 2012; Shen & Moore, 2014). These models have recently made a revival as it has been shown that by mixing the outer He layer with a small amount of C alters the burning network from slow triple-$\alpha$ to the fast ${}^{12}C(\alpha,\gamma)^{16}O$ channel (Shen & Moore, 2014). This reduces the mass of the He shell required to form a sustained nuclear detonation by an order of magnitude compared to previous work (e.g.,  Woosley & Weaver, 1994; Livne & Arnett, 1995). The resulting density structures are close to spherical with small polarization (Bulla et al., 2016b), and to first order, the outer layers hardly affect the light curves (LCs) beyond $\approx 1$ week after the explosion (Polin et al., 2018). However, it is not clear whether the reduced He mass can trigger a detonation in the C/O layers of the WD. In sub-$M_{Ch}$ models the luminosity of the SN is correlated with the ejecta mass of the explosion, where less luminous objects have a smaller WD mass (MWD); (Sim et al., 2010; Blondin et al., 2017).

The luminosity of a SN Ia is dependent on the amount of 56Ni synthesized in the explosion. More luminous SNe produce larger amounts of 56Ni (e.g.,  Arnett, 1982; Stritzinger et al., 2006a; Mazzali et al., 2007; Childress et al., 2015). Furthermore, events which are more luminous also have broader light curves, which is the underlying basis for the luminosity-width-relation (LWR) (Phillips, 1993; Phillips et al., 1999). The LWR can be understood in terms of opacities, where brighter objects have more 56Ni, produce more heating, are dominated by doubly ionized species, and therefore have slowly evolving light curves. Whereas, the faintest SNe Ia have less 56Ni, less heating, are dominated by singly ionized species, and have faster light curves. (Nugent et al., 1997; Umeda et al., 1999; Kasen et al., 2009; Hoeflich et al., 2017).

At the faint end of the LWR, there are sub-luminous SNe Ia (1991bg-like; Filippenko et al., 1992; Leibundgut et al., 1993; Turatto et al., 1996). The literature contains a number of different scenarios accounting for the origins of sub-luminous SNe Ia. These extend from the dynamical merger of two WDs111For normal SNe Ia this is currently out of favour as it is inconsistent with low polarization observations (Patat et al., 2012; Bulla et al., 2016a). (García-Berro & Lorén-Aguilar, 2017), to sub-$M_{Ch}$ explosions (Scalzo et al., 2014; Blondin et al., 2017, 2018), and to $M_{Ch}$ DDT explosions (Hoeflich et al., 2002, 2017). A key to understanding SNe Ia is to address the on-going question of whether normal, transitional222Transitional SNe Ia are thought to be the link between the normal and sub-luminous populations (e.g., see Hsiao et al. 2015; Ashall et al. 2016a, b; Gall et al. 2018). and sub-luminous SNe Ia are separate groups, form a continuum, or are a mixture of diverse scenarios. This work aims to address this question.

The ongoing discussion about explosion scenarios in SNe Ia physics may, at least in part, be attributed to different assumptions in the modeling process. These assumptions include LTE population levels (e.g.,  Goldstein & Kasen, 2018), a small atomic network of isotopes (e.g.,  Polin et al., 2018), and that $M_{Ch}$ explosions do not have a central core of electron capture elements. On the other hand, we know non-LTE effects are important, and, even full non-LTE simulations result in different conclusions on the nature of sub-luminous SNe Ia, see Hoeflich et al. (2002); Blondin et al. (2017); Hoeflich et al. (2017); Blondin et al. (2018).

NIR spectroscopy offers a promising way to investigate the physics of SNe Ia. In the $H$ band, between maximum light and +10d, an emission component is formed by blends of a large number of emission lines above the photosphere (Wheeler et al., 1998; Hoeflich et al., 2002; Hsiao, 2009; Hsiao et al., 2013). This $H$-band feature consists of a multiplet of many allowed Fe ii/Co ii/Ni ii lines formed within the 56Ni-rich layers and is thus correlated with the luminosity of the SN (Hsiao et al., 2013; Ashall et al., 2019).

One of the main objectives of the Carnegie Supernova Project II (CSP-II; Phillips et al. 2019) was to obtain a large sample of NIR spectra of SNe Ia (Hsiao et al., 2019). Using this data, Ashall et al. (2019) found a correlation between the outer blue-edge velocity, $v_{edge}$, of the $H$-band break region and the optical light-curve shape for normal, transitional and sub-luminous SNE Ia. 333We note that this work does not include analysis of other SNe Ia sub-types objects such as SN 1991T or SNe Iax. Here, we explain this correlation in terms of SNe Ia physics and models. We compare the data to both spherical sub-$M_{Ch}$ and $M_{Ch}$ explosion models with non-LTE light curves and spectra published in the literature. We use 1D calculations as they artificially suppress mixing and produce a chemically layered structure. This is in line with both observations (Fesen et al., 2007; Maeda et al., 2010; Diamond et al., 2018; Dhawan et al., 2018) and inferred abundance stratification results (e.g.,  Stehle et al., 2005; Tanaka et al., 2011; Ashall et al., 2016b). This suppressed mixing may be due to high magnetic fields (Hristov et al., 2018).

In this work, we show how $v_{edge}$ varies among different explosion models, even for SNe with similar 56Ni masses. Therefore $v_{edge}$ provides new information beyond the total amount of 56Ni  synthesized in the explosion, which has been classically used to analyze SNe Ia (Stritzinger et al., 2006a; Childress et al., 2015; Dhawan et al., 2017; Scalzo et al., 2019). A second goal of this paper is to put $v_{edge}$  into context with the classical characterization using SNe Ia light curve shape and absolute magnitude, and to show an example of how this combined information can be used to distinguish between explosion models.

2 Measurements of Spectra and Light Curve Parameters

In this work, the $H$-band break is represented by the Doppler shift, $v_{edge}$, of the bluest component of the $H$-band multiplet at 1.57 $\mu$m (Fig. 1). In a SN Ia explosion, the photosphere recedes through the IME layers before it reaches the 56Ni region. Once this 56Ni-rich region is exposed, the Fe/Co/Ni emission in the $H$-band begins to emerge. Ashall et al. (2019) found that at about +10 days relative to $B$-band maximum was a good time to measure $v_{edge}$, because during this phase the spectra are dominated by single ionized iron group elements, and by then the feature has emerged in all SNe Ia. However, note that at earlier times, $v_{edge}$ may be affected by lines of IME. Whereas, significantly beyond +10d the emission from forbidden lines starts to dominate. In fact, the ideal time to measure $v_{edge}$ may be right after the H-band feature can be clearly distinguished from other blends. However, this would require daily observations which are not currently available.

We measure $v_{edge}$ by the method outlined in Ashall et al. (2019) in both the observed and theoretical spectra. Briefly, $v_{edge}$ is measured by fitting a Gaussian profile to the minimum of the blue-edge of the $H$-band break. The fit is produced over a fixed wavelength range and iterated to find convergence. The model spectra are based on detailed non-LTE radiation transport explosions for normal, transitional and sub-luminous SNe Ia (Hoeflich et al., 2002, 2017; Blondin et al., 2018). For all models close to $M_{Ch}$, $v_{edge}$ corresponds to the region where the 56Ni mass fraction, $X_{\rm{Ni}}$, is approximately 0.02-0.03. For the sub-luminous sub-$M_{Ch}$ model with low mass ($M_{WD}$=0.9$M_{\odot}$) (Blondin et al., 2018) the corresponding abundance is slightly higher ($X_{\rm{Ni}}$ is $\approx$ 0.10)444We note that Blondin et al. (2018) use incomplete nuclear reaction networks, and complete networks predict a larger 56Ni production (Shen et al., 2018a). However, the velocity where $X_{\rm{Ni}}\approx$ 0.10 in the complete networks is consistent with that of Blondin et al. (2018) within 500km s*-1*.. This is because the lower mass produces a smaller optical depth. For all models, regardless of $M_{WD}$, the value of $X_{\rm{Ni}}$ is small. Therefore, we use the H-band break as an indicator for the outer edge of the 56Ni region. For a discussion of the uncertainties in the fluxes see the references for each of the models.

In Fig. 1, we present observations of the spectral evolution for a normal-bright and a sub-luminous SN Ia. For bright SNe Ia, the iron-peak feature emerges at about +3 days past maximum light when the overall NIR continuum is dominated by Thomson scattering and, thus, will have a very little line blending beyond the blue edge. For the sub-luminous example, blends of other transitions are obvious in both the observations and the theoretical model, because they have a lower ionization state and less Thomson scattering (Fig. 2). These blends can be seen until about +10 days, and can be attributed to lines of singly-ionized iron-group elements, Ca ii and Ca i (1.46, 1.51 $\mu$m) and neutral IME elements, namely Si i (1.52 $\mu$m), S i (1.46, 1.54 and 1.52, 1.57 $\mu$m) (Hoeflich et al., 2002). Therefore, for sub-luminous SNe Ia spectra should be inspected to make sure there are no blends of IMEs555We note that as these IME lines are very sensitive to the ionization state, whether epochs earlyer than +10 d can be used to measure $v_{edge}$ can be decided by the spectra of each specific SN.. However, even for the least luminous objects at +10 d after maximum it is predicted that there will be virtually no contamination by IMEs. Hence, this is a good time to measure $v_{edge}$. The effect of blends as a function of time can be seen in Figs. 1, 2, and 3. For sub-luminous SNe Ia, these blends mimic a rapid drop in the evolution of $v_{edge}$  before +10 d as can be seen in the velocity evolution of SN 1999by (see Fig. 3). For transitional SNe Ia, such as SN 2011iv, SNhunt281, and iPTF13ebh, the observed rapid drop in $v_{edge}$ as a function of time can be understood as follows: $v_{edge}$ is larger and is thus at lower densities compared to sub-luminous supernovae. These densities are close to the critical density ($\leq 10^{7...8}particles/cm^{3}$) which marks the transition from allowed to forbidden lines. This results in a rapid drop of $v_{edge}$. We note that, in normal-bright SNe, the temperatures are higher resulting is larger collisional rates which de-populate the levels responsible for forbidden lines and delay the rapid drop in $v_{edge}$.

In this work, we use the sBV parameter defined in Burns et al. (2014) and used in Ashall et al. (2019), as well as $\Delta{\rm m_{15,s}(B)}$ and $\Delta{\rm m_{15,s}(V)}$ . $\Delta{\rm m_{15,s}(B)}$ and $\Delta{\rm m_{15,s}(V)}$ (Höflich et al., 2010; Hoeflich et al., 2017), are modified decline rate parameters produced for a time base of $t=15~{}d*s$, with $s$ being the stretch parameter of the light curve as defined in Goldhaber et al. (2001). They can be parameterized by $\Delta{\rm m_{15,s}(B)}={\rm\frac{\Delta m_{(15\times s)}}{s}}$. Normal SNe Ia have an $\Delta{\rm m_{15,s}(B)}$ and $\Delta{\rm m_{15,s}(V)}$ of $\sim$1.2 mag and 0.7 mag, respectively. We chose to use $\Delta{\rm m_{15,s}(B)}$ and $\Delta{\rm m_{15,s}(V)}$ as they can be theoretically interpreted as a measurement of diffusion time scales (Hoeflich et al., 2017), whereas the theoretical interpretation of $s$ or sBV is not straight forward but a combination of effects.

3 Brief Description of the Explosion Scenarios and their relation to light curve parameters

Normal, transitional, and sub-luminous SNe Ia can be thought to be in three different regimes. In the case of normal SNe Ia the opacity of the ejecta is dominated by doubly ionized species throughout most of the envelope, leading to a slowly declining post-maximum light curve. Transitional SNe Ia are located in the area of rapidly decreasing luminosity in the LWR (Hoeflich et al., 2002, 2017; Ashall et al., 2018)). This is the region where a SN Ia enters the regime of a quickly decreasing opacity in the envelope soon after maximum, due to an earlier onset of the recombination front that results in a faster release of stored energy. Whereas, sub-luminous SNe Ia are dominated by singly ionized species, which leads to a fast declining post-maximum light curve.

Within spherical DDT models, for the brightest SNe Ia most of the 56Ni is made in the detonation phase. Furthermore, in these models the 56Ni production is an almost smooth function of $\rho_{tr}$, but its relative change is quick between models. This is reflected by a rapidly decreasing luminosity over the transitional regime, which, to first order, is $\propto M($56Ni$)$. In other words, the 56Ni production during the detonation phase changes from $\sim$0.3 to 0 $M_{\odot}$, as $\rho_{tr}$ decreases from $\sim$1.8$\times$107g cm*-3* to 0.8$\times$107g cm*-3* over the transitional regime towards sub-luminous SNe Ia. For sub-luminous SNe Ia almost all of the 56Ni is produced during the deflagration burning phase, which happens over a \sim$$0.25\,M_{\odot} range. This leads to a close to constant luminosity for sub-luminous SNe Ia and a 56Ni production of 0.1-0.2 $M_{\odot}$ (e.g.,  Stritzinger et al., 2006b), which is dependent on the rate of electron capture (Hoeflich et al., 2017; Gall et al., 2018).

In the sub-$M_{Ch}$ models considered here a central explosion is triggered in a static WD (Blondin et al., 2018). These types of models are used as a proxy for a helium detonation, as well as the dynamically driven double-degenerate double-detonation scenario (Shen et al., 2018b). In these classes of models the mass of the WD is correlated to the luminosity and the light-curve shape. Lower mass WDs produce less effective burning, less nuclear statistical equilibrium (NSE) elements, less 56Ni, less heating, lower opacities, and a faster light curves.

4 The relation of $v_{edge}$ and light curve parameters

4.1 $v_{edge}$ versus sBV

Here, we discuss $v_{edge}$ as a function of light-curve color-stretch parameter, sBV. As shown in Ashall et al. (2019) and as illustrated in the upper left panel of in Fig. 4, SNe with a larger sBV are found to exhibit higher values of $v_{edge}$.

A measure of $v_{edge}$ serves as a powerful diagnostic tool because it is robust due to Doppler shifts that are well measured by spectra at the onset of the emergence of the $H$-band feature. For a comparison between data and theory, in Fig. 4 we plot results obtained from non-LTE, spherical $M_{Ch}$ DDT models (Hoeflich et al., 2017), as the open green diamonds. Note that in this figure the values of $v_{edge}$ are obtained from four different DDT models (Model 23, 20, 12, and 8) of Hoeflich et al. (2017), whereas in Figure 3, one sub-luminous DDT model and one sub-luminous sub-$M_{Ch}$ model are plotted as a function of time.

A comparison with the abundance structures of the DDT models shows that $v_{edge}$  measures the point in velocity space where $X_{Ni}$ falls to the order of 0.02 to 0.03, for the entire range from normal-bright to sub-luminous SNe Ia.

For burning to NSE, the density must be larger than $\approx 2\times 10^{7}g/cm^{3}$. In the DDT scenario, for normal-bright SNe Ia, $v_{edge}$ corresponds to a region where the 2 to 3 % of 56Ni required to form the emission feature is located between $\approx-13,000$ to $-10,000$ km s*-1*. Because there is little mass involved in these layers, the change in the total amount of 56Ni produced over this $v_{edge}$ range is small and, consequently, the luminosity difference between normal SNe Ia is little.

There is a fast drop in $v_{edge}$  over an sBV range of $\sim$0.6 to 0.45, the mean $v_{edge}$ drops from \sim$$-11,500 km s*-1* to \sim$$-5,500 km s*-1*. This is because the change in mass per unit velocity rapidly decreases with increasing velocity. From DDT models, the bright-transitional SNe Ia are characterized by an NSE production that is dominated by detonation burning, whereas the less luminous transitional supernovae are dominated by deflagration burning. Because the densities are still low, a large relative change in the total NSE mass coincides with a large shift in $v_{edge}$. Therefore, for transitional SNe Ia a uniform change in $\rho_{tr}$ produces a significant drop in $v_{edge}$. This drop in $v_{edge}$  happens in the same regime as the luminosity drop in the LWR. However, these processes are not causally linked. Hence, $v_{edge}$ is a complementary measurement of the SNe Ia physics and scenario, and it is not a proxy for the light-curve shape. This is because the diffusion time scales depend on the 56Ni distribution, 56Ni mass, and ejecta mass., whereas, $v_{edge}$  only depends on the outer edge of the 56Ni region, and measures the specific $E_{\rm kin}$. Finally, for sub-luminous SNe Ia, most of the NSE burning takes place during the deflagration phase in the inner, high-density regions, which produces a minimum value of $v_{edge}$ at \sim$$-5,500 km s*-1*.

Two of the SN (SN 2011fe and SN 2011iv) in our work have been analyzed using the abundance stratification technique (Mazzali et al., 2014; Ashall et al., 2018). This method reverse engineers the abundance structure in the ejecta, using a comparison between observed and synthetic spectra (e.g.,  Stehle et al., 2005; Ashall et al., 2014, 2016b). We also plot the velocity of 56Ni at 3% in abundance from these models (see the blue open diamonds in Fig. 4). There is a good agreement, within $\sim$1000 km s*-1*  with the abundance stratification results, non-LTE DDT models, and observations. It should be noted that by definition the abundance stratification models have the same luminosity as their corresponding SNe.

4.2 SN 1999by

One supernova from the sample, SN 1999by a sub-luminous SN, has been modeled by both $M_{Ch}$ (Hoeflich et al., 2002) and sub-$M_{Ch}$ (Blondin et al., 2018) explosions, which makes it a perfect test case for this work. Therefore, included in Fig. 4 is the sub-$M_{Ch}$ model (SCH2p0) for the sub-luminous SN 1999by from Blondin et al. (2018) (open black square). The SCH2p0 ($M_{WD}$=0.9$M_{\odot}$) sub-$M_{Ch}$ model has a velocity of $-$7,000 km s*-1*, which is much larger, by 1,500 km s*-1*, than the observations. In this model $v_{edge}$ corresponds to the region where the $X_{\rm{Ni}}$ is 0.10. This is due to a combination of geometric dilution and the lower mass meaning that a larger mass fraction is required to form the emission feature.

Fig. 5 presents the spectra of SN 1999by (black) as well as the $M_{Ch}$ DDT model (blue) from Hoeflich et al. (2002), and the sub-$M_{Ch}$ model, SCH2p0, (red), from Blondin et al. (2018), all of which are at $\sim$+11d relative to $B$-band maximum. Note that the models from Hoeflich et al. (2002) and Blondin et al. (2018) have similar $B$-band rise times of 14.5 d and 14.6 d, respectively. It is clear that the $M_{Ch}$ DDT model produces the correct velocity compared to the observations, whereas the sub-$M_{Ch}$ model has a value of $v_{edge}$ which is too large.

As shown by both the $M_{Ch}$ and sub-$M_{Ch}$ models, $v_{edge}$ is determined by the interplay between the mass of 56Ni and the mass of IME formed in the explosion, because there is a steep drop in the abundances during the transition between nuclear statistical equilibrium and incomplete Si-burning. This makes $v_{edge}$  a stable measurement regardless of the explosion scenario. In effect, $v_{edge}$ is a measure of the specific kinetic energy, $E_{\rm kin}$  $(M($56Ni$)/M_{WD}$), of the region. For a given M(56Ni), models with lower values of $M_{WD}$ will result in systematically higher values of $v_{edge}$.

4.3 $v_{edge}$ & $\Delta{\rm m_{15,s}(B)}$, $\Delta{\rm m_{15,s}(V)}$

sBV measures the timing of the turnover in the color curves of SNe Ia, but as we will see below, may mask diversity. Therefore, we use the light curve parameters $\Delta{\rm m_{15,s}(B)}$ and $\Delta{\rm m_{15,s}(V)}$ to analyze the correlation. The bottom panels in Fig. 4 present $v_{edge}$ as a function of $\Delta{\rm m_{15,s}(B)}$ and $\Delta{\rm m_{15,s}(V)}$. Once again the DDT models fit the data well, where as the sub-$M_{Ch}$ model is not close to SN 1999by.

Nine out of ten SNe Ia in the plot are consistent with $M_{Ch}$ DDT models. However, SN 2015bo, which has a high-cadence light curve, with pre-maximum coverage, and will be the subject of a future individual analysis, has a large $\Delta{\rm m_{15,s}(B)}$=1.89 mag but a small $\Delta{\rm m_{15,s}(V)}=1.09$ mag. The combined values of light curve shape and $v_{edge}$ for SN 2015bo are inconsistent with both the DDT and sub-$M_{Ch}$ models. For a $v_{edge}$=$-$7000km s*-1* DDT models predict a $\Delta{\rm m_{15,s}(B)}=\sim 1.85$ mag and $\Delta{\rm m_{15,s}(V)}=\sim 1.32$ mag, and sub-$M_{Ch}$ models predict $\Delta{\rm m_{15,s}(B)}=\sim 1.73$ mag and $\Delta{\rm m_{15,s}(V)}=\sim 1.22$ mag. It is difficult to reconcile this deviation from the data and models by varying the physics of sub-$M_{Ch}$ or $M_{Ch}$ DDT explosions.

Dynamical merger models are characterized by red colors at maximum light, a slowly evolving $V$-band light curve, a fast declining $B$-band light curve, and have 56Ni located at low velocities (Hoeflich & Khokhlov, 1996; García-Berro & Lorén-Aguilar, 2017). These properties make dynamical mergers a viable scenario for SN 2015bo, and demonstrate that there may be different explosion mechanisms and progenitor scenarios present within the sub-luminous SNe Ia population.

We note that, on its own, $v_{edge}$ may not be able to distinguish between different models. The method here also requires accurate high-cadence light curves. For example, in the case of SN 2013ay where the photometric coverage begins at $\sim$+10 d, we cannot rule out either the sub-$M_{Ch}$ or $M_{Ch}$ mass models, despite having high signal-to-noise NIR spectra.

The diversity among SNe Ia is apparent when we combine $\Delta m_{15,s}$ and $v_{edge}$, and extra information may be obtained if the absolute luminosity ($M_{V,B}$) of the supernova is also utilized. Therefore, for further insights, we suggest that all three parameters are used. In the one supernova, SN 1999by, where we can accurately determine the absolute magnitude from observations, it is clear that the $M_{Ch}$ model is favored. Combining absolute magnitude, with $v_{edge}$ and light-curve shape will allow for future work to more accurately discriminate between explosion scenarios and models. For example, a transitional $M_{Ch}$ explosion might have the same value of $v_{edge}$ as a sub-luminous sub-$M_{Ch}$ explosion, and both objects may have similar light-curve shapes, but different 56Ni masses and absolute magnitudes. Therefore, combining all parameters will provide additional information.

5 Conclusion

Using the correlation found between light-curve shape and $v_{edge}$ in Ashall et al. (2019), we have demonstrated that $v_{edge}$ is a new comprehensive way to measure the outer edge of the 56Ni region in SNe Ia ejecta. Brighter SNe Ia have 56Ni located at higher velocities than sub-luminous SNe Ia (see Fig. 4). This is consistent with previous results obtained by nebular phase spectral modelling (Mazzali et al., 1998; Botyánszki & Kasen, 2017). Using SN 1999by as an example, we have also demonstrated that a combination of $v_{edge}$, $\Delta m_{15,s}$, and absolute magnitude can be a new method to probe for diversity and explosion scenarios in SNe Ia.

$v_{edge}$ is a stable measurement which is determined by the interplay between the mass of 56Ni and the mass of IME formed in the explosion, and corresponds to the sharp transition in the ejecta between complete and incomplete Si-burning regions.

$v_{edge}$ measures the point in velocity space where $X_{Ni}$ falls to the order of 0.03-0.10, and is dependent on the ejecta mass of the explosion. $v_{edge}$ is stable when compared to models because it measures a Doppler shift which is dependent on the underlying abundance structure. $v_{edge}$ only depends on the presence of Fe ii and Co ii at the edge of the 56Ni region rather than the correct absolute flux in the model. Furthermore, $v_{edge}$ is an important tool for studying SNe Ia physics as it is independent of distance.

For sub-luminous SNe Ia, at early times, the edge of the $H$-band break may be contaminated by Ca ii, Ca i, Si i, and S i lines. Therefore, we urge that caution should be used when choosing an epoch to measure $v_{edge}$, because these line blends may artificially increase its value. The supernova needs to be in the 56Ni-rich region, and not affected by line blends before measuring $v_{edge}$. For the least luminous SNe Ia, this occurs around +10d. Although, this value can be sensitive to the amount of mixing of 56Ni in the explosion, and each time series of spectra should be examined on a case-by-case basis.

Within the framework of DDT models, the quick drop in the $v_{edge}$ vs. sBV correlation for transitional SNe Ia is due to the rapidly changing expansion velocity as a function of mass. We demonstrated that $M_{Ch}$ models may be able to reproduce the evolution of $v_{edge}$ over the entire luminosity range of SNe Ia. However, the value of $v_{edge}$ obtained from the sub-$M_{Ch}$ model of the sub-luminous SN 1999by differs from the observations by $\sim$1,500km s*-1*. This strongly favors high-mass explosions for the very sub-luminous SNe Ia.  However, we cannot conclude that all of the sub-luminous SNe are inconsistent with sub-$M_{Ch}$ models. It is possible that future sub-$M_{Ch}$ models will overcome their current problems.

Additional distance-independent information can be obtained using $\Delta{\rm m_{15,s}(B)}$ and $\Delta{\rm m_{15,s}(V)}$ in combination with $v_{edge}$. This allows for the diversity amongst low-luminosity SNe Ia to be probed. We find that SN 2015bo is inconsistent with both the spherical $M_{Ch}$ DDT models and sub-$M_{Ch}$ models considered here, but may have characteristics of a dynamical merger models, which adds to the evidence for diversity among the sub-luminous population.

One of the limitations of our study includes the fact that we have only used published non-LTE models with NIR spectra. However, in the future, the method presented here should be applied to a diverse set of explosion scenarios and models, as well as observations of SNe Ia which belong to different areas of the luminosity width relation.

Our analysis here favors high $M_{WD}$ explosions, for all of the supernovae examined. However, it appears that there could be multiple explosion mechanisms amongst sub-luminous SNe Ia, and NIR spectra can reveal this diversity. We are beginning to obtain a full view of the SNe Ia phenomenon ranging from early time studies (e.g.,  Hosseinzadeh et al., 2017; Stritzinger et al., 2018), to mid-infrared studies (e.g.,  Telesco et al., 2015; Hoeflich et al., 2018), and NIR studies (e.g.,  Hsiao et al., 2019; Ashall et al., 2019). Simultaneously examining all of these phenomena and comparing them to modern explosion models may let us unlock the mysteries about what SNe Ia are, which will enable us to improve upon the utility of these objects as distance indicators. Looking towards the future, we belive that $v_{edge}$ will help us understand the physical nature of the SNe Ia and how they explode.

6 Acknowledgements:

In memory of Alexei Khokhlov, a great scientist and friend from whom we learned so much over the years. We want to thank many colleagues for helpful discussions. This work has been supported in part by NSF awards AST-1008343 & AST-1613426 (PI: M.M. Phillips), AST-1613472 (PI: E.Y Hsiao), AST-1715133 (PI: P. Hoeflich), AST-1613455 (PI:N. Suntzeff), and in part by a Sapere Aude Level II grant (PI: M.D. Stritzinger) provided by the Danish National Research Foundation (DFF). M.D. Stritzinger is funded by a research grant (13261) from the VILLUM FONDEN. EB acknowledges partial support from NASA Grant NNX16AB25G. EB and MDS thank the Aarhus University Research Fund (AUFF) for a Sabbatical research grant. Research by DJS is supported by NSF grants AST-1821967, 1821987, 1813708 and 1813466. NBS and KK also thank George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy for support during this research.

In Figure A1 we present the $v_{edge}$ fits of the $M_{Ch}$, sub-$M_{Ch}$ and data of SN 1999by, the fitting procedure from Ashall et al. (2019) was followed. The small differences between the minima of the fits and the data/models are on the order of 200 km s*-1*, which is well with the error bars, and therefore does not affect the conclusions above.