# All known Type Ia supernovae models fail to reproduce the observed
bolometric luminosity-width correlation

Amir Sharon1 and Doron Kushnir1
1Dept.of Particle Phys. & Astrophys., Weizmann Institute of Science, Rehovot
76100, Israel
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Type Ia supernovae (SNe Ia) are widely believed to arise from thermonuclear
explosions of white dwarfs (WDs). However, ongoing debate surrounds their
progenitor systems and the mechanisms triggering these explosions. Recently,
Sharon & Kushnir showed that existing models do not reproduce the observed
positive correlation between the $\gamma$-ray escape time, $t_{0}$, and the
synthesized 56Ni mass, $M_{\mathrm{Ni56}}$. Their analysis, while avoiding
complex radiation transfer (RT) calculations, did not account for the viewing-
angle dependence of the derived $t_{0}$ and $M_{\mathrm{Ni56}}$ in multi-
dimensional (multi-D) models during pre-nebular phases, where most
observations performed. Here, we aim to identify an observational
width–luminosity relation, similar to the $t_{0}$–$M_{\mathrm{Ni56}}$ relation
to constrain multi-D models during pre-nebular phases while minimizing RT
calculation uncertainties. We show that the bolometric luminosity at $t\leq
30$ days since explosion can be accurately computed without non-thermal
ionization considerations, which are computationally expensive and uncertain.
We find that the ratio of the bolometric luminosity at 30 days since explosion
to the peak luminosity, $L_{30}/Lp$, correlates strongly with $t_{0}$. Using a
sample of well-observed SNe Ia, we show that this parameter tightly correlates
with the peak luminosity, $L_{p}$. We compare the observed $L_{30}/Lp$–$L_{p}$
distribution with models from the literature, including non-spherical models
consisting of head-on WD collisions and off-centered ignitions of sub-
Chandrasekhar mass WDs. We find that all known SNe Ia models fail to reproduce
the observed bolometric luminosity-width correlation.

###### keywords:

methods: data analysis – supernovae: general.

††pubyear: 2024††pagerange: All known Type Ia supernovae models fail to
reproduce the observed bolometric luminosity-width correlation–D

##  1 Introduction

Type Ia supernovae (SNe Ia) are widely accepted to be the result of
thermonuclear explosions of white dwarfs (WDs), but their progenitor systems
and explosion mechanism remain under debate (for a review, see, e.g., Maoz et
al., 2014). Several theoretical scenarios for the progenitor systems have been
suggested, including a Chandrasekhar mass (Chandra) or sub-Chandrasekhar mass
(sub-Chandra) WD that ignites due to some external interaction and direct WD
collisions.

Predictions from these models have demonstrated agreement with certain
observational characteristics of SNe Ia in specific instances. One particular
set of characteristics frequently employed for model-observation comparisons
is the Phillips relation (Phillips, 1993; Phillips et al., 1999), which
relates the maximum flux with the width of the light curve in a specific
wavelength band. This relation holds significance in cosmology, and its
parameters are relatively straightforward to derive from observations,
rendering it a widely adopted method for such comparisons (Kasen, 2006; Sim et
al., 2010; Blondin et al., 2017; Shen et al., 2021a; Collins et al., 2022).
However, deriving these parameters from theoretical models and simulations is
quite challenging. One of the key challenges in the calculation of these
parameters and of most observable quantities is radiative-transfer (RT)
calculations, which often prohibit a robust comparison to observations (see
reviews, e.g. Hillebrandt & Niemeyer, 2000; Noebauer & Sim, 2019).

While advances in computational capabilities and theoretical understanding
have significantly improved RT simulations, the large number of physical
processes involved, particularly opacity from thousands of atomic transitions
with both absorptive and scattering characteristics, prohibit comprehensive 3D
calculations based on first principles. Various physical approximations — in
particular, different treatments of the significant deviations from local
thermodynamic equilibrium (LTE) — are employed by different RT codes to
calculate the properties of the plasma and the radiation field. Additionally,
approximate treatment of atomic physics is required due to the partially
calibrated atomic data.

Recently, Blondin et al. (2022) compared various radiative transfer (RT) codes
applied to identical benchmark models (the StaNdaRT public electronic
repository111https://github.com/sn-rad-trans). Among the codes considered,
only CMFGEN (Hillier & Miller, 1998; Dessart & Hillier, 2010) performs full
non-local thermodynamic equilibrium (NLTE) calculations from the onset of the
explosion. The properties obtained from CMFGEN, such as light curves,
temperature, and ionization profiles, behave differently than those from LTE
codes after a few tens of days post-explosion, highlighting the importance of
NLTE treatment during these phases. However, CMFGEN is computationally
expensive and thus limited to 1D profiles.

One approach to bypass the radiative transfer (RT) calculation challenges is
determining the $\gamma$-ray escape time, $t_{0}$, defined by (Jeffery, 1999)

$f_{\textrm{dep}}(t)=\frac{t_{0}^{2}}{t^{2}},\>\>\>f_{\textrm{dep}}\ll 1,$ (1)

where $t$ is the time since explosion and $f_{\textrm{dep}}$ is the
$\gamma$-ray deposition fraction, representing the portion of the generated
$\gamma$-ray energy that is deposited in the ejecta. This quantity can be
estimated directly from the ejecta or using simple $\gamma$-ray radiation
transfer simulations (Stritzinger et al., 2006; Scalzo et al., 2014; Wygoda et
al., 2019a; Guttman et al., 2024). To measure $t_{0}$ in observations of SNe,
the ejecta must be sufficiently optically thin so that the bolometric
luminosity, $L(t)$, equals the instantaneous deposited energy from radioactive
decay:

$L(t)=Q_{\text{dep}}(t).$ (2)

For SNe Ia, this condition is typically met at $t\gtrsim 60$ days post-
explosion for most SNe (Wygoda et al., 2019a; Sharon & Kushnir, 2020b). The
$\gamma$-ray escape time and the synthesized 56Ni mass have recently been used
to compare models with observations for various types of SNe (Sharon &
Kushnir, 2020b, a, 2023). In Sharon & Kushnir (2020a), the
$t_{0}$–$M_{\mathrm{Ni56}}$ distribution of SNe Ia was compared with model
predictions from the literature. It was found that none of the models
reproduced the observed relation, which showed a strong, positive correlation
between the two parameters. This positive correlation, similar to the Phillips
relation, implies that bright SNe are characterized by slowly evolving light
curves and vice versa. Additionally, there is some scatter in the observed
$t_{0}$–$M_{\mathrm{Ni56}}$ relation, which could be attributed to multi-
dimensional (multi-D) effects from non-spherical explosions.

In multi-D models, the luminosity can be unevenly distributed across different
viewing angles, even when the total luminosity, integrated over all angles,
satisfies Equation (2). This uneven distribution might cause the inferred
$t_{0}$ and $M_{\mathrm{Ni56}}$ values to vary between viewing angles,
deviating from their true values. Only at very late times ($t\gtrsim$150 day
for typical SNe Ia), when the ejecta is in the deep nebular phase and
sufficiently optically thin, does the emitted radiation become completely
isotropic regardless of the ejecta’s structure. However, obtaining the
bolometric luminosity at these late times is quite challenging because the
supernova has significantly faded, and a large portion of the flux shifts to
the near- and mid-infrared wavelengths. Only a few supernovae have the
required temporal and wavelength coverage, such as the recent SN 2021aefx,
having JWST observations at $t>200$ days post-explosion (Kwok et al., 2023;
Chen et al., 2023). Considering multi-D effects could explain the observed
variability in the $t_{0}$–$M_{\mathrm{Ni56}}$ distribution, allowing for more
stringent constraints on non-spherical models. However, radiative transfer
(RT) calculations are necessary for this purpose, and, as previously
mentioned, these calculations contain significant uncertainties.

One might argue that to address RT uncertainties and calculate $t_{0}$ for
multi-D models, it is possible to extract the escape time from the calculated
bolometric light curve by solely using times when Equation (2) holds. This
method was applied to 1D ejecta in Kushnir et al. (2020), and the resulting
$t_{0}$ values were consistent with those obtained directly from the ejecta or
through $\gamma$-ray RT simulations. However, a multi-dimensional structure
within the ejecta complicates the use of this method to mitigate inaccuracies
in bolometric light curve calculations performed by RT codes. The challenge
arises because, although Equation (2) might hold for the angle-averaged
luminosity at certain times, the distribution of luminosity among different
observed angles remains unknown and cannot be verified by the $\gamma$-ray
deposition. Therefore, the luminosity of a non-spherical ejecta at a given
observed angle cannot be determined free of the uncertainties associated with
RT.

Figure 1: The $L_{30}/L_{p}$–$L_{p}$ distribution of SNe Ia, along with 1D and
2D explosion models. Observed SNe, depicted as black circles, form a tight
relation, wherein $L_{30}/L_{p}$ increases monotonically with $L_{p}$
throughout the entire luminosity range. 1D models consist of the Chandra
explosion of Dessart et al. (2014, green line) and the sub-Chandra detonations
of Kushnir et al. (2020, red line). The 2D models include direct WD collisions
(Kushnir et al., 2013) and off-centered sub-Chandra ignitions (Schinasi-
Lemberg & Kushnir, 2024). For these models, shown are the results for
different lines of sight, uniformly distributed in $\cos{\theta}$, and their
angle-averaged values are indicated by a larger symbol with the same shape and
a similar color, and, for the off-centered sub-Chandra models, a black-colored
edge. The masses of each collision configuration are denoted in solar masses
alongside their respective angle-averaged values or connected by lines. The
off-centered detonations are marked according to the ignition location
parameter, with red, teal, orange, and purple squares, indicating
$z_{\text{ig}}=0,0.1,0.25,0.5$, respectively. The progenitor masses are
displayed below their angle-averaged values. The $z_{\text{ig}}=0$ results
slightly deviate from their 1D counterparts due to 2D instabilities in the
thermonuclear detonation wave (see Schinasi-Lemberg & Kushnir, 2024, for a
detailed discussion). None of the models can account for the observed
$L_{30}/L_{p}$–$L_{p}$ distribution. A version of this figure in magnitude
space is provided in Appendix C.

In this paper, we seek an observational "width–luminosity" relation, similar
to the $t_{0}$–$M_{\mathrm{Ni56}}$ relation, that would allow to constrain
multi-D models at pre-nebular phases while minimizing the inherent
uncertainties of RT calculations. We derive a relation that accomplishes this
by imposing two constraints on the observational data. The first is that only
the bolometric light curve is considered instead of observing specific bands.
The second constraint is that the analysis is limited to early times, $t<30$
days since the explosion. We discuss the motivation for these constraints in
Section 3.

Previous studies have examined the early-time bolometric properties of SNe Ia
and explosion models (Contardo et al., 2000; Stritzinger et al., 2006;
Phillips et al., 2006; Scalzo et al., 2014, 2019; Wygoda et al., 2019a; Gronow
et al., 2021; Shen et al., 2021b). Notably, the decline in bolometric
magnitude during the first 15 days after the peak, $\Delta
M_{\text{bol}}(15)$, was found to correlate well with the $\gamma$-ray escape
time (Stritzinger et al., 2006) and the peak luminosity, $L_{p}$ (Scalzo et
al., 2019), although this relation was not as tight as in individual bands
(i.e., the Phillips relation). The bolometric magnitude decline rate was also
used in Shen et al. (2021b) and Gronow et al. (2021) to compare simulations of
non-spherical sub-Chandra explosions with SNe Ia observations from Scalzo et
al. (2019). Their models broadly agreed with observations but failed to
reproduce the bright segment of the observed relations, and the variations due
to viewing angle were more pronounced than observed.

In this study, we utilize the early-time bolometric light curve to establish a
new "width–luminosity" relation with several novel features: (1) We select the
luminosity ratio 30 days post-explosion to the peak luminosity,
$L_{30}/L_{p}$, as our shape parameter. This parameter, which is independent
of the supernova’s brightness and distance, offers better model-constraining
power and corresponds to earlier times when RT simulation results are more
reliable, compared to the commonly used $\Delta M_{\text{bol}}(15)$; (2) We
analyze several different SNe Ia explosion models from the literature,
including 1D models of Chandra and sub-Chandra explosions and non-spherical 2D
models of head-on collisions and off-centered sub-Chandra ignitions; (3) We
utilize a high-quality sample of well-observed SNe Ia from Sharon & Kushnir
(2020b) and supplement it with additional objects. By focusing on early-time
observables, we require less stringent observational requirements than those
needed for determining $t_{0}$ and $M_{\mathrm{Ni56}}$, necessitating
continuous observations for at least ${\approx}100$ days post-explosion. As a
result, early-time observables can be measured for a larger fraction of
supernovae, significantly increasing the sample size.

We find a tight relation between the $L_{30}/L_{p}$–$L_{p}$ of our sample,
shown as black symbols in Figure 1. This relationship exhibits a monotonic
increase over the entire luminosity range, resembling the
$t_{0}$–$M_{\mathrm{Ni56}}$ correlation shown in Figure 2. Upon comparing
these findings to models from the literature, we observe that none of the
models reproduce the observed $L_{30}/L_{p}$–$L_{p}$ relation, paralleling the
results found in Sharon & Kushnir (2020a).

The paper is organized as follows: In Section 2, we describe our observational
data, the sample of SNe Ia, and the explosion models we have incorporated.
Section 3 outlines our motivation for using the early bolometric light curve
to constrain models. The resulting $L_{30}/L_{p}$–$L_{p}$ relation is
presented in Section 4 and illustrated in Figure 1. In Section 5, we examine
the opacity of the 1D models to account for the short rise times of the low-
luminosity models. Our findings are discussed and summarized in Section 6.
Appendix A estimates when the ionization levels of SNe Ia ejecta depart from
LTE. Appendix B compares the rise time of our models to a simplified
analytical solution (Kushnir & Katz, 2019) to evaluate the effects of the
ejecta opacity. In Appendix C, we provide some of our results in magnitude
space for completeness. Tables containing the parameters of both the observed
sample and the models are provided in Appendix D.

##  2 Observed data and models

###  2.1 Observations

The observational data comprises the SNe Ia sample from Sharon & Kushnir
(2020b) and Sharon & Kushnir (2023), with some modifications222Several SNe
were excluded upon reevaluation due to incomplete light curve coverage at
peak, and the distance to SN 2011fe was updated using the latest Cepheid-based
distance from Riess et al. (2022)., augmented with SNe lacking late-time
observations required for measuring $t_{0}$ and $M_{\mathrm{Ni56}}$, yet
possessing sufficient early-time data for this analysis. These additional SNe,
sourced from the Carnegie Supernova Project (Contreras et al., 2010;
Stritzinger et al., 2011; Krisciunas et al., 2017; Burns et al., 2018), expand
the sample size to 47 SNe. The photometry, distance, extinction values, and
explosion time are taken from the literature. All SNe in the sample have near-
infrared (NIR) photometry, contributing ${\approx}20-40$% of the total flux at
times considered in this analysis (Scalzo et al., 2014; Sharon & Kushnir,
2023). Bolometric light curves are constructed using the methods described by
Sharon & Kushnir (2020b), and are available in the online supplementary
material.

Following the approach of Sharon & Kushnir (2020a), we construct the
$t_{0}$–$M_{\mathrm{Ni56}}$ distribution of the sample (only for objects with
adequate observations) using the methods of Sharon & Kushnir (2020b). The
results, shown in Figure 2 as black symbols, align closely with the
distribution in Sharon & Kushnir (2020a), as most of the samples overlap. This
comparison is instructive for contrasting the bolometric width–luminosity
relation (Figure 1).

Utilizing the sample of bolometric light curves, the analysis parameters for
each SN are calculated as follows. The peak luminosity and time are determined
by fitting a low-order polynomial to all luminosity measurements
$\\{L_{i}\\}_{i=1}^{N}$ with $L_{i}>0.85\cdot\max(L_{i})$. The uncertainty in
peak luminosity includes statistical errors and several systematic terms, such
as uncertainties in distance, extinction, and missing UV flux (missing flux
from longer wavelengths is negligible due to the required NIR coverage), all
of which are propagated to the peak luminosity error. The value of $L_{30}$ is
obtained by linear interpolation. The error in the $L_{30}/L_{p}$ parameter
(see Section 3) is unaffected by distance uncertainty and only minimally
influenced by uncertainties in extinction and missing UV flux. However, the
error includes a contribution from the unknown explosion time, estimated by
assuming the explosion time uncertainty is 5% of the time to the first
measurement, $\Delta t_{\mathrm{exp}}=0.05(t_{1}-t_{\mathrm{exp}})$, where
$t_{1}$ and $t_{\mathrm{exp}}$ are the times of the first luminosity
measurement and the estimated explosion time, respectively.

The derived parameters for the SNe sample are presented in Table 1.

###  2.2 Models

The models ejecta analyzed in this work include 1D and 2D models from the
literature. The 1D models encompass explosions of Chandra WDs (Dessart et al.,
2014), as well as centrally ignited sub-Chandrasekhar CO WDs with masses
ranging from $0.8$ to $1.1\,M_{\odot}$ (Kushnir et al., 2020). The 2D models
include head-on WD collisions (Kushnir et al., 2013) and off-centered ignited
sub-Chandra CO WDs (Schinasi-Lemberg & Kushnir, 2024). These 2D models are
axisymmetric, with their luminosity depending on the observed inclination
angle $\theta$, measured relative to the symmetry axis ($\theta=0,180\degree$
along the symmetry axis). The off-centered ignited sub-Chandra models have the
same mass range as the 1D models and include a parameter
$z_{\text{ig}}=0,0.1,0.2,0.5$, representing the ignition position along the
symmetry axis normalized by the WD radius ($z_{\text{ig}}=0$ corresponds to
central ignition).

In all models, the explosion was simulated until the ejecta reached a state of
free streaming, resulting in a homologously expanding profile. At this stage,
the models’ $\gamma$-ray escape time, $t_{0}$, can be computed through simple
$\gamma$-ray transfer simulations or direct analytical calculations (Wygoda et
al., 2019a). For consistency with Sharon & Kushnir (2020a), we present the
results using the analytical calculation in this work, noting that the
differences compared to simulations are within a few percent. The $t_{0}$
results and the synthesized 56Ni mass from the explosion are provided in Table
2 and displayed in Figure 2, replicating the findings of Sharon & Kushnir
(2020a); Schinasi-Lemberg & Kushnir (2024). Note that the low-luminosity sub-
Chandra $z_{\text{ig}}=0$ ignitions show slight deviations from their 1D
counterparts due to 2D instabilities in the thermonuclear detonation wave (see
Schinasi-Lemberg & Kushnir, 2024, for a detailed discussion). The conclusion
that none of the models accurately reproduce the observed distribution remains
unchanged.

To determine the models’ bolometric width–luminosity relation, we conducted RT
simulations with the provided ejecta profiles to compute their bolometric
light curves. These simulations were performed using the URILIGHT code (Wygoda
et al., 2019a, b), a Monte Carlo code based on the approximations used in the
SEDONA program (Kasen et al., 2006). The key assumptions include homologous
expansion, an LTE configuration for the ions’ ionization state and energy
level occupation, and expansion opacities with optical depths calculated using
the Sobolev approximation. The thermalization parameter $\epsilon$, which
defines the probability that a photon absorbed in a given transition is re-
emitted in a different transition, was set to 0.8.

Figure 2: The distribution of the 56Ni mass, $M_{\mathrm{Ni56}}$, and the
$\gamma$-ray escape time, $t_{0}$, for the observed sample and explosion
models. The symbols are the same as in Figure 1. The lowest and highest sub-
Chandra progenitor mass is displayed near their respective values. Our results
replicate the findings of Sharon & Kushnir (2020a); Schinasi-Lemberg & Kushnir
(2024). Note that the low-luminosity sub-Chandra $z_{\textrm{ig}}=0$ ignitions
show slight deviations from their 1D counterparts due to 2D instabilities in
the thermonuclear detonation wave (see Schinasi-Lemberg & Kushnir, 2024, for a
detailed discussion). The conclusion that none of the models accurately
reproduce the observed distribution remains unchanged.

##  3 The bolometric light curve shape parameter

This section presents the rationale for selecting the $L_{30}/L_{p}$ shape
parameter as an observable that can robustly constrain models. In Section 3.1,
we identify the observed quantities RT simulations can produce with minimal
uncertainty, limiting our analysis to these results. Next, we identify
observables within these constraints to compare models and observations
effectively (Section 3.2).

###  3.1 Identifying Minimally Uncertain Observed Quantities

The first constraint we impose is limiting the analysis to the bolometric
light curve rather than comparing specific bands. The reason is that the
simulated flux of a specific band is significantly more sensitive to
uncertainties in the opacities of bound-bound transitions. These uncertainties
can arise from incomplete atomic data, the opacity calculation method, and the
material’s ionization state. The re-emission of absorbed photons redistributes
the flux to other wavelengths, typically from the UV to the IR, making the
total flux less susceptible to these uncertainties (Gronow et al., 2021; Shen
et al., 2021a). To illustrate this claim, we use the StaNdaRT public
electronic repository, focusing on the toy06 benchmark model, a $1\,M_{\odot}$
model with characteristic Ia SNe 56Ni mass ($0.6\,M_{\odot}$) and kinetic
energy ($E_{\text{kin}}=10^{51}\,\textrm{erg}$) values, and analytic density
and composition profiles.

Figure 3: Comparison of multi-band light curves generated by various RT codes
with respect to the URILIGHT code, performed on the toy06 model from Blondin
et al. (2022). All data are taken from the StaNdaRT public electronic
repository. Plotted is the code’s deviation from URILIGHT, in flux space, for
the $B$, $V$, $I$, and $H$ bands. All bands except the $V$ band show
deviations of tens of percents. The $V$ band light curves are in better match,
with deviations of up to $25$ percent for all the codes except STELLA. After
30 days, deviations in the $B$ and $H$ bands substantially increase.

In Figure 3, we compare the multi-band light curves (in flux space) of the
toy06 model from various RT codes333The RT codes are: ARTIS (Shingles et al.,
2020), CMFGEN (Hillier & Miller, 1998), CRAB (Utrobin, 2004), KEPLER (Woosley
et al., 2002), SEDONA (Kasen et al., 2006), STELLA (Blinnikov & Bartunov,
1993), SUMO (Jerkstrand, 2011), and SuperNu (Wollaeger et al., 2013). against
the results of the URILIGHT code. In most bands, differences are tens of
percent at most times and can exceed 50 percent. Apart from the STELLA code,
variations are less pronounced in the $V$ band but can still reach
${\approx}25$ percent. Generally, codes that show over-luminous light curves
in one band tend to have under-luminous light curves in another band due to
flux redistribution. Figure 4 illustrates the same comparison for the
bolometric luminosity. As shown in the figure, the deviations in bolometric
luminosity are smaller than in individual bands and, apart from the initial
${\approx}10$ days, are mostly within ${\approx}15$ percent.

Figure 4: Same as Figure 3 but for the bolometric luminosity, and the
comparison is against a larger set of models. Deviations here are smaller than
individual bands an, apart from the initial ${\approx}10$ days, are mostly
within ${\approx}15$ percent.

Another effect observable in Figure 3 is that starting from $\approx 30$ days
post-explosion, the deviations significantly increase in certain bands.
Additionally, the light curves of CMFGEN, the only NLTE code used during non-
nebular times, deviate noticeably from the other codes. We, therefore,
attribute this increased deviation at later times to the impact of NLTE
effects, which become more pronounced as the density decreases. The deviation
from LTE is driven by ionization and excitation caused by the thermalization
of high-energy (up to a few MeV) non-thermal leptons produced in the 56Ni
decay chain, either as decay products or as electrons up-scattered by
$\gamma$-rays from the decay (Axelrod, 1980; Kozma & Fransson, 1992; Baron et
al., 1996). This deviation significantly affects the material’s opacity and
the SNe light curves. Non-thermal ionization is known to be the dominant
ionization process in the nebular phase and, as we demonstrate below, is also
significant at earlier times.

To evaluate the significance of NLTE ionization, we again utilize the StaNdaRT
repository to compare the evolution of ionization structures between URILIGHT
(LTE) and CMFGEN (NLTE). Figure 5 shows the results for the toy06 model,
focusing on the fractional ionization levels of Co and Fe averaged across the
entire ejecta. Up to ${\approx}30$ days post-explosion, the ionization levels
from both codes are in good agreement. However, after this period, the
ionization levels begin to diverge, with the CMFGEN ejecta becoming
significantly more ionized. This divergence marks the onset of significant
ionization from non-thermal leptons. Note that the abundance of Co V is lower
compared to Fe V in the CMFGEN simulation. This discrepancy arises because
non-thermal ionization is not included for Co IV due to a lack of atomic data
444John Hillier, private communication.. This limitation further demonstrates
the uncertainties inherent in these calculations.

In Appendix A, we use simple analytical considerations to determine when non-
thermal ionization becomes significant. Our calculations show that this epoch
is around 30 days post-explosion, consistent with previous findings.
Consequently, we limit our analysis to the early bolometric light curve,
$t\leq 30$ days from the explosion, corresponding to times when the LTE
assumption holds. Our approach aligns with the findings of Shen et al.
(2021a), who compared light curves of sub-Chandrasekhar explosions using both
LTE and NLTE RT codes. They observed that 15 days past peak light, significant
discrepancies arise between the two methods in the observed magnitudes across
most bands, while differences at peak light are much smaller. Moreover, the
bolometric luminosities were roughly similar during the considered times, up
to approximately 30 days since peak light. They also examined the effect of
the absorption parameter, $\epsilon$, in LTE simulations and found that
altering its value had minimal impact on the bolometric light curve, though
individual magnitudes varied starting from 15 days after peak light.

Figure 5: Temporal evolution of the mean Co (top) and Fe (bottom) ionization
levels in the toy06 model, depicted for URILIGHT (solid lines) and CMGFEN
(dashed lines). The ionization levels show good agreement up to ${\approx}30$
days from the explosion, after which they diverge. This is attributed to the
epoch when ionization from non-thermal leptons becomes significant.

###  3.2 Selecting Observables for Model Comparison

We aim to construct a width–luminosity relation under the constraints from
above, similar to the Phillips relation and the
$t_{0}\text{--}M_{\mathrm{Ni56}}$ distribution. The luminosity parameter we
choose is, naturally, the peak bolometric luminosity. In Figure 6 we compare
$L_{p}$ with the late-time luminosity parameter, $M_{\mathrm{Ni56}}$.
Consistent with previous works (e.g., Scalzo et al., 2019), these parameters
for the observed SNe Ia, shown as black symbols, display an almost perfect
correlation, with a Pearson correlation coefficient of $\rho=0.984$.

The selection of the shape parameter is less trivial and we choose the
luminosity ratio 30 days after the explosion to the peak luminosity,
$L_{30}/L_{p}$. In Figure 7, we compare $L_{30}/L_{p}$ with the late-time
shape parameter, $t_{0}$. Here, the correlation of the observed sample is also
prominent but to a lesser extent, with a Pearson correlation coefficient of
$\rho=0.872$. We also fit a linear regression model between the two
parameters, represented by $L_{30}/L_{p}=a\cdot t_{0}+b$. The results, shown
by a dashed red line, have a slope of $a\approx 0.022\,\text{day}^{-1}$. This
approach aims to compare the ability to constrain models with the
$L_{p+15}/L_{p}$ parameter (see below). Given the strong correlations evident
in Figures 6 and 7, we expect the positive correlation observed in the
$t_{0}$–$M_{\mathrm{Ni56}}$ relation to hold for the bolometric
width–luminosity relation as well.

Figures 6 and 7 also present the same comparisons for the explosion models.
For non-spherical models, the $L_{p}$ and $L_{30}/L_{p}$ results are shown for
different values of the inclination angle $\theta$, evenly distributed in
$\cos{\theta}$, along with their angle-averaged values. A strong correlation
is evident within the models’ $M_{\mathrm{Ni56}}$–$L_{p}$ distribution, and
they align with the observed relation (Figure 6). However, the behavior of the
shape parameters in Figure 7 is more complex. While the models generally
follow the observed relation, the collision and Chandra models represent the
slowly evolving SNe Ia, whereas the sub-Chandra models only cover the quickly
evolving region (this is also evident in Figure 2). Notably, the
$t_{0}$–$L_{30}/L_{p}$ relation of the sub-Chandra models is non-monotonic.
This occurs because low-mass WD progenitors exhibit short rise times (from
explosion to peak light) despite having longer $\gamma$-ray escape times than
higher-mass progenitors. We discuss this behavior further in Section 5. a
$1\,M_{\odot}$ model with characteristic Ia SNe 56Ni mass ($0.6\,M_{\odot}$)
and kinetic energy ($E_{\text{kin}}=10^{51}\,\textrm{erg}$) Finally, we
compare the $L_{p}$–$L_{30}/L_{p}$ relation of the models and RT codes in the
StaNdaRT repository. Figure 8 displays the observed SNe Ia relation along with
the RT code results for the four benchmark ejecta models in the StaNdaRT
repository. These models include, in addition to the toy06 model described
above, the toy01 model, which is identical to the toy06 model except for a
lower 56Ni yield of $0.1,M_{\odot}$. They also include two models from the
Chandra WD explosions of Dessart et al. (2014), the ddc10 and ddc25 models,
with 56Ni yields of $0.12$ and $0.52,M_{\odot}$, respectively (note that not
all of the RT results are available for all models). We calculate the scatter
in the $L_{30}/L_{p}$ parameter for each explosion model among the RT codes,
excluding the ARTIS results due to their significant deviation from the
others. The scatter for the ’normal’ luminosity models, toy06 and ddc10, is
${\approx}9$ and ${\approx}6$ percent, respectively. For the low-luminosity
models, toy01 and ddc25, the scatter is ${\approx}15$ and ${\approx}6$
percent, respectively. The small scatter in $L_{30}/L_{p}$ between models
supports our choice for the shape parameter.

We also explore replacing $L_{30}$ with the luminosity 15 days post-peak,
$L_{p+15}$, similar to the bolometric magnitude decline rate, $\Delta
M_{\text{bol}}(15)=-2.5\log(L_{p+15}/L_{p})$, used in previous studies
(Stritzinger et al., 2006; Scalzo et al., 2019; Shen et al., 2021b). In
Stritzinger et al. (2006), for a sample of 16 SNe Ia, this parameter was found
to correlate well with $t_{0}$. In Scalzo et al. (2019), it was shown to
correlate with $M_{\mathrm{Ni56}}$ and $L_{p}$. Figure 9 presents the
$L_{p+15}/L_{p}$–$t_{0}$ distribution of our sample. We use $L_{p+15}/L_{p}$
instead of the more commonly used form of $\Delta M_{\text{bol}}(15)$ to
facilitate better comparison with the $L_{30}/L_{p}$ results. The $\Delta
M_{\text{bol}}(15)$ –$t_{0}$ distribution is shown in Appendix C. Similar to
Figure 7, the observed sample (black symbols) exhibits a strong correlation
between these parameters, and the models show similar qualitative behavior in
both relations. As with the $L_{30}/L_{p}$–$t_{0}$ relation, we fit a linear
regression model between these parameters and find a slope of
${\approx}0.011\,\text{day}^{-1}$, about half of the value when using
$L_{30}/L_{p}$. This result demonstrates the advantage of the $L_{30}/L_{p}$
parameter in distinguishing SNe, as it changes more rapidly with respect to
$t_{0}$ and spans a broader range of values than the $L_{p+15}/L_{p}$
parameter.

Explosion models are also better separated in Figure 7 compared to Figure 9.
In the latter, high-luminosity sub-Chandra and collision models have similar
$L_{p+15}/L_{p}$ values despite the notable differences in their $t_{0}$
values. Another drawback of using $L_{p+15}$ is that the peak time of SNe Ia
can slightly exceed 15 days, causing $L_{p+15}$ to correspond to times later
than 30 days post-explosion when NLTE effects become significant (Figure 5).
Nonetheless, calculating $L_{p+15}$ does not require explosion time, reducing
the associated uncertainty compared to $L_{30}/L_{p}$. For completeness and to
facilitate comparison with previous works, we also include $L_{p+15}/L_{p}$ in
our analysis.

Figure 6: Comparison between the 56Ni mass, $M_{\mathrm{Ni56}}$, to the peak
luminosity, $L_{p}$, for the observed sample and explosion models. The symbols
are the same as in Figure 1. The two parameters have an almost perfect
correlation.  Figure 7: Same as Figure 6 but for the $\gamma$-ray escape time,
$t_{0}$, and $L_{30}/L_{p}$. For the sub-Chandra models, the lowest and
highest progenitor mass is displayed near their respective values. The
parameters exhibit strong correlations in both the observations and the
models, except for the sub-Chandra detonations. These models show a decrease
of $L_{30}/L_{p}$ for the high $t_{0}$ values, corresponding to low-mass WD
progenitors. This drop is related to the short rise times of these models; see
the main text for a detailed discussion. We also fit a linear regression model
to the observed sample (dashed red line), and we find a slope of $\approx
0.022\,\text{day}^{-1}$. Figure 8: The $L_{p}$–$L_{30}/L_{p}$ relation of the
observed sample (black symbols) and the models from the StaNdaRT repository.
The StaNdaRT results are represented by the colored symbols, with the colors
and shapes corresponding to different explosion models and RT codes,
respectively. The scatter in the $L_{30}/L_{p}$ parameter between the RT codes
for each explosion model (excluding ARTIS; see text for details) is provided
in the legend.  Figure 9: Same as Figure 7 but for the
$L_{p+15}/L_{p}$–$t_{0}$ relation. A version of this figure in magnitude space
is provided in Appendix C.

##  4 The $L_{30}/L_{p}$–$L_{p}$ distribution

This section compares the $L_{30}/L_{p}$–$L_{p}$ distribution of observed SNe
Ia to various explosion models. The results are given in Appendix D and shown
in Figure 1. A version of Figure 1 in magnitude space is provided in Appendix
C.

Similar to the Phillips relation (Phillips, 1993; Phillips et al., 1999) and
to the $t_{0}$–$M_{\mathrm{Ni56}}$ distribution (Figure 2), the observed
bolometric width-luminosity relation shows a strong correlation between the
shape and luminosity of SNe Ia, where bright SNe are slower and vice versa.
Our results are broadly consistent with Figure 10 of Scalzo et al. (2019),
where this trend is also noticeable. However, our sample covers a wider range
of luminosities, and our relation using the $L_{30}/L_{p}$ parameter exhibits
greater precision, within ${\approx}10$ percent in either the luminosity or
$L_{30}/L_{p}$, compared to their relation using $\Delta M_{\text{bol}}(15)$,
which displays a significantly more scatter, extending to tens of percents.
The greater precision of our sample is partly due to the choice of the shape
parameter and partly due to the data quality and analysis, as our relation
using $\Delta M_{\text{bol}}(15)$ (see Section 4.1) is tighter than the
relation in Scalzo et al. (2019).

The results of the models resemble those of the $t_{0}$–$M_{\mathrm{Ni56}}$
distribution (Figure 2), where none of the models reproduce the positive
correlation in the $L_{30}/L_{p}$–$L_{p}$ relation across the entire
luminosity range.

The Chandra models, represented by the green line, overlap with the
observations only in a limited region. Unlike the observed relation, these
models exhibit a distinct negative correlation across most luminosity ranges.
The faint models are much slower than observed, while the bright models are
faster.

The 1D sub-Chandra models, depicted by the red line, align well with low-
luminosity SNe. However, the $L_{30}/L_{p}$ values for the more luminous SNe
show significant tension with the observations. Notably, the more luminous SNe
Ia, corresponding to 56Ni masses of $M_{\mathrm{Ni56}}\gtrsim 0.5\,M_{\odot}$
and $L_{p}\gtrsim 1.2\times 10^{43}\,\text{erg}\,\text{s}^{-1}$, encompass the
vast majority of events (Sharon & Kushnir, 2022). The results for the off-
centered ignitions of sub-Chandra WDs are shown for WD masses of
$0.8,0.85,0.9,0.95,1,1.1\,M_{\odot}$ and ignition locations of
$z_{\text{ig}}=0,0.1,0.25,0.5$. These results are presented for different
values of the inclination angle $\theta$, distributed evenly in
$\cos{\theta}$, along with the angle-averaged values. The $z_{\text{ig}}=0$
configurations slightly deviate from their 1D counterparts due to 2D
instabilities in the thermonuclear detonation wave (see Schinasi-Lemberg &
Kushnir, 2024, for a detailed discussion).

The models with $z_{\text{ig}}>0$ show significant variations due to the
inclination angle, with a spread of up to ${\approx}25$ percent in
$L_{30}/L_{p}$ and ${\approx}50$ percent in $L_{p}$ for the most asymmetric
configurations. These variations are more pronounced for high masses,
displaying a clear anti-correlation between the parameters. For $M\lesssim
0.85,M_{\odot}$, the observables are only marginally influenced by the
inclination angle. Our results generally align with the non-spherical sub-
Chandra explosions presented by Shen et al. (2021b). However, their findings
show larger variations in the shape parameter, particularly for low-mass
configurations. Compared to the 1D sub-Chandra models, the off-centered
detonation results cover a larger fraction of the $L_{30}/L_{p}$–$L_{p}$
distribution and account for some of the observed spread. Nonetheless, they
fail to reproduce the $L_{30}/L_{p}$–$L_{p}$ positive correlation for the
luminous SNe, where the predicted $L_{30}/L_{p}-L_{p}$ is smaller by
${\approx}25$ percent compared to observations.

The head-on WD collisions, observed at various inclination angles evenly
distributed in $\cos{\theta}$, are represented by blue lines and circles, with
dark blue circles indicating the angle-averaged values. The masses of each
configuration are denoted in solar masses alongside their respective angle-
averaged values or connected by lines. Configurations with equal masses
exhibit reflection symmetry with respect to the plane perpendicular to the
symmetry axis. Consequently, observables at angles $\theta$ and $-\theta$ are
expected to align. Figure 1 confirms this alignment, albeit with slight
variations due to Monte Carlo noise. These models can reproduce only the
bright segment of the observed distribution. While models with intermediate
masses achieve luminosities and shape parameters roughly consistent with
observations, their luminosity varies significantly across different $\theta$
values, often exceeding observed luminosities. The shape parameter of these
models is less sensitive to variations in the observed angle. The low-
luminosity models exhibit high values of $L_{30}/L_{p}$ compared to
observations at similar luminosities and fail to match the observed relation
across all $\theta$ values. In these models, the observed inclination angle
primarily affects $L_{30}/L_{p}$, altering it by up to ${\approx}20$ percent,
which is insufficient to agree with the observed values. The relationship
between the observed angle variations and the ejecta properties is a topic for
future study.

In summary, none of the models can account for the observed
$L_{30}/L_{p}$–$L_{p}$ distribution.

###  4.1 The $L_{p+15}/L_{p}$–$L_{p}$ relation

In Figure 10, we present the observed distribution of $L_{p+15}/L_{p}$ and
$L_{p}$. The general trend that lower luminosity SNe Ia evolve more rapidly is
evident in this relationship. However, as noted in Section 3.2, the
$L_{p+15}/L_{p}$ parameter covers a smaller range of values, and the relation
is not strictly monotonic, with some intermediate-luminosity objects
displaying very high shape parameters.

The models’ behavior mirrors that of the $L_{30}/L_{p}$–$L_{p}$ relation, but
the separation between the models is less pronounced. This is most apparent in
the overlap between the high-luminosity sub-Chandrasekhar models and the
collision models, which is also evident in the $L_{p+15}/L_{p}$–$t_{0}$
relation (Figure 9). Here, sub-Chandrasekhar models exhibit $L_{p+15}/L_{p}$
values that are higher than observed for SNe with low to intermediate
luminosities, whereas the $L_{30}/L_{p}$–$L_{p}$ relation better represented
these SNe with the sub-Chandrasekhar models. Conversely, sub-Chandrasekhar
models align more closely with the high luminosity regime, where most SNe Ia
are situated. Results for viewing angles opposite the explosion point of more
massive progenitors align with the most luminous objects. However, for viewing
angles near the explosion point, the models predict too high luminosities that
evolve more rapidly than observed data. A version of this figure in magnitude
space is provided in Appendix C.

For the Chandra models, our findings are broadly consistent with those of
Kasen & Woosley (2007), who derived light curves for Chandrasekhar-mass WD
explosions and observed nearly constant $\Delta M_{\text{bol}}(15)$ values
across a $M_{\mathrm{Ni56}}$ range of $0.35$–$0.7\,M_{\odot}$. In this
luminosity range, our $\Delta M_{\text{bol}}(15)$ results exhibit similar
values but show variations of ${\approx}15$ percent.

Figure 10: Same as Figure 1, but for the $L_{p+15}/L_{p}$–$L_{p}$
distribution. Compared to Figure 1, the $L_{p+15}/L_{p}$ shape parameter
covers a smaller range of values and is not strictly monotonic with the
luminosity. Additionally, the models are not as well separated, which is most
evident in the overlap between the sub-Chandrasekhar and collision models. A
version of this figure in magnitude space is provided in Appendix C.

##  5 Opacity of low-luminosity models

We demonstrated in Section 3 that the sub-Chandra models exhibit a peculiar
behavior in the low-luminosity region regarding the two shape parameters
considered in this work, $t_{0}$ and $L_{30}/L_{p}$. While $t_{0}$ increases
monotonically as the luminosity decreases, $L_{30}/L_{p}$ does not follow a
monotonic trend across all luminosities and reverses for the two faintest
models. This anti-correlation is evident in Figure 7, where $L_{30}/L_{p}$ of
the sub-Chandra models drops for high values of $t_{0}$, corresponding to the
fainter models of the $0.8$ and $0.85\,M_{\odot}$ WD progenitors. The drop in
$L_{30}/L_{p}$ is attributed to the short rise time evident in these models.
Since both the rise time and the $\gamma$-ray escape time are expected to
increase with the ejecta’s mass and decrease with the kinetic energy of the
ejecta and the extent of the 56Ni distribution within it, this suggests that
the anti-correlation is likely due to opacity differences.

As discussed in the preceding sections, the opacity of optical light, which
depends on the temperature, density, and composition of the matter,
significantly impacts the evolution of the SNe light curve. In contrast, the
opacity for $\gamma$-ray transfer is almost constant for the ejecta considered
here. Therefore, we evaluate and compare the characteristic optical opacity of
the models. This is not straightforward, as optical opacity is wavelength-
dependent, varies throughout the ejecta, and evolves over time. To address the
wavelength dependency, we use the Rosseland mean opacity since the photons
travel through diffusion during the early stages of the SN. For each epoch, we
calculate the mass-averaged Rosseland mean opacity:

$\left<\kappa_{\textrm{R}}(t)\right>=\frac{1}{m_{*}}\int_{0}^{m_{*}(t)}\kappa_{\text{R}}dm,$
(3)

where $m_{*}(t)$ is the enclosed mass within which the diffusion approximation
is valid, determined by an optical depth from the ejecta outer edge of
$\tau>2$.

The temporal evolution of the mass-averaged Rosseland mean opacity for several
selected models is shown in Figure 11, with the epoch of peak light for each
model marked by a black point. The sub-Chandra models, represented by solid
lines, exhibit a significant decrease in average opacity for the low-mass
progenitors compared to other models. This decrease is particularly pronounced
in the $0.8\,M_{\odot}$ model, which has substantially lower opacities –
nearly half as much around peak light – compared to the other models. The
opacities of the faintest and brightest Chandra models, indicated by the
dashed lines, also show significant variation.

The opacity variations observed for both the sub-Chandra and Chandra models
are consistent with their behavior in Figure 7. These variations explain the
short rise times of the low-luminosity sub-Chandra models despite their
relatively high $t_{0}$ values. In Appendix B, we provide a quantitative
analysis of these results and directly relate the models’ rise time to their
average Rosseland mean opacity.

Figure 11: Mass-averaged Rosseland mean opacity as a function of time for
several selected models. The models include the 1D sub-Chandra configurations
(solid lines) and the two Chandra models with the faintest and brightest
luminosities (dashed lines). The black point on each curve indicates the epoch
of peak light. Averaging was performed over all masses with an optical depth
from the outer edge of the ejecta satisfying $\tau>2$. The opacity of the
$0.8\,M_{\odot}$ WD sub-Chandra model is significantly lower than the other
models, consistent with its short rise time.

##  6 Summary and Conclusions

In this study, we aimed to identify an observational width–luminosity
relation, similar to the $t_{0}$–$M_{\mathrm{Ni56}}$ relation, that would
allow to constrain multi-D models at pre-nebular phases while minimizing the
inherent uncertainties of RT calculations. We argued that the bolometric light
curve up to 30 days from the explosion can be computed without accounting for
NLTE effects, which are inherently uncertain and difficult to compute (Section
3). Subsequently, we introduced a width-luminosity relation for SNe Ia, where
the peak luminosity $L_{p}$ is compared with the ratio of the luminosity at 30
days from the explosion to the peak luminosity, $L_{30}/L_{p}$. The
distribution of these parameters for a sample of 47 SNe Ia is presented in
Figure 1, along with the results from 1D and 2D explosion models (see Section
4).

In previous work, we compared the observed $t_{0}$–$M_{\mathrm{Ni56}}$
relation to known SNe Ia models from the literature (Sharon & Kushnir, 2020a,
also see Figure 2 for an updated relation). Although that analysis bypassed RT
calculations, it could not account for multi-D effects and required stringent
observational conditions, including late-time observations up to ${\sim}100$
days. While the current analysis relies on RT calculations, it does not face
these limitations. The conclusion of this work aligns with that of Sharon &
Kushnir (2020a): None of the known SNe Ia models can reproduce the observed
width-luminosity distribution of SNe Ia, even when considering multi-D
effects.

Several avenues could potentially align the models more closely with the
observations. One possibility involves adjusting the initial composition of
the sub-Chandra WDs, which evolve too rapidly compared to the bulk of the
observations, from CO towards heavier elements. This adjustment would reduce
the available thermonuclear energy of the WD, resulting in lower velocities of
the ejected material and a slower evolution of the light curve. However, a
heavier composition is expected only for very massive WDs (Lauffer et al.,
2018, $M\gtrsim 1.1,M_{\odot}$). We are currently investigating this
possibility, and the findings will be published in a subsequent work.

Another avenue is to perform more accurate calculations of the collision
model. The calculations by Kushnir et al. (2013) were carried out with
relatively low resolution and a simplified 13-isotope reaction network.
Preliminary results from more accurate calculations do not show a substantial
change in the $L_{30}/L_{p}\text{--}L_{p}$ distribution and will be reported
in a future paper. Additionally, 3D calculations of collisions with non-zero
impact parameters could yield different results; however, such high-resolution
calculations are currently beyond our computational capabilities.

Throughout the paper, we have mentioned non-thermal processes as significant
sources of uncertainty in RT modeling. These processes substantially impact
the SN’s evolution, yet their full extent and impact remain poorly understood.
Traditionally, non-thermal effects are primarily considered during the nebular
phase; however, Dessart et al. (2012) demonstrated that non-thermal excitation
alters the spectra of Type Ib SNe, even at earlier phases, during the
photospheric phase. Therefore, it is crucial to understand their impact on
pre-nebular times and the SN light curves. Furthermore, parameters associated
with non-thermal processes contain considerable uncertainties. For example,
electron-impact ionization cross-sections are often unknown, and the absence
of such cross-sections for Co III ionization in CMFGEN causes substantial
discrepancies in Co and Fe ionization levels, as illustrated in Figure 5.
Future work should study in more detail the effect non-thermal processes have
on the SNe observables and their sensitivity to the non-thermal parameters.

## Acknowledgements

We thank Boaz Katz for offering this work and for the useful discussions. We
thank Stéphane Blondin and John Hillier for their assistance regarding CMFGEN.
DK is supported by a research grant from The Abramson Family Center for Young
Scientists, and by the Minerva Stiftung.

## Data Availability

The data underlying this article are available in the article and in its
online supplementary material.

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##  Appendix A Emergence of Non-thermal ionization

In this appendix, we use simple analytic arguments to estimate the epoch at
which the ionization rate from non-thermal leptons becomes significant,
leading to a departure from LTE for the ionization levels. The ionization
fraction $x_{i}$ for a given element at ionization level $i$ is determined
from the equation (Jerkstrand, 2011)

$\frac{dx_{i}}{dt}=\Gamma_{\text{ion},i-1}x_{i-1}+\Gamma_{\text{rec},i+1}x_{i+1}-(\Gamma_{\text{ion},i}x_{i}+\Gamma_{\text{rec},i}x_{i}),$
(4)

where $\Gamma_{\text{ion},i}$ and $\Gamma_{\text{rec},i}$ are the total
ionization and recombination rates per particle from level $i$, respectively.
Since the ionization and the recombination time scales are much shorter than
the dynamical time, we look for the steady-state solution:

$\Gamma_{\text{ion},i}x_{i}=\Gamma_{\text{rec},i+1}x_{i+1}.$ (5)

At early times, when non-thermal effects are still negligible and the
ionization levels are at their LTE values, the ionization rate is primarily
due to photoionization. As the ejecta expands and cools, non-thermal
ionization caused by the impact of fast leptons will start to dominate once
its rate becomes comparable to the recombination rate.

To estimate the time when non-thermal ionization becomes significant, we
examine the state of the iron-group elements in the ejecta’s core, composed of
Ni, Co, and Fe. For simplicity, we consider only Fe ionization levels in what
follows. We assume that the iron in the ejecta is composed only of Fe III and
Fe IV, with relative abundances $x_{{}_{\textrm{III}}}$ and
$x_{{}_{\textrm{IV}}}=1-x_{{}_{\textrm{III}}}$, respectively. These are the
two dominant ions after 20 days from the explosion, as seen in Figure 5. The
epoch $t_{\text{eq}}$ when the non-thermal ionization rate becomes significant
is estimated by

$\Gamma^{\text{nt}}_{\text{ion}}(t_{\text{eq}})x_{{}_{\textrm{III}}}(t_{\text{eq}})=\Gamma_{\text{rec}}(t_{\text{eq}})x_{{}_{\textrm{IV}}}(t_{\text{eq}}),$
(6)

where $\Gamma^{\text{nt}}_{\text{ion}}$ is the non-thermal ionization rate per
particle for Fe III, $\Gamma_{\text{rec}}$ is the recombination rate per
particle for Fe IV, and the ionization levels correspond to their LTE
evolution within the ejecta. We now estimate the recombination and ionization
rate per particle of these ions.

Assuming that recombination is primarily from free electrons within the
considered temperature range (Jerkstrand, 2011), the recombination rate is
given by

$\Gamma_{\text{rec}}(t)=\alpha n_{e}(t),$ (7)

where $\alpha$ is the temperature-dependent recombination coefficient and
$n_{e}$ is the electron density. At a temperature of
$T{\sim}10^{4}\,\text{K}$, typical for the ejecta environment ${\approx}30$
days after explosion, $\alpha\sim 1\text{--}5\times
10^{-12}\,\text{cm}^{3}\,\text{s}^{-1}$ for doubly- or triply-ionized iron-
group ions (Shull & van Steenberg, 1982; Nahar, 1996; Jerkstrand, 2011). The
electron density is given by

$n_{e}=\chi_{e}n_{\text{ion}}=\chi_{e}\frac{1}{A}\frac{M_{\text{core}}}{m_{p}}\frac{3}{4\pi(v_{\text{ej}}\cdot
t)^{3}},$ (8)

where $\chi_{e}$ is the mean number of free electrons per atom,
$n_{\text{ion}}$ is the ion density, $A=56$ is the atomic mass number, $m_{p}$
is the proton mass, $M_{\text{core}}$ is the total mass of the core, and
$v_{\text{ej}}$ is the characteristic velocity of the ejecta’s core. Assuming
the degree of ionization is similar for different elements in the ejecta, then
$\chi_{e}=2x_{{}_{\textrm{III}}}+3x_{{}_{\textrm{IV}}}$. Unless large amounts
of non-radioactive isotopes (e.g., 54Fe) are produced in the explosion,
$M_{\text{core}}\approx M_{\mathrm{Ni56}}$, so we use $M_{\mathrm{Ni56}}$ to
denote the core mass in what follows.

To estimate the non-thermal ionization rate, we assume that a fraction
$\eta_{\text{ion}}$ of the energy of the fast leptons is converted into
ionization of the ions, while the remaining energy goes into heating of the
free electrons and excitation of bound electrons. The value of
$\eta_{\text{ion}}$ depends primarily on the fraction of free electrons and
the composition and only weakly on the number densities. For SNe Ia ejecta,
$\eta_{\text{ion}}\approx 0.02-0.2$ (Kozma & Fransson, 1992; Dessart et al.,
2012; Li et al., 2012; Shingles et al., 2022). Assuming, for simplicity, that
the energy for non-thermal ionization is distributed among the elements and
ions in proportion to their relative abundance due to similar cross sections
(Arnaud & Rothenflug, 1985), the non-thermal ionization rate per particle will
be

$\Gamma^{\text{nt}}_{\text{ion}}(t)=\eta_{\text{ion}}\frac{Q_{\text{dep}}(t)}{I_{{}_{\textrm{III}}}\cdot
N_{\text{ion}}}=\eta_{\text{ion}}\frac{Q_{\text{dep}}(t)}{M_{\mathrm{Ni56}}}\frac{m_{p}A}{I_{{}_{\textrm{III}}}},$
(9)

where $Q_{\text{dep}}(t)$ is the total energy deposition of the radioactive
decay, $I_{{}_{\textrm{III}}}=30.65\,\text{eV}$ is the Fe III ionization
energy, $N_{\text{ion}}$ is the total number of ions, and it is assumed that
all radioactive energy is deposited in the core. The term
$\widetilde{Q}_{\text{dep}}(t)\equiv Q_{\text{dep}}(t)/M_{\mathrm{Ni56}}$ is
independent of $M_{\mathrm{Ni56}}$, varying only due to differences in the
$\gamma$-ray deposition fraction. For the observed range of $\gamma$-ray
escape times of SNe Ia, $t_{0}\approx 30-45$ day (Sharon & Kushnir, 2020b),
the range of $\widetilde{Q}_{\text{dep}}(30\,\textrm{day})$ varies by at most
$\approx 30$ percent. Plugging in the expressions for the rates in Equation
(6), and using $x_{{}_{\textrm{III}}}=1-x_{{}_{\textrm{IV}}}$, we get

$t_{\text{eq}}^{3}\widetilde{Q}_{\text{dep}}(t_{\text{eq}})=\frac{3}{4\pi
v_{\text{ej}}^{3}}\frac{\alpha
M_{\mathrm{Ni56}}I_{{}_{\textrm{III}}}}{\eta_{\text{ion}}}\left(\frac{1}{A\,m_{p}}\right)^{2}\left(2+x_{{}_{\textrm{IV}}}\right)\frac{x_{{}_{\textrm{IV}}}}{1-x_{{}_{\textrm{IV}}}}.$
(10)

We next demonstrate that Equation (10) is consistent with
$t_{\text{eq}}\approx 30\,\rm{days}$ observed in Section 3 for the toy06 model
from Blondin et al. (2022). For the toy06 model, we find
$\widetilde{Q}_{\text{dep}}(t)\approx\widetilde{Q}_{\text{dep}}(t=30\,\text{day})\approx
1.03\times 10^{43}\,\frac{\text{erg}\,\text{s}^{-1}}{M_{\odot}}$, with
${\approx}90$ percent of this energy deposited in the core. Using
$\alpha=5.1\times 10^{-12}\,\text{cm}^{3}\,\text{s}^{-1}$ for Fe IV
recombination (Nahar, 1996), we get

$\displaystyle t_{\text{eq}}\approx$
$\displaystyle\left(\frac{0.05}{\eta_{\text{ion}}}\right)^{1/3}\left(\frac{M_{\mathrm{Ni56}}}{0.6\,M_{\odot}}\right)^{1/3}\left(\frac{0.55\times
10^{9}\,\text{cm}\,\text{s}^{-1}}{v_{\text{ej}}}\right)$ (11)
$\displaystyle\times\frac{1}{0.38}\left[\left(1+\frac{x_{{}_{\textrm{IV}}}}{2}\right)\frac{x_{{}_{\textrm{IV}}}}{1-x_{{}_{\textrm{IV}}}}\right]^{1/3}\times
31.5\,\text{day},$

where we use the core mass ($M_{\mathrm{Ni56}}=0.6\,M_{\odot}$) and ejecta
velocity values ($v=0.55\times 10^{9}\,\text{cm}\,\text{s}^{-1}$) of the toy06
model, and normalize for $x_{{}_{\textrm{IV}}}=0.05$ (the term in the square
brackets and the value preceding it cancel out for this value), which marks
the onset where Fe ionization in the toy06 model diverges from LTE (Figure 5).
The good agreement with Figure 5 supports our simplified description of the
deviation from LTE. Note the weak dependence of $t_{\textrm{eq}}$ to the model
parameters, except for $v_{\textrm{ej}}$ with linear dependence.

For a more accurate treatment, we present in Figure 12 solutions of Equation
(10) for several $\eta_{\text{ion}}$ values, shown as red lines. The 56Ni
mass, velocity, and deposition fraction are that of the toy06 model. The
temperature is assumed to remain constant at $10^{4}\,$K, thereby maintaining
the recombination coefficient constant. The solid and dashed black curves
trace the evolution of $x_{{}_{\textrm{IV}}}$ in the URILIGHT and CMFGEN
codes, respectively, for the toy06 model. The intersection between the
solution of Equation (10) and the LTE evolution of $x_{{}_{\textrm{IV}}}$
marks the epoch where non-thermal ionization equals the recombination rate,
becoming non-negligible. As illustrated in Figure 12, $x_{{}_{\textrm{IV}}}$
drops rapidly within a narrow time range, occurring between 25 to 35 days from
the explosion. Consequently, the value of $t_{\text{eq}}$ at the intersection
has a weak dependence on the parameters of Equation (10). Figure 12 also shows
that these intersection times are consistent with the departure of
$x_{{}_{\textrm{IV}}}$ from LTE value, as evidenced by its evolution simulated
with CMFGEN.

Figure 12: The evolution of the Fe IV fraction, $x_{\textrm{IV}}$, over time
where non-thermal electron ionization equals the recombination rate (red
lines), calculated usingEquation (10) with parameters specific to the toy06
model. Various line styles represent different values of $\eta_{\text{ion}}$,
governing the ionization fraction of non-thermal electron energy loss. The Fe
IV evolution for the toy06 model simulated with URILIGHT and CMFGEN is
indicated by solid and dashed black lines, respectively. The intersection
between the solution of Equation (6) and the simulated evolution marks the
epoch when the non-thermal ionization rate begins to dominate over the thermal
ionization rate.

For completeness, we perform the same analysis for the low-luminosity model
toy01. We replace the model parameters such that the 56Ni mass and velocity
are $M_{\mathrm{Ni56}}=0.1\,M_{\odot}$, $v=0.24\times
10^{9}\,\text{cm}\,\text{s}^{-1}$. In addition, due to the model’s low core
mass, the energy deposition $\widetilde{Q}_{\text{dep}}$ takes into account
that a lower fraction of the energy deposition goes into the core, around
${\approx}65$ percent, than in the toy06 model. The results for the toy01
model are shown in Figure 13, and are also consistent with the departure of
$x_{{}_{\textrm{IV}}}$ from its LTE value slightly before $30\,\rm{days}$.

Figure 13: Same as Figure 12 but for the toy01 model. The results for this
model are also consistent with the departure of $x_{{}_{\textrm{IV}}}$ from
its LTE value occurring slightly before $30\,\rm{days}$.

##  Appendix B Comparison of the rise time to analytic results

In this appendix, we relate the rise time of our 1D models to the opacity of
the ejecta. For this purpose, we use the analytical results of (Kushnir &
Katz, 2019, hereafter analytical results), which provide a solution for
diffusion in an isotropic, homologously expanding ejecta, under radiation-
dominated pressure, uniform density and opacity, $\kappa$, and local energy
deposition from radioactive decay. Following the methods in Kushnir & Katz
(2019), the luminosity emitted from the ejecta can be calculated for a given
ejecta mass $M$, outer velocity $v_{o}$, opacity $\kappa$, and the extent of
the energy generation region $x_{s}$, such that

$\epsilon(v,t)=\frac{Q(t)}{\frac{4\pi}{3}(x_{s}v_{o}t)^{3}}\times\begin{cases}1&v<x_{s}v_{o}\\\
0&\text{elsewhere}\end{cases},$ (12)

where $\epsilon$ is the energy generation rate per unit volume and $Q(t)$ is
the total energy generation rate at time $t$. The resulting luminosity for an
impulse of energy $E_{\delta}$ occurring at time $t_{\delta}$ is given by

$\displaystyle\frac{L_{\delta}(t;t_{\delta})}{E_{\delta}}=$
$\displaystyle\frac{t_{\delta}}{t_{\text{diff}}^{2}}\cdot\frac{6}{x_{s}^{3}}\sum_{n=1}^{\infty}\left(-1\right)^{n+1}\left(\frac{\sin{(n\pi
x_{s})}}{n\pi}-x_{s}\cos{(n\pi x_{s})}\right)$ (13) $\displaystyle\quad\times
e^{-\frac{(n\pi)^{2}}{2t_{\text{diff}}^{2}}(t^{2}-t_{\delta}^{2})}\cdot
u(t-t_{\delta}),$

where $t_{\text{diff}}^{2}=9\kappa M/4\pi v_{o}c$ and $u$ is the Heaviside
step function. The luminosity for a given energy generation rate $Q(t)$ is

$L(t)=\int_{0}^{t}dt^{\prime}Q(t^{\prime})L_{\delta}(t;t^{\prime}).$ (14)

We proceed by comparing the analytical results with those obtained from our
radiative transfer (RT) code, hereafter referred to as numerical or simulated
results. Initially, we validate the analytical outcomes by simulating ejecta
with uniform opacity and density, a constant volumetric energy generation
rate, and local energy deposition without $\gamma$-ray transfer. Subsequently,
we conduct simulations incorporating density profiles and 56Ni distributions
specific to our models, incorporating $\gamma$-ray transfer while maintaining
a uniform and constant opacity of $0.2\,\text{g}\,\text{cm}^{-2}$. To ensure
consistency between the simulated and analytical models, we adjust the
parameters of the analytical models —- mass, velocity, and $x_{s}$ -— so that
the ejecta mass, kinetic energy, and 56Ni mass remain conserved. For sub-
Chandra models, $x_{s}$ is determined such that the mass enclosed within
$x_{s}$ matches the 56Ni mass, given that these models exhibit a 56Ni
distribution extending outward from the center. In contrast, for Chandra
models where the central ejecta comprises non-radioactive intermediate-mass
elements (IGEs) and 56Ni becomes dominant farther from the center, a single
parameter $x_{s}$ cannot fully characterize the 56Ni distribution. Therefore,
we sum the total mass of IGE in the central regions until the radius where
56Ni becomes the dominant ion. We then determine the extent of the IGE in the
analytical model, $x_{\text{IGE}}$, such that the enclosed mass within
$x_{\text{IGE}}$ equals the IGE mass at the center of the ejecta profile. The
analytical model calculation involves energy generation occurring in the shell
between $x_{\text{IGE}}$ and $x_{s}$, ensuring that this shell’s mass
corresponds to the ejecta profile’s 56Ni mass.

Figure 14 shows the rise times obtained from constant-opacity simulations
(black lines) and analytical models (red lines) following the prescription
above. As seen in the figure, the agreement between numerical and analytical
results holds within a $10$ percent margin for both sub-Chandra (solid lines)
and Chandra (dashed lines) models. This robust agreement underscores the
effectiveness of the analytical model. Additionally, the peak time
consistently decreases with the SN luminosity across the entire range,
mirroring the behavior of $t_{0}$ for these models.

Figure 14: Epoch of peak bolometric luminosity against the synthesized 56Ni
mass, for configurations with a constant opacity of
$0.2\,\text{g}\,\text{cm}^{-2}$. Black lines represent simulation results,
while results from the analytic model of Kushnir & Katz (2019) are depicted in
red. Solid lines correspond to sub-Chandra models, whereas dashed lines
correspond to Chandra models. The parameters used in the analytical models are
detailed in the text. For both models, the peak time shows a monotonic
decrease with SN luminosity, consistent with the behavior of $t_{0}$ for these
models.

We proceed to compare the full numerical calculations with the analytical
model, accounting for varying opacity in time and space within the
simulations. To align the analytical model appropriately, we employ the mass-
averaged Rosseland mean opacity (Equation (3)) at peak light,
$\left<\kappa_{\textrm{R}}(t_{p})\right>$. The resulting opacities range from
$0.1$ to $0.18\,\text{g}\,\text{cm}^{-2}$ for sub-Chandra models, and from
$0.12$ to $0.18\,\text{g}\,\text{cm}^{-2}$ for Chandra models. In both models,
the opacity increases monotonously with luminosity. The rise times derived
from the analytical models using this approach are presented in Figure 15,
alongside the simulation results. For sub-Chandra models, the analytical
results (solid red line) closely match numerical calculations (solid black
line) across the entire luminosity range, with deviations up to $\approx 3$
percent. This robust agreement further underscores the influence of opacity
variation on rise times and the $L_{30}/L_{p}$ parameter. Meanwhile, the
analytical results (dashed red line) for Chandra models also align well with
numerical calculations (dashed black line), albeit with deviations slightly
larger, up to $\approx 10$ percent.

Figure 15: Same as Figure 14, but for full simulations, where opacity varies
and is computed dynamically during the simulation. The opacity used in the
analytical model is the mass-averaged Rosseland mean opacity derived from
Equation (3), evaluated at peak light.

##  Appendix C Some results in magnitude space

Throughout this work, we used linear scaling for the luminosity and shape
parameters we have presented. Here, we include additional graphs of some of
our relations in magnitude space for completeness and comparison purposes.
These are the $\Delta M_{\text{bol}}(15)$-$t_{0}$ relation, shown in Figure
16, and the $L_{30}/L_{p}$–$L_{p}$ and $L_{p+15}/L_{p}$ relations, shown in
Figures 17 and 10, respectively.

Figure 16: Same as Figure 9 but $L_{p+15}/L_{p}$ is replaced with $\Delta
M_{\text{bol}}(15)=-2.5\log(L_{p+15}/L_{p})$. Figure 17: Same as Figure 1, but
in magnitude space. Figure 18: Same as Figure 10, but in magnitude space.

##  Appendix D Parameters of the SNe sample and of the models

In this appendix, we outline the parameters utilized in our analysis. The
characteristics of the observed sample can be found in Table 1, while the
details of the models are provided in Table 2.

Table 1: Parameters of the SNe Ia sample.

name | $L_{p}$ | $L_{30}/L_{p}$ | $\Delta M_{15}(\textrm{bol})$ | $M_{\mathrm{Ni56}}$ | $t_{0}$ |  $t_{0}$ and $M_{\mathrm{Ni56}}$ ref.a
---|---|---|---|---|---|---
|  $(10^{43}$ $\textrm{erg}\,\text{s}^{-1})$ |  |  | $(M_{\odot})$ | (day) |
2003du | $1.36\pm 0.12$ | $0.54\pm 0.02$ | $0.82\pm 0.04$ | $0.61_{-0.15}^{-0.19}$ | $35.91_{-2.70}^{-2.35}$ | 1
2004ef | $1.10\pm 0.09$ | $0.44\pm 0.03$ | $1.00\pm 0.06$ | $-$ | $-$ | $-$
2004eo | $1.12\pm 0.13$ | $0.47\pm 0.03$ | $0.92\pm 0.08$ | $0.49_{-0.10}^{-0.12}$ | $37.07_{-1.89}^{-2.06}$ | 1
2004ey | $1.51\pm 0.16$ | $0.60\pm 0.03$ | $0.84\pm 0.05$ | $-$ | $-$ | $-$
2004gs | $0.92\pm 0.07$ | $0.43\pm 0.03$ | $0.97\pm 0.07$ | $0.41_{-0.08}^{-0.08}$ | $33.64_{-1.59}^{-1.81}$ | 1
2005A | $1.29\pm 0.15$ | $0.60\pm 0.03$ | $0.81\pm 0.05$ | $-$ | $-$ | $-$
2005M | $1.63\pm 0.16$ | $0.63\pm 0.04$ | $0.78\pm 0.06$ | $-$ | $-$ | $-$
2005cf | $1.43\pm 0.19$ | $0.55\pm 0.03$ | $0.89\pm 0.06$ | $0.65_{-0.14}^{-0.13}$ | $36.88_{-1.64}^{-1.96}$ | 1
2005el | $1.24\pm 0.48$ | $0.45\pm 0.03$ | $0.99\pm 0.06$ | $0.49_{-0.19}^{-0.20}$ | $33.33_{-1.63}^{-1.40}$ | 1
2005hc | $1.59\pm 0.14$ | $0.61\pm 0.04$ | $0.78\pm 0.07$ | $0.70_{-0.10}^{-0.17}$ | $43.25_{-1.81}^{-2.15}$ | 2
2005iq | $1.32\pm 0.12$ | $0.42\pm 0.03$ | $1.03\pm 0.07$ | $-$ | $-$ | $-$
2005kc | $1.31\pm 0.14$ | $0.48\pm 0.03$ | $0.97\pm 0.07$ | $-$ | $-$ | $-$
2005ke | $0.37\pm 0.07$ | $0.39\pm 0.02$ | $0.99\pm 0.04$ | $0.13_{-0.03}^{-0.02}$ | $34.08_{-1.00}^{-1.31}$ | 1
2005ki | $1.24\pm 0.12$ | $0.43\pm 0.03$ | $1.06\pm 0.07$ | $0.48_{-0.06}^{-0.05}$ | $33.13_{-1.18}^{-1.49}$ | 1
2006D | $1.11\pm 0.13$ | $0.42\pm 0.03$ | $1.01\pm 0.06$ | $0.41_{-0.04}^{-0.06}$ | $33.33_{-1.40}^{-0.97}$ | 1
2006ax | $1.44\pm 0.15$ | $0.53\pm 0.03$ | $0.85\pm 0.06$ | $-$ | $-$ | $-$
2006bh | $1.15\pm 0.13$ | $0.41\pm 0.03$ | $1.12\pm 0.07$ | $-$ | $-$ | $-$
2006et | $1.57\pm 0.14$ | $0.66\pm 0.03$ | $0.82\pm 0.06$ | $-$ | $-$ | $-$
2006kf | $1.02\pm 0.09$ | $0.38\pm 0.03$ | $1.06\pm 0.08$ | $0.38_{-0.04}^{-0.07}$ | $31.39_{-1.85}^{-1.31}$ | 1
2006mr | $0.13\pm 0.02$ | $0.34\pm 0.01$ | $1.14\pm 0.03$ | $0.04_{-0.01}^{-0.01}$ | $29.64_{-1.08}^{-0.78}$ | 2
2007N | $0.19\pm 0.02$ | $0.37\pm 0.03$ | $1.12\pm 0.04$ | $0.08_{-0.02}^{-0.02}$ | $32.61_{-2.10}^{-2.21}$ | 1
2007S | $1.62\pm 0.18$ | $0.65\pm 0.03$ | $0.79\pm 0.06$ | $-$ | $-$ | $-$
2007af | $1.02\pm 0.08$ | $0.46\pm 0.02$ | $0.96\pm 0.04$ | $0.42_{-0.07}^{-0.06}$ | $36.54_{-1.21}^{-1.63}$ | 1
2007ba | $0.60\pm 0.05$ | $0.39\pm 0.03$ | $1.01\pm 0.07$ | $-$ | $-$ | $-$
2007bc | $1.31\pm 0.13$ | $0.44\pm 0.04$ | $1.00\pm 0.08$ | $-$ | $-$ | $-$
2007bd | $1.28\pm 0.12$ | $0.46\pm 0.03$ | $1.11\pm 0.07$ | $-$ | $-$ | $-$
2007le | $1.40\pm 0.23$ | $0.64\pm 0.03$ | $0.82\pm 0.04$ | $0.64_{-0.12}^{-0.16}$ | $41.96_{-1.46}^{-1.76}$ | 1
2007on | $0.55\pm 0.18$ | $0.41\pm 0.02$ | $0.99\pm 0.06$ | $0.20_{-0.07}^{-0.08}$ | $30.98_{-1.35}^{-1.00}$ | 1
2008bc | $1.55\pm 0.15$ | $0.61\pm 0.03$ | $0.81\pm 0.05$ | $0.71_{-0.11}^{-0.19}$ | $38.78_{-3.01}^{-2.61}$ | 1
2008bf | $1.52\pm 0.15$ | $0.58\pm 0.03$ | $0.75\pm 0.05$ | $0.71_{-0.11}^{-0.07}$ | $40.36_{-2.44}^{-3.24}$ | 1
2008fp | $1.58\pm 0.25$ | $0.65\pm 0.04$ | $0.78\pm 0.05$ | $0.69_{-0.14}^{-0.15}$ | $40.04_{-1.68}^{-1.61}$ | 1
2008gp | $1.50\pm 0.15$ | $0.52\pm 0.03$ | $0.95\pm 0.07$ | $-$ | $-$ | $-$
2008hj | $1.52\pm 0.13$ | $0.56\pm 0.04$ | $0.89\pm 0.06$ | $-$ | $-$ | $-$
2008hv | $1.10\pm 0.09$ | $0.47\pm 0.03$ | $1.01\pm 0.05$ | $0.43_{-0.06}^{-0.07}$ | $34.34_{-1.67}^{-1.61}$ | 1
2009D | $1.55\pm 0.14$ | $0.63\pm 0.04$ | $0.83\pm 0.06$ | $-$ | $-$ | $-$
2009F | $0.24\pm 0.02$ | $0.34\pm 0.02$ | $1.13\pm 0.04$ | $-$ | $-$ | $-$
2009Y | $1.55\pm 0.17$ | $0.60\pm 0.04$ | $0.75\pm 0.05$ | $0.74_{-0.13}^{-0.12}$ | $43.74_{-1.42}^{-1.53}$ | 1
2009aa | $1.37\pm 0.12$ | $0.49\pm 0.03$ | $0.96\pm 0.07$ | $-$ | $-$ | $-$
2009ab | $1.29\pm 0.15$ | $0.47\pm 0.03$ | $0.94\pm 0.07$ | $-$ | $-$ | $-$
2009ad | $1.65\pm 0.16$ | $0.54\pm 0.04$ | $0.92\pm 0.08$ | $-$ | $-$ | $-$
2011fe | $1.22\pm 0.17$ | $0.63\pm 0.03$ | $0.70\pm 0.05$ | $0.58_{-0.09}^{-0.09}$ | $39.02_{-1.05}^{-1.22}$ | 1
2012fr | $1.40\pm 0.11$ | $0.53\pm 0.04$ | $0.83\pm 0.06$ | $0.62_{-0.09}^{-0.08}$ | $43.27_{-1.33}^{-1.54}$ | 1
2012ht | $0.65\pm 0.24$ | $0.43\pm 0.04$ | $1.01\pm 0.08$ | $0.24_{-0.09}^{-0.10}$ | $34.63_{-2.78}^{-2.09}$ | 1
2013aa | $1.51\pm 0.14$ | $0.55\pm 0.04$ | $0.86\pm 0.05$ | $0.64_{-0.08}^{-0.09}$ | $39.44_{-1.99}^{-2.17}$ | 2
2015F | $1.35\pm 0.14$ | $0.47\pm 0.06$ | $0.91\pm 0.14$ | $0.54_{-0.10}^{-0.15}$ | $38.42_{-2.34}^{-2.05}$ | 1
2015bp | $0.73\pm 0.28$ | $0.40\pm 0.02$ | $1.00\pm 0.06$ | $0.29_{-0.12}^{-0.12}$ | $31.94_{-2.04}^{-1.95}$ | 2
2017cbv | $1.55\pm 0.16$ | $0.62\pm 0.02$ | $0.75\pm 0.05$ | $0.76_{-0.17}^{-0.10}$ | $39.89_{-2.13}^{-4.20}$ | 2

  * a

Reference for the values of $t_{0}$ and $M_{\mathrm{Ni56}}$. (1) - Sharon &
Kushnir (2020b). (2) - this work.

Table 2: Parameters of the models. For 2D models, the values for $L_{p}$ and $L_{30}/L_{p}$ represent angle-averaged results. | name | $L_{p}$ | $L_{30}/L_{p}$ | $\Delta M_{15}(\textrm{bol})$ | $M_{\mathrm{Ni56}}$ | $t_{0}$
---|---|---|---|---|---|---
|  |  $(10^{43}$ $\textrm{erg}\,\text{s}^{-1})$ |  | (mag) | $(M_{\odot})$ | (d)
Chandra | DDC0_0p5d | 0.28 | 0.66 | 0.78 | 0.11 | 54.60
| DDC10_0p5d | 0.46 | 0.66 | 0.68 | 0.19 | 48.98
| DDC15_0p5d | 0.66 | 0.62 | 0.69 | 0.27 | 44.33
| DDC17_0p5d | 0.89 | 0.58 | 0.71 | 0.37 | 41.64
| DDC20_0p5d | 1.10 | 0.56 | 0.72 | 0.47 | 40.12
| DDC22_0p5d | 1.32 | 0.53 | 0.74 | 0.57 | 39.04
| DDC25_0p5d | 1.52 | 0.51 | 0.78 | 0.66 | 38.22
| DDC6_0p5d | 1.81 | 0.48 | 0.82 | 0.78 | 36.87
sub-Chandra, | M085_1Z | 0.08 | 0.32 | 1.17 | 0.03 | 34.98
central ignition | M08_1Z | 0.33 | 0.44 | 0.92 | 0.12 | 34.37
| M09_1Z | 0.66 | 0.45 | 0.88 | 0.26 | 32.96
| M10_1Z | 1.29 | 0.41 | 0.94 | 0.54 | 31.28
| M11_1Z | 1.79 | 0.40 | 0.94 | 0.79 | 30.46
sub-Chandra, | M08_zig0 | 0.12 | 0.29 | 1.19 | 0.04 | 33.30
off-centered ignitions | M08_zig01 | 0.11 | 0.29 | 1.20 | 0.03 | 33.60
| M08_zig025 | 0.13 | 0.31 | 1.19 | 0.04 | 33.80
| M08_zig05 | 0.16 | 0.36 | 1.09 | 0.06 | 36.10
| M085_zig0 | 0.25 | 0.38 | 1.04 | 0.09 | 33.90
| M085_zig01 | 0.27 | 0.40 | 1.01 | 0.10 | 34.00
| M085_zig025 | 0.32 | 0.42 | 0.97 | 0.12 | 33.90
| M085_zig05 | 0.36 | 0.45 | 0.93 | 0.14 | 35.30
| M09_zig0 | 0.60 | 0.46 | 0.88 | 0.24 | 33.00
| M09_zig01 | 0.59 | 0.46 | 0.88 | 0.24 | 33.00
| M09_zig025 | 0.64 | 0.46 | 0.89 | 0.25 | 32.70
| M09_zig05 | 0.65 | 0.45 | 0.90 | 0.26 | 33.00
| M095_zig0 | 0.94 | 0.45 | 0.86 | 0.39 | 32.00
| M095_zig01 | 0.94 | 0.45 | 0.87 | 0.39 | 32.00
| M095_zig025 | 0.97 | 0.44 | 0.89 | 0.40 | 31.80
| M095_zig05 | 0.95 | 0.44 | 0.89 | 0.39 | 31.90
| M1_zig0 | 1.26 | 0.43 | 0.88 | 0.54 | 31.20
| M1_zig01 | 1.26 | 0.43 | 0.88 | 0.54 | 31.20
| M1_zig025 | 1.28 | 0.43 | 0.90 | 0.54 | 31.00
| M1_zig05 | 1.27 | 0.42 | 0.93 | 0.53 | 31.10
| M11_zig0 | 1.76 | 0.43 | 0.85 | 0.80 | 30.30
| M11_zig01 | 1.78 | 0.42 | 0.87 | 0.80 | 30.30
| M11_zig025 | 1.79 | 0.41 | 0.89 | 0.80 | 30.20
| M11_zig05 | 1.79 | 0.40 | 0.92 | 0.79 | 30.00
Head-on collisions | M05-M05 | 0.30 | 0.43 | 0.95 | 0.11 | 37.79
| M055-M055 | 0.56 | 0.50 | 0.86 | 0.22 | 39.13
| M07-M05 | 0.64 | 0.51 | 0.86 | 0.26 | 42.22
| M06-M05 | 0.66 | 0.49 | 0.85 | 0.27 | 38.25
| M08-M05 | 0.73 | 0.56 | 0.83 | 0.29 | 43.35
| M06-M06 | 0.76 | 0.55 | 0.80 | 0.32 | 39.82
| M07-M06 | 0.87 | 0.57 | 0.79 | 0.38 | 40.66
| M08-M06 | 0.96 | 0.58 | 0.81 | 0.38 | 41.97
| M064-M064 | 0.95 | 0.57 | 0.78 | 0.41 | 39.58
| M08-M07 | 1.18 | 0.58 | 0.79 | 0.48 | 40.35
| M09-M07 | 1.24 | 0.61 | 0.77 | 0.51 | 41.87
| M09-M06 | 1.25 | 0.63 | 0.73 | 0.50 | 41.85
| M07-M07 | 1.24 | 0.58 | 0.76 | 0.56 | 38.52
| M09-M05 | 1.35 | 0.53 | 0.84 | 0.69 | 42.21
| M09-M08 | 1.27 | 0.62 | 0.74 | 0.54 | 40.66
| M08-M08 | 1.66 | 0.56 | 0.76 | 0.74 | 37.30
| M09-M09 | 1.74 | 0.64 | 0.70 | 0.78 | 39.24
| M10-M07 | 1.81 | 0.63 | 0.69 | 0.83 | 43.39
| M10-M08 | 1.88 | 0.66 | 0.67 | 0.81 | 42.03
| M10-M05 | 1.96 | 0.52 | 0.78 | 0.82 | 39.29
| M10-M06 | 2.03 | 0.58 | 0.74 | 0.88 | 40.78
| M10-M09 | 2.16 | 0.64 | 0.69 | 1.00 | 39.77
| M10-M10 | 2.63 | 0.62 | 0.68 | 1.25 | 39.04