#  Including a Luminous Central Remnant in Radiative Transfer Simulations for
Type Iax Supernovae

F. P. Callan,1 S. A. Sim,1 C. E. Collins,2 L. J. Shingles,2 F. Lach,3 F. K.
Röpke,3,4 R. Pakmor,5 M. Kromer3 and S. Srivastav1
1School of Maths and Physics, Queen’s University Belfast, University Road,
Belfast BT7 1NN, UK
2GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, 64291
Darmstadt, Germany
3Heidelberger Institut für Theoretische Studien, Schloss-Wolfsbrunnenweg 35,
D-69118, Heidelberg, Germany
4Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische
Astrophysik, Philosophenweg 12, D-69120 Heidelberg, Germany
5Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748,
Garching, Germany E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Type Iax supernovae (SNe Iax) are proposed to arise from deflagrations of
Chandrasekhar mass white dwarfs. Previous deflagration simulations have
achieved good agreement with the light curves and spectra of intermediate-
luminosity and bright SNe Iax. However, the model light curves decline too
quickly after peak, particularly in red optical and NIR bands. Deflagration
models with a variety of ignition configurations do not fully unbind the white
dwarf, leaving a remnant polluted with ${}^{56}\mathrm{Ni}$. Emission from
such a remnant may contribute to the luminosity of SNe Iax. Here we
investigate the impact of adding a central energy source, assuming
instantaneous powering by ${}^{56}\mathrm{Ni}$ decay in the remnant, in
radiative transfer calculations of deflagration models. Including the remnant
contribution improves agreement with the light curves of SNe Iax, particularly
due to the slower post-maximum decline of the models. Spectroscopic agreement
is also improved, with intermediate-luminosity and faint models showing
greatest improvement. We adopt the full remnant ${}^{56}\mathrm{Ni}$ mass
predicted for bright models, but good agreement with intermediate-luminosity
and faint SNe Iax is only possible for remnant ${}^{56}\mathrm{Ni}$ masses
significantly lower than those predicted. This may indicate that some of the
${}^{56}\mathrm{Ni}$ decay energy in the remnant does not contribute to the
radiative luminosity but instead drives mass ejection, or that escape of
energy from the remnant is significantly delayed. Future work should
investigate the structure of remnants predicted by deflagration models and the
potential roles of winds and delayed energy escape, as well as extend
radiative transfer simulations to late times.

###### keywords:

radiative transfer – white dwarfs – supernovae: general – transients:
supernovae – methods: numerical

††pubyear: 2023††pagerange:  Including a Luminous Central Remnant in Radiative
Transfer Simulations for Type Iax Supernovae–A

##  1 Introduction

Through modern transient surveys it has become clear that Type Ia supernovae
(SNe Ia) are a diverse class of objects: in addition to the “normal” SNe Ia,
which exhibit an empirical relation between their peak brightness and light
curve evolution (Phillips, 1993), there are multiple distinct sub-classes (see
e.g. Taubenberger 2017 for a review).

Type Iax supernovae (SNe Iax, Li et al. 2003; Foley et al. 2013) are the most
numerous peculiar sub-class of SNe Ia, recently estimated by Srivastav et al.
(2022) to make up $15^{+15}_{-9}\%$ of the total SNe Ia rate. Compared to
normal SNe Ia, SNe Iax show low absolute brightness at peak relative to their
light curve widths (e.g. McClelland et al. 2010; Stritzinger et al. 2014) and
thus fall outside the width-luminosity relation. They display a wide diversity
in their band light curve properties showing variations in peak brightness of
up to six magnitudes in certain bands and diversity in their light curve rise
times and decline rates (Magee et al. 2016; Srivastav et al. 2022).

SNe Iax are spectroscopically distinct from normal SNe Ia: the blue continua
and absorption from higher ionisation species observed in their early-time
spectra mean they show more similarities to the sub-classes of 1991T-like and
1999a-like SNe Ia, which exhibit hot photospheres (Foley et al. 2013; McCully
et al. 2014). Additionally, SNe Iax spectra show significantly lower expansion
velocities at peak compared to normal SNe Ia (Foley et al. 2013; Stritzinger
et al. 2015). The late time spectra of SNe Iax show a large number of
permitted lines from iron group elements (IGEs) along with some intermediate
mass elements (IMEs) and potential signatures of oxygen, while forbidden lines
of IGEs and IMEs are also present (Jha et al., 2006; Foley et al., 2016). As
such, SNe Iax never enter a fully nebular phase (unlike normal SNe Ia Graham
et al. 2017; Taubenberger et al. 2015). The SNe Iax sub-class also shows a
large spectroscopic diversity in line velocities, strengths and widths
particularly at later times (McCully et al. 2014; Stritzinger et al. 2015;
Yamanaka et al. 2015; Foley et al. 2016).

Multiple explosion scenarios have been suggested to explain SNe Iax, such as
the core collapse of massive stars (Foley et al. 2009; Valenti et al. 2009;
Moriya et al. 2010), pulsationally delayed detonations (Stritzinger et al.
2015), the failed detonation of an oxygen-neon (ONe) WD merging with a carbon-
oxygen (CO) secondary (Kashyap et al., 2018), a detonation ignited in a WD-
neutron star merger (Fernández & Metzger, 2013) and a detonation in a helium
shell surrounding a CO WD (Foley et al., 2009).

However, amongst the most widely discussed scenarios is that in which SNe Iax
are pure deflagrations of Chandrasekhar mass ($M_{\mathrm{Ch}}$) CO WDs
(Branch et al. 2004; Phillips et al. 2007), as the properties of such
explosions naturally reproduce many of the observed characteristics of SNe
Iax. In particular, simulations of pure deflagrations in $M_{\mathrm{Ch}}$ CO
WDs (Jordan et al., 2012; Kromer et al., 2013; Fink et al., 2014; Long et al.,
2014; Kromer et al., 2015; Leung & Nomoto, 2020; Lach et al., 2022) reproduce
many of the observed characteristics of SNe Iax, such as lower
${}^{56}\mathrm{Ni}$ masses and hence sub-luminous light curves as well as
lower expansion velocities. Furthermore, the pure deflagration model sequences
presented by Kromer et al. (2013); Fink et al. (2014) and Lach et al. (2022)
(hereafter K13, F14 and L22 respectively) are together able to reproduce
almost the full diversity in peak brightness of observed SNe Iax by varying
the initial geometry of the explosion simulations, failing to match only the
faintest observed SNe Iax. These models also produce light curves and spectra
in reasonably good agreement with bright and intermediate-luminosity SNe Iax,
particularly at earlier epochs.

The near-infrared light curves of SNe Iax further support the pure
deflagration scenario: the lack of a prominent secondary maximum that is
observed for normal SNe Ia (Li et al. 2003) points to a well-mixed ejecta
structure for SNe Iax (Jha et al. 2006; Phillips et al. 2007; Sahu et al.
2008), consistent with turbulent deflagration models (although see Stritzinger
et al. 2015 and Magee et al. 2022 for further discussion of stratification in
SNe Iax).

Despite the generally good agreement between $M_{\mathrm{Ch}}$ CO WD pure
deflagration models and observed SNe Iax, some systematic differences remain.
Specifically, the K13, F14 and L22 model light curves decline too quickly post
peak, particularly in the red and NIR bands, meaning the bolometric light
curves also decline too rapidly. This systematic difference worsens when
comparing to fainter SNe Iax where the decline post peak of the models is too
fast compared to observed SNe Iax in all bands. The faster decline for red,
optical, and NIR wavelengths after peak relative to other bands also makes the
model spectra too blue at later times relative to observed SNe Iax.

The outcome of pure deflagration simulations is sensitive to the initial
choice of ignition conditions (Jordan et al., 2012; Kromer et al., 2013; Fink
et al., 2014; Long et al., 2014; Kromer et al., 2015; Leung & Nomoto, 2020;
Lach et al., 2022). For a wide variety of ignition configurations, the
explosion is not energetic enough to fully unbind the WD (Jordan et al., 2012;
Kromer et al., 2013; Fink et al., 2014) potentially leaving behind a stellar
remnant polluted with burning products from the deflagration, including
${}^{56}\mathrm{Ni}$. The remnant is therefore expected to be luminous. Such a
luminous remnant may have been detected in late time observations ($\sim$ 4
years after explosion) of the faint SN Iax, SN 2008ha (Foley et al., 2014) and
bright SN Iax, SN 2012Z (McCully et al., 2022). Further evidence of a remnant
may be present in the slowly declining late time light curves of SN 2014dt
(Kawabata et al., 2018) and SN 2019muj (Kawabata et al., 2021) and the
peculiar late time spectra of SNe Iax that never become truly nebular (Kromer
et al. 2013; Foley et al. 2016; Kawabata et al. 2018, 2021; Maeda & Kawabata
2022).

Accounting for the remnant may be key to fully understanding the observed
properties of SNe Iax. An exploratory calculation by K13 found adding the
energy deposited by radioactive ${}^{56}\mathrm{Ni}$ in the remnant to the
light curve predicted for their N5def deflagration model significantly
improved the agreement with the bolometric light curve of SN 2005hk, a bright
SNe Iax, particularly at later times. However, their calculation simply adds
this energy to the lightcurve computed from the ejecta model and does not take
into account the interaction of this radiation with the ejecta. Therefore,
with this approach only a crude bolometric light curve can be calculated and
the early-time bolometric light curves will not be accurately represented.

Shen & Schwab (2017) also found their models of post SNe Iax remnant winds to
be consistent with late time photometric observations of SNe Iax.
Additionally, from analysis of the late-time spectra of the intermediate-
luminosity SNe Iax, SN 2019muj, Maeda & Kawabata (2022) find that a denser
inner ejecta component is present. They attribute this to a secondary mass
ejection from a remnant. The faint hybrid carbon-oxygen-neon (CONe) WD pure
deflagration model which Kromer et al. (2015) produced as an attempt to match
SN 2008ha also predicts a luminous remnant which they suggest may have a
significant impact on the optical display of the model.

The aim of this paper is to incorporate a central source representing the
remnant material into our 3D, time-dependant radiative transfer simulations in
order to predict how a luminous remnant impacts the light curves and spectra
predicted for $M_{\mathrm{Ch}}$ CO WD pure deflagration models covering a
range of peak luminosities. This improves on the exploratory calculation of
K13 since we can simulate the early time diffusion of this radiation through
the ejecta and investigate how it affects light curves, colours and spectra.
We take the pure deflagration models presented by L22 as a starting point.
Informed by the remnant compositions predicted by the hydrodynamic explosion
simulations of L22 we add a central ${}^{56}\mathrm{Ni}$ source representing a
luminous remnant to a sub-set of the standard L22 deflagration models. We
discuss the treatment we adopt for the remnant and summarise our radiative
transfer simulation set up in Section 2. In Section 3 we make detailed
comparisons between the light curves and spectra predicted by our radiative
transfer simulations and those of observed SNe Iax, before discussing the
implications of our results along with our conclusions in Section 4.

##  2 Numerical Methods

###  2.1 Explosion models

The sequence of $M_{\mathrm{Ch}}$ CO WD pure deflagration models presented by
L22 are the basis for this work. These models all adopt a single-spark
ignition, which has been argued to be a more realistic ignition configuration
than a multi-spark ignition as adopted by F14 (Kuhlen et al. 2006; Zingale et
al. 2009, 2011; Nonaka et al. 2012). The L22 sequence provides a good starting
point to explore the impact of including a luminous remnant in our
simulations: all models in this sequence produce explosions that are not
energetic enough to fully unbind the WD, leaving behind remnants with masses
greater than a solar mass that are polluted with burning products from the
deflagration including ${}^{56}\mathrm{Ni}$. Additionally, the L22 model
sequence covers almost the full peak brightness range of the SNe Iax class,
failing to reproduce only the very brightest and faintest observed SNe Iax. We
can therefore explore how a luminous remnant impacts our model comparisons
with a wide variety of observed SNe Iax from bright to faint events.

The leafs (Reinecke et al. 1999) hydrodynamic explosion simulations used to
calculate the L22 models adopted an expanding grid to track the ejecta (Röpke,
2005). This means the remnant is not well resolved at the end of the
hydrodynamic explosion simulations (at $t\,{=}\,100\,\mathrm{s}$). Therefore,
only basic properties of the remnant can be extracted: the overall mass and
composition at the end of the explosion simulations can be determined from the
tracer particles which remain bound (see L22 for details). We note however,
that while we can say the remnant is bound at $t\,{=}\,100\,\mathrm{s}$, this
does not necessarily mean that the material in the remnant will all remain
bound at later times. Indeed, the energy predicted to be deposited in the
remnants from the ${}^{56}\mathrm{Ni}$ decay chain indicates they are super-
Eddington by a factor of ${\sim}\,1000$ to $10000$ depending on the model and
time in the simulation. This means it is highly likely that some or all of the
remnant material becomes unbound, perhaps in the form of a radioactively
driven post supernova wind as proposed by Shen & Schwab (2017) or as a
secondary mass ejection as described by Maeda & Kawabata (2022). Therefore,
when we discuss the remnant in this work we are referring to any material
which remains bound up to the end of the hydrodynamic explosion simulation at
100 s and so would not be included as part of the ejecta in the radiative
transfer simulations for the standard L22 model sequence.

The remnants’ structure at the end of the L22 explosion simulations is that of
a puffed-up envelope of burning products from the deflagration, mixed with CO
material, which is settling onto a relatively dense CO core (see also K13 and
Jordan et al. 2012). The remnant contains a significant fraction of
${}^{56}\mathrm{Ni}$ (between 2 and 7% depending on the model) with this
${}^{56}\mathrm{Ni}$ predicted to be primarily found in the envelope (again
see L22 for details). For the models in this work, we calculated an estimate
of the maximum velocity the remnant material would approach if all the
radioactive energy deposited in the remnant over the simulation time was
converted into kinetic energy shared between all the remnant mass. The
estimated velocities (between $\sim$ 500 to 800 km s-1) are comparable to the
velocity of a single grid cell for all models. However, it is likely that only
the ${}^{56}\mathrm{Ni}$ rich part of the remnant envelope is ejected. The
Shen & Schwab (2017) remnant model with most similar mass to our model
remnants has an envelope mass of 0.1 $\mathrm{M}_{\odot}$ (see Section 4.2 for
more detailed discussion). If we assume all the radioactive energy is
deposited in the remnant envelope during the simulation time, the envelope
would reach an estimated velocity between $\sim$ 2000 to 2600 km s-1 depending
on the model. While more significant, these velocities are still only
comparable to the velocities of a small number of inner grid cells in our
simulations. As such, even if our model remnants become partially or fully
unbound they will likely remain small on the scale of the computational grid.

###  2.2 Incorporating the remnant in artis

To calculate synthetic observables (light curves and spectra) for our models,
to compare with observed SNe Iax, we use the 3D time-dependant Monte Carlo
radiative transfer code artis111https://github.com/artis-mcrt/artis/ (Sim,
2007; Kromer & Sim, 2009; Bulla et al., 2015; Shingles et al., 2020). In the
standard mode of operation, artis follows the propagation of $\gamma$-ray
photons emitted by radioactive decays and deposits energy in the supernova
ejecta before solving the radiative transfer problem self-consistently. For
this study we also include the contribution from a luminous remnant source in
our radiative transfer simulations by placing a ${}^{56}\mathrm{Ni}$ source in
the centre of the models with ${}^{56}\mathrm{Ni}$ mass informed by the
remnant ${}^{56}\mathrm{Ni}$ masses predicted by the L22 hydrodynamic
explosion simulations. The main properties of the remnant and ejected material
predicted by the L22 explosion simulations that form the basis of our new
models are summarised in Table 1. As we move to fainter deflagration models
the remnant ${}^{56}\mathrm{Ni}$ mass increases relative to the ejecta
${}^{56}\mathrm{Ni}$ mass. Therefore, if all energy released from the
${}^{56}\mathrm{Ni}$ decay chain in the remnant contributes to the optical
display of the models the remnant will have a larger impact on the observed
properties of faint models.

Model | $M_{\mathrm{ej}}$ | $M(^{56}\mathrm{Ni})_{\mathrm{ej}}$ | $M_{\mathrm{rem}}$ | $M(^{56}\mathrm{Ni})_{\mathrm{rem}}$
---|---|---|---|---
r10_d4.0_Z | 0.227 | 0.092 | 1.16 | 0.054
r48_d5.0_Z | 0.054 | 0.018 | 1.34 | 0.038
r114_d6.0_Z | 0.014 | 0.0058 | 1.38 | 0.030
Table 1: Summary of the total and ${}^{56}\mathrm{Ni}$ masses for the ejecta
and remnant material of the L22 models which our new models are based on.
Masses are given in solar masses.

artis is not configured to deal with a central luminous source in its standard
mode of operation. Therefore, to incorporate a treatment for luminous
remnants, we made use of developments made by Collins et al. (2022) which
allows more flexibility in the way energy is injected in our artis
simulations. We extended this method to treat energy due to radioactive decays
from the ${}^{56}\mathrm{Ni}$ decay chain in the remnant separately to energy
injected by radioactive decays in the ejecta. The rate and amount of radiation
escaping the remnant is affected by a number of factors including the
diffusion of the radiation through the remnant and potential mechanisms driven
by the radioactive energy deposited in the remnant such as winds in the
remnant envelope (Shen & Schwab, 2017) or the ejection of mass from the
remnant (Maeda & Kawabata, 2022). However, for simplicity, in our prescription
we assume the remnant emits instantaneously according to the
${}^{56}\mathrm{Ni}$ decay chain (i.e. assuming no significant delay due to
diffusion in the remnant itself). Due to the compactness of the remnant
material we expect the gamma rays emitted by the remnant to be trapped with
their energy used to heat the remnant (see also discussion in K13). We
therefore assume the remnant emits as a black body. Specifically, each time
there is a ${}^{56}\mathrm{Ni}$ decay in the remnant, a packet of UVOIR
radiation is emitted from the centre of the model grid and assigned a co-
moving frame frequency sampled from a black body distribution with the
temperature treated as a parameter of the simulation and representing the
remnant temperature at the decay time (see discussion below). Once emitted,
the UVOIR radiation packets travel through the ejecta and interact in the same
way as UVOIR radiation packets originating in the ejecta. This approach allows
us to obtain band light curves and spectra for our pure deflagration models
with the contribution of a luminous remnant included, which we can compare
directly to the light curves and spectra of observed SNe Iax.

In this work, we explore simulations first assuming a constant remnant
temperature. We also test the sensitivity of our findings to the remnant
temperature treatment adopted by exploring a temperature evolution of the
remnant consistent with black body emission from a surface of constant radius.
The SNe Iax we compare to throughout this study have temperature estimates
from observations which provide an approximate lower limit on the temperatures
of potential remnants present for these objects. These temperature estimates
guide the range of remnant temperatures we explore in this work: Sahu et al.
(2008) and McCully et al. (2014) estimate late time temperatures for the
bright SNe Iax, SN 2005hk of ${\lesssim}\,4500\,\mathrm{K}$ and ${\sim}\,4500$
to $7000\,\mathrm{K}$ respectively; Maeda & Kawabata (2022) estimate a
photospheric temperature for the intermediate-luminosity Type Iax, SN 2019muj
of 5500 K at 131 days after explosion and Srivastav et al. (2022) estimate the
temperature of the faint SNe Iax, SN 2020kyg varies from
${\sim}\,8000\,\mathrm{K}$ to $4000\,\mathrm{K}$ from g-band peak to 60 days
later. Additionally, models of post SNe Iax remnants by Shen & Schwab (2017)
with relatively similar remnant masses to those predicted for our models have
estimated temperatures of ${\sim}\,3000\,\mathrm{K}$ to $6000\,\mathrm{K}$ at
100 days after explosion.

Throughout this paper, the standard pure deflagration models retain their
names as introduced by L22 with the structure rX_dY_Z where r refers to the
ignition spark offset from the centre of the WD in km, d is the WD central
density in $10^{9}$ g $\mathrm{cm}^{-3}$ and Z the metallicity relative to the
solar value (all models presented in this work adopt solar metallicity). We
differentiate the models including a remnant contribution with an “R” in the
model name and also include information on the remnant temperature: for models
with fixed remnant temperature we include this temperature in Kelvin in the
model name and differentiate our model which adopts a temperature evolution
consistent with a black body of constant radius with the “const_rad” label.
Models which adopt remnant ${}^{56}\mathrm{Ni}$ masses less than those
predicted by the L22 hydrodynamic explosion simulations of the models are
labelled with the fraction of the predicted ${}^{56}\mathrm{Ni}$ mass that is
adopted for the remnant. For example model r48_d5.0_Z_R_6000K_0.33Ni is based
on the standard r48_d5.0_Z pure deflagration model but includes the
contribution from a remnant with a fixed characteristic temperature of 6000 K
and a ${}^{56}\mathrm{Ni}$ mass a third of that predicted by the L22
hydrodynamic explosion simulation of the standard deflagration model. We
summarise the assumptions of the remnants for the models presented in this
work in Table 2.

Model | $M(^{56}\mathrm{Ni})_{\mathrm{rem}}$ |  $T_{\mathrm{rem}}$ (K)
---|---|---
r10_d4.0_Z_R_2000K | 0.054 (1) | 2000
r10_d4.0_Z_R_8000K | 0.054 (1) | 8000
r10_d4.0_Z_R_15000K | 0.054 (1) | 15000
r10_d4.0_Z_R_const_rad | 0.054 (1) | -
r48_d5.0_Z_R_6000K_0.33Ni | 0.013 ($\frac{1}{3}$) | 6000
r114_d6.0_Z_R_4000K | 0.030 (1) | 4000
r114_d6.0_Z_R_6000K_0.1Ni | 0.003 ($\frac{1}{10}$) | 6000
Table 2: Summary of the assumed remnant properties for our new simulations.
The remnant ${}^{56}\mathrm{Ni}$ masses adopted are given in solar masses with
the fraction of the remnant ${}^{56}\mathrm{Ni}$ mass predicted by L22 that
these ${}^{56}\mathrm{Ni}$ masses represent included in brackets. The fixed
remnant temperature is also stated for applicable models.

###  2.3 Radiative transfer

artis follows the methods of Lucy (2002, 2003, 2005) and is based on dividing
the radiation field into indivisible energy packet Monte Carlo quanta. In this
work we adopt the NLTE approximation described by Kromer & Sim (2009) and as
used by F14 and L22. We do not make use of the full NLTE (non local
thermodynamic equilibrium) and non-thermal capabilities added to artis by
Shingles et al. (2020) which are required to accurately model the nebular
phase of SNe Ia (see, e.g., Dessart et al. 2014; Jerkstrand 2017; Shingles et
al. 2022). As such we do not extend the simulations presented in this work to
nebular phases and halt our analysis at 80 days post explosion. We mapped the
3D abundance and density structure of the unbound material at the end of the
L22 hydrodynamic simulations (by which point homologous expansion is a good
approximation Röpke 2005) to a $50^{3}$ Cartesian grid. 3D radiative transfer
simulations were then carried out for each model. In each simulation, $3\times
10^{7}$ energy packets were tracked through the ejecta for 200 logarithmically
spaced time steps between 0.3 and 100 days. We used the atomic data set
compiled by Gall et al. (2012) which is sourced from Kurucz & Bell (1995) and
Kurucz (2006). The much more extensive line list of Kurucz (2006) is used for
the important second and third ionisation stages of Fe, Co and Ni with the
Kurucz & Bell (1995) atomic data used for the remaining ions. We adopted a
grey approximation in optically thick cells (c.f. Kromer & Sim 2009) and at
times earlier than 0.4 days post explosion, local thermodynamic equilibrium
(LTE) was assumed.

##  3 Results

In the following section, we present comparisons between the bolometric and
band light curves and spectra predicted by radiative transfer simulations of
our new models with remnant contribution included and the standard L22
deflagration models along with observed SNe Iax. To investigate the impact of
including the remnant over a wide range of model brightnesses we include
comparisons between bright, intermediate-luminosity and faint models along
with corresponding observed SNe Iax (the light curve properties of our new
models and the L22 models we compare to are summarised in Tables 3 and 4). The
intermediate-luminosity and faint models with remnant contribution included
that produced best agreement with observed SNe Iax adopt remnant
${}^{56}\mathrm{Ni}$ masses less than those predicted by the hydrodynamic
explosion simulations of the models. We discuss this in detail in Section 4.
L22 found noticeable viewing angle effects for their standard pure
deflagration models but these effects were not so large that they altered any
of their findings. This is consistent with the viewing angle effects usually
exhibited by pure deflagration models which are generally less substantial
than those of other SNe Ia models due to their well-mixed ejecta structures
(see e.g. K13, F14 and L22). We find our models including central remnants do
not show significant viewing angle effects other than those already exhibited
by the standard L22 pure deflagration models. Therefore, we focus on
presenting angle-averaged results since these illustrate our main findings,
but we comment on observer orientation effects where relevant. We also include
the range in model parameters obtained from the line-of-site dependent light
curves alongside the angle averaged values in Tables 3 and 4 for reference.

###  3.1 Comparisons with bright SNe Iax

####  3.1.1 Light curves

Figure 1: Angle averaged bolometric light curves for the r10_d4.0_Z L22 model
and our new r10_d4.0_Z_R_8000K simulation. We also show the
${}^{56}\mathrm{Ni}$ decay energy deposition in the remnant (cyan), and the
light curve obtained by adding this to the r10_d4.0_Z model light curve (red).
The bolometric light curve of the bright SN Iax, SN 2005hk (Phillips et al.,
2007) is shown for comparison. Figure 2: Angle averaged bolometric and
UBVRIJH-band light curves for the r10_d4.0_Z L22 model and four new models
from this work in which we also treat energy injected by radioactive decays in
a luminous remnant in our radiative transfer simulations. Included for
comparison are light curves of SN 2005hk (Phillips et al., 2007), a bright SN
Iax. As Model r10_d4.0_Z_R_2000K did not provide a well converged temperature
solution at later times we only plot the light curves for this model up to 65
days. This poor convergence is a result of the low remnant temperature adopted
for this model.

Figure 1 shows comparisons between the bolometric light curves of the
brightest standard pure deflagration model from the L22 sequence, our new
r10_d4.0_Z_R_8000K model and the bright SN Iax, SN 2005hk. Also plotted is the
bolometric light curve produced if energy deposited by radioactive decays in
the remnant is instantaneously radiated (labelled “Energy_dep”). Following the
approach of Kromer et al. (2013) we also plot the bolometric light curve
produced if we simply add this remnant energy deposition to the bolometric
light curve of the r10_d4.0_Z model (labelled r10_d4.0_Z + Energy_dep). From
Figure 1 we see the r10_d4.0_Z + Energy_dep light curve produces good
agreement with the bolometric light curve of SN 2005hk at later times (after
${\sim}\,50$ days). However, at early times it significantly over-estimates
the brightness of the bolometric light curve until ${\sim}\,10$ days after
explosion and then declines too quickly until ${\sim}\,40$ days. The
r10_d4.0_Z_R_8000K bolometric light curve provides a good match to both the
rise, decline and peak bolometric brightness of the SN 2005hk light curve,
showing significantly improved agreement compared to the r10_d4.0_Z +
R_energy_dep bolometric light curve at earlier times. Therefore, including the
energy injected in the remnant directly in the radiative transfer simulations
is important for accurate predictions of the remnant contribution to the early
time bolometric light curves. Relative to Model r10_d4.0_Z the
r10_d4.0_Z_R_8000K bolometric light curve provides a better match to SN 2005hk
in rise and particularly in decline after peak also producing much better
agreement with the peak brightness of SN 2005hk. We also note that the
brightest viewing angles of Model r10_d4.0_Z_R_8000K reach the peak bolometric
luminosity of the brightest members of the SNe Iax class within their
uncertainties, something not previously achieved by the L22 pure deflagration
models that adopt a single-spark ignition.

Figure 2 shows angle-averaged bolometric and UBVRIJH-band light curves for the
r10_d4.0_Z model and four new models from this work in which energy injected
by radioactive decays in the remnant are also included in the radiative
transfer. The light curves of SN 2005hk are included for comparison. Each of
the four models with remnant contribution included adopt the full remnant
${}^{56}\mathrm{Ni}$ mass predicted by the hydrodynamic explosion simulations
of L22. However, the remnant temperature treatment is different for each model
so we can investigate the sensitivity of the synthetic observables obtained
from our radiative transfer simulations to what is assumed about the remnant’s
emission. The r10_d4.0_Z_R_2000K, r10_d4.0_Z_R_8000K and r10_d4.0_Z_R_15000K
models all have fixed characteristic black body temperatures while the
r10_d4.0_Z_R_const_rad model corresponds to the remnant emitting as a black
body surface of fixed radius (i.e. the remnant temperature decreases with
time). We discuss how these different choices of remnant temperature treatment
impact the band light curves of the models in detail below. However, we note
that provided the energy is injected centrally in the remnant, the optical
light curves are insensitive to different choices of remnant SED until $\sim$
30 days after explosion (see Figure 2).

From Figure 2 we see that Model r10_d4.0_Z is not bright enough at peak and
also too fast in both rise and decline in all bands to match the light curves
of SN 2005hk. By comparison the r10_d4.0_Z_R_8000K, r10_d4.0_Z_R_15000K and
r10_d4.0_Z_R_const_rad models provide a significantly better match to the peak
brightness, rise and decline of the band light curves of SN 2005hk (as was the
case for the bolometric comparisons). Specifically, we emphasise the extremely
good agreement between the R, I and H-band model light curves and those of SN
2005hk, in particular in decline after peak. Compared to bright SNe Iax such
as SN 2005hk, previous pure deflagration models (e.g. the N5def model
presented by K13) declined significantly too quickly after peak in these bands
even when the decline was well matched in blue bands. Including the remnant
contribution in our simulations significantly improves the agreement with the
red optical and NIR bands although there is still a deficit in J-band albeit
much reduced.

The band light curves of the r10_d4.0_Z_R_8000K and r10_d4.0_Z_R_15000K models
are very similar (see Figure 2). The biggest differences between the two
models is in J and H-band where the r10_d4.0_Z_R_8000K model produces better
agreement with SN 2005hk. This marginal improvement in agreement does not,
however, rule out the higher remnant temperature of $15000\,\mathrm{K}$.
Indeed, through additional simulations we have confirmed that a wide range of
fixed remnant temperatures from $6000\,\mathrm{K}$ to $15000\,\mathrm{K}$ (the
highest temperature we tested) produce model light curves in good agreement
with those of SN 2005hk. Therefore, our results are not excessively impacted
by different choices of fixed remnant temperature.

Model r10_d4.0_Z_R_const_rad allows us to test the impact of instead assuming
the remnant radius is fixed, meaning the remnant temperature evolves in time.
For Model r10_d4.0_Z_R_const_rad we adopt a remnant radius of $4.9\times
10^{14}\,\mathrm{cm}$. This is broadly consistent with the photospheric radius
Maeda & Kawabata (2022) derive for the proposed inner ejecta component of SN
2019muj, an intermediate luminosity SN Iax. The remnant radius of Model
r10_d4.0_Z_R_const_rad yields a black body temperature of 8000 K at 30 days
after explosion (i.e. the time the optical bands become sensitive to the
remnant temperature). We note that formally this radius is too large to be
consistent with central energy injection in the remnant for early times.
Nevertheless we continue to assume central energy injection noting the
emerging optical light curves are insensitive to the assumed remnant SED prior
to this time. Indeed, by the time the choice of remnant SED becomes relevant,
it is already contained within the inner 15% of the ejecta of Model
r10_d4.0_Z_R_const_rad which is consistent with our assumption of central
energy injection in the remnant. This is true for all our models presented in
this study that produce a reasonable match to observed SNe Iax.

Comparing band light curves for the r10_d4.0_Z_R_const_rad and
r10_d4.0_Z_R_8000K models, we see adopting an evolving remnant temperature
instead of a fixed temperature leads to only very small variations in the band
light curves, even at later times. The r10_d4.0_Z_R_const_rad model shows a
remnant effective temperature variation from ${\sim}\,12500\,\mathrm{K}$ to
$6000\,\mathrm{K}$ over the simulation time from 0.3 to 100 days. Our results
are therefore not overly sensitive to remnant temperature variations in this
range over the simulation time. The most noticeable differences are observed
in J and H-bands where the r10_d4.0_Z_R_const_rad model is slightly brighter
than the r10_d4.0_Z_R_8000K model from $\sim$ 50 days onwards. This is because
the r10_d4.0_Z_R_const_rad model has lower effective remnant temperature at
later times which increases the average wavelength of photon packets emitted
by the remnant and thus the brightness in NIR bands. Overall, any variations
in the band light curves between the r10_d4.0_Z_R_8000K and
r10_d4.0_Z_R_const_rad models are not significant in relation to comparisons
made with the light curves of SN 2005hk even in J and H-bands. Therefore,
within the time range of our simulations our conclusions are not sensitive to
whether the remnant has a fixed or evolving temperature and we make the choice
to primarily use remnants with fixed temperature for this study.

Of the models with remnant contribution included Model r10_d4.0_Z_R_2000K
produces by far the poorest agreement with SN 2005hk, declining much too
quickly post peak in the optical bands and much too slowly after peak in the
NIR bands. The poor agreement with the band light curves of SN 2005hk
effectively rules out a remnant temperature of 2000 K for our models. This
remnant temperature being excluded is consistent with the ejecta temperatures
estimated for SN 2005hk by Sahu et al. (2008) and McCully et al. (2014) of
$\lesssim$ 4500 K and $\sim$ 4500 to 7000 K respectively. These estimates do
not, however, rule out the remnant temperatures adopted for the remaining
models we compare to SN 2005hk in Figure 2.

####  3.1.2 Spectra

Figure 3: Absolute flux spectral comparisons between the angle averaged
r10_d4.0_Z_R_8000K model and r10_d4.0_Z L22 model along with SN 2005hk
(Phillips et al., 2007). Also plotted are the r10_d4.0_Z spectra scaled so
their peak flux matches that of the r10_d4.0_Z_R_8000K model spectra. Times
are relative to B-band peak. The spectra of SN 2005hk have been corrected for
reddening using the values from Phillips et al. (2007) and redshift corrected
using the values from Sahu et al. (2008).

Figure 3 shows spectral comparisons, at phases relative to B-peak, in absolute
flux, between Model r10_d4.0_Z_R_8000K, the r10_d4.0_Z L22 model and SN
2005hk. Also plotted are the r10_d4.0_Z spectra scaled by a constant factor
across all wavelengths such that the peak of the r10_d4.0_Z spectra matches
the peak of the r10_d4.0_Z_R_8000K model spectra at each epoch compared. All
spectra scaled throughout this work follow this scaling approach. We compare
relative to B-peak so the flux of the models is as similar as possible across
the epochs compared. Compared to the r10_d4.0_Z model, the r10_d4.0_Z_R_8000K
model spectra provide a significantly better match to the absolute flux of the
SN 2005hk spectra across all epochs. This was already clear from the band
light curve comparisons discussed above, but there are also some interesting
differences in spectral shape and features, which we discuss below.

The r10_d4.0_Z_R_8000K model shows its poorest agreement with the spectra of
SN 2005hk when comparing at the earliest epoch 8.4 days before B-peak.
Although the SED of the r10_d4.0_Z_R_8000K model provides a reasonable match
to SN 2005hk, particularly for wavelengths $\gtrsim 5500\,$Å, the spectrum is
too faint. This is a result of the rapid early-time light curve evolution.
When the spectra are instead compared at epochs relative to explosion
(corresponding to a spectral epoch 1.3 days later than that plotted for Model
r10_d4.0_Z_R_8000K) the flux match is significantly improved for this earliest
epoch. From Figure 3 we also see the r10_d4.0_Z_R_8000K model spectrum shows a
more noticeable lack of flux for wavelengths $\lesssim 4000\,$Å, with the
r10_d4.0_Z scaled model producing better agreement with SN 2005hk over these
wavelengths. This is a result of the shorter rise time to B-peak of the
r10_d4.0_Z model. This means the r10_d4.0_Z model is closer to explosion for
the comparison shown and thus has hotter ejecta and therefore a bluer
spectrum. When the models are instead compared relative to explosion they show
extremely similar spectral evolution until $\sim$ 10 days since explosion. For
this earliest epoch both models also show stronger spectral features for
wavelengths $\lesssim 5500\,$Å when comparing relative to explosion, improving
the spectral agreement with SN 2005hk.

For the epochs at 4.3 days before B-peak and 3.6 and 14.5 days after B-peak
the r10_d4.0_Z_R_8000K and r10_d4.0_Z scaled models display spectra which are
very similar across all wavelengths. Both models display a reasonable match to
the SED of SN 2005hk across these epochs. For the epochs at 4.3 days before
and 3.6 days after B-peak the agreement relative to the earliest epoch is
primarily improved for wavelengths $\lesssim 5500\,$Å. However, at 14.5 days
after B-peak, the epoch the models show best overall spectroscopic agreement
with SN 2005hk, there is an improved match to the SED of SN 2005hk across all
wavelengths. Additionally, over these epochs, both models show improved
agreement with the spectra of SN 2005hk relative to the earliest epoch shown
due to the increased number and strength of spectral features.

At 23.7 days after B-peak, the model spectra remain very similar although the
r10_d4.0_Z_R_8000K model matches the strengths of some of the absorption
features in the blue wavelengths observed for SN 2005hk better, such as the
strong absorption trough at $\sim 5200\,$Å primarily due to Fe ii. While the
overall spectral agreement with SN 2005hk is still good for this epoch, both
models show emission features (primarily attributed to Fe and Co ii in the
models) that are too weak relative to SN 2005hk, particularly for red
wavelengths. For the earlier epochs shown, the model spectral features are in
general more blue-shifted than those of SN 2005hk. However, for later epochs,
although there are still some mismatches, the model spectral features do not
show a preference for being blue-shifted or red-shifted relative to SN 2005hk.
This may suggest the outer regions of the model ejecta, where spectra are
formed at early times, have velocities in general too high compared to SN
2005hk, while the inner ejecta of the models, have velocities which provide a
better match to SN 2005hk. Another explanation for this behaviour may be
differences in the ionisation state between the outer ejecta of SN 2005hk and
the models leading to the early-time spectral features of the models generally
being formed at higher velocities.

The largest difference between the r10_d4.0_Z_R_8000K and r10_d4.0_Z scaled
model spectra is found for the latest epoch shown (39.5 days after B-peak). At
this epoch, the r10_d4.0_Z_R_8000K spectrum matches the SED of SN 2005hk well
across all wavelengths shown and reproduces the strength and velocities of
many of the spectral features across a wide range of wavelengths. The
r10_d4.0_Z scaled spectrum, on the other hand, has a noticeable lack of flux
for wavelengths longer than $\sim 5500\,$Å and has spectral features much
weaker than SN 2005hk for these redder wavelengths. This lack of flux for red
wavelengths at later times is a general issue for standard pure deflagration
models (see Section 1). Overall, the r10_d4.0_Z_R_8000K and r10_d4.0_Z scaled
model spectra are very similar for all but the latest epoch shown, producing
similarly good spectroscopic agreement with SN 2005hk over these epochs.

###  3.2 Comparisons with intermediate-luminosity SNe Iax

####  3.2.1 Light curves

Figure 4: Angle averaged UBVJHKs and griz band light curves for the
r48_d5.0_Z_R_6000K_0.33Ni, r114_d6.0_Z_R_4000K and r114_d6.0_Z_R_6000K_0.1Ni
models along with the L22 r48_d5.0_Z and r114_d6.0_Z models. The light curves
of SN 2019muj (Barna et al., 2021; Kawabata et al., 2021), an intermediate-
luminosity SN Iax and the faint SN Iax, SN 2020kyg (Srivastav et al., 2022)
are included for comparison. Note, the JHKs magnitudes are from the 2MASS
catalogue (Persson et al. 1998). The Ks filter is a K filter which has been
modified to reduce thermal background for ground based telescopes.

Figure 4 shows band light curves plotted for Model r48_d5.0_Z_R_6000K_0.33Ni
(which adopts a remnant ${}^{56}\mathrm{Ni}$ mass a third of that predicted by
L22, see Table 2 for summary of remnant properties), Model
r114_d6.0_Z_R_4000K, the L22 r48_d5.0_Z model and the intermediate-luminosity
SN Iax, SN 2019muj. Model r48_d5.0_Z_R_6000K_0.33Ni, our model with remnant
contribution included in best agreement with SN 2019muj, produces
significantly better agreement with the light curves of SN 2019muj than the
closest matching L22 model r48_d5.0_Z, matching the peak, rise and
particularly decline of SN 2019muj better in all bands (apart from $z$-band,
where the small number of photometric points for SN 2019muj makes the level of
agreement hard to judge between models). In particular, the remnant
contribution removes the systematic problem of the light curves declining too
rapidly. The improvement is again most dramatic in the red optical and NIR
bands. While the r48_d5.0_Z_R_6000K_0.33Ni model produces best agreement with
the optical band light curves of SN 2019muj, the agreement is still reasonably
good for all NIR bands shown, despite the rise being a bit slow relative to SN
2019muj in the H and Ks-bands. The remnant temperature adopted for Model
r48_d5.0_Z_R_6000K_0.33Ni is consistent with the photospheric temperature of
$\sim$ 5500 K estimated by Maeda & Kawabata (2022) for SN 2019muj at 131 days
after explosion. Overall, including energy injected by radioactive decays in
the remnant significantly improves the agreement of pure deflagration models
with the band light curves of intermediate-luminosity SNe Iax such SN 2019muj.

To match the optical luminosity of SN 2019muj for models including the entire
remnant ${}^{56}\mathrm{Ni}$ mass predicted by L22 we require remnant
temperatures lower than that adopted for Model r48_d5.0_Z_R_6000K_0.33Ni. Of
the models we investigated that adopted the entire L22 ${}^{56}\mathrm{Ni}$
mass, Model r114_d6.0_Z_R_4000K produces best agreement with SN 2019muj.
Comparing the light curves of this model and Model r48_d5.0_Z_R_6000K_0.33Ni
to SN 2019muj we see both models produce good agreement with the BVgr band
light curves of SN 2019muj although Model r48_d5.0_Z_6000K_0.33Ni provides an
overall better match in these bands. Model r48_d5.0_Z_6000K_0.33Ni also
provides a reasonably good match to SN 2019muj in all other bands shown.
However, Model r114_d6.0_Z_R_4000K is too bright at peak in all remaining
bands relative to SN 2019muj. Additionally, all the NIR light curves of Model
r114_d6.0_Z_R_4000K are at least a magnitude too bright compared to SN 2019muj
from peak until the end of the simulation at 100 days. Model r114_d6.0_Z is
the faintest from the L22 sequence meaning it is not possible to reproduce the
correct brightness for any observed SNe Iax fainter than SN 2019muj if we
assume the entire remnant ${}^{56}\mathrm{Ni}$ mass contributes directly to
the observed luminosity of the model. Therefore, to produce good agreement
with both intermediate-luminosity and faint SNe Iax, we must adopt remnant
${}^{56}\mathrm{Ni}$ masses in our models less than those predicted by L22. We
discuss the implications of this finding in more detail in Section 4.

####  3.2.2 Spectra

Figure 5: Absolute flux spectral comparisons between the angle averaged
r48_d5.0_Z_6000K_0.33Ni model, r48_d5.0_Z L22 model and SN 2019muj (Barna et
al., 2021). Also plotted are the r48_d5.0_Z model spectra scaled so their peak
flux matches that of Model r48_d5.0_Z_6000K_0.33Ni. Times are relative to
B-band peak. The spectra of SN 2019muj have been corrected for red shift and
reddening using the values from Barna et al. (2021). The observed spectra
shown here were taken from WISeREP (Yaron & Gal-Yam 2012).

Figure 5 shows absolute flux spectral comparisons between Model
r48_d5.0_Z_6000K_0.33Ni, Model r48_d5.0_Z, Model r48_d5.0_Z scaled to match
the peak of r48_d5.0_Z_6000K_0.33Ni and SN 2019muj. Model
r48_d5.0_Z_6000K_0.33Ni provides a significantly better overall flux match to
the spectra of SN 2019muj at all epochs compared to the r48_d5.0_Z model.
Therefore, following the same approach as Section 3.1.2 we primarily focus on
the r48_d5.0_Z scaled flux model alongside Model r48_d5.0_Z_6000K_0.33Ni for
our comparisons to SN 2019muj so we can evaluate how including the remnant
contribution impacts the spectral evolution of intermediate-luminosity
deflagration models.

The r48_d5.0_Z_6000K_0.33Ni model provides a very good match to the overall
SED and the strengths and velocities of a significant number of the spectral
features of SN 2019muj, across all epochs shown. Relative to the
r48_d5.0_Z_6000K_0.33Ni model, the r48_d5.0_Z scaled model produces a
similarly good match to the spectral features of SN 2019muj except for the
latest epoch shown, where the r48_d5.0_Z spectral features appear too weak for
wavelengths longer than $\sim 6500\,$Å primarily due to the lack of flux for
the model at these red wavelengths. The r48_d5.0_Z scaled model produces a
significantly worse match to the colour evolution of SN 2019muj: while the
model matches the first epoch shown (5 days before B-peak), it becomes too red
at 2 and 11 days after peak. At 19.7 days after peak the model then exhibits
an improved match to the colours of SN 2019muj before showing spectra which
become increasingly too blue for the latest two epochs shown at 27.7 and 50.1
days after peak. Compared to bright models, including the remnant contribution
for intermediate-luminosity models leads to a greater change in spectral
properties and bigger overall improvement in the agreement with observed SNe
Iax. This suggests the energy injected into the ejecta due to the radiation
emitted from the remnant has a greater impact on the ejecta conditions of
intermediate-luminosity deflagration models.

###  3.3 Comparisons with faint SNe Iax

####  3.3.1 Light curves

From Figure 4 we can see band light curve comparisons between Model
r114_d6.0_Z_6000K_0.1Ni (which adopts a remnant ${}^{56}\mathrm{Ni}$ mass a
tenth of that predicted by L22), the r114_d6.0_Z L22 model and the faint SNe
Iax, SN 2020kyg. From fitting the SED of SN 2020kyg, Srivastav et al. (2022)
find a black body temperature which varies from $\sim$ 8000 K at g band peak
to $\sim$ 4000 K at 60 days after g band peak, which is consistent with the
remnant temperature we adopt for our model. L22 predict a remnant
${}^{56}\mathrm{Ni}$ mass $\sim$ 5 times greater than the ejecta
${}^{56}\mathrm{Ni}$ mass for Model r114_d6.0_Z. However, the ejecta component
of Model r114_d6.0_Z is already brighter than SN 2020kyg at peak in most
bands. Therefore, it follows that to provide a reasonable match to the peak
brightness of SN 2020kyg our model with remnant contribution included must
adopt a ${}^{56}\mathrm{Ni}$ mass only a small fraction of that predicted by
L22 (see Section 4 for discussion).

The r114_d6.0_Z_6000K_0.1Ni model matches the decline of SN 2020kyg in all
bands better than the standard r114_d6.0_Z model, which declines much too
rapidly in all bands. In particular the r114_d6.0_Z_6000K_0.1Ni model quite
successfully matches the decline after peak in the optical bands until $\sim$
80 days after explosion (the latest photometric point of SN 2020kyg
available). However, the r114_d6.0_Z_6000K_0.1Ni model is too bright at peak
by up to half a magnitude in the optical bands and a magnitude or more in the
NIR bands, while also declining too slowly after peak in the NIR bands.
Although we expected the r114_d6.0_Z_6000K_0.1Ni model would be too bright
compared to SN 2020kyg, the more significant overproduction of flux at NIR
wavelengths suggests our adopted remnant temperature may be too low.

To test this we carried out another simulation also based on Model r114_d6.0_Z
but with an increased fixed remnant black body temperature of
$8000\,\mathrm{K}$ and a further reduced ${}^{56}\mathrm{Ni}$ mass a twentieth
of that predicted by L22. This model did produce light curves which were
slightly fainter at peak, up to $\sim$ 0.2 magnitudes in optical bands and as
much as $\sim$ 0.4 in NIR bands, leading to closer agreement with SN 2020kyg.
This model also produced NIR band light curves that matched the decline SN
2020kyg more successfully. However, this came at the expense of significantly
reducing the agreement of the model decline in the r, i and z-bands and in
general providing a worse match to the band light curves of SN 2020kyg
compared to the r114_d6.0_Z_6000K_0.1Ni model. We also note that as we move to
faint deflagration models in which we include the remnant contribution, the
models become more sensitive to the choice of remnant temperature. This may be
a result of the ejecta becoming optically thin more rapidly for faint models
meaning the remnant radiation has a larger contribution to the synthetic
observables at earlier times for faint models.

Overall, we have been able to achieve a significant improvement in agreement
with the band light curves of SN 2020kyg, particularly in decline, by
including the remnant radiation in our simulations. However, there is no
obvious way using our approach to produce a simulation based on the standard
L22 pure deflagration models with remnant contribution included that can
produce good agreement with the light curves of SN 2020kyg across all bands
(see Section 4 for further discussion).

####  3.3.2 Spectra

Figure 6: Absolute flux spectral comparisons between the angle averaged
r114_d6.0_Z_6000K_0.1Ni model, r114_d6.0_Z L22 model and SN 2020kyg (Srivastav
et al., 2022). Also plotted are the r114_d6.0_Z spectra scaled so their peak
flux matches that of Model r114_d6.0_Z_6000K_0.1Ni where required, to allow
clearer comparison of spectral features. Times are relative to g-band peak.
The spectra of SN 2020kyg have been corrected for red shift and reddening
using the values from Srivastav et al. (2022).

Figure 6 shows spectroscopic comparisons between Model
r114_d6.0_Z_6000K_0.1Ni, Model r114_d6.0_Z, Model r114_d6.0_Z scaled to match
Model r114_d6.0_Z_6000K_0.1Ni at peak and SN 2020kyg. For the earliest epoch
shown (3.1 days before g peak) both models are spectroscopically very similar
and provide a good match in absolute flux to the spectrum of SN 2020kyg. At
the next epoch shown (3.5 days after peak) the r114_d6.0_Z model still shows a
good match to the absolute flux of SN 2020kyg while the
r114_d6.0_Z_6000K_0.1Ni model becomes too bright. For the last 3 epochs shown
(16.4, 24.4 and 53.8 days after g peak) the r114_d6.0_Z_6000K_0.1Ni model
provides a significantly better match to the absolute flux of SN 2020kyg
compared to the r114_d6.0_Z model which becomes much too faint. Additionally,
while the spectra of the r114_d6.0_Z_6000K_0.1Ni model are also slightly too
blue they provide a significantly better match to the overall colours: the
r114_d6.0_Z model is significantly too blue. Therefore, as was the case for
intermediate-luminosity deflagration models, including the remnant radiation
for faint deflagration models improves agreement with observed SNe Iax because
of a wavelength dependant increase in flux for the r114_d6.0_Z_6000K_0.1Ni
model, producing a colour evolution more similar to SN 2020kyg.

The r114_d6.0_Z_6000K_0.1Ni model is able to reproduce the strength and
location of many of the spectral features of SN 2020kyg for the earlier epochs
shown. However, for the latest two epochs shown, while the model is still
reproducing some of the spectral features, they are often too weak. This is
particularly clear for the last epoch shown at wavelengths redder than $\sim
5500\,$Å where the r114_d6.0_Z_6000K_0.1Ni model spectra struggles to match
the large number of strong sharp features exhibited by the spectrum of SN
2020kyg. This is likely because the spectra of SN 2020kyg at these later
epochs show an increasing number of features due to forbidden transitions
(Srivastav et al., 2022) which can not be treated accurately using the atomic
data set and approximate NLTE treatment we have adopted in our radiative
transfer simulations for this work. Adopting the full NLTE treatment and more
extensive atomic dataset of Shingles et al. (2020) may produce model spectra
which more successfully reproduce these spectral features at later times.

Focusing again on the latest two epochs shown we see interesting differences
in the evolution of the Ca ii NIR triplet. In the spectral comparison at 24.4
days after peak the absorption feature at $\sim 8400\,$Å, which we have
confirmed to be the most blue-ward absorption feature of the Ca ii NIR triplet
in the models, matches the velocity of the feature in SN 2020kyg well.
However, for the spectral comparison at 53.8 days after peak while this
absorption feature has remained at the same velocity for the model it has
shifted significantly to the red for SN 2020kyg. Additionally, comparing the
overall profile of the Ca ii NIR triplet in the latest epoch shown we see that
it is both bluer and broader for the deflagration models compared to SN
2020kyg. This explains the lack of emission from the 8542 Å component of the
Ca ii NIR triplet for the models as it is suppressed by the blue shifted
absorption tail of the 8662 Å component, something which Maeda & Kawabata
(2022) also observed for their spectral models of SN 2019muj. These
comparisons of the Ca ii NIR triplet suggest there is a low velocity, higher
density region of the ejecta in SN 2020kyg, where spectral features are
forming at later epochs, which is not accounted for in our models. Such an
ejecta structure has already been proposed for other observed SNe Iax (see
e.g. Sahu et al. 2008) while Foley et al. (2016) and Shen & Schwab (2017)
suggest such an ejecta structure may be consistent with radioactively driven
post SNe Iax remnant winds. Alternatively, Maeda & Kawabata (2022) have
suggested that a scenario involving a secondary higher density ejecta
component which is ejected from the remnant envelope $\sim$ a month after
explosion may explain this spectral behaviour at later times.

##  4 Discussion and Conclusions

###  4.1 Summary

We have presented a new set of radiative transfer simulations for models based
on the $M_{\mathrm{Ch}}$ CO WD pure deflagration models of L22 in which we
also include energy injection from a luminous remnant, as predicted in the
explosion models and possibly observed for the SNe Iax, SN 2008ha (Foley et
al., 2014), SN 2012Z (McCully et al., 2022), SN 2014dt (Kawabata et al., 2018)
and SN 2019muj (Kawabata et al., 2021). While our models require an assumption
about the SED of photons from the remnant we found no significant differences
in the synthetic observables of the models where we assumed a black body SED
with either constant radius or temperature. Therefore, our models are not
overly sensitive to the specific choice of effective remnant temperature at a
given time. Moreover, our models which produced best agreement with observed
SNe Iax all adopted remnant temperatures consistent with late time temperature
estimates of observed SNe Iax. We note, however, that if we were to extend our
simulations to later times, the remnant temperature predicted for a model with
fixed remnant radius will increasingly diverge from a fixed remnant
temperature making this choice more relevant. Additionally, at later times the
ejecta become significantly more optically thin meaning radiation emitted from
the remnant will be less impacted by interaction with ejecta material and thus
will likely become more sensitive to the remnant temperature treatment
utilised.

Including the remnant ${}^{56}\mathrm{Ni}$ mass predicted for the brightest
pure deflagration model from the L22 sequence increases the peak bolometric
magnitude of the model by $\sim$ half a magnitude. This means the brightest
viewing angles of this model can match the peak bolometric luminosity of the
brightest SNe Iax within their uncertainties, something not previously
possible for the L22 single-spark pure deflagration models. Additionally, the
light curves are in good agreement with the bright SNe Iax, SN 2005hk in peak
magnitude, rise and decline in all bands. This represents a significant
improvement as previous bright deflagration models presented by K13, K14 and
L22 exhibited declines systematically too rapid compared to bright SNe Iax in
the red optical and NIR bands. The inclusion of the remnant also improved
spectroscopic agreement with SN 2005hk, primarily due to the better overall
flux agreement, although for the latest epoch compared the redder spectrum
formed also improved the match with the spectral colours of SN 2005hk.
However, including the remnant contribution causes a more pronounced change in
ejecta conditions for intermediate-luminosity and faint models leading to
bigger improvements in their spectroscopic agreement with observed SNe Iax
relative to standard deflagration models.

Of the models investigated in this work, intermediate-luminosity deflagration
models with remnant contribution included display the largest improvement in
agreement with observed SNe Iax relative to standard deflagration models.
Including the remnant produces model light curves in good agreement with the
intermediate-luminosity SN Iax, SN 2019muj in terms of peak magnitude, rise
and decline in all bands. This again represents a significant improvement
compared to standard deflagration models, which show light curves that decline
systematically too fast after peak in all bands relative to observed SNe Iax.
Additionally, including the remnant contribution leads to significantly
improved spectral agreement with SN 2019muj due to the better match with the
evolution of the SED.

Including the remnant contribution also significantly improves the agreement
with the light curve evolution of the faint SNe Iax, SN 2020kyg. This
improvement is primarily due to the slower light curve decline after peak in
all bands compared to faint standard deflagration models which show declines
that are significantly too fast relative to SN 2020kyg. Additionally, as was
the case for intermediate-luminosity models, including the remnant
contribution leads to significantly better spectral agreement with SN 2020kyg
due to the improved match to the evolution of its SED. The inclusion of the
remnant does, however, make the model light curves too bright at peak relative
to SN 2020kyg in all bands and significantly too bright over the whole
simulation time in NIR bands. This is to be expected as the faintest member of
the L22 model sequence is already slightly brighter than SN 2020kyg at peak.
Therefore, to achieve agreement with the faintest observed SNe Iax for
deflagration models with remnant contribution included we require models with
an ejecta component $\sim$ a magnitude or more fainter than the faintest model
from the L22 sequence. A future investigation of whether it is possible to
produce even fainter pure deflagration models, something L22 suggested can be
achieved for single-spark deflagration models, would therefore be of interest.

###  4.2 Implications for the remnant

While our models which produce good agreement with the bright SNe Iax, SN
2005hk, adopt the full remnant ${}^{56}\mathrm{Ni}$ mass predicted by L22 our
models in best agreement with the intermediate-luminosity SNe Iax, SN 2019muj
and faint SNe Iax, SN 2020kyg, had remnant ${}^{56}\mathrm{Ni}$ masses one
third and one tenth of those predicted by L22 respectively (see Table 2). From
Table 1 we see that moving to fainter deflagration models the remnant
${}^{56}\mathrm{Ni}$ mass predicted increases relative to the ejecta
${}^{56}\mathrm{Ni}$ mass. Adopting the entire remnant ${}^{56}\mathrm{Ni}$
mass predicted by L22 and assuming all energy emitted from the
${}^{56}\mathrm{Ni}$ decay chain in the remnant contributes to the optical
display of the faintest L22 deflagration model (r114_d6.0_Z) produces an event
of comparable brightness to SN 2019muj (see Section 3.2.1). L22 suggest it
should be possible to produce models fainter than r114_d6.0_Z. However,
utilising the entire predicted remnant ${}^{56}\mathrm{Ni}$ mass in our
intermediate-luminosity and faint deflagration models results in a very large
change in the overall properties of the models leading to poor agreement with
observed SNe Iax. This suggests not all the energy injected in the remnant
from radioactive decays can be used to power the light curves of intermediate-
luminosity and faint models. This finding is consistent with what Kromer et
al. (2015) found for their faint hybrid CONe WD pure deflagration model they
compared to the faint SNe Iax, SN 2008ha.

Models of SNe Iax remnants by Shen & Schwab (2017) that predict some of the
radioactive energy released powers a wind in the outer envelope of the remnant
provide a potential explanation for this. If a large enough fraction of the
energy released in the remnant is used to power such a wind, the energy
contributing to the light curve will be significantly reduced. Alternatively,
Maeda & Kawabata (2022) have proposed a mechanism in which energy from
radioactive decays causes hot ${}^{56}\mathrm{Ni}$ rich regions of the remnant
to be ejected on the timescale of approximately a month after explosion, again
a fraction of the energy injected in the remnant is required to power this
process. As we do not take into account either of these mechanisms in our
simulations these scenarios are roughly equivalent to having a reduced remnant
${}^{56}\mathrm{Ni}$ mass for our models (although time variations in the
fraction of deposited energy driving these mechanisms will not be captured).

To estimate the potential impact of remnant winds, we calculated the
approximate kinetic energy required to drive winds in the envelope of the
remnant for the r48_d5.0_Z and r114_d6.0_Z models from the L22 sequence, using
the SNe Iax remnant wind models computed by Shen & Schwab (2017) as a guide.
The Shen & Schwab (2017) models with greatest remnant masses have core and
envelope masses of 0.9 and 0.1 $\mathrm{M}_{\odot}$ respectively, giving
slightly lower but similar total remnant masses to the L22 remnant masses
predicted for the models used in this study (1.16 to 1.38
$\mathrm{M}_{\odot}$). Although we are unable to differentiate the remnant
core and envelope from the L22 simulations we do have approximate yields of
the IGEs and IMEs in the remnant which should primarily be found in the
envelope and make up a sizeable fraction of the total envelope mass. The IGEs
and IMEs predicted for the remnants sum to 0.074 and 0.053
$\mathrm{M}_{\odot}$ for the r48_d5.0_Z and r114_d6.0_Z models respectively.
As some of the envelope will be unburnt CO material, which we can not
determine the mass of, an envelope mass of 0.1 $\mathrm{M}_{\odot}$ seems
reasonable for these models and we utilise this envelope mass as an initial
guess for the mass driven by the radioactive wind in this calculation. The
Shen & Schwab (2017) models have average winds speeds up to $\sim$ 1100 km s-1
which we adopt for our calculation here to obtain an estimated upper limit on
the energy used to drive winds in the remnant envelope.

The total energy deposited by the ${}^{56}\mathrm{Ni}$ decay chain over the
simulation time is $4.85\times 10^{48}$ ergs for Model
r48_d5.0_Z_R_6000K_0.33Ni and $3.83\times 10^{48}$ ergs for Model
r114_d6.0_Z_R_6000K_0.1Ni. Models r48_d5.0_Z_R_6000K_0.33Ni and
r114_d6.0_Z_R_6000K_0.1Ni require that two thirds and nine tenths of this
energy contributes to processes other than powering the model light curves,
respectively. From the calculation discussed above we estimate the total
kinetic energy required to drive the remnant wind over the simulation time is
$1.20\times 10^{48}$ ergs. For both models, this accounts for just over a
third of the discrepancy between the energy we inject in the model remnants
and the energy we predict would be deposited in the remnants from the L22
remnant ${}^{56}\mathrm{Ni}$ mass estimates. For envelope masses higher than
the 0.1 $\mathrm{M}_{\odot}$ we estimated for these calculations an even
greater proportion of the energy injected from radioactive decays in the
remnant could be driving winds. It is therefore plausible that energy used to
drive such winds can explain why only a fraction of the total energy injected
in the remnants from radioactive decays is required to power our
r48_d5.0_Z_6000K_0.33Ni and r114_d6.0_Z_6000K_0.1Ni model light curves.

Shen & Schwab (2017) also suggest the hot relatively high density regime of
the remnant may be so highly ionised that ${}^{56}\mathrm{Ni}$ and
${}^{56}\mathrm{Co}$ have radioactive decay rates lower than those we
typically use in our simulations, which are appropriate for atomic or low
ionisation species. This reduction in decay rate is expected to be relevant in
highly ionised regimes with densities $\lesssim$ $10^{5}$ g cm-3 (see Shen &
Schwab 2017 for details). The remnants predicted by the pure deflagration
simulations of L22 have maximum central densities of $\sim$ $10^{5}$ g cm-3.
Therefore, it is plausible that regions of the remnant may inhabit this regime
of reduced decay rates providing another possible explanation for our
requirement to adopt lower remnant ${}^{56}\mathrm{Ni}$ masses than those
predicted by L22.

Maeda & Kawabata (2022) alternatively suggest that diffusion timescales in the
remnant are long enough that they lead to significant delays in the radiation
contributing to the optical display of SNe Iax. In their proposed scenario
${}^{56}\mathrm{Ni}$ rich material is ejected from the remnant around a month
after explosion. They estimate diffusion in this material will become
efficient by around 20 days since explosion allowing radiation originating
there to begin to contribute to the optical display. However, they suggest it
could be $\sim$ a year before radiation produced from the decay of
${}^{56}\mathrm{Ni}$ and ${}^{56}\mathrm{Co}$ in the remaining bound material
can diffuse efficiently and thus contribute to the optical display of the SNe
Iax. As we assume no significant diffusion time in the remnant in our models,
this scenario could also explain the reduced remnant ${}^{56}\mathrm{Ni}$
masses we must adopt for intermediate luminosity and faint models.

As discussed above, adopting reduced ${}^{56}\mathrm{Ni}$ masses in our
simulations can provide a rough approximation of the behaviour expected in our
models at early times if delayed radioactive decays (Shen & Schwab, 2017) or
significant diffusion times in the remnant (Maeda & Kawabata, 2022) are
considered. However, relative to these cases, reducing the remnant
${}^{56}\mathrm{Ni}$ mass will underestimate the total radiation emitted by
the remnant over time. Additionally, the contribution of the remnant to the
optical display at later times will be significantly underestimated.
Determining how these processes would influence the synthetic observables
predicted by our simulations requires detailed modelling of the remnant
structure and evolution, which is outside the scope of this study. However, we
speculate that if we were to account for delayed radioactive decays or
significant diffusion times in the remnant in our simulations, the remnant
would likely have a reduced impact on the model light curves around peak but a
greater influence on the light curve decline at later times relative to the
simulations presented here.

We finally note that it is also possible that the L22 simulations overestimate
the remnant ${}^{56}\mathrm{Ni}$ masses. For fainter deflagration models, the
mass of burnt material relative to the total WD mass is much smaller, meaning
the yields of the burnt material are more uncertain for these models than for
brighter more energetic models which burn more material. However, these
uncertainties would have to account for very large reductions in the remnant
${}^{56}\mathrm{Ni}$ masses predicted by L22 for the models (up to a factor of
10) and it is therefore highly unlikely these uncertainties are relevant here.

###  4.3 Future work

Including the energy injected from a luminous remnant in our radiative
transfer simulations leads to significantly improved agreement with the
spectra and light curves of observed SNe Iax, strengthening the case of pure
deflagration models for SNe Iax. However, outstanding questions remain.
Firstly, it is unclear how the assumptions of the radiative transfer impact
our results. In this work, we have adopted the NLTE approximation described by
Kromer & Sim (2009). However, including a full NLTE treatment of the plasma
conditions has been shown to have important effects on the synthetic
observables produced by radiative transfer simulations of SNe Ia, especially
at later times (Blondin et al., 2013; Dessart et al., 2014; Shingles et al.,
2020; Shen et al., 2021; Shingles et al., 2022; Collins et al., 2023). In
future work we will utilise the full NLTE treatment of Shingles et al. (2020)
to investigate the importance of NLTE effects. In particular, adopting the
full NLTE treatment will enable us to extend our simulations to the late
phases where the remnant contribution becomes more prominent as the ejecta
become increasingly optically thin. This will allow us to investigate if the
remnant is required to explain the peculiar late time spectra of observed SNe
Iax which never become truly nebular, as suggested by Foley et al. (2016).

Important questions also remain around the nature of the remnant. Hydrodynamic
explosion simulations of pure deflagrations following the remnants more
closely would provide detailed remnant structures and compositions which would
improve our understanding of the remnant structure providing insight into its
likely evolution and emission. This is highly relevant to determining if the
energy injection we utilise in this study is reasonable, particularly for our
intermediate-luminosity and faint models that require remnant
${}^{56}\mathrm{Ni}$ masses lower than those predicted by L22. As such, it
would also be useful to further investigate the possibility of radioactively
driven winds and delayed radioactive decays as suggested by Shen & Schwab
(2017) and the second ejecta component proposed to be ejected from the remnant
envelope by Maeda & Kawabata (2022), in particular focusing on how introducing
such mechanisms impacts the overall properties of pure deflagration models and
their agreement with observed SNe Iax.

## Acknowledgements

FPC and SAS, acknowledge funding from STFC grant ST/X00094X/1. This work used
the DiRAC Data Intensive service (CSD3 [*]) at the University of Cambridge,
managed by the University of Cambridge University Information Services on
behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The DiRAC component
of CSD3 at Cambridge was funded by BEIS, UKRI and STFC capital funding and
STFC operations grants. DiRAC is part of the UKRI Digital Research
Infrastructure. The authors gratefully acknowledge the Gauss Centre for
Supercomputing e.V. (www.gauss-centre.eu) for funding this project by
providing computing time on the GCS Supercomputer JUWELS at Jülich
Supercomputing Centre (JSC). CEC acknowledges funding by the European Union
(ERC, HEAVYMETAL, 101071865). Views and opinions expressed are however those
of the author(s) only and do not necessarily reflect those of the European
Union or the European Research Council. Neither the European Union nor the
granting authority can be held responsible for them. LJS acknowledges support
by the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (ERC Advanced Grant KILONOVA No. 885281). LJS
acknowledges support by Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) - Project-ID 279384907 - SFB 1245 and MA 4248/3-1. This work was
supported in part by the European Union (ChETEC-INFRA, project no. 101008324),
NSF/IReNA and Tschira Foundation. NumPy and SciPy (Oliphant, 2007), Matplotlib
(Hunter, 2007) and artistools222https://github.com/artis-mcrt/artistools/
(Shingles et al., 2023) were used for data processing and plotting.

## Data Availability

The light curves and spectra presented here will be made available online.

## References

  * Barna et al. (2021) Barna B., et al., 2021, MNRAS, 501, 1078
  * Blondin et al. (2013) Blondin S., Dessart L., Hillier D. J., Khokhlov A. M., 2013, MNRAS, 429, 2127
  * Branch et al. (2004) Branch D., Baron E., Thomas R. C., Kasen D., Li W., Filippenko A. V., 2004, PASP, 116, 903
  * Bulla et al. (2015) Bulla M., Sim S. A., Kromer M., 2015, MNRAS, 450, 967
  * Collins et al. (2022) Collins C. E., Bauswein A., Sim S. A., Vijayan V., Martínez-Pinedo G., Just O., Shingles L. J., Kromer M., 2022, arXiv e-prints, p. arXiv:2209.05246
  * Collins et al. (2023) Collins C. E., Sim S. A., Shingles L. J., Gronow S., Röpke F. K., Pakmor R., Seitenzahl I. R., Kromer M., 2023, MNRAS, 524, 4447
  * Dessart et al. (2014) Dessart L., Hillier D. J., Blondin S., Khokhlov A., 2014, MNRAS, 441, 3249
  * Fernández & Metzger (2013) Fernández R., Metzger B. D., 2013, The Astrophysical Journal, 763, 108
  * Fink et al. (2014) Fink M., et al., 2014, MNRAS, 438, 1762
  * Foley et al. (2009) Foley R. J., et al., 2009, AJ, 138, 376
  * Foley et al. (2013) Foley R. J., et al., 2013, ApJ, 767, 57
  * Foley et al. (2014) Foley R. J., McCully C., Jha S. W., Bildsten L., Fong W.-f., Narayan G., Rest A., Stritzinger M. D., 2014, ApJ, 792, 29
  * Foley et al. (2016) Foley R. J., Jha S. W., Pan Y.-C., Zheng W. K., Bildsten L., Filippenko A. V., Kasen D., 2016, MNRAS, 461, 433
  * Gall et al. (2012) Gall E. E. E., Taubenberger S., Kromer M., Sim S. A., Benetti S., Blanc G., Elias-Rosa N., Hillebrandt W., 2012, MNRAS, 427, 994
  * Graham et al. (2017) Graham M. L., et al., 2017, MNRAS, 472, 3437
  * Hunter (2007) Hunter J. D., 2007, Computing in Science and Engineering, 9, 90
  * Jerkstrand (2017) Jerkstrand A., 2017, in Alsabti A. W., Murdin P., eds, , Handbook of Supernovae. p. 795, doi:10.1007/978-3-319-21846-5_29
  * Jha et al. (2006) Jha S., Branch D., Chornock R., Foley R. J., Li W., Swift B. J., Casebeer D., Filippenko A. V., 2006, AJ, 132, 189
  * Jordan et al. (2012) Jordan IV G. C., Perets H. B., Fisher R. T., van Rossum D. R., 2012, ApJ, 761, L23
  * Kashyap et al. (2018) Kashyap R., Haque T., Lorén-Aguilar P., García-Berro E., Fisher R., 2018, The Astrophysical Journal, 869, 140
  * Kawabata et al. (2018) Kawabata M., et al., 2018, PASJ, 70, 111
  * Kawabata et al. (2021) Kawabata M., et al., 2021, PASJ, 73, 1295
  * Kromer & Sim (2009) Kromer M., Sim S. A., 2009, MNRAS, 398, 1809
  * Kromer et al. (2013) Kromer M., et al., 2013, MNRAS, 429, 2287
  * Kromer et al. (2015) Kromer M., et al., 2015, MNRAS, 450, 3045
  * Kuhlen et al. (2006) Kuhlen M., Woosley S. E., Glatzmaier G. A., 2006, ApJ, 640, 407
  * Kurucz (2006) Kurucz R. L., 2006, in Stee P., ed., EAS Publications Series Vol. 18, Radiative Transfer and Applications to Very Large Telescopes. EDP Sciences, pp 129–155, http://adsabs.harvard.edu/abs/2006EAS....18..129K
  * Kurucz & Bell (1995) Kurucz R., Bell B., 1995, Atomic Line Data, Kurucz CD-ROM No. 23., Cambridge, Mass.: Smithsonian Astrophysical Observatory., http://kurucz.harvard.edu/cdroms.html
  * Lach et al. (2022) Lach F., Callan F. P., Bubeck D., Röpke F. K., Sim S. A., Schrauth M., Ohlmann S. T., Kromer M., 2022, A&A, 658, A179
  * Leung & Nomoto (2020) Leung S.-C., Nomoto K., 2020, The Astrophysical Journal, 900, 54
  * Li et al. (2003) Li W., et al., 2003, PASP, 115, 453
  * Long et al. (2014) Long M., et al., 2014, The Astrophysical Journal, 789, 103
  * Lucy (2002) Lucy L. B., 2002, A&A, 384, 725
  * Lucy (2003) Lucy L. B., 2003, A&A, 403, 261
  * Lucy (2005) Lucy L. B., 2005, A&A, 429, 19
  * Maeda & Kawabata (2022) Maeda K., Kawabata M., 2022, ApJ, 941, 15
  * Magee et al. (2016) Magee M. R., et al., 2016, A&A, 589, A89
  * Magee et al. (2022) Magee M. R., Gillanders J. H., Maguire K., Sim S. A., Callan F. P., 2022, MNRAS, 509, 3580
  * McClelland et al. (2010) McClelland C. M., et al., 2010, ApJ, 720, 704
  * McCully et al. (2014) McCully C., et al., 2014, ApJ, 786, 134
  * McCully et al. (2022) McCully C., et al., 2022, ApJ, 925, 138
  * Moriya et al. (2010) Moriya T., Tominaga N., Tanaka M., Nomoto K., Sauer D. N., Mazzali P. A., Maeda K., Suzuki T., 2010, ApJ, 719, 1445
  * Nonaka et al. (2012) Nonaka A., Aspden A. J., Zingale M., Almgren A. S., Bell J. B., Woosley S. E., 2012, ApJ, 745, 73
  * Oliphant (2007) Oliphant T. E., 2007, Computing in Science & Engineering, 9, 10
  * Persson et al. (1998) Persson S. E., Murphy D. C., Krzeminski W., Roth M., Rieke M. J., 1998, AJ, 116, 2475
  * Phillips (1993) Phillips M. M., 1993, ApJ, 413, L105
  * Phillips et al. (2007) Phillips M. M., et al., 2007, PASP, 119, 360
  * Reinecke et al. (1999) Reinecke M., Hillebrandt W., Niemeyer J. C., 1999, A&A, 347, 739
  * Röpke (2005) Röpke F. K., 2005, A&A, 432, 969
  * Sahu et al. (2008) Sahu D. K., et al., 2008, ApJ, 680, 580
  * Shen & Schwab (2017) Shen K. J., Schwab J., 2017, ApJ, 834, 180
  * Shen et al. (2021) Shen K. J., Blondin S., Kasen D., Dessart L., Townsley D. M., Boos S., Hillier D. J., 2021, ApJ, 909, L18
  * Shingles et al. (2020) Shingles L. J., et al., 2020, MNRAS, 492, 2029
  * Shingles et al. (2022) Shingles L. J., Flörs A., Sim S. A., Collins C. E., Röpke F. K., Seitenzahl I. R., Shen K. J., 2022, MNRAS, 512, 6150
  * Shingles et al. (2023) Shingles L. J., Collins C. E., Holas A., Callan F. P., Sim S. A., 2023, artistools, doi:10.5281/zenodo.7942849, https://github.com/artis-mcrt/artistools
  * Sim (2007) Sim S. A., 2007, MNRAS, 375, 154
  * Srivastav et al. (2022) Srivastav S., et al., 2022, MNRAS, 511, 2708
  * Stritzinger et al. (2014) Stritzinger M. D., et al., 2014, A&A, 561, A146
  * Stritzinger et al. (2015) Stritzinger M. D., et al., 2015, A&A, 573, A2
  * Taubenberger (2017) Taubenberger S., 2017, Handbook of Supernovae, pp 317–373
  * Taubenberger et al. (2015) Taubenberger S., et al., 2015, MNRAS, 448, L48
  * Valenti et al. (2009) Valenti S., et al., 2009, Nature, 459, 674
  * Yamanaka et al. (2015) Yamanaka M., et al., 2015, ApJ, 806, 191
  * Yaron & Gal-Yam (2012) Yaron O., Gal-Yam A., 2012, PASP, 124, 668
  * Zingale et al. (2009) Zingale M., Almgren A. S., Bell J. B., Nonaka A., Woosley S. E., 2009, ApJ, 704, 196
  * Zingale et al. (2011) Zingale M., Nonaka A., Almgren A. S., Bell J. B., Malone C. M., Woosley S. E., 2011, ApJ, 740, 8

##  Appendix A Light Curve parameter tables

Model | $t_{rise}^{bol}$ | $M_{peak}^{bol}$ | ${\Delta}m_{15}^{bol}$ | $t_{rise}^{u}$ | $M_{peak}^{u}$ | ${\Delta}m_{15}^{u}$ | $t_{rise}^{g}$ | $M_{peak}^{g}$ | ${\Delta}m_{15}^{g}$
---|---|---|---|---|---|---|---|---|---
r10_d4.0_Z | $11.52^{12.00}_{10.93}$ | $-17.35^{-17.65}_{-17.23}$ | $1.09^{1.26}_{0.97}$ | $9.30^{10.20}_{8.05}$ | $-16.70^{-17.24}_{-16.41}$ | $3.48^{3.95}_{3.01}$ | $11.08^{11.69}_{10.47}$ | $-17.49^{-17.68}_{-17.30}$ | $1.71^{1.83}_{1.60}$
r10_d4.0_Z_R_2000K | $14.11^{14.76}_{13.69}$ | $-17.69^{-18.03}_{-17.52}$ | $0.77^{1.06}_{0.62}$ | $12.99^{13.80}_{12.32}$ | $-17.09^{-17.71}_{-16.72}$ | $3.20^{3.71}_{2.81}$ | $14.28^{14.93}_{13.50}$ | $-17.93^{-18.34}_{-17.65}$ | $1.58^{1.81}_{1.42}$
r10_d4.0_Z_R_8000K | $14.08^{14.59}_{13.60}$ | $-17.70^{-18.04}_{-17.51}$ | $0.84^{1.07}_{0.66}$ | $12.99^{13.74}_{12.00}$ | $-17.10^{-17.72}_{-16.73}$ | $3.13^{3.53}_{2.84}$ | $14.21^{14.66}_{13.57}$ | $-17.94^{-18.34}_{-17.65}$ | $1.51^{1.65}_{1.39}$
r10_d4.0_Z_R_15000K | $14.08^{14.59}_{13.63}$ | $-17.70^{-18.04}_{-17.52}$ | $0.84^{1.07}_{0.71}$ | $12.99^{13.57}_{11.94}$ | $-17.10^{-17.72}_{-16.73}$ | $3.09^{3.57}_{2.77}$ | $14.25^{14.69}_{13.74}$ | $-17.94^{-18.35}_{-17.68}$ | $1.47^{1.64}_{1.34}$
r10_d4.0_Z_R_const_rad | $14.08^{14.85}_{13.48}$ | $-17.70^{-18.05}_{-17.52}$ | $0.83^{1.07}_{0.70}$ | $12.99^{13.57}_{12.26}$ | $-17.10^{-17.74}_{-16.70}$ | $3.13^{3.57}_{2.78}$ | $14.25^{14.69}_{13.70}$ | $-17.93^{-18.34}_{-17.66}$ | $1.51^{1.65}_{1.39}$
r48_d5.0_Z | $7.38^{8.29}_{6.70}$ | $-15.81^{-16.22}_{-15.59}$ | $1.34^{1.62}_{1.18}$ | $5.53^{6.21}_{4.49}$ | $-15.29^{-15.98}_{-14.81}$ | $4.27^{4.77}_{4.04}$ | $7.20^{7.75}_{6.56}$ | $-15.95^{-16.39}_{-15.69}$ | $1.96^{2.15}_{1.80}$
r48_d5.0_Z_R_6000K_0.33Ni | $9.07^{9.46}_{8.63}$ | $-16.33^{-16.66}_{-16.09}$ | $1.07^{1.38}_{0.87}$ | $8.42^{8.89}_{8.03}$ | $-15.92^{-16.47}_{-15.52}$ | $4.01^{4.61}_{3.68}$ | $9.24^{9.52}_{8.87}$ | $-16.58^{-16.88}_{-16.32}$ | $1.79^{1.97}_{1.65}$
r114_d6.0_Z | $4.36^{4.72}_{4.04}$ | $-14.91^{-14.99}_{-14.79}$ | $2.43^{2.51}_{2.36}$ | $2.93^{3.64}_{2.50}$ | $-14.62^{-14.78}_{-14.37}$ | $4.85^{5.35}_{4.45}$ | $4.62^{4.81}_{4.33}$ | $-15.11^{-15.20}_{-15.01}$ | $2.30^{2.42}_{2.18}$
r114_d6.0_Z_R_4000K | $6.94^{7.59}_{4.78}$ | $-16.74^{-16.89}_{-16.40}$ | $1.14^{1.23}_{0.88}$ | $6.08^{6.75}_{4.93}$ | $-16.78^{-16.92}_{-16.44}$ | $4.64^{5.07}_{4.37}$ | $7.64^{8.49}_{6.35}$ | $-16.52^{-16.61}_{-16.34}$ | $1.81^{1.95}_{1.67}$
r114_d6.0_Z_R_6000K_0.1Ni | $5.09^{5.36}_{4.88}$ | $-15.27^{-15.35}_{-15.07}$ | $1.68^{1.78}_{1.59}$ | $4.12^{4.36}_{3.80}$ | $-15.02^{-15.16}_{-14.65}$ | $5.02^{5.39}_{4.72}$ | $5.61^{5.64}_{5.29}$ | $-15.53^{-15.61}_{-15.34}$ | $2.24^{2.32}_{2.11}$
Model | $t_{rise}^{r}$ | $M_{peak}^{r}$ | ${\Delta}m_{15}^{r}$ | $t_{rise}^{i}$ | $M_{peak}^{i}$ | ${\Delta}m_{15}^{i}$ | $t_{rise}^{z}$ | $M_{peak}^{z}$ | ${\Delta}m_{15}^{z}$
r10_d4.0_Z | $14.05^{14.75}_{13.42}$ | $-17.86^{-17.97}_{-17.80}$ | $1.25^{1.41}_{1.13}$ | $16.24^{18.07}_{13.82}$ | $-17.47^{-17.53}_{-17.43}$ | $1.10^{1.28}_{0.92}$ | $18.03^{18.93}_{16.86}$ | $-17.71^{-17.79}_{-17.65}$ | $1.20^{1.33}_{1.09}$
r10_d4.0_Z_R_2000K | $16.51^{17.68}_{15.42}$ | $-18.13^{-18.22}_{-18.07}$ | $0.94^{1.07}_{0.82}$ | $21.55^{23.50}_{15.06}$ | $-17.57^{-17.63}_{-17.51}$ | $0.86^{1.02}_{0.36}$ | $22.72^{23.85}_{21.51}$ | $-17.91^{-18.02}_{-17.85}$ | $1.00^{1.12}_{0.87}$
r10_d4.0_Z_R_8000K | $16.78^{18.19}_{15.42}$ | $-18.14^{-18.23}_{-18.07}$ | $0.79^{0.89}_{0.70}$ | $21.97^{23.58}_{20.61}$ | $-17.62^{-17.68}_{-17.58}$ | $0.64^{0.78}_{0.52}$ | $22.91^{23.93}_{21.43}$ | $-17.93^{-17.99}_{-17.87}$ | $0.73^{0.87}_{0.58}$
r10_d4.0_Z_R_15000K | $16.55^{17.95}_{15.61}$ | $-18.13^{-18.23}_{-18.07}$ | $0.72^{0.83}_{0.62}$ | $21.78^{23.97}_{16.63}$ | $-17.58^{-17.65}_{-17.52}$ | $0.62^{0.79}_{0.39}$ | $22.95^{23.85}_{21.74}$ | $-17.91^{-17.98}_{-17.86}$ | $0.69^{0.81}_{0.59}$
r10_d4.0_Z_R_const_rad | $16.70^{17.68}_{15.61}$ | $-18.14^{-18.24}_{-18.08}$ | $0.77^{0.90}_{0.68}$ | $21.97^{23.54}_{20.33}$ | $-17.61^{-17.76}_{-17.56}$ | $0.63^{0.79}_{0.51}$ | $22.87^{23.93}_{21.55}$ | $-17.93^{-18.03}_{-17.86}$ | $0.72^{0.90}_{0.58}$
r48_d5.0_Z | $10.16^{10.95}_{9.11}$ | $-16.24^{-16.41}_{-16.10}$ | $1.50^{1.67}_{1.37}$ | $11.83^{13.16}_{10.43}$ | $-15.97^{-16.09}_{-15.87}$ | $1.38^{1.49}_{1.23}$ | $12.17^{12.95}_{11.56}$ | $-16.12^{-16.29}_{-16.01}$ | $1.24^{1.35}_{1.09}$
r48_d5.0_Z_R_6000K_0.33Ni | $12.85^{14.59}_{9.82}$ | $-16.58^{-16.72}_{-16.47}$ | $1.05^{1.22}_{0.86}$ | $15.64^{16.80}_{14.28}$ | $-16.20^{-16.30}_{-16.14}$ | $0.79^{0.90}_{0.68}$ | $15.95^{16.83}_{15.00}$ | $-16.42^{-16.54}_{-16.36}$ | $0.83^{0.97}_{0.65}$
r114_d6.0_Z | $6.15^{6.39}_{5.64}$ | $-15.24^{-15.30}_{-15.17}$ | $2.73^{2.83}_{2.55}$ | $6.92^{7.60}_{6.11}$ | $-14.93^{-15.00}_{-14.84}$ | $2.78^{2.96}_{2.59}$ | $7.47^{7.75}_{7.07}$ | $-15.09^{-15.15}_{-14.96}$ | $2.00^{2.14}_{1.86}$
r114_d6.0_Z_R_4000K | $8.26^{8.70}_{7.24}$ | $-16.41^{-16.56}_{-16.14}$ | $0.99^{1.12}_{0.84}$ | $9.75^{11.52}_{8.53}$ | $-16.37^{-16.61}_{-15.84}$ | $0.65^{0.82}_{0.44}$ | $10.78^{14.15}_{8.67}$ | $-16.58^{-16.82}_{-15.99}$ | $0.63^{0.90}_{0.31}$
r114_d6.0_Z_R_6000K_0.1Ni | $7.23^{7.66}_{6.79}$ | $-15.52^{-15.59}_{-15.40}$ | $1.66^{1.89}_{1.52}$ | $8.88^{9.64}_{8.25}$ | $-15.13^{-15.22}_{-14.99}$ | $1.37^{1.50}_{1.27}$ | $9.41^{10.17}_{8.91}$ | $-15.36^{-15.45}_{-15.15}$ | $1.35^{1.48}_{1.26}$
Table 3: Bolometric and ugriz Sloan band light curve properties for our new models and the L22 models we compare to in this work. For each quantity the value determined from the angle-averaged light curve is quoted along with the range obtained from the model line-of-site dependent light curves (displayed as a ± range on each angle averaged value). Model | $t_{rise}^{U}$ | $M_{peak}^{U}$ | ${\Delta}m_{15}^{U}$ | $t_{rise}^{B}$ | $M_{peak}^{B}$ | ${\Delta}m_{15}^{B}$ | $t_{rise}^{V}$ | $M_{peak}^{V}$ | ${\Delta}m_{15}^{V}$ | $t_{rise}^{R}$ | $M_{peak}^{R}$ | ${\Delta}m_{15}^{R}$
---|---|---|---|---|---|---|---|---|---|---|---|---
r10_d4.0_Z | $9.42^{10.14}_{8.34}$ | $-17.57^{-18.09}_{-17.30}$ | $3.38^{3.71}_{3.00}$ | $10.40^{11.15}_{9.65}$ | $-17.28^{-17.70}_{-17.07}$ | $2.07^{2.21}_{1.96}$ | $12.85^{13.46}_{12.18}$ | $-17.79^{-18.00}_{-17.68}$ | $1.34^{1.45}_{1.24}$ | $14.09^{14.67}_{13.46}$ | $-17.94^{-18.04}_{-17.89}$ | $1.18^{1.34}_{1.08}$
r10_d4.0_Z_R_2000K | $13.08^{13.86}_{12.44}$ | $-17.98^{-18.57}_{-17.61}$ | $3.12^{3.51}_{2.82}$ | $13.77^{14.55}_{12.95}$ | $-17.72^{-18.20}_{-17.38}$ | $1.93^{2.16}_{1.73}$ | $15.54^{16.08}_{14.87}$ | $-18.20^{-18.41}_{-18.07}$ | $1.18^{1.31}_{1.04}$ | $16.51^{17.64}_{15.38}$ | $-18.18^{-18.26}_{-18.13}$ | $0.87^{0.98}_{0.76}$
r10_d4.0_Z_R_8000K | $13.08^{13.69}_{12.06}$ | $-17.99^{-18.59}_{-17.61}$ | $3.04^{3.36}_{2.75}$ | $13.70^{14.28}_{13.06}$ | $-17.73^{-18.20}_{-17.39}$ | $1.85^{2.00}_{1.68}$ | $15.57^{16.04}_{15.06}$ | $-18.20^{-18.42}_{-18.05}$ | $1.06^{1.17}_{0.94}$ | $16.86^{18.34}_{15.38}$ | $-18.19^{-18.27}_{-18.13}$ | $0.72^{0.83}_{0.64}$
r10_d4.0_Z_R_15000K | $13.08^{13.63}_{12.06}$ | $-17.98^{-18.58}_{-17.62}$ | $3.00^{3.38}_{2.73}$ | $13.74^{14.25}_{12.99}$ | $-17.72^{-18.20}_{-17.39}$ | $1.81^{1.97}_{1.65}$ | $15.47^{16.31}_{14.95}$ | $-18.20^{-18.43}_{-18.08}$ | $0.99^{1.13}_{0.89}$ | $16.51^{18.07}_{15.49}$ | $-18.18^{-18.27}_{-18.13}$ | $0.66^{0.76}_{0.56}$
r10_d4.0_Z_R_const_rad | $13.11^{13.66}_{12.41}$ | $-17.99^{-18.60}_{-17.58}$ | $3.05^{3.30}_{2.76}$ | $13.74^{14.28}_{13.09}$ | $-17.73^{-18.21}_{-17.39}$ | $1.85^{2.04}_{1.71}$ | $15.54^{16.04}_{14.91}$ | $-18.20^{-18.43}_{-18.07}$ | $1.05^{1.17}_{0.87}$ | $16.74^{18.03}_{15.49}$ | $-18.19^{-18.28}_{-18.14}$ | $0.70^{0.82}_{0.62}$
r48_d5.0_Z | $5.63^{6.28}_{4.98}$ | $-16.14^{-16.80}_{-15.68}$ | $4.17^{4.59}_{3.96}$ | $6.69^{7.24}_{6.11}$ | $-15.76^{-16.27}_{-15.45}$ | $2.39^{2.59}_{2.21}$ | $8.46^{9.28}_{7.51}$ | $-16.17^{-16.44}_{-16.01}$ | $1.55^{1.68}_{1.43}$ | $10.43^{11.18}_{9.24}$ | $-16.34^{-16.51}_{-16.21}$ | $1.46^{1.60}_{1.33}$
r48_d5.0_Z_R_6000K_0.33Ni | $8.47^{8.84}_{8.13}$ | $-16.79^{-17.30}_{-16.40}$ | $3.89^{4.31}_{3.58}$ | $8.94^{9.35}_{8.60}$ | $-16.42^{-16.76}_{-16.12}$ | $2.22^{2.39}_{2.05}$ | $9.86^{10.67}_{9.31}$ | $-16.68^{-16.87}_{-16.52}$ | $1.22^{1.37}_{1.11}$ | $13.43^{14.83}_{9.92}$ | $-16.66^{-16.79}_{-16.56}$ | $0.99^{1.13}_{0.82}$
r114_d6.0_Z | $3.12^{3.61}_{2.72}$ | $-15.45^{-15.58}_{-15.21}$ | $4.74^{5.13}_{4.38}$ | $4.26^{4.47}_{4.07}$ | $-14.98^{-15.07}_{-14.84}$ | $2.81^{2.94}_{2.69}$ | $5.45^{5.86}_{5.11}$ | $-15.22^{-15.28}_{-15.14}$ | $2.16^{2.26}_{2.06}$ | $6.15^{6.42}_{5.61}$ | $-15.34^{-15.39}_{-15.27}$ | $2.71^{2.80}_{2.55}$
r114_d6.0_Z_R_4000K | $6.08^{6.73}_{5.09}$ | $-17.57^{-17.71}_{-17.24}$ | $4.46^{4.88}_{4.21}$ | $7.24^{7.95}_{5.50}$ | $-16.51^{-16.62}_{-16.33}$ | $2.37^{2.55}_{2.19}$ | $7.71^{8.43}_{6.90}$ | $-16.52^{-16.62}_{-16.35}$ | $1.32^{1.49}_{1.17}$ | $8.56^{9.04}_{7.44}$ | $-16.55^{-16.72}_{-16.24}$ | $0.90^{1.01}_{0.75}$
r114_d6.0_Z_R_6000K_0.1Ni | $4.26^{4.50}_{3.93}$ | $-15.86^{-15.99}_{-15.50}$ | $4.91^{5.25}_{4.62}$ | $5.23^{5.45}_{4.99}$ | $-15.40^{-15.49}_{-15.16}$ | $2.79^{2.91}_{2.65}$ | $6.45^{6.70}_{6.23}$ | $-15.58^{-15.67}_{-15.45}$ | $1.78^{1.90}_{1.67}$ | $7.32^{7.75}_{6.88}$ | $-15.60^{-15.65}_{-15.48}$ | $1.58^{1.78}_{1.47}$
Model | $t_{rise}^{I}$ | $M_{peak}^{I}$ | ${\Delta}m_{15}^{I}$ | $t_{rise}^{J}$ | $M_{peak}^{J}$ | ${\Delta}m_{15}^{J}$ | $t_{rise}^{H}$ | $M_{peak}^{H}$ | ${\Delta}m_{15}^{H}$ | $t_{rise}^{K}$ | $M_{peak}^{K}$ | ${\Delta}m_{15}^{K}$
r10_d4.0_Z | $16.86^{18.15}_{15.34}$ | $-18.04^{-18.11}_{-17.99}$ | $1.23^{1.37}_{1.10}$ | $11.49$ | $-17.14$ | $0.68$ | $21.22$ | $-17.58$ | $1.10$ | $21.00$ | $-17.45$ | $1.25$
r10_d4.0_Z_R_2000K | $21.74^{23.30}_{20.57}$ | $-18.19^{-18.27}_{-18.12}$ | $0.98^{1.11}_{0.89}$ | - | - | - | - | - | - | - | - | -
r10_d4.0_Z_R_8000K | $21.97^{23.26}_{20.76}$ | $-18.22^{-18.29}_{-18.17}$ | $0.74^{0.88}_{0.64}$ | $10.96$ | $-17.21$ | $0.23$ | $26.77$ | $-17.78$ | $0.54$ | $27.21$ | $-17.58$ | $0.52$
r10_d4.0_Z_R_15000K | $21.86^{23.30}_{20.22}$ | $-18.20^{-18.28}_{-18.14}$ | $0.70^{0.87}_{0.62}$ | $11.05$ | $-17.20$ | $0.40$ | $26.11$ | $-17.69$ | $0.59$ | $26.07$ | $-17.48$ | $0.65$
r10_d4.0_Z_R_const_rad | $21.97^{23.73}_{20.73}$ | $-18.22^{-18.30}_{-18.17}$ | $0.73^{0.87}_{0.61}$ | $10.96$ | $-17.20$ | $0.23$ | $26.69$ | $-17.77$ | $0.51$ | $27.70$ | $-17.57$ | $0.47$
r48_d5.0_Z | $11.80^{12.51}_{11.12}$ | $-16.51^{-16.65}_{-16.40}$ | $1.41^{1.52}_{1.30}$ | $10.30$ | $-15.68$ | $0.93$ | $13.96$ | $-16.00$ | $1.21$ | $14.09$ | $-15.91$ | $1.59$
r48_d5.0_Z_R_6000K_0.33Ni | $15.51^{16.39}_{14.72}$ | $-16.77^{-16.87}_{-16.68}$ | $0.89^{1.01}_{0.78}$ | $14.09$ | $-15.83$ | $0.21$ | $21.31$ | $-16.33$ | $0.51$ | $22.94$ | $-16.26$ | $0.50$
r114_d6.0_Z | $7.11^{7.50}_{6.70}$ | $-15.51^{-15.58}_{-15.40}$ | $2.50^{2.63}_{2.34}$ | $4.76$ | $-14.69$ | $2.17$ | $9.07$ | $-14.98$ | $3.09$ | $8.90$ | $-14.97$ | $3.78$
r114_d6.0_Z_R_4000K | $9.99^{12.51}_{8.90}$ | $-16.89^{-17.16}_{-16.37}$ | $0.67^{0.83}_{0.42}$ | $11.67$ | $-17.49$ | $0.56$ | $12.19$ | $-17.9$ | $0.63$ | $14.09$ | $-18.03$ | $0.52$
r114_d6.0_Z_R_6000K_0.1Ni | $9.06^{9.74}_{8.25}$ | $-15.73^{-15.81}_{-15.55}$ | $1.48^{1.62}_{1.40}$ | $5.54$ | $-14.81$ | $0.46$ | $11.49$ | $-15.56$ | $1.18$ | $11.90$ | $-15.42$ | $1.13$
Table 4: Same as Table 3 for the Bessel UBVRIJHK bands. Note, the lower
relative flux of the models in the JHK-bands meant reliable values could not
be determined for the line-of-sight dependant light curve quantities due to
the significant Monte Carlo noise. Therefore only the angle averaged values
are quoted for these bands. Additionally, we do not include any JHK-band light
curve quantities for Model r10_d4.0_Z_R_2000K due to its peculiar light curve
behaviour in these bands which we believe to be unphysical.