11institutetext:  Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
11email<EMAIL_ADDRESS>22institutetext: Unidad Mixta Internacional
Franco-Chilena de Astronomía, CNRS/INSU UMI 3386 and Instituto de Astrofísica,
Pontificia Universidad Católica de Chile, Santiago, Chile 33institutetext: NRC
“Kurchatov institute”, Institute for Theoretical and Experimental Physics
(ITEP), Moscow 117218, Russia 44institutetext: Sternberg Astronomical
Institute, Moscow State University, Moscow 119234, Russia 55institutetext:
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo Institutes for Advanced Study,
The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
66institutetext: Astrophysics Research Centre, School of Mathematics and
Physics, Queen’s University Belfast, Belfast BT7 1NN, Northern Ireland, UK
77institutetext: GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße
1, 64291 Darmstadt, Germany 88institutetext: Institut d’Astrophysique de
Paris, CNRS-Sorbonne Université, 98 bis boulevard Arago, 75014, Paris, France
99institutetext: Center for Theoretical Astrophysics, Los Alamos National
Laboratory, Los Alamos, NM, 87545, USA 1010institutetext: Theoretical
Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
1111institutetext: Department of Physics and Astronomy, Michigan State
University, East Lansing, MI 48823, USA 1212institutetext: Department of
Physics and Astronomy & Pittsburgh Particle Physics, Astrophysics, and
Cosmology Center (PITT PACC),
University of Pittsburgh, 3941 O’Hara Street, Pittsburgh, PA 15260, USA
1313institutetext: The Oskar Klein Centre, Department of Astronomy, Stockholm
University, AlbaNova, SE-10691 Stockholm, Sweden 1414institutetext:
Departments of Physics and Astronomy, University of California Berkeley and
Lawrence Berkeley National Laboratory, USA 1515institutetext: Dept. of
Particle Phys. & Astrophys., Weizmann Institute of Science, Rehovot 76100,
Israel 1616institutetext: Department of Computational Mathematics, Science,
and Engineering, Michigan State University, East Lansing, MI 48824, USA
1717institutetext: Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-
Straße 1, 85748 Garching bei München, Germany 1818institutetext: Facultad de
Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, La
Plata, B1900, Argentina 1919institutetext: Department of Astronomy and
Theoretical Astrophysics Center, University of California, Berkeley, CA 94720,
USA 2020institutetext: Institute of Astronomy, Russian Academy of Sciences,
Pyatnitskaya St. 48, 119017 Moscow, Russia 2121institutetext: Department of
Physics, New York University, New York, NY, 10003, USA 2222institutetext:
Computer, Computational, and Statistical Sciences Division, Los Alamos
National Laboratory, Los Alamos, NM, 87545, USA 2323institutetext: Department
of Physics & Astronomy, Louisiana State University, Baton Rouge, LA, 70803,
USA 2424institutetext: Department of Astronomy and Astrophysics, University of
California, Santa Cruz, CA 95064, USA 2525institutetext: Department of
Physics, NRCN, Beer-Sheva 84190, Israel 2626institutetext: ARTIS Collaboration
2727institutetext: CMFGEN Collaboration 2828institutetext: CRAB Collaboration
2929institutetext: KEPLER Collaboration 3030institutetext: SEDONA
Collaboration 3131institutetext: STELLA Collaboration 3232institutetext: SUMO
Collaboration 3333institutetext: SuperNu Collaboration 3434institutetext:
TARDIS Collaboration 3535institutetext: URILIGHT Collaboration

# StaNdaRT: A repository of standardised test models and outputs for supernova
radiative transfer

Stéphane Blondin (CMFGEN) 11222727 Sergei Blinnikov (STELLA) 3344553131
Fionntan P. Callan (ARTIS) 662626 Christine E. Collins (ARTIS) 66772626 Luc
Dessart (CMFGEN) 882727 Wesley Even (SuperNu) 9910103333 Andreas Flörs
(TARDIS) 773434 Andrew G. Fullard (TARDIS) 11113434 D. John Hillier (CMFGEN)
12122727 Anders Jerkstrand (SUMO) 13133232 Daniel Kasen (SEDONA) 14143030 Boaz
Katz (URILIGHT) 15153535 Wolfgang Kerzendorf (TARDIS) 111116163434 Alexandra
Kozyreva (STELLA) 17173131 Jack O’Brien (TARDIS) 11113434 Ezequiel A. Pássaro
(TARDIS) 18183434 Nathaniel Roth (SEDONA) 3030 Ken J. Shen (SEDONA) 19193030
Luke Shingles (ARTIS) 66772626 Stuart A. Sim (ARTIS) 662626 Jaladh Singhal
(TARDIS) 3434 Isaac G. Smith (TARDIS) 11113434 Elena Sorokina (STELLA)
33443131 Victor P. Utrobin (CRAB) 3320202828 Christian Vogl (TARDIS) 17173333
Marc Williamson (TARDIS) 21213434 Ryan Wollaeger (SuperNu) 222223233333 Stan
E. Woosley (KEPLER) 24242929 Nahliel Wygoda (URILIGHT) 25253535

(Received 27 May 2022; accepted 22 September 2022)

We present the first results of a comprehensive supernova (SN) radiative-
transfer (RT) code-comparison initiative (StaNdaRT), where the emission from
the same set of standardised test models is simulated by currently used RT
codes. We ran a total of ten codes on a set of four benchmark ejecta models of
Type Ia SNe. We consider two sub-Chandrasekhar-mass ($M_{\mathrm{tot}}=1.0$
$\rm{M}_{\odot}$) toy models with analytic density and composition profiles
and two Chandrasekhar-mass delayed-detonation models that are outcomes of
hydrodynamical simulations. We adopt spherical symmetry for all four models.
The results of the different codes, including the light curves, spectra, and
the evolution of several physical properties as a function of radius and time
are provided in electronic form in a standard format via a public repository.
We also include the detailed test model profiles and several Python scripts
for accessing and presenting the input and output files. We also provide the
code used to generate the toy models studied here. In this paper, we describe
the test models, radiative-transfer codes, and output formats in detail, and
provide access to the repository. We present example results of several key
diagnostic features.

###### Key Words.:

supernovae: general – Radiative transfer

##  1 Introduction

Accurate radiative-transfer (RT) calculations remain a key challenge in the
study of astronomical transients such as supernovae (SNe). While advances in
computational capabilities and theoretical understanding have allowed great
progress in the ability to simulate radiation transport, the large number of
physical processes involved, in particular opacity from thousands of atomic
transitions with a mixed absorptive and scattering character, prohibit
comprehensive 3D calculations based on first principles. Several physical
approximations of different forms —in particular different treatments of the
significant deviations from local thermodynamic equilibrium (LTE)— are
employed by different RT codes to calculate the properties of the gas and of
the radiation field. Approximate treatment of atomic physics is also required
due to the partially calibrated atomic data.

The back-reaction of radiation on the hydrodynamics provides an additional
challenge, requiring the solution of hydrodynamic equations coupled to the RT
solution. However, in many cases, and in particular for Type Ia SNe (SNe Ia)
at phases beyond several days, which we focus on here, the radiation carries a
negligible fraction of the energy and the ejecta are freely expanding
homologously. The RT problem in these cases is decoupled from the
hydrodynamics problem, the latter providing the initial ejecta profiles
(‘ejecta models’ hereafter). The ejecta profiles include density, composition,
and initial temperature as a function of position. The initial time (of order
1 day) is such that on the one hand it is much larger than the explosion
timescale (of order 1 second) so that the expansion is nearly homologous and
on the other hand sufficiently early such that radiation has hardly diffused
across the ejecta and the only evolution is due to adiabatic expansion and
radioactive decay.

Table 1: Summary of ejecta conditions. The yields for representative species corresponds to the start of the simulations in our model set (2 d post explosion for the toy models and $\sim$1 d post explosion for the DDC models). The 56Ni mass is given prior to any decay. Model | $M_{\rm ej}$ | $E_{\rm kin}$ |  M(56Ni)t=0 | M(Fe) | M(Ca) | M(S) | M(Si) | M(O) | M(C)
---|---|---|---|---|---|---|---|---|---
|  ($\rm{M}_{\odot}$) |  ($10^{51}$ erg) |  ($\rm{M}_{\odot}$) |  ($\rm{M}_{\odot}$) |  ($\rm{M}_{\odot}$) |  ($\rm{M}_{\odot}$) |  ($\rm{M}_{\odot}$) |  ($\rm{M}_{\odot}$) |  ($\rm{M}_{\odot}$)
toy06 | 1.00 | 1.00 | 0.600 | 0.001 | 0.040 | 0.140 | 0.220 | 0.000 | 0.000
toy01 | 1.00 | 1.00 | 0.100 | 0.000 | 0.090 | 0.315 | 0.495 | 0.000 | 0.000
DDC10 | 1.42 | 1.51 | 0.620 | 0.112 | 0.041 | 0.166 | 0.257 | 0.101 | 0.002
DDC25 | 1.39 | 1.18 | 0.115 | 0.098 | 0.024 | 0.237 | 0.478 | 0.282 | 0.022

Comparisons of the results of different RT codes for the same ejecta models
play an important role in estimating the accuracy of different approximations
and can be used to validate new codes. The number and sophistication of RT
codes has significantly developed in recent years, increasing the need for
diverse benchmark ejecta models that will allow detailed and careful
comparisons. In this paper, we describe the first results of a collaborative
effort of ten groups around the world that are developing existing RT codes
(in alphabetical order: ARTIS, CMFGEN, CRAB, KEPLER, SEDONA, STELLA, SUMO,
SuperNu, TARDIS, and URILIGHT; see Sect. 3 for descriptions and references) to
create a systematic code-comparison framework. As a first important step, all
groups agreed on a set of four test model ejecta and standardised output
formats. Each group calculated the resulting radiative display with their
respective codes for the same ejecta models and shared the results in a new
public electronic repository on GitHub111https://github.com/sn-rad-trans. We
did not attempt to agree on a specific setup for each code, nor did we
synchronise atomic data between codes.

The structure of the paper is as follows. In Sect. 2, the benchmark models are
described. These include two simplistic sub-Chandrasekhar-mass (sub-$M_{\rm
Ch}$) toy models with profiles that are defined analytically and two more
realistic models that result from hydrodynamical simulations of the $M_{\rm
Ch}$ delayed-detonation scenario. All the models considered here are
spherically symmetric (1D). We give short descriptions of the RT codes that
were employed in this first comparison in Sect. 3, with emphasis on the main
physical approximations that are used in each. In Sect. 4 we describe the
publicly available repository of results. In particular, descriptions are
provided of the output files and of Python codes that are included for reading
them. In Sect. 5, several example comparisons of the results of the different
codes are provided in order to illustrate the available outputs. We
voluntarily make no attempt to analyse the sources of discrepancies. While the
comparisons focus on observable aspects of the emission, comparisons to
observations and conclusions regarding the implications for the applicability
of the codes are intentionally not addressed in order to keep the focus of the
paper on the description of the comparison. Finally, we outline future plans
for this comparison project in Sect. 6.

##  2 Test models

The code-comparison test suite consists of four SN Ia models. Two are
sub-$M_{\rm Ch}$ models with analytic density and composition profiles (Sect.
2.1; ‘toy’ models), and the remaining two are $M_{\rm Ch}$ models resulting
from hydrodynamical simulations (Sect. 2.2; DDC models). The models were set
up or selected based on their ${}^{56}\rm{Ni}$ yield, in order to have two
models corresponding to ‘normal’ SNe Ia (toy06 and DDC10 with $\sim$0.6
$\rm{M}_{\odot}$ of ${}^{56}\rm{Ni}$) and two low-luminosity models (toy01 and
DDC25 with $\sim$0.1 $\rm{M}_{\odot}$ of ${}^{56}\rm{Ni}$). We present the toy
and DDC models in turn in the following sections, and summarise their
properties in Table 1. The density profiles at a reference time of 1 d post
explosion and initial composition profiles are shown in Figs. 1 and 2,
respectively.

###  2.1 Toy models

Figure 1: Density profiles of our model set at an adopted time of 1 d post
explosion.

Figure 2: Composition profiles at the start of our simulations in our model
set (2 d post explosion for the toy models and $\sim$1 d post explosion for
the DDC models). For the toy models, this represents the full set of species
present (${}^{56}\rm{Ni}$ and decay products ${}^{56}\rm{Co}$ and
${}^{56}\rm{Fe}$, as well as IMEs: Ca S, Si), while for the DDC models only a
subset of species are shown for illustration. The ${}^{56}\rm{Ni}$ mass
fraction is given at the time of explosion. We also show the total IGE mass
fraction (from Sc to Ni; dashed line) and the total IME mass fraction (from Na
to Ca; dotted line). The total IGE mass fraction coincides with the
${}^{56}\rm{Ni}_{0}$ line in the toy models and is not shown for sake of
clarity.

The toy models were generated using the `mk_snia_toy_model.py` script (see
Sect. 4) using the `--highni` (for the normal SN Ia model) and `--lowni` (for
the low-luminosity SN Ia model) options. Both models have a total mass
$M_{\mathrm{ej}}=1.0$ $\rm{M}_{\odot}$, a kinetic energy
$E_{\mathrm{kin}}=10^{51}$ erg, and are calculated at the time $t_{f}=2$ d
post explosion. The models have an exponential density profile (e.g. Jeffery,
1999),

$\rho=\rho_{\mathrm{c}}e^{-v/v_{e}},$ (1)

where

$v_{e}=\sqrt{\frac{E_{\mathrm{kin}}}{6M_{\mathrm{ej}}}}\approx
2895\mathrm{\,\rm{km\,s}^{-1}}$ (2)

is the e-folding velocity, and

$\rho_{\mathrm{c}}=\frac{M_{\mathrm{ej}}}{8\pi v_{e}^{3}t_{f}^{3}}\approx
6.32\times 10^{-10}\mathrm{\,\rm{g\,cm}^{-3}}$ (3)

is the central density at the chosen time.

The ejecta consist of $n$ spherical shells indexed by $i=1,\ldots,n$. We use a
uniform velocity grid with width $\Delta v=50$ $\rm{km\,s}^{-1}$. Each shell
then has an inner and outer velocity coordinate given by: $v_{0,i}=(i-1)\Delta
v$ and $v_{1,i}=v_{0,i}+\Delta v$.

As in Jeffery (1999), we define the dimensionless radial coordinate
$z=v/v_{e}$ which we use to compute the mass of each shell as:

$M_{i}=\int\rho(z)\ \mathrm{d}V=\rho_{\mathrm{c}}\int e^{-z}\ \mathrm{d}V,$
(4)

where the integral runs from $z_{0,i}$ to $z_{1,i}$ and the volume element is
$dV=4\pi r^{2}dr=4\pi v_{e}^{3}t_{f}^{3}z^{2}dz$, where we assume a
homologously expanding ejecta ($r=vt=v_{e}zt$).

Likewise, each shell volume $V_{i}$ is computed from the inner and outer radii
($r_{i,\\{0,1\\}}=v_{i,\\{0,1\\}}t_{f}$), which results in the mean density of
each shell:

$\rho_{i}=\frac{M_{i}}{V_{i}}=3\rho_{c}\frac{e^{-z_{0,i}}(z_{0,i}^{2}+2z_{0,i}+2)-e^{-z_{1,i}}(z_{1,i}^{2}+2z_{1,i}+2)}{z_{1,i}^{3}-z_{0,i}^{3}}.$
(5)

Starting from the central shell, we keep adding successive shells until the
total mass is within 0.01% of the required ejecta mass. For the 1.0
$\rm{M}_{\odot}$ toy models considered here, this results in 807 shells, where
the last shell has an outer velocity of 40 350 $\rm{km\,s}^{-1}$. We note that
not all codes use this velocity grid; in particular, the number of shells and
maximum velocity vary significantly among codes (see Sect. 3).

We assume the ejecta consist of a central region composed of iron-group
elements (IGEs), here only ${}^{56}\rm{Ni}$ and its decay products
${}^{56}\rm{Co}$ and ${}^{56}\rm{Fe}$, and an outer region composed of the
intermediate-mass elements (IMEs) Ca, S, and Si with constant mass fractions
throughout the layer: $X(\mathrm{Ca})=0.1$, $X(\mathrm{S})=0.35$, and
$X(\mathrm{Si})=0.55$. These values were chosen based on the delayed-
detonation model DDC10 (see Sect. 2.2 below). Our ‘normal’ toy model consists
of 0.6 $\rm{M}_{\odot}$ of ${}^{56}\rm{Ni}$ and 0.4 $\rm{M}_{\odot}$ of IMEs,
while our low-luminosity toy model consists of 0.1 $\rm{M}_{\odot}$ of
${}^{56}\rm{Ni}$ and 0.9 $\rm{M}_{\odot}$ of IMEs. The ${}^{56}\rm{Ni}$ and
IME composition profiles are smoothly connected using an analytic function
(here a cosine bell) over a mass interval $\Delta M_{\mathrm{trans}}$ (set to
0.2 $\rm{M}_{\odot}$ for the low-luminosity model and 0.4 $\rm{M}_{\odot}$ for
the regular model). At a given mass coordinate $m$, the ${}^{56}\rm{Ni}$ mass
fraction is set to:

$X_{{}^{56}\rm{Ni}}(m)=\begin{cases}1&\text{for $m<m_{1}$},\\\
1-f_{\mathrm{cos}}(m)&\text{for $m_{1}\leq m\leq m_{2}$},\\\ 0&\text{for
$m>m_{2}$},\end{cases}$ (6)

where $m_{1}=M(^{56}\rm{Ni})-\Delta$$M_{\mathrm{trans}}/2$,
$m_{2}=M(^{56}\rm{Ni})+\Delta$$M_{\mathrm{trans}}/2$, and

$f_{\mathrm{cos}}(m)=\dfrac{1}{2}\left\\{1-\cos\left[\left(\dfrac{m-m_{2}}{\Delta
M_{\mathrm{trans}}}\right)\pi\right]\right\\}.$ (7)

The IME mass fraction is then simply set to
$X_{\mathrm{IME}}(m)=1-X_{{}^{56}\rm{Ni}}(m)$. Our toy models therefore
consist of only six chemical species or isotopes (${}^{56}\rm{Ni}$,
${}^{56}\rm{Co}$, ${}^{56}\rm{Fe}$, 40Ca, 32S, 28Si): this was a voluntary
choice in order to keep the model as simple as possible while retaining the
defining characteristics of a SN Ia.

The initial temperature profile at $t_{f}=2$ d is determined by solving the
first law of thermodynamics assuming a radiation-dominated gas, local energy
deposition from ${}^{56}\rm{Ni}$ decay, and no diffusion (i.e. the temperature
in each zone is solved independently of the adjacent zones). Given these
assumptions, the temperature at $t_{f}$ can be determined analytically by
noting that the time-weighted internal energy, $tE(t)$, is equal to the time-
integrated time-weighted decay energy deposition rate,
$\int{tQ(t)\mathrm{d}t}$, as noted by Katz et al. (2013):

$T_{i}=\left(\frac{\rho_{i}X_{i}(\mathrm{{}^{56}\rm{Ni}})_{0}\int_{0}^{t_{f}}tq_{\rm
Ni}(t)\mathrm{d}t}{a\leavevmode\nobreak\ t_{f}}\right)^{1/4},$ (8)

where $X_{i}(^{56}\rm{Ni})_{0}$ is the ${}^{56}\rm{Ni}$ mass fraction at
$t\approx 0$ in the i’th cell, $a$ is the radiation constant, and $q_{\rm
Ni}(t)$ is the energy release rate per unit mass (ignoring neutrinos) of the
${}^{56}\rm{Ni}$$\rightarrow$${}^{56}\rm{Co}$$\rightarrow$${}^{56}\rm{Fe}$
decay chain. In this formulation, we ignore the time-weighted internal energy
shortly after explosion, $E(t_{0})t_{0}$.

It is clear from Eq. 8 that the temperature is predicted to be zero in zones
devoid of ${}^{56}\rm{Ni}$ ($\gtrsim 12000$ $\rm{km\,s}^{-1}$). We therefore
impose a constant temperature floor in these zones of 5000 K. The resulting
excess internal energy is quickly radiated away because these layers have a
relatively low optical depth, such that the impact on the longer-term
radiative display is negligible.

###  2.2 Delayed-detonation models

In addition to the two toy models above, we consider two $M_{\rm Ch}$ delayed-
detonation models resulting from hydrodynamical simulations. These were chosen
from the DDC model grid presented in Blondin et al. (2013) in order to closely
match the ${}^{56}\rm{Ni}$ yields of the toy models: Our low-luminosity model
DDC25 yields $\sim 0.12$ $\rm{M}_{\odot}$ of ${}^{56}\rm{Ni}$ (cf. 0.1
$\rm{M}_{\odot}$ for the toy01 model), and our ‘normal’ DDC10 model yields
$\sim 0.52$ $\rm{M}_{\odot}$ of ${}^{56}\rm{Ni}$ (cf. 0.6 $\rm{M}_{\odot}$ for
the toy06 model). We refer the reader to Blondin et al. (2013) for a detailed
description of the DDC models.

The outputs of the hydrodynamical modelling correspond to 30-60 s post
explosion, by which time the ejecta have reached a state of homologous
expansion. We applied a small amount of radial mixing to the hydrodynamical
input with a characteristic velocity width $\Delta v_{\rm mix}=400$
$\rm{km\,s}^{-1}$ to smooth sharp variations in composition. The ejecta were
then evolved to 0.5 d post explosion by solving the energy equation given by
the first law of thermodynamics, assuming the plasma is radiation dominated
and neglecting diffusion. Apart from the two-step
${}^{56}\rm{Ni}$$\rightarrow$${}^{56}\rm{Co}$$\rightarrow$${}^{56}\rm{Fe}$
decay chain, we also treat eight additional two-step decay chains associated
with 37K, 44Ti, 48Cr, 49Cr, 51Mn, 52Fe, 55Co, and 57Ni, and a further six one-
step decay chains associated with 41Ar, 42K, 43K, 43Sc, 47Sc, and 61Co (see
Dessart et al., 2014).

The ejecta at 0.5 d are then remapped onto the 1D, non-LTE, radiative-transfer
code CMFGEN of Hillier & Dessart (2012) and evolved until $\sim 1$ d post
explosion (0.976 d for the DDC10 model and 1.3 d for the DDC25 model; see
Sect. 3.2 for details), at which point the ejecta serve as initial conditions
for the other radiative-transfer codes presented in Sect. 3.

##  3 Radiative-transfer codes

Table 2: Physical ingredients and approximations used in each code for the test models in this paper. Code | RT method | Homologous |  $\gamma$-ray | Non-thermal | Excitation | Ionisation | Radiation | Line Opacity | Thermalisation
---|---|---|---|---|---|---|---|---|---
|  | expansion | deposition | deposition |  |  |  field $J_{\nu}$ | $\kappa_{\nu}$ |  parameter $\epsilon$
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
ARTIS | MC | yes | MC | $\cdots$ |  LTE($T_{R}$) |  approx. $dn/dt=0$ |  scaled LTE($T_{R}$) | Sobolev | $\cdots$
ARTIS nebular | MC | yes | MC | Spencer-Fano | $dn/dt=0$ | $dn/dt=0$ | $dJ/dt$ | Sobolev | $\cdots$
CMFGEN | RTE-CMF | yes | MC | Spencer-Fano | $dn/dt$ | $dn/dt$ | $dJ/dt$ | $\kappa_{\nu}$ | $\cdots$
CRAB | RH-1G | no | grey | Kozma/Fransson |  LTE($T_{R}$) |  LTE($T_{R}$) | $dJ/dt$ | Expansion | 0.9
KEPLER | FLD | no | grey | $\cdots$ | $\cdots$ |  LTE($T_{e}$) |  LTE($T_{e}$) | $\kappa=\mathrm{const.}$ | $\cdots$
SEDONA | MC | yes | MC | $\cdots$ |  LTE($T_{e}$) |  LTE($T_{e}$) | $dJ/dt$ | Expansion | 0.8 or 1.0
SUMO | MC | yes | grey | Spencer-Fano | $dn/dt=0$ | $dn/dt=0$ | $dJ/dt=0$ | Sobolev | $\cdots$
STELLA | RH-MG | no | grey | $\cdots$ |  LTE($T_{e}$) |  LTE($T_{e}$) | $dJ/dt$ | Expansion | 0.9
SuperNu | MC | yes | MC grey | $\cdots$ |  LTE($T_{e}$) |  LTE($T_{e}$) | $dJ/dt$ | $\kappa_{\nu}$ | 1.0
TARDIS | MC | yes | $\cdots$ | $\cdots$ |  scaled LTE($T_{R}$) |  scaled LTE($T_{R}$) |  scaled LTE($T_{R}$) | Sobolev | $\cdots$
URILIGHT | MC | yes | MC | $\cdots$ |  LTE($T_{e}$) |  LTE($T_{e}$) | $dJ/dt$ | Expansion | 0.8

In the following subsections, each group provides a brief description of their
code, highlighting the specific setup used in the calculations for this paper.
Each code subsection follows a similar structure: brief description of the
code (and whether it assumes a homologous velocity law); treatment of
$\gamma$-ray energy deposition and non-thermal effects; computation of the
temperature structure; treatment of excitation and ionisation; evaluation of
the radiation field; sources of opacity and atomic data; resolution (spatial
and frequency) and typical runtime. Table 2 summarises the physical
ingredients and approximations used in each code for the test models
considered here.

###  3.1 ARTIS

ARTIS222Source code available at https://github.com/artis-mcrt/artis is a
Monte Carlo radiative transfer code that uses the indivisible energy packet
method of Lucy (2002). The code was originally described by Sim (2007) and
Kromer & Sim (2009), with later improvements presented by Bulla et al. (2015)
and Shingles et al. (2020). The code is three-dimensional and follows the
time-evolution of the radiation field and state of the gas. It assumes a
strictly homologous velocity law.

Injection of energy into the ejecta is calculated by following the deposition
of $\gamma$-ray packets that are injected in accordance with the radioactive
decays of 56Ni and 56Co, following Lucy (2005). Additional decay channels have
been included in the studies of specific models. The simulated $\gamma$-ray
transport is non-grey and takes into account Compton scattering, photoelectric
absorption, and pair-creation opacities. In our standard runs, the code does
not include the effects of excitation or ionisation by non-thermal particles.
However, Shingles et al. (2020) presented updates to the code that include a
Spencer-Fano treatment of non-thermal ionisation as required for late-phase
modelling. Results obtained with this improved version (artisnebular) are
included for late phases for the models in this study.

The electron temperature in each grid zone is estimated by balancing of
heating and cooling rates (accounting for $\gamma$-ray and positron
deposition, bound-bound, bound-free, and free-free processes). In its regular
mode of operation (artis), the code uses an approximate non-LTE treatment to
estimate the ionisation state in the ejecta (based on Monte Carlo
photoionisation estimators; see Kromer & Sim 2009) and an LTE treatment of
excitation is adopted. This approach has been used in most of our published
studies, and is also used in most of the artis calculations presented here.
However, this method has limitations that become increasingly important at
later phases (e.g. it neglects non-thermal heating and ionisation and tends to
overestimate the plasma temperature at low densities due to incomplete
treatment of cooling by forbidden lines). To improve these issues, Shingles et
al. (2020) presented updates to the code that include a full non-LTE
population solver (together with the Spencer-Fano solver mentioned above).
Results obtained with this improved version (artisnebular) are included for
late-phase calculations here.

Monte Carlo estimators are used to track the radiation field in each grid
cell. In general, we use volume-based estimators (see Lucy 1999 or Noebauer &
Sim 2019) to extract radiation-field-dependent quantities from the flight
histories of our Monte Carlo quanta. In its standard mode of operation, artis
uses detailed Monte Carlo estimators to obtain photoionisation rates from the
radiation field, but relies on an estimated scaling for excited-state
photoionisation and on a dilute black-body radiation field model for bound-
bound excitation (see Kromer & Sim 2009 for details). However, the improved
artisnebular version uses a more detailed frequency binned representation of
the radiation field (see Shingles et al. (2020) for details). The code has the
capacity to iterate on each time step with the aim of achieving consistency
between the radiation field estimates and the packet transport in each step.
However, in practise we find that this iteration is not required and we
therefore generally simply use the radiation field quantities extracted from
the previous time step to estimate the radiative rates that are needed for the
current step.

In simulating ultraviolet to infrared photon transport, the code accounts for
electron scattering, bound-bound, bound-free, and free-free processes. Bound-
free and bound-bound processes are treated using the _Macro Atom_ approach of
Lucy (2002, 2003) and adopting the Sobolev approximation for line opacity. The
code does not use an expansion opacity (or similar) but treats line opacity
based on a frequency-ordered list of transitions treated in the Sobolev limit
(i.e. no line overlap is taken into account).

In our simulations, atomic data are primarily drawn from the Kurucz atomic
line lists (see Kromer & Sim 2009) – in Appendix A.1 we summarise the ions and
numbers of levels and lines that we include. The photon transport is carried
out on a 3D Cartesian grid that co-expands with the ejecta. The artis
simulations included here were carried out on a 1003 grid. The resolution
therefore corresponds to around 500 – 1000 km s-1, depending on the model. The
simulations are typically run on 1000 computer cores for one to two days.

###  3.2 CMFGEN

CMFGEN is a 1D, non-LTE, time-dependent radiative-transfer code that solves
the transfer equation in the co-moving frame of spherical outflows. Details
about the code, techniques, and approximations can be found in Hillier &
Miller (1998), Hillier (2003), Hillier (2012), and (for SN calculations) in
Hillier & Dessart (2012)333CMFGEN, with documentation, is available at
www.pitt.edu/~hillier. The velocity law for the outflow is in general
monotonic (but see e.g. Dessart et al. 2015 in the case of interacting SNe)
and is assumed here to be homologous (such that $\partial v/\partial r=v/r$).

In the present calculations, non-local energy deposition from radioactive
decay is treated using a Monte-Carlo approach for $\gamma$-ray transport
(Hillier & Dessart, 2012). Non-thermal processes associated with the high-
energy electrons produced by Compton scattering and photoelectric absorption
of these $\gamma$ rays are accounted for through a solution of the Spencer-
Fano equation (Spencer & Fano, 1954; Li et al., 2012).

The temperature structure is constrained through the energy equation that has
the form:

$\rho{De\over Dt}-{P\over\rho}{D\rho\over Dt}=4\pi\int
d\nu(\chi_{\nu}J_{\nu}-\eta_{\nu})+\dot{\epsilon}_{\rm decay}\,,$ (9)

where $D\over Dt$ is the Lagrangian derivative, $e$ is the internal energy per
unit mass, $P$ is the gas pressure, and $\dot{\epsilon}_{\rm decay}$ is the
energy absorbed per second per unit volume (see Hillier & Dessart 2012 for
further details). We verify the accuracy of the solution by examining a global
energy constraint (equivalent to conservation of flux in a static atmosphere;
see Hillier & Dessart 2012 for details), and an equation describing energy
conservation as applied to the heating and cooling of free electrons. These
two equations are related to the above Eq. 9 via the transfer equation, and
the rate equations, respectively (e.g. Hillier, 2003). Because of model
assumptions (e.g. the use of super-levels) these two equations are not
satisfied exactly, but the errors (typical at the 1% level or smaller) are too
small to affect the validity of the models. Processes contributing to electron
heating and cooling include bound-free ionisation and recombination,
collisional ionisation and recombination, collisional excitation and de-
excitation, free-free emission, Auger ionisation, charge exchange reactions
(primarily with H i and He i, and hence negligible in SN Ia ejecta), net
cooling from non-thermal processes, and heating by radioactive decay.

Atom and ion-level populations are determined through a solution of the time-
dependent rate equations, coupled to the above energy equation and the zeroth
and first moments of the radiative-transfer equation (see below). We consider
the following processes: bound–bound processes, bound–free processes,
collisional ionisation and recombination, collisional excitation and de-
excitation, Auger ionisation (Hillier, 1987; Hillier & Miller, 1998), and non-
thermal excitation and ionisation (Li et al., 2012). We additionally consider
further processes involving H and He (two-photon decay, charge-exchange
reactions, and Rayleigh scattering), although these are negligible here (and
completely absent from the toy models, which contain no H or He). To ease the
solution of the rate equations, atomic levels are grouped into super levels
(see Hillier & Miller 1998 for details).

The frequency-dependent mean intensity $J_{\nu}$ is obtained via a solution of
the time-dependent transfer equation in the co-moving frame to first order in
$v/c$. More specifically, we solve the zeroth and first moment equations,
which are closed using so-called Eddington factors $f_{\nu}=K_{\nu}/J_{\nu}$,
where $K_{\nu}$ is the second moment of the specific intensity (related to the
radiation pressure). The Eddington factors are obtained from a formal solution
of the time-independent transfer equation.

We consider the following sources of opacity: electron scattering, bound-free
(including photoionisation from excited states), bound-bound444In these
calculations we assume a Doppler profile with a constant effective Doppler
width (including both thermal and turbulent velocities) of 50
$\rm{km\,s}^{-1}$. In more general SN modelling, the effective Doppler width
is varied to test its effect on model results., free-free, and Auger
ionisation. As noted earlier, Rayleigh scattering and two-photon processes
(for H and He only) are also part of the global opacity budget but are
negligible here.

A description of the sources of atomic data can be found in the Appendix
(Sect. A.2). The number of levels (both super-levels and full levels) and
corresponding number of bound-bound transitions are given in Tables 6-11. We
ignore weak transitions with a $gf$ value555This value is the product of the
statistical weight $g$ of the lower level and the oscillator strength $f$ of
an atomic transition. below some cutoff, typically set to $10^{-4}$. For the
toy models, the following ions were included: Si ii–iv, S ii–iv, Ca ii–iv, Fe
i–v, Co ii–vii, and Ni ii–v. For the delayed-detonation DDC models the
following ions were included: C i–iv, O i–iv, Ne i–iii, Na i, Mg ii–iii, Al
ii–iii, Si ii–iv, S ii–iv, Ar i–iii, Ca ii–iv, Sc ii–iii, Ti ii–iii, Cr ii–iv,
Mn ii–iii, Fe i–vii, Co ii–vii, and Ni ii–vii (we also include the ground
states of Cl iv, K iii, and V i for the sole purpose of tracking changes in
the abundances of radioactive isotopes). For all the aforementioned ions, we
also consider ionisations to and recombinations from the ground state of the
next ionisation stage (e.g. Ni viii in the case of Ni). As time proceeds and
the temperature in the spectrum-formation region drops, the highest ionisation
stages have a negligible impact on the RT solution. When this occurs, smaller
model atoms are used for these ions, or the ions are removed altogether from
the atomic model set.

The toy models were remapped onto a coarser spatial grid with 100 depth
points. No remapping was performed for the DDC models. Typical wall-clock
runtimes are of the order of 24h per time step on a single computing node with
8-12 CPUs, thus taking 2-3 months to complete for a sequence covering the
first 200 d or so post explosion.

###  3.3 CRAB

CRAB is a 1D, implicit, Lagrangian radiation hydrodynamics code developed to
study the light curves during SN outbursts and the corresponding outflows
evolved from hydrostatic state up to homologous expansion (Utrobin, 2004).
Non-local energy deposition of $\gamma$ rays from radioactive decay is
determined by solving the corresponding one-group $\gamma$-ray transport with
the approximation of an effective absorption opacity of 0.06 $Y_{\mathrm{e}}$
cm2 g-1. Positrons are assumed to deposit their energy locally. This energy
deposition produces non-thermal ionisation and heating, the rates of which are
taken from Kozma & Fransson (1992).

The radiation hydrodynamic equations include a time-dependent energy equation,
which is based on the first law of thermodynamics and determines the gas
temperature structure. In the outer, transparent and semitransparent layers of
the SN ejecta, the local energy balance is in control of a net balance between
heating and cooling processes, while in the inner, optically thick layers, it
is determined by the diffusion of equilibrium radiation.

The code has two options for the treatment of atom and ion level populations.
In option A, the elements H, He, C, N, O, Ne, Na, Mg, Si, S, Ar, Ca, and Fe,
and the negative hydrogen ion H- are included in the non-LTE ionisation
balance. All elements but H are treated with three ionisation stages. The
ionisation balance is controlled by the following elementary processes:
photoionisation and radiative recombination, electron ionisation and three-
particle recombination, and non-thermal ionisation. In option B, we use an LTE
treatment of ionisation and excitation for elements from the option A list or
all elements from H to Zn modelled with an arbitrary number of ionisation
stages. Atomic and ionic level populations are determined using the Boltzmann
formulae and the Saha equations for an element mixture with the local electron
or non-equilibrium radiation temperature.

The time-dependent radiation transfer is treated in a one-group (grey)
approximation in the outer, transparent and semitransparent layers of the SN
envelope, while in the inner, optically thick layers where thermalisation of
radiation takes place and LTE applies, the diffusion of equilibrium radiation
is described in the approximation of radiative heat conduction. The bolometric
luminosity of the SN is calculated by including retardation and limb-darkening
effects.

In the inner, optically thick layers, the Rosseland mean opacity is evaluated
for the local electron temperature, while in the outer, transparent and
semitransparent layers non-LTE effects are taken into account when determining
the mean opacities and the thermal emission coefficient. The mean opacities
include processes of photoionisation, free-free absorption, Thomson scattering
on free electrons, and Rayleigh scattering on neutral hydrogen. The
contribution of lines to the expansion opacity is evaluated by the generalised
formula of Castor et al. (1975) or by the formula of Blinnikov (1996) using
the Sobolev approximation for line opacity. The expansion opacity in an
expanding medium is treated with a thermalisation parameter set to 0.9 as
recommended by Kozyreva et al. (2020) to model SNe Ia.

The sources of atomic data for processes in continuum can be found in
(Utrobin, 2004). Oscillator strengths of lines are taken from the Kurucz line
database666http://kurucz.harvard.edu/cdroms.html containing nearly 530 000
lines. Energy level data are from the atomic spectra database of the National
Institute of Standards and Technology.

The zoning of model toy06 with 808 zones is adequate for modelling a light
curve. For this model, a typical runtime is of the order of 5 h on one CPU for
the first 140 d.

###  3.4 KEPLER

KEPLER is a one-dimensional, implicit Lagrangian hydrodynamics code with
appropriate physics for the study of massive stellar evolution and SNe (Weaver
et al., 1978; Woosley et al., 2002). The radiation transport is flux-limited
diffusion using a single temperature for the matter and radiation.

An important difference between KEPLER and some of the other codes used here
is that KEPLER does not assume a coasting configuration. At early times in
particular, when the matter is very optically thick, energy input from 56Ni
and 56Co decay will both heat the matter and accelerate it. The correction to
the kinetic energy is small for the cases studied here, but the integral of
the emitted light will be less than the integral of the decay energy that is
deposited (minus adiabatic losses).

Our approach to $\gamma$-ray energy deposition is discussed in the Appendix of
Ensman & Woosley (1988). Since the early 1990s, a value of $\kappa_{\gamma}$ =
0.054 cm-2 g-1 has been used for the global opacity parameter in KEPLER’s
leakage scheme to model SNe Ia and that is the value used here. A better value
can be obtained by comparing with Monte Carlo calculations for a given class
of model, such as SN IIP, SN Ib, and so on. The value 0.054 cm2 g-1 is larger
than the standard local grey opacity $(0.06\pm 0.01)Y_{e}$ cm2 g-1 (e.g.
Swartz et al., 1995), where $Y_{e}\approx 0.5$, as it is used to calculate the
effective column depth from the outer edge of a spherical zone to the surface.
For all points except the geometric centre of the explosion, the angle-
averaged column depth would be greater than along this radial line. The
averaging is therefore approximated by taking a larger opacity. In reality,
this number would vary with the radial distribution of 56Ni and would be
smaller if all the 56Ni were at the centre.

The temperature structure is computed by solving the hydrodynamics equations
including the effects of expansion and acceleration with energy input by
radioactive decay and transported by radiative diffusion. Because of the way
KEPLER handles flux-limited transport using a single temperature for the
radiation and matter, the pressure in the outermost zones can be
overestimated. This can lead to a small (of order 10%) overestimate of the
conversion of radiation to kinetic energy in those zones at late times. To
alleviate this problem, the luminosity is taken to be the maximum of the value
at the (electron-scattering) photosphere and the zone that includes 95% of the
mass. The former dominated the light curve until after peak.

An important parameter of the calculation is the atomic opacity used to
transport thermal radiation. In KEPLER, the total ‘optical’ opacity consists
of two parts: (a) electron scattering, which is calculated using a full Saha
solve for all ionisation stages of all 19 species present; and (b) a constant
additive opacity, $\kappa_{a}$, taken to reflect the effect of Doppler-
broadened lines. The electron-scattering opacity is temperature-, density-,
and composition dependent and therefore varies with location and time. The
additive opacity is a constant everywhere for all time. Traditionally, we have
used a value of 0.1 cm2 g-1 in our studies of Type Ia light curves, but the
best value will depend on SN type. A comparison of SN Ib models calculated
using CMFGEN and KEPLER (Ertl et al., 2020) suggested a value of 0.03 cm2 g-1
and we regard this as a lower limit for the average. The actual value varies
with location and time in a complicated way. Here we adopt $\kappa_{a}=0.1$
cm2 g-1 for the results presented in this paper.

KEPLER carries a nuclear network of 19 species (Weaver et al., 1978) which do
not perfectly correspond to the species in the initial models provided. Care
was taken to translate the 56Ni and 56Co abundances given to the zero-age 56Ni
and stable iron mass fractions which are used by KEPLER for energy generation.
The species 12C and 16O were mapped without change. Other species such as
20Ne, 28Si, and so on were slightly augmented where necessary by adding in
nearby odd-Z abundances; that is, 20Ne included 20Ne and 23Na, 24Mg included
24Mg and 27Al, and so on. As KEPLER does not compute spectra, this should have
negligible consequences.

The four models were mapped into KEPLER. The total mass, kinetic energy, and
56Ni mass were preserved to 0.05%. Because KEPLER is not a special
relativistic code, zones with velocities greater than $0.1c$ were trimmed from
the input. High-velocity zones would also cause difficulty with editing the
luminosity at late times if the light-crossing time ceased to be negligible.
Removing this high-velocity material decreased the kinetic energy of all
models by about 2%. This should have negligible effects on the light curve.
The zoning of the DDC models was relatively coarse by traditional KEPLER
standards. After trimming the high-velocity zones, only 80 zones remained. The
zoning of the toy models was better with roughly 600 zones remaining. Rezoning
was not carried out. For a light curve with no nuclear burning, the zoning is
adequate. All calculations took at most a few minutes on a laptop.

###  3.5 SEDONA

SEDONA is a time-dependent, multi-frequency Monte Carlo radiative transfer
code originally developed to study SN light curves, spectra, and polarisation
(Kasen et al., 2006). For this comparison study, the gas properties were
tracked on 1D spherical Lagrangian zones and are evolved under the assumption
of homologous expansion.

The radioactive decays of ${}^{56}\rm{Ni}$ and ${}^{56}\rm{Co}$ were tracked
and used to source $\gamma$-ray packets. For the DDC10 model, radioactive
decays of 48Cr and 48V were also included in addition to the ${}^{56}\rm{Ni}$
decay chain. More detailed radioactive decay networks can be implemented in
SEDONA, but only the previously mentioned radioactive-decay processes were
used in this comparison study.

The $\gamma$-ray packets were transported using the Monte Carlo procedure
subject to simplified treatments of bound-free absorption and Compton down-
scattering. The $\gamma$-ray interactions were treated as heating terms that
entered into the thermal heating balance that sourced a population of thermal
photon packets.

The temperature in each shell was computed assuming radiative equilibrium. In
more detail, the LTE assumption let us set the gas emissivity to the Planck
function multiplied by the frequency-dependent line opacity. The temperature
was then iteratively adjusted until the frequency-integrated emissivity was
equal to the total radiative energy from both $\gamma$-rays and re-emitted
photon packets absorbed over the previous particle propagation step, as
measured in the co-moving frame of the fluid.

For most of its published applications, SEDONA assumes LTE in order to compute
opacities, although non-LTE capabilities have been developed and were recently
applied (Roth et al., 2016; Shen et al., 2021). For this comparison study,
only the LTE opacity mode was used in order to compute the excitation and
ionisation states of the gas. More precisely, this means that ionisation
fractions for each element were computed by simultaneously solving the Saha
equation using a local temperature, and the charge conservation equation
across all elements and isotopes present. Meanwhile, the bound-electron level
populations within each ionisation stage were set by the Boltzmann factors
given by the local temperature.

Photon packets were transported in three dimensions as a direct solution to
the time-dependent radiative transfer equation. For all interactions with the
gas, the packets were mapped to the 1D Lagrangian zones.

Once the level populations had been computed, the bound-bound opacity was
computed using the expansion opacity formalism, as described by Eastman &
Pinto (1993) and Kasen et al. (2006). For the toy01 and toy06 models, we only
included the bound-bound opacities and electron scattering opacity, using the
Thomson cross section. For DDC10, simplified bound-free and free-free
opacities were also included in addition to the previously mentioned
opacities, although they did not have a noticeable effect during the early
stages of the explosion, which is what we wish to compare using these
calculations.

A thermalisation parameter $\epsilon$ of 1.0 was used for the toy01 and toy06
models. This means that all photon packets (other than $\gamma$-ray packets)
that were absorbed were immediately thermalised, so that each absorption event
was followed by re-emission of a photon packet with a frequency sampled from
the thermal emissivity. For the DDC10 model, $\epsilon$ was set to 0.8, so in
that case 20% of absorbed photon packets (not including the $\gamma$-ray
packets) were coherently scattered instead of having their frequency re-
sampled.

The atomic data used for the bound-bound transitions were taken from Kurucz CD
1. This is a larger set of atomic data than CD 23. The details are described
in Appendix A.3.

For this comparison study, the co-moving frequency grid for the thermalised
photons (i.e. not the $\gamma$-rays) used 17664 bins, with equal logarithmic
spacing, ranging from $10^{14}$ Hz to $2\times 10^{16}$ Hz. The output spectra
used 1044 frequency bins over the range $1.1\times 10^{14}$ Hz to $2\times
10^{16}$ Hz ($\sim 150$ Å to $\sim 2.7$ $\mu$m). The time steps for the
homologous expansion and radioactive decay began at approximately a few hours
at the start of the calculation, and grew to no longer than 1 day, with a
maximum time-step growth rate of 10%. One million $\gamma$-ray packets were
emitted at each time step. As these $\gamma$ rays deposited their energy,
their packets were discarded, while one thermally sourced photon packet was
emitted for each discarded $\gamma$-ray packet. With all of the settings
described above, a calculation run to 100 days post-explosion required about
10 CPU hours, and could be efficiently performed in parallel across several
hundred processors, reducing the elapsed wall time to less than one hour.

###  3.6 STELLA

The multi-group radiation hydrodynamics code STELLA (Blinnikov & Bartunov,
1993; Blinnikov et al., 1998, 2006) is capable of computing the evolution of
the radiation field coupled to the hydrodynamics, as well as the bolometric
light curve, spectral energy distribution, and resulting broad-band magnitudes
and colours. Therefore, STELLA does not require the condition of homologous
expansion and is capable of treating shock propagation and any dynamical
processes taking place in the SN ejecta.

Energy deposition from ${}^{56}\rm{Ni}$ and ${}^{56}\rm{Co}$ radioactive decay
is treated in a one-group diffusion approximation with an absorption opacity
of 0.05 $Y_{\mathrm{e}}$ cm 2 g${}^{\,-1}$ according to Swartz et al. (1995).
The energy of positrons is thermalised locally.

The ionisation and level populations of a limited set of species (H, He, C, N,
O, Ne, Na, Mg, Al, Si, S, Ar, Ca, stable Fe, stable Co, and stable Ni) is
treated in the LTE approximation. Radiation is not treated in equilibrium with
the matter. The colour temperature is estimated as a black body temperature
via the least-squares method.

The opacity includes photoionisation, free–free absorption, and electron
scattering processes assuming LTE for the plasma and line interactions.
Radioactive ${}^{56}\rm{Ni}$ and ${}^{56}\rm{Co}$ contribute to the stable Fe
when the line opacity is calculated. The expansion opacity is calculated
according to (Eastman & Pinto, 1993). The thermalisation parameter for the
line opacity treatment is set to 0.9 (Kozyreva et al., 2020). The line opacity
is calculated using a data base of 153 441 spectral lines partly from Kurucz &
Bell (1995), Verner & Yakovlev (1995), and Verner et al. (1996b).

The spectral energy distribution is computed in the wavelength range from 1 Å
to 50 000 Å. The frequency range is divided into 129 bins with equal
logarithmic spacing, in which the radiative transfer equations are solved at
every time step. For the toy06 model, a higher resolution simulation with 629
frequency bins was run, which is the version used in the following sections
for this model. The overall opacity might be slightly underestimated in the
simulations with 629 frequency bins compared to simulations involving 129
bins, because the expansion opacity is calculated for the lines in the given
bin and is not extended to another bin even if the velocity gradient is very
high. The final (pseudo-)bolometric light curve is obtained by integrating the
spectra over frequency.

To avoid numerical artefacts, ejecta models closer in time to the explosion
were used (all are included in the data repository): 1 h post explosion for
both toy models, and $\sim 29$ s and $\sim 63$ s post explosion for models
DDC10 and DDC25, respectively. The toy models were computed on a lower-
resolution grid (with 202 zones), while the DDC models were computed at a
higher resolution (with 399 zones). The typical runtime on a single processor
is a few hours at most.

###  3.7 SUMO

SUMO (Jerkstrand, 2011; Jerkstrand et al., 2011, 2012) is a homologous non-LTE
code with radiative transfer. It is specialised to calculate spectra and light
curves in the post-peak phases of the SN. The code is written in Fortran and
is parallelised with MPI.

Gamma-ray transfer is done by ray tracing using a grey opacity of $0.06Y_{e}$
cm2 g-1. Positrons can be transferred (using an effective opacity of 8.5 times
the $\gamma$-ray one) but here they were assumed to be locally trapped. The
cascade of non-thermal electrons following the scattering of gamma rays and
positrons is solved for with the Spencer-Fano method (Kozma & Fransson, 1992).

Zone temperatures are solved from the first law of thermodynamics considering
heating by non-thermal processes and photoionisation, and cooling by net
collisional line excitation, free-free emission, and recombination. The
temperature is solved either in steady state (heating = cooling) (Jerkstrand
et al., 2012) or time-dependently (Pognan et al., 2022). Here, the steady-
state mode was used.

The rate equations are solved by considering spontaneous radiative decay with
Sobolev escape probabilities (assuming homology) modified to include continuum
processes and strong line overlaps, thermal and non-thermal collisions
(excitations and ionisations), photoexcitation and de-excitation,
photoionisation, and recombination. Options exist to also include charge
transfer processes and molecular chemistry (Liljegren et al., 2020), but these
were not considered here. The rate equations can be solved either in steady
state (inflow = outflow) (Jerkstrand et al., 2012) or time dependently (Pognan
et al., 2022). Here, the steady-state mode was used.

Radiative transfer is calculated with a Monte Carlo simulation that is
iterated with the solvers for temperature, ionisation, and excitation. During
the transfer, photon packets can experience electron scattering, bound-free
and free-free absorption, and line absorption. The line transfer is resolved
line by line (with Sobolev formalism) rather than using an expansion opacity
formalism. Photoexcitation is either fully coupled to the non-LTE solutions
(for low- and mid-lying levels), or decoupled (for high-lying levels), instead
giving a fluorescence cascade on the spot. The radiation field is computed in
steady state.

The atomic data come from a variety of sources, mostly described in Jerkstrand
et al. (2011) and Jerkstrand et al. (2012). Model atoms have LS-states
resolved, that is they do not use super levels. In models computed up to 2022
(including the ones here), ions up to and including doubly ionised were
included, although higher ions are now being included (Pognan et al., 2022).
For the toy models computed here, the model atoms are Fe i (496 levels), Fe ii
(578 levels), Fe iii (600 levels), Ni i (136 levels), Ni ii (500 levels), Ni
iii (8 levels), Co i (317 levels), Co ii (108 levels), Co iii (306 levels), Si
i (494 levels), Si ii (77 levels), Si iii (2 levels), S i (123 levels), S ii
(5 levels), S iii (6 levels), Ca i (198 levels), Ca ii (69 levels), and Ca iii
(1 level)777For Ni iii Si iii, S ii and Ca iii the low number of levels is
sufficient as the next state energies are at 52 152, 52 853, 79 395 and 203
373 cm-1, respectively..

For the runs here, the ejecta were resampled to $\Delta v=500$
$\rm{km\,s}^{-1}$, and truncated at $v=30000$ $\rm{km\,s}^{-1}$. The radiative
transfer was computed on a wavelength grid from 400 Å to 25000 Å, with a
logarithmic resolution $\mathrm{d}\lambda/\lambda=10^{-3}$. Typical wall-clock
runtimes for a single epoch are a few hours on a typical 128 core setup.

###  3.8 SuperNu

SuperNu is a multi-group LTE radiative-transfer code that employs Implicit
Monte Carlo (IMC) and Discrete Diffusion Monte Carlo (DDMC) (Wollaeger et al.,
2013; Wollaeger & van Rossum, 2014). IMC solves the thermal radiative-transfer
equations semi-implicitly by treating some absorption and emission as
instantaneous effective scattering (see e.g. Fleck & Cummings, 1971). DDMC
optimises IMC in optically thick regions of space (Densmore et al., 2012) and
ranges of wavelength (Abdikamalov et al., 2012; Densmore et al., 2012) by
replacing many low mean-free-path scattering events with single leakage
events. SuperNu can apply IMC and DDMC in both static and homologous, semi-
relativistically expanding atmospheres. The code has been verified by analytic
and semi-analytic radiative transfer tests (Wollaeger et al., 2013) and on the
W7 model of SNe Ia (Nomoto et al., 1984; Wollaeger & van Rossum, 2014).

For the $\gamma$-ray transfer, SuperNu employs a constant absorption opacity
of 0.06 $Y_{e}$ cm2 g-1, where $Y_{e}$ is ejecta gas electron fraction,
following the prescription of Swartz et al. (1995). The $\gamma$-ray packets
in SuperNu are not directly converted to optical packets, but instead are used
to tally the total $\gamma$-ray energy deposition per spatial cell. The
deposition energy values are then added to the thermal source for optical
packets.

The ejecta gas temperature is calculated using the standard IMC semi-implicit
linearisation (Fleck & Cummings, 1971) of the comoving internal energy
equation (Wollaeger et al., 2013): internal energy is recast to gas
temperature using the standard relation $\partial e=c_{v}\partial T$, where
$e$ is internal energy, $c_{v}$ is heat capacity at constant volume, and $T$
is gas temperature. Due to the IMC time linearisation, energy deposited from
gamma-rays and (locally) from beta particles appears simultaneously in both
the comoving internal energy equation and the radiation equation.

Ionisation and excitation are both treated with Saha-Boltzmann statistics
evaluated at the gas density and temperature at the beginning of the time
step. The multi-element system is solved iteratively by converging the free
electron number. The resulting population densities are then used to calculate
opacities. The radiation field is represented by fully time-dependent Monte
Carlo packets, which are sourced from the LTE emissivity. The radiation field
is not constrained to be in equilibrium with the gas, and so in general the
system is ‘two-temperature’.

Opacity is discretised into groups via direct integration over co-moving
wavelength (but see Fontes et al. 2020 for a study with SuperNu using
expansion opacity). Opacity in SuperNu includes free-free (Sutherland, 1998),
and bound-free (Verner & Yakovlev, 1995; Verner et al., 1996a) processes, as
well as the bound-bound opacities from the Kurucz line
lists888http://kurucz.harvard.edu/atoms.html, and the standard elastic Thomson
scattering opacity. Weak lines in the Kurucz data set are omitted from the
SuperNu line list where the opacity is dominated by stronger lines. The total
number of available lines for the present simulations is 591 288. This list
was motivated by studies using the PHOENIX code in the work of van Rossum
(2012), using the full line list (¿107 lines) as a benchmark. For the
simulations in this work, opacity is computed in 1000 logarithmic wavelength
groups from 100 to 32000 Å.

Each simulation presented for SuperNu uses 4 194 304 source packets per time
step, with maximum active packet populations of between 100 and 200 million.
The wavelength group structure that the packets are tracked through is the
same as the opacity group structure (though for the simulations here, the
wavelength bounds for the flux tallies are 1000 and 32,000 Å). The DDC and toy
models use 115 and 202 velocity space cells, respectively. For the time grid,
the DDC and toy models use 200 logarithmically increasing time steps out to
day 80, from about day 1 (DDC models) or day 2 (toy models) post explosion.

SuperNu has MPI+OpenMP parallelisation. The SuperNu simulations of the toy01,
toy06, DDC10, and DDC25 models presented here cost 190, 320, 398, and 400
core-hours, respectively, each using 16 MPI ranks and 8 OpenMP threads per
rank.

###  3.9 TARDIS

TARDIS is an open-source steady-state 1D radiative-transfer code that uses
indivisible energy packets as its transport quanta following the methods in
Abbott & Lucy (1985), Lucy & Abbott (1993), Mazzali & Lucy (1993), Lucy
(2002), Lucy (2003), and Lucy (2005). Kerzendorf & Sim (2014) describes the
initial version of the code which was primarily used to model SNe Ia.
Subsequently, the code has been significantly enhanced to include a non-
thermal approximation treatment for helium (Boyle et al., 2017) and the
continuum processes and relativity treatments required for hydrogen-rich SNe
(Vogl et al., 2019). TARDIS has been continuously enhanced since then (see
e.g. Kerzendorf et al., 2022, similarly gamma-ray energy deposition is a new
module of TARDIS, but has not been used for the models in this work). Full
documentation and an extended physics walk-through for TARDIS can be found
online999https://tardis-sn.github.io/tardis.

TARDIS assumes a steady-state homologously expanding SN envelope and injects
Monte Carlo packets —randomly sampled from a given distribution (by default a
black-body)— from an inner boundary. The code supports bound-bound, bound-
free, free-free, and Thomson opacities with several redistribution schemes
from simple scattering to a macro-atom (Lucy, 2002, 2003). Summary packet
statistics are used to estimate radiation field quantities (temperature,
dilution factors, mean intensities, heating- and photo-ionisation rates, and
line source functions). The estimated quantities are used to calculate the
ionisation and excitation populations in steady state with a choice between
LTE and several formulations of non-LTE (nebular approximation of Abbott &
Lucy 1985 for ionisation, and dilute-LTE and full non-LTE for excitation).
TARDIS then calculates Sobolev optical depths for line interaction, and
opacities for continuum processes. These values are then used in the
subsequent Monte Carlo step, which produces summary statistics for updating
the opacities. The other convergence criterion is a match of the integrated
packet output luminosity and the requested luminosity, which is achieved
through iterative adjustments of the inner temperature.

To handle atomic data, the TARDIS collaboration has developed an additional
package named Carsus (available at https://github.com/tardis-sn/carsus;
Pássaro et al., 2019) with documentation available online
101010https://tardis-sn.github.io/carsus. Carsus can read atomic data (masses,
ionisation energies, levels, lines, photoionisation cross sections, and
collisional cross sections) from NIST (Kramida & Ralchenko, 1999), Chianti
(Dere et al., 2019; Dere, K. P. et al., 1997), CMFGEN (Hillier & Miller, 1998;
Dessart & Hillier, 2010), and Kurucz (Kurucz, 2009). For the models in this
paper, we used Kurucz CD 23 as the source of atomic data (see Appendix A.4 for
the full description).

The setup files for TARDIS that are used in this work are available
online111111https://github.com/tardis-sn/tardis-
setups/tree/master/2022/sn_radtrans_compare using an atomic data set from
Kurucz CD 23. In the setup used for the code comparison work, we run TARDIS in
a mode that self-consistently finds a temperature stratification given an
inner boundary velocity and output luminosity. We used the mean bolometric
luminosity of the other comparison codes for our output luminosity. The
structure of the toy01 and toy06 models was set at 520 shells between 9000 and
35000 km s-1. The DDC models were truncated to 40 shells between 9000 and
35000 km s-1. The inner boundary velocity is found by iterating until the
dilution factor is close to 0.5 in the innermost zone. Table 3 shows the inner
boundary velocity and requested output luminosity for the different epochs.

Table 3: Velocity and temperature at the inner boundary given the requested output luminosity for the TARDIS calculation of the toy06 model. Time | $\log_{10}L_{\mathrm{req}}$ | Velocity at IB | Temperature at IB
---|---|---|---
(days) |  (erg s-1) |  ($\rm{km\,s}^{-1}$) | (K)
5 | 42.48 | 20500 | 8805
10 | 42.95 | 17000 | 8870
15 | 43.04 | 10000 | 11525
20 | 43.00 | 5500 | 15513

We used the nebular approximation for ionisation and the dilute-lte
approximation for excitation. We used $5\times 10^{5}$ packets for estimating
our radiation field and 30 iterations for convergence. In the final iteration,
we estimated the source functions with $5\times 10^{5}$ packets and then used
the formal integral to synthesise a spectrum. For the comparison, we use line
opacities and Thomson opacities with the macroatom interaction scheme. The
resolution of the output spectra was uniform from 500 to 20000 Å with 2 Å bin
width. The models were run on one CPU with runtimes of less than one hour.

###  3.10 URILIGHT

URILIGHT is a time-dependent Monte-Carlo code written in Fortran 90 by Yoni
Elbaz based on the approximations that are used in SEDONA (see Sect. 3.5
above, Kasen et al. 2006 and references therein), in particular assuming
homologous expansion. A detailed description of this program and previous
comparisons to other published radiative-transfer codes for several benchmark
problems are presented in Wygoda et al. (2019). The code is publicly available
and can be downloaded from
https://www.dropbox.com/sh/kyg1z1xwi0298ru/AAAqzUMbr6AkoVfkSVIYChTLa?dl=0.

Energy deposition resulting from the decay of radioactive isotopes is
calculated by a Monte-Carlo solution of the $\gamma$-ray transport, for which
interaction with matter is included through Compton scattering and
photoelectric absorption. For the calculations in this work, only 56Ni and
56Co decay were included. The temperature is iteratively solved for in each
cell by requiring that the total emissivity be equal to the total absorbed
energy. LTE is assumed for calculating the ionisation and excitation states:
ionisation is obtained by solving the Saha equation, and excitation levels are
set by the Boltzmann-distribution.

Opacities include bound-bound and free-free absorption and Thomson scattering
off free electrons. The atomic line data for the bound-bound transitions,
which constitutes the main and most important source of opacity in SNe Ia, are
taken from the extended set of lines by Kurucz121212CDs 1 and 23 from
http://kurucz.harvard.edu/cdroms.html. Following a bound-bound interaction,
most photons (a fraction $\epsilon$, which is a global parameter of the
simulation; see Kasen et al. 2006) are thermalised and re-emitted at a
different wavelength with a distribution set by the emissivity. The rest of
the photons (fraction $1-\epsilon$) are re-emitted at the same wavelength
(within the line width). As in Wygoda et al. (2019), we used $\epsilon=0.8$ in
the runs performed here.

The runs here use 162 spatial cells for the toy01 and toy06 models and 115
spatial cells for the DDC10 and DDC25 models. All models are run with 128 time
steps logarithmically spaced between 2 and 210 days, and a uniform spectrum
resolution of 10 Å between 100 and 30000 Å. Each run typically took 30 hours
on a single core, or approximately one hour when parallelised.

Table 4: Code outputs and computed models. Code | RT | Bolometric | Spectrum | Early | Nebular | Computed Models
---|---|---|---|---|---|---
Name | Method | Flux | or SED | Times | Times | toy06 | toy01 | DDC10 | DDC25
ARTIS | MC | calculated | ✓ | ✓ | ✓ | ✓ | ✗ | ✓ | ✗
CMFGEN | RTE-CMF | calculated | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
CRAB | RH-1G | calculated | ✗ | ✓ | ✗ | ✓ | ✗ | ✗ | ✗
KEPLER | FLD | calculated | ✗ | ✓ | ✗ | ✓ | ✓ | ✓ | ✓
SEDONA | MC | calculated | ✓ | ✓ | ✗ | ✓ | ✓ | ✓ | ✗
STELLA | RH-MG | calculated | ✓ | ✓ | ✗ | ✓ | ✓ | ✓ | ✓
SUMO | MC | calculated | ✓ | ✗ | ✓ | ✓ | ✓ | ✗ | ✗
SuperNu | MC | calculated | ✓ | ✓ | ✗ | ✓ | ✓ | ✓ | ✓
TARDIS | MC | input | ✓ | ✓ | ✗ | ✓ | ✓ | ✓ | ✓
URILIGHT | MC | calculated | ✓ | ✓ | ✗ | ✓ | ✓ | ✓ | ✓

##  4 Data repository of test models and standardised outputs

The ejecta models and output files from the RT simulations of the different
codes are provided in a new data repository, which is publicly available and
can be accessed at https://github.com/sn-rad-trans/data1.

Descriptions of the files available in the repository are provided below,
including ejecta model files (Sect. 4.1), output files (Sect. 4.2), and Python
codes that were used to create the analytic toy model ejecta and codes for
reading the output files (Sect. 4.3).

###  4.1 Ejecta model files

As described in Sect. 2, the code comparison is performed using four SN Ia
models (main parameters in Table 1). The RT input files that were distributed
among the groups are provided in the repository, including two toy-model files
snia_toy01_2d.dat and snia_toy06_2d.dat and the two delayed-detonation model
files DDC10_0.976d.dat and DDC25_1.300d.dat.

####  4.1.1 Toy model files

The two toy-model files, snia_toy01_2d.dat and snia_toy06_2d.dat, represent a
snapshot of the ejecta at 2.0 days post explosion and include 807 shells
(rows) with the following 21 columns131313For the STELLA runs, the files
snia_toy01_1h_lowres.dat and snia_toy06_1h_lowres.dat represent a snapshot of
the ejecta at 1 h post explosion and include 202 shells (rows) with the same
structure.:

  1. (1)

Shell index (1-807)

  2. (2)

Velocity at shell centre ($\rm{km\,s}^{-1}$)

  3. (3)

Shell mass ($\rm{M}_{\odot}$)

  4. (4)

Lagrangian mass coordinate at the outer shell boundary ($\rm{M}_{\odot}$)

  5. (5)

Pre-decayed ($t=0$) stable IGE mass fraction (0 for all shells)

  6. (6)

Pre-decayed ($t=0$) 56Ni mass fraction

  7. (7)

IME mass fraction (of which 10% is Ca, 35% is S, and 55% is Si by mass)

  8. (8)

Ti mass fraction (0 for all shells)

  9. (9)

Unburnt C+O mass fraction (0 for all shells)

  10. (10)

Radius at shell centre (cm) = velocity at shell centre $\times$ 2 days
(homologous expansion)

  11. (11)

Mean density over shell ($\rm{g\,cm}^{-3}$), not density at shell centre

  12. (12)

Temperature (K)

  13. (13)-(21)

Mass fractions of 56Ni, 56Co, 56Fe, Ca, S, Si, O, and C at 2 days post
explosion

####  4.1.2 Delayed-detonation model files

The two delayed-detonation model files DDC10_0.976d.dat and DDC25_1.300d.dat
represent a snapshot of the ejecta at 0.976 days and 1.3 days, respectively,
with 115 shells with the following 50 columns141414For the STELLA runs, the
files DDC10_29.29s_highres.dat and DDC25_62.60s_highres.dat represent a
snapshot of the ejecta at $\sim 29$ s and $\sim 63$ s post explosion,
respectively, with 399 shells. The structure is the same but includes four
additional columns giving the elemental mass fractions of H, He, N, and P.:

  1. (1)

Velocity at shell centre ($\rm{km\,s}^{-1}$)

  2. (2)

Radius at shell centre (cm) = velocity at shell centre $\times$ the time post
explosion

  3. (3)

Shell volume (cm3)

  4. (4)

Density at zone centre ($\rm{g\,cm}^{-3}$), not mean density over shell

  5. (5)

Shell mass estimate (g) = shell volume $\times$ density at zone centre

  6. (6)

Temperature (K)

  7. (7)-(26)

Elemental mass fractions at snapshot time of C, O, Ne, Na, Mg, Al, Si, S, Cl,
Ar, K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, and Ni

  8. (27)-(50)

isotopic mass fractions of the following radioactive nuclei at snapshot time:
56Ni, 56Co, 57Ni, 57Co, 48Cr, 48V, 49Cr, 49V, 51Mn, 51Cr, 55Co, 55Fe, 37K,
37Ar, 52Fe, 52Mn, 44Ti, 44Sc, 41Ar, 42Ar, 42K, 43K, 47Sc, and 61Co

###  4.2 RT output files

For each simulation by a specific group, six output file types are generated
for each of the ejecta models.

####  4.2.1 Output file names

The name of each file has the following structure:

        <output type>_<model>_<code name>.txt

where

  * •

`<output type>` represents the type of output (described below) and can take
one of six values: lbol_edep, edep, phys, ionfrac_<element> (where <element>
is the name of a given element, e.g. ca for calcium), spectra, and
wsynphot_mags,

  * •

`<model>` represents one of the models and can take one of four values: toy06,
toy01, ddc10, and ddc25,

  * •

`<code name>` represents the code name with an optional additional descriptor
(useful to distinguish between different code settings when applied to a given
model) and can take one of 12 values: artis, artisnebular, cmfgen, crab,
kepler, sedona, stella, stella_fr600 (for the STELLA runs for the toy06 models
that use 629 frequency bins instead of the default 129), supernu, sumo,
tardis, and urilight.

In principle, there are 12 code names $\times$ 4 ejecta models $\times$ 5
output files (excluding the ionfrac_<element> files) = 240 files and an
additional 12 code names $\times$ (2 toy models $\times$ 6 elements + 2 DDC
models $\times$ 20 elements) = 624 ionfrac_<element> files. This results in a
total of 864 files, although in practice not all files are available for
various reasons: (a) a code was not applied to a given model (it was agreed
that all groups should at least compute the toy06 model, but the other three
were considered optional); (b) a given code cannot produce the specified
output (e.g. SEDs for grey codes); (c) a given code does not provide the
desired output quantities by default (i.e. modification of the source code
would be necessary). Table 4 summarises the outputs and computed models for
each code.

####  4.2.2 Six output file types

The six types of output files include:

  1. 1.

Pseudobolometric (UVOIR) luminosity and global energy deposition as a function
of time.

File name: `lbol_edep_<model>_<code name>.txt`

Header: after some optional comment lines, the following two lines give the
number of epochs (NTIMES, 100 in this example) and the column headings:

            #<optional comment lines>
        #NTIMES: 100
        #time[d] Lbol[erg/s] Edep[erg/s]

Contents: NTIMES rows with the following three columns:

    1. (1)

Time since explosion in days

    2. (2)

Pseudobolometric (UVOIR) luminosity in erg s-1

    3. (3)

Global energy deposition by $\gamma$ rays and positrons in erg s-1

  2. 2.

Energy deposition

File name: `edep_<model>_<code name>.txt`

Header: after some optional comment lines, the following four lines give the
number of epochs (NTIMES, 100 in this example), the number of cells (NVEL, 200
in this example), the list of epochs in days with all NTIMES values (the ‘…’
should correspond to actual values), and finally the column headings (here the
‘…’ can be used as is):

            #<optional comment lines>
        #NTIMES: 100
        #NVEL: 200
        #TIMES[d]: 2.0 3.0 ... 100.0
        #vel_mid[km/s] Edep_t0[erg/s/cm^3]
         Edep_t1[erg/s/cm^3] ... Edep_tn[erg/s/cm^3]

Contents: NVEL rows with the following NTIMES+1 columns (101 in this example):

    1. (1)

Velocity at the centre of each cell in $\rm{km\,s}^{-1}$

    2. (2)-(101)

Total $\gamma$-ray + positron energy deposition rate at each of the NTIMES
epochs within the corresponding cell in erg s-1 cm-3

  3. 3.

Physical conditions

File name: `phys_<model>_<code name>.txt`

Header: after some optional comment lines, the following three lines give the
number of epochs (NTIMES, 100 in this example), the list of epochs with all
NTIMES values (the ‘…’ should correspond to actual values), and finally one
empty comment line:

            #<optional comment lines>
        #NTIMES: 100
        #TIMES[d]: 2.0 3.0 ... 100.0
        #

Contents: NTIMES blocks (one for each epoch), each containing a block header
with three lines giving the epoch (in days, 2.0 d in this example), the number
of cells saved for this epoch (NVEL, 200 in this example), and finally the
column headings:

            #TIME: 2.0
        #NVEL: 200
        #vel_mid[km/s] temp[K] rho[gcc] ne[/cm^3]
         natom[/cm^3]

Each block content consists of NVEL rows with the following five columns:

    1. (1)

Velocity at the centre of each cell in $\rm{km\,s}^{-1}$

    2. (2)

Temperature in K

    3. (3)

Density in $\rm{g\,cm}^{-3}$

    4. (4)

Free electron density in cm-3

    5. (5)

Total atom density in cm-3

  4. 4.

Ionisation fraction (one file per element)

File name:
`ionfrac_<element>_<model>_<code name>.txt`

Header: after some optional comment lines, the following four lines give the
number of epochs (NTIMES, 100 in this example), the number of ionisation
stages (NSTAGES, 6 in this example, starting at neutral and up to
$\texttt{NSTAGES}-1$ times ionised), the list of epochs with all NTIMES values
(the ‘…’ should correspond to actual values), and finally one empty comment
line:

            #<optional comment lines>
        #NTIMES: 100
        #NSTAGES: 6
        #TIMES[d]: 2.0 3.0 ... 100.0
        #

Contents: NTIMES blocks (one for each epoch), each containing a block header
with three lines giving the epoch (in days, 2.0 d in this example), the number
of cells saved for this epoch (NVEL, 200 in this example), and finally the
column headings (where we consider the element Fe in this example):

            #TIME: 2.0
        #NVEL: 200
        #vel_mid[km/s] fe0 fe1 fe2 fe3 fe4 fe5

Each block content consists of NVEL rows with the following NSTAGES+1 columns
(7 in this example):

    1. (1)

Velocity at the centre of each cell in $\rm{km\,s}^{-1}$

    2. (2)-(7)

Fraction of ions (dimensionless) in the corresponding cell (fe0 = Fe i, fe1 =
Fe ii etc.). The sum of the fractions in each of the NVEL rows is expected to
be unity.

  5. 5.

Spectral sequence

File name: `spectra_<model>_<code name>.txt`

Header: after some optional comment lines, the following four lines give the
number of epochs (NTIMES, 100 in this example), the number of wavelengths
(NWAVE, 2000 in this example), the list of epochs in days with all NTIMES
values (the ‘…’ should correspond to actual values), and finally the column
headings (here the ‘…’ can be used as is):

            #<optional comment lines>
        #NTIMES: 100
        #NWAVE: 2000
        #TIMES[d]: 2.0 3.0 ... 100.0
        #wavelength[Ang] flux_t0[erg/s/Ang]
         flux_t1[erg/s/Ang] ... flux_tn[erg/s/Ang]

Contents: NWAVE rows with the following NTIMES+1 columns (101 in this
example):

    1. (1)

Wavelength in Å

    2. (2)-(101)

Fluxes at each of the NTIMES epochs at the corresponding wavelength in erg s-1
Å-1

  6. 6.

Synthetic photometry

File name: `wsynphot_mags_<model>_<code name>.txt`

Header: after some optional comment lines, the following three lines give the
number of epochs (NTIMES, 100 in this example), the number of photometric
bands (NBANDS, 8 in this example), and finally the column headings (giving
each band name, $UBVRIJHK$ in our example):

            #<optional comment lines>
        #NTIMES: 100
        #NBANDS: 8
        #time[d] U B V R I J H K

Contents: NTIMES rows with the following NBANDS+1 columns (9 in our example):

    1. (1)

Time since explosion in days

    2. (2)-(9)

Absolute magnitudes in each band at the corresponding time

###  4.3 Useful Python codes

All the figures presented in this paper are automatically generated and
accessible in a Python notebook `all_plots.ipynb` which is available in the
data repository. A separate Python notebook `photometry.ipynb` in the `code-
comparison1/` directory is used to generate the synthetic photometry
(`wsynphot_mags` files) from the `spectra` files on the fly. Three further
useful Python codes can be used to:

  1. 1.

Generate the toy models

Location:
`data1/input_models/mk_snia_toy_model.py`

Calling syntax:

            python mk_snia_toy_model.py --highni
        python mk_snia_toy_model.py --lowni

This code generates the toy models used in this paper, as explained in Sect.
2.1. The upper command generates the toy06 model (`snia_toy06_2d.dat`), while
the lower line generates the toy01 model (`snia_toy01_2d.dat`). Comments
inside the code provide information on how to create new similar test models
with different parameters.

  2. 2.

Read the input files

Location: `data1/input_models/read_inputs.py`

Calling syntax:

            python read_inputs.py

This code reads the input files of the test models and creates the following
files corresponding to Figs. 1 and 2 as well as Table 1: `density_profile.pdf`
(Fig. 1), `composition_profile_<model>.pdf` (where <model> is one of toy06,
toy01, ddc10, and ddc25; Fig. 2), and models_summary.tex (Table 1). The code
also includes the functions read_snia_toy_model() and read_ddc_model() to read
the files into Python variables.

By default, the code is expected to be executed from within the
`data1/input_models/` directory, although the path to the data1/ directory can
be specified using the `--path2data` option.

  3. 3.

Read the output files

Location: `data1/read_outputs.py`

Calling syntax:

            python read_outputs.py file1.txt file2.txt
        python read_outputs.py /path/to/file*.txt

This code reads the six output file types (see 4.2.2) and produces
corresponding plots. The code also includes the functions: read_lbol_edep(),
read_spectra(), read_edep(), read_phys(), read_ionfrac(), and read_mags() to
read the output files into Python variables.

The code can be executed from any directory because the full path of each file
can be specified (wildcards are also accepted). When uploading new output
files to the data repository users are required to ensure they match the
expected format exactly. This can be achieved using the `--checkformat` option
(and optionally the `--noplot` option to disable the plotting functionality
when checking a large number of files).

##  5 Example results

Figure 3: Pseudobolometric (UVOIR) light curves for model toy06. The inset
shows a zoom into the maximum-light epoch (the estimated time of maximum light
is indicated with a ‘+’ sign).  Figure 4: Peak pseudobolometric (UVOIR)
luminosities vs. rise times (from explosion to peak) for all four test models.
The data points corresponding to the CMFGEN, CRAB, and SuperNu calculations of
the toy06 model are difficult to distinguish.  Figure 5: Total energy
deposition rate from $\gamma$ rays and positrons for model toy06, normalised
to the analytic function given by Eq. 10 (the normalisation allows differences
to be seen more clearly).

Figure 6:  Left: Instantaneous ratio of the luminosity and the total energy
deposition rate for model toy06. Right: Ratio of the time-weighted integrals
of the luminosity and energy deposition rate for model toy06 (here we do not
show the results for the SUMO code because the calculation starts at 75 d post
explosion). Both ratios are expected to reach unity at late times.

Figure 7: Ejecta (gas) temperature and ionisation profiles as a function of
velocity for the toy06 model. Left column: Temperature at 15 days (top), 40
days (middle), and 200 days (bottom) post explosion. Right column: Mean
ionisation of cobalt at 15 days (top), 40 days (middle), and 200 days (bottom)
post explosion. We restrict the abscissa range of the ionisation plots to
$v\leq 12000$ $\rm{km\,s}^{-1}$ since the Co abundance drops to 0 at larger
velocities.  Figure 8: Multi-band ($UBVRIJHK$) light curves for the toy06
model. We note the different ordinate range for the $U$ band.

In this section we provide example results that are extracted from the outputs
of the different simulations. The purpose of this section is to illustrate the
contents of the data repository and comparisons that can be made using it. No
comparison to observations or in-depth investigations of the sources of
differences in results are made here. We note that while some groups provided
a few sets of results for different physical approximations that appear in the
repository, only one set of results is shown for each code in this set of
examples. One exception is ARTIS, for which both the regular (artis) and
nebular (artisnebular) versions are included for each model in order to
present results for this code from early to late times.

The results are provided in the following subsections. In Sect. 5.1,
bolometric properties of the emission are shown for simulations of the toy06
model, including the pseudobolometric (UVOIR) light curve and the energy
deposition rate as a function of time. Rise times and peak luminosities are
provided for the toy06 and toy01 models. In Sect. 5.2, the resulting profiles
of temperature and mean ionisation state of cobalt are shown at selected
times. In Sect. 5.3, multi-band light curves and colour curves are shown for
model toy06. $B$-band rise times are provided for all models where available.
Example spectra at selected early and late times are shown in Sect. 5.4.
Throughout the section, each figure provides a legend with the colours or
symbols associated with each code. A uniform coding scheme is used, such that
a given code can be systematically identified in all the figures.

###  5.1 Bolometric (UVOIR) evolution

In this subsection, bolometric properties are shown based on the `lbol_edep`
output files. In Fig. 3, the UVOIR light curves for model toy06 are shown for
the first 100 days after explosion for the different codes. The rise times and
peak luminosities are shown in Fig. 4 for all four test models. The peak
luminosities were obtained by a parabolic fit to light curve around maximum
light.

In Fig. 5, the total energy deposition from $\gamma$ rays and positrons based
on the same files is shown as a function of time for the first 200 days after
explosion for the toy06 model. In order to highlight the differences among
codes, all results are normalised to the same analytic approximation for the
deposition:

$\displaystyle Q_{\rm dep,norm}\equiv$ $\displaystyle\frac{M_{\rm
Ni}}{M_{\odot}}\left(0.97\left[1-e^{-(40d/t)^{2}}\right]+0.03\right)\times$
(10) $\displaystyle\left(6.45\leavevmode\nobreak\
e^{-t/8.8d}\leavevmode\nobreak\ +\leavevmode\nobreak\ 1.45\leavevmode\nobreak\
e^{-t/111.3d}\leavevmode\nobreak\ \right)\leavevmode\nobreak\
10^{43}\leavevmode\nobreak\ \rm erg\leavevmode\nobreak\ s^{-1}$ (11)

where $M_{\rm Ni}=0.6$ $\rm{M}_{\odot}$ is the total (undecayed) 56Ni mass in
the toy06 model151515The prefactor in parenthesis in the first line of Eq.
(10), is equivalent to a simplistic approximation in which a fraction of 0.03
of the energy is emitted in positron kinetic energy which is immediately
deposited in the ejecta while the rest is emitted in gamma-rays with a purely
absorptive optical depth given by $\tau_{\gamma}=t_{0}^{2}/t^{2}$ with a
gamma-ray escape time (e.g. Jeffery, 1999) of $t_{0}=40$d..

In Fig. 6, two physical diagnostics of the relation between the energy
deposition and the bolometric light curves are shown for the toy06 model (both
expected to approach unity at late times): the instantaneous ratio of the
energy deposition rate and the luminosity (left) and the ratio of cumulative
time-weighted integrals of the energy deposition rate and luminosity (right).

###  5.2 Temperature and ionisation profiles

In Fig. 7 the thermodynamic structure of the ejecta as a function of velocity
is shown based on the `phys` and `ionfrac_co` output files for model toy06.
Three different times of particular interest are shown: 15 days post
explosion, close to the peak of the light curve (upper panels), 40 days post
explosion, close to the break in the $B-V$ colour curve (middle panels), and
200 days post explosion, during the nebular phase (lower panels). For each of
these times, two profiles are shown: the (gas) temperature profile (left
panels), and the mean ionisation level of Co, defined as
$\sum\nolimits_{i=0}i\cdot f_{i}$, where $f_{i}$ is the fraction of Co ions
ionised $i$ times (right panels). Co and Fe dominate the opacity at these
times and the ionisation profiles of Fe are very similar to those of Co (not
shown here).

###  5.3 Multi-band light curves and colours

In this section, multi-band properties are shown for the different codes for
different models. The photometry is extracted from the spectra reported in the
`spectra` output files using the wsynphot
package161616https://github.com/starkit/wsynphot (using the Vega calibration
spectrum
`alpha_lyr_stis_003.fits`171717https://archive.stsci.edu/hlsps/reference-
atlases/cdbs/calspec/alpha_lyr_stis_003.fits and a third-order spline
interpolation). In Fig. 8, the $UBVRIJHK$ light curves are shown for model
toy06 up to 100 days after explosion. The $B$-band rise times (from explosion
to peak) are extracted by fitting a high-order polynomial around maximum light
and shown for all models in Fig. 9. The $B-V$ and $V-R$ colour curves for
model toy06 are shown in Fig. 10.

###  5.4 Spectroscopic evolution

Figure 9: Peak magnitude vs. rise time in the $B$ band for all four test
models.

Figure 10: $B-V$ and $V-R$ colour evolution for the toy06 model.

Figure 11: Spectra at 5 d post explosion and at the time of peak UVOIR
luminosity (the caption indicates the time of the spectrum computed closest to
peak) for the toy06 model.

Figure 12: Spectra at 50 d and 200 d post explosion for the toy06 model. The
colour coding for the 50 d spectra is the same as in Fig. 11 and is not
repeated here for sake of clarity.

Spectra obtained for the toy06 model at selected times are shown in Figs. 11
and 12 based on the `spectra` output files. The wavelength range is restricted
to 3000-10000 Å to allow direct comparison. This does not represent the full
range available in the output files, which differs between different codes. In
Fig. 11, early-time spectra (5 days post explosion and around peak luminosity)
are shown while in Fig. 12 we show spectra at later times (50 days and 200
days post explosion). We note that nebular times require specialised treatment
in non-LTE which is only implemented in the ARTIS, CMFGEN, and SUMO codes for
this paper181818SEDONA also has non-LTE capabilities but these were not used
here..

##  6 Discussion and outlook

In this paper, we present an online public data repository that includes test
models for radiative-transfer (RT) calculations of emission from SNe and
standardised simulation outputs by ten different groups that allows direct
comparison. Python scripts that generate the analytic toy models and read the
different formats are provided as well.

The main purpose of the repository is to allow studies of the different
physical approximations involved in the different codes and to assess the
robustness of different predictions of radiation transfer. In addition, the
repository can be used for finding bugs in the codes or can provide checks in
the development of new codes.

We plan to extend the set of test models to include other SN types, as well as
multi-dimensional models. In addition, we intend to produce more specialised
test cases for which exact solutions can be found and agreed upon; these will
provide benchmarks for RT code development. We also aim to include models with
standardised atomic data sets that will allow the study of the effects of
atomic physics on the emission for the different approximations.

Other groups that are developing RT codes are encouraged to add their results
to the repository in the standardised format. Information for this purpose is
provided in the README.md file in the repository.

###### Acknowledgements.

The results presented in this paper are based on work performed before Feb
24th 2022\. We thank the Schwartz/Reisman Institute for Theoretical Physics at
the Weizmann institute of science and the Max Plank Institute for Astrophysics
for hosting the workshops during 2018-2019 which lead to this collaboration.
This work was supported by the ‘Programme National de Physique Stellaire’
(PNPS) of CNRS/INSU co-funded by CEA and CNES. This research was supported by
the Excellence Cluster ORIGINS which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC-2094-390783311. SB acknowledges support from the ESO
Scientific Visitor Programme in Garching. CC acknowledges support by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme under grant agreement No. 759253. AJ
acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 Research and Innovation Program, ERC Starting
Grant 803189 – SUPERSPEC. Lawrence Livermore National Laboratory is operated
by Lawrence Livermore National Security, LLC, for the U.S. Department of
Energy, National Nuclear Security Administration under Contract DE-
AC52-07NA27344. AF and LS acknowledge support by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation program
(ERC Advanced Grant KILONOVA No. 885281). The reported study was funded by the
Russian Scientific Foundation (RSF), project number 19-12-00229, and the
Russian Foundation for Basic Research (RFBR) and the Deutsche
Forschungsgemeinschaft (DFG), project number 21-52-12032. SAS acknowledges
funding from the UKRI STFC Grant ST/T000198/1. Part of this work was performed
using the Cambridge Service for Data Driven Discovery (CSD3), part of which is
operated by the University of Cambridge Research Computing on behalf of the
STFC DiRAC HPC Facility (www.dirac.ac.uk). The DiRAC component of CSD3 was
funded by BEIS capital funding via STFC capital grants ST/P002307/1 and
ST/R002452/1 and STFC operations grant ST/R00689X/1. DiRAC is part of the
National e-Infrastructure. M.W. acknowledges support from the NASA Future
Investigators in NASA Earth and Space Science and Technology grant
(80NSSC21K1849) and support from the Thomas J. Moore Fellowship at New York
University. Los Alamos National Laboratory is operated by Triad National
Security, LLC, for the National Nuclear Security Administration of U.S.
Department of Energy (Contract No. 89233218CNA000001). RTW also acknowledges
Daniel van Rossum for the development of SuperNu that enabled this work.

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##  Appendix A Atomic data

In this Appendix we provide a more detailed overview of the atomic data used
in the calculations done with the ARTIS (Sect. A.1), CMFGEN (Sect. A.2),
SEDONA (Sect. A.3), and TARDIS (Sect. A.4) codes, along with appropriate
references. We refer the reader to Sect. 3 for a more succinct presentation of
the atomic data used by other groups and codes.

###  A.1 ARTIS

Table 5 lists the elements with numbers of ions, levels, and bound-bound
transitions used by the ARTIS code. These are drawn from the Kurucz atomic
line lists (see Kromer & Sim 2009).

Table 5: Summary of atomic data used in ARTIS simulations. Element | Ion stages | Levels | Lines
---|---|---|---
He | I-III | 723 | 1 487
C | I-V | 1 032 | 7 925
N | I-V | 655 | 8 991
O | I-III | 381 | 3 388
F | I-V | 752 | 7 603
Ne | I-V | 740 | 10 462
Na | I-V | 209 | 1 249
Mg | I-V | 724 | 4 141
Al | I-V | 556 | 4 147
Si | I-V | 1 207 | 8 984
P | I-V | 436 | 2 639
S | I-V | 324 | 1 805
Cl | I-V | 469 | 5 739
Ar | I-V | 665 | 11 629
K | I-V | 167 | 1 031
Ca | I-V | 613 | 13 003
Sc | I-VII | 692 | 11 863
Ti | I-VII | 993 | 22 139
V | I-VII | 1 343 | 32 273
Cr | I-VII | 1 777 | 53 557
Mn | I-VII | 1 726 | 38 041
Fe | I-VII | 2 760 | 111 890
Co | I-VII | 1 529 | 34 098
Ni | I-VII | 2 078 | 54 990
Cu | I-V | 360 | 5 072
Zn | I-V | 313 | 3 178
Total |  | 23 224 | 461 324

###  A.2 CMFGEN

Tables 6-11 give the number of levels (both super-levels and full levels; see
Hillier & Miller 1998 and Dessart & Hillier 2010 for details) for the model
atoms used in the radiative-transfer calculations presented in this paper.
$N_{\mathrm{SL}}$ refers to the number of super levels used for the solution
of the rate equations, and $N_{\mathrm{full}}$ refer the number of full levels
used to solve the transfer equation and compute the observed spectrum. We
report the uppermost level for each ion treated in the fourth column. ‘W’
refers to states in which higher $\ell$ states (usually f or higher) have been
combined into a single level. In the last column we give the number of bound-
bound transitions in the model ion taking into account all $N_{\mathrm{full}}$
levels. Ions for which $N_{\mathrm{SL}}=N_{\mathrm{full}}=1$ (with no entries
for the last level configuration or number of lines) correspond to the final
ionisation stage of a given element, for which ionisations to and
recombinations from the ground state are considered. The ions Cl iv, K iii,
and V i were included for the sole purpose of tracking changes in abundance of
radioactive isotopes. The entries for those ions in Tables 6-7 also have
$N_{\mathrm{SL}}=N_{\mathrm{full}}=1$, but we specify the configuration of the
ground state in the last level column and set the number of lines to zero.

Oscillator strengths for CNO elements were originally taken from Nussbaumer &
Storey (1983, 1984). These authors also provide transition probabilities to
states in the ion continuum. The largest source of oscillator data is from
Kurucz (2009)191919Data are available online at http://kurucz.harvard.edu; its
principal advantage over many other sources (e.g. Opacity Project) is that LS
coupling is not assumed. More recently, non-LS oscillator strengths have
become available through the Iron Project (Hummer et al. 1993), and work done
by the atomic-data group at Ohio State University (Nahar 2010). Other
important sources of radiative data for Fe include Becker & Butler (1992,
1995a, 1995b); Nahar (1995). Energy levels have generally been obtained from
the National Institute of Standards and Technology. Collisional data are
sparse, particularly for states far from the ground state. The principal
source for collisional data among low-lying states for a variety of species is
the tabulation by Mendoza (1983); other sources include Berrington et al.
(1985), Lennon et al. (1985), Lennon & Burke (1994), Shine & Linsky (1974),
Tayal (1997a, b), Zhang & Pradhan (1995a, b, 1997). Photoionisation data is
taken from the Opacity Project (Seaton 1987) and the Iron Project (Hummer et
al. 1993). Unfortunately Ni and Co photoionisation data are generally
unavailable, and we used crude approximations. Charge exchange cross-sections
are from the tabulation by Kingdon & Ferland (1996).

Table 6: Model atoms used in CMFGEN calculations for the toy06 model. Ion | $N_{\mathrm{SL}}$ | $N_{\mathrm{full}}$ | Last level | $N_{\mathrm{lines}}$
---|---|---|---|---
Si ii | 32 | 62 | 3s2 7g 2G | 1 032
Si iii | 33 | 61 | 3s 5g3G | 615
Si iv | 37 | 48 | 10f 2Fo | 816
Si v | 1 | 1 |  |
S ii | 56 | 324 | 3s 3p3(5So)4p 6P | 16 965
S iii | 48 | 98 | 3s 3p2(2D)3d 3P | 1 723
S iv | 27 | 67 | 3s 3p(3Po)4p 2D | 1 091
S v | 1 | 1 |  |
Ca ii | 44 | 77 | 3p6 30w 2W | 3 365
Ca iii | 16 | 40 | 3s2 3p5 5s 1Po | 210
Ca iv | 18 | 69 | 3s 3p5(3Po)3d 4Do | 646
Ca v | 1 | 1 |  |
Fe i | 44 | 136 | 3d6(5D)4s 4p x5Fo | 3 934
Fe ii | 228 | 2 698 | 3d5(4F)4s 4p b4Go | 1 060 528
Fe iii | 83 | 698 | 3d5(2H)4d 1K | 73 419
Fe iv | 100 | 1 000 | 3d4(3G)4f 4Po | 144 005
Fe v | 47 | 191 | 3d3(4F)4d 5F | 7 892
Fe vi | 1 | 1 |  |
Co ii | 136 | 2 747 | 3d7(2D)6p 3Po | 1 186 076
Co iii | 124 | 3 917 | 3d6(3D)6d 4P | 1 357 405
Co iv | 37 | 314 | 3d5(2P)4p 3Po | 17 952
Co v | 32 | 387 | 3d4(3F)4d 2H | 27 046
Co vi | 1 | 1 |  |
Ni ii | 59 | 1 000 | 3d8(3F)7f 4Io | 103 224
Ni iii | 47 | 1 000 | 3d7(2D)4d 3Sb | 132 677
Ni iv | 28 | 254 | 3d6(1G1)4p 2Go | 12 512
Ni v | 46 | 183 | 3d5(2D3)4p 3Fo | 6 033
Ni vi | 1 | 1 |  |
Total | 1 328 | 15 377 |  | 4 159 166
Table 7: Model atoms used in CMFGEN calculations for the toy01 model. Ion | $N_{\mathrm{SL}}$ | $N_{\mathrm{full}}$ | Last level | $N_{\mathrm{lines}}$
---|---|---|---|---
Si ii | 32 | 62 | 3s2 7g 2G | 1 032
Si iii | 33 | 61 | 3s 5g3G | 615
Si iv | 37 | 48 | 10f 2Fo | 816
Si v | 1 | 1 |  |
S ii | 56 | 324 | 3s 3p3(5So)4p 6P | 16 965
S iii | 48 | 98 | 3s 3p2(2D)3d 3P | 1 723
S iv | 27 | 67 | 3s 3p(3Po)4p 2D | 1 091
S v | 1 | 1 |  |
Ca ii | 44 | 77 | 3p6 30w 2W | 3 365
Ca iii | 16 | 40 | 3s2 3p5 5s 1Po | 210
Ca iv | 18 | 69 | 3s 3p5(3Po)3d 4Do | 646
Ca v | 1 | 1 |  |
Fe i | 44 | 136 | 3d6(5D)4s 4p x5Fo | 3 934
Fe ii | 228 | 2 698 | 3d5(4F)4s 4p b4Go | 1 060 528
Fe iii | 83 | 698 | 3d5(2H)4d 1K | 73 419
Fe iv | 100 | 1 000 | 3d4(3G)4f 4Po | 144 005
Fe v | 47 | 191 | 3d3(4F)4d 5F | 7 892
Fe vi | 1 | 1 |  |
Co ii | 44 | 162 | 3d6(5D)4s 4p 7Fo | 6 475
Co iii | 33 | 220 | 3d6(3F)4p 4Do | 9 836
Co iv | 27 | 164 | 3d5(4D)4p 3Fo | 5 759
Co v | 32 | 387 | 3d4(3F)4d 2H | 27 046
Co vi | 1 | 1 |  |
Ni ii | 27 | 177 | 3d7(4P)4s 4p(3P) 4So | 5 757
Ni iii | 20 | 107 | 3d7(2H)4p 3Io | 2 228
Ni iv | 20 | 130 | 3d6(3F2)4p 4Do | 3 375
Ni v | 46 | 183 | 3d5(2D3)4p 3Fo | 6 033
Ni vi | 1 | 1 |  |
Total | 1 068 | 7 105 |  | 1 382 750

-.1 Model atoms used in CMFGEN calculations up until 10.56 d post explosion for the DDC10 and DDC25 models. Ion $N_{\mathrm{SL}}$ $N_{\mathrm{full}}$ Last level $N_{\mathrm{lines}}$
Table 7. continued.

Ion $N_{\mathrm{SL}}$ $N_{\mathrm{full}}$ Last level $N_{\mathrm{lines}}$
lrrlr C i & 14 26 2s 2p3 3Po 229
C ii 14 26 2s 2s4d2D 159
C iii 62 112 2s 8f1Fo 1 759
C iv 59 64 30 2 798
C v 1 1
O i 19 51 2s2 2p3(4So)4f 3F 401
O ii 30 111 2s2 2p2(3P)4d 2D 2 242
O iii 50 86 2p 4f1D 1 270
O iv 53 72 2p 2p3p”2Po 1 614
O v 1 1
Ne i 70 139 2s2 2p5(2P¡3/2¿)6d 25/2o 3 126
Ne ii 22 91 2s2 2p4(3P)4d 2P 2 142
Ne iii 23 71 2s2 2p3(2Do)3d 3So 884
Ne iv 1 1
Na i 22 71 30w2W 3 128
Na ii 1 1
Mg ii 22 65 30w2W 2 810
Mg iii 31 99 2p5 7s 1Po 1 526
Mg iv 1 1
Al ii 26 44 3s 5d1D 333
Al iii 17 45 10z2Z 699
Al iv 1 1
Si ii 31 59 3s2(1S)7g2G 683
Si iii 33 61 3s 5g1G 614
Si iv 37 48 10f2Fo 781
Si v 1 1
S ii 56 324 3s 3p3(5So)4p 6P 16 346
S iii 48 98 3s 3p2(2D)3d 3P 1 629
S iv 27 67 3s 3p(3Po)4p 2D 760
S v 1 1
Cl iv 1 1 3p2 3P 0
Cl v 1 1
Ar i 56 110 3s2 3p5(2P¡3/2¿)7p 23/2 3 030
Ar ii 134 415 3s2 3p4(3P¡1¿)7i 26 40 224
Ar iii 32 346 3s2 3p3(2Do)8s 1Do 13 677
Ar iv 1 1
K iii 1 1 3p5 2Po 0
K iv 1 1
Ca ii 21 77 3p6 30w2W 3 365
Ca iii 16 40 3s2 3p5 5s 1Po 210
Ca iv 18 69 3s 3p5(3Po)3d 4Do 646
Ca v 1 1
Sc ii 38 85 3p6 3d 4f 1Po 1 905
Sc iii 25 45 7h 2Ho 454
Sc iv 1 1
Ti ii 37 152 3d2(3F)5p 4Do 6 173
Ti iii 33 206 3d 6f 3Ho 9 392
Ti iv 1 1
V i 1 1 3d3 4s2 a4F 0
V ii 1 1
Cr ii 28 196 3d4(3G)4p x4Go 7 193
Cr iii 30 145 3d3(2D2)4p 3Do 4 661
Cr iv 29 234 3d2(3P)5p 4Po 12 569
Cr v 1 1
Mn ii 25 97 3d4(5D)4s2 c5D 464
Mn iii 30 175 3d4(3G)4p y4Ho 6 292
Mn iv 1 1
Fe i 44 136 3d6(5D)4s 4p x5Fo 3 731
Fe ii 275 827 3d5(6S)4p2(3P)4P 89 426
Fe iii 69 607 3d5(4D)6s3D 19 483
Fe iv 100 1 000 3d4(3G)4f 4Po 144 005
Fe v 47 191 3d3(4F)4d 5F 7 892
Fe vi 44 433 3p5(2P)3d4(1S) 2Pco 27 983
Fe vii 29 153 3p5(2P)3d3(b2D) 1Po 3 414
Fe viii 1 1
Co ii 34 144 3d6(5D)4s 4p 7Do 4 112
Co iii 37 361 3d6(5D)5p 4Po 21 716
Co iv 37 314 3d5(2P)4p 3Po 17 220
Co v 32 387 3d4(3F)4d 2H 27 046
Co vi 28 323 3d3(2D)4d 1S 19 180
Co vii 31 319 3p5(2P)d4(3F) 2Do 18 016
Co viii 1 1
Ni ii 19 93 3d7(4F)4s 4p 6Do 1 639
Ni iii 15 67 3d7(4F)4p 3Do 724
Ni iv 36 200 3d6(3D)4p 2Do 8 066
Ni v 46 183 3d5(2D3)4p 3Fo 6 033
Ni vi 37 314 3d4(5D)4d 4F 18 976
Ni vii 37 308 3d3(2D)4d 3P 18 364
Ni viii 1 1
Total 2 338 10 605 613 214

Notes: Due to a $gf$ cut (level dependent, $gf>2\times 10^{-3}$) only 163 452
lines were included in the non-LTE calculations of the level populations. 308
846 lines were include when computing the observed spectrum.

Table 8: Large Co model atoms used in CMFGEN calculations for the DDC10 (between 11.62 d and 15.47 d post explosion) and DDC25 (between 11.62 d and 17.02 d post explosion) models. Ion | $N_{\mathrm{SL}}$ | $N_{\mathrm{full}}$ | Last level | $N_{\mathrm{lines}}$
---|---|---|---|---
Co ii | 81 | 1 000 | 3d7(4P)4f 5Fo | 123 533
Co iii | 85 | 1 016 | 3d6(5D)5f 6Do | 139 700
Co iv | 56 | 1 000 | 3d5(2D)5s 1D | 138 508
Total | 2 134 | 10 816 |  | 861 562
Table 9: Model atoms used in CMFGEN calculations from 17.02 d post explosion for the DDC10 model and from 18.72 d post explosion for the DDC25 model. Ion | $N_{\mathrm{SL}}$ | $N_{\mathrm{full}}$ | Last level | $N_{\mathrm{lines}}$
---|---|---|---|---
C ii | 14 | 26 | 2s2 4d 2D | 181
C iii | 62 | 112 | 2s 8f1Fo | 1 788
O i | 29 | 75 | 2s2 2p3(4So)5f 3F | 837
O ii | 63 | 143 | 2s2 2p2(3P)5p 2Po | 3 650
O iii | 44 | 86 | 2s 2p2(4P)3p 3Po | 1 013
Ne ii | 22 | 91 | 2s2 2p4(3P)4d 2P | 2 143
Ne iii | 24 | 56 | 2s2 2p3(4So)4p 5P | 457
Mg ii | 31 | 80 | 30w 2W | 3 863
Al iii | 31 | 80 | 30w 2W | 3 892
Si iii | 33 | 61 | 3s 5g3G | 615
S ii | 56 | 324 | 3s 3p3(5So)4p 6P | 16 843
S iii | 48 | 98 | 3s 3p2(2D)3d 3P | 1 633
S iv | 27 | 67 | 3s 3p(3Po)4p 2D | 761
Ar iii | 32 | 346 | 3s2 3p3(2Do)8s 1Do | 13 681
Ti ii | 61 | 1 000 | 3d2(3F)9p 4Fo | 185 756
Cr ii | 28 | 196 | 3d4(3G)4p x4Go | 8 249
Fe iii | 83 | 698 | 3d5(2H)4d 1K | 73 419
Co ii | 109 | 948 | 3d7(4P)6s 3P | 189 440
Co iii | 85 | 1 016 | 3d6(5D)5f 6Do | 139 700
Co iv | 56 | 1 000 | 3d5(2D)5s 1D | 139 240
Ni ii | 59 | 1 000 | 3d8(3F)7f 4Io | 103 224
Ni iii | 47 | 1 000 | 3d7(2D)4d 3Sb | 132 677
Ni iv | 54 | 1 000 | 3d6(5D)6p 6Fo | 145 527
Total | 2 351 | 14 434 |  | 1 539 740
Table 10: Large Co model atoms used in CMFGEN calculations for the DDC10 and DDC25 models from 33.15 d post explosion onward. Ion | $N_{\mathrm{SL}}$ | $N_{\mathrm{full}}$ | Last level | $N_{\mathrm{lines}}$
---|---|---|---|---
Co ii | 136 | 2 747 | 3d7(2D)6p 3Po | 1 186 076
Co iii | 124 | 3 917 | 3d6(3D)6d 4P | 1 357 405
Total | 2 292 | 18 373 |  | 3 713 110
Table 11: Reduced Fe iv, Co iv, and Ni iv model atoms used in CMFGEN calculations for the DDC10 and DDC25 models from 40.11 d post explosion onward. Ion | $N_{\mathrm{SL}}$ | $N_{\mathrm{full}}$ | Last level | $N_{\mathrm{lines}}$
---|---|---|---|---
Fe iv | 35 | 176 | 3d4(3G)4p 4Ho | 6 595
Co iv | 37 | 314 | 3d5(2P)4p 3Po | 17 952
Ni iv | 28 | 254 | 3d6(1G1)4p 2Go | 12 512
Total | 2 182 | 16 117 |  | 3 321 397

###  A.3 SEDONA

SEDONA used the Kurucz CD 1 line list to compute the bound-bound opacities.

Table 12: Bound-bound atomic data information from the Kurucz CD 1 line list for the toy06 and toy01 models computed with SEDONA Element $Z$ | Element $A$ | $N_{\mathrm{stages}}$ | $N_{\mathrm{lines}}$
---|---|---|---
28 | 56 | 6 | 7 609 586
28 | 58 | 6 | 7 609 586
27 | 56 | 6 | 8 024 034
26 | 56 | 6 | 6 620 297
20 | 40 | 6 | 632 282
16 | 32 | 6 | 2 289
14 | 28 | 6 | 8 709
Total |  |  | 22 897 197
Table 13: Bound-bound atomic data information from the Kurucz CD 1 loneliest for the DDC10 model computed with SEDONA Element $Z$ | Element $A$ | $N_{\mathrm{stages}}$ | $N_{\mathrm{lines}}$
---|---|---|---
28 | 56 | 5 | 5 543 966
28 | 58 | 5 | 5 543 966
27 | 57 | 5 | 7 066 365
27 | 56 | 5 | 7 066 365
26 | 56 | 5 | 6 175 428
24 | 52 | 5 | 3 010 793
24 | 48 | 5 | 3 010 793
23 | 48 | 5 | 2 226 269
22 | 48 | 5 | 939 560
22 | 44 | 5 | 939 560
20 | 40 | 4 | 101 996
16 | 32 | 5 | 2 142
14 | 28 | 5 | 8 705
12 | 24 | 3 | 3 127
10 | 20 | 3 | 11 428
8 | 16 | 6 | 8 062
6 | 12 | 6 | 8 139
Total |  |  | 25 105 980

###  A.4 TARDIS

TARDIS used the Kurucz CD 23 line list to compute the bound-bound opacities.
The atomic data file was generated with the Carsus package on August 24, 2017,
named `kurucz_cd23_chianti_H_He.h5` and signed with UUID1
`6f7b09e887a311e7a06b246e96350010` and MD5 `864f1753714343c41f99cb065710cace`.

Table A.4 tabulates the total number of levels ($N_{\mathrm{levels}}$), meta-
stable levels ($N_{\mathrm{meta}}$), and lines ($N_{\mathrm{lines}}$) for the
atoms used in the four models. Atoms present in the atomic data file but not
used by the models were not listed.

-.1 Bound-bound atomic data information from the Kurucz CD 23 line list for all models computed with TARDIS. Ion $N_{\mathrm{levels}}$ $N_{\mathrm{meta}}$ $N_{\mathrm{lines}}$
Table A.4. continued.

Ion $N_{\mathrm{levels}}$ $N_{\mathrm{meta}}$ $N_{\mathrm{lines}}$
lrrr C i & 833 603 3249
C ii 86 6 374
C iii 81 6 388
C iv 36 1 192
O i 150 9 854
O ii 173 6 1374
O iii 141 9 766
O iv 146 19 465
O v 97 5 459
Ne i 284 23 2422
Ne ii 283 9 3468
Ne iii 64 9 269
Ne iv 99 12 340
Ne v 64 13 164
Na i 58 1 334
Na ii 35 3 171
Na iii 69 4 353
Na iv 46 5 110
Na v 71 10 187
Mg i 552 366 1580
Mg ii 75 2 510
Mg iii 93 4 704
Mg iv 54 5 169
Mg v 53 5 132
Al i 273 160 482
Al ii 197 10 2602
Al iii 58 1 342
Al iv 31 3 142
Al v 56 9 77
Si i 558 230 3856
Si ii 100 13 567
Si iii 169 10 1248
Si iv 52 1 307
Si v 35 3 125
S i 153 12 727
S ii 85 7 500
S iii 58 22 170
S iv 28 5 50
S v 19 5 41
Cl i 229 23 2542
Cl ii 128 21 973
Cl iii 78 13 431
Cl iv 33 5 121
Cl v 27 5 43
Ar i 215 8 2397
Ar ii 314 26 4567
Ar iii 96 16 655
Ar iv 39 7 104
Ar v 22 6 49
K i 94 15 575
K ii 22 4 66
K iii 40 5 192
K iv 24 5 57
K v 33 9 75
Ca i 198 5 2906
Ca ii 93 3 752
Ca iii 150 13 1766
Ca iv 70 26 122
Ca v 39 5 91
Sc i 272 11 4221
Sc ii 168 14 2215
Sc iii 43 3 217
Sc iv 127 13 953
Sc v 22 6 29
Ti i 441 38 8771
Ti ii 204 37 2597
Ti iii 199 13 2289
Ti iv 39 3 139
Ti v 51 13 331
V i 502 60 6995
V ii 323 75 4545
V iii 299 35 5304
V iv 98 13 995
V v 64 3 335
Cr i 394 83 4172
Cr ii 733 97 17224
Cr iii 214 71 2122
Cr iv 154 35 1717
Cr v 46 13 220
Mn i 322 52 3023
Mn ii 569 94 8362
Mn iii 391 95 5848
Mn iv 103 40 677
Mn v 84 35 602
Fe i 848 68 22905
Fe ii 796 85 21753
Fe iii 566 97 9860
Fe iv 276 99 3559
Fe v 180 71 1865
Co i 317 46 5298
Co ii 256 47 2853
Co iii 213 58 2247
Co iv 296 96 4092
Co v 267 94 3542
Ni i 180 17 2671
Ni ii 717 25 17150
Ni iii 344 47 5456
Ni iv 235 70 2712
Ni v 323 101 4733
Total 19135 3819 243353