# Critical ingredients of Type Ia supernova radiative-transfer modelling

Luc Dessart, D. John Hillier, Stephane Blondin, Alexei Khokhlov

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Luc Dessart, D. John Hillier, Stephane Blondin, Alexei Khokhlov. Critical ingredients of Type Ia supernova radiative-transfer modelling. Monthly Notices of the Royal Astronomical Society, 2014, 441 (4), pp.3249–3270. 10.1093/mnras/stu789. hal-01441015

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# Critical ingredients of Type Ia supernova radiative-transfer modelling

Luc Dessart,<sup>1,2*</sup> D. John Hillier,<sup>3</sup> Stéphane Blondin<sup>1</sup> and Alexei Khokhlov<sup>4</sup>

Accepted 2014 April 17. Received 2014 April 16; in original form 2013 August 28

## ABSTRACT

We explore the physics of Type Ia supernova (SN Ia) light curves and spectra using the 1D non-local thermodynamic equilibrium (non-LTE) time-dependent radiative-transfer code CMFGEN. Rather than adjusting ejecta properties to match observations, we select as input one ‘standard’ 1D Chandrasekhar-mass delayed-detonation hydrodynamical model, and then explore the sensitivity of radiation and gas properties of the ejecta on radiative-transfer modelling assumptions. The correct computation of SN Ia radiation is not exclusively a solution to an ‘opacity problem’, characterized by the treatment of a large number of lines. We demonstrate that the key is to identify and treat important atomic processes consistently. This is not limited to treating line blanketing in non-LTE. We show that including forbidden-line transitions of metals, and in particular Co, is increasingly important for the temperature and ionization of the gas beyond maximum light. Non-thermal ionization and excitation are also critical since they affect the colour evolution and the $\Delta M_{15}$ decline rate of our model. While impacting little the bolometric luminosity, a more complete treatment of decay routes leads to enhanced line blanketing, e.g. associated with $^{48}$Ti in the *U* and *B* bands. Overall, we find that SN Ia radiation properties are influenced in a complicated way by the atomic data we employ, so that obtaining converged results is a real challenge. Nonetheless, with our fully fledged CMFGEN model, we obtain good agreement with the golden standard Type Ia SN 2005cf in the optical and near-IR, from 5 to 60 d after explosion, suggesting that assuming spherical symmetry is not detrimental to SN Ia radiative-transfer modelling at these times. Multi-D effects no doubt matter, but they are perhaps less important than accurately treating the non-LTE processes that are crucial to obtain reliable temperature and ionization structures.

**Key words:** radiative transfer – supernovae: general – supernovae: individual: 2005cf – white dwarfs.

## 1 INTRODUCTION

Over the last two decades Type Ia supernovae (SNe Ia) have become important tools for measuring basic cosmological parameters and the energy content of the Universe (Riess et al. 1998; Perlmutter et al. 1999). SNe Ia are likely the explosions of carbon-oxygen de-generate stars in binary systems (Hoyle & Fowler 1960). However, the evolutionary channels leading to SN Ia events are only crudely understood, and the physical state of the progenitor star and details of the explosion mechanism(s) are still debated.

In a single-degenerate scenario, the white dwarf (WD) evolves towards explosion by accreting hydrogen or helium from a non-degenerate stellar companion (Whelan & Iben 1973; Nomoto 1982), whereas in a double-degenerate scenario the explosion is caused by a merger of two degenerate stars (Iben & Tutukov 1984; Web-bink 1984). At present, the delayed-detonation models with varying deflagration-to-detonation transition density in a Chandrasekhar-mass WD offer good agreement with SN Ia observations, e.g. for the range of luminosities and the stratification of chemical elements (Khokhlov, Mueller & Hoeflich 1993; Hoeflich et al. 1996).

The study of the photometric and spectroscopic properties of SNe Ia requires numerical radiative-transfer tools. The early work of Arnett (1982) and Pinto & Eastman (2000a) used analytic modelling to extract a basic understanding of their bolometric light curve and

<footnote n="1"><sup>1</sup>Aix Marseille Université, CNRS, LAM (Laboratoire d'Astrophysique de Marseille) UMR 7326, F-13388 Marseille, France</footnote>

<footnote n="2"><sup>2</sup>Laboratoire Lagrange, UMR7293, Université Nice Sophia-Antipolis, CNRS, Observatoire de la Côte d'Azur, F-06300 Nice, France</footnote>

<footnote n="3"><sup>3</sup>Department of Physics and Astronomy and Pittsburgh Particle physics, Astrophysics, and Cosmology Center (PIIT PACC), University of Pittsburgh, Pittsburgh, PA 15260, USA</footnote>

<footnote n="4"><sup>4</sup>Department of Astronomy and Astrophysics, the Enrico Fermi Institute, and the Computational Institute, The University of Chicago, Chicago, IL 60637, USA</footnote>

<footnote>*E-mail: Luc.Dessart@oca.eu</footnote>
to estimate the ejecta kinetic energy and $^{56}$Ni mass. Unfortunately, this approach does not yield constraints on important ejecta properties, such as chemical composition and stratification, and lacks information on colour evolution.

There are a number of approaches for doing the radiative transfer of SNe Ia more accurately. One approach is to treat the photospheric layers exclusively and assume steady state. Radiation transport is undertaken using the Monte Carlo technique (e.g. Mazzali & Lucy 1993) or by solving the transfer equation (e.g. Nugent et al. 1995; Pauldrach et al. 1996; Baron et al. 1996; Blondin et al. 2006; Branch et al. 2006; Pauldrach, Hoffmann & Hultzsch 2013). The advantages of this approach are computational speed, and the ease with which model parameters and abundances can be altered to fit models to observations. The drawback is that model adjustments may be used erroneously to overcome missing physics (either in the atomic data or in the model). One major disadvantage of the method is that the infrared spectral range is optically thin even before the B-band maximum, and the concept of a well-defined photosphere becomes meaningless.

Another approach is to perform time-dependent radiation transport and model the entire SN ejecta. This has been done using grey or multigroup radiative transfer (see, e.g. Hoeflich, Mueller & Khokhlov 1993; Hoeflich & Khokhlov 1996; Blinnikov et al. 1998, 2006). The gas is treated in local thermodynamic equilibrium (LTE) and the opacities are approximated using the formalism of Karp et al. (1977) and Pinto & Eastman (2000b). Continuum and line opacity contributions are summed over an energy bin and the transfer is solved for each bin. One alternative uses the Monte Carlo technique (Lucy 2005; Kasen, Thomas & Nugent 2006; Sim 2007; Kromer & Sim 2009; Sim et al. 2013). The benefit is the possibility of extension to 3D but the drawback is again the approximate treatment of the thermodynamic state of the gas.

Whether we consider SNe or stellar atmospheres, the leakage of radiative energy through the photosphere is known to drive the material out of LTE (Mihalas 1978). However, because of the fast expansion and small mass of SNe Ia, the low ejecta density prevents LTE conditions even at depth as early as the peak of the bolometric light curve. There is thus much interest in designing radiative-transfer tools that explicitly treat the non-LTE aspects of the problem, i.e. by solving the statistical-equilibrium equations directly (Baron et al. 1996), while solving simultaneously for the radiation transport problem time dependently (Höflich et al. 2002; Jack, Hauschildt & Baron 2011; Hillier & Dessart 2012).

In this work, we discuss our own efforts, started in 2008, to model SNe Ia with CMFGEN. In its present form, the code contains a number of important improvements implemented in recent years, primarily for the modelling of core-collapse SNe. The line-blanketed aspects are discussed in Hillier & Miller (1998) in the context of hot star winds; the modifications to treat SN atmospheres are discussed in Dessart & Hillier (2005a,b); the extension for the time-dependent treatment of the statistical-equilibrium equations is presented in Dessart & Hillier (2008); the philosophy of the full time-dependent approach for both the gas and the radiation is given in Dessart & Hillier (2010), with details given in Hillier & Dessart (2012). Our approach is non-LTE, time dependent and solves for the gas and radiation properties at all depths, from the innermost to the outermost ejecta mass shells.

The non-LTE treatment applies to the full radiative-transfer problem, hence allows the same level of sophistication for the computation of the light curves and the spectra. The multiband light curves are computed by direct integration of the emergent wavelength-dependent flux computed by the non-LTE time-dependent solver

along the time sequence. Our non-LTE approach conserves energy and provides a physical solution to multiband light curves and spectral evolution simultaneously. The interaction between radiation and matter is solved exactly, i.e. without any ad hoc prescription for the nature of opacity and emissivity sources.

In Blondin et al. (2013), we presented the results for a set of delayed-detonation models and compared their radiative properties to observed SNe Ia at bolometric maximum. Here, we discuss the technical aspects of SN Ia radiative transfer modelling using the hydrodynamical delayed-detonation model DDC10. In a future paper, we will cover in greater depth the properties of the radiative transfer in SN Ia ejecta and photospheres, discussing the departures from LTE, the thermalization/scattering character of spectral lines, as well as spectrum formation. We delay to subsequent papers the discussion of dependencies of the SN Ia radiation on ejecta properties, in particular on the abundance of $^{56}$Ni synthesized in the explosion.

An important message from our work on SNe Ia is that with detailed non-LTE radiative transfer, we can reproduce the fundamental SN Ia light curve and spectral properties with the basic delayed-detonation scenario, even with the assumption of spherical symmetry. Although this represents a very important result, it has limited value if we do not understand why or how it works. Until recently, we were unable to reproduce the fundamental radiative properties of SNe Ia. The reasons for this failure were related to assumptions in the modelling, rather than issues with the properties of the progenitor and the explosion model. So, rather than only presenting the properties of the radiative-transfer model that works, we also present the original models we ran and describe how they failed (Section 3). We describe the numerous attempts to solve the discrepancies, roughly in a chronological order, and present some ingredients that solve these problems (Section 3.3). We then discuss the impact of non-local energy deposition and γ-ray escape on SN Ia properties (Section 4), as well as the influence of non-thermal processes (Section 5). Having covered the various ingredients controlling SN Ia radiation, we study the origin of the secondary maximum observed in near-IR SN Ia light curves (Section 6). We present our conclusions in Section 7.

## 2 NUMERICS

All simulations presented in this work start from the same delayed-detonation model DDC10. We refer the reader to Blondin et al. (2013) for a description of the hydrodynamical model and the basic set up for the radiative transfer calculations.

CMFGEN works in the same standard form for any SN ejecta. Hence, the simulations we present here are all carried out with the standard line-blanketed non-LTE time-dependent radiative-transfer technique presented in Hillier & Dessart (2012). What differentiates one SN simulation from another is the hydrodynamical input (composition, etc.) and the model atoms employed. Because the SN Ia ejecta are thin early on, we allow for non-local energy deposition in all models beyond 10 d after explosion, unless stated otherwise. As discussed in the appendix of Hillier & Dessart (2012), we use a Monte Carlo approach for the γ-ray transport. Similarly, given the large abundance of unstable nuclei, we treat non-thermal processes, with the method presented in Li, Hillier & Dessart (2012). In its original form, the non-thermal solver included excitation rates for all ions but ionization rates only for the ions specified in an input file. In the course of this work, we realized that not all intermediate-mass elements (IMEs) and iron-group elements (IGEs) were included in this file. We had C, O, Ne, Na, Mg, Al, Si, S, Ar, Ca, Fe and Ni,
but by mistake did not have entries for Ti, Cr and Co. This was cor-
rected in model DDC10_A4D1 using additional cross-sections from
Mazzotta et al. (1998), to complement those provided by Arnaud &
Rothenflug (1985).

The flexibility in CMFGEN permits the testing of numerous effects,
in particular the influence of model atoms whose characteristics
can be easily adjusted (e.g. number of levels, number of transitions,
superlevel assignments, source of atomic data; see Appendix A). In
our simulations, we include 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, V I, Cl IV,
K III, Cr II–IV, Mn II–III, Fe I–VII, Co II–VII and Ni II–VII. When a given
ion becomes sub-dominant at all depths because of recombination,
its impact on the radiative transfer becomes negligible and it is
conveniently excluded from the computation at all subsequent times.

Species V, Cl, K are only included if/because they belong to decay
routes that we wanted to incorporate in the calculation (see below).
However, because we are not attempting to describe accurately their
potential impact on the transfer (and also because we do not have
a satisfactory set of model atoms for the corresponding ions), we
only include the ground state of a single ionization state for each of
these species (i.e. V I, Cl IV and K III).

We discuss the properties of SN Ia radiation at times that en-
compass the initial brightening of the object (from day one to the
light-curve peak), the maximum light properties (see also Blondin
et al. 2013) and the transition to the nebular phase until 60 d after ex-
plosion. This bridges very diverse conditions – from optically thick
to optically thin and with different processes dominant in different
density regimes. The code handles this evolution in a smooth fash-
ion. In practice, the inner boundary for the radiative transfer is the
same as the inner boundary of the ejecta, i.e. we do time-dependent
simulations for the full ejecta at all times. In model DDC10, this
inner boundary is at a velocity V₀ of 490 km s⁻<fnref n="1" />. Because there is
no dynamics in our simulations, the velocity distribution in mass
space is fixed throughout a CMFGEN sequence. When the ejecta is
optically thick, we impose a zero-flux condition at V₀, i.e. H_v = 0 at
all frequencies v, and the incoming intensity I_v⁺ is set equal to the
local Planck function at v. When the ejecta turns thin, we impose a
nebular condition at V₀, i.e. I_v⁺ = I_v⁻, where v and v' are shifted to
account for the Doppler shift along the ray (see Hillier & Dessart
2012 for details). It is important to realize that, in this approach,
the entire ejecta is modelled at all times by CMFGEN, i.e. that the
radiative transfer is solved at all depths with merely a change in
inner boundary condition at V₀ when the ejecta turns nebular. This
transition occurs earlier in the near-IR than in the UV, so we tend to
switch to the nebular condition when the Rosseland-mean optical
depth is still well above unity.

To begin a sequence, we map the explosion structure into CMFGEN
although we needed to impose a floor temperature of ~6000 K.<fnref n="1" />
This artificially stores energy in the corresponding layers. Hence,
in all time sequences, the ejecta needs to first relax by radiating
away this excess energy. This usually takes only a few time steps
since these layers have a relatively low optical depth. We generally
exclude such results from the presentation and focus on times suf-
ficiently advanced that this initial tinkering bears no impact on the
ejecta and radiation properties. The general lack of SN Ia observa-
tions prior to ≲1 d and computational tractability motivates a start
time at about 1 d after explosion, which is our standard choice here.
This time is early enough so that the initial evolution until 1 d after
explosion can be done assuming no diffusion (we do this with a

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Top: illustration of the ejecta chemical stratification in velocity and mass space (top axis) for model DDC10 and for representative species including C/O/Ne, and representative IMEs and IGEs. We also overplot the $^{56}$Ni distribution as a dotted line. Note the absence of $^{56}$Ni in the inner ejecta layers. The time is 29 s after explosion. Middle: ejecta gas temperature at 1 d after explosion. Note the enforced floor temperature in the outer ejecta, which is necessary for the initial relaxation of the first model in any time sequence. Bottom: same as middle, but now for the mass density.</figcaption></figure>

separate program; see Dessart et al. 2011). The standard procedure
in CMFGEN is to adopt a time step equal to 10 per cent of the current
time.

We show various ejecta properties in Fig. 1. In the top panel,
we plot the chemical stratification versus velocity and Lagrangian
mass at the end of the hydrodynamical simulation – the time is
then 29 s after the start of the combustion in the WD. We only

<footnote n="1">¹ Newer models can use a substantially lower floor temperature.</footnote>
<table><caption>**Table 1.** Summary of model assumptions for our SN Ia radiative-transfer calculations with CMFGEN. All radiative-transfer simulations are based on the delayed-detonation model named DDC10, which is characterized by an initial $^{56}$Ni mass of 0.65 M$_{\odot}$ (see Blondin et al. 2013 for details), and start at one day after explosion. Simulations include either one- ($^{56}$Ni) or all two-step decay chains presented in Table B1. γ-ray energy deposition is treated as local ('L') or solved for using a Monte Carlo transport approach ('NL'). Atom refers to the characteristics of the model atoms used for the CMFGEN calculations.</caption><thead><tr><th>Model</th><th>Decays</th><th>E<sub>dep</sub></th><th>Non-thermal</th><th>Atom</th></tr></thead><tbody><tr><td>DDC10_A0</td><td><sup>56</sup>Ni</td><td>NL</td><td>Yes</td><td>Small</td></tr><tr><td>DDC10_A1</td><td><sup>56</sup>Ni</td><td>NL</td><td>Yes</td><td>Big</td></tr><tr><td>DDC10_A2<sup>a</sup></td><td><sup>56</sup>Ni</td><td>NL</td><td>Yes</td><td>Huge</td></tr><tr><td>DDC10_A1D1</td><td>Two-step</td><td>NL</td><td>Yes</td><td>Big</td></tr><tr><td>DDC10_A3</td><td><sup>56</sup>Ni</td><td>NL</td><td>Yes</td><td>Big + [Co III]</td></tr><tr><td>DDC10_A3D1</td><td>Two-step</td><td>NL</td><td>Yes</td><td>Big + [Co III]</td></tr><tr><td>DDC10_A3L</td><td><sup>56</sup>Ni</td><td>L</td><td>Yes</td><td>Big + [Co III]</td></tr><tr><td>DDC10_A3T</td><td><sup>56</sup>Ni</td><td>NL</td><td>No</td><td>Big + [Co III]</td></tr><tr><td>DDC10_A4D1<sup>b</sup></td><td>Two-step</td><td>NL</td><td>Yes</td><td>Huge + [IME,IGE]</td></tr></tbody></table>

<sup>a</sup>For model DDC10_A2, we only perform a few calculations at selected post-explosion times, rather than computing a full sequence (Section 3.2).

<sup>b</sup>The model atom for DDC10_A4D1 is the same as for model DDC10_A2, but also includes forbidden-line transitions of all metals. Furthermore, unlike previous simulations, non-thermal ionization is included for all IMEs and IGEs.

show representative species, namely C (unburnt), O (unburnt or produced by C burning), the most abundant IMEs, and the IGEs Fe and Ni. Particularly striking is the low $^{56}$Ni abundance at velocities less than 2000 km s$^{-1}$ (termed the nickel hole). This 'hole' is a signature of 1D Chandrasekhar-mass SN Ia explosions and it stems from the relatively high central density of such massive WDs.<fnref n="2" /> The lower two panels describe the temperature and the mass density at 1 d after explosion. The gridding of the CMFGEN calculation, which is approximately equally spaced on a logarithmic optical-depth scale (but with constraints on the change in velocity across grid points), is shown with symbols. We typically use 110 depth points in our radiative-transfer simulations, but this number can increase or decrease by ~10 per cent depending on ejecta conditions (e.g. formation of steep ionization fronts). As time proceeds, the spectrum formation region recedes to deeper layers so we tend to reduce the maximum radius (or velocity) with time, while keeping the same number of depth points; the resolution thus improves as we progress along a time sequence.

In the following sections, we describe the various models we have computed (see also the summary given in Table 1). For model DDC10_A2, which treats nearly two million lines in non-LTE, we only do a few simulations at selected times.

## 2.1 Variations on adopted model atoms

We have used five different sets, including a small (suffix A0), a big (A1, A3) and a huge model atom (A2, A4). We have also run sets of sequences in which the Co III model atom was modified to include forbidden-line transitions (A3). The most complete model atom is A4, which we use in one sequence after bolometric maximum. It includes a huge model atom of the same size as A2 (with emphasis on Co II and Co III), but forbidden-line transitions are also included

for all ions associated with IMEs and IGEs. In model A4, we also updated several atomic models. A description of the model atoms is provided in Appendix A. The specific model names are DDC10_A0, DDC10_A1, DDC10_A2, DDC10_A3 and DDC10_A4.

Extensive testing has shown that employing a large model atom for iron is necessary, even when that species is not very abundant, as in SNe II. So, all sets of model atoms include a large model atom for Fe II to Fe IV (Dessart & Hillier 2010; Li et al. 2012; Dessart et al. 2013a,b) – other ionization stages of Fe are given a modest-size model atom because they do not dominate and are only present at times when and locations where radiative diffusion is very inefficient due to the small photon mean-free-path in the corresponding ejecta regions.

## 2.2 Local versus non-local energy deposition

Explosions of Chandrasekhar-mass WDs yielding ejecta with a kinetic energy on the order of 1 B become thin to γ-rays as early as two weeks after explosion (see, e.g. Hoeflich, Khokhlov & Mueller 1992). To document the implications on both SN Ia spectra and light curves, we have computed a sequence where local energy deposition is assumed (model DDC10_A3L) – all other simulations are performed with allowance for non-local energy deposition past 10 days after explosion, using the γ-ray Monte Carlo transport code described in the appendix of Hillier & Dessart (2012).

## 2.3 Influence of decay chains included

Because of the prevalent role of $^{56}$Ni and $^{56}$Co in controlling SN Ia radiative properties, the general custom is to include only that decay chain. In reality, these explosions produce a variety of unstable nuclei, either IMEs or IGEs, that take part in two-step or one-step decay chains. These nuclei have a range of lifetimes, from less than a day to years, and can thus influence SN Ia ejecta on very different time-scales.

In this paper, we thus explore the effect associated with the treatment of additional two-step decay chains (Table B1; additional one-step decay chains, with an especially important influence at early times, is discussed in Dessart et al. 2014b). Besides their impact on the internal energy of the gas, these decays modify the composition and hence can alter the predicted spectra through changes in line blanketing.

Some variants of model DDC10 are thus run with different assumptions regarding nuclear decay – some isotopes may be treated as stable even if unstable by nature. In CMFGEN, physically unstable isotopes are treated as unstable only if the associated decay route is considered in the calculation. By default, simulations include only the $^{56}$Ni two-step decay chain. Simulations that include all two-step decay chains described in Appendix B have suffix 'D1'.

## 2.4 Influence of non-thermal processes

In Li et al. (2012) and Dessart et al. (2012), we have presented our treatment of non-thermal processes arising from γ-ray emission of unstable nuclei. We test their importance by running the model sequence DDC10_A3T in which these non-thermal processes are ignored – all other model sequences include non-thermal processes.

In the course of this work, we realized that all models and their variants up to DDC10_A3 (see Table 1) did not include non-thermal ionization for Ti, Cr and Co, but treated non-thermal excitation as expected. We thus run the new model DDC10_A4 with non-thermal ionization accounted for all species/ions. Non-thermal processes are

<footnote n="2">² This feature, however, does not seem to persist in 3D simulations of delayed detonations (Seitenzahl et al. 2013).</footnote>
consequently stronger in model DDC10\_A4 than in DDC10\_A3, implying that the differences we discuss with model DDC10\_A3T are in reality even larger. We do not show that specific comparison because model DDC10\_A4 differs with model DDC10\_A3T in more ways than the non-thermal treatment alone. For historical reasons, the sequence using model atom A4 also included all two-step decay chains, so this model is called DDC10\_A4D1.

## 2.5 Comparison to observations

Throughout this paper, we test the compatibility of our synthetic spectra and multiband light curves against the well observed SN 2005cf. We use the optical spectra published in Garavini et al. (2007) and Bufano et al. (2009). We use near-IR spectra published in Gall et al. (2012), from which we also adopt the *B*-band maximum time of JD 245 3534.0. We also use photometry from Pastorello et al. (2007). As in Blondin et al. (2013), we adopt a distance modulus of 32.17 mag and a total reddening (Milky Way and host galaxy) $E(B-V)$ of 0.22 mag (Wang et al. 2009). We use $R_V = 3.1$ and the extinction law of Cardelli, Clayton & Mathis (1989).

We use SN 2005cf primarily to test how our model DDC10, which has a $^{56}$Ni mass similar to that inferred for SN 2005cf (Blondin et al. 2013), compares with its multi-epoch multiwavelength observations. For each comparison, we take the model that is the closest to the observation time, implying an offset of $\lesssim 1$ d around bolometric maximum and $\sim 2$ d at late times. A detailed discussion of the match to specific line features, line identifications, line widths, colours, decline rates, etc. is left to a future paper on the specific modelling of SN 2005cf.

Throughout this paper, we use a single ejecta model, i.e. DDC10, without altering any of its properties. In this sense, it would be fortuitous if our synthetic spectra matched every spectral feature and if our multiband light curves matched all the photometric properties of this SN. A corollary is that model DDC10, with its 0.65 $M_{\odot}$ of $^{56}$Ni, cannot be used to compare with any sub-luminous or superluminous SN Ia, for obvious reasons.

# 3 REPRODUCING THE FUNDAMENTAL PROPERTIES OF AN SN Ia

In this section, we present in a chronological way the work we have done. For a number of years, we failed to reproduce the most fundamental colour properties of SNe Ia beyond the peak of the light curve, for reasons that became clear only recently.

## 3.1 Statement of the problem

Previous studies have emphasized the difficulty of modelling SNe Ia – the ejecta are rich in metals and the problem is time dependent (Pinto & Eastman 2000a,b). While some invoke the need for millions to billions of lines to model the transport adequately (Kasen et al. 2008), some attempts in the mid-1990s, which obtained a satisfactory match to observations, employed not even a million lines (Hoflich 1995). Although Baron et al. (1996) emphasized the importance of non-LTE effects, many SN Ia simulations assume either full LTE for the gas state (populations, ionization) or use a nebular approximation for the ionization together with LTE for the level populations. Since reasonable fits have been obtained with a variety of techniques, it is still unclear what are the critical ingredients for modelling SNe Ia.

Over the years, we have explored the influence of model atoms on our results in the context of core-collapse SNe (Dessart &

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Top: bolometric light curve for model DDC10\_A0. Middle: same as top, but now showing the evolution of the fractional luminosity falling over the photometric passbands *UBVRI*. For comparison, we show the corresponding luminosity for SN 2005cf inferred from its *UBVRI* photometry. Bottom: same as in middle panel, but now showing a comparison of model DDC10\_A0 photometry and counterparts for SN 2005cf – the size of filled-dots is $\sim 0.25$ mag. While the *V* band is matched at all epochs, there is very poor agreement beyond maximum for the *UBJHK* bands. We use a distance modulus of 32.17 mag, an $E(B-V)$ of 0.22 mag, and an $R_V$ of 3.1.</figcaption></figure>

Hillier 2010, 2011; Dessart et al. 2011, 2013a; Li et al. 2012). This revealed the critical importance of including large model atoms for iron, primarily Fe I and Fe II in the low-ionization conditions of SNe II/Ib/Ic. The ionization conditions in SN Ia ejecta are typically much larger, so we started with large model atoms for Fe II–IV. The complete model atom for model DDC10\_A0 is given in Table A1. Starting at day one, we evolved the ejecta up until 60 d after explosion and show the resulting bolometric light curve in the top panel of Fig. 2. We obtain a rise time of 17.43 d to a maximum bolometric luminosity of $1.45 \times 10^{43}$ erg s$^{-1}$.

Since it is not possible to directly compare the bolometric luminosity to observations, we confront the fractional luminosity falling
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Comparison between the spectral evolution of model DDC10\_A0 (red) and the observations of SN 2005cf (black) – we show $F_\lambda$ on a linear scale. Times are given with respect to B-band maximum. Synthetic spectra have been reddened, redshifted and scaled to match the distance to SN 2005cf. Spectra are also scaled vertically for better visibility; the label on the right gives the true B-band magnitude offset between model and observations at each date. After good agreement up to bolometric maximum, models DDC10\_A0 and SN 2005cf eventually disagree, the model retaining a blue colour becomes more and more discrepant.</figcaption></figure>

over the bands *UBVRI* in model DDC10\_A0 and infer the corresponding quantity for SN 2005cf. The agreement is satisfactory around the peak of the light curve, but the agreement becomes poorer as time progresses into the nebular phase (middle panel of Fig. 2). This disagreement could arise in two ways – we have too much late time energy deposition in the model, or alternatively, there is too much model flux coming out in the *UBVRI* bands. In model DDC10\_A0, the bulk of the flux comes out in the range 3000 Å–1 µm. This fraction rises steadily from 70 per cent at 1 d to 90 per cent at the peak of the light curve, and it stays at 90 per cent until the end of the simulation at 60 d. The flux falling shortward of 3000 Å is typically a few per cent, and that falling in the near-IR is typically half that.

To understand the colour evolution, we compare the multi-band light curves of model DDC10\_A0 with those observed for SN 2005cf (bottom panel of Fig. 2). While we obtain a good match to the V-band light curve at all times, the match to other bands is satisfactory only up to the peak. Beyond the peak, the model is obviously too blue, showing excess flux in the U and B bands, and a flux deficit in the near-IR.

Spectroscopically, the mismatch between model DDC10\_A0 and the observations of SN 2005cf is striking (Fig. 3). Up to the peak, the model shows the very standard SN Ia signatures, although the

spectral energy distribution (SED) is somewhat too red initially. The basic morphology of line profiles is also well matched. However, as time proceeds beyond bolometric maximum, our synthetic spectra are systematically too blue. This occurs in spite of the strong fading of the SN (well reproduced by our model), which is due to both the decreasing radioactive decay energy that is released and the increasing fraction of γ-rays that escape the ejecta.

So, the problem with model DDC10\_A0 is not with the bolometric luminosity, or the rate at which radiant energy leaks out of the ejecta, but instead with the computed colour evolution. Although a very basic property, SN colour is one of the hardest property to get right because it is sensitive to temperature, ionization, opacity, etc. For example, increasing the size of the Fe I model atom in an SN II-P simulation yielded a fading of 2 mag of our synthetic U-band magnitude (Dessart & Hillier 2011; Dessart et al. 2013a). Such a sensitivity is very problematic for the convergence of radiative-transfer results.

The present problem is not limited to model DDC10\_A0. Earlier calculations with CMFGEN, using the hydrodynamical inputs of Kasen & Woosley (2007), yielded the same discrepancy. We also explored whether other delayed-detonation models showed the same discrepancy. We tried a new version of model N32 of Hoeflich & Khokhlov (1996), as well as the full series of models of Blondin et al. (2013) which cover a factor of >5 in $^{56}$Ni mass, and found that all resulting time sequences eventually develop this colour problem after the peak. The issue is probably related to the extraordinary conditions existing in SN Ia ejecta. In the following sections, we explore various routes to solve the problem.

## 3.2 An opacity problem?

A fundamental property of SN Ia ejecta that sets them aside from other SN ejecta is their unique composition, split between IMEs and IGEs. For once ionized atoms, the number of free electrons per nucleon goes from 1 for hydrogen, to 1/4 for helium, to only 1/56 for $^{56}$Fe. Thus, the electron scattering opacity per unit mass is much lower, by typically more than an order of magnitude, than in SN II ejecta.

SNe Ia also have a representative ejecta kinetic energy of 1 B for about a tenth of the ejecta mass of SNe II. Compared to SNe II, SN Ia ejecta have faster expansion rates and are characterized by lower densities early on. Consequently, because of the weakening of the electron scattering opacity per unit mass and the low ejecta density, the continuum mass absorption coefficient is reduced in Type Ia compared to Type II SNe. However, in SNe Ia, metals, with their large mass fraction, are a strong source of additional opacity. The complex atomic structures of metals, with their unfilled 3d/4s shell, leads to the presence of millions of lines, with those from iron and cobalt (e.g. Fe II, Fe III, Co II and Co III) being of greater importance.

In model DDC10\_A0, we include all metal line transitions with a *gf* value greater than 0.002 (but also limited by our adopted model atom).<fnref n="3" /> This cut only applies to elements whose atomic-mass number is greater than 20 (i.e. Ne), does not apply to the lowest *n* levels (*n* is typically 9) and a transition is omitted only when there are at least *m* (*m* is typically 9) stronger downward transitions from the level. Thus, this procedure does not cut important transitions to ground levels, and forbidden and semiforbidden transitions among low-lying states. With the model atoms employed in model

<footnote n="3">³ There is no cut in the *gf* value when we compute the final spectrum.</footnote>
DDC10\_A0 (Table A1), we include a total of 8370 (1773) full (super) levels (see Hillier & Miller 1998 for a description of superlevels in CMFGEN), which corresponds to 174 674 bound-bound transitions.

We have a colour problem with model DDC10\_A0, but in spite of the order of $10^5$ lines included in the simulation, which is rather small compared to the millions or billions of lines often invoked (Kasen et al. 2008), the total *UBVRI* flux and the V-band light curves are nonetheless well matched. This suggests that the bulk of the energy diffusing out of the ejecta and producing the SN Ia bolometric luminosity is not critically sensitive to the opacity, i.e. a reasonable description as in model DDC10\_A0 is sufficient to capture the bolometric evolution. Paradoxically, the discrepancy with observations occurs at later times when the ejecta turns optically thin, and thus when one would naively think the opacity should matter less.

We investigate the effect of increasing the size of model atoms on model colours. With model DDC10\_A1, we employ a larger model atom for Co\_II-IV as well as Ni\_II-IV (Table A2) for the modelling of near-peak and post-peak epochs. We also lower the *gf* cut from 0.002 to 0.0001. Model A1 includes 13 959 (2149) full (super) levels, which corresponds to 629 396 transitions, which come primarily from Fe, Co and Ni. Despite such improvements, the radiative properties of models DDC10\_A0 and DDC10\_A1 remain very similar (Fig. 4). The bolometric luminosity of each model agrees to within a few per cent at all times (shown further below), confirming that employing huge model atoms to solve for the SN Ia bolometric luminosity is not critical. Enhanced opacity leads to enhanced blanketing in model DDC10\_A1, which leads to a mild reddening of the colours, — the brightness decreases in the blue and augments in the red, in particular in the near-IR (top panel of Fig. 4). The impact on optical synthetic spectra remains small (bottom panel of Fig. 4). While 90 per cent of the flux falls within the range 3000–1 µm after bolometric maximum, as in model DDC10\_A0, the UV and near-IR contributions are now at the same level (~5 per cent).

With model DDC10\_A2, we increase further the model atoms for Co\_II and Co\_III, with, respectively, 2747 (136) and 3917 (315) full (super) levels. The total number of full (super) levels is 17 533 (2338) and the total number of bound-bound transitions is 1738 088. Comparing these models at a few epochs, we find that the difference is small (Fig. 5), comparable to or weaker than the change obtained between models A0 and A1. This again is too small a change to resolve the colour discrepancy.

The colour problem we face concerns primarily the *U* and *B* bands. Rather than increasing the opacity in these spectral regions by making the model atoms of IGEs more complete, we finally investigated if a change in composition could help. An obvious source of opacity in the *U* and *B* bands is Ti\_II. Interestingly, $^{48}$Cr is an unstable isotope at the origin of the chain $^{48}$Cr $\rightarrow$ $^{48}$V $\rightarrow$ $^{48}$Ti. The first step has a half-life of 0.898 33 d, and the second step has a half-life of 15.9735 d, hence comparable to the typical rise time of SNe Ia. Although the total mass of synthesized $^{48}$Cr is only on the order 0.001 $M_{\odot}$, its mass fraction in the region 5000–10 000 km s$^{-1}$ is $10^{-3}$ while the mass fraction of Ti is $10^{-5}$ (dominated by $^{44}$Ti). Hence, by allowing for this decay chain, the Ti mass fraction will rise by two orders of magnitude in this velocity range by the time the SN gets to bolometric maximum. In model DDC10\_A1D1, we thus repeat the model sequence DDC10\_A1 from scratch but now include all two-step decay chains compiled in Table B1. As for other time sequences, this simulation took another 2–3 months. We show a spectral comparison between models DDC10\_A1 and DDC10\_A1D1 in Fig. 6. In the latter, the additional decay energy leads to an increase in luminosity at the few per cent level (shown

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Comparison between models DDC10\_A0 and DDC10\_A1 for the multiband light curves (top) and spectral evolution (bottom). Labels at left give the post-explosion time in days, which increases from top to bottom.</figcaption></figure>

further down). More importantly, the spectral differences remain confined to the 3000–4000 Å region where the Ti\_II opacity is the strongest (Filippenko et al. 1992; Nugent et al. 1995; Blondin et al. 2013). At the light-curve peak, Ti is however three times ionized in those regions, interior to 10 000 km s$^{-1}$, where its abundance was most enhanced through $^{48}$Cr/$^{48}$V decay.

## 3.3 Accounting for critical coolants

Part of the ambiguity with the colour problem diagnosed above is that various processes can lead to a change of colour. Historically, much of the colour evolution of SNe Ia has been associated with the redistribution of flux from the UV and blue part of the optical
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Comparison at selected post-explosion times of the gas temperature (top) and synthetic spectra (bottom) between model DDC10_A1 and DDC10_A2. The increase in the total number of transitions from 629 396 to 1 738 088 has essentially no impact on the spectra while the effect on the temperature is weak.</figcaption></figure>

where the opacity is large to the near-IR where the opacity is low. This fluorescence process has been associated with an opacity issue – the more complete the treatment of line opacity, the larger the number of transitions, the stronger the redistribution and the redder the SED (Hoeflich et al. 1993; Pinto & Eastman 2000b).

However, the colour of the emergent radiation is also related to the temperature and ionization state of the gas. The hotter the gas, the bluer the SED. The thermodynamic state of the gas also con-

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Illustration of the impact on synthetic spectra of treating only one decay chain (associated with $^{56}$Ni; model DDC10_A1, red), or allowing for all two-step decay chains presented in Appendix B (model DDC10_A1D1, blue). Note the influence of additional Ti II opacity at ~20 d in the *U* band, but the weak influence of these decayed species throughout the DDC10_A1D1 spectrum at nebular times.</figcaption></figure>

trols what ions contribute to the opacity – this matters since higher ionization stages tend to have their opacity at shorter wavelengths.

In Fig. 7, we show the evolution of the gas temperature in model DDC10_A1D1. Crudely, the ejecta can be broken into three distinct regions, each showing its own temperature evolution. In the outer region (above 17 000 km s⁻¹), the gas is initially cool due to the rapid expansion of the ejecta. However, it gets hotter with time due to the non-local energy deposition. These fast expanding layers have a very low density, hence cool very inefficiently.

In the intermediate region (4000–17 000 km s⁻¹), where $^{56}$Ni is the most abundant, there is initially a strong temperature gradient. With time the gas cools, the gradient decreases, and the temperature levels off to a value of ≳10 000 K. From day 30 to 60 the temperature remains almost constant.

In the nickel hole, the temperature initially is much lower than that in the intermediate region. Due to diffusion, and later on the deposition of energy by γ-rays, its cooling is slower so that by day 30 the temperature across the hole, and out to 10 000 km s⁻¹, is roughly constant. However, unlike the intermediate region, the ejecta continues to cool producing a large temperature jump at ~3000 km s⁻¹.
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Evolution of the gas temperature in model DDC10_A1D1 from 1 to 64 d after explosion. Note the development of a strong temperature jump at the inner edge of the $^{56}$Ni-rich region, while the temperature in that region, between 3000 and 15 000 km s$^{-1}$ retains a near constant value at all times after bolometric maximum.</figcaption></figure>

It now becomes clear that the colour problem we face comes from the overestimated temperature in the spectrum forming region after the light-curve peak. The material stays hot, the ionization high and the SED appears blue. Models DDC10_A0, DDC10_A1, former attempts with model n32n, as well as other delayed-detonation models eventually form that temperature plateau and no longer cool. This does not affect their bolometric luminosity, which follows the decay energy deposition rate. What seems fundamentally discrepant is the cooling rate.

What held us away from the solution for a long time is that the ejecta, while turning nebular, has high densities. In the region 5000–10 000 km s$^{-1}$, the free-electron density is of the order of $10^9$ cm$^{-3}$ at 30 d, and hence, we would not expect cooling by forbidden lines to be important since the typical critical density for [Co II] and [Co III] lines is 2–3 orders of magnitude lower. Consequently (and especially since many atoms were developed for stellar atmosphere calculations), most model atoms for IMEs and IGEs did not originally include forbidden-line transitions (but with the exception of Fe II, Co II, Si II, Si III, S II and S III). High temperature

and ionization conditions being generally met at high density and optical depth, model atoms for ions like Fe III, Fe IV, Co III or Co IV did not include forbidden-line transitions in models DDC10_A0, DDC10_A1 and DDC10_A2. We experimented their potential role by doing a new model sequence named DDC10_A3D1, which is identical to DDC10_A1D1 apart from the treatment of [Co III] lines.

The impact on the evolution of the gas temperature and ionization is drastic (Fig. 8). Prior to day 20, the difference in temperature between models DDC10_A1D1 and DDC10_A3D1 remains small, and limited to the ejecta regions above 10 000 km s$^{-1}$. As time progresses, the ejecta expands and thins out, and the temperature contrast between the two models grows. Instead of forming a temperature plateau, the new model DDC10_A3D1 continuously cools. At 20–30 d after explosion, the main coolants at $\sim 20\ 000\ \text{km s}^{-1}$ are [S III] 9533 Å ($3p^2\ ^3\text{P}-3p^2\ ^1\text{D}$ transition), [Si II] 2334–2335 Å ($3p^2\ ^2\text{P}-3p^2\ ^4\text{P}$ transitions), Si III 1206 Å (transition 3s-3p) and [S II] 4069–4077 Å ($3p^3\ ^4\text{S}-3p^3\ ^2\text{P}$ transitions). As we progress deeper, e.g. at 11 000 km s$^{-1}$, the main coolants are [S II] 4069–4077 Å, [Co III] 5888 Å ($3d^7\ ^4\text{F}-3d^7\ ^2\text{G}$), Si III 1206 Å, [S III] 9533 Å. Deeper still, e.g. at 3800 km s$^{-1}$, the main coolants are [Co III] 5888 Å, [S III] 9533 Å, Si III 1206 Å, Fe III 1914 Å ($3d^5\ ^4\text{s}\ ^7\text{S}-3d^5\ ^4\text{p}\ ^7\text{P}$). In each case, the quoted line on its own represents 10–50 per cent of the total line-cooling rate for the corresponding ion, and whatever the depth, it is generally the same line that dominates. It is in stark contrast with the notion that millions of lines would be needed to model SN Ia radiation accurately. At 60d after explosion and in regions $\lesssim 10\ 000\ \text{km s}^{-1}$ where the spectrum forms, the main coolants are [Co III] 5888 Å, Co II 2286 Å (not a forbidden line, but instead a strong 4s–4p transition), [Fe II] 12570 Å and [S III] 9533 Å. A description of forbidden-line transitions is given in Hansen, Raassen & Uylings (1984) and Quinet (1998).

This cooling leads to a progressive recombination of the ejecta, in particular from Co III to Co II in model DDC10_A3D1 (right-hand panel of Fig. 8). Cobalt represents 70 per cent of the total mass fraction around 5000–10 000 km s$^{-1}$, and so this ionization change eventually makes Co II the primary source of line blanketing, in particular in the blue part of the spectrum. The combined effects of enhanced cooling (i.e. cooler photosphere) and enhanced blanketing (i.e. strengthening of Co II opacity) leads to a significant reddening of the emergent radiation (Fig. 9). The process is a runaway since

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Left: evolution of the ejecta gas temperature with velocity (depth) and post-explosion time for SN Ia models DDC10_A1D1(solid) and DDC10_A3D1 (dashed). The only difference between the two simulations is the treatment of forbidden-line transitions in model DDC10_A3D1, while they are ignored in model DDC10_A1D1. Right: same as left, but now showing the ionization state for Co. Note the progressive recombination from Co III at light-curve peak to Co II–Co III in the spectrum formation region at 60 d after explosion.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Illustration of the impact on synthetic spectra of introducing forbidden-line transitions in the Co II model atom (model DDC10\_A3D1, blue), all else being the same as in model DDC10\_A1D1 (red). While the bolometric luminosity is identical between the two, the extra cooling reduces the temperature and the ionization in the spectrum formation region, leading to large differences in colour and spectral morphology.</figcaption></figure>

more cooling induces more recombination, stronger optically-thin line emission by Co II in the near-IR, which induces further cooling and recombination, etc.

So, our colour problem was fundamentally associated with the inadequate handling of coolants rather than insufficient opacity. Even well above the critical density, forbidden-line transitions can act as efficient coolants as soon as the time of bolometric maximum in SN Ia ejecta. In hindsight, this is not surprising. Below the critical density, the cooling due to collisionally excited lines scales as the density squared (assuming the ion is dominant) while above the critical density the cooling scales only linearly with the density. As the electron density increases above the critical density the relative importance of the line falls relative to other processes which are still scaling with the square of the density. However, in SNe Ia, Co and Fe are not impurity species, thus enhancing their importance for the energy balance, and allowing forbidden lines to be important coolants well above their critical density.

Fig. 10 clearly shows that the new model DDC10\_A3D1 does a very good job at reproducing the *UBVRI* luminosity of SN 2005cf, where other models DDC10\_A0, DDC10\_A1 and DDC10\_A1D1 yielded essentially the same discrepancy. Comparing observed to synthetic light curves, we find model DDC10\_A3D1 yields a very

<figure><img src="image_12.png" /><figcaption>**Figure 10.** Left: *UBVRI* luminosity for models A0, A1, A1D1, A3, together with the corresponding inferred luminosity for SN 2005cf. Right: comparison between multiband light curves of SN 2005cf and model DDC10\_A3D1. The synthetic photometry has been corrected for extinction, redshift and distance dilution.</figcaption></figure>

satisfactory match to the SN 2005cf colour evolution. Prior to peak, the mismatch appears to be related to a global flux offset (at the few tens of per cent level) since the colours agree well between the two. This offset may stem from the overestimated kinetic energy of our delayed-detonation models (Blondin et al. 2013).

More spectacular is the spectroscopic match to the observations of SN 2005cf (Fig. 11). The model matches the colour evolution, line-profile morphology, in particular as we proceed through bolometric maximum and progressively evolve from thick to thin conditions. The spectrum is well matched throughout the optical. A noticeable exception is the 8500 Å region where the Ca II triplet is overestimated in our simulations. We find that the Ca II line strength is quite sensitive to overlapping Co II line emission. The calcium lines form outside of the cobalt emitting region, so that the strength of the Ca II triplet depends sensitively on this background flux controlled by Co line emission. This discrepancy is also at the origin of the excess I-band flux at $\gtrsim 30$ d after explosion (bottom panel of Fig. 10).

After much exploration to identify the origin of this last problem, we realized that the non-thermal ionization routine did not include all species (the non-thermal excitation routine was fine). Namely, while non-thermal excitation rates were computed for all ions and levels, we did not have any entry for Ti, Cr and Co for the non-thermal ionization cross-sections. Using data from Mazzotta et al. (1998), we updated these rates and reran our SN Ia model DDC10. Non-thermal processes have little impact before the peak, so we restarted the sequence DDC10\_A3D1 at bolometric maximum
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Same as Fig. 3, but now showing a comparison between model DDC10\_A3D1 (red) and the observations of SN 2005cf (black). Contrary to model DDC10\_A0, model DDC10\_A3D1 includes both big model atoms, [Co III] lines, and all two-step decay chains presented in Table B1. Apart from the 8500 Å region after bolometric maximum, the agreement with observations is very satisfactory.</figcaption></figure>

using this updated non-thermal solver. Unfortunately, we also found
that using a huge model atom for Co II and Co III produced a redder
SED, so we needed to redo the sequence to address this.

We compare the resulting multiband light curves of model DDC10\_A4D1 with SN 2005cf in Fig. 12. Overall, the colour agreement in all optical and near-IR bands is good. We show the comparison with optical and near-IR spectra in Figs 13 and 14. The labels appearing on the right indicate the magnitude offset with the contemporaneous *B*-band or *K*-band magnitude. This offset is typically of the order of 0.1 mag, with only a few larger offsets of ≲0.3 mag at some epochs. Hence, in model DDC10\_A4D1, the flux offset in the red part of the optical is gone, primarily because allowance of non-thermal ionization for Co leads to enhanced ionization overall, and in particular for Co. The general agreement with observations is improved, although we obtain a poorer fit in the 5500 Å region. In the near-IR, the agreement with SN 2005cf spectra is satisfactory (Fig. 14), although some features do not follow the same evolution as observed (they appear too soon, or are too pronounced, etc.). One concern is also the reliability of observed near-IR spectra, in particular for the relative flux. The remaining mismatches with SN 2005cf could be due to the choice of the delayed-detonation model but might also indicate that further improvement in the adopted atomic models and processes are needed.

The critical role of forbidden-line transitions in SNe Ia, in par-
ticular their influence on the gas and radiation properties as early
as the peak of the light curve does not seem documented in the

literature. Early works on SN Ia radiation modelling did emphasize
the critical role of forbidden-line transitions, but the focus of these
studies was on nebular times exclusively (Axelrod 1980; Eastman
& Pinto 1993; Kuchner et al. 1994) – here, we demonstrate their
importance as early as bolometric maximum. Kasen et al. (2006)
do not include forbidden-line transitions in their radiative-transfer
simulations of the SN Ia near-IR secondary maximum that takes
place ~40 d after explosion, nor in their subsequent study of the
width–luminosity relation (Kasen & Woosley 2007).

Selecting lines based on their oscillator strength is fundamentally inadequate since the forbidden lines that are so critical for cooling the gas have very low oscillator strengths and are generally optically thin. They are not critical for trapping photons, but they are key for cooling the gas, controlling its temperature and ionization state, and thus determining what ions provide opacity sources. To some extent, this suggests that the SN Ia radiation properties are not exclusively controlled by opacity and fluorescence/branching, but also by the way the ejecta cool through thin lines. These lines may be intrinsically optically thin, but the photons they radiate may nonetheless be scattered or absorbed if they overlap with other lines.

4 LOCAL VERSUS NON-LOCAL ENERGY
DEPOSITION

The low mass and high expansion rate of SN Ia ejecta, com-
bined with the presence of <sup>56</sup>Ni nuclei at large ejecta velocities (or
Lagrangian mass), allow &gamma;-ray escape as early as 10–15 d after ex-
plosion. This affects the SN properties at all times beyond the peak
of the light curve (Fig. 15), by a magnitude that supersedes any
variation in model atoms we have tested in this study. &gamma;-ray escape
is in fact one of the key ingredients shaping SN Ia bolometric light
curves.

By enforcing local energy deposition in model DDC10\_A3L, we obtain gas temperatures and ionization states that are much higher than in model DDC10\_A3. The temperature in the <sup>56</sup>Ni-rich region stays high. Interestingly, the temperature in the <sup>56</sup>Ni hole becomes much lower than in the shells above it at nebular times, suggesting that radiative cooling completely inhibits the diffusion of heat to deeper layers. Importantly, the gas ionization stays high, despite the treatment of [Co III] lines. For example, Co does not recombine any more after the light-curve peak but remains as Co<sup>2+</sup> in the region 3000–15000 km s<sup>-1</sup>. Although the spectrum reddens because of intense blanketing, it remains bluer than in standard SNe Ia. Rather than developing strong Fe II lines, e.g. at 5169 Å, the model shows very strong [Co III] lines. For the last time displayed, we overlay the synthetic spectrum when these forbidden-line transitions are omitted in the calculation (green line). As in model DDC10\_A3, [Co III] lines are very strong coolants in SNe Ia, but here because of the much larger energy deposition (i.e. no &gamma;-ray escape) and no recombination (Co III is the dominant Co ion), these lines play an even stronger role. Because it will be the subject of a forthcoming paper, we note only in passing that the SN Ia feature that forms &le;6000 Å at ~10 d after the light-curve peak is associated with [Co III] &ndash; the association with Na I is unfounded on numerous grounds (Dessart et al. 2014a).

We note that model DDC10\_A3L does not show any bump in the
post-maximum bolometric luminosity, nor any bump in the near-IR
light curves, despite the strong cooling through [Co III]. Both are in
fact intimately related, as we discuss in Section 6.
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Comparison between multiband light curves of SN 2005cf and model DDC10_A4D1. The synthetic photometry has been corrected for extinction, redshift and distance dilution. The size of the dots in the figure is approximately 0.3 mag.</figcaption></figure>

**5 NON-THERMAL PROCESSES ASSOCIATED WITH RADIOACTIVE DECAY**

All simulations described so far in this paper include a treatment of non-thermal processes, following the procedure presented in Li et al. (2012). However, to assess the effect of non-thermal processes on the gas and radiation properties, we have run the sequence DDC10_A3T, identical to DDC10_A3, but forcing all decay energy to be deposited as heat, i.e. forcing non-thermal rates associated with radioactive decay to zero.

Without detailed non-LTE simulations, it is difficult to guess the importance of such non-thermal processes in SNe Ia. Indeed, the large abundance of unstable nuclei (in particular $^{56}$Ni and $^{56}$Co), should favour the strength of non-thermal processes. However, the large ionization of SN Ia ejecta makes the electron density relatively high, a property that tends to quench non-thermal ionization and excitation (Xu & McCray 1991; Dessart et al. 2012). Furthermore, since $^{56}$Ni is produced primarily at depth in the ejecta (how deep depends on mixing during the explosion phase), in layers moving at $\lesssim 15\,000\,\text{km s}^{-1}$, non-thermal processes are irrelevant at early times when the spectrum formation region is located in the faster moving outer ejecta layers, i.e. non-thermal processes are confined at such times to layers where thermalization is secured by the large ejecta optical depth.

In Fig. 16, we show the impact of these non-thermal processes when they start becoming visible around the peak of the bolometric light curve. The thermal model DDC10_A3T indeed starts appearing bluer at $\sim 20\,\text{d}$ after explosion, but the effect is rather small. This occurs because model DDC10_A3T has a larger ejecta temperature and ionization in the originally $^{56}$Ni-rich ejecta layers (i.e. more energy is deposited in the form of heat). The main effect is to make the forbidden-line transition [Co III] 5888 Å stronger. However, as the ejecta becomes optically thin, the thermal model becomes somewhat cooler, but more importantly, it becomes significantly less ionized, making $\text{Co}^{+}$ dominate throughout the ejecta. In contrast, non-thermal processes maintain an equal share of $\text{Co}^{+}$ and $\text{Co}^{2+}$ for Co (we find the same holds for Fe). At the peak of the light curve, of the total decay energy deposited, we find that

the fraction going into non-thermal ionization is 5–10 per cent at all depths, and systematically about 1–2 times larger than that going into non-thermal excitation. At 60 d after explosion, this fraction is 5 per cent for both non-thermal ionization and excitation. Although small, these fractions correspond to large non-thermal ionization and excitation contributions, given that modest energies are needed to alter the thermodynamic state of the gas.

As time progresses further into the nebular phase, non-thermal processes maintain this high ionization, although the decay of $^{56}$Co into $^{56}$Fe eventually makes iron dominate over cobalt. We in fact find that the Co and Fe ionization, as well as the ejecta temperatures at $\lesssim 15\,000\,\text{km s}^{-1}$, remain roughly constant at $\sim 7500\,\text{K}$ from 60 d until 200 d after explosion. Our delayed-detonation models are eventually primarily cooled through few forbidden-line transitions of Fe III and Fe II, in particular transitions like [Fe III] 4658 Å, which are connected to the ground state (for a discussion, see Axelrod 1980; Kuchner et al. 1994; Maurer et al. 2011). A forthcoming study will present these results in detail.

As we mentioned earlier, the non-thermal solver did not treat non-thermal ionization for Ti, Cr and Co in model DDC10_A3D1. When we include the associated rates in model DDC10_A4D1, the ejecta ionization and temperature increase a little, exacerbating the contrast with model DDC10_A3T. However, the enhanced blanketing we obtain in DDC10_A4D1, caused by the huge Co II and Co III model atoms employed, leads only to a modest hardening of the DDC10_A4D1 synthetic spectra compared to those obtained for model DDC10_A3D1 (see Section 3.3).

**6 NEAR-IR SECONDARY MAXIMUM**

The near-IR bump seen in SNe Ia has been studied in the past by, e.g. Hoflich, Khokhlov & Wheeler (1995), and more recently by Kasen (2006) and Jack, Hauschildt & Baron (2012), who find that it stems from an ionization effect associated with cobalt primarily. Here, we investigate the photometric properties of our various DDC10 time sequences and discuss what controls the behaviour, in our simulations, of the near-IR light curves. This also serves as an additional test for CMFGEN.
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Comparison between model DDC10_A4D1 and the observed spectra of SN 2005cf. We correct the synthetic flux to account for the distance, redshift and extinction of 0.5cf. Spectra are scaled vertically for convenience, although the label on the right gives the true B-band magnitude offset between model and observations at each date – this offset is typically small.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Figure 14.** Same as Fig. 13, but now showing the near-IR range. We show log F<sub>λ</sub> for better visibility. For each spectrum, we quote the true magnitude offset in the K band, while the vertical positioning is adjusted for optimal visibility.</figcaption></figure>

While optical bands show a single pronounced peak around bolo-
metric maximum, near-IR bands show two peaks, one before the
bolometric maximum and one after. At early times, when a well-
defined photosphere exists, the brightness in any given spectral

region is controlled primarily by the radius and temperature at
the photosphere. On the rise to light-curve peak, the photosphere
dramatically expands but heats up moderately. At such times, the
diffusion of heat from greater depth is strongly degraded by ejecta
<figure><img src="image_17.png" /><figcaption>**Figure 15.** Left-hand column, top: bolometric luminosity for models DDC10_A0, DDC10_A1, DDC10_A1D1, DDC10_A3D1 and DDC10_A3L. Modifications in the model atom, and hence opacity, lead to very modest changes in the bolometric luminosity. However, if we prevent γ-ray escape (model DDC10_A3L), there is a dramatic change in the bolometric luminosity. Left-hand column, bottom: spectral comparison between model DDC10_A3 (red; non-local energy deposition and γ-ray escape are allowed for) with model DDC10_A3L (blue) in which we assume local energy deposition at all times. The green curve corresponds to the spectrum obtained by taking out all the forbidden-line transitions of Co III treated in model DDC10_A3L. Right-hand column, top: Same as top/left, but now showing the light-curve evolution in the *U*, *V* and *K* bands. Right-hand column, middle and bottom: snapshots of the gas temperature and Co ionization state at selected post-explosion times. Local energy deposition maintains a higher (lower) temperature in the $^{56}$Ni-rich (poor) regions, which directly impacts the gas ionization. For local energy deposition, Co remains as Co III in the region 3000–15 000 km s$^{-1}$.</figcaption></figure>

expansion. Hence, early on, the SN brightens in all bands. However, as time proceeds, heating is strong enough to raise the temperature in the spectrum formation region, causing the SED to shift to the blue, with a peak around 3000 Å at bolometric maximum. The hardening of the radiation is so strong that, despite the fast expansion of the photosphere, the flux in the red decreases. The resulting first peak in red bands occurs in all models (DDC10_A0, DDC10_A1,

DDC10_A3, etc.), irrespective of model atoms used, at about 14.0 (I band), 11.0 (J band), 9.7 (H band) and 9.8 d (K band). This time differs by a few 0.1 d between models at most. The bolometric maximum occurs at 17.8 d (again, a difference of ±0.2 d is seen between models). In our delayed-detonation model DDC10, the optical colours continuously change prior to peak, thus making the SN behave quite differently from a 'fireball' (Nugent et al. 2011).
<figure><img src="image_18.png" /><figcaption>**Figure 16.** Same as Fig. 15, but this time showing the differences in bolometric luminosity, colours and spectra between the model sequence DDC10_A3, in which non-thermal processes are treated, and model sequence DDC10_A3T, in which all decay energy deposited in the ejecta is treated as heat.</figcaption></figure>

All our models discussed above and presented in Table 1 show a secondary bump in the near-IR light curves, but this bump is pronounced and mimics a secondary maximum only in models DDC10_A3, DDC10_A3D1 and DDC10_A4D1. As shown in the left-hand panel of Fig. 17,<fnref n="4" /> the relative flux emitted outside of the optical range remains ≲30 per cent beyond bolometric maximum,

leading to a mild re-brightening in the near-IR. The larger that fraction, the stronger the near-IR ‘maximum’. Strictly speaking, and as evidenced in the right-hand panel of Fig. 17, only models DDC10_A3 and DDC10_A3D1 exhibit a genuine secondary maximum. For model DDC10_A3D1, the specific numbers for the corresponding post-explosion times are 43.3 d (−18.95 mag), 46.4 d (−18.19 mag), 41.2 d (−18.96 mag) and 42.32 d (−18.87 mag) in the *I*, *J*, *H* and *Ks* bands.

Interestingly, our model bolometric light curves sometimes show an excess at about 40 d after explosion, although this excess is entirely absent in all models with the overionization and spurious temperature jump. In other words, the only models that exhibit a late bump in bolometric luminosity treat [Co III] lines and recombine to

<footnote n="4">⁴ The glitches in the light curves correspond to times when we vary radiative-transfer ‘ingredients’, i.e. when we alter model atom characteristics or introduce non-thermal processes. In the future, we will need to try to incorporate all the necessary microphysics at the start to avoid such unwanted variations, although it can be hard to do due to memory or CPU constraints.</footnote>
<figure><img src="image_19.png" /><figcaption>**Figure 17.** Left: illustration of the relative fraction of the flux that contributes to the *UBVRI* brightness for models DDC10_A0, DDC10_A1, DDC10_A1D1, DDC10_A3, DDC10_A3D1 and DDC10_A4D1. Right: same as left, but now showing the light-curve evolution in the *U*, *V* and *K* bands. Dots correspond to the actual times the CMFGEN models are computed.</figcaption></figure>

Co II. Recombination energy of 1 M⊙ of Co from twice to once ionized liberates about $5 \times 10^{44}$ erg, which, if radiated over a week, produces a meagre $1.6 \times 10^6$ L⊙, i.e. too little to cause the bump. However, SN ejecta are radiation dominated, so that the energy held up in trapped radiation completely overwhelms what may be stored in excitation and ionization energy. Assuming that the gas and the trapped radiation are in equilibrium, a decrease by 1000 K at 10 000 K in region between $[r_0, r_1]$ of [3,5] $\times 10^{15}$ cm liberates an energy of $a\Delta T^4\Delta V$, where *a* is the radiation constant, $\Delta V \sim 4(r_1^3 - r_0^3)$. Radiated over 10 d, the corresponding power is $10^{42.5}$ erg s$^{-1}$. In practice, the radiation and the gas are not in equilibrium, i.e. the mean intensity drops below the Planck source function, but this suggests that the release of trapped radiation is indeed large enough to produce the bump seen and observed.

There is no doubt the bump in bolometric luminosity is associated with a change in ionization (Hoflich et al. 1995), which comes with a large change in temperature. This ionization change affects primarily cobalt, which goes from Co III to Co II soon after the bolometric maximum in our delayed-detonation model. While the main coolant for Co III are forbidden transitions in the optical (primarily at ~5900 Å), where the opacity from overlapping lines is relatively large (especially at $\lesssim 5000$ Å), the main coolants for Co II are forbidden transitions in the near-IR associated with the states $4s^2-4p^1$, As shown in Fig 18, besides a general increase in the overall flux, strong lines develop in model DDC10_A3D1 at 1.6–1.8 µm (model DDC10_A4D1 has similar near-IR spectral properties), while they are weak or absent in model DDC10_A1D1. These lines are primarily due to permitted transitions of Co II. To conclude, both the bump in bolometric luminosity and in near-IR light curves stem from an ionization shift and the sudden strengthening of Co II emission.

Kasen (2006) model the near-IR light curves of SNe Ia and reproduces the basic morphology, including the secondary maximum. Surprisingly, he does not account for forbidden-line transitions, which we demonstrate here are key for getting the proper ejecta ionization state (i.e. Co III versus Co II, Fe III versus Fe II), and associated emission features.

# 7 DISCUSSION AND CONCLUSIONS

In this work, we have utilized a single delayed-detonation explosion model of a Chandrasekhar mass WD as input for spectral calculations of SNe Ia. Such explosion models, although very at-

<figure><img src="image_20.png" /><figcaption>**Figure 18.** Comparison between synthetic spectra of models DDC10_A1D1 and DDC10_A3D1 from 20 to 40 d after explosion, i.e. when the near-IR generally presents a secondary maximum. The same scaling is applied to both sets of synthetic spectra, so the offset between them is genuine (see Fig. 17).</figcaption></figure>

tractive for their ejecta properties and radiative properties, need to be further studied to understand the growing lack of evidence for companion stars, giving further support to the notion that many SNe Ia may arise from the coalescence of two WDs. Despite the
limitations of the progenitor model, it is important to build confi-
dence in the radiative-transfer modelling of SNe Ia, in particular
be able to accurately compute the colour and spectral evolution of
such models and understand the dependencies of synthetic observ-
ables. In this and future studies, we wish to investigate how well 1D
delayed-detonation models can reproduce standard SN Ia spectra
and multiband light curves, i.e. whether their spectral signatures
support the delayed-detonation explosion scenario for SNe Ia. Can
a 1D treatment, as employed in CMFGEN, be at all successful in that
task?

Unfortunately, accurate modelling of SN Ia radiation, in par-
ticular of their spectra, is very difficult. The large abundance of
IMEs and IGEs, and the lack of hydrogen, means that the contin-
uum opacity is small and that the opacity, in stark contrast to SNe
II, is dominated by line opacity at most wavelengths. In addition,
the small ejecta masses, and low densities, mean that departures
from LTE are large, and significantly influence spectral formation.
Since non-LTE effects are important, we need accurate atomic data
(not just line opacities) which for IMEs and IGEs is often lacking,
or of insufficient quality. Further, the scale of the problem means
that approximate techniques are often used to simplify the radiative
transfer and/or the determination of the thermodynamic state of the
gas.

In this work, we have tried to overcome many of the limitations so
that we can accurately model the spectra of 1D delayed-detonation
models. Using CMFGEN we have undertaken time-dependent radiative
transfer and statistical-equilibrium calculations to model Ia spectra.
We make no assumptions about how photons are thermalized, and
we also utilize the same model for comparison of spectra at differ-
ent epochs. The latter minimizes the influence of free-parameters.
However, these calculations have limitations. While we use large
model atoms, these remain of limited size to match with the current
computer capabilities (each time step takes about 2 d and requires
10 Gb of RAM). We also use a moderately large turbulent velocity
of 50 km s⁻¹, which enhances intrinsic line overlap, and we have
used superlevels to help facilitate the solution of the rate equations.
While we have made some tests of the influence of these assump-
tions, such tests have not been exhaustive.

Using our DDC10 model, we were initially unable to match the gross properties of SN Ia spectra after bolometric maximum. Various tests were undertaken to determine the cause of the discrepancy. Surprisingly, the mismatch in spectra was not due to missing opacity. Increasing the size of the model atoms and the number of lines treated did not solve the problem. Rather, the spectral mismatch was due to the neglect of [Co III] lines in the Co III model atom. These lines provide crucial cooling, even at densities well above their critical density, which shifts the Co ionization from Co III towards Co II. This, in turn, enhances the opacity in the *U* band leading to better agreement with observations. Using the same hydrodynamical model, we are now able to match the spectrum and multiband light curves of SN 2005cf from pre-maximum (−12 d) to well beyond maximum (+40 d). Thus, we can reproduce the basic light curve and fundamental spectral properties of SNe Ia with the standard delayed-detonation scenario, even with the assumption of spherical symmetry.

Of all processes we allow to vary in this work, the most important
one that controls the light-curve morphology of SNe Ia is γ-ray es-
cape. Indeed, the low mass of SNe Ia ejecta causes a huge leakage of
energy. It starts to be visible about a week after explosion through an
increase in the luminosity; the non-local energy deposition ‘speeds’
up the diffusion of radiant energy. Beyond bolometric maximum,
non-local energy deposition is superseded by γ-ray escape so that

models that treat γ-ray transport, rather than assuming full trapping,
fade significantly faster. In nature, this leakage is function of the
trapping efficiency and should thus vary with ejecta mass, expansion
rate, and $^{56}$Ni distribution (see, e.g. Pinto & Eastman 2000a).

Despite the ionized conditions in SN Ia ejecta, we find that non-
thermal processes play an important role, even at bolometric maxi-
mum. Earlier on, decay energy is deposited at high optical depths,
which inhibits non-thermal effects in the spectrum formation region.
However, as the ejecta thins out, non-thermal processes maintain
a much higher ionization in simulations that include them, allow-
ing for the concomitant presence of Co II and Co III instead of Co II
alone. This alters the colour evolution after light-curve peak. The
magnitude of the effect varies subtly with the optical depth, since
thermal excitation/ionization can also occur if the heat deposited
is large enough to cause a significant temperature difference. From
60 to 200 d after explosion, the spectrum formation, located at
$\lesssim 10$ 000 km s$^{-1}$, retains a fairly constant temperature of $\sim 7500$ K
and a stable ionization with equal fractions for Co$^{+}$–Co$^{2+}$ and Fe$^{+}$–
Fe$^{2+}$. We expect a modulation in the magnitude of such non-thermal
effects in SN Ia ejecta endowed with different initial masses of $^{56}$Ni.
We will explore this issue in a forthcoming study.

Our radiative-transfer simulations cover from the UV to the far-
IR at all epochs computed. The near-IR light curves for all models,
even those characterized by a small model atom, develop a bump af-
ter bolometric maximum. However, only models that reproduce the
basic colour evolution in the optical (and thus capture adequately
the shift in ionization) develop a pronounced secondary maximum
in the near-IR, as observed. In fact, both the bump in bolometric
luminosity and near-IR brightness are tied to an ionization shift, in
particular from Co III to Co II, which causes enhanced ejecta cool-
ing, and thus an increase in bolometric luminosity. This escaping
radiation appears primarily through [Co II] emission, which takes
place primarily in the near-IR.

The intense development and testing we have performed with CMFGEN for SN Ia calculations in the last five years suggests that it is in fact possible to reproduce with unprecedented fidelity the fundamental radiative properties of SNe Ia, both at pre-peak, peak and post-peak epochs. This suggests that while multidimensionality may play a role, it is not paramount for the radiative transfer solution. Far more important is to treat explicitly non-LTE, line blan-keting, non-thermal processes, and to include all relevant radiative and collisional processes. Although generally neglected, forbidden-line transitions are found to play an essential role in controlling the ejecta ionization and temperature as early as the peak of the light curve. Following these fruitful benchmarking and educational ex-plorations, we are now in a position to investigate the physics of SN Ia explosions and search for clues about the progenitor systems.

ACKNOWLEDGEMENTS

We acknowledge useful discussions with Carles Badenes, and in-
teraction with Monique Arnaud and Pasquale Mazzotta on ioniza-
tion cross-sections. We also thank Daniel Kasen for his insightful
comments regarding the two emission features near 1.7 µm. LD
and SB acknowledge financial support from the European Com-
munity through an International Re-integration Grant, under grant
number PIRG04-GA-2008-239184, and from ‘Agence Nationale de
la Recherche’ grant ANR-2011-Blanc-SIMI-5-6-007-01. DJH ac-
knowledges support from STScI theory grant HST-AR-12640.01,
and NASA theory grant NNX10AC80G. This work was also sup-
ported in part by the National Science Foundation under Grant no.
PHYS-1066293 and benefited from the hospitality of the Aspen
Center for Physics. AK acknowledges the NSF support through the NSF grants AST-0709181 and TG-AST090074. This work was granted access to the HPC resources of CINES under the allocation c2013046608 made by GENCI (Grand Equipement National de Calcul Intensif).

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## APPENDIX A: MODEL ATOMS

The model atoms adopted for all simulations in this work are essentially the same origin as those used in Dessart & Hillier (2011). The sources of atomic data are varied, and in many cases multiple data sets for a given ion are available. In some cases, these multiple data sets represent an evolution in data quality and/or quantity, while in other cases they represent different sources and/or computational methods. Comparisons of models calculated with different data sets and atomic models potentially provide insights into the sensitivity of our results to the adopted model atoms and hydrodynamical inputs (although such calculations have yet to be undertaken for SNe).

Oscillator strengths for CO 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); 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,b), Nahar (1995). Atomic data from the opacity project comes from TOPBASE (Cunto et al. 1993). Energy levels have generally been obtained from 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). Photoionization data are taken from the Opacity Project (Seaton 1987; Cunto et al. 1993), the Iron Project (Hummer et al. 1993; Nahar & Pradhan 1996) and Nahar & Pradhan (1993). Unfortunately, Ni and Co photoionization data are generally unavailable so we have utilized crude approximations, except for photoionization from the ground state for which we use data from Verner & Yakovlev (1995). Charge exchange cross-sections are from the tabulation by Kingdon & Ferland (1996). Atomic data for C IV was obtained from Leibowitz (1972) and Peach, Saraph & Seaton (1988), and for the carbon isoelectronic sequence from Luo & Pradhan (1989). Collision strengths for Ar II are from Tayal & Henry (1996). The LS Ne I photoionization cross-sections were modified according to Seaton (1998). The same procedure was applied to using Ar I mixing coefficients computed at http://aphysics2.lanl.gov/tempweb/lanl. Additional data for Ne I were obtained from the MCHF/MCDHF website: http://nlte.nist.gov/MCHF.

For the modelling of SNe Ia, the main issues concern the cobalt atomic data, and in particular Co II and Co III. For these, accurate photoionization cross-sections and collisional rates are needed to improve the accuracy of the radiation transfer modelling.

## APPENDIX B: DECAY ROUTES

Energy wise, the $^{56}$Ni decay chain is the most important for SNe Ia calculation. However, other chains may need to be considered if they are associated with unstable isotopes present in regions where $^{56}$Ni is absent (for example in the outer ejecta), or if these chains produce isotopes with a potentially strong line blanketing power. This is the case of Ti II, whose mass fraction can be increased by two orders of magnitude through the decay of $^{48}$Cr and $^{48}$V.

We have modified CMFGEN to handle multiple decay chains, either two-step like $^{56}$Ni → $^{56}$Co → $^{56}$Fe, or one-step. In this work, we

<table><caption>**Table A1.** Summary of the model atom A0, referred to as 'Small' in Table 1. The source of the atomic data sets is given in Dessart & Hillier (2010) and in Appendix A. $N_f$ ($N_s$) refers to the number of full (super) levels, and $N_{trans}$ to the corresponding number of bound-bound transitions. The last column refers to the upper level for each ion treated. In this configuration, the total number of full (super) levels treated is 8370 (1773), which corresponds to 174 674 bound-bound transitions.</caption><thead><tr><th>Species</th><th>N<sub>f</sub></th><th>N<sub>s</sub></th><th>N<sub>trans</sub></th><th>Upper level</th></tr></thead><tbody><tr><td>C I</td><td>26</td><td>14</td><td>120</td><td>2s2p<sup>3</sup> <sup>3</sup>P<sup>0</sup></td></tr><tr><td>C II</td><td>26</td><td>14</td><td>87</td><td>2s<sup>2</sup>4d <sup>2</sup>D<sup>5/2</sup></td></tr><tr><td>C III</td><td>112</td><td>62</td><td>891</td><td>2s8f <sup>1</sup>F<sup>0</sup></td></tr><tr><td>C IV</td><td>64</td><td>59</td><td>1446</td><td>n=30</td></tr><tr><td>O I</td><td>51</td><td>19</td><td>214</td><td>2s<sup>2</sup>2p<sup>3</sup> <sup>4</sup>S)4f <sup>3</sup>F<sup>3</sup></td></tr><tr><td>O II</td><td>111</td><td>30</td><td>1157</td><td>2s<sup>2</sup>2p<sup>2</sup> <sup>3</sup>P)4d <sup>2</sup>D<sup>5/2</sup></td></tr><tr><td>O III</td><td>86</td><td>50</td><td>646</td><td>2p4f <sup>1</sup>D</td></tr><tr><td>O IV</td><td>72</td><td>53</td><td>835</td><td>2p<sup>2</sup>(<sup>3</sup>P)3p <sup>2</sup>P<sup>0</sup></td></tr><tr><td>Ne I</td><td>139</td><td>70</td><td>1587</td><td>2s<sup>2</sup>2p<sup>5</sup> <sup>2</sup>P<sup>3/2</sup>)6d <sup>2</sup>[5/2]<sup>1/3</sup></td></tr><tr><td>Ne II</td><td>91</td><td>22</td><td>1106</td><td>2s<sup>2</sup>2p<sup>4</sup> <sup>3</sup>P)4d <sup>2</sup>P<sup>3/2</sup></td></tr><tr><td>Ne III</td><td>71</td><td>23</td><td>460</td><td>2s<sup>2</sup>2p<sup>3</sup> <sup>2</sup>D<sup>0</sup>)3d <sup>3</sup>S<sup>1</sup></td></tr><tr><td>Na I</td><td>71</td><td>22</td><td>1614</td><td>30w <sup>2</sup>W</td></tr><tr><td>Mg II</td><td>65</td><td>22</td><td>1452</td><td>30w <sup>2</sup>W</td></tr><tr><td>Mg III</td><td>99</td><td>31</td><td>775</td><td>2p<sup>5</sup>7s <sup>1</sup>P<sup>0</sup></td></tr><tr><td>Al II</td><td>44</td><td>26</td><td>171</td><td>3s5d <sup>1</sup>D<sup>2</sup></td></tr><tr><td>Al III</td><td>45</td><td>17</td><td>362</td><td>10z <sup>2</sup>Z</td></tr><tr><td>Si II</td><td>59</td><td>31</td><td>354</td><td>3s<sup>2</sup>(<sup>1</sup>S)7g <sup>2</sup>G<sup>7/2</sup></td></tr><tr><td>Si III</td><td>61</td><td>33</td><td>310</td><td>3s5g <sup>1</sup>Ge<sup>4</sup></td></tr><tr><td>Si IV</td><td>48</td><td>37</td><td>405</td><td>10f <sup>2</sup>F<sup>0</sup></td></tr><tr><td>S II</td><td>324</td><td>56</td><td>8208</td><td>3s3p<sup>3</sup> <sup>5</sup>S<sup>0</sup>)4p <sup>6</sup>P</td></tr><tr><td>S III</td><td>98</td><td>48</td><td>837</td><td>3s3p<sup>2</sup>(<sup>2</sup>D)3d <sup>3</sup>P</td></tr><tr><td>S IV</td><td>67</td><td>27</td><td>396</td><td>3s3p<sup>(3</sup>P<sup>0</sup>)4p <sup>2</sup>D<sup>5/2</sup></td></tr><tr><td>Ar I</td><td>110</td><td>56</td><td>1541</td><td>3s<sup>2</sup>3p<sup>5</sup> <sup>2</sup>P<sup>3/2</sup>)7p <sup>2</sup>[3/2]<sup>2</sup></td></tr><tr><td>Ar II</td><td>415</td><td>134</td><td>20 197</td><td>3s<sup>2</sup>3p<sup>4</sup> <sup>3</sup>P<sup>1</sup>)7i <sup>2</sup>[6]<sup>11/2</sup></td></tr><tr><td>Ar III</td><td>346</td><td>32</td><td>6898</td><td>3s<sup>2</sup>3p<sup>3</sup> <sup>2</sup>Do)8s <sup>1</sup>Do</td></tr><tr><td>Ca II</td><td>77</td><td>21</td><td>1736</td><td>3p<sup>6</sup>30w <sup>2</sup>W</td></tr><tr><td>Ca III</td><td>40</td><td>16</td><td>108</td><td>3s<sup>2</sup>3p<sup>5</sup>5s <sup>1</sup>P<sup>0</sup></td></tr><tr><td>Ca IV</td><td>69</td><td>18</td><td>335</td><td>3s3p<sup>5</sup>(<sup>3</sup>P<sup>0</sup>)3d <sup>4</sup>D<sup>0</sup><sup>1/2</sup></td></tr><tr><td>Sc II</td><td>85</td><td>38</td><td>979</td><td>3p<sup>6</sup>3d4f <sup>1</sup>P<sup>1</sup></td></tr><tr><td>Sc III</td><td>45</td><td>25</td><td>235</td><td>7h <sup>2</sup>H<sup>0</sup><sup>11/2</sup></td></tr><tr><td>Ti II</td><td>152</td><td>37</td><td>3134</td><td>3d<sup>2</sup>(<sup>3</sup>F)5p <sup>4</sup>D<sup>0</sup><sup>7/2</sup></td></tr><tr><td>Ti III</td><td>206</td><td>33</td><td>4735</td><td>3d6f <sup>3</sup>H<sup>0</sup></td></tr><tr><td>Cr II</td><td>196</td><td>28</td><td>3629</td><td>3d<sup>4</sup>(<sup>3</sup>G)4p x<sup>4</sup>G<sup>0</sup><sup>11/2</sup></td></tr><tr><td>Cr III</td><td>145</td><td>30</td><td>2359</td><td>3d<sup>3</sup>(<sup>2</sup>D2)4p <sup>3</sup>D<sup>0</sup></td></tr><tr><td>Cr IV</td><td>234</td><td>29</td><td>6354</td><td>3d<sup>2</sup>(<sup>3</sup>P)5p<sup>4</sup>P<sup>0</sup><sup>5/2</sup></td></tr><tr><td>Mn II</td><td>97</td><td>25</td><td>236</td><td>3d<sup>4</sup>(<sup>5</sup>D)4s2 <sup>5</sup>D<sup>4</sup></td></tr><tr><td>Mn III</td><td>175</td><td>30</td><td>3173</td><td>3d<sup>4</sup>(<sup>3</sup>G)4p y<sup>4</sup>H<sup>0</sup><sup>13/2</sup></td></tr><tr><td>Fe I</td><td>136</td><td>44</td><td>1900</td><td>3d<sup>6</sup>(<sup>5</sup>D)4s4p x<sup>5</sup>F<sup>0</sup></td></tr><tr><td>Fe II</td><td>115</td><td>50</td><td>1437</td><td>3d<sup>6</sup>(<sup>1</sup>G1)4s d<sup>2</sup>G<sup>7/2</sup></td></tr><tr><td>Fe III</td><td>477</td><td>61</td><td>6496</td><td>3d<sup>5</sup>(<sup>4</sup>F)5s <sup>5</sup>F<sup>1</sup></td></tr><tr><td>Fe IV</td><td>294</td><td>51</td><td>8068</td><td>3d<sup>4</sup>(<sup>5</sup>D)4d <sup>4</sup>G<sup>5/2</sup></td></tr><tr><td>Fe V</td><td>191</td><td>47</td><td>3977</td><td>3d<sup>3</sup>(<sup>4</sup>F)4d <sup>5</sup>F<sup>3</sup></td></tr><tr><td>Fe VI</td><td>433</td><td>44</td><td>14 103</td><td>3p5<sup>(2</sup>P)3d4<sup>(1</sup>S) <sup>2</sup>Pc<sup>0</sup><sup>3/2</sup></td></tr><tr><td>Fe VII</td><td>153</td><td>29</td><td>1753</td><td>3p5<sup>(2</sup>P)3d<sup>3</sup> (b<sup>2</sup>D) <sup>1</sup>P<sup>1</sup></td></tr><tr><td>Co II</td><td>144</td><td>34</td><td>2088</td><td>3d<sup>6</sup>(<sup>5</sup>D)4s4p <sup>7</sup>D<sup>0</sup></td></tr><tr><td>Co III</td><td>361</td><td>37</td><td>10 937</td><td>3d<sup>6</sup>(<sup>5</sup>D)5p <sup>4</sup>P<sup>3/2</sup></td></tr><tr><td>Co IV</td><td>314</td><td>37</td><td>8684</td><td>3d<sup>5</sup>(<sup>2</sup>P)4d <sup>3</sup>P<sup>0</sup></td></tr><tr><td>Co V</td><td>387</td><td>32</td><td>13 605</td><td>3d<sup>4</sup>(<sup>3</sup>F)4d <sup>2</sup>H<sup>9/2</sup></td></tr><tr><td>Co VI</td><td>323</td><td>28</td><td>9608</td><td>3d<sup>3</sup>(<sup>2</sup>D)4d <sup>1</sup>S<sup>0</sup></td></tr><tr><td>Co VII</td><td>319</td><td>31</td><td>9096</td><td>3p5<sup>(2</sup>P)3d4<sup>(3</sup>F) <sup>2</sup>D<sup>3/2</sup></td></tr><tr><td>Ni II</td><td>93</td><td>19</td><td>842</td><td>3d<sup>7</sup>(<sup>4</sup>F)4s4p <sup>6</sup>D<sup>0</sup><sup>1/2</sup></td></tr><tr><td>Ni III</td><td>67</td><td>15</td><td>379</td><td>3d<sup>7</sup> <sup>4</sup>F 4p <sup>3</sup>D<sup>0</sup></td></tr><tr><td>Ni IV</td><td>200</td><td>36</td><td>4085</td><td>3d<sup>6</sup>(<sup>3</sup>D)4p <sup>2</sup>D<sup>5/2</sup></td></tr><tr><td>Ni V</td><td>183</td><td>46</td><td>3065</td><td>3d<sup>5</sup>(<sup>2</sup>D3)4p <sup>3</sup>F<sup>0</sup></td></tr><tr><td>Ni VI</td><td>314</td><td>37</td><td>9569</td><td>3d<sup>4</sup>(<sup>5</sup>D)4d <sup>4</sup>F<sup>9/2</sup></td></tr><tr><td>Ni VII</td><td>308</td><td>37</td><td>9225</td><td>3d<sup>3</sup>(<sup>2</sup>D)4d <sup>3</sup>P<sup>2</sup></td></tr></tbody></table>
<table><caption>**Table A2.** Same as for Table 1, but now showing the bigger Fe/Co/Ni atoms used for the model sequence DDC10_A1 (the model atom for other species is kept the same and the details about these are not repeated here). Because the larger model atom only starts at later times when the ejecta has cooled, some high-ionization stages for IGEs are excluded. The total number of full (super) levels treated is 13 959 (2149), which corresponds to 629 396 bound-bound transitions. Model atom A3 is identical to A1 except that it includes forbidden-line transitions for Co III.</caption><thead><tr><th>Species</th><th>N<sub>f</sub></th><th>N<sub>s</sub></th><th>N<sub>trans</sub></th><th>Upper level</th></tr></thead><tbody><tr><td>Fe I</td><td>136</td><td>44</td><td>1900</td><td>3d<sup>6</sup>(<sup>5</sup>D)4s4p x<sup>5</sup>F<sup>0</sup><sub>3</sub></td></tr><tr><td>Fe II</td><td>827</td><td>275</td><td>44 831</td><td>3d<sup>5</sup>(<sup>6</sup>S)4p<sup>2</sup>(<sup>3</sup>P)<sup>4</sup>P<sup>1/2</sup></td></tr><tr><td>Fe III</td><td>607</td><td>69</td><td>9794</td><td>3d<sup>5</sup>(<sup>4</sup>D)6s<sup>3</sup>D<sub>2</sub></td></tr><tr><td>Fe IV</td><td>1000</td><td>100</td><td>72 223</td><td>3d<sup>4</sup>(<sup>3</sup>G)4f<sup>4</sup>P<sup>0</sup><sub>5/2</sub></td></tr><tr><td>Fe V</td><td>191</td><td>47</td><td>3977</td><td>3d<sup>3</sup>(<sup>4</sup>F)4d<sup>5</sup>F<sub>3</sub></td></tr><tr><td>Fe VI</td><td>433</td><td>44</td><td>14 103</td><td>3p5(2P<sup>0</sup>)3d<sup>4</sup>(<sup>1</sup>S)<sup>2</sup>Pc<sup>0</sup><sub>3/2</sub></td></tr><tr><td>Co II</td><td>1000</td><td>81</td><td>61 986</td><td>3d<sup>7</sup>(<sup>4</sup>P)4f<sup>5</sup>F<sup>0</sup><sub>4</sub></td></tr><tr><td>Co III</td><td>1000</td><td>72</td><td>68 462</td><td>3d<sup>6</sup>(<sup>5</sup>D)5f<sup>4</sup>F<sup>0</sup><sub>9/2</sub></td></tr><tr><td>Co IV</td><td>1000</td><td>56</td><td>69 425</td><td>3d<sup>5</sup>(<sup>2</sup>D)5s<sup>1</sup>D<sub>2</sub></td></tr><tr><td>Co V</td><td>387</td><td>32</td><td>13 605</td><td>3d<sup>4</sup>(<sup>3</sup>F)4d<sup>2</sup>H<sup>9/2</sup></td></tr><tr><td>Co VI</td><td>323</td><td>28</td><td>9608</td><td>3d<sup>3</sup>(<sup>2</sup>D)4d<sup>1</sup>S<sub>0</sub></td></tr><tr><td>Ni II</td><td>1000</td><td>59</td><td>51 707</td><td>3d<sup>8</sup>(<sup>3</sup>F)7f<sup>4</sup>I<sup>0</sup><sub>9/2</sub></td></tr><tr><td>Ni III</td><td>1000</td><td>47</td><td>66 486</td><td>3d<sup>7</sup>(<sup>2</sup>D)4d<sup>3</sup>Sb<sub>1</sub></td></tr><tr><td>Ni IV</td><td>1000</td><td>54</td><td>72 898</td><td>3d<sup>6</sup>(<sup>5</sup>D)6p<sup>6</sup>F<sup>11/2</sup></td></tr><tr><td>Ni V</td><td>183</td><td>46</td><td>3065</td><td>3d<sup>5</sup>(<sup>2</sup>D3)4p<sup>3</sup>F<sub>3</sub></td></tr><tr><td>Ni VI</td><td>314</td><td>37</td><td>9569</td><td>3d<sup>4</sup>(<sup>5</sup>D)4d<sup>4</sup>F<sup>9/2</sup></td></tr></tbody></table>

<table><caption>**Table A3.** Same as for Table 1, but now showing the huge Co II and Co III model atom A2 used in model DDC10_A2 – all other ions have the same characteristics as in DDC10_A1 with the exception of Fe vi, Co vi and Ni vi, which are excluded in DDC10_A2 because of the lower ionization of the ejecta at the times we perform our tests. Model atom A4 is identical to A2 except that it also treats all important forbidden-line transitions of metal ions – this model atom is used in the sequence DDC10_A4D1. With this configuration, the total number of full (super) levels treated is now only 17 553 (2338), which corresponds to 1738 088 bound-bound transitions.</caption><thead><tr><th>Species</th><th>N<sub>f</sub></th><th>N<sub>s</sub></th><th>N<sub>trans</sub></th><th>Upper level</th></tr></thead><tbody><tr><td>Co II</td><td>2747</td><td>136</td><td>593 140</td><td>3d<sup>7</sup>(<sup>2</sup>D)6p<sup>3</sup>P<sup>0</sup><sub>1</sub></td></tr><tr><td>Co III</td><td>3917</td><td>315</td><td>679 280</td><td>3d<sup>6</sup>(<sup>3</sup>D)6d<sup>4</sup>P<sup>3/2</sup></td></tr></tbody></table>

generally treat only the <sup>56</sup>Ni decay chain, but investigate at times the impact of having all two-step decay chains presented in Table B1. The role of one-step decay chains is discussed in Dessart et al. (2014b). The Monte Carlo transport code that calculates the non-local γ-ray energy deposition was modified to handle the same chains.

Nuclear masses are taken from Audi, Wapstra & Thibault (2003), while decay products (γ-ray lines, electrons/positrons and neutrons) and energies are taken from http://www.nddc.bnl.gov/chart.

For an illustration, we show in Fig. A1 the effect on the bolometric light curve of introducing all two-step decay chains rather than <sup>56</sup>Ni decay chain only. The effect is at most of ≲10 per cent, and limited to early, i.e. pre-peak, times.

<table><caption>**Table B1.** Summary of the two-step decay chains used. For all chains, we give the characteristics for each of the two steps, starting with the half-life, the total energy radiated in γ-rays $Q_v$, and the total energy liberated in the form of particles $Q_{th}$. We then list the main γ-ray lines emitted in each decay together with their probability.</caption><thead><tr><th colspan="2">56Ni → 56 Co → 56 Fe</th><th colspan="2">56Co → 56 Fe</th></tr><tr><th>t<sub>1/2</sub> = 6.075 d</th><th>t<sub>1/2</sub> = 77.233 d</th><th>t<sub>1/2</sub> = 6.075 d</th><th>t<sub>1/2</sub> = 77.233 d</th></tr><tr><th>Q<sub>γ</sub> = 1.718 MeV</th><th>Q<sub>γ</sub> = 3.633 MeV</th><th>Q<sub>γ</sub> = 1.718 MeV</th><th>Q<sub>γ</sub> = 3.633 MeV</th></tr><tr><th>Q<sub>th</sub> = 0.000 MeV</th><th>Q<sub>th</sub> = 0.116 MeV</th><th>Q<sub>th</sub> = 0.000 MeV</th><th>Q<sub>th</sub> = 0.116 MeV</th></tr><tr><th>E<sub>γ</sub></th><th>Prob.</th><th>E<sub>γ</sub></th><th>Prob.</th></tr></thead><tbody><tr><td>0.158</td><td>98.8</td><td>0.511</td><td>38.0</td></tr><tr><td>0.270</td><td>36.5</td><td>0.847</td><td>100.0</td></tr><tr><td>0.480</td><td>36.5</td><td>0.977</td><td>1.4</td></tr><tr><td>0.750</td><td>49.5</td><td>1.038</td><td>14.0</td></tr><tr><td>0.812</td><td>86.0</td><td>1.175</td><td>2.3</td></tr><tr><td>1.562</td><td>14.0</td><td>1.238</td><td>67.6</td></tr><tr><td></td><td></td><td>1.360</td><td>4.3</td></tr><tr><td></td><td></td><td>1.771</td><td>15.7</td></tr><tr><td></td><td></td><td>2.015</td><td>3.1</td></tr><tr><td></td><td></td><td>2.035</td><td>7.9</td></tr><tr><td></td><td></td><td>2.598</td><td>17.3</td></tr><tr><td></td><td></td><td>3.010</td><td>1.0</td></tr><tr><td></td><td></td><td>3.202</td><td>3.2</td></tr><tr><td></td><td></td><td>3.253</td><td>7.9</td></tr><tr><td></td><td></td><td>3.273</td><td>1.9</td></tr></tbody></table>

<table><thead><tr><th colspan="2">57Ni → 57 Co → 57 Fe</th><th colspan="2">57Co → 57 Fe</th></tr><tr><th>t<sub>1/2</sub> = 1.483 d</th><th>t<sub>1/2</sub> = 271.740 d</th><th>t<sub>1/2</sub> = 1.483 d</th><th>t<sub>1/2</sub> = 271.740 d</th></tr><tr><th>Q<sub>γ</sub> = 1.937 MeV</th><th>Q<sub>γ</sub> = 0.122 MeV</th><th>Q<sub>γ</sub> = 1.937 MeV</th><th>Q<sub>γ</sub> = 0.122 MeV</th></tr><tr><th>Q<sub>th</sub> = 0.154 MeV</th><th>Q<sub>th</sub> = 0.000 MeV</th><th>Q<sub>th</sub> = 0.154 MeV</th><th>Q<sub>th</sub> = 0.000 MeV</th></tr><tr><th>E<sub>γ</sub></th><th>Prob.</th><th>E<sub>γ</sub></th><th>Prob.</th></tr></thead><tbody><tr><td>0.127</td><td>16.7</td><td>0.014</td><td>9.2</td></tr><tr><td>0.511</td><td>87.0</td><td>0.122</td><td>85.6</td></tr><tr><td>1.378</td><td>81.7</td><td>0.137</td><td>10.7</td></tr><tr><td>1.758</td><td>5.8</td><td></td><td></td></tr><tr><td>1.919</td><td>12.3</td><td></td><td></td></tr></tbody></table>

<table><thead><tr><th colspan="2">48Cr → 48 V → 48 Ti</th><th colspan="2">48V → 48 Ti</th></tr><tr><th>t<sub>1/2</sub> = 0.898 d</th><th>t<sub>1/2</sub> = 15.973 d</th><th>t<sub>1/2</sub> = 0.898 d</th><th>t<sub>1/2</sub> = 15.973 d</th></tr><tr><th>Q<sub>γ</sub> = 0.432 MeV</th><th>Q<sub>γ</sub> = 2.910 MeV</th><th>Q<sub>γ</sub> = 0.432 MeV</th><th>Q<sub>γ</sub> = 2.910 MeV</th></tr><tr><th>Q<sub>th</sub> = 0.002 MeV</th><th>Q<sub>th</sub> = 0.145 MeV</th><th>Q<sub>th</sub> = 0.002 MeV</th><th>Q<sub>th</sub> = 0.145 MeV</th></tr><tr><th>E<sub>γ</sub></th><th>Prob.</th><th>E<sub>γ</sub></th><th>Prob.</th></tr></thead><tbody><tr><td>0.112</td><td>96.0</td><td>0.511</td><td>99.8</td></tr><tr><td>0.308</td><td>100.0</td><td>0.944</td><td>7.8</td></tr><tr><td>0.511</td><td>3.2</td><td>0.984</td><td>100.0</td></tr><tr><td></td><td></td><td>1.312</td><td>97.5</td></tr><tr><td></td><td></td><td>2.240</td><td>2.4</td></tr></tbody></table>

<table><thead><tr><th colspan="2">49Cr → 49 V → 49 Ti</th><th colspan="2">49V → 49 Ti</th></tr><tr><th>t<sub>1/2</sub> = 0.029 d</th><th>t<sub>1/2</sub> = 330.000 d</th><th>t<sub>1/2</sub> = 0.029 d</th><th>t<sub>1/2</sub> = 330.000 d</th></tr><tr><th>Q<sub>γ</sub> = 1.055 MeV</th><th>Q<sub>γ</sub> = 0.000 MeV</th><th>Q<sub>γ</sub> = 1.055 MeV</th><th>Q<sub>γ</sub> = 0.000 MeV</th></tr><tr><th>Q<sub>th</sub> = 0.598 MeV</th><th>Q<sub>th</sub> = 0.000 MeV</th><th>Q<sub>th</sub> = 0.598 MeV</th><th>Q<sub>th</sub> = 0.000 MeV</th></tr><tr><th>E<sub>γ</sub></th><th>Prob.</th><th>E<sub>γ</sub></th><th>Prob.</th></tr></thead><tbody><tr><td>0.062</td><td>16.4</td><td>0.511</td><td>0.0</td></tr><tr><td>0.091</td><td>53.2</td><td></td><td></td></tr><tr><td>0.153</td><td>30.3</td><td></td><td></td></tr><tr><td>0.511</td><td>186.0</td><td></td><td></td></tr></tbody></table>
<table><caption>Table B1 – continued</caption><tbody><tr><td colspan="4"><sup>51</sup>Mn → <sup>51</sup>Cr → <sup>51</sup>V</td></tr><tr><td colspan="2"><sup>51</sup>Mn → <sup>51</sup>Cr</td><td colspan="2"><sup>51</sup>Cr → <sup>51</sup>V</td></tr><tr><td colspan="2"><table><tbody><tr><td>t<sub>1/2</sub> = 0.032 d</td><td>t<sub>1/2</sub> = 27.700 d</td></tr><tr><td>Q<sub>γ</sub> = 0.992 MeV</td><td>Q<sub>γ</sub> = 0.032 MeV</td></tr><tr><td>Q<sub>th</sub> = 0.933 MeV</td><td>Q<sub>th</sub> = 0.000 MeV</td></tr><tr><td>E<sub>γ</sub></td><td>Prob.</td><td>E<sub>γ</sub></td><td>Prob.</td></tr><tr><td>0.511</td><td>194.2</td><td>0.320</td><td>9.9</td></tr></tbody></table></td></tr><tr><td colspan="4"><sup>55</sup>Co → <sup>55</sup>Fe → <sup>55</sup>Mn</td></tr><tr><td colspan="2"><sup>55</sup>Co → <sup>55</sup>Fe</td><td colspan="2"><sup>55</sup>Fe → <sup>55</sup>Mn</td></tr><tr><td colspan="2"><table><tbody><tr><td>t<sub>1/2</sub> = 0.730 d</td><td>t<sub>1/2</sub> = 1002.200 d</td></tr><tr><td>Q<sub>γ</sub> = 1.943 MeV</td><td>Q<sub>γ</sub> = 0.000 MeV</td></tr><tr><td>Q<sub>th</sub> = 0.430 MeV</td><td>Q<sub>th</sub> = 0.000 MeV</td></tr><tr><td>E<sub>γ</sub></td><td>Prob.</td><td>E<sub>γ</sub></td><td>Prob.</td></tr><tr><td>0.477</td><td>20.2</td><td>0.511</td><td>0.0</td></tr><tr><td>0.511</td><td>152.0</td><td></td><td></td></tr><tr><td>0.931</td><td>75.0</td><td></td><td></td></tr><tr><td>1.317</td><td>7.1</td><td></td><td></td></tr><tr><td>1.370</td><td>2.9</td><td></td><td></td></tr><tr><td>1.408</td><td>16.9</td><td></td><td></td></tr></tbody></table></td></tr><tr><td colspan="4"><sup>37</sup>K → <sup>37</sup>Ar → <sup>37</sup>Cl</td></tr><tr><td colspan="2"><sup>37</sup>K → <sup>37</sup>Ar</td><td colspan="2"><sup>37</sup>Ar → <sup>37</sup>Cl</td></tr><tr><td colspan="2"><table><tbody><tr><td>t<sub>1/2</sub> = 1.226 s</td><td>t<sub>1/2</sub> = 35.040 d</td></tr><tr><td>Q<sub>γ</sub> = 1.072 MeV</td><td>Q<sub>γ</sub> = 0.000 MeV</td></tr><tr><td>Q<sub>th</sub> = 2.347 MeV</td><td>Q<sub>th</sub> = 0.000 MeV</td></tr><tr><td>E<sub>γ</sub></td><td>Prob.</td><td>E<sub>γ</sub></td><td>Prob.</td></tr><tr><td>0.511</td><td>199.8</td><td>0.003</td><td>5.5</td></tr><tr><td>2.796</td><td>1.8</td><td></td><td></td></tr><tr><td>3.601</td><td>0.0</td><td></td><td></td></tr></tbody></table></td></tr><tr><td colspan="4"><sup>52</sup>Fe → <sup>52</sup>Mn → <sup>52</sup>Cr</td></tr><tr><td colspan="2"><sup>52</sup>Fe → <sup>52</sup>Mn</td><td colspan="2"><sup>52</sup>Mn → <sup>52</sup>Cr</td></tr><tr><td colspan="2"><table><tbody><tr><td>t<sub>1/2</sub> = 0.345 d</td><td>t<sub>1/2</sub> = 0.015 d</td></tr><tr><td>Q<sub>γ</sub> = 0.751 MeV</td><td>Q<sub>γ</sub> = 2.447 MeV</td></tr><tr><td>Q<sub>th</sub> = 0.191 MeV</td><td>Q<sub>th</sub> = 1.113 MeV</td></tr><tr><td>E<sub>γ</sub></td><td>Prob.</td><td>E<sub>γ</sub></td><td>Prob.</td></tr><tr><td>0.169</td><td>99.2</td><td>0.511</td><td>190.0</td></tr><tr><td>0.378</td><td>1.6</td><td>1.434</td><td>98.3</td></tr><tr><td>0.511</td><td>112.0</td><td>2.965</td><td>1.0</td></tr><tr><td></td><td></td><td>3.129</td><td>1.0</td></tr></tbody></table></td></tr><tr><td colspan="4"><sup>44</sup>Ti → <sup>44</sup>Sc → <sup>44</sup>Ca</td></tr><tr><td colspan="2"><sup>44</sup>Ti → <sup>44</sup>Sc</td><td colspan="2"><sup>44</sup>Sc → <sup>44</sup>Ca</td></tr><tr><td colspan="2"><table><tbody><tr><td>t<sub>1/2</sub> = 21 915.000 d</td><td>t<sub>1/2</sub> = 0.165 d</td></tr><tr><td>Q<sub>γ</sub> = 0.000 MeV</td><td>Q<sub>γ</sub> = 2.136 MeV</td></tr><tr><td>Q<sub>th</sub> = 0.000 MeV</td><td>Q<sub>th</sub> = 0.596 MeV</td></tr><tr><td>E<sub>γ</sub></td><td>Prob.</td><td>E<sub>γ</sub></td><td>Prob.</td></tr><tr><td>0.511</td><td>0.0</td><td>0.511</td><td>188.5</td></tr><tr><td></td><td></td><td>1.157</td><td>99.9</td></tr></tbody></table></td></tr></tbody></table>

<figure><img src="image_21.png" /><figcaption>Figure A1. Illustration of the bolometric light curve for model with (DDC10_A3D1) and without (DDC10_A3D0) the additional two-step decay chains. To better reveal the small contrast, we show the ratio with the DDC10_A3D0 used as reference.</figcaption></figure>

This paper has been typeset from a TeX/LATeX file prepared by the author.