11institutetext: Universidad Nacional de Río Negro. Sede Andina, Mitre 630
(8400) Bariloche, Argentina 22institutetext: Consejo Nacional de
Investigaciones Científicas y Técnicas (CONICET), Argentina 33institutetext:
Instituto de Astrofísica de La Plata (IALP), CCT-CONICET-UNLP. Paseo del
Bosque S/N (B1900FWA), La Plata, Argentina 44institutetext: Kavli Institute
for the Physics and Mathematics of the Universe (WPI), The University of
Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan 55institutetext:
Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La
Plata, Paseo del Bosque S/N, B1900FWA La Plata, Argentina

# Supernovae double-peaked light curves from double-nickel distribution

Mariana Orellana 1122 Melina C. Bersten 334455<EMAIL_ADDRESS>
(Received 26 May 2022; accepted 26 August 2022)

Among supernovae (SNe) of different luminosities, many double-peaked light
curves (LCs) have been observed, representing a broad morphological variety.
In this work, we investigate which of these can be modelled by assuming a
double-peaked distribution of their radioactive material, as originally
proposed for SN2005bf. The inner zone corresponds to the regular explosive
nucleosynthesis and extends outwards, according to the usual scenario of
mixing. The outer 56Ni–rich shell may be related to the effect of jet-like
outflows that have interacted with more distant portions of the star before
the arrival of the SN shock. As the outer layer is covered by matter that is
optically less thick, its energy emerges earlier and generates a first peak of
radiation. To investigate this scenario in more detail, we have applied our
hydrodynamic code that follows the shock propagation through the progenitor
star and takes into account the effect of the $\gamma$-ray photons produced by
the decay of the radioactive isotopes. We present a simple parametric model
for the 56Ni abundance profile and explore the consequences on the LC of
individually varying the quantities that define this distribution, setting our
focus onto the stripped-envelope progenitors. In this first study, we are
interested in the applicability of this model to SNe that have not been
classified as superluminous, thus, we have selected our parameter space
accordingly. Then, within the same mathematical prescription for the 56Ni-
profile, we revisited the modelling process for a series of objects: SN2005bf,
PTF2011mnb, SN2019cad, and SN2008D. In some cases, a decrease in the gamma ray
opacity is required to fit the late time observations. We also discuss the
other cases in which this scenario might be likely to explain the LC
morphology. A steep initial decline in the observed bolometric LC within less
than few days after the explosion becomes less feasible for this model,
because it requires a large abundance of 56Ni near the stellar surface,
indicating a strongly inverted distribution. An initial bolometric rise before
the two peaks seems more favourable for the double-nickel case, particularly
as it can be difficult to explain through other scenarios, unless a
combination of power sources is invoked.

###### Key Words.:

supernovae: general — supernovae: individual (SN2005bf, PTF2011mnb, SN2019cad,
SN2008D)

††offprints: M. Orellana

##  1 Introduction

The observed shape of the LC of H-poor core collapse SNe (or stripped envelope
supernovae) usually exhibits only one maximum in the bolometric luminosity.
This maximum is well understood as being powered by the decay of radioactive
material generated during the explosion

The ability to detect SNe at increasingly early times has been evolving a
great deal in recent years (see e.g. projects such as ZTF, KISS, and HITS,
among others). Thus, there is also a higher fraction of SNe with early
detection. From theoretical grounds an early cooling phase is predicted,
consisting of a decrease in bolometric luminosity prior to the main peak. The
duration of this phase depends on the progenitor structure, for a more compact
star, the faster the associated cooling phase (see e.g. Bersten et al., 2012).
Therefore, the detectability of the cooling phase depends on the SNe type. For
example, for type IIb SNe that are expected to retain some relatively extended
H-rich envelope the cooling phase has been more easily observed as opposed to
SNe Ib or Ic for whom this phase has been elusive. In very few cases, some
emission for SNe Ib or Ic has been reported and generally the detection is
only in one or two photometric bands (see e.g. the case of SN 1999ex,
Stritzinger et al. 2002, or iPTF15dtg, Taddia et al. 2016). In addition to the
cooling emission there are some cases where two clear peak has been
discovered. Currently, there is a growing variety of double-peaked supernovae
that have been reported, however, they are still uncommon.

The peculiar types of SNe, along with the basic premises of how they can be
distinguished, are presented in Kasen (2017). Naturally, more rare events have
been discovered since the advent and growth of wide-field surveys and rapid-
response follow-up facilities (see Modjaz et al., 2019, and references
therein).

Throughout this work, we will refer to the peaks of luminosity that occur
apart from the spike of the shock breakout (SBO). The SBO is expected to
happen even before the cooling emission and is harder to detect (see e.g. the
SBO detection of SN 2016gkg, Bersten et al. 2018).

The luminosities of double-peaked light curves (LCs) have shown a wide range,
including superluminous SNe. The frequency and detectability of double-peaked
SLSNe was examined by Nicholl & Smartt (2016) based on the flux excess from
polynomial fits at the earliest epochs. After an intensive literature search,
we do not know of a similar study for double-peaked SNe with normal
luminosities.

The ideas to explain the double maxima of the LC mainly comprise the
interaction between the ejecta and either a modified structure at the outskirt
of the progenitor or with unbounded circumstellar matter (CSM) surrounding the
progenitor (Moriya et al., 2013; Piro, 2015; Ertl et al., 2020; Jin et al.,
2021) and with different extents (Bersten et al., 2012; Nakar & Piro, 2014;
Taddia et al., 2016). This interpretation works fairly well when the observed
brightness starts by decreasing (cooling phase) before rising to the
radioactive peak. However, that scenario faces a challenge in cases where the
luminosity is initially increasing to a first maximum that takes place later
than about 10 days since the explosion.

For certain LCs, a promising alternative involve nickel plus a tunable central
engine, which is likely a magnetar that produces a delayed injection of energy
or models in which the progenitor star expelled shells of material years
before the explosion and get detached from the star, so the interaction
produces an important rebrightening at late phases (see e.g. Li et al. 2020).
Another possibility that was proposed and calculated a decade ago concerns an
early feature that may result from the ejecta colliding with a binary
companion, granted it is a system with a suitable inclination (Kasen, 2010).

The unprecedented LC morphology among Type Ib/Ic of SN2005bf (Tominaga et al.,
2005), with a first maximum about 16 days after explosion and a second
brighter peak 24 days later, drives the proposal of an unusual 56Ni
distribution in the model developed by Folatelli et al. (2006). That concept
inspired an alternative explanation for SN2008D with the scenario consisting
of 56Ni-rich material deposited in the outer layers of the ejecta, carried out
by a jet-like phenomenon (Bersten et al., 2013), which is quite consistent
with the X-ray emission early detected in association with this SN (Soderberg
et al., 2008; Mazzali et al., 2008). Later, LSQ13abf was found to resemble
SN2008D and was subject to a similar analysis with regard to the justification
of the shape of the LC (Stritzinger et al., 2020). PFT11mnb and SN2019cad
showed stronger analogies with the LC of SN2005bf and, therefore, the model
with a double 56Ni profile was also proposed for them (Taddia et al., 2018;
Gutiérrez et al., 2021). We also refer to Magee & Maguire (2020), who recently
explored the case of 56Ni layers as a source of early light curve bumps in
type Ia supernovae. The external nickel that forms a high-velocity blob is the
source of extra energy needed to explain the first peak. This model has the
advantage of having a single power source, but at the cost of considering a
more elaborate distribution. However, aside from a few references, its
potential has not been deeply explored and this is the motivation for the
present study.

This paper is organized as follows: first, we present the numerical setup of
the 1D radiation-hydrodynamic simulations in Section 2 and apply it to a fixed
progenitor. The resulting parameter exploration is included in Section 3. We
then present the information recollected from observed SNe with early
detections in Section 4. In Section 4.1, we present the modelling of our
selected family, along with a brief comparison to previous results. A more
extreme example is presented in Section 4.2, demonstrating an earlier first
peak. We conclude our paper with a discussion in Section 5.

##  2 Numerical models

We used a one-dimensional (1D) local thermal equilibrium (LTE) hydrodynamical
code to perform our numerical exploration. The code assumes the diffusion
approximation for optical photons. For the gamma rays produced through the
radioactive decay, we applied a gray approximation. A constant value of
$\kappa_{\gamma}=0.03$ cm2/g was adopted for the gamma opacity. That quantity
was modified in some cases in order to model the last decline in the LC
(discussed in Section 4.1). All the details on the equations of state and
opacities tables adopted in the code can be found in Bersten et al. (2011).
The explosion is simulated by injecting energy near to the progenitor core
which produces a powerful shock wave that propagates inside the ejecta until
its arrives to the surface where the photons can begin to propagate out,
producing the first electromagnetic signal of the SN. Therefore, the code
computes the whole evolution of the LC in a self consistent way up to the
nebular phase. However, due to the crude treatment of the radiation transfer,
even if the broadband photometry can be estimated by assuming a blackbody
emission, only the bolometric luminosity is confidently calculated. Thus, our
analysis is focussed on the properties of bolometric LCs.

An initial configuration in hydrostatic equilibrium was used to initiate the
hydrodynamical calculations. In this study, we employed progenitor models from
stellar evolutionary calculations (more details given below), however, the
distribution of the radioactive material, which is in principle a consequence
of the explosive burning, is treated in a parametric way and assumed to be an
initial condition in our calculation. This has been extensively applied in the
literature where the nickel is usually mixed out by hand, justified by the
effects of instabilities which cannot be properly taken into account in 1D
calculations. Since our code allows for any distribution of this material
inside the ejecta, here, we go one step forward and we assume that the nickel
can have two separate rich regions: an inner component which is typically
assumed in SN models and an external component. Here, we are interested in
exploring the LC morphology produced with this type of double-nickel profile
in a parametric and systematic way.

Although a self-consistent model is beyond the scope of this study, the
double-peaked 56Ni distribution is well motivated. It has proven to be
successful in modelling a few SNe and suggested for some others (see Section
4) It is interesting to have them re-investigated within a matching
parameterization for the 56Ni-profile. The framework is established by the
hypothetical presence of outflows involved in the explosion, as in the case of
gamma-ray burst (see e.g. MacFadyen et al. 2001; Woosley & Bloom 2006;
Banerjee & Mukhopadhyay 2013). One possible effect of the jet-like outflows is
to trigger the nucleosynthesis of radioactive elements somewhere at the outer
layers of the ejecta before the shock front of the SN arrives (Nishimura et
al., 2015). Alternatively, perhaps the jets can transport out some material
mixed with radioactive elements. We speculate that such a displacement could
be more efficient than hydrodynamical instabilities usually invoked for the
mixing, which justifies the hypothesis of a separate nickel-rich layer. The
presence of jet-like outflows does not imply an energetic photon detection.
The high-energy emission might not occur because the jets do not break the
stellar surface, although, alternatively, even if they do so, the energetic
photons may not be detected for geometric reasons (Margalit et al., 2018). In
fact, a GRB is thought to be produced in the unusual case, where the jet
tunnels throughout the entire star and is viewed on-axis (Sobacchi et al.,
2017).

Details of the jet dynamics, the interaction with the stellar matter, and
nucleosynthesis are not specified in our model. Throughout this work, we
assume that the external M(${}^{56}\mathrm{Ni}$) component is a fraction of
the one sited at the inner layers. Also, we restrain the amount of 56Ni within
the range of previously published mass values in normal stripped-envelope SNe,
namely, a mean value of 0.293 M⊙ and ranging between $0.03-2.4$ M⊙ (Anderson,
2019). We consider the upper part of this range to be an extreme overestimate
for the normal SNe, as it is more than $3\sigma$ aside from the mean value. We
also note that this author has remarked on the questionable accuracy of
popular methods used to estimate M(${}^{56}\mathrm{Ni}$).

A double-peaked nickel distribution was assumed for the first time to model
the unusual LC morphology of SN 2005bf. The two components presented in that
study follows a continuous profile with abundance values always greater than
zero and denoted as a mildly inverted distribution (see details in Folatelli
et al. (2006)). Here, we preferred to assume a more simplistic profile with
two boxes for the nickel abundance, with a mass fraction denoted by $X$. In
our numerical scheme, $X$ is a function of Lagrangian mass coordinate
($M_{r}$), which is directly mapped to the mass fraction, $f$, a more general
variable that allows for comparisons between different progenitors. Although
we decided to use a simplified nickel profile, we explored more complex
profiles to facilitate the interpretation of the effect of each parameter on
the LC. In particular, we tested the effect of assuming smoother 56Ni
profiles, but we have not found any noticeable difference in the resulting LC,
as we discuss in Gutiérrez et al. (2021).

In the next section (Section 3), we present and discuss our results for a
fixed progenitor star with a pre-SN mass of 11 M⊙ (denoted He11) and a radius
of $\sim 5$ R⊙. This structure corresponds to a main sequence star of 30 M⊙
and its evolution was computed with the public stellar evolution MESA code
(Paxton et al., 2011) with no rotation and other setup as described in
Martinez et al. (2020).

##  3 Simple double 56Ni parameter space

Figure 1 displays the scheme of the nickel distribution analyzed here. The
profile has an inner 56Ni-rich layer and an outer component, both with
constant nickel abundances denoted by $X_{\rm in}$ and $X_{\rm out}$,
respectively. These layers are located at separate zones, with no nickel in
the middle, and the spread is regulated by fractions, $f,$ of the total mass,
$M,$ of the progenitor. The abundances of all the other elements in a given
shell add $(1-X)$ to fulfill the normalization. Once the SN reaches the
homologous expansion, this distribution in mass could be expressed as a
function of velocity, as done in Bersten et al. (2013), with the notion of a
slower (for the inner) and a faster (for the external) 56Ni-rich shells. We
considered an inner component close to the compact remnant, namely, extending
from the inner border of the ejecta, $f_{0}$, up to a fraction, $f_{1}$, that
has the usual partial-mixing sense when the nickel abundance profile is a
boxcar function (as in Bersten et al. 2011, though different from e.g. Jin et
al. 2021 or Sharon & Kushnir 2020). The external 56Ni component is placed from
$f_{2}$, to $f_{3}$, so it can be truncated below the surface, or reach it if
$f_{3}=1$. The total mass of 56Ni varies as result of the change of any $X$ or
$f$ parameter, therefore, we decided to indicate its value in parentheses in
the figures presenting the LC that result from the individual parameter
variation. In this section, we discuss the morphology of the first and second
peaks, but without mentioning the earlier initial spike due to the shock
breakout, since it is much brighter and faster than the peaks discussed here.

Figures 2, 4, and 5 show the effect on the bolometric LC of the variation of
each of the parameters that define our nickel distribution, keeping the others
fixed. For these calculations, we used He11 as the initial configuration, with
explosion energy of 2 foe (foe$\equiv$ 1051 erg) and a compact remnant of 2.15
M⊙ (except in Fig. 4, where we allow for a variation of the remnant mass). A
variety of two peaked LC are produced, ranging from curves with two clearly
separate peaks to cases where both peaks begin to merge. The luminosity of the
first (second) peak are mainly dominated by the external (internal) abundance,
although the extension of the nickel-rich region also has an important effect.
We note that when $f_{1}$ is adjusted closer to $f_{2}$, the peaks blend into
a more subtle double-hump morphology. The timescale of the first rise is also
related to $f_{3}$ that controls the outermost border of the 56Ni
distribution. As found by Noebauer et al. (2017), the rising phase duration is
related to the optical depth of the material placed on top of the border that
we denote $f_{3}$. The rate of $L$ decline after the first peak depends on
$f_{2}$ (as can be see in Figure 5).

In a previous analysis performed by Orellana & Bersten (2021), we study a
double-nickel distribution for a less massive progenitor, (He4). From that
analysis, together with the results presented here, we found that a large
temporal departure between the light curve maxima is a consequence of
depositing the inner and outer nickel at a larger distance in mass coordinate,
that is, if $f_{2}$ is much greater than $f_{1}$. Furthermore, the temporal
separation of the peaks seems more prominent for a larger ejected mass,
produced for a more massive progenitor. For a fixed progenitor, the mass of
the ejecta is controlled by $f_{0}$ and the effect of its variation can be
seen in Figure 4.

The effect of the $X_{\rm out}$ variation is presented in Figure 2. It has a
clear correspondence with the luminosity of the first peak in the LC. We show
also an accompanying plot of the evolution of the temperature and velocity at
the thermalization depth (Fig 3). As suggested by Ensman & Burrows (1992), the
’thermalization’ depth can be estimated as the layer where the continuum is
actually formed, given by the condition $\tau_{\rm abs}\tau_{\rm sct}\sim 1$,
where $\tau_{\rm sct}$ is the optical depth for scattering and $\tau_{\rm
abs}$ is the optical depth for absorption.

The first rise in the LC after the shock breakout results from the heating
power provided by the radioactive decay and this seems to be a distinctive
feature of the scenario, although it is subtle if the external
M(${}^{56}\mathrm{Ni}$) is small. In the first few days (around day five since
the explosion), the temperature $T_{\rm th}$ may even rise for a while before
decreasing again. An increase in temperature should have an effect on the
broadband photometry, producing a differential change in flux depending on the
wavelength. This is probably attributable to a color effect, although this
change may be slight, especially if the external mass of nickel is small. If
such a behaviour were to be found in a given SN along with the first peak in
the LC, this is an strong sign of an extended distribution of radioactive
material in the outer ejecta regions, as in the scenario we are studying here.
As part of the energy transforms to kinetic, it is also accompanied by an
increase in velocity. Both panels include the case without external 56Ni for
comparison.

Figure 1:  Schematic abundance profile of 56Ni as a function of the interior
mass fraction in the star. The parameters used to describe this distribution
are indicated in the axes. The color of each arrow and corresponding label is
used in next figures to facilitate the interpretation.

Figure 2:  Effect of the variation in the abundance of 56Ni, with the other
parameters fixed. All these calculations correspond to the He11 progenitor,
with $E_{\rm exp}=2.0$ foe, a $2.15$ M⊙ compact object (then $f_{0}=0.199$),
$f_{1}=0.25$, $f_{2}=0.91$ and $f_{3}=0.991$. Total M(${}^{56}\mathrm{Ni}$)
given in units of solar masses is indicated in parentheses. Left: Effect of
variation of the inner abundance $X_{\rm in}$. Right: Same effect but related
to the external abundance $X_{\rm out}$. In the left panel, the outer
abundance of 56Ni is set to $X_{\rm out}=0.01$ and $X_{\rm in}=0.047$ in the
right panel.

Figure 3: Effect of the external abundance of nickel on the temperature and
velocity evolution. The other parameters have the same values as in the right
panel of Figure 2. The curves show a zoom on the temperature and velocity at
the color depth (also known as the thermalization depth) before the second
peak of the LC. Afterwards, they all behave similar to the case $X_{\rm
out}=0$.

Figure 4: Effect of the variation in the parameter $f_{0}$ that, for the fixed
progenitor (here He11), defines both the mass of the ejecta and of the compact
object. The numbers in parentheses are the total M(${}^{56}\mathrm{Ni}$) in
M⊙. In this case $f_{1}=0.4,\,f_{2}=0.85,\,f_{3}=0.95,\,X_{\rm
in}=0.047,\,X_{\rm out}=0.01$.

Figure 5: Variation of the fractions $f_{i}$ for the He11 progenitor, with
$E_{\rm exp}=2$ foe, a fixed $2.15$ M⊙ compact object (then $f_{0}=0.20$),
abundances $X_{\rm in}=0.05$ and $X_{\rm out}=0.01$, and the varying parameter
indicated in each panel. For the left panel: $f_{2}=0.85$ and $f_{3}=0.95$.
For the middle panel: $f_{1}=0.30$ and $f_{3}=0.95$. In the rigth panel:
$f_{1}=0.30$ and $f_{2}=0.70$. With the abundances fixed, the mass of 56Ni
change according to the variation of the distribution parameters. The total
M(${}^{56}\mathrm{Ni}$) is indicated in parentheses.

##  4 Published double-peaked SNe

In recent years, thanks to great observational efforts, the number of SNe that
present an emission prior to the main maximum of their light curve has notably
increased. Currently, this early emission has been detected in almost all type
of SNe, although the detection frequency is dependent on the type of
supernovae, which may be related to their origin. The list in Table 1 is a
collection of reported SNe, including a few superluminous SNe that have shown
early emission. They illustrate the underlying diversity in how the
designation of double-peaked SNe is applied. Figure 6 shows the data set from
this list of references and allows us to make some comparisons. First, we note
that there is a wide range of luminosities, timescales, and, hence, slopes in
the LCs. Our intention has been to remain thorough with regard to the
population of normal SNe having an initial rise followed by a double peak in
the observed bolometric data.

In this selection, we excluded all the SNe that manifest only an excess of
emission in the bolometric LC, despite of having two distinct light curve
components in one or more photometric bands. Sometimes that excess is only
modest and noticeable when one focus on the luminosity data at epochs around
the main peak. For example, SN2013ge displays such a morphology, mentioned as
a shoulder in the LC description (Drout et al., 2016). Interestingly, in order
to explain this case, the authors have considered a model with outwardly mixed
56Ni and incomplete gamma-ray trapping to explain the rapid decline.

Type IIb SN1993J has attracted great attention (e.g. Chevalier & Fransson
1994; Woosley et al. 1994; Nakar & Piro 2014, and references) and it has been
studied for many years and in different wavelengths (in radio, e.g. Marcaide
et al. 2009; Martí-Vidal et al. 2011; and most recently published in IR by
Zsíros et al. 2022). Among the few SNe of type Ib included in the list, we aim
to dedicate more attention to SN2008D (in Section 4.2). The SLSN2006oz
presents a plateau-like morphology, with a duration of 6–10 days before the
rise of the main peak, which is more evident when the axis of time has linear
scale. Ouyed et al. (2016) set up a specific scenario to explain the origins
of this LC, which consists of the transition from a neutron star to a quark
star. The more conventional optional models are not well constrained in this
case, due to the lack of post-maximum observations (Leloudas et al., 2012). We
refer to Smith et al. (2016) for the SLSNe that have well-measured pre-
explosion photometry, some of them with double peaks. Next, we focus onto the
SNe that that have not been classified as superluminous 111We do not simply
refer to a ’normal’ supernovae because SN2005bf is somewhat brighter than the
mean SNe..

A different but fascinating case (albeit excluded from our list) is offered by
the hydrogen-rich SN2009ip and its analog SN2010mc Margutti et al. (2014).
Their status of true SNe has been questioned, suggesting that they may
represent another type of violent, non-terminal event.

The calcium-strong transients, such as SN2019ehk or SN2021gno, are SNe that
present atypically high calcium-to-oxygen nebular line ratios. They have
generated a consistently inconclusive debate with regard the preferred
scenario (see Jacobson-Galán et al., 2021; Shen et al., 2019; Jacobson-Galán
et al., 2022, and references therein); thus, the origin of these events is
unclear. It is worth mentioning that these cases show a fast evolution of the
early bolometric LC and their modelling with a double 56Ni profile could be
comparable to the case we discuss in Subsection 4.2.

Table 1: Light curve references. When not stated, there is a reasonably accurate estimate of the time of explosion, $t_{0}$. Name | Type | Reference | Comment
---|---|---|---
SN1993J | IIb | Ray et al. (1993) |
SN1999ex | Ib/c | Stritzinger et al. (2002) |
SN2005bf | Ib/c | Folatelli et al. (2006) |
SN2006oz | SLSN I | Leloudas et al. (2012) | days since discovery
SN2008D | Ib | Modjaz et al. (2009) | XRT080109
|  | Bersten et al. (2013) | earliest $L_{\rm bol}$ data
|  | Tanaka et al. (2009c) | latest data
PTF11mnb | Ic | Taddia et al. (2018) | Stripped envelope
PTF12dam | SLSN I | Vreeswijk et al. (2017) |  $t_{0}=70$ d, slow decline, no double
LSQ13abf | Ib | Stritzinger et al. (2020) |
iPTF13dcc | SLSN I | Vreeswijk et al. (2017) |  $t_{0}=89$ d
LSQ14bdq | SLSN Ic | Nicholl et al. (2015) |
DES14X3taz | SLSN I | Smith et al. (2016) |
iPTF14gqr | Ic | De et al. (2018) | SN 2014ft
iPTF15dtg | Ic | Taddia et al. (2016) |
SN2019cad | Ic | Gutiérrez et al. (2021) | err $\Delta t_{0}=4$ d
SN2019dge | Ib | Yao et al. (2020) | first data with large error
SN2019ehk | Ib | Jacobson-Galán et al. (2020, 2021) | Ca-rich. err $\Delta t_{0}=0.1$ d
| IIb | De et al. (2021) |
SN2019stc | SLSN I | Gomez et al. (2021) |
SN2020bvc | Ic-BL | Ho et al. (2020) | similar to SN2006aj, X-ray detection
SN2020faa | SLSNII | Yang et al. (2021) | iPTF14hls-like

Figure 6: Data from the literature, as detailed in Table 1, displays a variety
of double-peaked SNe with early detections. For visualization, the error bars
are not included. For objects that present a clear double-peaked LC, we see
that the morphology in both peaks is quite diverse and it is likely that the
physical explanation of these is varied as well. Some SLSNe are also shown
and, in addition, PTF12dam, which is not double-peaked, is included as a
reference of the SLSN.

Figure 6 shows that the SNe that has early detections are more often
discovered as they pass through a decline in the bolometric luminosity. This
pattern does not necessarily exclude a previous rise. It is then convenient to
clarify what we mean for the early phases when describing the observational
$L_{\rm bol}$ data of double-peaked SNe. The early phase consists of the
epochs between the discovery and the first time the luminosity trend changes,
either at a local minimum (so there was an early decline only detected) or at
a local maximum (an early rise was observed). When a given SN exhibits an
early decline, there are two possibilities: either we are seeing the end of
the cooling phase or this decline follows the end of a non-detected first
maximum.

###  4.1 A selected family

From all the events shown in Figure 6, we selected a group that share the
observational characteristics of the two maxima clearly present in the
bolometric LC with an initial rise in luminosity prior to the first maximum.
We restricted our study to objects that have not been classified as SLSNe. We
are left with the few cases that show analogies in the LC to SN2005bf. This SN
has very accurate measurements, except for the first and the very late ($t>70$
d) bolometric points (Folatelli et al., 2006). There are other two SNe that
clearly resemble SN2005bf; PTF11mnb (Taddia et al., 2018) and SN2019cad
(Gutiérrez et al., 2021). For the latter, we considered the zero–extinction
data. These three LCs can be compared in more detail in Figure 7, which
includes the available error bars. In each case, a double 56Ni profile was
considered in order to model the LC. Our new contribution is to use a unique
function for the nickel abundance profile $X(f)$, the one presented in Section
2.

Table 2 presents our fitting parameters for this selected family of SNe. The
resulting LC are shown in Figure 8. We also show a panel with the velocity
evolution, but we notice the models are not meant to explain photospheric
velocities. However, the comparisons with velocities give reasonably good
results, especially for PTF11mn and SN2019cad.

For SN2005bf and SN 2019cad, the hypothesis of a constant standard gamma-ray
opacity does not accommodate the complete observed LC, as was found in the
aforementioned references. A modified leakage is invoked at late times. Our
hydrodynamic code is not well designed for the phases with optically thin
conditions, this fact could help explain the need for an adaptation of
$\kappa_{\gamma}$values.

According to our results, for SN2005bf and SN2019cad, the turn-down of the
gamma-ray opacity required to match the fast decline in the LC after the main
peak is a decrease of $\kappa_{\gamma}$ by a factor of 10 and 60,
respectively. In support of this hypothesis, we note that Tanaka et al.
(2009a) found polarimetric evidence for an aspherical, possibly unipolar,
explosion in the case of SN2005bf. A significant change of the gamma-ray
leakage could be expected for SN2019cad given the LC analogy with SN2005bf.

Instead, PTF11mnb does not require an adaptation with reduced gamma-ray
opacity, but we note the large error bars in the bolometric data at epochs
after $\sim 60$ days. In this case, the alternative scenario provided by
Taddia et al. (2018) consist of a hybrid combination with a smaller
M(${}^{56}\mathrm{Ni}$)$\approx 0.11$ M⊙ from Arnett model, plus a magnetar
with $B=5\times 10^{14}$ G and $P=18.1$ ms. However, the authors disfavour
this magnetar because nickel stands to offer a simpler explanation for the
overall LC.

A similar proposal has been discussed for SN2005bf (Maeda et al., 2007), but
with a relatively large braking index as an alternative to full energy
trapping. A magnetar also works for SN2019cad (Gutiérrez et al., 2021) with
the 56Ni already accommodated into a two-zone profile. We note that the
formation of a magnetar and the eventual presence of outflows that are thought
to reshape the nickel profile could, in principle, coexist (Shankar et al.,
2021).

Figure 7: Three objects selected to apply the proposed double-peaked model.
The presence of a raising phase before the first peak and the time of this
peak were a critical characteristic for our sample selection. We also note
that the time of the second peak is at an epoch much later that what has
typically been observed for SESNe.

Figure 8: Fitting models for our selected sample, SN2005bf, PTF11mnb, and
SN2019cad, within our parametric prescription for the 56Ni double distribution
profile. The late light curve decline slope for SN2005bf and SN2019cad is
reproduced when passing from a fixed $\kappa_{\gamma}$ opacity (dashed line)
to one adjusted to account for the gamma-ray leakage (solid line).

The need to tune the value of $\kappa_{\gamma}$ can be thought as a drawback
for the double 56Ni model, but it can be roughly justified in relation to the
asymmetries of the explosion. For a sample of normal SNe, Wheeler et al.
(2015) found very different late-time tails and they analysed the source of
their heterogeneous slopes in relation to the trapping of gamma-ray energy
from radioactive decay. It is interesting to mention that although (Chen et
al., 2015) postulated (in another context) that to model the late epochs
($t>200$ d) of PTF12dam, a modest increase in the escape of gamma-ray
radiation was needed. We note that for the magnetar scenario, the energies of
the gamma-ray photons can be quite different than those from the radioactive
decay, therefore entering into a range where it is valid to suggest that the
opacity for the high-energy radiation is dominated by pair production and
becomes one order of magnitude lower than in previous phases. A similar
treatment of the late-time leakage of hard radiation was invoked by other
authors (e.g. Wang et al. 2015; Nicholl et al. 2017; Cartier et al. 2022);
whereas, recently, a more consistent approach was investigated by Vurm &
Metzger (2021).

For SN2019cad, we show a model very similar than previously obtained in
Gutiérrez et al. (2021). In the present case, the inner 56Ni component has a
flat abundance, while in the previous study, $X_{\text{in}}$ exhibited a
slight outward increase with mass coordinates to mimic the abundances of Si-
Ar-Ca. Here, the LC shape of SN2019cad is reproduced by a pre-SN mass of 11
M⊙, and an explosion energy of 4 foe, which is slightly greater than the 3.5
foe reported in Gutiérrez et al. (2021). The double-peaked 56Ni distribution
has an external component of 0.049 M⊙and an internal component of 0.28 M⊙. The
LC after the main peak drops steeply. To model this effect, an enhancement of
the gamma-ray leakage is assumed at around the date of $L$-main peak: a
constant $\kappa_{\gamma}=0.03$ is switched to $\kappa_{\gamma}=0.001$ cm2
g-1. As in the model presented by Gutiérrez et al. (2021), the external nickel
needs to be close but below the surface to fit the time scale of the first
peak of the light curve. For the cases of SN2005bf and PTF2011mnb, it is not
straightforward to establish a detailed comparison between our results and the
models of Folatelli et al. (2006) and Taddia et al. (2018) because of the
different 56Ni profile. In Taddia et al. (2018), the nickel profile for
PTF11mnb is scaled from SN2005bf. The published values of total
M(${}^{56}\mathrm{Ni}$) are 0.6 and 0.59 M⊙, respectively, while we obtained
0.448 M⊙ for SN2005bf and 0.499 M⊙ for PTF11mnb.

Table 2: Model parameters

Parameter | SN2005bf | PTF11mnb | SN2019cad 1
---|---|---|---
$M_{\rm ej}$ |  $6.1$ M⊙ |  $6.1$ M⊙ |  $9.55$ M⊙
$M_{\rm preSN}$ |  $8$ M⊙ |  $8$ M⊙ |  $11$ M⊙
$E_{\rm exp}$ |  $1.7\times 10^{51}$ erg |  $1.5\times 10^{51}$ erg |  $4\times 10^{51}$ erg
$\kappa_{\gamma}$ [cm2/g] | 0.03 $t\leq 65$ d | 0.03 all epochs | 0.03 $t\leq 45$ d
| 0.0018 $t>65$ d |  | 0.001 $t>45$ d
$f_{0},\,f_{1}$ | 0.2, 0.247 | 0.2, 0.259 | 0.13, 0.154
$f_{2},\,f_{3}$ | 0.524, 0.99 | 0.563, 1 | 0.646, 0.947
$X_{\text{ in}}$, $X_{\text{ out}}$ | 0.952, 0.029 | 0.960, 0.029 | 0.99, 0.015
M(${}^{56}\mathrm{Ni}$) in, out a | 0.352, 0.096 M⊙ | 0.395, 0.104 M⊙ | 0.283, 0.049 M⊙
M(${}^{56}\mathrm{Ni}$) total | 0.448 M⊙ | 0.499 M⊙ | 0.332 M⊙

  * a M(${}^{56}\mathrm{Ni}$) is afterwards computed, not an initial parameter.

  * 1 For the data with host-galaxy $E(U-B)=0$ in Gutiérrez et al. (2021).

###  4.2 An extreme example

A very interesting feature noted for SN 2008D was the early detection in
X-rays and multiband follow-up (Modjaz et al., 2009), whose combined
interpretation has been debated in the past (see also Branch & Wheeler 2017
for a summary of the open issues). In Figure 9, we show our modelling of
SN2008D light curve. This SN has a fast evolution in the LC with the main peak
(the second) at a timescale of $\sim 17$ d. We note this is similar to the
time when the SNe of the sample of Section 4.1 present the first and weakest
peak. That early peak can be therefore interpreted as indicative for a 56Ni
profile with rather different values of the parameters than those treated in
the previous section, which makes this SN an intriguing extreme object.

This morphology is particularly challenging for other models, since a more
promising case is represented by the explosion of a Wolf Rayet star through a
thick wind (see discussions by Dessart et al. 2018; Rabinak & Waxman 2011, and
references). Models with an envelope give a reasonable fit to the early
observations with the exception of the earliest data point, that was
considered in Bersten et al. (2013) analysis, but not in other LC models for
SN2008D. For the main peak, Tanaka et al. (2009b) discussed the degeneracy of
the parameters between the possible progenitors, with the
M(${}^{56}\mathrm{Ni}$)$\sim 0.05-0.07$ M⊙and the 56Ni efficiently mixed into
the outer ejecta layers.

After considering different envelopes and thick wind configurations to fit all
of the early SN2008D data, Bersten et al. (2013) assumed a double-nickel
distribution as an alternative and found that it improved the agreement with
the data. Here, we considered, for SN2008D, a progenitor star with main
sequence mass of 18 $M_{\odot}$, producing, at the moment of the explosion, a
Helium core of $\sim 5$ M⊙, denoted by He5. The evolution of this progenitor
model was calculated by Nomoto & Hashimoto (1988). To fit the first peak, the
double profile of nickel is pushed to the extreme case where it reaches the
surface of the star and the abundance of radioactive material is particularly
high there ($X_{\rm out}=0.71$), a value that is probably unphysical. The
model shown here has M(${}^{56}\mathrm{Ni}$) of 0.074 M⊙ in the inner region
and 0.018 M⊙ at the outer layer, that is larger than 0.01 M⊙ for the external
M(${}^{56}\mathrm{Ni}$) in Bersten et al. (2013), but we note that also the
exploded progenitor star is different (He8 in their case). In our scheme of
parameters, the internal component extends from $f_{0}=0.36$ up to
$f_{1}=0.93,$ with $X_{\rm in}=0.026;$ while the outer one goes from
$f_{2}=0.995$ to $f_{3}=1$.

We simulated the explosion with a typical value $E_{\rm exp}=1$ foe, rather
closer to the energy in the simple model from Chevalier & Fransson (2008). In
particular, SN2008D does show a less flat velocity evolution than the SNe in
the previous section. If the velocity data were considered when building the
model, it is likely that a more energetic explosion would be favoured to
explain the fast ejecta, as found by Bersten et al. (2013). Similarly to our
approach to modelling SN 2005bf and SN 2019cad, a change in the trapping of
the gamma-rays, is needed to improve the match with the LC at late epochs
($t>40$ d). Specifically, we introduce a slight change, lowering the value of
$\kappa_{\gamma}$ by a factor of 2.

Figure 9: SN2008D requires a strongly inverted 56Ni profile, with the external
abundance of $X_{\rm out}=0.71$ in a layer that reaches the surface; in this
case, the progenitor model is He5 with $E_{\rm exp}=1$ foe. The late LC
decline is reproduced when passing from a fixed $\kappa_{\gamma}=0.03$
cm${}^{2}/g$ opacity (dashed line) to half this value at $t>40$ d (solid
line). The lower panel shows the evolution of the photospheric velocity,
although we did not use observational velocities to constrain the model. The
velocities of the photosphere are taken from Tanaka et al. (2009b).

This model is still simple and we cannot make further comments on the
spectroscopic features. Dessart et al. (2018) modelled SN2008D prior to the
re-brightening as the explosion of a lower mass (2.73 M⊙) star with an
extended and tenuous envelope. The authors most vocal critique with regard to
the external 56Ni is that it should amplify non-thermal effects in the outer
layers, generating very broad HeI absorption lines at one week – which were
not found in the observational data. However, these authors were unable to
properly model all of the existing early data. The newest models for the type
Ic supernovae LSQ13abf and SN 2008D are presented by Woosley et al. (2021),
including a consideration of their progenitor evolution in close binaries.

##  5 Discussion and conclusions

The possibility of using an arbitrary distribution of 56Ni in our code offers
the chance to investigate the effects of any spherically symmetric profile. In
the scheme we considered in this work, the alleged presence of jet-like
outflows in the collapse of a massive star could drive the presence of
external radioactive material that could well be responsible for the doubled
distribution of 56Ni. A more consistent justification of this scenario would
emerge from a thorough understanding of the explosion mechanism itself and the
development of calculations that (following the propagation of the outflows)
could properly locate the nucleosynthetic yields of radioactive elements into
the stellar interior. Studies in this direction were made by Khokhlov et al.
(1999), while a discussion on the radioactive isotope yields from various jet-
like explosions of magnetorotational core-collapse SNe was initiated by
Nishimura et al. (2015). The case of a jet choked inside of the star is of
particular interest. Related studies have been presented by Nakar (2015) and
Senno et al. (2016). In addition, an analysis by Soker & Gilkis (2017),
focussed on SLSNe, called for a paradigm shift from neutrino-driven to jet-
induced explosion models of all core collapse supernovae.

The calculated LCs occur with different timescales depending on several
parameters beyond the radioactive content – be it stellar- or explosion-
related. The strongest conclusion from our results regards the SNe whose data
resemble the SN2005bf morphology, namely, those with an early rise before the
first peak of the LC. In this scenario, the first peak after shock breakout in
the bolometric LC would be potentiated by the outer 56Ni. This is reasonable
since the energy coming from the outer radioactive material is covered by an
optically less thick layer, then its energy emerges sooner than the outcome
from the inner 56Ni. Instead of accommodating the outer nickel as part of one
distribution strongly mixed outwards and a different power source for the main
peak, we have two separated layers that are radioactively enriched. We re-
estimated the nucleosynthetic yields within this prescription for the selected
sample of reported SNe that are SN2005bf-like, plus the case of SN2008D, which
has less available data on an early rise. For SN2008D, that early rise was
only once modelled within this scenario by Bersten et al. (2013). If the model
depicted here is considered acceptable for the case of SN2008D, it could be
noted that it would thus exhibit one of the shortest timescales for the first
peak. This encourages the proposal that the model can also accommodate other
SNe as well. However, the contrast in luminosity at the earliest phases is
also important. We cannot place a strong statement on the remaining SNe shown
in Section 4 because of all the other variables that need to be further
explored and that are beyond the scope of the current work.

We have designed a systematic study to explore the applicability of a simple
56Ni distribution to model some of the double-peak SNe. We have considered
that the outer mass of nickel is in all cases lower than the inner one. This
is regulated through the abundance and extent in mass coordinate. We
considered a fixed H-poor progenitor (He11) here, whereas for a less massive
case, we previously presented a brief study in Orellana & Bersten (2021). Our
prior results are consistent with the exploration of the parameter space
presented here. Based on a direct comparison, the less massive star produces a
more rapid evolution of the overall LC, but a simple relation between the
timescales for the peaks and the 56Ni profile does not appear to be
straightforward.

The subsequent decline after the first peak does not seem entirely consistent
with the rapid early decline observed in many SNe. If the first luminosity
data fall very steeply, this is not what we would expect from nickel power,
unless it has an extreme distribution. Therefore, the observed early decline
slope of the LC could minimize doubts about a prior, non-detected rise. For a
given SN, this comparison is valid if the explosion date is well determined,
however, because of the inherent difficulties of the discovery at the earliest
stages after the explosion, sometimes this information is not available. We
cannot rule out that our selection of SNe with an initial bolometric rise may
suffer from an observational bias that makes the sample so small.

Diffusing the radioactive energy out of the ejecta is a mechanism that
combined with the radioactive decay timescale cannot produce an arbitrarily
early first peak. Our exploration indicates that around one day offers only a
limiting case. In the extreme distribution, we fit to the LC of SN2008D, the
first peak happens at around $\sim 1.3$ days as a consequence of the external
nickel located in a very thin, superficial layer, with very high 56Ni
abundance. Noebauer et al. (2017) explored the effect of other short-living
isotopes that could be important into the shaping of the early phases of the
LC. That additional energy source, provided by 52Fe and 48Cr decay channels,
is not yet implemented in our code.

In the cases where the outer 56Ni is deeper and less concentrated, the rise
takes few days since the explosion and, therefore, the first peak is produced
later. This is related to the decay half-life of the radioactive elements we
are considering. Moreover, if the first peak occurs within $\sim 10-20$ days
after the explosion epoch as in the case of SN2005bf-like objects, the outer
abundance found is moderate ($\sim 0.1$ at most) and the radial extent in mass
is wider (that is, $f_{2}$ lower). We note that other models face difficulties
in producing an early rise with the first peak within this timescale, which is
coincidental with the time when other SNe exhibit their maximum (see for
instance González-Gaitán et al. 2015). In the cases we investigate here, well-
covered photometry prior to the first peak was helpful in constraining how
deeply the external 56Ni-rich material should be placed. The radioactive 56Ni
that powers the second peak of these objects is concentrated close to the
compact object.

The so-called hybrid models containing different sources of energy can work
for many SNe. The question of whether this model, coupled with a doubled
distribution of 56Ni, is a good alternative seems to be related to the first
slope of the observed bolometric LC, with an early rise preferred over an
early decline. The timescale of the minimum between the peaks is somehow
related to the progenitor mass; however, to put quantitative constrains, the
whole parameter space should be further explored, namely, the nickel profile
varied in combination with other physical quantities that we have considered
to be fixed here (the explosion energy, $E_{\rm exp}$, the stellar
characteristics such as mass and radius, as well as the leakage capacity for
the gamma rays or other transport parameters). Any improvement on the model
should be also accompanied by the inclusion of the velocity data into the
estimation of the goodness of the fit (see discussion in Martinez et al. 2020
and references therein), which we have not done here.

In order to explain the observed late-time slope of the LC, we had to abandon
the idea of the complete trapping of gamma-rays and thermalization of the
energy deposited in the ejecta. Instead, a few days after the second maximum,
this was accounted by introducing a change in $\kappa_{\gamma}$. This may
emulate the fact that even if a gamma-ray photon is absorbed, a fraction of
its energy does not thermalize and is allowed to leak out. This change is
phenomenologically motivated and case-dependent. We cannot provide a rigorous
quantitative justification of this assumption. However, this is part of a
long-known problem that produces great heterogeneity in the late decline in
the LC (Clocchiatti & Wheeler 1997, and references). A variable gamma-opacity
has been suggested to result from asymmetries in the ejecta (Branch & Wheeler,
2017) and the gamma-ray leakage could also be enhanced due to the presence of
small-scale inhomogeneities, for instance, low-density holes or clumps in the
ejecta that we cannot catch in our 1D code. We note that 3D calculations with
the pertinent details have been developed by Jerkstrand et al. (2020) for the
modelling of decay gamma-ray photons that are not captured by the ejecta.
Their detection points to a large degree of 56Ni-mixing and confirms that some
of the high-energy photons leak out. In an independent way, the gamma-ray
transport was simulated through Monte Carlo techniques by Wilk et al. (2019),
who suggested a time and spatially varying gray opacity factor could improve
results within the approximation of Swartz et al. (1995), namely, the one we
apply in this work.

Nowadays, early detections of SNe are becoming more frequent and there are
many cases where the detected emission poses a challenge to existing models.
Therefore, it is plausible that more exotic explanations need to be explored
to explain these LCs. In this sense, calculations such as those presented in
this study can help elucidate the underlying physical scenario.

###### Acknowledgements.

We thank the anonymous referee for the comments and suggestions that have
helped to improve the paper. This research was partially fund by UNRN PI2020
40B885, and grant PICT-2020-SERIEA-01141. We also acknowledge the support of
CONICET through project PIP 112-202001-10034.

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