# The $\gamma$-ray deposition histories of Calcium-rich supernovae

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

###### Abstract

Calcium-rich supernovae (Ca-rich SNe) are faint, rapidly evolving transients
whose progenitor system is yet to be determined. We derive the $\gamma$-ray
deposition histories of five Ca-rich SNe from the literature in order to place
constraints on possible progenitor systems. We find that the $\gamma$-ray
escape time, $t_{0}$, of the Ca-rich SNe sample is $\approx
35\text{--}65\,\rm{d}$, within the unoccupied region between Type Ia SNe and
stripped envelope supernovae (SESNe). The $t_{0}\text{--}M_{\mathrm{Ni56}}$
distribution of these SNe, where $M_{\mathrm{Ni56}}$ is the synthesised 56Ni
mass in the explosion, creates a continuum between the Type Ia and SESNe
$t_{0}\text{--}M_{\mathrm{Ni56}}$ distribution, hinting at a possible
connection between all the events. By comparing our results to models from the
literature, we were able to determine that helium shell detonation models and
core-collapse models of ultra-stripped stars are unlikely to explain Ca-rich
SNe, since the gamma-ray escape time in these models is smaller than the
observed values. Models that agree with the observed
$t_{0}\text{--}M_{\mathrm{Ni56}}$ distribution are explosions of low mass,
$M\approx 0.75\text{--}0.8\,M_{\odot}$, white dwarfs and core-collapse models
of stripped stars with an ejecta mass of $M\approx 1\text{--}3\,M_{\odot}$.

###### keywords:

supernovae: general -methods: data analysis

††pubyear: 2021††pagerange: The $\gamma$-ray deposition histories of Calcium-
rich supernovae–The $\gamma$-ray deposition histories of Calcium-rich
supernovae

##  1 Introduction

Calcium-rich supernovae (Ca-rich SNe) are faint, rapidly evolving transients
with features of [Ca II] emission in their nebular phase spectra (see, e.g.,
Perets et al., 2010; Kasliwal et al., 2012; De et al., 2020). Their light
curves are characterised by relatively short rise and decline times and peak
magnitudes of $-14$ to $-16.5$, which are fainter than type Ia SNe and most
core-collapse (CC) SNe. They are usually detected in old stellar environments
in the far outskirts of their host galaxies, suggesting that they originate
from old progenitors that have travelled a great distance from their birth
site, or that they occur in very faint systems such as globular clusters
(although recent works have failed to detect an underlying host system Lyman
et al., 2014; Lunnan et al., 2017). The exact progenitor systems of Ca-rich
SNe remain unknown.

Several models have been suggested for their origin and explosion mechanism,
such as helium-shell detonations on white dwarfs (WDs; Sim et al., 2012;
Waldman et al., 2011), a tidal disruption of a WD by a neutron star (NS) or an
intermediate-mass black hole (Rosswog et al., 2008; Metzger, 2012), a CC of
ultra-stripped stars (Tauris et al., 2015; Moriya et al., 2017), and a merger
of a WD with another WD or a NS (Zenati et al., 2019, 2020; Pakmor et al.,
2021; Jacobson-Galán et al., 2022).

One approach to constrain the progenitor system is to compare photometric and
spectroscopic observations to radiation-transfer calculations of different
progenitor models. Dessart & Hillier (2015) performed simulations of WD
helium-shell detonations and found photometric and spectroscopic similarities
with several Ca-rich SNe, although the simulated light curves evolved faster
than observations. Moriya et al. (2017) studied the CC of ultra-stripped stars
(with ejecta mass $<0.2\,M_{\odot}$), and found that the simulated light
curves are consistent with several Ca-rich SNe, although the spectra agreed
with only part of the objects. Polin et al. (2021) calculated the nebular
spectra of double-detonation sub-Chandra Type Ia explosions, and found that
the results of the low-mass progenitors are similar to Ca-rich events, with a
high ratio of observed [Ca II]/[Fe III], despite the small amount of
synthesised Ca in the simulation. However, the photospheric phase spectra did
not match well with the masses considered in their work ($M_{\text{WD}}\geq
0.7\,M_{\odot}$).

Another approach is to analyse the environments and kinematics of Ca-rich SNe.
The conclusion of these studies is that Ca-rich SNe originate from WDs (Foley,
2015; Perets & Beniamini, 2021), or that the CC of massive stars cannot be
their only explosion channel (Dong et al., 2022). Shen et al. (2019) proposed
that the progenitors of Ca-rich SNe are either old, metal-poor stars (with an
unknown explosion mechanism) or binary systems with at least one white dwarf.
Binary systems dynamically form in a globular cluster and are then ejected
from the cluster and explode - either due to a helium-shell detonation caused
by a merger (for a double-WD binary), helium shell deflagration (for a WD+He-
burning star companion), or a tidal disruption (for a WD+NS binary).

The volumetric rate of Ca-rich SNe may also be used to constrain their
progenitor system. While the sample of Ca-rich events is quite small, it has
increased significantly in the last decade due to large-scale surveys. Using a
sample of three events from the Palomar Transient Factory, Frohmaier et al.
(2018) calculated a rate of $1.21_{-0.39}^{+1.13}\times
10^{4}\,\text{Gpc}^{-3}\,\text{yr}^{-1}$. The ZTF CLU campaign (De et al.,
2020) found eight new sources with peak $r$ band magnitudes $-17<M_{r}<-15.5$,
and estimated their volumetric rate to be much smaller,
$2.21^{+1.01}_{-0.67}\times 10^{3}\,\text{Gpc}^{-3}\text{yr}^{-1}$. De et al.
(2020) also distinguished between two sub-classes of Ca-rich SNe based on the
spectroscopic similarity at peak light: Type Ib/c SNe (Ca-Ib/c) and SN 1991bg-
like Type Ia SNe (Ca-Ia). They further claimed that the peak light
spectroscopic properties form a continuum between the two subtypes.

Figure 1: Distribution of the $\gamma$-ray escape time, $t_{0}$, and
$M_{\mathrm{Ni56}}$, for the Ca-rich SNe analysed in this work, compared with
other types of SNe. Ca-rich SNe (black symbols) have $t_{0}$ values between
$35\text{-}60\,\rm{d}$. The blue shaded regions show the
$t_{0}$-$M_{\mathrm{Ni56}}$ range for different host extinctions
($0<E(B-V)<0.5\,\text{mag}$); for SN 2005E, SN 2012hn, and SN 2021gno, a de-
reddened NIR correction function was used to correct their bolometric light-
curves. The other types of SNe, taken from Sharon & Kushnir (2020), include
Type Ia (blue symbols), Type IIb (orange symbols), and Type Ib/c (yellow
symbols). Note that the Ca-rich sample is found in a unique region and does
not overlap with the other SN types.

In this paper, we adopt an alternative approach. Namely, we constrain the
progenitor systems of Ca-rich SNe by studying $t_{0}$, the $\gamma$-ray escape
time of radioactive products through the ejected material, defined by (see
Jeffery, 1999):

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

where $t$ is the time since explosion and $f_{\text{dep}}(t)$ is the
$\gamma$-ray deposition function, which describes the fraction of the
generated $\gamma$-ray energy that is deposited in the ejecta. At late enough
times, the ejecta becomes optically thin and the luminosity equals the
instantaneous deposition:

$Q_{\text{dep}}(t)=Q_{\gamma}(t)f_{\text{dep}}(t)+Q_{\text{pos}}(t),$ (2)

where $Q_{\text{dep}}(t)$ is the deposited energy in the ejecta from
radioactive decay (in this work, we mostly consider 56Ni and its products),
$Q_{\gamma}(t)$ is the radioactive energy generated from $\gamma$-ray photons
and $Q_{\text{pos}}(t)$ is the kinetic energy of positrons. For a small enough
$\gamma$-ray optical depth, each $\gamma$-ray photon has a small chance of
colliding with matter, such that the deposition function is proportional to
the column density, which scales as $t^{-2}$.

The $\gamma$-ray escape time, together with the synthesized 56Ni mass, can be
accurately measured from bolometric light curves of SNe, and can be easily
calculated for a given ejecta model, without radiation transfer calculations
(Wygoda et al., 2019). In Sharon & Kushnir (2020), it was shown that the
observed $t_{0}-M_{\mathrm{Ni56}}$ distribution can be used to classify
different types of SNe. Type Ia SNe have $t_{0}\approx 30\text{--}45$, while
stripped envelope (SE) SNe have $t_{0}\approx 80\text{--}170$.

Here we use the methods of Sharon & Kushnir (2020) to calculate the 56Ni mass
and the $\gamma$-ray escape time for Ca-rich SNe. Due to the rarity and dim
luminosity of Ca-rich SNe, our sample size is limited and contains only five
SNe: SN 2005E, SN 2012hn, SN 2016hnk, SN 2019ehk, and SN 2021gno. The main
results are shown in Figure 1, where we compare our derived values of
$M_{\mathrm{Ni56}}$ and $t_{0}$ for Ca-rich SNe to other types of SNe. As can
be seen in the figure, the Ca-rich sample is located in a region devoid of any
other SN types. This unique region forms a continuous bridge between Ia SNe
and Type IIb SNe. The $t_{0}$ values of the Ca-rich sample partially overlap
with Ia SNe, with the largest $t_{0}$ values similar to those of the lowest
Type IIb SNe. The $M_{\mathrm{Ni56}}$ values of the Ca-rich sample are, in
general, lower than those of other SNe, but the highest $M_{\mathrm{Ni56}}$
values agree with those of Type IIb SNe and with the lowest Ia SNe
$M_{\mathrm{Ni56}}$ values. Our findings raise the possibility that Ca-rich
SNe are related to either Type IIb SNe or Ia SNe (or both).

The paper is organised as follows: We present the sample of Ca-rich SNe in
Section 2. In Section 3, we briefly describe the methods of our analysis and
the $t_{0}\text{--}M_{\mathrm{Ni56}}$ distribution of our sample. In Section
4, we compare our results with models from the literature. We conclude in
Section 5.

##  2 The Ca-rich SN Sample

We analyse five well-observed Ca-rich events - SN 2005E, SN 2012hn, SN
2016hnk, SN 2019ehk, and SN 2021gno. We compute their bolometric light curves
using published photometry and estimated reddening and distances, using the
methods described in Sharon & Kushnir (2020). In brief, we first construct
light curves in all available bands, where we interpolate and extrapolate for
any missing epochs. The sources of photometry are Perets et al. (2010, SN
2005E), Valenti et al. (2014, SN 2012h), Galbany et al. (2019, SN 2016hnk);
Jacobson-Galán et al. (2020a, SN 2016hnk), Jacobson-Galán et al. (2020b, SN
2019ehk); Nakaoka et al. (2020, SN 2019ehk), and Jacobson-Galán et al. (2022,
SN2021gno)111 Jacobson-Galán et al. (2022) also provides the photometric
observations of the Ca-rich SN 2021inl, but these observations are not
sufficient to construct an accurate light curve with our methods.. The light
curves are converted to flux densities at their effective wavelengths to
create a spectral energy distribution (SED) for each epoch. The flux density
of wavelengths longer than the band with the longest effective wavelength is
estimated with a blackbody (BB) fit, and for short wavelengths it is linearly
extrapolated to zero flux at $2000\mbox{\AA}$. Since a significant fraction of
the total flux is emitted in near-infrared (NIR) wavelengths (see Section
2.1), NIR photometry is required to calculate the bolometric light curve
accurately. As SN 2005E, SN 2012hn, and SN 2021gno lack NIR observations, we
calculate their pseudo-bolometric light-curve
($2000<\lambda<8000\,\mbox{\AA}$) and correct for the missing NIR flux with
the NIR flux fraction of SN 2016hnk, which has the longest time span of NIR
observations (see Section 2.1 for a detailed discussion).

It is difficult to estimate the extinction correction of the host galaxy in
the case of Ca-rich SNe, a limitation that contributes significantly to the
uncertainty of the derived $M_{\mathrm{Ni56}}$ ($t_{0}$ is less sensitive to
the adopted extinction, see Section 3). We choose a favoured host extinction
value for each SNe based on previous works, but we consider for each SNe a
wide host reddening range, $0<E(B-V)<0.5\,\text{mag}$, with a Milky Way
extinction law of $R_{V}=3.1$. Finally, the spectral energy distributions are
integrated to obtain the bolometric luminosity. The obtained bolometric light
curves, using the favoured extinction values, are shown in Figures 5-9. The
photometry, the processed photometry (after interpolation, extrapolation, and
de-reddening) and the bolometric luminosity of the SNe are included in the
supplementary material.

We next describe in more detail each of the SNe in our sample, and in Section
2.1 we discuss the missing NIR flux correcting method.

### SN 2005E

SN 2005E, the Ca-rich SN prototype (Perets et al., 2010), exploded at a
projected distance of ${\sim}23$ kpc from the centre of the S0/a galaxy NGC
1032. Spectroscopy at peak light showed similarities with Type Ib SNe. We use
the $BVRI$ photometry from Perets et al. (2010), and we correct for the
missing NIR flux using the NIR flux evolution of SN 2016hnk. Following Perets
et al. (2010); Waldman et al. (2011), we choose zero host extinction as the
favoured value.

### SN 2012hn

SN 2012hn exploded at a projected distance of 6.2 kpc from the centre of the
E/S0-type galaxy NGC 2272. We use photometry from Valenti et al. (2014), which
includes observations from the $U$ to the $I$ bands, and we correct for the
missing NIR flux using the NIR flux evolution of SN 2016hnk. Valenti et al.
(2014) estimated a host extinction of $E(B-V)_{\text{h}}=0.2\,\text{mag}$,
based on the equivalent width (EW) of Na I D lines, which we adopt as the
favoured value.

### SN 2016hnk

SN 2016hnk was located at a projected distance of 3.71 kpc from the centre of
the SBa type galaxy MCG-01-06-070 (Galbany et al., 2019; Jacobson-Galán et
al., 2020a). The SN showed spectroscopic similarities to SN 1991bg at peak
light, but its luminosity decline is slower than Ia SNe and other Ca-rich SNe.
It was observed in the optical and NIR wavelengths by Galbany et al. (2019)
and Jacobson-Galán et al. (2020a). On the one hand, Galbany et al. (2019)
reported a host extinction of $E(B-V)_{\text{h}}=0.45\,\text{mag}$, based on
comparisons to Ia SNe and using the observed ratio of H$\alpha$ and H$\beta$
fluxes from host-galaxy spectra to estimate the Balmer decrement. Jacobson-
Galán et al. (2020a), on the other hand, did not correct for host extinction
since there was no evidence for Na I D absorption in any spectra. Given that
the intrinsic color of Ca-rich SNe is unknown, and given the large
uncertainties of the Balmer decrement method of Galbany et al. (2019), we
adopt the value of $E(B-V)_{\text{h}}=0\,\text{mag}$ as the favoured value.

### SN 2019ehk

SN 2019ehk exploded close to the core of the star-forming galaxy M100
(Jacobson-Galán et al., 2020b; Nakaoka et al., 2020). It featured a double-
peaked light curve, with the first peak interpreted as the expansion and
cooling of a shocked envelope or as a CSM interaction, or a combination
thereof (Jacobson-Galán et al., 2020b). SN 2019ehk had extensive ground-based
optical observations up to ${\sim}100$ days after the explosion, which were
later supplemented with additional ground-based and HST observations
(Jacobson-Galán et al., 2021; De et al., 2021). NIR observations in the $JHK$
bands are available up to ${\sim}25$ days from the explosion. SN 2019ehk
spectra showed a very deep Na I D absorption line, with both Jacobson-Galán et
al. (2020b) and Nakaoka et al. (2020) estimating a host extinction of
$E(B-V)_{\text{h}}=0.5\,\text{mag}$, which we adopt as the favoured value. De
et al. (2021) observed the SN ${\sim}280\,$ days from peak light in the $g$
and $I$ bands. Using these measurements to calibrate the late-time spectrum in
Jacobson-Galán et al. (2020a), they measured the [Ca II] and [O I] line fluxes
and interpreted the SN as a core collapse of a ${\approx}9.5\,M_{\odot}$
progenitor that was stripped of most of its envelope mass by a companion.

### SN 2021gno

SN 2021gno exploded in the SBa type galaxy NGC 4165, at a projected distance
of 3.6 kpc from its centre (Jacobson-Galán et al., 2022). Similar to SN
2019ehk, the light curves of SN 2021gno showed two peaks, the first was
interpreted to be the result of shock cooling emission and/or a CSM
interaction (Jacobson-Galán et al., 2022). It was observed from ${\approx}0.6$
to ${\approx}90$ days after the explosion with UV and optical filters, but had
no NIR observations. We account for the missing flux using the NIR flux of SN
2016hnk. Jacobson-Galán et al. (2022) did not correct for host extinction
since none of the spectra exhibited Na I D absorption, and we also adopt zero
reddening as our favoured value.

###  2.1 NIR fraction

Figure 2 shows the ratio of the NIR flux ($\lambda>8000\,\mbox{\AA}$) to the
total flux as a function of time since peak bolometric light for SN 2016hnk
(solid, blue line) and SN 2019ehk (solid, red line). Since the host extinction
estimates are highly uncertain, and in order to demonstrate the effect of the
extinction correction, we also plot the NIR fraction of SN 2016hnk with
$E(B-V)_{\text{h}}=0.5\,\text{mag}$ (dashed, blue line), and of SN 2019ehk
with zero host extinction (dashed, red line). The NIR flux ratios of Ia SNe
(solid, teal lines) and SESNe (dashed, black lines) from the sample of Sharon
& Kushnir (2020) are shown as well. The NIR flux ratio of SN 2016hnk reaches a
maximal value of ${\approx}0.5$, similarly to SESNe. De-reddening with
$E(B-V)_{h}=0.5\,\text{mag}$ reduces the maximal value to ${\approx}0.3$, in
agreement with Type Ia SNe (similar to the
$E(B-V)_{\text{h}}=0.45\,\text{mag}$ determined by Galbany et al., 2019, to
match the SN 2016hnk color curve to that of Ia SNe). SNe 2019hnk’s NIR ratio,
de-reddened with $E(B-V)_{\text{h}}=0.5\,\text{mag}$, is higher than all other
SNe, but seems consistent with SN 2016hnk, assuming no host extinction places
SN 2016hnk well above the other SNe. Assuming that SN 2016hnk and SN 2019ehk
share a similar NIR fraction leads to the conclusion that their NIR fraction
is similar to SESNe, and that $E(B-V)_{\text{h}}\approx 0(0.5)\,\text{mag}$
for SN 2016hnk (SN 2019ehk). This conclusion justifies the adopted NIR
correction with the (zero host extinction) NIR flux evolution of SN 2016hnk.

Figure 2: Ratio of the NIR flux ($\lambda>8000\,\mbox{\AA}$) to the total flux
as a function of time since peak bolometric light. Blue lines: SN 2016hnk with
$E(B-V)_{\text{h}}=0\,\text{mag}$ (solid, preferred value) and with
$E(B-V)_{\text{h}}=0.5\,\text{mag}$ (dashed). Red lines: SN 2019ehk with
$E(B-V)_{\text{h}}=0.5\,\text{mag}$ (solid, preferred value) and with
$E(B-V)_{\text{h}}=0\,\text{mag}$ (dashed). The NIR flux ratios of Ia SNe
(solid, teal lines) and SESNe (dashed, black lines) from the sample of Sharon
& Kushnir (2020) are shown as well. Assuming that SN 2016hnk and SN 2019ehk
share a similar NIR fraction leads to the conclusion that their NIR fraction
is similar to SESNe, and that $E(B-V)_{\text{h}}\approx 0(0.5)\,\text{mag}$
for SN 2016hnk (SN 2019ehk).

##  3 The $t_{0}$$-M_{\text{Ni}}$ relation of Ca-rich SNe

In this section, we calculate $t_{0}$ and $M_{\mathrm{Ni56}}$ from the
bolometric light curves constructed in Section 2. The calculation is based on
the Katz integral (Katz et al., 2013; Shussman et al., 2016; Nakar et al.,
2016), described in detail in Wygoda et al. (2019) and Sharon & Kushnir
(2020). In brief, the Katz integral is given by

$\begin{split}&QT=LT-ET,\\\
&QT\equiv\int_{0}^{t}Q_{\text{dep}}(t^{\prime})t^{\prime}dt^{\prime},\>\>\>&LT\equiv\int_{0}^{t}L(t^{\prime})t^{\prime}dt^{\prime},\end{split}$
(3)

where $Q_{\text{dep}}(t)$ is defined in Equation (2), $L(t)$ is the bolometric
luminosity, and $ET$ is the integrated time-weighted luminosity that would be
emitted if no 56Ni was produced. To describe the deposition fraction at all
times, we use the following interpolating function, which connects the
expected behaviour at early and late times:

$f_{\text{dep}}(t)=\frac{1}{\left(1+\left(t/t_{0}\right)^{n}\right)^{\frac{2}{n}}},$
(4)

where $n$ is a parameter that controls the smoothness of the interpolation and
is determined in the fitting process. The quality and time span of the current
Ca-rich SN sample are insufficient to accurately determine $ET$, so we assume
$ET=0$ in what follows. This choice has a negligible impact on $t_{0}$ and
$M_{\mathrm{Ni56}}$. We further omit the first peaks of SN 2019ehk and
SN2021gno ($t<10$ days since explosion) from our calculations, since it is
unlikely that these peaks are 56Ni-powered (Jacobson-Galán et al., 2020a;
Nakaoka et al., 2020; Jacobson-Galán et al., 2022).

The fit is performed by minimising the expression

$\frac{N_{\text{bins}}}{N_{\text{obs}}}\sum_{t_{i}\in
t_{L=Q}}\left[\left(\frac{L(t_{i})}{LT(t_{i})}-\frac{Q_{\text{dep}}(t_{i})}{QT(t_{i})}\right)\frac{LT(t_{i})}{L_{err}(t_{i})}\right]^{2},$
(5)

where $L_{\text{err}}$ is the luminosity error, $N_{\text{obs}}$ is the number
of observations, and $N_{\text{bins}}$ is the number of independent time bins,
defined as the number of times that $Q_{\text{dep}}$ changes by $10\%$ over
the time range of each SN. The ratio $N_{\text{bins}}/N_{\text{obs}}$ affects
only the uncertainty of the parameters (and not the best-fit values, see
below). The time range $t_{L=Q}=[t_{\text{min}},t_{\text{max}}]$ accounts for
the times where the assumption of $L=Q_{\text{dep}}$ is valid. The upper
limit, $t_{\text{max}}$, is determined by the latest epoch where the
observations follow the deposition model. In all of the SNe in our sample
except SN 2019ehk, $t_{\text{max}}$ is set to the last phase. SN 2019ehk
exhibits a substantial deviation from the deposition model at $t>80\,$day
since explosion (see Figure 8. The lower limit, $t_{\text{min}}$, is the
earliest epoch at which the fit procedure would result in a deviation of the
fit from the observations that is centred around zero. $t_{\text{min}}$ and
$t_{\text{max}}$ (if different from the last epoch) are indicated in Figures
5-9 by vertical dashed-dotted lines. The advantage of using the Katz integral
is that Equation (5) does not depend on $M_{\mathrm{Ni56}}$ and on distance,
so the fit is performed over $t_{0}$ and $n$ alone. $M_{\mathrm{Ni56}}$ is
then found by comparing the luminosity in the fitted range to the deposited
radioactive energy.

The uncertainty of the parameters are estimated by performing a Markov Chain
Monte Carlo (MCMC) algorithm using the MCMCSTAT Matlab
package222https://mjlaine.github.io/mcmcstat/, where the likelihood function
is Equation (5) and the priors are uniformly distributed over reasonable
domains.

The inferred $t_{0}$ and $M_{\mathrm{Ni56}}$ values are given in Table 1 and
shown in Figure 1 (black squares). The best-fits to each object are shown in
Figures 5-9. We find that the Ca-rich SNe occupy a small region in the
$t_{0}\text{--}M_{\mathrm{Ni56}}$ plane, with $t_{0}$ values in the range of
$35\text{-}65\,\rm{d}$ and $M_{\mathrm{Ni56}}$ values in the range of
$(1-5)\times 10^{-2}\,M_{\odot}$. When comparing our results for SN 2016hnk
with the results of Jacobson-Galán et al. (2020a), we find that the $t_{0}$
values are in good agreement, but that our $M_{\mathrm{Ni56}}$ is
${\approx}50$ per cent higher. This difference is expected, since the analysis
in Jacobson-Galán et al. (2020a) is based on a $3000\text{--}9000\,\mbox{\AA}$
pseudo-bolometric light-curve, and the NIR flux of this SN is
${\approx}40{--}50\%$ of its total flux (see Figure 2).

Table 1: The bolometric light curve parameters, derived using the Katz
integral method. The values of the derived parameters are the median values of
the posterior distribution, together with the $68\%$ confidence levels.

Name |  $\mu$a |  $E(B-V)_{\text{MW}}$b |  $E(B-V)_{\text{host}}$c | $R_{V}^{\text{host}}$ |  $M_{\text{Ni}56}$ $[M_{\odot}]$ |  $t_{0}\,$[day]
---|---|---|---|---|---|---
2005E | 32.78$\,\pm\,$0.06 | 0.03 | 0.00$\,\pm\,$0.00 | - | $0.009^{+0.001}_{-0.001}$ | $34^{+3}_{-3}$
2005E, $E(B-V)=0.5$ | 32.78$\,\pm\,$0.06 | 0.03 | 0.50$\,\pm\,$0.00 | 2 | $0.026^{+0.002}_{-0.002}$ | $32^{+2}_{-2}$
2012hn | 32.14$\,\pm\,$0.15 | 0.10 | 0.20$\,\pm\,$0.05 | 3.1 | $0.009^{+0.001}_{-0.001}$ | $48^{+3}_{-3}$
2012hn, $E(B-V)=0.5$ | 32.14$\,\pm\,$0.15 | 0.10 | 0.50$\,\pm\,$0.00 | 3.1 | $0.020^{+0.003}_{-0.003}$ | $46^{+3}_{-3}$
2016hnk | 34.17$\,\pm\,$0.03 | 0.02 | 0.00$\,\pm\,$0.00 | - | $0.044^{+0.005}_{-0.004}$ | $60^{+7}_{-5}$
2016hnk, $E(B-V)=0.5$ | 34.17$\,\pm\,$0.03 | 0.02 | 0.50$\,\pm\,$0.10 | 3.1 | $0.122^{+0.032}_{-0.032}$ | $75^{+13}_{-10}$
2019ehk | 31.05$\,\pm\,$0.13 | 0.02 | 0.47$\,\pm\,$0.10 | 3.1 | $0.024^{+0.007}_{-0.007}$ | $55^{+7}_{-4}$
2019ehk, $E(B-V)=0$ | 31.05$\,\pm\,$0.13 | 0.02 | 0.00$\,\pm\,$0.00 | - | $0.012^{+0.002}_{-0.002}$ | $62^{+4}_{-3}$
2021gno | 32.42$\,\pm\,$0.40 | 0.03 | 0.00$\,\pm\,$0.00 | - | $0.013^{+0.005}_{-0.005}$ | $43^{+2}_{-2}$
2021gno, $E(B-V)=0.3$ | 32.42$\,\pm\,$0.40 | 0.03 | 0.30$\,\pm\,$0.00 | 3.1 | $0.032^{+0.012}_{-0.012}$ | $41^{+2}_{-2}$

  * a

Distance modulus

  * b

Galactic extinction towards the SN

  * c

Host extinction

Because of the large uncertainty of the host extinction, we repeat the
calculations with reddening values in the range of $0<E(B-V)<0.5\,\text{mag}$
with a Milky Way extinction law of $R_{V}=3.1$. We also test a different NIR
correction for SN 2005E, SN 2012hn, and SN 2021gno, by using the de-reddened
NIR fraction of SN 2016hnk. The range of considered host extinctions and NIR
corrections change $M_{\mathrm{Ni56}}$ by up to a factor of ${\sim}3$ and
$t_{0}$ by up to $25$ per cent from the favoured values (blue-shaded regions
in Figure 1).

We compare in Figure 1 the derived $t_{0}$ and $M_{\mathrm{Ni56}}$ values of
the Ca-rich SNe sample to Ia SNe (blue symbols), Type IIb SNe (red symbols),
and Type Ib/c SNe (yellow symbols) samples from Sharon & Kushnir (2020)333A
few additional SNe have been added to the sample: SN 2011fu (IIb), SN 2013aa
(Ia), SN 2015bp (Ia), SN 2017cbv (Ia), and SN 2021acat (IIb). Details
regarding their analysis will be published in the future.. The Ca-rich sample
is found in a unique region, unoccupied by any other SNe types. This unique
region forms a continuous bridge between Ia SNe and Type IIb SNe. The $t_{0}$
values of the Ca-rich sample partially overlap with those of Ia SNe, with the
largest $t_{0}$ values similar to the lowest Type IIb SNe values. The
$M_{\mathrm{Ni56}}$ values of the Ca-rich sample are, in general, lower than
the other SNe, but the highest $M_{\mathrm{Ni56}}$ values are in agreement
with those of Type IIb SNe and with the lowest Ia SNe $M_{\mathrm{Ni56}}$
values. Our findings raise the possibility that Ca-rich SNe are related to
either Type IIb SNe or Ia SNe, or both. We discuss a possible connection
between these types of SNe in Section 5.

##  4 Comparison with models

In this section, we compare our observations to models from the literature. We
consider both models that were proposed to explain Ca-rich SNe and models of
other SNe explosions, specifically Ia SNe and SESNe models. The
$M_{\mathrm{Ni56}}$ values of the models are provided in the original works,
and we calculate the $t_{0}$ values of the models by way of $\gamma$-ray MC
simulations, using the method in Sharon & Kushnir (2020)444The explosion
ejecta profiles were kindly provided by the authors of the considered
publications.. The results of the Ca-rich models and non-Ca-rich models are
presented in Figure 3 and Figure 4, respectively.

We next describe in more detail each of the models.

###  4.1 Models of Ca-rich SNe

Figure 3: Same as Figure 1 but superimposed on models of Ca-rich SNe. The
models include the helium shell detonations of Waldman et al. (2011) (magenta
star), the collapse of ultra-stripped stars of Moriya et al. (2017) (downward-
facing, turquoise triangle), the helium shell double-detonations of Sim et al.
(2012) (upward-facing, green triangles) and the companion-triggered explosion
of a hybrid HeCO WD Pakmor et al. (2021) (rightward-facing, orange triangle).
For the model of Waldman et al. (2011) and two of the models of Sim et al.
(2012), $M_{\mathrm{Ni56}}$ is replaced with
$M_{\textrm{Cr48}}+M_{\textrm{V48}}$, since the 48Cr decay chain dominates the
energy deposition in these models at $t\lesssim 100$ days after the explosion.

### The helium shell model of Waldman et al. (2011); Dessart & Hillier (2015)

Waldman et al. (2011); Dessart & Hillier (2015) considered a helium shell
detonation at the surface of a low-mass C/O WD (model CO.45HE.2 of Waldman et
al., 2011). This model has an ejecta mass of $0.2\,M_{\odot}$, a kinetic
energy of $0.178\times 10^{51}\,\text{erg}$, and its main power source is the
decay chain 48Cr$\rightarrow^{48}$V$\rightarrow^{48}$Ti (the first step with a
half-life of $<1\,\rm{d}$ is not important in what follows). The products of
48V decay and of 56Co decay have similar $\gamma$-ray energies, so the
$\gamma$-ray escape opacities are also similar. For a given density profile,
we calculate a deviation of ${\approx}5$ per cent in the value of $t_{0}$
between 48V and 56Co. We find for the considered model a small value of
$t_{0}\approx 15\,\text{day}$ (magenta star in Figure 3), as a result of the
low ejecta mass. This $t_{0}$ value is much smaller than the observed $t_{0}$
values of the Ca-rich SNe sample, so it is likely not a viable explanation for
this class of SNe.

We can also rule out as an explanation for Ca-rich SNe all the explosive shell
models that rely on the 48Cr decay chain. Since the half-life of 48V,
${\approx}15.96\,\rm{d}$, is approximately five times shorter than that of
56Co, the bolometric luminosity of such models drops much faster than all
other bolometric light-curves considered in this work, regardless of the
$\gamma$-ray deposition histories. As a result, a fit of the bolometric light
curve to 48Cr decay, if such a fit were possible, would lead to a much larger
$t_{0}$ than the value obtained for 56Ni decay. For example, a full
$\gamma$-ray deposition is required at all times (up to ${\approx}67$ days
after the explosion), for SN 2005E, which results in $t_{0}\gtrsim
150\,\text{day}$. However, the small ejecta mass of explosive shell models
yields a much shorter $\gamma$-ray escape time.

### The double-detonation of the helium shell of low-mass white dwarfs of Sim
et al. (2012)

Sim et al. (2012) simulated the detonations of an accreted helium layer of
${\sim}0.21\,M_{\odot}$ on low-mass carbon-oxygen (CO) WDs. They considered
three cases that follow the ignition of the helium shell: a converging-shock
double-detonation (CSDD), where the helium detonation compresses the WD core
and triggers its detonation near the centre; an edge-lit double-detonation
(ELDD), where the CO core is directly ignited at its surface; and a shell-only
detonation (HeD), where the CO core fails to ignite. For each type, they
considered two initial profiles, ’S’ and ’L’, with total masses of $0.79$ and
$0.66\,M_{\odot}$, respectively. Sim et al. (2012) mention that the helium
shell detonation of the L-model is improbable. The simulations’
nucleosynthesis yield includes radioactive nuclei other than 56Ni , mainly
48Cr and 52Fe, which can overcome the 56Ni as the primary power source. This
is the case for the ELDD-L and the HeD-L models, where the energy deposition
is dominated by 48Cr decay for the relevant times ($t\lesssim 100$ days since
explosion). For each of these models, we calculate the escape time with
respect to the 48Cr distribution, although the results are very similar to the
escape time with respect to the 56Ni distribution.

The $t_{0}$ values of these models are between ${\sim}14$ to
${\sim}28\,\text{day}$ (upward-facing, green triangles in Figure 3). The
highest two values belong to the CSDD models and are close to the lower end of
the Ca-rich SNe sample, while the other models have $t_{0}\lesssim
20\,\text{day}$, which are much smaller than the observations. However, the
56Ni mass of the CSDD-S model is higher by several factors than those in all
the Ca-rich SNe in the sample, and is an order of magnitude greater than
the56Ni mass in SNe with low $t_{0}$. Additionally, the ELDD-L and HeD-L
models are primarily powered by 48Cr decay, which further challenges the
feasibility of these models. Therefore, out of the six models of Sim et al.
(2012), we find that only the CSDD-L model is somewhat near the observations.

### The ultra-stripped star model of Moriya et al. (2017)

We consider the simulated CCSN of an ultra-stripped star with an ejecta mass
of $0.2\,M_{\odot}$ and a kinetic energy of $0.25\times 10^{51}\,\text{erg}$
from (Moriya et al., 2017). We find a small value of $t_{0}\approx
15\,\text{day}$ (downward-facing, turquoise triangle in Figure 3), as a result
of the low ejecta mass. The $t_{0}$ value of this model is much smaller than
the observed $t_{0}$ values of the Ca-rich SNe sample, so it is ruled out as a
viable explanation for this class of SNe.

### The thermonuclear explosion of a massive hybrid HeCO WD triggered by a
companion of Pakmor et al. (2021)

Pakmor et al. (2021) performed a 3D simulation to study the interaction of a
He-rich hybrid $0.69\,M_{\odot}$ HeCO WD with a more massive $0.8\,M_{\odot}$
CO WD just before they merge. The accretion from the hybrid WD to the CO WD
resulted in a helium detonation that caused the full detonation and disruption
of the hybrid WD. A total 56Ni mass of $0.018\,M_{\odot}$ is synthesised in
the explosion, out of which $0.013\,M_{\odot}$ is in the $0.6\,M_{\odot}$
ejecta. The profile we obtained depicts the entire $0.69\,M_{\odot}$ detonated
WD, although some of the ejecta would be caught on the unbound, primary WD. We
do not attempt to account for this effect, so our results may differ by
${\sim}10\%$ from the actual value. Additionally, the 56Ni mass of the profile
is lower than that in Pakmor et al. (2021), ${\approx}0.009\,M_{\odot}$, which
is probably due to remapping between different simulations555R. Pakmor,
private communication.. The $t_{0}$ value of this model is
${\approx}28\,\textrm{day}$ (right-facing, orange triangle in Figure 3), which
is higher than that of most WD shell explosion models, but is still somewhat
lower than the observations. The synthesised 56Ni is in agreement with the
fainter Ca-rich SNe.

###  4.2 Non-Ca-rich models

Figure 4: Same as Figure 1 but superimposed on models of non-Ca-rich SNe. The
models consist of Type Ia SNe models; The sub-Chandra detonation of Kushnir et
al. (2020) with a progenitor mass range of $0.7\text{--}1.1\,M_{\odot}$ (red
line; progenitor masses are indicated on the plot), Chandrasekhar-mass
explosions of Dessart & Hillier (2015) (green line), and WD collisions of
Kushnir et al. (2013) (leftward-facing brown triangles); SESNe: IIb/Ib/Ic
explosions of Dessart et al. (2016) (upward-facing, dark-green triangles) and
Ib/Ic explosions of stripped helium stars of Woosley et al. (2021) (downward-
facing, brown triangles).

### Comparison to Ia SN models

We consider several Type Ia explosion models, shown in Figure 4: the sub-
Chandra detonation of Kushnir et al. (2020) (red line), Chandrasekhar-mass
explosions of Dessart & Hillier (2015) (green line), and WD collisions of
Kushnir et al. (2013) (leftward-facing, brown triangles). The Chandrasekhar-
mass explosions and WD collisions do not agree with the Ca-rich observations,
but the lower end of the sub-Chandra detonations, with WD masses
${\approx}0.7\text{--}0.8\,M_{\odot}$, is consistent with the low-$t_{0}$ Ca-
rich SNe.

### Stripped envelope CCSNe models

Figure 4 also displays stripped-envelope SNe models - IIb/Ib/Ic explosions of
Dessart et al. (2016) (upward-facing, dark-green triangles) and Ib/Ic
explosions of stripped helium stars of Woosley et al. (2021) (downward-facing
brown triangles). Both works span a wide range of ejecta masses and explosion
energies. The models of Dessart et al. (2016) have typically higher $t_{0}$
values than the observations, but there is some overlap of the lower end of
the models, corresponding to ejecta masses of
${\approx}1.7\text{--}3.2\,M_{\odot}$, with the higher end of the
observations. The models of Woosley et al. (2021) cover a large region of
$\gamma$-ray escape times, including the entire range of the observed Ca-rich
$t_{0}$ values. The profiles that are within the range of the Ca-rich SNe
region have helium star masses (ejecta masses) of
${\approx}2.5\text{--}5.5\,M_{\odot}$
(${\approx}0.75\text{--}2.6\,M_{\odot}$).

To summarise, we find that the $t_{0}$ values of most Ca-rich models are too
low compared to, and in tension with, the observations, a result of their
small ejecta masses. We also find that some non-Ca-rich SNe models agree with
the observations. Of the models we have considered, we find that low-
luminosity thermonuclear WD sub-Chandra detonations (Kushnir et al., 2020),
with progenitor masses ${\lesssim}0.8\,M_{\odot}$, and stripped envelope CCSNe
(Dessart & Hillier, 2015; Woosley et al., 2021) with ejecta masses of
${\approx}0.5\text{--}3\,M_{\odot}$, have $t_{0}$ and $M_{\mathrm{Ni56}}$
values that are compatible with the observations.

##  5 Discussion

In this work, we calculated the $\gamma$-ray deposition histories of Ca-rich
SNe. The $\gamma$-ray escape time, $t_{0}$, and the 56Ni mass were previously
shown to have different characteristic values for different SN Types (Sharon &
Kushnir, 2020). The results of this work, shown in Figure 1, place Ca-rich SNe
in a region unoccupied by other SN types in terms of the
$t_{0}$–$M_{\mathrm{Ni56}}$ distribution. Their $t_{0}$ values fall between
those of Ia SNe and SESNe, bridging the gap between the two SN types, though
the $M_{\mathrm{Ni56}}$ values of most of the SNe in the sample are lower than
those of Type Ia SNe and SESNe.

One of the models of Ca-rich SNe is the explosive burning of He shells on WDs,
which was proposed by De et al. (2020) to be the strongest candidate for these
events due to the early type hosts and high volumetric rates of Ca-rich SNe.
By analysing their spectroscopic and photometric properties, they further
claimed that Ca-Ia and red Ca-Ib/c events are consistent with a double-
detonation of a He shell that ignites the entire star, while green Ca-Ib/c are
the results of a shell-only detonation. The red Ca-Ib/c progenitors have a
lower total mass and a thicker shell compared to those of Ca-Ia events, and
their explosion would result in Fe group elements in the shell and
intermediate mass elements in the core. The gamma-ray deposition histories
impose several constraints on some of these models. The $t_{0}$ values of the
shell-only models of Waldman et al. (2011) and Sim et al. (2012) are
inconsistent with observations. In addition, most of them are powered by the
48Cr decay chain, where the half-life of its products is much shorter than the
56Ni decay chain. He shell detonations were also studied by Sim et al. (2012),
and we find that only a small part of the models is somewhat close to the
observations. We conclude that He shell models are unlikely to explain Ca-rich
SNe, since the gamma-ray escape time for these models is smaller than the
observed values. Note that unlike optical light-curve comparisons,
$\gamma$-ray deposition comparisons rely on known, simple physics.

Explosions of low-mass ($M\approx 0.7\text{--}0.8\,M_{\odot}$) WDs are in
agreement with the low $t_{0}$ part of the Ca-rich SNe (see Figure 1). Yet the
higher $t_{0}$ events, i.e., SN 2016hnk and SN 2019ehk, which also have larger
$t_{0}$ values, do not agree with these models. Some of the properties of SN
2019ehk and SN 2021gno, which are unrelated to our analysis, support a massive
star origin for these explosions: SN 2019ehk exploded close to the core of a
star-forming galaxy, and both are spectroscopically similar to type Ib and
have a double-peaked light curve. In addition, some of the SESN models we have
tested are in agreement with the higher $t_{0}$ events.

Despite the possibility of multiple progenitors, there seems to be a
continuity in the $t_{0}$–$M_{\mathrm{Ni56}}$ distribution, and the location
of the SNe within the distribution is correlated with the continuum of
spectroscopic properties, as derived by De et al. (2020). Three of the SNe in
our sample - SN 2005E, SN 2012hn, and SN 2016hnk - were analysed in De et al.
(2020). SN 2005E, showing weak Si II lines, strong He I lines and no line
blanketing, is located at one end of the spectroscopic analysis sequence (Ca-
Ib/c green objects), and is also at the edge of the
$t_{0}$–$M_{\mathrm{Ni56}}$ distribution, having the lowest $t_{0}$ value. SN
2016hnk, showing strong Si II lines, no He I lines and a line-blanketed
continuum, is located at the other end of the sequence of De et al. (2020)
(Ca-Ia objects), and is at the other end of the $t_{0}$–$M_{\mathrm{Ni56}}$
distribution, having the highest $t_{0}$ and $M_{\mathrm{Ni56}}$ values. SN
2012hn is located in the middle of the sequence (Ca-Ib/c red). The peak
spectrum of SN 2019ehk, being similar to that of iPTF12bho (Jacobson-Galán et
al., 2020b), places the SN in the Ca-Ib/c red region. This is also the case
for SN 2021gno, as its spectrum is most similar to that of 2019ehk (Jacobson-
Galán et al., 2022). All three SNe (i.e., SN 2012hn, SN 2019ehk, and
SN2021gno) lie in the middle of the $t_{0}$–$M_{\mathrm{Ni56}}$ distribution
of our analysis. To summarise, the $t_{0}$ values of our analysis and the
position in the classification of De et al. (2020) are highly correlated.
However, the SN classified as a Ca-Ia object, SN 2016hnk, is closer to the
SESNe distribution than the rest of our sample, while the Ca-Ib/c-classified
SNe are farther away from it.

The continuity in the Ca-rich parameters, which seem to connect the Type Ia
and IIb SNe, raises the possibility that the explosion mechanism of these
events is similar. Despite the differences between the progenitors of the two
types (WD explosions for Type Ia SN vs. the collapse of stripped, massive
stars for Type IIb SNe), the energy source of the SN of both types could be
the same. While it is well established that the energy source of Type Ia is
thermonuclear, the energy source of CCSN is under debate, with the main
candidates being gravitational (Woosley & Janka, 2005; Janka, 2012) or
thermonuclear (Burbidge et al., 1957; Kushnir, 2015). The bridge formed by the
Ca-rich SNe that links between Type Ia and IIb SN hints that they might be the
result of a similar, though not identical, process.

The newly discovered SN 2022oqm (Irani et al., 2022) poses additional
challenges for the origin of Ca-rich SNe. Classified as a Ic SNe, its spectrum
at ${\sim}60$ days since explosion displays strong C II [and Ca II] emissions
with no detectable [O I], marking it as a Ca-rich event. However, Irani et al.
(2022) found $t_{0}$ and $M_{\mathrm{Ni56}}$ values of ${\approx}36$ day and
$0.12\,M_{\odot}$, respectively, placing it within the Type Ia region of the
$t_{0}$–$M_{\mathrm{Ni56}}$ distribution, despite its spectral classification.
A massive star origin raises additional difficulties, such as its explosion
site properties, though a WD origin faces some challenges as well, and none of
the scenarios can be ruled out at present (Irani et al., 2022).

We have also tried to calculate the luminosity function (LF) of Ca-rich SNe
using the results from De et al. (2020), and compared it to the Ia SN LF in
Sharon & Kushnir (2021). The LFs do not overlap, since the Ia LF has a peak
$r$ magnitude $M_{r}<-17.5$, brighter than all of the Ca-rich sample.
Comparing the rates of the low end Ia LF with the rates of the high end of the
Ca-rich LF might strengthen or rule out the connection between the events.
However, we did not find strong evidence in favour of either side, as the
rates at both ends have large uncertainties due to the low number of events.
Additional events would further constrain the rates at these luminosities.

Uncovering the origins of Ca-rich SNe is a difficult task, as these objects
are faint, have lower rates compared to Type Ia and CCSNe, and might be
composed of several progenitor systems. The $\gamma$-ray deposition histories
analysis might help in this task, and this work has shown that some of the
models for the Ca-rich progenitors can indeed be ruled out. However, our
analysis is limited due to its small sample size, and might be biased towards
luminous events. Additional objects might help reveal the entire range of
$t_{0}$ and $M_{\mathrm{Ni56}}$ of these SNe, allowing to place further
constraints on their origin and explosion mechanism.

## Acknowledgements

We thank Boaz Katz for useful discussions. DK is supported by a research grant
from The Abramson Family Center for Young Scientists, an ISF grant, and by
Minerva Stiftung. This work made use of the Heidelberg Supernova Model Archive
(HESMA)666https://hesma.h-its.org. We thank Luc Dessart, Takashi Moriya,
Stuart Sim and Stan Woosley for sharing their ejecta profiles with us.

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Figure 5: The best-fit results of the Katz integral method for SN 2005E. The
median values and 68% confidence level of the parameters are presented in
Table 1. Left panels: Comparison of the observed bolometric light curves to
the best-fit model (with the parameters given in the boxes, solid line) and to
the radioactive energy generation rates (same as assuming $f_{\text{dep}}=1$,
dashed line). The distance and extinction estimates are given in the boxes as
well. The errors represent the total errors (statistical and systematic).
Right panels: Comparison of the deposition function, $f_{\text{dep}}$, that
corresponds to the best-fit model (solid line) to the ratio
$(L-Q_{\text{pos}})/Q_{\gamma}$. This ratio corresponds to
$L_{\gamma}/Q_{\gamma}$ for $t\in t_{L=Q}$, where we use the observed $L$ and
the derived $Q_{\text{pos}},Q_{\gamma}$. The total errors (statistical and
systematic) are indicated by grey bars, while the (photometric) statistical
errors are indicated by black bars. In both panels, the epochs of
$t_{\text{min}}$ and $t_{\text{max}}$ (if different from the last phase) are
indicated by vertical dashed-dotted lines. Figure 6: Same as Figure 5 for SN
2012hn. Figure 7: Same as Figure 5 for SN 2016hnk. Figure 8: Same as Figure 5
for SN 2019ehk. Figure 9: Same as Figure 5 for SN 2021gno.