# SDSSJ0018-0939: A Clear Signature of Sub-Chandrasekhar Mass Type 1a
Supernova

S. K. Jeena  Department of Physics
Indian Institute of Technology Palakkad, Kerala, India  Projjwal Banerjee
Department of Physics
Indian Institute of Technology Palakkad, Kerala, India

###### Abstract

Very metal-poor (VMP) stars (${\rm[Fe/H]}\leq-2$) that have sub-solar values
of ${\rm[X/Fe]}$ for $\alpha$ elements such as Mg, Si, and Ca, are referred to
as $\alpha$-poor VMP stars. They are quite rare among VMP stars and are
thought to have formed from gas enriched predominantly by a single Type Ia
supernovae (SN1a) in contrast to most VMP stars which are $\alpha$-enhanced
and usually associated with core-collapse supernovae. The observed abundance
pattern in such stars can provide a direct way to probe the nucleosynthesis in
individual SN1a. Although the abundance patterns in some $\alpha$-poor VMP
stars have been shown to be consistent with SN1a ejecta, a clear
nucleosynthetic signature for SN1a resulting from the explosion of a near
Chandrasekhar mass (near-${\rm M_{Ch}}$) or a sub-Chandrasekhar mass
(sub-${\rm M_{Ch}}$) white dwarf, has not been unambiguously detected. We
perform a detailed analysis of various formation channels of VMP stars and
find that the $\alpha$-poor VMP star SDSSJ0018-0939, which was earlier
reported as a star with potential pair-instability supernova origin, provides
almost a smoking-gun signature of a sub-${\rm M_{Ch}}$ SN1a resulting from He
detonation. We find that compared to other $\alpha$-poor VMP stars that were
previously identified with SN1a, SDSSJ0018-0939 is the only star that has a
clear and unambiguous signature of SN1a. Interestingly, our results are
consistent with constraints on SN1a from recent galactic chemical evolution
studies that indicate that sub-${\rm M_{Ch}}$ SN1a account for $\sim
50\hbox{--}75\,\%$ of all SN1a and are possibly the dominant channel in the
early Galaxy.

Chemically peculiar stars(226) — Stellar nucleosynthesis(1616) — Type Ia
supernovae(1728) — Population II stars(1284) — Population III stars(1285)

##  1 Introduction

Studying very metal-poor (VMP) stars with metallicity
$[\text{Fe}/\text{H}]<-2$ is crucial for understanding the evolution of
elements in the early Galaxy during the first billion years after the Big Bang
(Beers & Christlieb, 2005; Frebel & Norris, 2015). The majority of VMP stars
exhibit an enhanced abundance of $\alpha$ elements such as C, O, Mg, Si, and
Ca with super-solar values of $[\text{X}/\text{Fe}]$ (Cayrel et al., 2004).
These stars are thought to have formed from gas polluted by core-collapse
supernovae (CCSNe), and their detailed abundance pattern can be explained by
the mixing and fallback of CCSN ejecta (Umeda & Nomoto, 2003, 2005; Heger &
Woosley, 2010; Tominaga et al., 2014). On the other hand, VMP stars that have
$[\text{X}/\text{Fe}]<0$ for Mg and most $\alpha$ elements, are known as
$\alpha$-poor VMP stars and are quite rare among VMP stars. Their peculiar
abundance pattern is typically associated with gas polluted by SN1a (Ivans et
al., 2003; Li et al., 2022) as nucleosynthesis in SN1a models naturally
produces negligible Mg along with sub-solar values of $[\text{X}/\text{Fe}]$
for $\alpha$ elements.

In this regard, it is important to note that the nature of progenitors as well
as the mechanism of explosion for SN1a are subjects of ongoing debate, as
successful explosions from first principles are still lacking in numerical
simulations. Broadly, the scenarios proposed are associated with the explosion
of a white dwarf (WD) in a binary configuration with a red giant star (single
degenerate) or with another WD (double degenerate). The proposed explosion
mechanism includes pure detonation, pure deflagration, and delayed detonation
(deflagration to detonation) for near-Chandrasekhar mass
(near-$\mathrm{M_{Ch}}$) models along with double detonation for sub-
Chandrasekhar mass (sub-$\mathrm{M_{Ch}}$) models. The explosion mechanisms
and nucleosynthesis resulting from the various scenarios have been extensively
studied over the last several decades (Nomoto et al., 1984; Woosley et al.,
1986; Khokhlov, 1991; Iwamoto et al., 1999; Woosley & Kasen, 2011; Seitenzahl
et al., 2013b; Fink et al., 2014; Leung & Nomoto, 2018; Shen et al., 2018;
Bravo, 2019; Bravo et al., 2019; Leung & Nomoto, 2020; Lach et al., 2020;
Gronow et al., 2021). Among the various explosion mechanisms and scenarios
that have been proposed, delayed detonation for near-$\mathrm{M_{Ch}}$ and
double detonation for sub-$\mathrm{M_{Ch}}$ SN1a is currently considered to be
the dominant channels (Shen et al., 2018; Kirby et al., 2019). The relative
frequency of near-$\mathrm{M_{Ch}}$ and sub-$\mathrm{M_{Ch}}$ models and their
contribution to the galactic chemical evolution is one of the crucial
questions that several recent studies have tried to constrain. By considering
the chemical evolution of key elements such as Mn and Ni, several studies have
found that a significant fraction of the contribution comes from
sub-$\mathrm{M_{Ch}}$ SN1a resulting from He detonations accounting for
$50\hbox{--}70\,\%$ of all SN1a (Seitenzahl et al., 2013a; Kirby et al., 2019;
de los Reyes et al., 2020; Kobayashi et al., 2020; Eitner et al., 2020, 2023).
Furthermore, some of these also suggest that sub-$\mathrm{M_{Ch}}$ SN1a could
even be more dominant in the early Galactic evolution with
near-$\mathrm{M_{Ch}}$ SN1a dominating in the later phases (Kirby et al.,
2019).

An important implication of such findings is that many of the $\alpha$-poor
VMP stars should carry a clear signature of individual sub-$\mathrm{M_{Ch}}$
SN1a. Detailed abundance in $\alpha$-poor VMP stars can thus be used to
confirm this and also provide direct evidence of the existence of
sub-$\mathrm{M_{Ch}}$ SN1a. Even though abundance patterns in some
$\alpha$-poor VMP stars have been analysed in detail and compared to
theoretical SN1a yields, a clear unambiguous match to the observed pattern is
still lacking. The situation is made complicated by the fact that a recent
study by Jeena et al. (2024) has pointed out that CCSNe, that do not undergo
substantial fallback, can also result in patterns with sub-solar values of
$[\text{Mg}/\text{Fe}]$ that can potentially match the pattern observed in
some of the $\alpha$-poor VMP stars. This makes the unambiguous association of
an $\alpha$-poor VMP star to an SN1a even more challenging.

We identify a star named SDSSJ0018-0939, which exhibits the most prominent
signature of being born from gas polluted by sub-$\mathrm{M_{Ch}}$ SN1a
resulting from He detonation. SDSSJ0018-0939 is a well-known chemically
peculiar VMP star with metallicity $[\text{Fe}/\text{H}]=-2.5$, which has an
elemental abundance pattern that is distinct from other VMP stars and was
first reported by Aoki et al. (2014). The peculiar abundance pattern was
interpreted by Aoki et al. (2014) as a possible signature of a star formed
from a gas polluted by ejecta from a pair-instability SN (PISN). However, a
detailed analysis of PISN yields by Takahashi et al. (2018) found that PISN
can be ruled out as a source of elements in SDSSJ0018-0939. Our analysis also
confirms this where we find that PISN provides a very poor fit to the observed
abundance pattern and can be ruled out as the source of elements in
SDSSJ0018-0939. Interestingly, however, we find that SDSSJ0018-0939 stands out
as a unique star that shows a near smoking-gun signature of a
sub-$\mathrm{M_{Ch}}$ SN1a.

Our conclusion is based on a novel and detailed analysis that explores the
various scenarios of the origin of elements in VMP stars. This includes
enrichment from a single PISN, a single CCSN, a combination of ejecta from a
single CCSN and a single PISN, a combination of ejecta from two CCSNe, as well
as combinations of ejecta from a CCSN and ejecta from both
near-$\mathrm{M_{Ch}}$ and sub-$\mathrm{M_{Ch}}$ SN1a. We consider a wide
range of CCSN models that undergo mixing and fallback along with SN1a from
delayed detonation near-$\mathrm{M_{Ch}}$ models and double detonation
sub-$\mathrm{M_{Ch}}$ models and use a $\chi^{2}$ minimization to find the
best-fit models. We find that other $\alpha$-poor VMP stars such as COS171,
BD+80245, HE0533-5340, and SMSSJ034249-284216, which have recently been
identified with sub-$\mathrm{M_{Ch}}$ SN1a (McWilliam et al., 2018; Reggiani
et al., 2023), can also be fit well with purely CCSN models and thus cannot be
unambiguously associated with SN1a. In sharp contrast, SDSSJ0018-0939 has the
clearest signature of a star that is formed from gas polluted dominantly by a
sub-$\mathrm{M_{Ch}}$ SN1a resulting from a He detonation. We identify key
abundance features that make SDSSJ0018-0939 stand out among other
$\alpha$-poor VMP stars.

The structure of this paper is as follows: in Sec. 2, we discuss the
theoretical models used in this study; Sec. 3 we discuss the details of the
different scenarios for explaining abundance pattern in VMP stars and the
methods used to find best-fit models. The results of the analysis for
SDSSJ0018-0939 are discussed in Sec. 4 and the corresponding results for four
other $\alpha$-poor VMP stars in Sec. 5. Lastly, we finish with the summary
and conclusion in Sec. 6.

##  2 Theoretical Models and Abundance Patterns

Below, we describe the various theoretical models adopted in this study and
highlight the key features of the corresponding elemental abundance patterns.

###  2.1 PISN models

We adopt the PISN models from Heger & Woosley (2002) that include the fourteen
PISN yields from Pop III stars with He core masses ranging from
$65\hbox{--}130\,\mathrm{M}_{\odot}$ corresponding to zero-age main sequence
mass of $\sim 140\hbox{--}260\,\mathrm{M}_{\odot}$. Figure 1a shows the
corresponding abundance patterns for some of the PISN models covering the
progenitor mass range. As can be seen from the figure, the abundance pattern
varies substantially with the He core mass. Lower He core PISN models of
$\lesssim 90\,\mathrm{M}_{\odot}$ have a large overproduction of light and
intermediate elements relative to Fe with a less pronounced odd-even effect.
With increasing progenitor mass, the amount of Fe peak production increases
dramatically due to explosive Si burning that leads to lower
$[\text{X}/\text{Fe}]$ for elements lighter than Fe along with increased odd-
even effect with highly sub-solar values for odd elements.

Figure 1: Theoretical elemental pattern relative to Fe from various sources.
(a) PISN models of varying He core masses. (b) CCSN ejecta for z models of
$10.1$–$30\,\mathrm{M}_{\odot}$ without mixing and fallback, and have values
of $[\text{Mg}/\text{Fe}]\lesssim 0.0$. (c) SN1a models, where the black solid
line corresponds to a typical near-$\mathrm{M_{Ch}}$ model, and the dotted and
dash-dotted lines represent sub-$\mathrm{M_{Ch}}$ models with varied core and
shell masses.

###  2.2 CCSN Models

We use a large adaptive co-processing network with reaction rates based on
Rauscher et al. (2002) and follow the evolution of the star from birth to
death via CCSN using the 1D hydrodynamic stellar evolution code Kepler (Weaver
et al., 1978; Rauscher et al., 2003). We consider non-rotating models of mass
ranging from $10.1$–$30\,\mathrm{M}_{\odot}$, as detailed in Jeena et al.
(2024). The initial composition of these models is taken from Cyburt et al.
(2002) that corresponds to primordial Big Bang nucleosynthesis. We refer to
these as the z models.

The explosion is simulated by driving a piston from a mass coordinate, $M_{\rm
cut,ini}$, below which all the matter is assumed to fall back to the proto-
neutron star. We consider two different choices of $M_{\rm cut,ini}$ in this
study, which is identical to Heger & Woosley (2010). In the first case,
$M_{\rm cut,ini}$ is chosen to be the mass coordinate where the entropy per
baryon $S$ exceeds $4\,k_{\rm B}$, which typically coincides with the base of
the oxygen shell. In the second case, $M_{\rm cut,ini}$ is chosen to be the
mass coordinate at the edge of the iron core where there is a $Y_{\rm e}$
jump. We refer to the former as $S_{4}$ models and the latter as $Y_{\rm e}$
models. We use a nomenclature where we use the mass followed by the nature of
the model ($S_{4}$ or $Y_{\rm e}$) to clearly identify the various CCSN
models. For example, a $15\,\mathrm{M}_{\odot}$ progenitor from $S_{4}$ model
is referred to as z15-$S_{4}$. We consider explosion energies of $1.2\times
10^{51}\,\mathrm{erg}$ and $1.2\times 10^{52}\,\mathrm{erg}$ for all mass
models. Lower mass progenitors of $\sim 10$–$12\,\mathrm{M}_{\odot}$ are found
to have explosion energies of $\lesssim 10^{51}\,\mathrm{erg}$ from detailed
3D simulations (Müller et al., 2019). To explore this, we also consider
explosion energies of $0.3\times 10^{51}\,\mathrm{erg}$ and $0.6\times
10^{51}\,\mathrm{erg}$ for all progenitors of initial mass
$<12\,\mathrm{M}_{\odot}$. The ejecta for each SN explosion is calculated
using the mixing and fallback model identical to Jeena et al. (2023) and Jeena
et al. (2024) which are based on the models described in Tominaga et al.
(2007); Ishigaki et al. (2014). In this model, following the explosive
nucleosynthesis by the SN shock, all material above a mass coordinate $M_{\rm
cut,fin}$ is fully ejected, whereas a fraction $f_{\rm cut,}$ of the material
between $M_{\rm cut,ini}$ and $M_{\rm cut,fin}$ is ejected. $M_{\rm cut,fin}$
and $f_{\rm cut}$, are treated as free parameters, whereas the value of
$M_{\rm cut,ini}$ is fixed as mentioned above. The value of $M_{\rm cut,fin}$
is varied in steps of $0.1\,\mathrm{M}_{\odot}$, starting from a minimum value
of $M_{\rm cut,ini}$ to a maximum value that corresponds to the base of the H
envelope. The value of $f_{\rm cut}$ varied between 0 to 1. Thus, the amount
of any isotope ejected by the SN for each model depends on the values of
$M_{\rm cut,fin}$ and $f_{\rm cut}$. In this case, $\Delta M_{\rm
fb}=(1-f_{\rm cut})\Delta M_{\rm cut}$ is the amount of matter that falls back
to the compact remnant, where $\Delta M_{\rm cut}=M_{\rm cut,fin}-M_{\rm
cut,ini}$.

The nucleosynthesis of elements in massive stars involves a complex sequence
of nuclear reactions during various stages of stellar evolution along with
explosive nucleosynthesis during SN explosion as discussed in detail in
Woosley et al. (2002); Heger & Woosley (2010) and more recently in Jeena et
al. (2024). The final abundance pattern depends on the explosion energy and
mass of the progenitor along with the details of mixing and fallback. However,
as shown in Jeena et al. (2024), sub-solar values of $[\text{X}/\text{Fe}]$
for $\alpha$ elements result from models with negligible fallback that is
particularly relevant for $\alpha$-poor VMP stars. We thus show a few
representative CCSN models without fallback that have
$[\text{Mg}/\text{Fe}]\lesssim 0.0$ in Fig. 1b.

###  2.3 SN1a Models

In this study, we consider the delayed detonation model for
near-$\mathrm{M_{Ch}}$ and the double detonation model for
sub-$\mathrm{M_{Ch}}$ for SN1a as these are currently considered to be the
dominant channels as mentioned in the introduction. In both cases, we adopt
the nucleosynthetic yields obtained from high-resolution 3D studies for the
explosion of WDs of low metallicity. In near-$\mathrm{M_{Ch}}$ models, we
adopt the nucleosynthetic yields derived from the delayed detonation model
N100_Z0.01 by Seitenzahl et al. (2013b), corresponding to a WD with a central
density of $2.9\times 10^{9}\,\mathrm{g}\,\mathrm{cm}^{-3}$ and an initial
metallicity of $0.01\,{\rm Z_{\odot}}$. For sub-$\mathrm{M_{Ch}}$ models, we
adopt the yields from eleven double detonation models by Gronow et al. (2021)
that have an initial metallicity of $0.001\,{\rm Z_{\odot}}$ with CO core
masses ranging from $0.8$–$1.1\,\mathrm{M}_{\odot}$ that have central
densities of $\lesssim 2.5\times 10^{9}\,\mathrm{g}\,\mathrm{cm}^{-3}$ with He
shell masses of $0.02$–$0.1\,\mathrm{M}_{\odot}$. The sub-$\mathrm{M_{Ch}}$
models are labelled with the CO core mass ($M_{\rm CO}$) and He shell mass
($M_{\rm He}$), e.g., M09_05 corresponds to a WD with $M_{\rm
CO}=0.9\,\mathrm{M}_{\odot}$ and $M_{\rm He}=0.05\,\mathrm{M}_{\odot}$.

Figure 1c shows the abundance pattern of the adopted SN1a models. In both
near-$\mathrm{M_{Ch}}$ and sub-$\mathrm{M_{Ch}}$ models, there is negligible
production of elements lighter than Si. However, the abundance pattern from Si
to Fe peak varies significantly among the various sub-$\mathrm{M_{Ch}}$ models
that are also distinct from the near-$\mathrm{M_{Ch}}$ model. In the
near-$\mathrm{M_{Ch}}$ model, the abundance of $\alpha$ elements is sub-solar
($[\alpha/{\rm Fe}]<0$) for elements up to Ti (see Fig. 1c). The
sub-$\mathrm{M_{Ch}}$ yield patterns, on the other hand, depend on both
$M_{\rm CO}$ and $M_{\rm He}$. The lowest mass $M_{\rm CO}\sim
0.8\,\mathrm{M}_{\odot}$ can even produce $\alpha$-enhanced ($[\alpha/{\rm
Fe}>0$) abundance pattern from Si–Cr resulting from incomplete Si burning of
both the CO core and He shell. With higher $M_{\rm CO}\gtrsim
1.0\,\mathrm{M}_{\odot}$ and $M_{\rm He}\sim 0.1\,\mathrm{M}_{\odot}$, the
$\alpha$-poor feature, that is usually associated with SN1a, is seen for
elements from Si to Ca although Ti to Cr continues to be enhanced with super-
solar values of $[\text{X}/\text{Fe}]$ in some of the models. The enhanced
Ti–Cr in such models is exclusively due to the contribution from He shell
burning that invariably undergoes incomplete Si burning. The contribution of
He shell burning to the final ejecta is particularly high for $M_{\rm
CO}\lesssim 1\,\mathrm{M}_{\odot}$ with intermediate values of $M_{\rm He}\sim
0.05\,\mathrm{M}_{\odot}$. Thus, the combination of sub-solar values of
$[\text{X}/\text{Fe}]$ from Si to Ca along with super-solar values from Ti-Cr
is a unique feature found only in some of the sub-$\mathrm{M_{Ch}}$models that
is otherwise not found in either CCSN or near-$\mathrm{M_{Ch}}$models.
Additionally, solar $[\text{Mn}/\text{Fe}]\sim 0$, usually associated with
near-$\mathrm{M_{Ch}}$ models, is also seen in the lower CO core mass models
whereas, for CO core masses of $\gtrsim 1\,\mathrm{M}_{\odot}$,
$[\text{Mn}/\text{Fe}]$ is always sub-solar. Lastly, $[\text{Ni}/\text{Fe}]$
is sub-solar in all sub-$\mathrm{M_{Ch}}$ models which is distinct from
near-$\mathrm{M_{Ch}}$ model that has $[\text{Ni}/\text{Fe}]\sim 0.1$.

In comparison to the CCSN models, the major difference is that SN1a models
have negligible production of elements lighter than Si and mostly have sub-
solar values of $[\text{X}/\text{Fe}]$ for $\alpha$ elements from Si to Ca. We
note that CCSN models that do not undergo fallback can also have similar
features as discussed in Sec. 2.2. However, the unique feature seen in some of
the sub-$\mathrm{M_{Ch}}$ SN1a models from Ti to Cr mentioned above is not
found in any of the CCSN models. Even after accounting for mixing and
fallback, all CCSN models have sub-solar values of $[\text{X}/\text{Fe}]$ for
Ti to Cr. This is because elements from Ti to Fe are mostly produced in the
same region within the CCSN ejecta and are thus insensitive to the details of
mixing and fallback. Similarly, $[\text{Mn}/\text{Fe}]\sim 0$ is not found in
any of the CCSN models but is a key signature of the near-$\mathrm{M_{Ch}}$
model. We note that $[\text{Mn}/\text{Fe}]\gtrsim 0$ can also be found in some
of the lighter CO core mass sub-$\mathrm{M_{Ch}}$ models but the combination
of sub-solar values of $[\text{X}/\text{Fe}]$ for Mg to Ca along with
$[\text{Mn}/\text{Fe}]\sim 0$ is only found in the near-$\mathrm{M_{Ch}}$
model.

##  3 Matching the Abundance pattern of SDSSJ0018-0939

In order to match the observed abundance pattern of SDSSJ0018-0939, we
consider six distinct scenarios for the origin of elements in VMP stars formed
in the early Galaxy. In the first case, we assume all elements are exclusively
produced by a single PISN. In the second case, we assume all elements are
exclusively produced by a single CCSN undergoing mixing and fallback. Next, we
consider the mixing of the ejecta from a single CCSN with that of either a
near-$\mathrm{M_{Ch}}$ or a sub-$\mathrm{M_{Ch}}$ SN1a. We then consider the
mixing of ejecta of two CCSNe. Finally, we consider the mixing of the ejecta
from a single PISN and a single CCSN.

###  3.1 Best-fit models from a single PISN

In order to find the best-fit model, we follow the same procedure as detailed
in Jeena et al. (2023). For each element ${\rm X}_{i}$, we define the number
yield $Y_{{\rm X}_{i}}=\Sigma_{j}m({\rm X}_{i}^{j})/A_{j}$, where $m({\rm
X}^{j}_{i})$ is the yield of the $j^{\rm th}$ isotope and $A_{j}$ is the
corresponding mass number. The ratio of the number abundance $N$ of any
element $X_{i}$ relative to a reference element $X_{\rm R}$ can be written as

$\frac{N_{{\rm X}_{i}}}{N_{\rm X_{R}}}=\frac{Y_{{\rm X}_{i}}}{Y_{\rm X_{R}}}.$
(1)

Following this, the best-fit model parameters are determined using a
$\chi^{2}$ prescription as discussed in Heger & Woosley (2010) and more
recently in Jeena et al. (2023, 2024). In this prescription, $\chi^{2}$ is
defined as the square of the deviation between the observed and predicted
values summed over all elements. The square of the inverse of the observed
uncertainty ($\sigma_{i}$) is used as the weights for each element where we
set $\sigma_{i}=\max(\sigma_{i},0.1)$ to ensure that $\chi^{2}$ is not
excessively sensitive to elements with extremely low values of $\sigma_{i}$.
Because PISNe have large explosion energies of $\gtrsim
10^{52}\,\mathrm{erg}$, we impose a minimum dilution of
$M{\rm{}_{dil,PISN}^{min}}=10^{5}\,\mathrm{M}_{\odot}$ for all models to be
consistent with the level of dilution expected from such high energy
explosions (Chiaki et al., 2018).

###  3.2 Best-fit models from a single CCSN

The procedure for finding the best-fit model for a single CCSN is similar to
the one for PISN described above. In this case, however, both $Y_{{\rm
X}_{i}}$ and $Y_{{\rm X_{R}}}$ depend on the two free parameters $M_{\rm
cut,fin}$ and $f_{\rm cut}$. For every CCSN progenitor, we consider all
possible combinations of the two parameters to generate thousands of models.
The process is then repeated for each CCSN progenitor. The best-fit model is
again determined by the $\chi^{2}$ formulation mentioned above. In order to be
consistent with the dilution found in simulations of metal mixing from CCSN,
we only consider solutions for which the effective dilution mass from CCSN is
greater than $M{\rm{}_{dil,CCSN}^{min}}=10^{4}\,\mathrm{M}_{\odot}$ for
explosion energies of $\leq 1.2\times 10^{51}\,\mathrm{erg}$ and
$M{\rm{}_{dil,CCSN}^{min}}=10^{5}\,\mathrm{M}_{\odot}$ for $1.2\times
10^{52}\,\mathrm{erg}$. These values correspond roughly to the minimum value
of the effective dilution mass for the CCSN ejecta from the detailed
simulation of inhomogeneous metal mixing in the early Galaxy (Chiaki et al.,
2018; Magg et al., 2020).

We note here that detailed 3D simulations have shown that Sc is dominantly
produced by neutrino-processed proton-rich ejecta that are not modelled in our
1D calculations. Sc yield in 3D models, that include detailed neutrino
interaction, can be a factor of $\gtrsim 10$ higher than the corresponding 1D
model (Sieverding et al., 2020; Wang & Burrows, 2024) at solar metallicity.
This implies that almost all of the Sc at zero metallicity will be from the
neutrino-processed material. Because of this reason, in our analysis, we treat
the measured abundance of Sc (if available) as an upper limit. The same is
true for K which has negligible production in the ejecta of Pop III stars but
will be produced entirely in the neutrino-processed material that is not
included in our models. We also treat K as an upper limit whenever they are
detected.

###  3.3 Best-fit models from the mixing of CCSN and SN1a ejecta

In order to determine the best-fit model based on the mixing of ejecta from
two distinct sources, we consider the prescription used in the recent work
Jeena et al. (2023). In this prescription, the mixing of the ejecta of the two
sources, i.e., CCSN and SN1a, is parameterized by a single parameter $\alpha$
given by

$\alpha=\frac{M{\rm{}_{dil,CCSN}}}{M{\rm{}_{dil,CCSN}}+M{\rm{}_{dil,1a}}},$
(2)

where $M{\rm{}_{dil,CCSN}}$ and $M{\rm{}_{dil,1a}}$ are the effective dilution
masses from CCSN and SN1a, respectively. Following the mixing procedure, the
number abundance of any element ${\rm X}_{i}$ relative to a reference element
${\rm X_{R}}$ can be written as

$\frac{N_{{\rm X}_{i}}}{N_{\rm X_{R}}}=\frac{Y_{{\rm X}_{i}}}{Y_{\rm
X_{R}}}=\frac{{\alpha Y_{{{\rm X}_{i}},{\rm
1a}}+\left(1-\alpha\right)\,Y_{{{\rm X}_{i}},{\rm CCSN}}(M_{\rm
cut,fin},f_{\rm cut})}}{{\alpha
Y_{\rm{X_{R},1a}}+\left(1-\alpha\right)\,Y_{\rm X_{R},CCSN}(M_{\rm
cut,fin},f_{\rm cut})}},$ (3)

where the subscript ‘1a’ and ‘CCSN’ are used for the number yields
corresponding to SN1a and CCSN, respectively. In this case, the final
abundance pattern depends on the parameter $\alpha$ in addition to $M_{\rm
cut,fin}$ and $f_{\rm cut}$. The three parameters are then varied to find the
best-fit model using the same $\chi^{2}$ prescription used for PISN and CCSN.
For any given fit, the relative contribution from each source to an element
${\rm X}_{i}$, can be quantified by computing the fraction $\eta({\rm X}_{i})$
of the total elemental yield $Y_{{\rm X}_{i}}$. We compute $\eta({\rm X}_{i})$
for SN1a and CCSN, i.e., $\eta_{\rm 1a}({\rm X}_{i})$ and $\eta_{\rm
CCSN}({\rm X}_{i})$, where $\eta_{\rm 1a}({\rm X}_{i})\,+\,\eta_{\rm
CCSN}({\rm X}_{i})=1$.

Depending on the type of SN1a, we consider two different mixing scenarios:

  * •

CCSN+near-$\mathrm{M_{Ch}}$: we consider mixing of CCSN ejecta with the ejecta
from the near-$\mathrm{M_{Ch}}$ model from Seitenzahl et al. (2013b).

  * •

CCSN+sub-$\mathrm{M_{Ch}}$: we consider mixing of CCSN ejecta with the ejecta
from sub-$\mathrm{M_{Ch}}$ models from Gronow et al. (2021) for the various CO
core and He shell masses as discussed in Sec. 2.3.

In both cases, we consider mixing and fallback of CCSN models as discussed in
Sec. 2.2. Thus, CCSN+near-$\mathrm{M_{Ch}}$ and CCSN+sub-$\mathrm{M_{Ch}}$
include the mixing between all the combinations of CCSN ejecta from mixing and
fallback with SN1a ejecta. The mixing parameter $\alpha$ controls the mixing
of the CCSN and SN1a ejecta, where the values are varied from a lowest value
of $10^{-7}$ (negligible SN1a contribution) to a maximum value of $1$
(exclusively SN1a contribution). In both scenarios, we find the best-fit
models by minimising $\chi^{2}$ as mentioned earlier, where we again impose
the criteria that both $M{\rm{}_{dil,1a}}$ and $M{\rm{}_{dil,CCSN}}$ are
greater than a minimum dilution, where we adopt
$M{\rm{}_{dil,1a}^{min}}=10^{4}\,\mathrm{M}_{\odot}$ and the same values of
$M{\rm{}_{dil,CCSN}^{min}}$ as mentioned in Sec.3.2. Similar to the single
CCSN model, we treat the observed values of Sc and K as upper limits.

Figure 2: Top: The elemental abundance pattern of SDSSJ0018-0939 compared with
the best-fit models from various scenarios: PISN (magenta dotted line), CCSN
(black dashed line), 2CCSNe (cyan dashed line), CCSN+near-$\mathrm{M_{Ch}}$
(red dash-dotted line), CCSN+sub-$\mathrm{M_{Ch}}$ (blue dash-dotted line),
and CCSN+PISN (green dashed line). Bottom: The fraction $\eta$ for all
elements produced by near-$\mathrm{M_{Ch}}$ in CCSN+near-$\mathrm{M_{Ch}}$
scenario (red), sub-$\mathrm{M_{Ch}}$ in CCSN+sub-$\mathrm{M_{Ch}}$ scenario
(blue), and PISN in CCSN+PISN scenario (green).

###  3.4 Best-fit models from the mixing of ejecta from two CCSNe

It is important to note that because $\alpha\approx 0$ essentially corresponds
to zero contribution from SN1a, implies that all possible single CCSN models
are subsets of the set of models for the CCSN+SN1a scenario. This guarantees
that the $\chi^{2}$ from the best-fit CCSN+SN1a is always lower than the
$\chi^{2}$ from a single CCSN scenario. Thus, in order to evaluate whether an
observed abundance pattern can be fit better by CCSN relative to CCSN+SN1a, we
consider the possibility of mixing the ejecta from two different CCSN.
Unfortunately, each CCSN progenitor has thousands of possible ejecta
corresponding to the different combinations of $M_{\rm cut,fin}$ and $f_{\rm
cut}$. This makes the total number of ejecta models, from all combinations of
parameters for combining the ejecta from two CCSN, exceedingly large ($\gtrsim
10^{11}$ models) and computationally prohibitively expensive. We thus explore
the limited parameter space of combining the ejecta from mixing and fallback
from a CCSN with that of another CCSN that undergoes no fallback, i.e.,
$f_{\rm cut}=1$. Although this explores a tiny fraction of the total parameter
space of the mixing between two CCSN, the combinations cover the most
important part relevant for $\alpha$-poor VMP stars that tend to prefer CCSN
models with negligible fallback as fallback invariably increases $[\alpha/{\rm
Fe}]$. For brevity, we refer to this scenario as 2CCSNe from hereon. In order
to find the best-fit model, we follow the same procedure as used for the
mixing between CCSN and SN1a as discussed in Sec. 3.3 where SN1a is replaced
by the second CCSN ejecta with no fallback. As in the case of the single CCSN
scenario, we treat the observed values of Sc and K as upper limits.

###  3.5 Best-fit models from the mixing of ejecta from a PISN and a CCSN

Finally, we consider the mixing of the ejecta between a single PISN and a
single CCSN model. The procedure is identical to the one listed in Sec.3.3.
Similar to the CCSN+SN1a scenarios, the CCSN models explore all possible
combinations of mixing and fallback for each mass model that is combined with
all possible PISN models. Similar to single PISN and single CCSN scenarios, we
impose the same minimum dilution mass for all models and treat Sc and K as
upper limits instead of using the observed values.

##  4 Best-fit Models for SDSSJ0018-0939

Table 1: Best-fit models and parameters along with the dilution mass for
SDSSJ0018-0939.

Scenario | Model name | $E_{\rm exp}$ | $\chi^{2}$ | $\alpha$ | $\Delta M_{\rm cut}$ | $\Delta M_{\rm fb}$ | $M{\rm{}_{dil}}$
---|---|---|---|---|---|---|---
|  | ($\times 10^{51}\,\mathrm{erg}$) |  |  | ($\mathrm{M}_{\odot}$) | ($\mathrm{M}_{\odot}$) | ($\times 10^{4}\,\mathrm{M}_{\odot}$)
PISN |  $120\,\mathrm{M}_{\odot}$ He core | 71.0 | 10.57 | – | – | – | $4.0\times 10^{4}$
CCSN |  z11.8-$Y_{\rm e}$ | 1.2 | 3.33 | – | 0.11 | 0.07 | 1.91
CCSN+PISN |  z11.8-$Y_{\rm e}$ + | 1.2 | 3.33 | $10^{-7}$ | 0.11 | 0.07 | 1.9
$120\,\mathrm{M}_{\odot}$ He core | 71.0 | – | – | $1.9\times 10^{7}$
2CCSNe |  z11.3-$Y_{\rm e}+$ | 1.2 | 3.30 | 0.25 | 0.00 | 0.00 | 7.85
z11.8-$Y_{\rm e}$ | 1.2 | 0.11 | 0.06 | 2.62
CCSN+near-$\mathrm{M_{Ch}}$ |  z20-$S_{4}+$ | 12.0 | 2.43 | 0.33 | 1.07 | 0.00 | 13.70
N100_Z0.01 | – | – | – | 27.80
CCSN+sub-$\mathrm{M_{Ch}}$ |  z30-$Y_{\rm e}+$ | 1.2 | 0.72 | 0.57 | 4.23 | 1.60 | 24.60
M10_05 | – | – | – | 18.50

Figure 2 shows the best-fit models from all possible combinations, i.e.,
exclusively from a single PISN (magenta dotted line), a single CCSN (black
dashed line), 2CCSNe (cyan dashed line), CCSN+near-$\mathrm{M_{Ch}}$ SN1a (red
dash-dotted line), CCSN+sub-$\mathrm{M_{Ch}}$ SN1a (blue dash-dotted line),
and CCSN+PISN (green dashed line). The fraction $\eta$ for all elements
produced by near-$\mathrm{M_{Ch}}$in CCSN+near-$\mathrm{M_{Ch}}$scenario
(red), sub-$\mathrm{M_{Ch}}$in CCSN+sub-$\mathrm{M_{Ch}}$scenario (blue), and
PISN in CCSN+PISN scenario (green) is also shown in the bottom panel of Fig.
2. The best-fit models and the corresponding parameters are listed in Table 1.

As can be seen from Fig. 2, the best-fit PISN model provides a very poor fit
with $\chi^{2}=10.57$ and fails to fit multiple elements with particularly
large deviation from the observed values for Na, Al, Si, Ca, Ti and V.
Compared to PISN models, the best-fit CCSN model provides a better fit with
$\chi^{2}=3.33$, and can fit most of the elements from Na to Ca and Cr to Ni.
However, even after accounting for the $1\sigma$ uncertainty, it fails to fit
the observed abundances of several elements such as C, Si, Ti, V, Mn, and Co
with large deviations of $\sim 0.5\,{\rm dex}$ from the observed values for C,
Ti, and V. The best-fit CCSN+PISN model provides the exact same solution as
the best-fit single CCSN with effectively zero contribution from PISN with
$\eta_{\rm PISN}\lesssim 10^{-4}$ ($\lesssim 0.01\%$) for all elements
confirming the lack of any discernible PISN features in the star. The
situation does not improve when mixing from 2CCSNe is considered with a
negligible improvement in the quality of fit with a $\chi^{2}=3.30$. The
particularly poor fit for C, Ti, and V for the best-fit models from both
single CCSN and 2CCSNe scenarios strongly indicates that the observed pattern
cannot be explained by CCSN ejecta alone. Here, it is important to note that
because of the high value of $\log g=5$, the highly sub-solar value of
$[\text{C}/\text{Fe}]\sim-1$ cannot be attributed to the destruction of C due
to internal mixing in SDSSJ0018-0939. Thus, the very low
$[\text{C}/\text{Fe}]$ in this star is inherited from the birth material and a
strong indicator of additional Fe contribution from SN1a. This can be seen
clearly from the best-fit CCSN+near-$\mathrm{M_{Ch}}$ model that provides a
slightly better fit with a lower $\chi^{2}=2.43$ which is largely due to a
much better fit to C. Compared to CCSN models, higher [Mn/Fe] from
near-$\mathrm{M_{Ch}}$ models leads to a perfect fit for Mn, but is
counterbalanced by a worse fit to Al. Similar to the best-fit CCSN models, the
best-fit CCSN+near-$\mathrm{M_{Ch}}$ model cannot fit C, Ti, V, and Co, with a
particularly large deviation from the observed values for Ti and V.

In contrast, the best-fit model from the CCSN+sub-$\mathrm{M_{Ch}}$, resulting
from the mixing of the ejecta from a $30\,\mathrm{M}_{\odot}$ CCSN and a
sub-$\mathrm{M_{Ch}}$ SN1a with $M_{\rm CO}=1\,\mathrm{M}_{\odot}$ and $M_{\rm
He}=0.05\,\mathrm{M}_{\odot}$, provides an excellent fit to the observed
abundance pattern with the lowest $\chi^{2}=0.72$. In particular, the best-fit
CCSN+sub-$\mathrm{M_{Ch}}$ model can explain almost all the elements except
for Ni and can perfectly fit C along with all elements from Ti–Fe.
Importantly, the near perfect fit for Ti to Cr is entirely due to the
contribution for sub-$\mathrm{M_{Ch}}$ SN1a as is evident from the fact that
$\eta_{\rm 1a}\sim 1$ (see the bottom panel of Fig. 2) from Sc to Mn for the
best-fit model. We note that the particular feature of
$[\text{X}/\text{Fe}]\gtrsim 0$ for Ti, V, and Cr with
$[\text{Ti}/\text{Cr}]\gtrsim 0$ can only be fit by sub-$\mathrm{M_{Ch}}$
models that arise from the incomplete Si burning of the He shell as discussed
earlier in Sec. 2.3. Ejecta from sub-$\mathrm{M_{Ch}}$ SN1a also contributes
majorly to all elements from Si–Ca as well as Fe–Ni with $\eta_{\rm 1a}\gtrsim
0.5$. The CCSN ejecta, on the other hand, accounts for almost all the elements
lighter than Si but is a subdominant contributor to elements from S–Fe. For
elements such as Si, Co, and Ni, both CCSN and sub-$\mathrm{M_{Ch}}$ ejecta
make comparable contributions. Because the contribution from CCSN ejecta to
the Fe group is non-negligible, this forces the CCSN model to have a very low
fallback of the innermost material containing Fe group elements in order to
fit the sub-solar values of $[\text{X}/\text{Fe}]$ from C to Si.

It is important to note that in the CCSN+sub-$\mathrm{M_{Ch}}$ scenario,
although the contribution of both CCSN and sub-$\mathrm{M_{Ch}}$ SN1a is
required to fit the full observed abundance pattern, the primary reason for
the excellent fit is driven by the crucial contribution of
sub-$\mathrm{M_{Ch}}$ SN1a that can fit the unique pattern from Ti–Cr. We find
that in addition to the $30\,\mathrm{M}_{\odot}$ CCSN model that provides the
best fit, many other CCSN models that have a low fallback give comparable fits
with very low $\chi^{2}\lesssim 0.8$ when their ejecta is mixed with the
sub-$\mathrm{M_{Ch}}$ SN1a model. Similar to the best-fit
CCSN+sub-$\mathrm{M_{Ch}}$ model, all such models have $\eta_{\rm 1a}\gtrsim
1$ for elements from Ti–Cr, confirming the fact that the primary reason for
the excellent fit is due to sub-$\mathrm{M_{Ch}}$ SN1a contribution.

Figure 3: Same as Fig. 2, but for stars COS171, BD+80245, HE0533–5340, and
SMSSJ034249–284216.  Table 2: Same as Table. 1, but for stars COS171,
BD+80245, HE0533-5340, and SMSSJ03429-284216.

Star | Scenario | Model name | $E_{\rm exp}$ | $\chi^{2}$ | $\alpha$ | $\Delta M_{\rm cut}$ | $\Delta M_{\rm fb}$ | $M{\rm{}_{dil}}$
---|---|---|---|---|---|---|---|---
|  |  | ($\times 10^{51}\,\mathrm{erg}$) |  |  | ($\mathrm{M}_{\odot}$) | ($\mathrm{M}_{\odot}$) | ($\times 10^{4}\,\mathrm{M}_{\odot}$)
COS171 | PISN |  $120\,\mathrm{M}_{\odot}$ He core | 71.0 | 44.46 | – | – | – | $2.1\times 10^{3}$
CCSN |  z23-$Y_{\rm e}$ | 1.2 | 4.59 | – | 0.11 | 0.11 | 1.10
CCSN+PISN |  z23-$Y_{\rm e}+$ | 1.2 | 4.41 | 0.004 | 0.11 | 0.11 | $1.33$
$130\,\mathrm{M}_{\odot}$ He core | 87.3 | – | – | $3.3\times 10^{2}$
2CCSNe |  z11.6-$Y_{\rm e}+$ | 1.2 | 3.95 | 0.57 | 0.00 | 0.00 | 1.35
z23-$Y_{\rm e}$ | 1.2 | 0.21 | 0.20 | 1.02
2CCSNe |  z11.5-$Y_{\rm e}+$ | 1.2 | 3.36 | 0.50 | 0.00 | 0.00 | 0.50
($M_{\rm dil,CCSN}^{\rm min}=5\times 10^{3}\,\mathrm{M}_{\odot}$) |  z14.1-$Y_{\rm e}$ | 1.2 | 0.20 | 0.19 | 0.50
CCSN+near-$\mathrm{M_{Ch}}$ |  z23-$Y_{\rm e}+$ | 1.2 | 4.58 | 0.02 | 0.11 | 0.11 | 1.14
N100_Z0.01 | – | – | – | 55.70
| CCSN+sub-$\mathrm{M_{Ch}}$ |  z23-$Y_{\rm e}+$ | 1.2 | 3.09 | 0.32 | 0.68 | 0.47 | 1.41
| M11_05 | – | – | – | 2.99
BD+80245 | PISN |  $125\,\mathrm{M}_{\odot}$ He core | 78.8 | 45.7 | – | – | – | $4.3\times 10^{3}$
CCSN |  z20.5-$Y_{\rm e}$ | 1.2 | 8.55 | – | 1.45 | 0.29 | 1.00
($M_{\rm dil,CCSN}^{\rm min}=5\times 10^{3}\,\mathrm{M}_{\odot}$) |  z20.5-$Y_{\rm e}$ | 1.2 | 8.37 | – | 2.11 | 1.28 | 0.51
CCSN+PISN |  z20.5-$Y_{\rm e}+$ | 1.2 | 8.53 | 0.002 | 1.54 | 0.34 | 1.00
$70\,\mathrm{M}_{\odot}$ He core | 8.9 | – | – | $5.0\times 10^{2}$
2CCSNe |  z10.1-$Y_{\rm e}+$ | 1.2 | 7.31 | 0.50 | 0.00 | 0.00 | 1.00
z20.5-$Y_{\rm e}$ | 1.2 | 1.45 | 0.51 | 1.00
2CCSNe |  z10.1-$Y_{\rm e}+$ | 1.2 | 6.58 | 0.50 | 0.00 | 0.00 | 0.51
($M_{\rm dil,CCSN}^{\rm min}=5\times 10^{3}\,\mathrm{M}_{\odot}$) |  z20.5-$Y_{\rm e}$ | 1.2 | 2.21 | 1.63 | 0.51
CCSN+near-$\mathrm{M_{Ch}}$ |  z17.2-$S_{\rm 4}+$ | 1.2 | 7.63 | 0.19 | 1.26 | 0.03 | 1.09
N100_Z0.01 | – | – | – | 4.64
| CCSN+sub-$\mathrm{M_{Ch}}$ |  z28-$Y_{\rm e}+$ | 1.2 | 4.18 | 0.24 | 4.53 | 3.67 | 1.01
| M10_10 | – | – | – | 3.18
HE0533-5340 | PISN |  $110\,\mathrm{M}_{\odot}$ He core | 56.4 | 33.83 | – | – | – | $1.72\times 10^{4}$
CCSN |  z22-$Y_{\rm e}$ | 1.2 | 5.94 | – | 3.08 | 2.68 | 1.05
CCSN+PISN |  z20.5-$Y_{\rm e}$ + | 1.2 | 4.93 | 0.002 | 2.59 | 2.15 | 1.67
$100\,\mathrm{M}_{\odot}$ He core | 41.9 | – | – | $8.3\times 10^{2}$
2CCSNe |  z20.5-$Y_{\rm e}+$ | 1.2 | 5.71 | 0.13 | 0.00 | 0.00 | 1.07
z13.2-$Y_{\rm e}$ | 1.2 | 0.60 | 0.59 | 7.16
CCSN+near-$\mathrm{M_{Ch}}$ |  z22-$Y_{\rm e}$ | 1.2 | 3.91 | 0.23 | 0.68 | 0.40 | 8.54
N100_Z0.01 | – | – | – | 28.60
| CCSN+sub-$\mathrm{M_{Ch}}$ |  z22-$Y_{\rm e}$ | 1.2 | 4.91 | 0.28 | 0.68 | 0.58 | 7.77
| M09_03 | – | – | – | 20.00
SMSSJ034249-284216 | PISN |  $130\,\mathrm{M}_{\odot}$ He core | 87.3 | 27.48 | – | – | – | $1.3\times 10^{4}$
CCSN |  z22-$S_{\rm 4}$ | 1.2 | 6.60 | – | 1.74 | 0.00 | 2.00
CCSN+PISN |  z27-$Y_{\rm e}$ + | 12.0 | 4.98 | 0.04 | 3.10 | 0.96 | 10.0
$120\,\mathrm{M}_{\odot}$ He core | 71.0 | – | – | $2.4\times 10^{2}$
2CCSNe |  z22-$S_{\rm 4}+$ | 1.2 | 4.73 | 0.24 | 0.00 | 0.00 | 1.01
z12.7-$S_{\rm 4}$ | 1.2 | 0.40 | 0.18 | 3.19
CCSN+near-$\mathrm{M_{Ch}}$ |  z17.2-$S_{\rm 4}+$ | 1.2 | 6.17 | 0.14 | 0.88 | 0.00 | 2.10
N100_Z0.01 | – | – | – | 12.80
| CCSN+sub-$\mathrm{M_{Ch}}$ |  z22-$Y_{\rm e}$ | 1.2 | 4.13 | 0.23 | 0.88 | 0.66 | 2.70
| M10_10 | – | – | – | 9.10

##  5 Signature of SN1a in Other Stars?

As mentioned in the introduction, although $\alpha$-poor VMP stars have been
associated with SN1a, very few studies have attempted to identify the observed
pattern in individual stars with theoretical yields from SN1a. An exception to
this has been the work by McWilliam et al. (2018) where conclusive evidence of
sub-$\mathrm{M_{Ch}}$ SN1a origin for the star COS171 in Ursa Minor was
reported. Instead of considering the contribution of CCSN directly, the
abundance of a reference VMP star in Ursa Minor was used as a proxy for the
composition of the gas polluted by CCSN which was then combined with SN1a
ejecta. Recently, Reggiani et al. (2023) has also identified three
$\alpha$-poor stars that show possible signatures of sub-$\mathrm{M_{Ch}}$
SN1a contribution. However, only pure SN1a ejecta was considered in the
analysis and CCSN contribution was neglected. Among the three stars, BD+80245
has the highest metallicity of $[\text{Fe}/\text{H}]=-1.73$ whereas
SMSSJ034249–284216 and HE 0533–5340 have metallicities of
$[\text{Fe}/\text{H}]=-1.97$ and $-2.44$, respectively. Below we analyse these
stars for the different scenarios considered for SDSSJ0018-0939 to see if the
observed abundance patterns can be explained by PISN or CCSN, or whether SN1a
contribution from either near-$\mathrm{M_{Ch}}$ or sub-$\mathrm{M_{Ch}}$
models is required.

###  5.1 COS171

COS171 is an $\alpha$-poor star with $[\text{Fe}/\text{H}]=-1.35$ which
happens to be one of the highest metallicity stars in Ursa Minor (Cohen &
Huang, 2010). Because this star has a low $\log g=0.8$, the initial C in the
star would have been considerably depleted (Placco et al., 2014). For this
reason, we treat the observed value of C in this star as a lower limit. We
analysed this star for all the possible scenarios and the best-fit models are
shown in Fig. 3a and the corresponding details are listed in Table 2.

The best-fit PISN model provides an extremely poor fit with $\chi^{2}=44.46$
and clearly cannot explain the abundance pattern. For a single CCSN, the best-
fit model is from a $23\,\mathrm{M}_{\odot}$ progenitor with negligible
fallback with a $\chi^{2}=4.59$ and can fit all elements from Na to Ca and V
to Mn. However, the quality of fit is particularly poor for Ti and all
elements from Fe to Cu as the fit is well beyond the observed $1\sigma$
uncertainty. Additionally, it fails to fit O and Zn. The best-fit CCSN+PISN
model has slightly lower $\chi^{2}=4.41$ with PISN contributing $\sim
20\hbox{--}30\%$ to elements above Si. However, as can be seen from Fig. 3a,
the quality of fit is indistinguishable from a single CCSN indicating that
including PISN does not improve the quality of fit. The best-fit model from
2CCSNe provides a slightly better fit with a $\chi^{2}=3.95$ with somewhat
better fits for O and Zn. However, similar to the single CCSN best-fit model,
it gives a poor fit for Ti and all elements from Fe to Cu. The quality of fit
remains unchanged for the best-fit CCSN+near-$\mathrm{M_{Ch}}$ model as the
best-fit corresponds to the combination where there is negligible contribution
from SN1a as is evident from $\eta_{\rm 1a}$ from Fig. 3a. Lastly, for the
best-fit CCSN+sub-$\mathrm{M_{Ch}}$ model, the quality is marginally better
than the best-fit 2CCSNe model with a $\chi^{2}=3.09$. This model can
additionally fit O, Cu, and Zn along with a slightly better fit for Ti at the
cost of a poorer fit for V and Mn. Surprisingly, the primary reason for the
slightly better fit for the best-fit CCSN+sub-$\mathrm{M_{Ch}}$ model is that
it can perfectly fit Cu even though SN1a contribution to Cu is negligible with
$\eta_{\rm 1a}\sim 0.2$ (see the bottom panel in Fig. 3a).

We note here that COS171 is not a VMP star. Thus, if the origin of elements in
COS171 is entirely from CCSN, it is likely to have come from ejecta from
several events. However, as mentioned earlier, combining ejecta from multiple
sources is not possible as it is computationally expensive. Thus, in order to
test whether multiple CCSN ejecta can match the observed abundance pattern we
lower the $M{\rm{}_{dil,CCSN}^{min}}$ to $5\times 10^{3}\,\mathrm{M}_{\odot}$
for a single CCSN so that it can match the overall high metallicity in
addition to the relative abundance pattern. Interestingly, we find that even
with a lower value of minimum dilution, the best-fit model for a single CCSN
remains unchanged with a $M{\rm{}_{dil,CCSN}=1.1\times
10^{4}\,\mathrm{M}_{\odot}}$ and a $\chi^{2}=4.59$. The best-fit model for
2CCSNe does slightly better and can additionally fit O and Zn but continues to
give a poor fit for Ti and elements from Fe to Cu. The corresponding value of
$\chi^{2}=3.36$ is comparable to the best-fit CCSN+sub-$\mathrm{M_{Ch}}$
model. The reason for the overall poor fit is primarily due to the peculiar
abundance pattern from Fe to Zn. In particular, highly sub-solar
$[\text{Ni}/\text{Fe}]\sim-0.7$ along with super-solar
$[\text{Co}/\text{Ni}]\sim 0.5$ cannot be produced by either CCSN or SN1a.

In summary, even though the best-fit CCSN+sub-$\mathrm{M_{Ch}}$ model provides
the overall best fit with the lowest $\chi^{2}$, the quality of fit is
comparable to the best-fit 2CCSNe model while the overall quality of fit is
poor for both. Thus, the observed abundance pattern in COS171 cannot be
strongly attributed to SN1a in sharp contrast to SDSSJ0018-0939. We note,
however, that given the relatively high metallicity of COS171 of
$[\text{Fe}/\text{H}]-1.35$ and the fact that it has the highest metallicity
among the stars observed in Ursa Minor (Cohen & Huang, 2010), SN1a could be a
natural solution from the point of view of galactic chemical evolution (Kirby
et al., 2019). Our analysis, on the other hand, is based entirely on the
observed abundance pattern where a clear SN1a signature is absent.

###  5.2 BD+80245

The best-fit for BD+80245 for the different scenarios is shown in Fig. 3b with
the corresponding details listed in Table 2. Similar to COS171, PISN provides
an extremely poor fit with $\chi^{2}=45.7$. The quality of fit for a single
CCSN and 2CCSNe are comparable with best-fit $\chi^{2}$ of 8.55 and 7.31,
respectively, with an overall poor fit as they cannot match the abundance of
multiple elements with particularly large deviation from the observed
abundances for Mg, Al, Ti, Ni, and Zn. The quality of fit from the best-fit
CCSN+PISN model is almost identical to the best-fit model from the single CCSN
scenario with very little PISN contribution of $\lesssim 5\%$ for Mg and Si
and almost near zero contribution from PISN for all other elements. In fact,
the CCSN model in the best-fit CCSN+PISN model is also identical to the single
CCSN case indicating that no PISN contribution is required to explain the
abundance pattern in this star. The best-fit CCSN+near-$\mathrm{M_{Ch}}$ model
provides a similar quality of fit with a $\chi^{2}=7.63$. The best-fit
CCSN+sub-$\mathrm{M_{Ch}}$ model gives the lowest $\chi^{2}=4.18$ even though
the overall quality of fit is only slightly better. While it does provide a
better fit for Ti, Ni and Zn, the fit is quite poor for Mg, Al, Si, Cr, and Co
with large deviations from the observed values well beyond the $1\sigma$
uncertainty.

Because BD+80245 has a metallicity of $[\text{Fe}/\text{H}]=-1.73$, we also
considered the possibility of reducing $M{\rm{}_{dil,CCSN}^{min}}=5\times
10^{3}\,\mathrm{M}_{\odot}$ similar to COS171. In this case, the quality of
fit for the best-fit model from a single CCSN remains almost unchanged with
$\chi^{2}=8.37$, the value is slightly better for the best-fit model from
2CCSNe with $\chi^{2}=6.58$. Despite the lower $\chi^{2}$, the quality of fit
remains poor as these models cannot match the abundance of any additional
elements. The reason for the overall poor fit is primarily due to Al and Mg
which are produced exclusively by CCSN. In particular, it has an extremely
sub-solar value of $[\text{Al}/\text{Fe}]\sim-1.7$ leading to highly super-
solar $[\text{Mg}/\text{Al}]\sim 1.2$. This cannot be matched by any of the
CCSN models which leads to an overall poor fit for all models. We note that
the best-fit PISN model can match the low Al along with the highly super-solar
$[\text{Mg}/\text{Al}]$ but gives an extremely poor fit for almost all other
elements.

Thus, similar to COS171, among all the scenarios, although
CCSN+sub-$\mathrm{M_{Ch}}$ provides the best fit, the quality of fit is poor
and it is difficult to claim a clear signature for SN1a contribution for
BD+80245 in contrast to SDSSJ0018-0939.

###  5.3 HE0533-5340

The best-fit models for HE0533-5340 are shown in Fig. 3c and the details of
the models and best-fit parameters are listed in Table 2. Similar to other
stars, the best-fit PISN model provides an extremely poor fit with a
$\chi^{2}=33.8$. Unlike COS171 and BD+80245, the best-fit model from the
CCSN+near-$\mathrm{M_{Ch}}$ provides the best overall fit with $\chi^{2}=3.9$.
The primary reason for CCSN+near-$\mathrm{M_{Ch}}$ being the best fit is the
elevated $[\text{Mn}/\text{Fe}]\sim 0.1$ which is a key signature of the
near-$\mathrm{M_{Ch}}$ model. The overall quality of fit, however, is poor as
the best-fit model cannot match multiple elements such as Mg, Al, Si, Ti, Cr,
and Ni. The best-fit model from CCSN+sub-$\mathrm{M_{Ch}}$ provides a
comparable quality of fit with $\chi^{2}=4.91$. In this case, although the
$\chi^{2}$ is slightly higher for the best-fit CCSN+sub-$\mathrm{M_{Ch}}$
model, it can match more elements such as Ti and Ni within the observed
$1\sigma$ uncertainty compared to the best-fit CCSN+near-$\mathrm{M_{Ch}}$
model. Interestingly, the best-fit CCSN+PISN model provides a comparable fit
with a $\chi^{2}=4.93$ where PISN contributes to $\sim 50\%$ to the even $Z$
elements from Si to Cr. The quality of fit is slightly better than both the
single CCSN and 2CCSNe models that have $\chi^{2}$ of 5.94 and 5.71,
respectively. The best-fit CCSN+PISN and 2CCSNe models can both fit Mg within
the $1\sigma$ uncertainty in addition to all the elements that can be fit by
the CCSN+near-$\mathrm{M_{Ch}}$ model. We note that even though this star can
potentially have a PISN origin, it cannot be distinguished from any of the
other scenarios. The overall poor quality of fit for the observed pattern in
HE0533-5340 across all scenarios is primarily due to the peculiar abundance
features from Mg–Si and Cr–Mn. In particular, similar to BD+80245, the highly
sub-solar value of $[\text{Al}/\text{Fe}]\sim-1.4$ with super-solar
$[\text{Mg}/\text{Al}]\sim 1.1$ cannot be fit by any CCSN model. Additionally,
high super-solar $[\text{Si}/\text{Fe}]\sim 0.5$ cannot be simultaneously fit
with extreme sub-solar $[\text{Al}/\text{Fe}]$. Lastly, sub-solar
$[\text{Cr}/\text{Mn}]$ cannot be produced by either CCSN or SN1a leading to a
poor fit for Cr for all models.

In summary, even though CCSN+near-$\mathrm{M_{Ch}}$ provides the best fit, the
overall quality of fit remains poor. Additionally, the quality of fit from
scenarios that do not include SN1a such as CCSN+PISN and 2CCSNe is comparable
and can even match more elements within the observed $1\sigma$ uncertainty.
Consequently, the observed abundance pattern in HE0533-5340 cannot be
unambiguously attributed to SN1a.

Figure 4: Same as Fig. 3, but Al is neglected.

###  5.4 SMSSJ034249-284216

The best-fit model for SMSSJ034249-284216 from all the scenarios is shown in
Fig. 3d. The situation is very similar to COS171, with the best-fit
CCSN+sub-$\mathrm{M_{Ch}}$ model providing the overall best fit with
$\chi^{2}=4.13$ and a comparable quality of fit from the best-fit 2CCSNe model
with $\chi^{2}=4.73$. Among these two models, the best-fit
CCSN+sub-$\mathrm{M_{Ch}}$ model can match the abundance of Ca, Ti, and Mn
whereas the best-fit 2CCSNe model can match Si and Zn and vice-versa. Compared
to the best-fit CCSN+sub-$\mathrm{M_{Ch}}$ model, the best-fit models from a
single CCSN and CCSN+near-$\mathrm{M_{Ch}}$ provide a slightly worse fit
whereas a single PISN provides an extremely poor fit. Similar to HE0533-5340,
the best-fit CCSN+PISN model can provide a fit comparable to the 2CCSNe case
and has major contributions of up to $\sim 80\%$ to most elements from Mg to
Ni with a CCSN model that is distinct from both the single CCSN and 2CCSNe
models. This star can potentially have a PISN origin, which, however, cannot
be distinguished from other scenarios as is the case for HE0533-5340 discussed
above. The overall quality of fit from all scenarios is poor which is
primarily because they completely fail to match the abundances of Mg, Al and
Cr. Similar to BD+80245, this star has an extremely sub-solar value of
$[\text{Al}/\text{Fe}]\sim-1.4$ leading to highly super-solar
$[\text{Mg}/\text{Al}]\sim 1.2$ that cannot be matched by any of the CCSN
models. Similar to HE0533-5340, sub-solar $[\text{Cr}/\text{Mn}]$ cannot be
produced by any source yielding a poor fit for Cr for all models.

Thus, the situation is similar to COS171 where although the best-fit
CCSN+sub-$\mathrm{M_{Ch}}$ model provides the overall best fit with the lowest
$\chi^{2}$, the quality of fit is comparable to the best-fit 2CCSNe and
CCSN+PISN models while the overall quality of fit is poor for both. Thus, the
observed abundance pattern in SMSSJ034249-284216 cannot be strongly attributed
to SN1a.

###  5.5 Best-fit without Al for BD+80245, HE0533-5340, and SMSSJ034249-284216

As discussed in the above analysis, the extremely sub-solar value of
$[\text{Al}/\text{Fe}]$ is common to all three stars and is one of the primary
reasons for the overall poor fit. We note, however, that a substantially
higher abundance of $[\text{Al}/\text{Fe}]$ has been reported for both
BD+80245 (Ivans et al., 2003) and HE0533-5340 (Cohen et al., 2013) with
$\Delta[\text{Al}/\text{Fe}]$ of $+0.41$ and $+0.68$ compared to Reggiani et
al. (2023). Additionally, Marino et al. (2019) has found similar values of
$[\text{Al}/\text{Fe}]$, they have reported extremely high $\sigma$ of $0.49$
for SMSSJ034249-284216. Because the Al abundance is potentially uncertain and
given that it has a large effect on the quality of fit, we repeat the analysis
for the three stars by neglecting Al. The best-fit results are presented in
Fig. 4. Overall, neglecting Al does lead to a substantial improvement in the
quality for all stars. As before, the overall best-fit model for BD+80245 and
HE0533-5340 comes from CCSN+sub-$\mathrm{M_{Ch}}$ and
CCSN+near-$\mathrm{M_{Ch}}$, respectively, with much improved $\chi^{2}$ of
1.75 and 2.78, respectively. However, in both cases, the quality of fit from
the best-fit 2CCSNe model is comparable with $\chi^{2}$ of 2.31 and 3.03 for
BD+80245 and HE0533-5340, respectively. Interestingly, for SMSSJ034249-284216,
the overall best-fit model is from 2CCSNe with a $\chi^{2}$ of 1.6 which is
closely followed by CCSN+near-$\mathrm{M_{Ch}}$with $\chi^{2}$ of 1.7.

Thus, in summary, the results clearly show that the abundance pattern in
BD+80245, HE0533-5340, and SMSSJ034249-284216 cannot be strongly associated
with SN1a as ejecta from CCSNe alone can also provide comparable fit.

##  6 Summary and Conclusions

SDSSJ0018-0939 is an $\alpha$-poor VMP star that was previously identified as
a star with a possible signature from PISN by Aoki et al. (2014). However, we
find that none of the theoretical PISNe abundance patterns is remotely
compatible with the observed pattern as the best-fit PISN pattern gives an
extremely poor fit with a high $\chi^{2}$ which is consistent with the results
found by Takahashi et al. (2018). Even when CCSN+PISN scenario is considered,
the best-fit solution is found to be the one with zero contribution from PISN
which indicates that the abundance pattern does not have any PISN features.
Instead, the peculiar abundance pattern observed in SDSSJ0018-0939 can be fit
perfectly by the mixing of the ejecta between a $30\,\mathrm{M}_{\odot}$ CCSN
with negligible fallback and a sub-$\mathrm{M_{Ch}}$ SN1a resulting from He
detonation of a WD with $M_{\rm CO}=1\,\mathrm{M}_{\odot}$ and $M_{\rm
He}=0.05\,\mathrm{M}_{\odot}$. In particular, we find that the unique pattern
around the Ti-Cr region with $[\text{X}/\text{Fe}]\gtrsim 0$ for Ti, V, and Cr
along with $[\text{Ti}/\text{Cr}]\gtrsim 0$ is the key signature of
sub-$\mathrm{M_{Ch}}$ SN1a resulting from contribution from He shell burning
that is distinct from any other source such as CCSNe or a combination of CCSN
and near-$\mathrm{M_{Ch}}$ SN1a. This, in combination with very low
$[\text{C}/\text{Fe}]\sim-1$ along with highly sub-solar values of
$[\text{X}/\text{Fe}]$ for Mg, Si, and Ca, point to a near smoking-gun
signature of a star formed from a gas strongly polluted by a
sub-$\mathrm{M_{Ch}}$ SN1a.

A clear signature of a sub-$\mathrm{M_{Ch}}$ SN1a in the early Galaxy is
particularly interesting as it is consistent with the findings from recent
studies that analyzed the observed evolution of $[\text{Mn}/\text{Fe}]$ and
$[\text{Ni}/\text{Fe}]$ in the Milky Way and dwarf galaxies (Seitenzahl et
al., 2013a; Kobayashi et al., 2020; Eitner et al., 2020, 2023). These studies
conclude that the contribution of sub-$\mathrm{M_{Ch}}$ SN1a is crucial and
could account for up to $\sim 50\hbox{--}75\%$ of the total SN1a. Furthermore,
there are indications that sub-$\mathrm{M_{Ch}}$ SN1a is dominant in the early
times in dwarf galaxies whereas near-$\mathrm{M_{Ch}}$ SN1a becomes important
at later times (Kirby et al., 2019; de los Reyes et al., 2020). Thus, it is
not surprising that a signature from a sub-$\mathrm{M_{Ch}}$ SN1a is more
likely to be found in VMP stars.

We also analyzed four stars, namely, COS171, BD+80245, HE0533-5340, and
SMSSJ034249-284216, that have been previously identified as stars that show a
possible signature of SN1a (McWilliam et al., 2018; Reggiani et al., 2023).
Overall, we find that none of the scenarios can provide a good fit to the
observed abundance pattern in these four stars. We find that even though the
best-fit models from scenarios involving CCSN+SN1a provide the best fit with
the lowest $\chi^{2}$, the best-fit models from 2CCSNe can always provide
comparable fits with a marginally higher $\chi^{2}$. Interestingly, we also
find comparable fits with CCSN+PISN scenario with a substantial contribution
from PISN for HE0533-5340 and SMSSJ034249-284216 although it is
indistinguishable from other best-fit scenarios. Thus, although the elements
observed in these stars could have originated from SN1a, the fact that CCSN
ejecta or a combination of CCSN and PISN ejecta can match the abundance
pattern almost equally well implies that the origin of elements in these stars
cannot be unambiguously attributed to SN1a. The conclusion remains unchanged
even when Al is neglected for the analysis even though the quality of fit
improves substantially for BD+80245, HE0533-5340, and SMSSJ034249-284216. In
this case, the quality of fit from the best-fit 2CCSNe model is again
comparable to the best-fit CCSN+SN1a models for BD+80245 and HE0533-5340. On
the other hand, the best-fit 2CCSNe model provides the overall best fit for
SMSSJ034249-284216. Thus, none of the four stars has a clearly distinguishable
SN1a signature in sharp contrast to SDSSJ0018-0939 where a clear signature of
an SN1a is established from the fact that a near-perfect fit is achieved only
from the best-fit CCSN+sub-$\mathrm{M_{Ch}}$ SN1a whereas all other scenarios
fail to provide a good fit.

The significance of SDSSJ0018-0939 as the star with the clearest signature of
SN1a among all currently known VMP stars is ultimately based on the robustness
of observed abundances of key elements such as C, Ti, V, and Cr. In this
regard, we note that large discrepancies in the estimated abundances of
elements are sometimes found when a given star is analyzed by different
groups. Large changes in the abundance of key elements can consequently affect
and even change their association with a particular source. A recent example
of this is the chemically peculiar star LAMOST J1010+2358 that was first
reported by Xing et al. (2023) whose estimated abundance pattern is found to
be considerably different in subsequent analysis by Skúladóttir et al. (2024)
and Thibodeaux et al. (2024). Thus, it is important that SDSSJ0018-0939 is
also analyzed by independent groups to confirm whether the observed abundance
pattern is robust.

Our results highlight the importance of $\alpha$-poor VMP stars as powerful
probes for gaining crucial insights about SN1a. Such stars provide the only
direct probe for detailed nucleosynthesis in SN1a that can differentiate
between near-$\mathrm{M_{Ch}}$ and sub-$\mathrm{M_{Ch}}$ models. However, we
also find that not all $\alpha$-poor VMP stars require SN1a contribution as
some of the CCSN models that do not undergo substantial fallback can also
match the observed abundance pattern equally well. Thus, a clear association
of SN1a with such stars should only be made after careful analysis.
Notwithstanding this, future detections of more $\alpha$-poor VMP stars with
detailed abundance patterns from multiple elements are required to find more
stars such as SDSSJ0018-0939 that can be used to gain crucial insights into
SN1a and their contribution to the early Galaxy.

## Data Availability

Data is available upon reasonable request.

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