# The Galactic Isotopic Decomposition for the Sculptor Dwarf Spheroidal Galaxy

Kanishk Pandey,1 Christopher West1,2
1Department of Physics and Astronomy, Carleton College, Northfield, MN, 55057,
USA
2Joint Institute for Nuclear Astrophysics—Center for the Evolution of the
Elements, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Stellar evolution models require initial isotopic abundance sets as input, but
these abundances are incomplete outside the solar neighborhood, are
challenging to infer from elemental observations, and are galaxy specific.
Compositions different from the Milky Way (MW) have distinct chemical
histories and are important to explore. We present an isotopic history model
for the Sculptor dwarf spheroidal galaxy (dSph) based on astrophysical
processes, using a complementary approach to GCE models, which can estimate
isotopic abundances for future nucleosynthesis studies. We approximated the
isotopic composition of Sculptor’s late stage evolution using the OMEGA
chemical evolution code and used Big Bang Nucleosynthesis (BBN) predictions as
the other boundary condition. Isotopic abundances were scaled from late stage
evolution to BBN values according to the astrophysical processes responsible
for their production. The isotopic abundances were summed into elemental
abundances and fit to observational Sculptor abundance data to tune the free
parameters. The completed model gives the average isotopic history of Sculptor
for massive star, Type Ia SNe, main $s$-process peak, and $r$-process
contributions. We find that Type Ia SNe contribute $\approx$ 86 per cent to
the late stage evolution Fe abundance, which agrees with other dSph chemical
evolution studies, and is greater than typical MW values of $\approx$ 70 per
cent found using a similar process. The model also finds that neutron star
mergers contribute $\approx$ 30 per cent to the late stage evolution Eu
abundance, suggesting that CCSNe may be the dominant $r$-process progenitors
in dSphs.

###### keywords:

Galaxy: abundances, evolution, dwarf — stars: abundances, massive, supernovae:
general

††pubyear: 2021††pagerange: The Galactic Isotopic Decomposition for the
Sculptor Dwarf Spheroidal Galaxy–The Galactic Isotopic Decomposition for the
Sculptor Dwarf Spheroidal Galaxy

##  1 Introduction

Stellar simulations require an initial isotopic composition, and this input is
crucial for obtaining the correct nucleosynthesis outputs. When modelling
compositions similar to the sun, this is aided by a robust solar isotopic
abundance composition. Many studies continue to refine this composition (e.g.,
Lodders, 2003; Lodders et al., 2009; Lodders, 2019; Asplund et al., 2005;
Asplund et al., 2009; Grevesse et al., 2005, 2007). However, isotopic
abundances are more challenging to determine for environments outside the
solar neighborhood due to the lack of isotopic data, and such abundances
cannot be inferred from elemental measurements.

Galactic chemical evolution (GCE) models are one possible method to find an
initial isotopic composition input. GCE models typically integrate stellar
yields across time-steps to build abundances from big bang nucleosynthesis
(BBN) to a final (e.g., solar) composition. In doing so, these models predict
average isotopic abundances which can, in part, facilitate our understandings
of stellar nucleosynthesis (Prantzos, 2008). GCE models must estimate
quantities which are not well known, such as SFR and infall/outflow rates, and
often do not address the dynamics within a galaxy. Usually, a galaxy is split
into zones as an approximation (e.g., Timmes et al., 1995). Other models, for
example, have calculated the evolution of heavy elemental abundances from C to
Zn in our solar neighborhood (e.g., Chiappini et al., 1997; Alibés et al.,
2001; Fenner & Gibson, 2003; Kobayashi et al., 2006).

Understanding the chemical evolution and isotopic histories of dwarf
spheroidal galaxies (dSphs) suffers additional challenges. These systems have
a low luminosity because they are gas-poor systems (Lanfranchi & Matteucci,
2003; Grebel et al., 2003) and have an older stellar population, possibly due
to being "fossils" from the pre-reionization era (Ricotti & Gnedin, 2005;
Theler et al., 2020). These characteristics make it difficult to obtain
observational data, which in turn makes it difficult to assess GCE model
yields for these galaxies. Furthermore, uncertainties in gas infall/outflow
parameterizations also more greatly impact results, as these systems are less
gravitationally bound (Lanfranchi & Matteucci, 2003; Calura & Menci, 2009).
However, past studies have compiled partial observational elemental abundances
for various dSphs (e.g., Kirby et al., 2009; Kirby et al., 2010; Kirby et al.,
2018; Tafelmeyer et al., 2010; Starkenburg et al., 2013; Jablonka et al.,
2015; Skúladóttir et al., 2017; Duggan et al., 2018; Hill et al., 2019;
Skúladóttir et al., 2019).

Despite these challenges, attempts have been made to model dSphs. One of the
first standardized models for an average dSph was devised by Lanfranchi &
Matteucci (2003), who developed a chemical evolution model for 8 dSphs in the
Local Group. Other substantial models have generally found that dSphs have a
low star formation efficiency, high wind efficiency, low gas content, and
therefore, a different chemical evolution history than that of the MW. (e.g.,
Recchi et al., 2001; Venn et al., 2004; Venn et al., 2012; Fenner et al.,
2006; Matteucci, 2008; Calura & Menci, 2009; Cohen & Huang, 2009, 2010;
Tolstoy et al., 2009; Kirby et al., 2009, 2011, 2020; Venn et al., 2012; de
Boer et al., 2012a, b; Romano & Starkenburg, 2013; Leaman et al., 2013; Homma
& Murayama, 2014; Vincenzo et al., 2014; Simon et al., 2015; Theler et al.,
2020; Reichert et al., 2020). Future models hope to provide a better
understanding of the chemical evolution differences between dSphs and spiral
galaxies, such as the time delay of the Type Ia onset and contributions from
neutron-capture processes. Additional interest in dSph modeling include, for
example, their role in larger galaxy formation (Lanfranchi & Matteucci, 2003;
Kirby et al., 2011). They are also believed to be the smallest dark matter
dominated systems in the Universe (Matteucci, 2008), and understanding dSphs
can further contribute to our knowledge of galactic dark matter profiles
(Mateo, 1998; Łokas, 2002; Łokas et al., 2005; Fenner et al., 2006; Gilmore et
al., 2007; Strigari, 2018). Due to their lack of dust and gas as well as large
separation between celestial objects, dSphs are also key environments for
x-ray and gamma ray studies (Jeltema & Profumo, 2008; Abdo et al., 2010).
Finally, dSphs are ideal environments to study the effects of Type Ia
Supernovae (Type Ia SNe) due to their dominance at higher metallicities (Kirby
et al., 2019).

In this work, we adopt a different approach for describing average isotopic
abundances than standard GCE modeling. We follow the methodology taken by West
& Heger (2013) to model the Sculptor dwarf spheroidal galaxy instead of the MW
(see Battaglia et al., 2008 for further background information on Sculptor).
The methods used are complementary to GCE models, and other such complimentary
models have been explored (e.g., Ting et al., 2012). The objective is to
describe an average isotopic history by scaling isotopic abundances according
to the astrophysical processes responsible for their production. We later
compare our Sculptor model against the model devised by West & Heger (2013)
for the MW. This is an improvement over previous studies that interpolate
between BBN scaled solar abundances (West & Heger, 2013) or use scaled
abundances with $\alpha$-enhancement (Limongi & Chieffi, 2018).

We emphasize that our methodology is distinct from a traditional GCE model.
This model can be used for future nucleosynthesis studies (for e.g., as inputs
for stellar evolution models). It is a scaling model tuned to data, and
provides average isotopic yields at any desired metallicity. In this respect,
GCE models are less helpful, as they require inputs from stellar models in the
first place and cannot effectively capture large scale stellar formation
(Prantzos, 2008).

Some benefits with using this approach over a traditional GCE model include:

  1. 1.

This model does not try to describe the more complex principles of GCE, such
as the slope of the high-mass end of the stellar IMF and total number of Type
Ia SNe, which inherently leads to large uncertainties Côté et al. (2016).
Consequently, this model does not introduce additional parameters to account
for the presence of winds, which is a fundamental characteristic for dSphs
(e.g., Lanfranchi & Matteucci, 2003). Rather, the impact of winds is
indirectly reflected when the model is fit to observed data and the best-
fitting parameter values are found.

  2. 2.

GCE models tend to also begin with large uncertainties as a consequence of the
first generation of stars not being homogeneously mixed (Prantzos, 2008).
Since GCE models try to improve upon earlier timesteps, they perform poorly at
BBN compositions and better near the late-stage evolution composition (West &
Heger, 2013; Welsh et al., 2020). This model begins where abundances are best
known and is then tuned to observed data, so it performs well in describing
environments outside the solar neighborhood if sufficient data exists.

  3. 3.

The model avoids the discrete, iterative time steps that define GCE models,
and is therefore a more comprehensive model throughout all metallicities.
Furthermore, this approach avoids the dynamic and evolutionary processes
employed in traditional GCE models (West & Heger, 2013). As stated in West &
Heger (2013), stellar yields that are calculated by GCE models are not just
based on metallicity but also environment, time, and space which is difficult
to assess.

Some limitations with using this approach over a traditional GCE model
include:

  1. 1.

Our model used the late-stage evolution (LSE) isotopic abundance pattern of
the studied system for a fixed point, and a complete set can be difficult to
obtain except for compositions similar to the Sun.

  2. 2.

The model does not attempt to do GCE calculations to evolve abundances, and
instead only uses the core principles of primary and secondary processes as a
motivation. Namely,

$\dot{X}_{\mathrm{\mathit{i,}\mathrm{p}}}=\text{const.}\Rightarrow
X_{i,\mathrm{p}}\propto Z,$ (1)
$\dot{X}_{\mathit{\mathit{i},}\mathrm{s}}\propto
X_{\mathit{\mathit{j},}\mathrm{p}}\Rightarrow X_{i,\mathrm{s}}\propto Z^{2},$
(2)

where $X_{i,\mathrm{p}}$ is the abundance of some isotope
$\mathrm{\mathit{i}}$ made in a primary process, the subscript $\mathrm{s}$
denotes secondary process, and $Z$ is the total metallicity. Since it does not
attempt to model dynamic quantities parameterized in GCE models, it avoids the
corresponding uncertainties, but at the cost of being able to offer
predictions. This more empirically-based method, in addition to isotopic
inputs, can still provide consistency checks and suggest trends within the
constraints of data to be investigated in future projects.

We have chosen the Sculptor dSph for this work due to its significant
observational datasets relative to other dSphs, especially in the metal-poor
regime (Romano & Starkenburg, 2013; Jablonka et al., 2015) and it has a SFH
that appears typical for dSphs (Fenner et al., 2006; Hill et al., 2019) so
that completed model may characterize the average isotopic history for dSphs.
Due to the abundant data, Sculptor is a prime example for dSph research as
well as galaxy formation and dark matter studies. Below, we summarize relevant
differences in evolution and nucleosynthesis trends between the MW and dSphs.

Past chemical evolution models suggest dSphs have a high wind efficiency
(Lanfranchi & Matteucci, 2003). This wind can transport enriched gas outside
the galaxy, thereby lowering the total gas content and the SFR. Furthermore,
the lower gas content suggests a lower metallicity distribution function (MDF)
for dSphs which can lead to a broad metal-poor tail (see Figure 3 of
Lanfranchi & Matteucci, 2003 and Figure 9 of Homma & Murayama, 2014).

Iron production begins to dominate with the onset of Type Ia SNe at an
observed metallicity of [Fe/H] $\approx-1$ in the MW (Smecker-Hane & Wyse,
1992; Venn et al., 2012; Kobayashi et al., 2011, 2020), which is also
consistent with results from West & Heger (2013). The Type Ia onset occurs at
lower metallicities ($-1.5\leq$ [Fe/H] $\leq-2.0$) in dSphs (e.g., Recchi et
al., 2001; Lanfranchi & Matteucci, 2003; Fenner et al., 2006; Tolstoy et al.,
2009; Cohen & Huang, 2009; Kirby et al., 2011; Venn et al., 2012; de Boer et
al., 2012a, b; Romano & Starkenburg, 2013; Vincenzo et al., 2014; Theler et
al., 2020; Reichert et al., 2020). Type Ia SNe have also been found to be more
dominant in dSphs at higher metallicities than the MW (Kirby et al., 2020),
which suggests that the decrease of the [$\alpha$/Fe] ratio is steeper than
the corresponding decrease in the MW (Recchi et al., 2001; Matteucci, 2008;
Vincenzo et al., 2014; Theler et al., 2020). Tolstoy et al. (2009) also found
a [Na/Fe] $-$ [Ni/Fe] correlation in dSphs which was first suggested by Nissen
& Schuster (1997) who discovered that stars in the MW halo that were deficient
in $\alpha$ elements were also deficient in Na, and to a lesser extent, Ni.

It has also been suggested that AGB stars begin dominating the chemical
evolution of dSphs at the same time as Type Ia SNe (Venn et al., 2012; Theler
et al., 2020; Skúladóttir et al., 2020). Furthermore, the contribution of AGB
stars is believed to be stronger in Sculptor than in the MW, especially at
higher metallicities (de Boer et al., 2012a; Skúladóttir et al., 2020). The
_r_ -process to _s_ -process transition has been proposed to begin at [Fe/H] =
$-$2.6 to [Fe/H] = $-$2 in the MW (Truran et al., 2002; Simmerer et al., 2004;
Kobayashi et al., 2011), but the same transition is still debated for dSphs.
Perhaps the most quantitative method to determine this transition was done by
Reichert et al. (2020) who found that [Eu/Ba] decreased at a metallicity of
[Fe/H] = $-$1.57 for Sculptor. The observed [Eu/Fe] data trends in Sculptor
have been shown to be similar to the MW, suggesting the _r_ -process
contribution is similar in these galaxies. In particular, [Eu/Fe] begins to
decrease at [Fe/H] $\geq$ $-$1, which is congruent with the Type Ia onset
value (Skúladóttir et al., 2020). Finally, there is evidence that suggests
that CCSNe, and not NSM, are the dominant _r_ -process progenitors in dSphs
(Tolstoy et al., 2009; Duggan et al., 2018; Skúladóttir et al., 2019; Reichert
et al., 2020; Skúladóttir et al., 2020). We later examine these proposed
differences between spiral and dSphs in this work.

This paper has the following outline: Section 2 highlights differences in the
implementation of this model to Sculptor compared to that done in West & Heger
(2013). The process for estimating Sculptor’s LSE isotopic abundance pattern
is also explained. In Section 3, the model is fit to available data and the
best fitting parameters are calculated. Section 4 presents the results
obtained from the elemental model and compares it with other dSph studies as
well as results from the MW model by West & Heger (2013). The results of the
model are also compared with results from a traditional GCE model. Finally,
Section 5 highlights possible conclusions of this work.

##  2 Model Description

We present a model that describes an average isotopic history of the Sculptor
dwarf spheroidal galaxy for contributions from massive stars, Type Ia SNe, and
select neutron-capture processes. We use a similar methodology to West & Heger
(2013) and describe below only the key differences for its adaptation to a
dSph. We also explain the process for estimating Sculptor’s LSE composition in
Section 2.2. More information on the methodology employed in this work can be
found in West & Heger (2013).

###  2.1 Model Differences

Here we summarize the differences between our method and that used by West &
Heger (2013). The isotopic solar abundance pattern used in this work was taken
from a new data set by Lodders (2019) which gives updated values relative to
their 2010 publication used in West & Heger (2013) (Lodders et al., 2009). We
also used theoretical BBN abundances of D $=2.59\cdot 10^{-5}$, 3He
$=0.996\cdot 10^{-5}$, 7Li $=4.648\cdot 10^{-10}$, and 6Li $=1.288\cdot
10^{-14}$ from Cyburt et al. (2016), which gives updated values relative to
their 2001 publication used in West & Heger (2013) (Cyburt et al., 2001). We
also used updated values for Y${}_{\text{P}}$ = 0.2470 from Pitrou et al.
(2018). We assumed that GCR, classical novae, and the _$\nu$_ -process behave
similarly in both the MW and Sculptor, as these processes have not been well-
explored in dSphs.

####  2.1.1 Massive Stars

We used yields from Heger & Woosley (2010) for an intermediary point at [Fe/H]
= $-$3 found with a Salpeter IMF, a low mixing parameter of 0.02512 used in a
running boxcar method, an explosion energy of E = 1051 erg, and stellar mass
bounds of $10-110\,\mathrm{M}_{\odot}$. Since the Heger & Woosley (2010)
simulation is typically used for MW studies, we briefly validate our choice
for using these massive star yields in this section, and examine the effect of
using a non-rotating model at low metallicities later in Section 2.3. Kirby et
al. (2019) compared massive star theoretical yields of 3 $\alpha$-elements
(Mg, Si, and Ca) and 3 iron-peak elements (Cr, Co, and Ni) from simulations by
Nomoto et al. (2006) and Heger & Woosley (2010) to their inferred yields in
five different dSphs. They determined that the inferred yields of the $\alpha$
and iron-peak elements generally agreed with both simulations for numerous
dSphs. Their findings on the Sculptor dSph are summarized below (for more
information, see Figure 7 of Kirby et al., 2019):

  1. 1.

The [Mg/Fe] value agrees with Nomoto et al. (2006), but this is expected as
the [Mg/Fe] ratio was chosen as the prior and the [Mg/Fe] yields from Nomoto
et al. (2006) normalized to quantify the fraction of elements produced in Type
Ia SNe

  2. 2.

The [Si/Fe] value generally agrees with both simulations

  3. 3.

The [Ca/Fe] value falls closer to Nomoto et al. (2006)

  4. 4.

The [Cr/Fe] ratio was in poor agreement with both simulations

  5. 5.

The [Co/Fe] value exceeds both simulations but is closer to Heger & Woosley
(2010)

  6. 6.

The [Ni/Fe] ratio is closer to Heger & Woosley (2010)

It is interesting to note that for the Sculptor dSph, the simulation by Nomoto
et al. (2006) seems to present more accurate results for the $\alpha$-elements
whereas the simulation by Heger & Woosley (2010) performs better for the iron-
peak elements. Both perform equally well for different elemental ratios;
however, we decided to use the simulation by Heger & Woosley (2010) since we
normalize the yields to the observed Fe abundance at [Fe/H] = $-$3.

We used the Salpeter IMF (with $\gamma$ = 1.35) as an input for the massive
star fixed point, which has found reasonable success in past chemical
evolution models for dSphs (Salpeter, 1955). Most notably, Lanfranchi &
Matteucci (2003) explored three distinct IMFs: the Salpeter, a flat Salpeter
($\gamma$ = $-$1.1), and Scalo and found that Salpeter fit the observed
properties of dSphs whereas the flat Salpeter and Scalo overproduced and
underproduced the alpha ratios, respectively. Romano & Starkenburg (2013) used
the Salpeter IMF and Chabrier-like IMF with $\gamma$ = 1.7 for m $>$ M⊙ from
Chabrier (2003). The Salpeter produced reasonable results and modifying the
model to the Chabrier IMF only produced miniscule secondary effects. Vincenzo
et al. (2014) adopted the Salpeter IMF for all their models and also tested an
IMF with a steeper slope but found negligible differences. The Salpeter IMF
has also been successful in describing observations in other studies (Wyse et
al., 1999; Recchi et al., 2001; Kirby et al., 2019).

It must be noted, however, that there is some degree of uncertainty with using
specific IMFs for galaxies other than the MW (Matteucci, 2008). Indeed, some
studies have not had success with the Salpeter IMF. In particular, Calura &
Menci (2009) found better agreement with an IMF slope of $\gamma$ = 1 for
numerous dSphs. Additionally, the three component IMF from Kroupa et al.
(1993) has also been used successfully by other groups (Fenner et al., 2006;
Kirby et al., 2011; Homma & Murayama, 2014).

In this work, we chose the Salpeter IMF due to its reasonable success with
past chemical evolution models, though we recognize that no firm consensus
exists for the best IMF in dSphs. We accept that there may be other IMFs that
provide better results, but currently see no firm consensus about the
preferred IMF of choice for dSphs.

####  2.1.2 Type Ia SNe

The theoretical Type Ia SNe yields and the dominant progenitor must accurately
model dSphs. Kirby et al. (2019) suggested that sub-Chandrasekhar mass white
dwarfs are the dominant Type Ia SNe progenitor in dSphs given the comparison
to [Ni/Fe] observations (only sub-Chandrasekhar predicted observed values of
[Ni/Fe] $<$ $-$0.01). Kirby et al. (2019) compared three sub-Chandrasekhar
models: Leung & Nomoto (2020), Shen et al. (2018), and Bravo et al. (2019).
[Ni/Fe] yields especially matched the sub-Chandrasekhar models of Shen et al.
(2018) and Leung & Nomoto (2020) with masses of 1$-$1.10 M⊙, as they were most
similar to the observed value of [Ni/Fe] $\approx$ $-$0.3 at [Fe/H] = $-$1.5
for Sculptor. We chose Leung & Nomoto (2020) Type Ia SNe yields for our model,
as Figure 6 of Kirby et al. (2019) shows that the standard model of Leung &
Nomoto (2020) fits Sculptor reasonably well for all elemental ratios. We used
their benchmark model 110$-$100$-$2$-$50 (X), corresponding to
M${}_{\text{WD}}$ = 1.1 M⊙, M${}_{\text{He}}$ = 0.1 M⊙, and Z = 0.02. The
existence of the sub-Chandrasekhar progenitor was found by McWilliam et al.
(2018) in the Ursa Minor dwarf galaxy (COS 171), and supported by Theler et
al. (2020) who showed that [Ni/Fe] declined past [Fe/H] $\geq$ $-$2 to
subsolar values, implying that the yield of Ni is lower than Fe in Type Ia
SNe.

Although the choice of the Type Ia progenitor is consistent with Sculptor’s
observational data, it is important to note that Type Ia progenitors likely
vary based on metallicity and SFH. Whereas sub-Chandrasekhar mass white dwarfs
may be the dominant progenitors of Type Ia SNe at low metallicities, near-
Chandrasekhar mass white dwarfs may dominate at higher metallicities.
Furthermore, [Ni/Fe] evolves differently in galaxies with an extended star
formation, such as MW, Fornax, and Leo 1. In such galaxies, it is predicted
that they could host Type Ia SNe that are delayed (Kirby et al., 2019). If
single-degenerate near-Chandrasekhar mass white dwarfs are delayed compared to
sub-Chandrasekhar mass white dwarfs, it is possible that galaxies with a
greater SFH have single-degenerate near-Chandrasekhar mass white dwarfs as
their dominant Type Ia progenitor, despite trends in [Ni/Fe] (see Kirby et
al., 2019 or Toonen et al., 2012 for further discussion).

####  2.1.3 Trans-Iron Isotopes

Elements that are heavier than iron are not formed by charged particle fusion
reactions because iron has the maximum value for the binding energy per
nucleon (Burbidge et al., 1957; Bertulani & Kajino, 2016). Ideally, all three
components of the _s_ -process (ls, hs, and "strong") would be treated
separately in the model. Due to the absence of _s_ -process contributions for
non _s_ -only isotopes in our decomposition of Sculptor’s LSE and due to lack
of observational data, we explicitly model the main _s_ -process only. All
three components of the main _s_ -process (ls, hs, and "strong" components)
are treated together and described as a power law:

$X^{\text{s}}_{i}(\xi)=X^{\text{s}}_{i,\text{LSE}}\cdot\xi^{h}$ (3)

where $X^{\text{s}}_{i}(\xi)$ is the _s_ -process abundance of isotope $i$ as
a function of the model parameter, $X^{\text{s}}_{i,\text{LSE}}$ is the _s_
-process abundance of isotope $i$ at Sculptor’s LSE, and $h$ is a free
parameter that describes $\xi$-dependence for a secondary process later fit to
observed data (see Section 3 and Table 2). Since the approximation in Equation
3 follows from lack of observational data to constrain each component
separately, results from fitting for the ls and "strong" components are
tentative and should be interpreted with care.

The weak _s_ -process should also be modelled separately from the main
components. Unfortunately, there is not a clear way to separate these
different contributions from the LSE composition given by OMEGA, which mixes
them together (see Section 2.2). One option may be to use the solar abundance
composition to approximate the contributions of these processes for all
elements (e.g., assume $\approx 10\%$ of Sr is made from the weak _s_ -process
in the MW, West & Heger, 2013). This method would poorly constrain a proposed
weak _s_ -process parameter since Sr is dominated by main $s$-process
contributions. Moreover, the parameter would only be fit to about $10$ Sr data
points. Furthermore, the elemental Sr data average has a high scatter (-2.2
$\leq$ [Sr/Fe] $\leq$ 0.5). We conclude that any resulting fit would likely
have negligible benefit given these compounded conditions and, therefore, we
choose not to model the weak _s_ -process in this work.

We split the _r_ -process function into two components to model contributions
from CCSNe and NSM. The CCSNe component is described as a power law and the
NSM component is described similar to the Type Ia SNe function. This choice
for NSM follows from an expected delay to contributions consistent with NS
merger events in binary systems (Mennekens & Vanbeveren, 2016). We define $g$
as the fraction that NSM contribute to the LSE abundance of the _r_ -process
isotopes:

$X^{\text{r}}_{i}(\xi,g)=g\cdot X^{\text{NSM}}_{i}(\xi)+(1-g)\cdot
X^{\text{CCSNe}}_{i}(\xi)$ (4)
$X^{\text{CCSNe}}_{i}(\xi)=X^{\text{r}}_{i,\text{LSE}}\cdot\xi^{p}$ (5)
$X^{\text{NSM}}_{i}(\xi)=X^{\text{r}}_{i,\text{LSE}}\cdot\xi\cdot\frac{[\text{tanh}(c\cdot\xi-d)+\text{tanh}(d)]}{[\text{tanh}(c-d)+\text{tanh}(d)]}$
(6)

where Equations 4, 5, and 6 follow the same convention as Equation 3. The
parameters $h$, $g$, $p$, $c$, and $d$ are later fit to data. Finally, the
$\gamma$-process was chosen to have an equal contribution from primary and
secondary seed nuclei:

$X^{\text{$\gamma$}}_{i}(\xi)=X^{\text{$\gamma$}}_{i,\text{LSE}}\cdot\xi^{\frac{{h+p}}{2}+1}$
(7)

(see West & Heger, 2013 for further discussion).

###  2.2 Late Stage Evolution Composition

Ideally, the LSE composition would come from observations. Since a complete
isotopic abundance set is not observable for Sculptor, we use the OMEGA GCE
code (Côté et al., 2017) to estimate the LSE composition. We used an initial
galaxy mass of 7.8x106 M⊙, a ratio between inflow and outflow rate of 1.02, a
ratio between outflow rate and SFR of 8, the number of NSM per stellar mass of
2.0x10-5, a mass ejected from NSM of 1.2x10-1 M⊙, and computed the GCE model
across 10,000 timesteps. The results delineated yields by process for further
analysis. We tested several AGB and massive star theoretical yields (with and
without rotation) to determine which final computed compositions provided the
best fit to Sculptor’s LSE, as well as an intermediary point at [Fe/H] = $-$3
(See Section 2.3). The best fit isotopic abundances were found using inputs
from Limongi & Chieffi (2018) (see Table 1) and a rotational velocity of 300
km s-1, which underproduced and overproduced the Ba LSE composition,
respectively. Thus, we computed a weighted average between the yields to fix
the Ba LSE composition. The AGB and massive star theoretical yields also
include contributions from winds. Additional inputs include the Sculptor SFH
from de Boer et al. (2012a), Type Ia SNe yields from Leung & Nomoto (2020)
(see Section 2.1.2), and a Salpeter IMF (See Section 2.1.1). For the LSE
yields, we extracted isotopic abundances from the GCE model at the timestep
corresponding to [Fe/H] = $-$1.25, which denotes the highest metallicity where
Sculptor data is plentiful.

The LSE composition of the MW and Sculptor and the ratio between them are
given in Figure 1. Both show known characteristic features of nucleosynthesis,
with Sculptor abundances consistently below MW abundances by $\approx$ 1.3 dex
on average. The abundances deviate most before the iron-peak, which is shown
by the larger ratios between A $\sim$ 40 $-$ 50\. While there is less
deviation between the abundances after the iron-peak, there are some notable
exceptions, such as 138La which is 2.3 dex above the MW composition. The weak
_s_ -process plateau is comparatively less defined for Sculptor yields. Note
that the Sculptor yields omit $p$-isotope contributions from photo-
disintegration events, which were not included in the OMEGA model.

Figure 1: Isotopic abundances for the Sun (Lodders, 2019) and the LSE of
Sculptor computed using OMEGA (Côté et al., 2017). The ratios between MW and
Sculptor abundances are also shown.

###  2.3 Impact of Rotation Using OMEGA

The LSE composition used rotating AGB and massive star yields from Limongi &
Chieffi (2018), but the intermediate point at [Fe/H] = $-$3 was found with
non-rotating massive star yields from Heger & Woosley (2010). In this section,
we validate our choice for using these yields with a different rotation
prescription by comparing OMEGA’s output from several rotating and non-
rotating theoretical yields (shown in Table 1) with Sculptor’s observed data
averages. Each theoretical model was used as input in OMEGA, and the
abundances of 14N, 22Ne, Na, Mg, and Ni were extracted from OMEGA at [Fe/H] =
$-$1.25 and [Fe/H] = $-$3\. In addition, [Ba/Fe] and [Eu/Fe] abundances were
also extracted at [Fe/H] = $-$1.25. These theoretical values were compared
with the Sculptor observed average at the same metallicity to compare
theoretical models. The rightmost columns of Figures 2 and 3 show the average
deviation between all theoretical values and the Sculptor observed average for
each model considered.

Figure 2 shows that the rotating models and a majority of the non-rotating
models are able to reproduce observed Sculptor trends for [Na/Fe] and [Mg/Fe],
although most models overproduce [Ni/Fe]. Both rotating and non-rotating
models are generally able to reproduce observed [Eu/Fe]. However, almost every
theoretical model underproduces [Ba/Fe], with only K10$\\_$LC18$\\_$R300
overproducing this value by $\approx$ 1.5 dex. We took the closest model to
the observed [Ba/Fe], K10$\\_$LC18$\\_$Ravg, and performed a weighted average
between it and K10$\\_$LC18$\\_$R300. The resulting AGB/massive star yields
were used as an input to determine the OMEGA LSE Model (LSEM). The yields from
LSEM at [Fe/H] = $-$1.25 determine the LSE isotopic composition.

The LSEM provides the smallest average deviation from Sculptor (Figure 2). It
should be noted, however, that LSEM does not reproduce [Ni/Fe] trends at
[Fe/H] = $-$1.25, although the other models considered do not either. It is
interesting to see that K10$\\_$LC18$\\_$R300 is the only model to accurately
reproduce the observed [Ni/Fe] Sculptor value, which suggests that rotation
may have been an integral part of Sculptor’s LSE. This contradicts previous
studies, however, that show that in the MW, rotating yields best fit the
observed data at the low metallicity region whereas non-rotating yields best
fit the data at the high metallicity region (Chiappini et al., 2006; Rizzuti
et al., 2019).

Figure 2 also shows that 14N and 22Ne are overproduced in rotating models, as
suggested from previous studies (Prantzos et al., 2018). However, since there
is no Sculptor data to compare the results from rotating and non-rotating
OMEGA simulations, it is difficult to discern whether a rotating or non-
rotating yield input is better for these ratios in a model that relies upon
fits to data.

Whereas Figure 2 shows that LSEM is more accurate than other theoretical
models at [Fe/H] = $-$1.25, this does not guarantee that the same model is
accurate at the intermediary fixed point ([Fe/H] = $-$3). Figure 3 compares
the same rotating and non-rotating theoretical models to the Sculptor data
averages of [Na/Fe], [Mg/Fe], and [Ni/Fe] at [Fe/H] = $-$3, along with the
addition of a theoretical non-rotating massive star simulation by Heger &
Woosley (2010). The right-most column shows that HW2010$\\_$Sim is the closest
model to the Sculptor data averages and performs significantly better than the
LSEM. Thus, we perform a massive star and Type Ia scaling at the intermediary
point of [Fe/H] = $-$3 with this model (see Section 2.4). Once again, these
results conflict with Chiappini et al. (2006); Rizzuti et al. (2019), as they
suggest that slow rotating yields best fit MW data at low metallicities,
though it is possible that rotation is different between the MW and dSphs. For
example, Tsujimoto (2006) predicted that massive stars in dSphs have a smaller
rotation than in the MW.

Table 1: Theoretical AGB and massive star theoretical models. All models can be found in the online NuPyCEE package (Côté et al., 2016), with the exception of LSEM. Model Acronym | Massive Star Yields | AGB Star Yields
---|---|---
LSEM | Weighted Averages from models K10$\\_$LC18$\\_$R300 and K10$\\_$LC18$\\_$Ravg | Karakas (2010)
HW2010$\\_$Sim | Heger & Woosley (2010) | N/A
K10$\\_$LC18$\\_$R300 |  Limongi & Chieffi (2018), with a rotational velocity of 300 km s-1 | Karakas (2010)
K10$\\_$LC18$\\_$R150 |  Limongi & Chieffi (2018), with a rotational velocity of 150 km s-1 | Karakas (2010)
K10$\\_$LC18$\\_$Ravg |  Limongi & Chieffi (2018), with an averaged velocity by Prantzos et al. (2018) | Karakas (2010)
K10$\\_$LC18$\\_$R000 |  Limongi & Chieffi (2018), with a rotational velocity of 0 km s-1 | Karakas (2010)
K10$\\_$K06$\\_$0.0HNe |  Kobayashi et al. (2006), with a 0.0 HNe fraction | Karakas (2010)
K10$\\_$K06$\\_$0.5HNe |  Kobayashi et al. (2006), with a 0.5 HNe fraction | Karakas (2010)
K10$\\_$K06$\\_$1.0HNe |  Kobayashi et al. (2006), with a 1.0 HNe fraction | Karakas (2010)
nugrid_FRUITY | Ritter et al. (2018) | Cristallo et al. (2009)
nugrid_K06 | Kobayashi et al. (2006) | Ritter et al. (2018)
nugrid_K10 | Ritter et al. (2018) | Karakas (2010)
nugrid_delay |  Ritter et al. (2018), with delay mass prescription from Fryer et al. (2012) | Ritter et al. (2018)
nugrid_delay_wind |  Ritter et al. (2018), with delay mass prescription and pre-explosion wind from Fryer et al. (2012) | Ritter et al. (2018)
nugrid_mix |  Ritter et al. (2018), with mix of delay and rapid mass prescriptions from Fryer et al. (2012) | Ritter et al. (2018)
nugrid_rapid |  Ritter et al. (2018), with rapid mass prescription from Fryer et al. (2012) | Ritter et al. (2018)
nugrid_ye |  Ritter et al. (2018), with mass-cut prescription based on the electronic fraction from Côté et al. (2016) | Ritter et al. (2018)
Figure 2: A comparison of OMEGA GCE models using rotating and non-rotating
theoretical models (described in Table 1) with the Sculptor observed average
at [Fe/H] = $-$1.25. The rightmost column depicts the average deviation of
each model from Sculptor observed averages for [Na/Fe], [Mg/Fe], [Ni/Fe],
[Ba/Fe], and [Eu/Fe]. Green and red markers indicate models that do and do not
consider rotation, respectively. Figure 3: A comparison of OMEGA GCE models
using rotating and non-rotating theoretical models with the Sculptor observed
average at [Fe/H] = $-$3\. The figure follows the same convention as Figure 2.

###  2.4 Massive and Type Ia SNe Scalings

We assume that both Type Ia SNe and massive stars (which include AGB yields
below the iron-peak) contribute a fraction of each intermediate-mass and iron-
peak isotope abundance at [Fe/H] = $-$3\. We use Heger & Woosley (2010) for
massive star yields and Leung & Nomoto (2020) for the Type Ia SNe yields (see
Sections 2.1.1 and 2.1.2 for specific reasoning behind this choice). We define
a fraction, $f$, to be the contribution of Type Ia SNe to the observed 56Fe
LSE abundance. Thus, we scale each LSE isotope in $12<A<70$ by the factor: $f$
$\cdot$ X${}^{\text{LSE}}_{56}$ $/$ X${}^{\text{Ia}}_{56}$, where
X${}^{\text{LSE}}_{56}$ is the observed LSE abundance of 56Fe in Sculptor and
X${}^{\text{Ia}}_{56}$ is the Type Ia contribution taken from the OMEGA
yields. This shifts the isotopic abundance pattern for Type Ia SNe at [Fe/H] =
$-$3 to fit the yield for 56Fe. The fraction $f$ is a free variable that is
later fit to observational data (see Table 2). The OMEGA LSE massive star
yields were scaled in a similar fashion by a factor (1 $-$ $f$) $\cdot$
X${}^{\text{LSE}}_{56}$ $/$ X${}^{\text{massive}}_{56}$, where
X${}^{\text{LSE}}_{56}$ is the OMEGA LSE abundance of 56Fe in Sculptor and
X${}^{\text{massive}}_{56}$ is the massive contribution. The effect of this
scaling is to normalize the Fe yields for both massive stars and Type Ia SNe
to its observed value.

An additional scaling was performed to ensure that all other isotopic
contributions from both massive stars and Type Ia SNe summed to the LSE
abundance for each isotope between $12\leq A\leq 70$:

$X^{\text{massive}}_{i,f}=\Bigg{(}\frac{\displaystyle
X^{\text{LSE}}_{i}}{\displaystyle
X^{\text{massive}}_{i,0}+X^{\text{a}}_{i,0}}\Bigg{)}X^{\text{massive}}_{i,0}$
(8)

$X^{\text{Ia}}_{i,f}=\Bigg{(}\frac{\displaystyle
X^{\text{LSE}}_{i}}{\displaystyle
X^{\text{massive}}_{i,0}+X^{\text{Ia}}_{i,0}}\Bigg{)}X^{\text{Ia}}_{i,0}$ (9)

where Xi,0 and Xi,f are the initial and scaled abundances of isotope i,
respectively, and $X^{\text{LSE}}_{i}$ is the observed LSE abundance of
isotope i given by OMEGA. Whereas the abundance pattern within each model is
distorted due to the scaling processes, the isotopic ratios are preserved
across each model. For example, the even-odd effect is still apparent, and the
iron-peaks for both massive (Figure 5) and Type Ia (Figure 4) can be seen.

Figure 4: The Type Ia yields from Leung & Nomoto, 2020 scaled to a nominal
value of 56Fe${}_{\text{Ia}}$/56Fe${}_{\text{LSE}}$ = 0.85. Shown are the
yields before (Stars) and after (Circles) fitting against massive star yields.
Figure 5: The Massive star yields from Heger & Woosley, 2010 fitted to
observed Sculptor data and scaled to a nominal value of
56Fe${}_{\text{massive}}$/56Fe${}_{\text{LSE}}$ = 0.15. Shown are the yields
before (Stars) and after (Circles) fitting against Type Ia yields.

##  3 Fitting Scaling Model to Observational Data

The isotopic scaling functions were summed into elemental scaling functions
and fit to Sculptor stellar abundance data to tune the free parameters. We
used data from Kirby et al. (2009) for $\alpha$ elements (Mg, Si, Ca, and Ti)
and Fe abundances for almost 400 stars measured using Keck/DEIMOS medium-
resolution spectra. We used data from Tafelmeyer et al. (2010) for iron-peak,
neutron-capture, and $\alpha$ elements for two metal-poor stars using high
spectroscopic resolution with VLT/UVES. We used data from Starkenburg et al.
(2013) for $\alpha$ and iron-peak elements for seven low-metallicity stars
using ESO VLT. We used data from Hill et al. (2019) for $\alpha$ elements (O,
Mg, Si, Ca, Ti), iron-peak elements (Sc, Cr, Fe, Co, Ni, Zn), and neutron-
capture elements (Ba, La, Nd, Eu) for 99 red-giant branch stars using high-
resolution VLT/FLAMES spectroscopy. We used data from Skúladóttir et al.
(2017) for Zn abundances for approximately 100 red giant branch stars using
ESO VLT/FLAMES/GIRAFFE spectra. We used data from Kirby et al. (2018) for
iron-peak elements (Cr, Co, Ni) for approximately 300 red giant stars using
Keck/DEIMOS medium-resolution spectra. We used data from Duggan et al. (2018)
for [Ba/Fe] for 120 red giant branch stars in Sculptor using medium-resolution
spectroscopy. We used data from Skúladóttir et al. (2019) for neutron-capture
elements Y, Ba, La, Nd, and Eu for 98 stars using ESO VLT/FLAMES spectre. We
used data from Jablonka et al. (2015) for iron-peak, neutron-capture, and
$\alpha$ elements for 5 metal-poor stars using the medium-resolution CaT from
Battaglia et al. (2008). All data is given in units of [X/Fe].

The fitting procedure was implemented as follows: the [Fe/H] axis was split
into 100 bins in the range $-$5 to $-$1\. Similarly, the [X/Fe] axis was split
into 100 bins in the range of the elemental data. Every data point was
assigned a Gaussian distribution using their respective observational
uncertainties:

$f_{i}=\text{exp}\Bigg{[}-0.5\cdot\Bigg{(}\frac{x_{i}-x_{0}}{\sigma_{x}}\Bigg{)}^{2}+-0.5\cdot\Bigg{(}\frac{y_{i}-y_{0}}{\sigma_{y}}\Bigg{)}^{2}\Bigg{]}$
(10)

where $f_{i}$ is the value of the distribution at bin ($x_{i}$, $y_{i}$) on
the [Fe/H] and [X/Fe] axes, $x_{0}$ and $y_{0}$ are the data point values, and
$\sigma_{x}$ and $\sigma_{y}$ are the associated uncertainties for each
$x_{0}$ and $y_{0}$. The distribution values were then averaged for each bin
and a standard deviation was found. A differential evolution algorithm (Storn
& Price, 1997) was implemented to minimize the $\chi^{2}$ between the average
observed curve and the functional curve while preserving the expected chemical
trends to avoid over-fitting. The free parameter values that resulted in the
smallest $\chi^{2}$ values for Mg, Ba, Eu, and Sr are shown in Table 2.

###  3.1 Type Ia Parameters

The Type Ia SNe parameters $a$, $b$, and the fraction of 56Fe${}_{\text{LSE}}$
attributed to Type Ia SNe, $f$, were found by using [Mg/Fe] due to the
prevalence of data available. Type Ia SNe and massive stars both contribute
significantly to the evolution of Mg and Fe (Heger & Woosley, 2010; Kirby et
al., 2019). The parameters were chosen to minimize the $\chi^{2}$ between the
[Mg/Fe] model and the observed [Mg/Fe] data averages. The best fitting
parameters were found to be $a$ = 1.628, $b$ = 1.073, and $f$ = 0.863 with a
$\chi^{2}$ of 0.359. The resulting model for [Mg/Fe] is shown in Figure 6,
along with data and the OMEGA evolution for comparison.

The model curve tracks the data average. At metallicities below Type Ia onset
([Fe/H] $<\approx$ $-$2.5), magnesium and iron scale together since only
massive stars contribute to their production. Here, the model line sits about
0.3 dex above the average line. This discrepancy is a consequence of the fixed
point obtained at [Fe/H] = $-$3 using Heger & Woosley (2010) (see Section
2.1.1). Whereas the OMEGA GCE evolution fits marginally closer to the data at
this metallicity, using it as the fixed point instead would drive the model
below the data average at higher metallicities. We choose to prioritize fits
to higher metallicities, where data is more plentiful. We also note the
paucity of data below [Fe/H] = $-$2.5. It is reasonable to predict that the
[Mg/Fe] trend would plateau in this regime, given their production mechanism,
though the value of that plateau is poorly constrained. Indeed, the model may
not be a good representation of the average chemical evolution as
metallicities decrease substantially below [Fe/H] $\approx$ $-$3\. The
chemical evolution may have been more stochastic in this regime, as some
studies have explored for the MW halo (Welsh et al., 2020).

The drop of the curve to LSE values at [Fe/H] $\approx$ $-$2.5 is caused by
the Type Ia onset which coincides with iron production from Type Ia SNe and
drives the ratio down to its late stage value. The model line mirrors the
average line after the drop up to [Fe/H] = $-$1.25, and the slopes of both
curves are similar during the onset.

Although the data average plateaus after LSE values ([Fe/H] $>$ $-$1.25), the
model does not. There is no metal-rich data available above [Fe/H] $\approx$
$-$1 to constrain [Mg/Fe], so caution should be used with the model in this
regime. If Sculptor continued to have active star formation, then the model
might capture an accurate data average trend for some finite range above
[Fe/H] = $-$1, as CCSNe continued to expel material that escapes the gravity
well, while Type Ia SNe injects the ISM with a lower [Mg/Fe] ratio.

Figure 6: The resulting model for [Mg/Fe] determined after finding the best-
fitting parameters. The dark shadow background shows the Gaussian
distributions of stellar data according to the uncertainties. The solid black
line gives the average curve, with the dotted black lines showing the standard
deviation curves. The model line is shown by the solid cyan line, and the
OMEGA GCE model is shown by the solid red line. The cyan background behind the
model line covers the range of possible model curves across the complete
parameter space for $a$, $b$, and $f$. Data points are taken from Kirby et
al., 2009 (green), Tafelmeyer et al., 2010 (yellow), Starkenburg et al., 2013
(brown), Jablonka et al., 2015 (pink), Kirby et al., 2018 (purple),
Skúladóttir et al., 2019 (maroon), and Hill et al., 2019 (blue).

###  3.2 _r_ -process and _s_ -process Parameters

Four parameters are required to describe the _r_ -process scaling: the primary
process exponent ($p$), the fraction that NSM contribute to _r_ -process
isotopes ($g$), as well as the scaling and shifting factors of the tanh
function to describe contributions from NSM ($c$ and $d$, respectively). Eu
was chosen for fitting these parameters as it is an _r_ -process peak element
with two isotopes, 151Eu and 153Eu, which both have dominant (85 per cent of
its solar value, West & Heger, 2013) contributions from the _r_ -process. The
best fitting parameter values were found to be $p$ = 0.391, $g$ = 0.303, $c$ =
5.343, and $d$ = 0.450 with $a$ $\chi^{2}$ = 0.022. The previously found best
fitting values for $a$, $b$, and $f$ (from Section 3.1) were used for Fe.
Whereas a nominal value of $h$ = 1.5 was used for the _s_ -process parameter
in West & Heger (2013), the LSE of Sculptor computed by OMEGA gives no _s_
-process contributions for Eu. The best fitting functional curve and average
curve for [Eu/Fe] are shown in Figure 7.

To determine the best value for the _s_ -process parameter $h$, Ba was chosen
as it has two s-only isotopes, 134Ba and 136Ba, along with three isotopes,
135Ba, 137Ba, and 138Ba, which have contributions from both the _s_ -process
and _r_ -process (West & Heger, 2013). The $\gamma$-process contributions from
the $p$ isotopes 130Ba and 132Ba were not considered for this fit, and are
negligible at all metallicities (West & Heger, 2013). The best fitting
parameter value was found to be $h$ = 1.798 with $a$ $\chi^{2}$ = 0.448. The
previously found best fitting values for $a$, $b$, $f$ were used for Fe, and
those for $p$, $g$, $c$, and $d$ were used for the _r_ -process contributions
to Ba. The best fitting model for [Ba/Fe] is shown in Figure 8.

Figure 7: The resulting model for [Eu/Fe] determined after finding the best-
fitting parameters. The figure follows the same convention as Figure 6, with
the addition of the CCSNe (green dashed line) and NSM (purple dashed line)
contributions to the model shown separately. The cyan background behind the
model line covers the range of possible model curves across the complete
parameter space for $c$, $d$, $p$, and $g$, and the darker purple hue shows
the subset of this range that is constrained by the parameter $g$. Plotted
data is from Skúladóttir et al., 2019 (maroon) and Hill et al., 2019 (blue).
Figure 8: The resulting model for [Eu/Fe] determined after finding the best-
fitting parameters. The cyan background behind the model line covers the range
of possible model curves across the parameter space for $h$. The figure
follows the same convention as Figure 6, with the addition of the _s_ -process
(green dashed line) and _r_ -process (purple dashed line) contributions to the
model shown separately. Plotted data is from Starkenburg et al., 2013 (brown),
Jablonka et al., 2015 (pink), Duggan et al., 2018 (orange), Skúladóttir et
al., 2019 (maroon), and Hill et al., 2019 (blue).

##  4 Results and Discussion

The complete elemental scaling model is given in Figure 9. Whereas the model
only considers s-process contributions to the s-only isotopes, s-process peaks
for Z $\approx$ 38, Z $\approx$ 56, and Z $\approx$ 82 (corresponding to Sr,
Ba, and Pb, respectively) can be seen. The odd-even effect can be seen below Z
$\approx$ 10, but this trend becomes less distinct at higher atomic numbers
(10 $\leq$ Z $\leq$ 30) compared to the MW (see Figure 9 of West & Heger
(2013)). Furthermore, the iron-peak seems to be less defined than the peak
observed in Figure 9 of West & Heger (2013). These two differences in these
trends occur due to the Type Ia progenitor used in this model, as theoretical
yields from sub-Chandrasekhar models tend to have more $\alpha$ element and
lower iron-peak abundances compared to deflagration models (see, for e.g.,
Nomoto et al., 1997, and Figure 6 of Kirby et al., 2019). The scalings for
H/He isotopes are approximate, since the LSE was computed using estimated
parameters which are reasonably chosen yet not perfectly known. Given the gas
exchange between dSphs and their environments, H and He may inherit larger
uncertainties from the chose parameterizations in OMEGA. Figure 10 shows all
the unique scaling functions against O considered in this work using the best-
fitting free parameters determined in Section 3.

Figure 9: The complete elemental scaling of the model. The abundances are
given relative to their MW solar values. The model only considers _s_ -process
contributions to the _s_ -process peak elements.  Figure 10: The scaling
functions of the model contributions relative to oxygen as functions of
metallicity. The “Massive” line shows the scaling of the massive category’s
contribution to 56Fe, and is normalized to the LSE contribution from this
category only (as in West & Heger, 2013). The total _r_ -process is shown in
addition to its modeled contributions from neutron star mergers and core-
collapse supernovae. The Type Ia onset metallicity, [Fe/H] = $-2.46$, is
identified by where the Type Ia scaling function drops to $1$ per cent its
initial value (blue dotted line).

The best-fitting free parameters offer useful comparisons between the chemical
evolution of the Sculptor dSph galaxy and the MW. As mentioned in Section 1,
iron production begins to dominate at an observed metallicity of [Fe/H]
$\approx-1$ in the MW. The value of the free parameter $b$, the shifting
factor for the Type Ia tanh function, is less for Sculptor ($b$ = 1.073) than
for the MW ($b$ = 2.722, West & Heger, 2013). This suggests that the Type Ia
onset occurs at a lower metallicity in dSphs than the MW and supports the
conclusions of past chemical evolution models. We define the onset to occur
when the Type Ia abundance drops to 1 per cent of its LSE value (in congruence
with West & Heger, 2013). We find the onset to occur at [Fe/H] = $-$2.46,
which is at a lower metallicity than the range $-$1.5 $<$ [Fe/H] $<$ $-$2.0
suggested by previous studies (eg. Fenner & Gibson, 2003 and Reichert et al.,
2020). The onset value occurs at a lower metallicity than the observed
metallicity of [Fe/H] $\approx-1$ in the MW (Smecker-Hane & Wyse, 1992; Venn
et al., 2012; Kobayashi et al., 2011, 2020), as well as the calculated value
of [Fe/H] = $-$1.1 in West & Heger (2013).

Our model finds that Type Ia SNe contribute more to the chemical evolution of
iron peak isotopes in dSph galaxies than the MW compared to massive star
contributions, in agreement with Lanfranchi & Matteucci (2003), Vincenzo et
al. (2014), and Kirby et al. (2019). In particular, we find the fraction of
LSE Fe from Type Ia SNe to be 0.86, which is greater than the MW (f = 0.69,
West & Heger, 2013). This is close to the value from Kirby et al. (2019) who
found f${}_{\text{Ia}}$ $\approx$ 0.8 during Sculptor’s LSE at [Fe/H] =
$-$1.25 (see Figure 1 of Kirby et al., 2019).

The decrease of the [$\alpha$/Fe] ratio after Type Ia onset has been predicted
to be steeper in dSphs than in the MW (Recchi et al., 2001; Matteucci, 2008;
Vincenzo et al., 2014; Theler et al., 2020). The onset slope in this model is
contained in the parameter $a$, the scaling factor for the Type Ia function.
We find a fit value ($a$ = 1.628) that is less than this parameter’s value for
the MW ($a$ = 5.024, West & Heger, 2013). This suggests the impact of Type Ia
SNe in driving the [Mg/Fe] ratio down to its LSE value occurred over a shorter
metallicity range in Sculptor. There is dispute over the existence of
[$\alpha$/Fe] low metallicity plateaus, however, so this conclusion should be
handled with care. Kirby et al. (2011) analyzed [$\alpha$/Fe] knees for
numerous dSphs and only found significant knees for the [Ca/Fe] ratio in
Sculptor and Ursa Minor but not for other $\alpha$-element ratios, such as
[Mg/Fe] or [Si/Fe]. An absence of low metallicity plateaus in the
[$\alpha$/Fe] ratios would suggest that Type Ia SNe were dominant mechanisms
throughout the entirety of the SFHs in dSphs (Kirby et al., 2011).
Clarification on this issue requires additional low metallicity data ([Fe/H]
$<-2.5$). Present data, though sparse, suggests a [Mg/Fe] average that indeed
plateaus, yet the metallicity which demarcates the [$\alpha$/Fe] knee is also
where the transition from sparse to more plentiful data occurs, which is
concerning.

Our results for the high metallicity behaviour of [Mg/Fe] are similar with
Kirby et al. (2011). They observed that the slope of [Mg/Fe] at [Fe/H] $>$
$-$1 is less than the slope at [Fe/H] $<$ $-$1.5, which suggests some
flattening of these ratios at higher metallicities. Our model shows that the
[Mg/Fe] slope decreases beyond [Fe/H] = $-$1.25.

Kirby et al. (2019) modelled the Type Ia onset ([Fe/H]${}_{\text{Ia}}$) for
various dSphs and determined a constant value of the ratio of an element
relative to Fe before the onset ([X/Fe]${}_{\text{CC}}$) using the Markov
Chain Monte Carlo method. They fixed the initial value of
[Fe/H]${}_{\text{Ia}}$ to $-$2 and calculated an angle and perpendicular
offset for each element X using a linear fit for [X/Fe] vs [Fe/H]. For
Sculptor, they calculated [Fe/H]${}_{\text{Ia}}$ = $-$2.12, which is 0.4 dex
above our calculated onset value of [Fe/H]${}_{\text{Ia}}$ = $-$2.5 relative
to the solar abundance decomposition. They also calculated
[Mg/Fe]${}_{\text{CC}}$ to be +0.56 $\pm$ 0.01, which is similar with our
model’s [Mg/Fe]${}_{\text{CC}}$ value of approximately 0.5 determined by OMEGA
(Figure 6). Our model’s [Mg/Fe]${}_{\text{CC}}$ value of 0.5 is also less than
the [Mg/Fe]${}_{\text{CC}}$ value for the MW which plateaus at approximately
0.57 (West & Heger, 2013). This supports the idea that a system with a low
[Mg/Fe]${}_{\text{CC}}$ value at low metallicities undergoes a slow, extended
SF (Fenner et al., 2006), since Sculptor’s SFH took place for an extended
period of time from 14 gyr to 7 gyr ago during a single formation event and
gradually decreased afterwards (de Boer et al., 2012a). Kirby et al. (2011)
also observed that the average value of [$\alpha$/Fe] drops from +0.4 at
[Fe/H] = $-$2.5 to 0.0 at [Fe/H] = $-$1.2. Our model also shows this trend,
but over predicts the [$\alpha$/Fe] value by about 1 dex at low metallicities.
Similarly, Mashonkina et al. (2017) noted that dSphs and the MW exhibit a
plateau of [$\alpha$/Fe] $\approx$ 0.3 at [Fe/H] $\leq$ $-$2.5, which is 2 dex
below our model’s plateau at the same metallicity range.

Figure 6 shows that the best-fitting model curve is at the edge of the
parameter space (shown as the cyan hue in the background). This is likely
because the Mg LSE value is below the data average by $\approx$ 0.15 dex,
which results in a parameter space that is slightly below the data average
from $-$2 $\leq$ [Fe/H] $\leq$ $-$1.25. Since the parameters $a$ and $b$ most
greatly influence the Mg model at this region, these parameters were shifted
to the top of the parameter space by the machine learning algorithm to best
match the data average. It is possible that our previous comparisons between
the best-fit values of $a$ and $b$ for Sculptor and the MW may change
significantly if we had a Mg LSE value that more closely matched the data
average. To test for such an effect, we recomputed the best-fit values for $a$
and $b$ assuming the LSE model gives the observed [Mg/Fe] value at [Fe/H] =
$-1.25$ and found that $a$ = 3.059 and $b$ = 0.898 (compared to $a=1.628$ and
$b=1.073$). The higher value of $a$ suggests a shallower slope for Type Ia,
and the lower value of $b$ suggests a lower metallicity for Type Ia onset.
Nevertheless, both sets of values are lower than the corresponding MW values,
hence the previous comparisons between the test for Sculptor and the MW models
hold.

The model predicts that [Ba/Fe] $\approx$ 0.1 at [Fe/H] = $-$1, which is above
the MW value by 0.4 dex. This supports the conclusions of previous studies
that AGB stars are the most likely sites for _s_ -process production in
Sculptor due to the large [Ba/Fe] ratios at high metallicities (Fenner et al.,
2006; Tolstoy et al., 2009; Cohen & Huang, 2009; Venn et al., 2012;
Skúladóttir et al., 2019; Skúladóttir et al., 2020; Theler et al., 2020;
Reichert et al., 2020). Whereas the model for Ba from West & Heger (2013) has
a value of [Ba/Fe] $\approx$ 0.2 at [Fe/H] = $-$1, which somewhat contradicts
this claim, their model slightly decreases after [Fe/H] $\geq$ $-$1, while our
model shows no such trend. Thus, our model does indeed predict higher [Ba/Fe]
values, only after [Fe/H] = $-$1.

The transition from _r_ -process to _s_ -process dominated contributions can
be identified by the intersection between the _s_ -process and _r_ -process
model lines. We find the intersection for [Ba/Fe] to occur at [Fe/H] $\approx$
$-$2.22, which is a bit lower than the value [Fe/H] = $-$2 suggested by
previous studies (Tolstoy et al., 2009; Venn et al., 2012; Skúladóttir et al.,
2019) and [Fe/H] = $-$1.57 suggested by Reichert et al. (2020). The difference
likely arises due to the different numerical methodologies in our work
compared or different definitions of the _r_ -process to _s_ -process
transitions. For example, Reichert et al. (2020) performed a least squares fit
on [Ba/H] to [Fe/H] and found that [Eu/Ba] decreased at a metallicity of
$-$0.57 for Sagittarius, $-$1.04 for Fornax, and $-$ 1.57 for Sculptor. This
suggests that more massive galaxies may have an _s_ -process onset that occurs
later than less massive galaxies. For the MW, West & Heger (2013) found this
transition to occur at [Fe/H] $\approx$ $-$1.9, which is consistent with
Reichert et al. (2020) and Cohen & Huang (2009) who suggested that the MW
should have an _s_ -process transition that is later than that of dSphs.

Previous studies have also suggested that AGB stars began dominating the
chemical evolution of dSphs on the same timescales as Type Ia SNe. Our model
finds a similar trend; the _s_ -process to _r_ -process transition ([Fe/H] =
$-$2.22) occurs near (though above) the Type Ia onset ([Fe/H] = $-$2.46)
relative to the solar abundance pattern (Lodders, 2019). They also seem to be
co-produced (to within 0.2 dex) after [Fe/H] $\approx$ $-$1.75 from Figure 10.
Another indication that AGB stars and Type Ia SNe dominate at similar
timescales is a flat [Ba/Fe] ratio at high metallicities (Venn et al., 2012;
Skúladóttir et al., 2020). The Ba model shows a flat ratio between $-2<$
[Fe/H] $<-1$ and increases after [Fe/H] $>$ $-$1, suggesting that AGB
enrichment dominates over Type Ia SNe enrichment in Sculptor at high
metallicities (see also Figure 8), though care should be taken since high
metallicity data is sparse. We also find that the main _s_ -process in AGB
stars have a somewhat stronger metallicity dependence in Sculptor than the MW,
as the best-fitting $h$-value of Sculptor ($h$ = 1.798) is greater than the
best-fitting $h$-value of the MW ($h$ = 1.509, West & Heger, 2013).

The [Eu/Fe] model ratio is above zero at low metallicities and seems to
plateau at a value of [Eu/Fe] $\approx$ 0.7. This is similar to the value of
[Eu/Fe] $\approx$ 0.8 for the MW (West & Heger, 2013), which supports previous
studies suggesting similar trends in both galaxies (Skúladóttir et al., 2020).
The [Eu/Fe] ratio begins to decline at [Fe/H] $\gtrsim$ $-$2.5, due to the
Type Ia onset.

Surprisingly, the primary process exponent parameter, $p$, was found to be
much lower ($p$ = 0.391) than the MW value ($p$ = 0.938, West & Heger, 2013).
A lower $p$-value corresponds to a larger decrease in the [Eu/Fe] ratio as
[Fe/H] increases beyond Type Ia onset. The value of the parameter $g$, the
fraction that NSM contributes to _r_ -process isotopes, suggests that NSM only
contributes about 30 per cent to the LSE of Eu, which agrees with previous
studies that propose CCSNe, rather than NSM, are the dominant _r_ -process
progenitors in dSphs (Tolstoy et al., 2009; Duggan et al., 2018; Skúladóttir
et al., 2019; Reichert et al., 2020; Skúladóttir et al., 2020).

Figure 11 and Figure 12 show the model for [Na/Fe] and [Ni/Fe], respectively.
Our model is in reasonable agreement to observed Na trends, with [Na/Fe]
increasing at low metallicities ([Fe/H] $<$ $-$2.5) and decreasing at higher
metallicities as suggested by Cohen & Huang (2009) and Tolstoy et al. (2009).
Pauce Na data, however, makes a detailed comparison difficult. Whereas the
model for Ni is similar to observed trends at low metallicities ([Fe/H] $\leq$
$-$3), it appears overproduced at higher metallicities ([Fe/H] $>$ $-$2). Both
Na and Ni models seem to perform at least as well as OMEGA in representing the
data averages.

Figure 11: The resulting model for [Na/Fe] determined after fitting
parameters. The figure follows the same convention as Figure 6. Plotted data
is from Tafelmeyer et al., 2010 (yellow), Starkenburg et al., 2013 (brown),
Jablonka et al., 2015 (pink), and Hill et al., 2019 (blue). Figure 12: The
resulting models for [Ni/Fe] determined after fitting parameters. Both sub-
Chandrasekhar (subC) and near-Chandrasekhar (nearC) model curves are shown.
The figure follows the same convention as Figure 6. Plotted data is from
Starkenburg et al., 2013 (brown), Jablonka et al., 2015 (pink), Tafelmeyer et
al., 2010 (yellow), Kirby et al., 2018 (purple), and Hill et al., 2019 (blue).

Nissen & Schuster (1997) noticed a Na/Ni correlation for MW halo stars, and
this correlation was also suggested to occur in dSphs by Tolstoy et al.
(2009). Figure 13 shows this correlation in Sculptor. Whereas Nissen &
Schuster (1997) determined the best-fitting line to be [Na/Fe] =
2.577$\cdot$[Ni/Fe] + 0.116 for the MW, we find the best-fitting line to be
[Na/Fe] = 3.663$\cdot$[Ni/Fe] + 0.670 during the early evolution of Sculptor
($-$0.23 $\leq$ [Ni/Fe] $\leq$ $-$0.075) with a $\chi^{2}$ = 0.002. We
constrain this fit to [Fe/H] $\leq$ $-$2.5 because after this point, the model
overproduces [Ni/Fe], as seen in the sharp downward trend in Figure 13 and
sharp upward trend in Figure 12. The larger slope for our best-fitting line
suggests that at low metallicities, Na and Ni were co-produced but Na was
produced at a higher rate in Sculptor than in the MW.

Figure 13: The best-fitting line for the Na-Ni correlation in Sculptor
compared to the the best-fitting line in the MW, determined by Nissen &
Schuster, 1997. The Sculptor best-fitting line was constrained to [Fe/H]
$\leq$ $-$2.5 ($-$0.23 $\leq$ [Ni/Fe] $\leq$ $-$0.075).

The overproduction of Ni at higher metallicities shown in Figure 12 is likely
due to the discrepancy between the observed [Ni/Fe] data average and the LSE
[Ni/Fe] value from OMEGA at [Fe/H] = $-$1.25, a difference of $\approx$ 0.5
dex. We find [Ni/Fe] $\approx$ 0.2 at [Fe/H] = $-$1.5, which is consistent
with Kirby et al. (2019) who found that the sub-Chandrasekhar yields by Leung
& Nomoto (2020) resulted in [Ni/Fe] $\approx$ 0.1 at [Fe/H] = $-$1.5. To see
if a different Type Ia progenitor improved our model for Ni, we also tested a
near-Chandrasekhar model with a deflagration-detonation transition (Leung &
Nomoto, 2018). Using this Type Ia progenitor, the [Ni/Fe] was _more_
overproduced ([Ni/Fe] $\approx$ 1) at [Fe/H] = $-$1.5. Although neither of
these Type Ia progenitors was able to reproduce observed [Ni/Fe] trends at
higher metallicities, the sub-Chandrasekhar yields produced better results
which agrees with Kirby et al. (2019).

The model for [C/Fe] is vastly overproduced at all metallicities. [Sc/Fe] fits
well with observed data, similar to [Na/Fe]. The model for [O/Fe] has a
similar shape with observed O abundances, but overproduces them by about 0.4
dex from $-$3 $<$ [Fe/H] $<$ $-$1\. [Si/Fe] shows similar trends, but is
overproduced by about 0.3 dex in the same metallicity range.

The model for iron-peak elements can also be evaluated. Co is the best fitting
iron-peak element compared to observed data throughout all metallicities.
[Co/Fe] slightly decreases at [Fe/H] $>$ $-$2.5, which was noted previously by
Cohen & Huang (2009, 2010); Kirby et al. (2019). This behaviour supports the
idea that Type Ia SNe are dominant in dSphs at higher metallicities, which is
why Fe production begins dominating the production of other iron-peak elements
(Kirby et al., 2019). The model overproduces [Cr/Fe] by about 0.75 dex at low
metallicities but is only slightly overproduced at higher metallicities by
about 0.2 dex ([Fe/H] $>$ $-$2.5). Both [Cr/Fe] and [Co/Fe] seem to plateau at
a value of 0 at higher metallicities, suggesting that Type Ia SNe produce Fe
and other iron-peak elements in roughly equal amounts. The model for [Mn/Fe]
slopes downward at all metallicities, which is the opposite trend than is
observed. However, Mn data is sparse and is only available at low
metallicities ($-$4 $\leq$ [Fe/H] $\leq$ $-$2.25). North et al. (2012) also
predicted Mn to be subsolar. The model for [Zn/Fe] is similar to Mn; it
decreases at all metallicities, as suggested by Cohen & Huang (2009);
Skúladóttir et al. (2017); Kirby et al. (2019).

It is useful to compare the effects of using rotating and non-rotating yields
on the [N/Fe] model at the intermediary point [Fe/H] = $-$3\. Figure 14 shows
the model and the OMEGA LSEM for N. Below [Fe/H] $<-3.3$, OMEGA produces more
N. Since the model’s intermediary point uses non-rotating yields and OMEGA
uses rotating yields, this trend is consistent with previous studies that
suggest that massive stars with faster rotation produce more N (Chiappini et
al., 2006). These studies have also shown that rotating yields for N are
strongly dependent on metallicity below [Z] $\leq$ $-$5, though our model is
not constrained by data at such low metallicities and hence does not seek to
describe an average chemical history there (if indeed an average is even well-
defined; see West & Heger, 2013 for further discussion). At [Fe/H] = $-$3, the
model gives [N/Fe] = 0.25 and OMEGA gives [N/Fe] = 0.0. These values suggest
that at the intermediary point, the choice of using non-rotating yields for
the model moderately impacts the production of N, although this choice
significantly impacts the production of other elements such as Na, Mg, and Ni
(see Figure 3). Figure 14 shows that below [Fe/H] = $-2.5$, the model line is
sloped due to the suggested secondary nature of 14N for non-rotating yields,
and the OMEGA curve is constant due to the suggested primary nature for
rotating yields (Prantzos et al., 2018). Whereas there is no N data to
determine which curve is more accurate, the discrepancies between the two
curves due to rotating and non-rotating yields suggests that the N model may
not be a good representation of the evolution of N in Sculptor. OMEGA is
likely more accurate in the low metallicity regime, where rotating stars
should increase the production of N (Chiappini et al., 2006). We choose to use
non-rotating yields from Heger & Woosley (2010) for the intermediary fixed
point, however, because it better models Na, Mg, and Ni (see Figure 3), and we
accept that the model may not be accurate for N.

Figure 14: The resulting model for [N/Fe] using the best-fitting parameters
found in Section 3. The model line is shown by the solid cyan line, and the
OMEGA GCE model is shown by the solid red line.

Finally, we compare our model to a traditional GCE model (see Figures 6, 7, 8,
11, 12). The GCE model used for comparison is the one also used to provide the
LSE abundance for Sculptor, hence our model values agree with OMEGA’s at
[Fe/H] $=-1.25$. Our Mg/Fe model tracks the data average more consistently
than OMEGA. Our Ba/Fe model and OMEGA both track the observed Ba trends,
though OMEGA likely underproduces the ratio at higher metallicities ([Fe/H]
$>$ $-$1.25), although the data is sparse. OMEGA does not fit the Eu/Fe ratio
at low metallicities, whereas our model performs significantly better in this
regime. Our model for Na/Fe roughly captures the observed average of Na,
though data is quite sparse. OMEGA shows a similar trend to our model at
higher metallicities ([Fe/H] $>$ $-$1.25), and it is difficult to assess from
available data which performs better. If Na is dominated by massive stars,
then one would expect the ratio against Fe to be above zero. It is not clear
why a [Na/Fe] ratio would be a decreasing function prior to Type Ia onset.
Both our model for Ni/Fe and OMEGA fail to reproduce observed Ni trends, as
both overestimate [Ni/Fe] at higher metallicities ([Fe/H] $>$ $-$2.5).
Finally, even though there is no N data to determine whether our N/Fe model or
OMEGA is more accurate, it is expected that OMEGA performs better, especially
in the low-metallicity regime ([Fe/H] $\leq$ $-2.5$). This is due to our model
using non-rotating yields for the intermediary fixed point which likely causes
an underproduction of N.

Table 2: Optimized Parameter Values Parameter | Best-fitting Value | Description
---|---|---
$a$ | 1.628 | Type Ia tanh scaling factor
$b$ | 1.073 | Type Ia tanh shifting factor
$f$ | 0.863 | Fraction of 56Fe${}_{\text{LSE}}$ from Type Ia
$p$ | 0.391 | Primary process exponent
$h$ | 1.798 |  _hs_ -process exponent
$c$ | 5.343 | NSM tanh scaling factor
$d$ | 0.450 | NSM tanh shifting factor
$g$ | 0.303 | Fraction of Eu${}_{\text{LSE}}$ from NSM

##  5 Conclusions

A model has been constructed that describes the average isotopic decomposition
of the Sculptor dwarf spheroidal galaxy for contributions from massive stars,
Type Ia SNe, main _s_ -process peak, and the _r_ -process, and the procedure
follows the methodology described in West & Heger (2013). Astrophysical
processes that dictate the chemical evolution of galaxies were discussed and
compared between the MW and dSphs. Parametric equations were assigned to these
processes and scaled between BBN and the Sculptor LSE decomposition, which was
computed using OMEGA at [Fe/H] = $-$1.25 (Côté et al., 2017). The isotopic
scalings were summed to elemental scalings and compared with Sculptor data to
fix the free parameters of the model. The result is a complete isotopic
abundance decomposition of Sculptor at any desired metallicity for the
processes considered.

While many assumptions and approximations were made when constructing the
model, this work still offers many comparisons between dSphs and spiral
galaxies, with Sculptor and the MW as candidates:

  * •

The Type Ia onset occurs at a lower metallicity in Sculptor than the MW.
Numerically, we define the onset to occur when the Mg abundance from Type Ia
SNe is 1 per cent of its Type Ia late-stage evolution value. This occurs at
[Fe/H] = $-$2.46, which is less than the observed Type Ia onset of the MW
([Fe/H] $\approx$ $-$1).

  * •

The model predicts that Type Ia SNe contribute $\approx$ 86 per cent to the
56Fe LSE abundance, which is greater than predicted values of $\approx$ 70 per
cent in the MW (West & Heger, 2013). Thus, Type Ia SNe contribute more to the
chemical evolution of dSphs compared to the MW, in agreement with past
chemical evolution models.

  * •

The slope of the [$\alpha$/Fe] ratio after the Type Ia onset is steeper in
Sculptor than the MW, suggesting that the impact of Type Ia SNe occurred over
a shorter metallicity range.

  * •

The model predicts that sub-Chandrasekhar Type Ia SNe are the dominant Type Ia
progenitor in Sculptor, in agreement with Kirby et al. (2019).

  * •

The _r_ -process to _s_ -process transition occurs at [Fe/H] $\approx$
$-$2.22, which disagrees with previous studies that suggest ranges from
$-$1.57 $<$ [Fe/H] $<$ $-$2.0. This metallicity is less than the MW transition
of [Fe/H] $\approx$ $-$1.9 (West & Heger, 2013), which is consistent with the
idea that a more massive galaxy should have a later transition.

  * •

The model suggests that AGB stars are more dominant in Sculptor compared to
the MW and are the most likely sites for _s_ -process production. AGB
enrichment also seems to occur at similar timescales as Type Ia SNe since the
_r_ -process to _s_ -process transition ([Fe/H] = $-$2.22) occurs at the same
time as the Type Ia onset ([Fe/H] = $-$2.5), as suggested by previous studies.
However, a [Ba/Fe] plateau is not observed at high metallicities since AGB
contribution is more pronounced than Type Ia SNe at this range.

  * •

The _r_ -process evolution in Sculptor appears to be similar to the evolution
in the MW, with similar [Eu/Fe] plateaus at low metallicities and a [Eu/Fe]
decline beginning at Type Ia onset.

  * •

The model shows that NSM only contribute $\approx$ 30 per cent to
Eu${}_{\text{LSE}}$, supporting previous studies that CCSNe, rather than NSM,
are the most dominant progenitor for _r_ -process production in Sculptor.

  * •

The model tracks the data average more closely than OMEGA for Mg, Ba, Eu, and
Na, and performs equally well in reproducing observed Ni trends. It is likely
that the model for N is not as accurate, since non-rotating yields were used
for the intermediary fixed point at [Fe/H] = $-$3.

In addition to offering comparisons between the chemical evolution of dSphs
and the MW, this model also provides isotopic inputs for future
nucleosynthesis studies of dSphs. Future works can explore this model for
other dSphs to provide further insight into the chemical evolution differences
between dSphs and spiral galaxies and ultimately further our understanding of
hierarchical galactic formation.

## Data Availability

The data that support the findings of this study are available in the public
domain. Table 3 lists the main data sources used in this work. Additional
results from this work (e.g., elemental model plots) are available from the
corresponding author upon request.

Table 3: Third-Party Data Sources Description | Publication | Data Link
---|---|---
Abundances of $\alpha$ elements for $\approx$ 400 stars | Kirby et al. (2009) | Link
Abundances of iron-peak, neutron-capture, and $\alpha$ elements for two stars | Tafelmeyer et al. (2010) |  Link (Table 7)
Abundances of iron-peak and $\alpha$ elements for seven stars | Starkenburg et al. (2013) |  Link (Table 6)
Abundances of iron-peak, neutron-capture, and $\alpha$ elements for 99 stars | Hill et al. (2019) | Link
Abundances of Zn for 100 stars | Skúladóttir et al. (2017) | Link
Abundances of iron-peak elements for $\approx$ 300 stars | Kirby et al. (2018) | Link
Abundances of Ba for 120 stars | Duggan et al. (2018) | Link
Abundances of neutron-capture elements for 98 stars | Skúladóttir et al. (2019) | Link
Abundances of iron-peak, neutron-capture, and $\alpha$ elements for 5 metal-poor stars | Jablonka et al. (2015) |  Link (Table 7)
OMEGA GCE code | Côté et al. (2017) | Link
Massive star theoretical yields | Heger & Woosley (2010) | Link
Type Ia SNe theoretical yields (sub-Chandrasekhar) | Leung & Nomoto (2020) |  Link (Table 7)
Type Ia SNe theoretical yields (near-Chandrasekhar) | Leung & Nomoto (2018) |  Link (Table 4)
AGB contributions | Karakas (2010) | Link
CCSNe contributions | Limongi & Chieffi (2018) | Link

##  6 Acknowledgements

We are grateful to the referee for several insightful comments and
suggestions. We also thank Benoit Côté and Amanda Karakas for their assistance
with this work.

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