# Assessing stellar yields in Galaxy chemical evolution: observational stellar
abundance patterns

Jinning Liang,1 Eda Gjergo,1,2,3 and XiLong Fan1
1School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072,
China
2School of Astronomy and Space Science, Nanjing University, Nanjing 210093,
People’s Republic of China
3Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210093, People’s Republic of China E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

One-zone Galactic Chemical Evolution (GCE) models have provided useful
insights on a great wealth of average abundance patterns in many environments,
especially for the Milky Way and its satellites. However, the scatter of such
abundance patterns is still a challenging aspect to reproduce. The leading
hypothesis is that dynamics is a likely major source of the dispersion. In
this work we test another hypothesis, namely that different assumptions on
yield modeling may be at play simultaneously. We compare whether the abundance
patterns spanned by the models are consistent with those observed in Galactic
data. First, we test the performance of recent yield tabulations, and we show
which of these tabulations best fit Galactic stellar abundances. We then group
the models and test if yield combinations match data scatter and standard
deviation. On a fixed Milky-Way-like parametrization of NuPyCEE, we test a
selection of yields for the three dominant yield sets: low-to-intermediate
mass stars, massive stars, and Type Ia supernovae. We also include the
production of r-process elements by neutron star mergers. We explore the
statistical properties spanned by such yields. We identify the differences and
commonalities among yield sets. We define criteria that estimate whether an
element is in agreement with the data, or if the model overestimates or
underestimates it in various redshift bins. While it is true that yields are a
major source of uncertainty in GCE models, the scatter of abundances in
stellar spectra cannot be explained by a simple averaging of runs across yield
prescriptions.

###### keywords:

galaxies: abundances – stars: abundances – Galaxy: evolution – methods: data
analysis

††pubyear: 2022††pagerange: Assessing stellar yields in Galaxy chemical
evolution: observational stellar abundance patterns–Assessing stellar yields
in Galaxy chemical evolution: observational stellar abundance patterns

##  1 Introduction

Galactic Archaeology, here used interchangeably with Galactic Chemical
Evolution (GCE), works on the premise that stellar surfaces encapsulate the
composition of their progenitor gas, so that little to no nucleosynthesis
reaches stellar surfaces via convection or other means. While this hypothesis
is simplistic for a range of stellar categories, works such as Hawkins et al.
(2020) have shown that most stellar wide binary pairs have indeed an extremely
homogeneous composition with a dispersion below 0.02 dex. Therefore, any
composition observed in the distribution of stellar abundance patterns should
reflect the past enrichment histories experienced by the progenitor gas.

GCE finds its roots in the landmark reviews by Burbidge et al. (1957), known
as the B2FH paper, and by Cameron (1957), written independently. Based on the
solar abundances compiled by Suess & Urey (1956), B2FH described the
nucleosynthesis processes that generate a wide array isotopic species,
including all the stable ones. Ever since B2FH, the field was successful in
solving a number of observational challenges.

One such achievement was the Tinsley-Wallerstein diagram (Wallerstein, 1962;
Tinsley, 1979), otherwise known as the [$\alpha$/Fe] vs [Fe/H] diagram. This
diagram is particularly sensitive to the rates of Type Ia supernovae (SNIa), a
source responsible of at least half of the iron enrichment (Matteucci et al.,
2009) ahead of core collapse supernovae (SNCC), and SNCC themselves which are
very effective at generating $\alpha$ species111$\alpha$ species are 4He, 12C,
16O, 20Ne, 24Mg, 28Si, 32S, 36Ar, and 40Ca. (Matteucci, 2001; Pagel, 2009).
Stars residing in different Galactic regions (specifically for the interests
of this paper, the halo, the thick disc and the thin disc) are clustered in
specific regions of the Tinsley-Wallerstein diagram, providing insights on
their past history: halo stars occupy a flat [$\alpha$/Fe]-enhanced,
[Fe/H]-poor region. The flat [$\alpha$/Fe] trend breaks at the age where SNIa
begin to enrich the interstellar medium (ISM), [Fe/H]$\geq-1$222The square
bracket notation represents logarithmic number densities of a species $A$,
relative to species $B$, and normalized to solar values:
$\mathrm{[A/B]}=\log\left(N_{\mathrm{A}}/N_{\mathrm{B}}\right)-\log\left(N_{\mathrm{A}}/N_{\mathrm{B}}\right)_{\odot}.$
, causing a downward [$\alpha$/Fe] slope toward zero (i.e., solar values) with
increasing iron abundance. This region after the onset of SNIa is where thick
disc stars are clustered. Lastly, thin disc stars tend to occupy a space right
below thick disc stars, on a flatter [$\alpha$/Fe] slope.

A recent investigation of the Tinsley-Wallerstein diagram on APOGEE-DR16 data
can be found in Spitoni et al. (2021), which uses MCMC methods to fit a multi-
zone GCE model. The original GCE models were one-zone, meaning that the galaxy
is treated as a point. In one-zone models, the metallicity and star formation
rate in a galaxy are homogeneously uniform as a function of time. These type
of models are still useful because of the insights they provide on the
essential constituents of galaxy evolution. In fact, more sophisticated models
should integrate back to the global properties that emerge from one-zone
models. For a recent extensive review on GCE, refer to Matteucci (2021).

Despite the many successes and predictions of GCE models, stellar abundances
are scattered across a wide range of values, and the exact origin of this
scatter is not yet completely understood for every chemical element. There
have been numerous attempts at reproducing these patterns, or at least at
constraining the conditions that may give rise to them. Most of the models
make the sensible assumption that dynamical effects are at play. Proposed
phenomena include stellar migration (Schönrich & Binney, 2009; Spitoni et al.,
2015; Johnson et al., 2021), gas radial flow (Andrews et al., 2017; Weinberg
et al., 2017; Trueman et al., 2022; Chen et al., 2022), the inclusion of a
variable IMF (Yan et al., 2019) or of a stochastic star formation rate (SFR,
Cescutti, 2008, with a specific application to heavy elements), or Bayesian
approaches (Côté et al., 2017; Rybizki et al., 2017; Spitoni et al., 2020).

In this work we explore an alternative route, namely we investigate whether
multiple enrichment phenomena as described by different yield models could
coexist in a Milky-Way like one-zone galaxy, and whether they may contribute
to the abundance patterns of a select number of elements. We follow Romano et
al. (2010), where they investigated existing yields in literature, and they
evaluated which yield in each enrichment channel best fits Galactic stellar
abundance data. In this work, we implement sets of more recent yields compared
to Romano et al. (2010), where they already investigated Karakas (2010) and
Kobayashi et al. (2006). We include also Limongi & Chieffi (2018); Nomoto et
al. (2013) for massive stars, Cristallo et al. (2011); Cristallo et al. (2015)
for AGBs, and Ritter et al. (2018) for both mass ranges. We overview the yield
mass-fraction, final mass, initial mass, initial metallicities. We re-
categorize the yields into 3 groups, according to analogous modeling
motivations: hypernovae fraction, massive star initial rotational velocity,
and delayed/rapid explosion prescriptions for massive stars. We then run on-
zone GCE simulations with the use of the OMEGA module in NuPyCEE (Côté et al.,
2017), using different yield combinations in each run. We therefore show the
resulting simulated abundance ratios as a function of iron abundance. We
lastly perform a statistical analysis on the aforementioned 3 groups of models
and on the data at different metallicity bins. We explore trends and
discrepancies.

This paper is structured as follows: Section 2 contains a description of the
observed stellar abundances (Section 2.1), of the stellar yields considered in
this work (Section 2.2), of the parameters chosen to run the NuPyCEE
simulations (Section 2.3.1), and of the statistical criteria (Section 2.3.2)
employed in the Results (Section 3). In Section 3 we also explore the
performance of the models in reproducing the abundance patterns of our 12
chosen elements. Our conclusions and discussions can be found in Section 4,
where the summary plots also appear.

Paper | C | N | O | Na | Mg | Al | Ca | Mn | Ni | Zn | Ba | Eu | Region of sampling
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Carretta et al. (2000) | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  |  |  |  |  |  | halo, thick and thin disc
Reddy et al. (2003) | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | thin disc
Cayrel et al. (2004) | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  | halo
Israelian et al. (2004) |  | $\times$ | $\times$ |  |  |  |  |  |  |  |  |  | halo
Bensby et al. (2005) |  |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | $\times$ | $\times$ | thick and thin disc
Reddy et al. (2006) | $\times$ |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | halo, thick and thin disc
Bensby et al. (2014) |  |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | $\times$ |  | thick and thin disc
Roederer et al. (2014) | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | halo
Zhao et al. (2016) | $\times$ |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  |  |  | $\times$ | $\times$ | halo, thick and thin disc
Mashonkina et al. (2017) |  |  |  | $\times$ | $\times$ | $\times$ | $\times$ |  | $\times$ |  | $\times$ | $\times$ | halo
Reggiani et al. (2017) |  |  |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  | halo
Table 1: Observational data, in chronological order, of the stellar elemental
abundances considered in the present work. The first column indicates the
references. In the subsequent 12 columns, the $\times$ marks whether a paper
contains that column’s element. The elements in order are C, N, O, Na, Mg, Al,
Ca, Mn, Ni, Zn, Ba, and Eu. The last column indicates in which region of the
Milky Way the sampled stars reside.

##  2 Methods

The Methods section is split among observational data, stellar yields
investigated, and theoretical models. Within the theoretical models section,
we include both a description of the GCE model and of the statistics criteria
employed in the analysis.

###  2.1 Observational Data

Model | Stellar Yields | Group | Z range | M range | Comments
---|---|---|---|---|---
| LIMs | Massive stars |  |  |  |
1 | F.R.U.I.T.Y. | N13 | A | [0.0001,0.05] | $\left[1.3,40.0\right]$ | $f_{\mbox{HNe}}=0.5$
2 | K10 | K06 | A | [0.0001,0.02] | $\left[1.0,40.0\right]$ | $f_{\mbox{HNe}}=0.0$
3 | K10 | K06 | A | [0.0001,0.02] | $\left[1.0,40.0\right]$ | $f_{\mbox{HNe}}=0.5$
4 | K10 | K06 | A | [0.0001,0.02] | $\left[1.0,40.0\right]$ | $f_{\mbox{HNe}}=1.0$
5 | K10 | LC18 | B | [3.236$\times 10^{-5}$,0.02] | $\left[1.0,120.0\right]$ | Set R, $V_{rot}=0.0$ km/s
6 | K10 | LC18 | B | [3.236$\times 10^{-5}$,0.02] | $\left[1.0,120.0\right]$ | Set R, $V_{rot}=150.0$ km/s
7 | K10 | LC18 | B | [3.236$\times 10^{-5}$,0.02] | $\left[1.0,120.0\right]$ | Set R, $V_{rot}=300.0$ km/s
8 | K10 | LC18 | B | [3.236$\times 10^{-5}$,0.02] | $\left[1.0,120.0\right]$ | Set R, $V_{rot,avg}$
9 | F.R.U.I.T.Y | NuGrid | C | [0.0001,0.02] | $\left[1.3,25.0\right]$ | MESA-only (delay)
10 | NuGrid | K06 | A | [0.001,0.02] | $\left[1.0,40.0\right]$ | $f_{\mbox{HNe}}=0.5$
11 | K10 | NuGrid | C | [0.0001,0.02] | $\left[1.0,25.0\right]$ | MESA-only (delay)
12 | NuGrid | NuGrid | C | [0.0001,0.02] | $\left[1.0,25.0\right]$ | MESA-only (delay)
13 | NuGrid | NuGrid | C | [0.0001,0.02] | $\left[1.0,25.0\right]$ | MESA-only (delay wind pre-exp)
14 | NuGrid | NuGrid | C | [0.0001,0.02] | $\left[1.0,25.0\right]$ | MESA-only (mix)
15 | NuGrid | NuGrid | C | [0.0001,0.02] | $\left[1.0,25.0\right]$ | MESA-only (rapid)
16 | NuGrid | NuGrid | C | [0.0001,0.02] | $\left[1.0,25.0\right]$ | MESA-only (classical)
Index | SNIa yields | Z range | Comments
a | I99 | 0.02 | CDD1
b | I99 | 0.02 | CDD2
c | I99 | 0.02 | W7
d | Ivo12 | [0.0002,0.02] | Stable
e | T86 | 0.02 |

Table 2: Legend for the models (_first column_) adopted in the present work.
Each model corresponds to a different yield variation. Numbers from 1 to 16
indicate combinations of yields coming from low-to-intermediate mass stars
(LIMs) and massive stars. The indices a to e identify the Type Ia SNe (SNIa)
yields paired with the above-mentioned models. The abbreviations stand for the
following: NuGrid (Ritter et al., 2018); F.R.U.I.T.Y. (Cristallo et al.,
2015); N13 (Nomoto et al., 2013); K06 (Kobayashi et al., 2006); K10 (Karakas,
2010); LC18 (Limongi & Chieffi, 2018); I99 (Iwamoto et al., 1999); Ivo12
(Seitenzahl et al., 2013); T86 (Thielemann et al., 1986). In the last column,
_Comments_ , we identify the various assumptions on the massive stellar yield
computations. More details on these assumptions can be found in the text.

In this section, we summarize the observational data that appear in the
present work. For our purposes, we have chosen to focus on the following
elements: C, N, O, Na, Mg, Al, Ca, Mn, Ni, Zn, Ba, and Eu. The yield
performance on each of these elements will be investigated in Section 3. All
abundances, whether theoretical or observational, are calibrated to
photospheric solar data (Asplund et al., 2009).

We consider the data-sets presented in Table 1. The last column includes the
Galactic environments – namely halo, think disc, thick disc, or solar
neighborhood and field stars – in which the sampled stars reside for each
study. Some of these studies sampled stars residing in multiple environments:

Figure 1: Logarithmic mass-fraction (defined as the yield mass over ejecta
mass) of the returned yields for 12 elements at solar metallicity, as a
function of logarithmic stellar mass. The "shift" label on the upper right
indicates, wherever necessary, the factor by which the mass-fractions have
been shifted in order to fit the same y-axis range (e.g., the Europium mass-
fraction ranges between a value of $10^{-9}$ and $10^{-10}$ while in the plot
it ranges between $10^{-2}$ and $10^{-3}$, a shift of +7 dex). The legends for
different stellar yields on the first row follow the description in Table 2.
The yields are styled in the following way: all of the low-to-intermediate
mass stars (LIMs) yields are represented with the dash-dotted lines; the
massive star yields can be represented by solid lines (Nomoto et al., 2013;
Kobayashi et al., 2006), by dotted lines (Limongi & Chieffi, 2018), or by
dashed lines (Ritter et al., 2018). Among the LIMs yields, the colorings are:
green (F.R.U.I.T.Y., Cristallo et al., 2011; Cristallo et al., 2015), included
in Models 1 and 9; blue (K10, Karakas, 2010) included in Models 2 to 8, and
black for Model 11; pink (NuGrid, Ritter et al., 2018) included in Model 10,
12, 13, 15 and 16, and yellow included in Model 14.  Figure 2: The ejected
stellar mass versus initial stellar mass for 16 yield tables. The colorbar of
the various curves represents different initial metallicities in the range of
their yield tables. Here we use a logarithmic scale in the color bar. The
lighter the color (yellower), the higher the initial metallicity. The blue
line provides a reference, where the ejected mass equals the initial stellar
mass. Groups and Models are described in Table 2.

  1. [i]

  2. 1.

Halo stars: Cayrel et al. (2004) selected 35 high signal-to-noise ratio giants
from the ESO Large Programme "First Stars", where they investigated 70 very
metal-poor dwarf and giants whose spectra were captured with the ESO VLT and
UVES spectrograph. 17 elemental abundances from C to Zn were reported in their
work, i.e. 10 out of the 12 elements we consider. Israelian et al. (2004), on
the other hand reports N and O only for 31 metal poor stars. Their spectra
were measured using multiple instruments. On top of finding a scaling relation
for [N/Fe] vs [Fe/H] in the range -3.1 < [Fe/H] < 0, they confirmed the
necessity of a primary component to the nitrogen enrichment. Roederer et al.
(2014) is one of two papers in our sample that reports abundances for all 12
elements. They report, in fact, 48 elemental abundances including an isotopic
breakdown for 5 additional species for 191 of the 313 stars observed over the
course of 10 years of the Magellan Inamori Kyocera Echelle campaign. Their
efforts focused primarily on the [Fe/H] $\leq$ -2.5 stellar sample. Mashonkina
et al. (2017) also focused on very metal-poor (-4 < [Fe/H] < -2) stars, a
selection of 59 stars residing primarily in the halo, but 7 of which were
found in dwarf spheroidal galaxies. We chose not to omit these stars. The
traced elements are a combination of $\alpha$\- and odd-Z-elements, as well as
Ni and the main neutron-capture species. They confirm their metal-poor sample
is $\alpha$-enhanced by a factor of 2 compared to solar $\alpha$/Fe
abundances. Reggiani et al. (2017) combines literature abundances with their
observations to constrain GCE over a wide range of metallicities (-3.6$\leq$
[Fe/H] $\leq$ -0.4). They find their results suggest strong inhomogeneous
hypernova enrichment. Group A in our yield set includes variations on this
enrichment mechanism. While they include Ba, they don’t report CNO elements or
Eu.

  3. 2.

Thin disc-dominated stars: Reddy et al. (2003) is the other reference from
Table 1 that investigated all the 12 elements analyzed in our work. They
provide abundances for 15 additional elements for 181 F and G dwarf stars, the
vast majority (Reddy et al., 2003, their Fig. 17) of which reside in the thin
disc. While this work reports stellar abundances from both thin and thick
disc, the prevalence of thin disc stars warrants placing them in a separate
category.

  4. 3.

Thin and thick disc stars: Also Bensby & Feltzing (2006) measured chemical
abundances in F and G dwarf stars, using the Coudé Echelle Spectrograph (CES),
with 35 and 16 stars residing in the thin and thick disc, respectively.
Similarly to Reddy et al. (2003), the thin/thick disc classification was based
on kinematic criteria. The work mainly focuses on carbon trends, but measures
the abundances of all of our chosen elements, with the exception of N. More
recently, Bensby et al. (2014) again investigated 714 F and G disc dwarf and
subgiant stars in the solar neighborhood. This work identifies the bimodality
of the [$\alpha$/Fe]-[Fe/H] relation for thin and thick disc stars, the latter
being the more $\alpha$-enhanced, despite a significant metallicity overlap in
the metal-poor to solar metallicity regime. Missing from the Bensby & Feltzing
(2006); Bensby et al. (2017) papers are C, N, and Mn and, in the latter, also
Eu.

  5. 4.

Halo, thick, and thin disc: On a sample of 19 extremely metal-poor stars
observed with CES, Carretta et al. (2000) expanded their selection with
literature data that measured high signal-to-noise (S/N > 150) spectra for 300
stars spanning a wide range of metallicities. They report abundances for light
intermediate elements (C, N, O, Na, and Mg). Reddy et al. (2006) then isolated
176 solar-neighborhood F and G dwarf thick-disc candidates. While 95 stars
were classified as thick disc stars, 48 could not be classified conclusively,
20 belonged to the halo while 13 to the thin disc. We therefore choose to
include this paper among the halo, thick, and thin disc group. Concerning our
12 elements, Reddy et al. (2006) provides a complete set with the exception of
nitrogen. Zhao et al. (2016) selects stars with the deliberate aim of
collecting spectra for thin, thick disc, and halo stars from $-2.62\leq$
[Fe/H] $+0.24$ and 17 their elemental abundances from Li to Eu. Of the
elements that concern our work, all are present but nitrogen and three species
around the iron peak (Mn, Ni, and Zn).

###  2.2 Stellar Yields

In this section, we describe and compare different stellar yields. All of the
yields are included in the NuPyCEE
database333https://github.com/NuGrid/NuPyCEE/tree/master/yield_tables. We
define mass-fraction as the ratio of the ejected mass for a specific chemical
species divided by the total ejecta mass. A comparison for different stellar
yields of 12 elements at solar metallicity is also shown in Fig. 1. The yields
we considered in this work and summarized in Table 1 are as follows:

  1. [i]

  2. 1.

For low and intermediate mass stars, the yields come from: Karakas (2010);
F.R.U.I.T.Y., i.e. Cristallo et al. (2011); Cristallo et al. (2015); and
NuGrid, i.e. Ritter et al. (2018).

  3. 2.

For massive stars, the yields come from: Nomoto et al. (2013), Kobayashi et
al. (2006), Limongi & Chieffi (2018), and (Ritter et al., 2018).

  4. 3.

For Type Ia Supernovae, the yields come from: Iwamoto et al. (1999),
Thielemann et al. (1986), and Seitenzahl et al. (2013).

Most of the analysis on yield variations in our work concern massive stars. Of
the four massive star yield tabulations we consider, two of them (Kobayashi et
al., 2006; Nomoto et al., 2013) are by the same authors and investigate the
impact to the chemical enrichment of the inclusion of hypernova yields in
massive stars. Hypernova episodes are set to occur for stars in the mass range
from 20 to 40 $M_{\odot}$. The authors define the fraction of stars,
$f_{HNe}$, that will experience a hypernova episode. $f_{HNe}$ varies from 0%
(Model 2) to 100% (Model 4). Three of the models (Model 1, 3, and 10 in Table
1) are set to $f_{HNe}=0.5$. We call this yield set “Group A”.

The next set (Models 5 to 8) includes only yield variations in “Set R” – the
recommended set – for Limongi & Chieffi (2018). Models 5, 6, and 7 assume that
all of the massive stars rotate at rotational velocities of 0, 150, and 300
km/s, respectively. Model 8 ($V_{rot,avg}$) represents an average of the 3
other rotation velocities. We call this yield set “Group B”.

Next come a series of NuGrid models computed with MESA (Paxton et al., 2015,
Models 9 and 11 to 16). In the _Comments_ column for such models, the term in
the parenthesis indicates the remnant mass prescription used for the core-
collapse explosion of massive stars as computed in Fryer et al. (2012).
“rapid” refers to the fast-convection model, i.e. it includes explosions
occurring within 0.250 s after bounce; while “delayed” refers to the delayed
convection model, which also include later explosions such as the ones
dominated by standing accretion shock instability. “mix” is a 50/50 mix of the
delayed and rapid models, while “delay wind pre-exp” adds pre-SN
nucleosynthesis and wind nucleosynthesis to the delayed model. Lastly, Model
16 is the classical approach as presented in Fryer et al. (2012). We call this
yield set “Group C”.

For SNIa yields, CDD1, CDD2, W7 are the 3 different models that represent the
SNIa detonation scenario from Iwamoto et al. (1999). “Stable” for Seitenzahl
et al. (2013) means that the SNIa yields contain only the stable isotopes.
Thielemann et al. (1986) reports only one SNIa yield set. The Type Ia SN
yields are shown on the bottom of Table 1 with model indices spanning from “a”
to “e”. Table 1 also reports the metallicity and mass range spanned by each
yield set. Among SNIa yields, only Seitenzahl et al. (2013) reports
metallicity-dependent yields in the very metal-poor vs solar-metallicity
regimes.

Each of these references computes yield tabulations on different metallicity
grids. The only shared metallicity is $Z=0.02$. In Fig. 1 we show the
logarithmic mass-fraction (namely the ratio of the yield expressed in solar
masses vs. the ejecta mass) as a function of logarithmic masses for all 12
elements and for the various yield prescriptions. Each panel refers to a
single element. Except for C, and Ba, LIMs yields have little to no dependence
on the stellar mass at these near-solar metallicities. Furthermore, these
models are in fairly good agreement with each other. We note that the AGB/LIMs
yield paper Karakas (2010) does not contain yields for Ca, Mn, Zn, Ba, Eu.

Figure 3: Logarithmic yields versus Atomic number for the 5 SNIa yield tables,
as indicated in Table 2. The atomic number range goes from 6(C) to 32(Ge)

When it comes to massive stars, there is often a clear dependence of the mass-
fraction on the initial stellar mass. We also notice greater variations across
models. Therefore, modeling assumptions have a greater impact on the
nucleosynthesis of each element. This is particularly evident in the 10-30
$M_{\odot}$ mass range. Only group B (Limongi & Chieffi, 2018) reports yields
for stars more massive than 40 $M_{\odot}$. Limongi & Chieffi (2018) accounts
for s-process production only, so there is no Eu from this set either. For the
lighter elements, as the atomic number increases, the mass-fraction on average
tends to decrease as a function of initial stellar mass – up to the model’s
limit for BH formation. The trend inverts as we approach and surpass the iron-
peak. We point out that the heavy elements Ba and Eu are produced at a
particularly low yield. Noting that the mass-fraction is normalized to ejected
masses of stars, it is also necessary to show the ejected mass of stellar
yields, which we do next.

In Fig. 2, we show the ejected mass as function of stellar mass of each model.
The color bar represents the logarithmic initial metallicity. For massive
stars, there is a strong dependence on metallicity when it comes to mass
ejecta. Generally, the higher initial stellar mass, the higher ejected mass.
But there are some exceptions, for example, Group B show a decreasing trend
when the stellar mass is larger than 25 $M_{\odot}$. In Group B (the Limongi &
Chieffi, 2018, yields, Set R), all stars above an initial mass of 25
$M_{\odot}$ collapse into black holes, so the yields above that mass limit
come from the ejected wind component. For LIMs, all the models do not show a
strong dependence of the ejecta mass on the initial stellar metallicity.
Models 9, 11, 12, and 13 show little dependence on Metallicity. Model 5, 6, 7,
and 8 show that the higher metallicity, the higher the ejected mass, while the
trend is opposite for the remaining models. Please note that the x-axes in
Figure 2 are not shared across the various columns, and the maximal stellar
mass reached by the yields varies between 25, 40, and 120 $M_{\odot}$.

In Fig. 3, we show logarithmic yields as function of atomic number ranging
from 6(C) to 32(Ge) for the 5 SNIa yields we considered. One can see from
Table 2, that most SNIa yields have no metallicity dependence, except for
Model d that differentiates between a low-metallicity and a solar-metallicity
regime. For the majority of the elements that we investigate in the present
work, none of the yield tabulations shown in Fig. 3 vary significantly. The
only exceptions are N, Na, and Zn. In the following analysis and predictions
we adopt only Model d as the SNIa yield prescription.

###  2.3 Theoretical model

In this section we describe the theoretical basis of both our chemical
evolution model and of our analysis. We provide the NuPyCEE prescription
employed in the runs which will be analysed in the remainder of this paper. We
fix the parametrization for chemical evolution across all runs, and we only
vary the yields for Models 1 through 16, as described in Table 1. Among the
enrichment channels, we include: massive stars, LIMs, SNIa, and also neutron
star mergers.

####  2.3.1 The NuPyCEE Simulation

This work is based on a one-zone chemical evolution prescription where only
the yields vary across models. We place particular importance to variations on
the yields for massive stars. Our aim is to investigate whether multiple one-
zone runs span the same abundance patterns of the data. In this study, we
exclusively use the chemical evolution code OMEGA (One-zone Model for the
Evolution fo GAlaxies Côté et al., 2016) implemented within the
NuPyCEE444https://github.com/NuGrid/NuPyCEE/ framework. A simplified model was
introduced in Côté et al. (2019) to investigate whether neutron star mergers
(NSM) may be the main contributors to the r-process enrichment (exemplified by
europium abundances) in the Milky Way. That specific prescription was applied
to all the runs in this paper, and will be described in the remainder of this
section555Special thanks to Benoit Coté for helping with the setup
reconstruction..

We keep OMEGA’s default initial mass function, the invariant canonical IMF
(Kroupa, 2001). Côté et al. (2019) simplified the star formation rate (SFR) to
a constant value. This assumption is quite typical and representative of
rotationally-supported galaxies (Schombert et al., 2019; Kroupa et al., 2020).
The star formation efficiency is set to 0.04. The fraction of the mass ejected
into the ISM from simple stellar populations (SSP) is 0.35.

In terms of inflow and outflow, we keep the default I/O Model: the mass mass-
loading factor $\eta$ (the ratio between outflow rate and SFR rate Martin,
1999; Veilleux et al., 2005) is non-zero and equal to 0.5, while the ratio
between the inflow rate and the outflow rate is 1.0. NuPyCEE treats the
evolution of dark matter(DM), which has an impact on the mass-loading factor
$\eta$. So the reported value of $\eta$ is its value at the final age of the
Galaxy. We also have the relation $\eta\propto v_{out}^{\gamma}$, where
$v_{out}$ refers to the velocity of the outflowing material (Murray et al.,
2005), which is proportional to the rotation velocity of galaxy. This velocity
is also proportional to the virial velocity. This implies that $\eta$ is
related to the evolving redshift and virial mass:
$\eta(t)=C_{\eta}M_{vir}(t)^{-\gamma/3}(1+z)^{-\gamma/2}$, where $C_{\eta}$ is
a normalization constant calculated in the simulation. We set the value of
$\gamma$ equal to 1.0.

For SNIa, the number of events per stellar mass formed in every SSP is set to
$1.2\times 10^{-3}$ [event/$M_{\odot}$], while for NSM, it is set to
$3.0\times 10^{-5}$. Furthermore, we set the mass ejecta per NSM event to
$2.5\times 10^{-2}M_{\odot}$. Both SNIa and NSM follow in OMEGA a power law
$\propto t^{-1}$ for the delayed time distribution of the detonation or
coalescence events, respectively. For NSM, the coalescence range is set from
10 Myr to 10 Gyr. These are the same parameters adopted by the authors in Côté
et al. (2019). Any other parameter was left unaltered to the default value as
provided in the NuPyCEE repository. In the present work, the only parameters
we change are the stellar yield tabulations.

####  2.3.2 Classification criteria

To quantify how well the yield models agree with observational data, we
present statistics criteria to investigate median, standard deviation, and
scatter. The criteria of our judgements are defined as:

$\displaystyle J_{M,i,j,k}=$
$\displaystyle\begin{cases}\text{Overestimated}&M_{i,j,k}-M_{\rm
dat,j,k}>\delta\\\ \text{In Agreement}&\left|\,M_{i,j,k}-M_{\rm
dat,j,k}\,\right|\leq\delta\\\ \text{Underestimated}&M_{i,j,k}-M_{\rm
dat,j,k}<-\delta\\\ \end{cases}$ (1a) $\displaystyle J_{\sigma,i,j,k}=$
$\displaystyle\begin{cases}\text{Overestimated}&{\sigma_{i,j,k}}/{\sigma_{\rm
dat,j,k}}-1>\delta\\\ \text{In
Agreement}&\left|\,{\sigma_{i,j,k}}/{\sigma_{\rm
dat,j,k}}-1\,\right|\leq\delta\\\
\text{Underestimated}&{\sigma_{i,j,k}}/{\sigma_{\rm dat,j,k}}-1<-\delta\\\
\end{cases}$ (1b) $\displaystyle J_{S,i,j,k}=$
$\displaystyle\begin{cases}\text{In
Disagreement}&{\mathscr{I}_{i,j,k}}/{S_{\rm dat,j,k}}=0\\\
\text{Overestimated}&{\mathscr{E}_{i,j,k}}/{S_{\rm dat,j,k}}>\delta\\\
\text{In Agreement}&({\mathscr{E}_{i,j,k}}/{S_{\rm dat,j,k}}\leq\delta\\\
&\cap\quad{\mathscr{I}_{i,j,k}}/{S_{\rm dat,j,k}}>1-\delta)\\\
\text{Underestimated}&({\mathscr{E}_{i,j,k}}/{S_{\rm dat,j,k}}\leq\delta\\\
&\cap\quad{\mathscr{I}_{i,j,k}}/{S_{\rm dat,j,k}}\leq 1-\delta)\\\
\end{cases}$ (1c)

where $J_{l,i,j,k}$ is the judgement. The subscript $l$ stands for either the
median $M$, the standard deviation $\sigma$, or the scatter $S$. The subscript
$i$ identifies the Group or group combination (A, B, C, AB, AC, BC, ABC) of
the various Models with their respective yield variations. The subscript $j$
refers to one of the 12 elements considered in this paper. Lastly, the
subscript $k$ identifies one of the four metallicity bins (-4.0 $\lesssim$
[Fe/H] $\lesssim$ -3.0, -3.0 $\lesssim$ [Fe/H] $\lesssim$ -2.0, -2.0
$\lesssim$ [Fe/H] $\lesssim$ -1.0, -1.0 $\lesssim$ [Fe/H] $\lesssim$ 0.0).
$M_{i,j,k},\sigma_{i,j,k},S_{i,j,k}$ indicate the median, standard deviation
and scatter of the Group $i$ of the element $j$ in the $k$th bin,
respectively. And $M_{\rm dat,j,k},\sigma_{\rm dat,j,k},S_{\rm dat,j,k}$ refer
to the median, standard deviation and scatter of data of the element $j$. The
scatter is defined in the last paragraph of this section. In Eq. 1, we set
$\delta=0.3$. Other nearby values were tested ($\delta=0.3\pm 0.05$) without
significant alterations to the classification outcome.

Figure 4: Schematic representation of the judgement for the scatter, as it is
introduced in Equation 1c, with 4 case examples. $\mathscr{I}_{i,j,k}$, the
intersection, is shown with the purple vertical segments.
$\mathscr{E}_{i,j,k}$, the excess, is shown with the yellow-orange vertical
segments. The four green lines are ad-hoc models created as an example for
this classification criterion. The red points are the median of the example
data. As for light blue error bars, they are scatter of the example data. The
maximum and minimum of the example models in each of the 4 metallicity bins
are shown by the dark green and light green dashed lines, respectively.

In Eq. 1c $\mathscr{I}_{i,j,k}$ is the intersection of Group $i$ and data for
the element $j$ which is the length of the overlap between data and group:

$\mathscr{I}_{i,j,k}=\\{S_{\rm i,j,k}\\}\cap\\{S_{\rm dat,j,k}\\},$ (2)

while $\mathscr{E}_{i,j,k}$ is the excess of Group $i$ and data for the
element $j$ which is the length of the non-overlap:

$\mathscr{E}_{i,j,k}=\\{S_{\rm i,j,k}\\}-\\{S_{\rm dat,j,k}\\}.$ (3)

Fig. 4 gives a schematic representation of the judgement for the scatter. Each
metallicity bin displays the value for the $\mathscr{I}_{i,j,k}$ and the
excess $\mathscr{E}_{i,j,k}$, as well as the value of the data scatter,
$S_{\rm dat,j,k}$, in light blue. Each metallicity bin represents a particular
scenario. The lowest metallicity bin (-4 < [Fe/H] < -3) shows the case where
the maximum of the example models is larger than the upper light blue error
bar – namely the upper bound of the data scatter – while the minimum of
example models is higher than the lower bound of the scatter. Due to the large
intersection and small excess, in this metallicity bin there is an agrement
between data and models. In the next metallicity bin (-3 < [Fe/H] < -2), the
scatter of the data is contained within the range spanned by the minimum and
maximum of the models. In this case, we are overestimating the scatter. The
very metal poor metallicity bin (-2 < [Fe/H] < -1) has a large data scatter
but low excess. This bin happens to be underestimated with the judgement for
the median but also with the judgement for the scatter. In fact, despite the
small excess, the scatter intersection is much smaller than the scatter of the
data. In the highest metallicity bin, data and models are in disagreement
because there is no intersection.

Theory and models will be more closely in agreement the more each of the
judgement criteria tends to the following values: $M_{i,j,k}-M_{\rm
dat,j,k}=0$, ${\sigma_{i,j,k}}/{\sigma_{\rm dat,j,k}}=1$ and
${\mathscr{I}_{i,j,k}}/{S_{\rm dat,j,k}}=1$. We therefore decide whether or
not a model is in good agreement with the estimate (and similarly, if it
overestimates or underestimates the abundance of an element) by the
appropriate choice of the arbitrary parameter $\delta$. In the present work,
as already stated, $\delta=30$% accomplishes quite well the task of
discriminating the performance of the models.

These statistics criteria employ median, standard deviation and scatter. This
information is presented in bin plots and box plots that appear throughout the
3 Section. For both observational data and groups, the median is defined as
the middle among sorte [X/Fe] values in specific metallicity bins for both
observational data and for groups. The standard deviation is defined as the
amount of variation or dispersion of a set of values in specific metallicity
bins, which takes the form of
$\sigma=\sqrt{\sum_{i=1}^{N}(x_{i}-\bar{x})/(N-1)}$. Where $x_{i}$ is the data
in specific metallicity bins and $\bar{x}$ is the mean for these data defined
as the sum of data values over the number of data, i.e.,
$\bar{x}=\sum_{i=1}^{N}x_{i}/N$. The median and standard deviation of the
observational data for each element can be calculated directly by the data we
collected from Table 1. For the models, first we interpolate the curves
linearly and then we evaluate the statistical properties in the same
metallicity bin. Lastly, we collect the statistical quantities from individual
models into groups or group combinations for each metallicity bin.

Some Figures (6, 9, 12, and 15) employ box plots which include 7 quantities:
the minimum, the first quartile ($Q1$), the median, the mean, the third
quartile ($Q3$), the maximum and the outliers. The difference between $Q3$ and
$Q1$ is the inter-quartile range ($IQR$). The maximum is defined as
$Q3+1.5IQR$ while minimum is defined as $Q1-1.5IQR$. The data points that
reside outside the minimum-maximum range are the outliers. We define the
subtraction of minimum from maximum as the scatter $S_{i,j,k}$ (see Eq. 1). In
the box plots of Sec. 3, the boxes represent $Q1$ and $Q3$ and use upper arms
and lower arms to represent maximum and minimum, respectively. The mean and
median are also shown inside the boxes. With that, we can investigate
succinctly the statistical features of the data, and we can check whether the
models follow similar trends. To compare the spread of the abundance data from
stellar spectra and model estimates, we create the pie-charts in Section 4 to
evaluate the comparisons.

Figure 5: The [X/Fe]-[Fe/H] relation in the Milky Way, where each column from
left to right represents C, N, and O respectively. Each row refers to a
different group. The models in the first row belong to Group A and are drawn
with a solid line. The models in the middle row belong to Group B and are
drawn with a dashed line. The models in the bottom row belong to Group C and
are drawn with a dotted line. Next we describe the color patterns. _first
row:_ Model 1 (black), Model 2 (dark red), Model 3 (dark blue), Model 4 (dark
green), Model 10 (purple); _second row:_ Model 5 (black), Model 6 (dark red),
Model 7 (dark blue), Model 8 (dark green); _third row:_ Model 9 (black), Model
11 (dark red), Model 12 (dark blue), Model 13 (dark green), Model 14 (purple),
Model 15 (olive), Model 16 (chocolate). The yield prescriptions of the models
are described in Table 2. Concerning the color patterns for the observational
data: halo stars are shown in royal blue, halo stars as well as thick and thin
disc stars are shown in light green, thick and thin disc stars are shown in
pink, thin disc-only stars are shown in gray. All of the data are shown with
empty markers. Specifically, the observational data sources are displayed as
follows: the halo stars in royal blue, the squares come from Cayrel et al.
(2004), the hexagons come from Roederer et al. (2014), while the circles from
Israelian et al. (2004). Light green halo and thin-and-thick disc stars are
either circles (Zhao et al., 2016), squares (Reddy et al., 2006) or hexagons
(Carretta et al., 2000). The pink thin and thick disc stars are either circles
(Bensby et al., 2005) or squares (Bensby et al., 2014). The gray circles of
thin disc stars come from Reddy et al. (2003).  Figure 6: Similarly to the
previous box plots, we show the [X/Fe]-[Fe/H] relation for the elements (C, N,
O in each column from left to right) and each group, with the addition of
Group ABC – the combination of all three base groups – on the top row. Unlike
Fig. 5, the data are grouped in four metallicity bin in the range -4 < [Fe/H]
< 0 and the box plot for each bin is shown in black. The median (dot-dashed
orange lines inside the boxes) and average (dot-dashed grey lines inside the
boxes) of the data are also shown. All of Group A models are shown in solid
blue lines, all of Group B models are shown in dashed red lines, while all of
Group C curves are shown in dotted green lines.

##  3 Results

In this section, we investigate the abundance patterns of each element in each
of the 16 models as a function of metallicity, using iron as the metallicity
indicator. We then employ the statistics criteria from Section 2.3.2 to
evaluate how closely the model groups match the data. Since different SNIa
yield tables show nearly the same result, we show the simulation with SNIa
yields “d” Seitenzahl et al. (2013), as labeled in the bottom of Table 2. We
recall that we have divided the 16 models into 3 groups: Group A includes
Model 1, 2, 3, 4, 10. Group B includes Model 5, 6, 7, 8. Group C includes
Model 9, 11, 12, 13, 14, 15, 16. The models in each group normally vary only
one parameter or nucleosynthetic prescription at a time – and primarily among
the yields of massive stars. Models in Group A vary the hypernova fraction
according to Kobayashi et al. (2006), models in Group B vary the rotational
velocity of massive stars according to Limongi & Chieffi (2018), while Group C
(NuGrid, Ritter et al., 2018) considers either fast- or delayed-convection
explosions (Fryer et al., 2012, fast-convection includes explosions occuring
250 ms after "bounce", while delayed convection explosoins extend beyond this
limit, including explosions dominated by the standing accretion shock
instability), as it is applied within the code Modules for Experiments in
Stellar Astrophysics (MESA, Paxton et al., 2011; Jones et al., 2015). We then
combine any 2 or 3 groups at a time in Group AB, Group AC, Group BC and Group
ABC.

Our results are divided into the following four sections: CNO elements in
Section 3.1, $\alpha$ and odd-Z elements in Section 3.2, iron-peak elements in
Section 3.3, and neutron capture elements in Section 3.4. For each of these
categories we will present three figures: a first plot that depicts the
[X/Fe]-[Fe/H] relation with the data scatter and the models where each row
refers to a different group while each column a different element; a second
plot, similar to the first plot, in which the data are sorted in 4 metallicity
bins and then represented by means of box-plots. In these plots, the top row
depicts all 16 models simultaneously; and a third plot, where in the same 4
metallicity bins, the median and standard deviation as defined in Section
2.3.2 are shown for the various groups and group combinations.

Figure 7: In metallicity bins consistent with the box-plots of Fig. 6 and Fig.
(18, 19), the median ($M_{i,j,k}$, _top row_) and standard deviation
($\sigma_{i,j,k}$_bottom row_) for each group and group combination. The black
points are shown in the middle of their metallicity bins (-4.0 dex $\lesssim$
[Fe/H] $\lesssim$ -3.0 dex, -3.0 dex $\lesssim$ [Fe/H] $\lesssim$ -2.0 dex,
-2.0 dex $\lesssim$ [Fe/H] $\lesssim$ -1.0 dex or -1.0 dex $\lesssim$ [Fe/H]
$\lesssim$ 0.0 dex). All of the panels adopt the same colors and styles: data
(grey lines), Group A (blue lines), Group B (red lines), Group C (green
lines), the combination of Group A and Group B (Group AB, purple lines), the
combination of Group A and Group C (Group AC, cyan lines), the combination of
Group B and Group C (Group BC, yellow lines), the combination of Group A and
Group B and Group C (Group ABC, black lines). A description of the statistics
criteria can be found in Section 2.3.2. The data and models used are
consistent with the previous figures, Fig. 5 and Fig. 6.

###  3.1 CNO elements

The CNO elements are the most abundant atomic species synthesized by stars.
Given that a wide range of nuclear processes can yield oxygen, carbon, and
nitrogen, their chemical evolution is rich and complex (for a recent review,
see Romano, 2022). In Salpeter (1952), it was found that the two main species
of carbon and oxygen, 12C and 16O, as well as 18O, could be produced during
He-burning (Hoyle, 1954; Freer & Fynbo, 2014) by either triple-$\alpha$ or
12C($\alpha$, $\gamma$)16O processes (Clayton, 2003). Other important isotopes
can instead form in the cold (13C, 14N, and 17O, Bethe, 1939) and hot (>
$10^{8}$ K 15N, Dearborn et al., 1978) CNO cycles during H-burning (for a
review, see Wiescher et al., 2010). Furthermore, experiments directly tailored
to measure key reactions involved with the CNO cycle have been conducted by
the LUNA Collaboration (Ananna et al., 2022).

Despite these advancements, one of the most significant open questions in
stellar nucleosynthesis concerns the Carbon-to-Oxygen abundance ratio at the
end of He burning (Buchmann & Barnes, 2006). The 12C($\alpha$, $\gamma$)16O
cross section in stellar environments are out of reach of current laboratory
sensitivities (Kunz et al., 2002), so the results need to be extrapolated from
experiments performed at higher energies (Aliotta et al., 2022). And similarly
for the triple-$\alpha$ reaction, Kibédi et al. (2020) found that improved
measurements lead to a 34% increase of its reaction rates, with unexplored
consequences on nucleosynthetic computations (a study on the impact of nuclear
reaction uncertainties be found in Fields et al., 2018).

####  3.1.1 Carbon

Carbon is the second most abundant metal in terms of cosmic as well as gas-
phase and stellar Galactic abundances (Nieva & Przybilla, 2012), right after
oxygen. Despite this, it still poses some challenges. The recent consensus is
that half of the carbon is synthesized by massive stars, while the remaining
half is produced by low-mass (1-4 $M_{\odot}$) AGB stars (Kobayashi et al.,
2011). In fact, new computations that consider both rotation and mass loss of
massive stars can yield a significant amount of 12C (Limongi & Chieffi, 2018;
Romano et al., 2020).

Figure 8: Similarly to Fig. 5, the [X/Fe]-[Fe/H] relation in the Milky Way,
where each column from left to right represents Na, Mg, Al, and Ca,
respectively. Each row refers to a different group. The models in the first
row belong to Group A and are drawn with a solid line. The models in the
middle row belong to Group B and are drawn with a dashed line. The models in
the bottom row belong to Group C and are drawn with a dotted line. Next we
describe the color patterns. _first row:_ Model 1 (black), Model 2 (dark red),
Model 3 (dark blue), Model 4 (dark green), Model 10 (purple); _second row:_
Model 5 (black), Model 6 (dark red), Model 7 (dark blue), Model 8 (dark
green); _third row:_ Model 9 (black), Model 11 (dark red), Model 12 (dark
blue), Model 13 (dark green), Model 14 (purple), Model 15 (olive), Model 16
(chocolate). The yield prescriptions of the models are described in Table 2.
Concerning the color patterns for the observational data: halo stars are shown
in royal blue, halo stars as well as thick and thin disc stars are shown in
light green, thick and thin disc stars are shown in pink, thin disc-only stars
are shown in gray. All of the data are shown with empty markers. Specifically,
the observational data sources are displayed as follows: . Of the halo stars
in royal blue, the squares come from Cayrel et al. (2004), the star symbols
come from Reggiani et al. (2017), the hexagons come from Roederer et al.
(2014), while the crosses come from Mashonkina et al. (2017). Light green halo
and thin-and-thick disc stars are either circles (Zhao et al., 2016), hexagons
(Carretta et al., 2000), or squares (Reddy et al., 2006). The pink thin and
thick disc stars are either circles (Bensby et al., 2005) or squares (Bensby
et al., 2014). The gray circles of thin disc stars come from Reddy et al.
(2003). The models are described in Table 2.

Fig. 5 shows on its first column the [C/Fe]-[Fe/H] relation for data and
theoretical models (Group A, B, and C in the first, second, and third row,
respectively). The observational data are color-coded according to their
region of sampling, as labeled in Table 1. We see that all models pass though
the halo stars. Fig. 6 shows that in the low-metallicity regime, Group A
slightly under-produces while Group B overproduces carbon with respect to the
interquartile. Both Group A (Kobayashi et al., 2006) and Group C (NuGrid
Ritter et al., 2018), while managing to reach solar abundances, are unable to
reproduce the slope of thin and thick disc when it comes to carbon abundances.
The only exception to this trend being Model 13 of Group C, where the massive
yields were computed including pre-SN nucleosynthesis and wind nucleosynthesis
to the delayed model. We note however that this model overestimates the
abundances of most halo stars. While both Group B and C show a decreasing
trend from the extremely metal-poor regime, reaching since early epochs a
super-solar enrichment, Group A grows from metal-poor [C/Fe] abundances and
grows in the range -4.5 dex $\lesssim$ [Fe/H] $\lesssim$ -3.5 dex and then
generally plateaus.

####  3.1.2 Nitrogen

Nitrogen is an element with one of the richest nucleosynthetic make ups (e.g.,
Vangioni et al., 2018; Romano et al., 2019). It is produced both as a primary
and as a secondary element666A primary element is synthesized from primordial
gas (H and He), while a secondary element is synthesized from metals. The
production of N as a primary element was at first proposed by Truran & Cameron
(1971) and Talbot & Arnett (1974). Later, Matteucci (1986) pointed out that in
order to explain Galactic halo nitrogen abundances, also massive stars should
have a primary N component. Maeder & Meynet (2000) and Meynet & Maeder (2002)
derived that if massive extremely metal-poor stars have a high initial
rotational speed, they can produce primary N. Nearly all of 14N (the most
abundant nitrogen isotope) forms during hydrostatic hydrogen burning, while
15N is synthesized during both the hot and cold CNO cycles (for a review, see
Wiescher et al., 2010). While 15N is only produced as secondary element, 14N
has a primary and a secondary component.

In Fig. 5 and Fig. 6, the impact of the primary component coming from rotating
low-metallicity massive stars is best exemplified by Group B. The black dashed
curve in Fig. 5 shows sub-solar N enrichment up to $\sim$ [Fe/H] < -2.5,
coming from the secondary component produced by massive stars. With increasing
metallicity, the onset of AGBs brings N up to solar abundances. All the
remaining yields in Group B contain a primary N component synthesized by
massive stars and hence display a plateau at low metallicities, in agreement
with halo stars. Group A yields generally struggle with reproducing low-
metallicity abundances, and Fig. 6 shows that models reside outside the
scatter of halo stars. The models catch up to solar N abundances for disc
stars, after the onset of AGB contributions. Group C models are generally
inconsistent with the data, aside for Model 13, which is the model that
includes pre-SN and wind nucleosynthesis to the yields of massive stars. From
Fig. 7 it becomes clear that none of the models can quite match the super-
solar and flat median of nitrogen abundances (even though Group B gets the
closest), but the combination of Group A and Group B generates a standard
deviation in excellent agreement with the observational data.

####  3.1.3 Oxygen

Oxygen is the better studied and better reproduced metal (Matteucci, 2012). It
is also the most abundant in nearly any astrophysical environment at scales
larger than individual stars (the circumstellar media of some carbon-rich AGB
stars can deplete all the available oxygen into CO molecules Gail et al.,
2009; Boyer et al., 2012). The reason why oxygen is so abundant is because it
is the natural end-product of He-burning (Salpeter, 1952; Clayton, 2003).

Among the 12 elements considered in this work, oxygen is the one best
reproduced by all yield models. Fig. 7 shows that the medians of models and
observational data alike are fairly consistent across metallicity bins.
However, the standard deviations of the original Groups (A, B, and C) are
generally small compared to the observational data values. A better
performance can be seen in group combinations, especially when Group C is
involved. From Fig. 5 it is possible to see that Group A converges to solar
oxygen abundances at solar metallicities, while the different initial
rotational velocities for massive stars in Group B cause the models to span
about 0.3 dex at solar metallicities.

###  3.2 Alpha and odd-Z elements

Figure 9: Similarly to Fig. 5 and Fig. 6, we show the [X/Fe]-[Fe/H] relation
for the elements (Na, Mg, Al and Ca in each column from left to right) and
each group, with the addition of Group ABC – the combination of all three base
groups – on the top row. Similarly to Fig. 6, the data are grouped in four
metallicity bin in the range -4 < [Fe/H] < 0 and the box plot for each bin is
shown in black. The median (dot-dashed orange lines inside the boxes) and
average (dot-dashed grey lines inside the boxes) of the data are also shown.
All of Group A models are shown in solid blue lines, all of Group B models are
shown in dashed red lines, while all of Group C curves are shown in dotted
green lines.

####  3.2.1 Sodium

Sodium, including 22Na and 23Na, are mainly produced by hydrostatic carbon
burning in massive stars dying as SNCC (Woosley & Weaver, 1995; Romano et al.,
2010). Other possible sources can be hydrogen envelope as consequence of the
Neon-Sodium cycle and s-process on 22Ne in Helium burning (Truran & Heger,
2003; Arcones & Thielemann, 2023). As for 22Na, it is the result of proton
capture on 21Ne in the carbon shell (Woosley & Weaver, 1995). Recent
observations based on the Gaia-ESO Survey (Gilmore et al., 2012) indicates
that the Na abundance in stars with mass below $\sim$2.0 $M_{\odot}$ seems to
be unaffected by internal mixing processes. For more massive stars instead,
there is an increasing trend of Na abundance with increasing stellar mass
(Smiljanic et al., 2016).

On the first column of Figure 8, only Group A is in agreement with halo and
disc data. Group B can fit the halo data well but underestimate the disc data
within the scatter but below the interquartile. As for Group C, all models can
predict the halo data but underestimate disc data except Model 13 and Model
16. Model 13 overestimates all data while Model 16 overestimates halo data
only. The scatter of data and groups for Na can be seen in Figure 9, first
column. The halo data displays a larger scatter than disc data. However, this
scatter can only be covered partially by groups except Group A at -4.0 dex
$\lesssim$ [Fe/H] $\lesssim$ -3.0 dex, -1.0 dex $\lesssim$ [Fe/H] $\lesssim$
0.0 dex and Group C at -2.0 dex $\lesssim$ [Fe/H] $\lesssim$ -1.0 dex, which
can be seen clearly from Figure 18. The median and standard deviation of data
and groups can be seen in Figure 10, the first column. One can see that no
group can predict the increasing trend of median of data. All models can
reproduce increasing trend for disc stars but only Group A or Group AB can
predict similar values. Similarly, no group can predict the decreasing trend
of the standard deviation for data, with the exception of Group B. However,
group B cannot predict the value of the standard deviation for halo stars.

Figure 10: In metallicity bins consistent with the rest of the manuscript, we
show the median ($M_{i,j,k}$, _top row_) and standard deviation
($\sigma_{i,j,k}$_bottom row_) for each group and group combination. The black
points are shown in the middle of their metallicity bins (-4.0 dex $\lesssim$
[Fe/H] $\lesssim$ -3.0 dex, -3.0 dex $\lesssim$ [Fe/H] $\lesssim$ -2.0 dex,
-2.0 dex $\lesssim$ [Fe/H] $\lesssim$ -1.0 dex or -1.0 dex $\lesssim$ [Fe/H]
$\lesssim$ 0.0 dex). All of the panels adopt the same colors and styles: data
(grey lines), Group A (blue lines), Group B (red lines), Group C (green
lines), the combination of Group A and Group B (Group AB, purple lines), the
combination of Group A and Group C (Group AC, cyan lines), the combination of
Group B and Group C (Group BC, yellow lines), the combination of Group A and
Group B and Group C (Group ABC, black lines). A description of the statistics
criteria can be found in Section 2.3.2. The data and models used are
consistent with the previous figures, Fig. 8 and Fig. 9.

####  3.2.2 Magnesium

Magnesium, including 24Mg, 25Mg and 26Mg, exists mainly owing to hydrostatic
carbon burning and explosive neon burning in massive stars (Woosley & Weaver,
1995; Romano et al., 2010). Some 24Mg can be produced by silicon burning
(Arcones & Thielemann, 2023). As for 25Mg and 26Mg, the other two Mg isotopes,
are secondary and therefore more sparsely produced. The helium burning phase
reaches sufficiently high temperatures to ignite 22Ne burning into 25Mg
(Truran & Heger, 2003). Then 25Mg captures a neutron and becomes 26Mg.
Nucleosynthesis of some 25Mg and 26Mg can occur in AGB stars. Similar to other
$\alpha-$elements, the abundance ratio of Magnesium provides an important
fossil signature in tracing different disc populations. Recent work suggests
that peculiar histories of star formation are needed to reproduce the
bimodality of abundance ratio of Magnesium shown in the data (Palla et al.,
2022).

In the second column of Figure 8, Group A is in agreement with disc data but
overestimates halo data. Group B underestimates the disc and halo data. Group
C except for model 13, which is in agreement with disc data but overestimates
halo data, can fit the halo data well. The scatter of data and groups for Mg
can be seen in Figure 9. Compared to Na, the scatter of halo data is much
smaller and similar to disc data. Since the scatter of data is very small, it
is hard for groups to be in agreement with data, which can be seen in Figure
18 where only median of data can be covered by Group A at all metallicities.
The median and standard deviation of data and groups can be seen in Figure 10.
One can see that all groups share similar decreasing trend with the median of
the data. Group AB is the closest one. For the standard deviation, only Group
A and Group B show a similar trend with data, but with some differences in the
standard deviation value.

####  3.2.3 Aluminium

Aluminum is mostly produced as a secondary product of hydrostatic carbon
burning and neon burning (Woosley & Weaver, 1995; Truran & Heger, 2003; Romano
et al., 2010). A small percentage of the 27Al is created instead in the
hydrogen burning shells of evolved stars by the MgAl chain (Romano et al.,
2010). As for 26A1, it is synthesized hydrostatically during stellar evolution
and explosively by the supernova shock in the neon or carbon burning phases
(Arcones & Thielemann, 2023). It is also possibly made in the hydrogen shell
and dredged up by the red giant envelope or in the outer part of the helium
core, which is the unburned residual of hydrogen burning (Woosley & Weaver,
1995). Recent observations based on the Gaia-ESO survey indicates that Al
abundance in stars with mass below $\sim$3.0 $M_{\odot}$ seem not to be
overabundant while giants in cluster more massive than this are Al-rich
(Smiljanic et al., 2016) and concludes that Al is underproduced by yields
(Kobayashi et al., 2006; Karakas, 2010; Ventura et al., 2013; Ventura et al.,
2014b, a) with large uncertainties.

In Figure 8, we compare the abundance data and groups for Al. There is a gap
between halo data and disc data at [Fe/H] $\sim$ -1.25. Group B and Group C
can only fit the halo data while Group A can only fit the disc data. Group B
and Model 13 from Group C overestimate the halo data slightly. Figure 9 shows
that this paucity in data causes a large scatter in the -2 < [Fe/H] < -1
metallicity bin compared to either metal-poorer or metal-richer regimes. Due
to the small scatter of Group A and Group B, they cannot cover the scatter of
data at all metallicities. The large Al abundances obtained in Model 13 cause
a large scatter in Group C, large enough to encompass the interquartile of the
data gap in the metal-poor bin. Figure 10, confirms that both median and
standard deviation are hardly reproduced.

Figure 11:  Similarly to Fig. 5, the [X/Fe]-[Fe/H] relation in the Milky Way,
where each column from left to right represents Mn, Ni, and Zn, respectively.
Each row refers to a different group. The models in the first row belong to
Group A and are drawn with a solid line. The models in the middle row belong
to Group B and are drawn with a dashed line. The models in the bottom row
belong to Group C and are drawn with a dotted line. Next we describe the color
patterns. _first row:_ Model 1 (black), Model 2 (dark red), Model 3 (dark
blue), Model 4 (dark green), Model 10 (purple); _second row:_ Model 5 (black),
Model 6 (dark red), Model 7 (dark blue), Model 8 (dark green); _third row:_
Model 9 (black), Model 11 (dark red), Model 12 (dark blue), Model 13 (dark
green), Model 14 (purple), Model 15 (olive), Model 16 (chocolate). The yield
prescriptions of the models are described in Table 2. Concerning the color
patterns for the observational data: halo stars are shown in royal blue, halo
stars as well as thick and thin disc stars are shown in light green, thick and
thin disc stars are shown in pink, thin disc-only stars are shown in gray. All
of the data are shown with empty markers. Specifically, the observational data
sources are displayed as follows: of the halo stars in royal blue, the squares
come from Cayrel et al. (2004), the star symbols come from Reggiani et al.
(2017), the hexagons come from Roederer et al. (2014), while the crosses come
from Mashonkina et al. (2017). Light green halo squares come from (Reddy et
al., 2006). The pink thin and thick disc stars are either circles (Bensby et
al., 2005) or squares (Bensby et al., 2014). The gray circles of thin disc
stars come from Reddy et al. (2003). The models are described in Table 2.

####  3.2.4 Calcium

Calcium and its isotopes are produced by a combination of hydrostatic oxygen
shell burning and explosive oxygen burning (Woosley & Weaver, 1995). It can
also be made by incomplete silicon burning processes (Romano et al., 2010).
Both 40Ca and 42Ca exist primarily by oxygen burning (Woosley & Weaver, 1995;
Truran & Heger, 2003; Arcones & Thielemann, 2023). 41Ca is primarily the
result of neutron capture by abundant 40Ca. 43Ca and 46Ca are made in the neon
shells and carbon shells. Specifically, 44Ca is made almost entirely in the
$\alpha$-rich freezeout, while 48Ca may be made in Type la supernova that
ignite a carbon deflagration very near the Chandrasekhar mass (Woosley &
Weaver, 1995).

Figure 12: Similarly to Fig. 5 and Fig. 6, we show the [X/Fe]-[Fe/H] relation
for the elements (Mn, Ni, and Zn in each column from left to right) and each
group, with the addition of Group ABC – the combination of all three base
groups – on the top row. Similarly to Fig. 6, the data are grouped in four
metallicity bin in the range -4 < [Fe/H] < 0 and the box plot for each bin is
shown in black. The median (dot-dashed orange lines inside the boxes) and
average (dot-dashed grey lines inside the boxes) of the data are also shown.
All of Group A models are shown in solid blue lines, all of Group B models are
shown in dashed red lines, while all of Group C curves are shown in dotted
green lines.

In Figure 8 we compare observational data for Ca and groups. Group A can
nicely predict the halo data and disc data, as can also be seen in Figure 18,
while Group B overestimates disc data due to an increase at -2.0 dex
$\lesssim$ [Fe/H] $\lesssim$ -0.0 dex. Except for Model 16, Group C
underestimates all data. The data scatter is very small at all metallicities
in Figure 9. Similar to Mg, the medians of all groups share similar trends
with the median of the data, as is shown in Figure 10. Group A and Group AB
are in excellent agreement. When comparing the standard deviation of groups
and data, it shows that only Group A can reproduce the standard deviation of
data at all metallicities.

###  3.3 Iron-peak elements

Depending on modeling assumptions such as IMF or SNIa rates, SNIa may be
responsible between 50% and 66% of the production of isotopes around the iron
peak (Prantzos et al., 2018; Matteucci, 2012).

Figure 13: In metallicity bins consistent with the rest of the manuscript, we
show the median ($M_{i,j,k}$, _top row_) and standard deviation
($\sigma_{i,j,k}$_bottom row_) for each group and group combination. The black
points are shown in the middle of their metallicity bins (-4.0 dex $\lesssim$
[Fe/H] $\lesssim$ -3.0 dex, -3.0 dex $\lesssim$ [Fe/H] $\lesssim$ -2.0 dex,
-2.0 dex $\lesssim$ [Fe/H] $\lesssim$ -1.0 dex or -1.0 dex $\lesssim$ [Fe/H]
$\lesssim$ 0.0 dex). All of the panels adopt the same colors and styles: data
(grey lines), Group A (blue lines), Group B (red lines), Group C (green
lines), the combination of Group A and Group B (Group AB, purple lines), the
combination of Group A and Group C (Group AC, cyan lines), the combination of
Group B and Group C (Group BC, yellow lines), the combination of Group A and
Group B and Group C (Group ABC, black lines). A description of the statistics
criteria can be found in Section 2.3.2. The data and models used are
consistent with the previous figures, Fig. 11 and Fig. 12.

####  3.3.1 Manganese

Manganese has only one stable isotope, that being 55Mn. It is primarily
synthesized during silicon burning and $\alpha$-rich freezeout (Romano et al.,
2010; Nissen, 2016). Like the other iron-peak elements, it is an important
feature of Type Ia supernovae, as shown in early spherically symmetric
simulations of the carbon-deflagration model W7 (Nomoto et al., 1984;
Thielemann et al., 1986). Manganese can also be produced in the inner regions
of explosive silicon burning in Chandrasekhar-mass deflagration models through
the decay of 55Co (Maeda et al., 2010; Travaglio et al., 2011). Silicon
burning also occurs in core collapse supernovae (Thielemann et al., 1996;
Woosley et al., 2002; Nomoto et al., 2013). At metal-poor regimes ([Fe/H] <
-1), SNCC will dominate the Mn enrichment, while at higher metallicities where
finally SNIa start occurring, both SNIa and SNCC yield roughly equal
contributions of Mn. This increasing trend can be seen in all models in Figure
11 and Figure 12, except for Model 7 and Model 13. Model 7 assumes the highest
initial rotational velocity for massive stars (300 km/s) for LC18, which is
unrealistic given that such conditions are more likely to occur at high
metallicities only. For roughly all models, the standard deviation is small,
but larger than the uncertainty of the data (Figure 12 and 13). As for the
median, with the exception of Group B, all models display a near-plateau in
the metal-poor regions, and then an increase with the onset of SNIa.

####  3.3.2 Nickel

Nickel comes in a variety of isotopes from an atomic mass of 56 to 64. Species
56 and 57 are radioactive $\gamma$-ray emitters in supernovae. These Ni
isotopes rapidly decay to their Fe isobars. 58Ni is the most abundant Ni
isotope, accounting for nearly 2/3 of all Ni species (Clayton, 2003). Ni
species are synthesized like the other iron-peak elements during the explosive
silicon burning and alpha-rich freezeout in supernovae. Both SNIa and SNCC
produce Ni, in roughly equal parts (e.g., Arcones & Thielemann, 2023;
Matteucci, 2012; Andrews et al., 2017).

In Figure 11, the yields of Group B have this striking flat [Ni/Fe] feature.
Prantzos et al. (2018) who uses the same yield set (Limongi & Chieffi, 2018)
for massive stars, obtains a moderate rise of [Ni/Fe] with metallicity. They
attribute the rise to the known overproduction of Ni in the W7 models (e.g.
Iwamoto et al., 1999, as seen in Figure 3). Our fiducial SNIa yield is
Seitenzahl et al. (2013), which indeed yields less Ni compared to W7.

####  3.3.3 Zinc

Zinc in synthesized both as an s-process element and through the alpha-rich
freezeout of SNCC (Fröhlich et al., 2006). The latter requires very high
electron fractions with an excess of 0.5 in the layers undergoing Si burning.
The standard deviation of Zn stellar abundances is quite small (Figure 12 and
Figure 13) and a modest decreasing trend toward solar values with increasing
metallicity can be appreciated in the data. Figure 11 shows that while Group A
seem to be in good agreement with the data, Group B and Group C fail at
reproducing the trends. Group B consistently underproduces Zn, while Group C
both wildly overproduces and underproduces Zn

###  3.4 Neutron-capture elements

Isotopes beyond the iron peak are primarily synthesized through neutron
capture. An isotope inundated by a neutron flux will have a probability to
capture a neutron which depends on the neutron density in a flux (detailed
calculations first proposed by Burbidge et al., 1957). The astrophysical sites
where neutron capture occurs are predominantly clustered around two peaks: one
originates from low neutron-density fluxes, so that the neutron capture occurs
slowly with respect to its beta decay (s-process) – and the other originates
from catastrophic events whose fluxes are very neutron rich. In the latter
case, the neutron capture occurs rapidly (r-process) relative to the isotope’s
beta decay timescale. The two principal sites of s-process enrichment are the
hydrostatic core He-burning and shell C-burning of both LIMs during their AGB
phase and of massive stars (Prantzos et al., 2020). There is yet no definitive
evidence on which are the dominant sites for r-process enrichment. NSM have
been the leading candidates for decades (Lattimer & Schramm, 1974; Thielemann
et al., 2017), and the gravitational wave event GW170817 (Abbott et al., 2017)
and the associated kilonova AT2017gfo (Smartt et al., 2017) confirmed for the
first time, as evidenced by the direct observation of strontium in day-1
spectra (Watson et al., 2019) that NSM do indeed produce r-process elements.
However, despite their potential as a significant r-process source, NSM alone
have repeatedly been shown to struggle in explaining the high r-process
enrichment of the most metal-poor stars, even when dynamics modeling is
included (Argast et al., 2004; van de Voort et al., 2020). Nevertheless, some
studies have argued the contrary (e.g., Shen et al., 2015). Within
hydrodynamical simulations, Haynes & Kobayashi (2019) suggest that NSM in
conjunction with magneto-hydrodynamical supernovae could describe the
abundance of r-process elements, as exemplified by europium abundances. Yet
the authors acknowledge that NSM alone are insufficient, a result later
confirmed by Kobayashi et al. (2023).

Figure 14: The [X/Fe]-[Fe/H] relation in the Milky Way, where each column from
left to right represents Ba and Eu, respectively. The color patterns for
observational data: data contained halo stars only are colored royal blue,
data contained thin disc only are colored gray, data contained both thick and
thin disc are colored pink, data contained halo, thick and thin disc are
colored light green. All of them are empty markers. Specifically, the
observational data sources are labeled as follows: pink empty circles Bensby
et al. (2005); pink empty squares Bensby et al. (2014); gray empty circles
Reddy et al. (2003); light green empty squares Reddy et al. (2006); royal blue
empty hexagons Roederer et al. (2014); light green empty circles Zhao et al.
(2016); royal blue empty stars Reggiani et al. (2017); royal blue empty
crosses Mashonkina et al. (2017). The models in the first row belong to Group
A and are drawn with a solid line. The models in the middle row belong to
Group B and are drawn with a dashed line. The models in the bottom row belong
to Group C and are drawn with a dotted lne. Next we describe the color
patterns. _first row:_ Model 1 (black), Model 2 (dark red), Model 3 (dark
blue), Model 4 (dark green), Model 10 (purple); _second row:_ Model 5 (black),
Model 6 (dark red), Model 7 (dark blue), Model 8 (dark green); _third row:_
Model 9 (black), Model 11 (dark red), Model 12 (dark blue), Model 13 (dark
green), Model 14 (purple), Model 15 (olive), Model 16 (chocolate). The models
are described in Table 2 Figure 15: Similarly to the previous box-plots, we
show the [X/Fe]-[Fe/H] relation for the elements (Ba and Eu in each column
from left to right) and each group, with the addition of Group ABC – the
combination of all three base groups – on the top row. Similarly to Fig. 6,
the data are grouped in four metallicity bin in the range -4 < [Fe/H] < 0 and
the box plot for each bin is shown in black. The median (dot-dashed orange
lines inside the boxes) and average (dot-dashed grey lines inside the boxes)
of the data are also shown. All of Group A models are shown in solid blue
lines, all of Group B models are shown in dashed red lines, while all of Group
C curves are shown in dotted green lines.

####  3.4.1 Barium

Both the s-process and the r-process yield a significant amount of Ba. The
s-process, occurring at later stages of galaxy evolution with the onset of AGB
enrichment, cannot account for the enrichment at low metallicities (below
[Fe/H] $\lesssim$ -2.5). Pagel & Tautvaisiene (1997) proposed that at the
earliest stages, Ba may be produced as an r-process element. The authors
suggested that massive stars may be the source of r-process enrichment. The
necessity for a fast enrichment component was confirmed in detailed GCE
studies (Cescutti et al., 2006). However, the standard evolution of massive
stars cannot reach the r-process regimes, therefore other processes associated
with massive stars such as magnetorotational hydrodynamic jets or collapsars
could be at play (Limongi & Chieffi, 2018; Siegel et al., 2019; Shibagaki et
al., 2016). The onset of NSM is delayed, therefore this mechanism also cannot
be responsible for early Ba enrichment (Kobayashi et al., 2020; Yamazaki et
al., 2022).

As seen in Figure 14, in the halo, or otherwise at very metal-poor
metallicities, the abundance of Ba is low, ranging from [Ba/Fe] = -2 to
moderately super solar abundances. In Group A, the variation among the models
is caused by the slight difference in iron production over time. None of the
models in Group A contain s-process enrichment, with the exception of Model 1.
The steep increase at high metallicities of the black curve is caused by the
AGB s-process enrichment from the F.R.U.I.T.Y. yields. Group B and group C
models also include the s-process. If rapidly rotating massive stars (Model 7
in dark blue of Group B) were to occur at high metallicities, they also could
cause a dramatic increase in Ba abundances through the s-process. The models
in all groups include r-process enrichment through NSM.

Figure 15 the subsolar interquartile range shows clearly that amid many
outliers, the standard deviation of Ba is large at low metallicities and small
at high metalliciites. None of the models display a particularly good fit to
the data, with the exception of Model 8 in Group B, i.e. the set averaged
across the models of varying rotational velocities. Figure 16 shows very poor
agreement between models and data. All models fall below the median values in
nearly all metallicity bins, and the scatter trends show generally little
correlation.

Figure 16: In metallicity bins consistent with the rest of the manuscript, we
show the median ($M_{i,j,k}$, _top row_) and standard deviation
($\sigma_{i,j,k}$_bottom row_) for each group and group combination. The black
points are shown in the middle of their metallicity bins (-4.0 dex $\lesssim$
[Fe/H] $\lesssim$ -3.0 dex, -3.0 dex $\lesssim$ [Fe/H] $\lesssim$ -2.0 dex,
-2.0 dex $\lesssim$ [Fe/H] $\lesssim$ -1.0 dex or -1.0 dex $\lesssim$ [Fe/H]
$\lesssim$ 0.0 dex). All of the panels adopt the same colors and styles: data
(grey lines), Group A (blue lines), Group B (red lines), Group C (green
lines), the combination of Group A and Group B (Group AB, purple lines), the
combination of Group A and Group C (Group AC, cyan lines), the combination of
Group B and Group C (Group BC, yellow lines), the combination of Group A and
Group B and Group C (Group ABC, black lines). A description of the statistics
criteria can be found in Section 2.3.2. The data and models used are
consistent with the previous figures, Fig. 14 and Fig. 15.

####  3.4.2 Europium

The two stable isotopes of Eu, 151 and 153, are produced exclusively through
r-process enrichment (Prantzos et al., 2020; Cowan & Thielemann, 2004). As
such, paired with Ba, Eu is often used as a discriminating indicator on
whether heavy elements in a star originate predominantly from s- or r-process
enrichment (e.g., Aguado et al., 2021). Subsolar [Ba/Eu] abundances would
indicate a predominance of r-process enrichment while supersolar [Ba/Eu]
abundances would instead favor s-process enrichment.

Figure 14 and 15 show that the models barely reach the lower data scatter for
Group A, or the 25th percentile for group B. Similar underproduction can also
be observed for Group C, with the exception of Model 13, where the pre-SN
nucleosynthesis and wind nucleosynthesis are added to the delayed model. Given
that none of the massive yields in our models produce r-process enrichment,
the abundance trends are exclusively dictated by the addition of NSM onto the
base GCE models. These results calibrated on a Milky-Way-like galaxy are in
agreement with literature results in that NSM occur too late in the chemical
enrichment history to explain the wide spread of the abundances for Eu
enhancement, especially at low metallicities.

###  3.5 Present-day abundance ratios

In this last result presented in Fig. 17, we show the abundance produced by 3
models selected from 3 different groups (Model 1, 8, 16) for all of the 12
elements. The abundances have been normalized to solar values. The horizontal
lines identify a range of $\pm$ 0.15 dex. These 3 models show the best
performance in their corresponding group. Model 1 underestimates Carbon
abundance and overestimates Oxygen and Sodium abundance. As for other
elements, the difference between reproduced abundance and solar abundance is
less than 0.15 dex. The other 2 models (Group B and Group C) underestimate the
abundance of Magnesium, Aluminium and Barium. Model 8 (Group B) overestimates
Nickel while Model 16 (Group C) overestimates Manganese, Nickel and Europium.

Figure 17: Abundance ratios over iron of the 12 elements investigated in the
present work normalized to solar values. The abundances are taken wherever the
model reaches [Fe/H] = [Fe/H]⊙. The x-axis indicates the atomic number. The
elements are highlighted by the vertical (dark-red dot-dashed) lines. The
black curves refer to the various models. We choose one representative model
from each group: Model 1 from Group A (solid line), Model 8 from Group B
(dashed line) and Model 16 from Group C (dotted line). The horizontal dark
blue dot-dashed lines mark a $\pm$ 0.15 dex deviation from the solar
normalization.

##  4 Discussion and Conclusions

In this paper we investigated the impact of the choice of yield tabulations on
chemical evolution runs produced by a single NuPyCEE parametrization, fine-
tuned to the Milky Way. We compared the model variations with observational
data coming from the major Galactic environments: thin disc, thick disc, and
halo – and combinations thereof. The summary of statistics for groups and
their combinations for all elements compared with the stellar abundance data
sorted in the same metallicity bins are shown in Fig. 18 and Fig. 19. The
yield variations in our study are divided into three groups: Group A, which
incorporates different hypernovae fractions; Group B, which varies the initial
rotational velocity of massive stars; and Group C, which explores variations
in the delayed/rapid explosion prescriptions for massive stars. The criteria
of our judgements for both figures are defined in Sec. 2.3.2.

Figure 18: For each of Group A, B, and C, the summary of the Judgement
criteria (Median, standard deviation, and scatter, Section 2.3.2) for the four
metallicity bins in the range -4 < [Fe/H] < 0\. Each pie chart is displayed in
increasing metallicity from left to right. Each shell refers to one of the 12
elements investigated in this work. From the inner to the outer shell, we show
carbon, nitrogen, oxygen, sodium, magnesium, aluminum, calcium, manganese,
nickel, zinc, barium and europium. The left slice in each pie chart depicts
Group A, the right slice Group B, while the bottom slice Group C. Median,
standard deviation, and scatter are shown in red, blue, and black,
respectively. If the shell slice has a solid color, it means that the
Judgement for the statistical quantity for the observed abundances is in
agreement with the models. If the shell slice is slightly fainter with a
dotted texture, it means that the data are overestimated by the models. For
the faintest shell slices with a striped texture, the data are underestimated
by the model. Figure 19: Similarly to Fig. 18, we show the pie charts for
group combinations: Group AB is represented in the top left slice, Group AC in
the top right slice, Group BC in the bottom right slice, and all the models
are combined in the bottom left slice with Group ABC.

Our main conclusions based on Fig. 18 and Fig. 19 can be summarized as
follows:

  1. 1.

The medians for single groups as well as group combinations is in good
agreement with the Galactic observational data for most elements up until
Zinc. However, neither the scatter nor the standard deviation of groups and
group combinations tend to correlate with observational trends. Specifically,
Group A and Group B underestimate most of the standard deviation and scatter
of most of the data while Group C overestimates them. As for the median, Group
A can predict nearly all of the data at all metallicities while Group B can
predict about half of the data at all metallicities. Group C can only predict
about half of the data at low metallicities and underestimates most of the
data at -2.0 dex $\lesssim$ [Fe/H] $\lesssim$ 0.0 dex.

     * $\bullet$

Combining 2 or 3 groups at a time, the underestimated standard deviation and
scatter of data only occupy a relatively small fraction at all metallicities.
Although these combinations increase the fraction of the reproduced data
scatter and reproduced standard deviation at low metallicities, they also
increase the fraction of overestimated scatter and standard deviation.
Standard deviation and scatter of data tend to be overestimated in the range
-2.0 dex $\lesssim$ [Fe/H] $\lesssim$ 0.0 dex.

  2. 2.

We draw the following conclusions regarding the breakdown of elemental groups:

     * $\bullet$

For CNO elements, Group A and Group B underestimate most of the standard
deviations of these 3 elements while Group C is in agreement with all of the
standard deviations at -2.0 dex $\lesssim$ [Fe/H] $\lesssim$ 0.0 dex.
Regarding the scatter, it appears that the majority of the estimates are too
small for all three groups.. Most of the standard deviation of these 3
elements can be reproduced by group combinations at all metallicities.

     * $\bullet$

In the case of Odd-Z elements (Na and Al in our sample), we found that the
median value of Na can be accurately reproduced by two of the groups. Group C
performs well at lower metallicities, while it underestimates Na at -2.0 dex
$\lesssim$ [Fe/H] $\lesssim$ 0.0 dex. However, Group C is in agreement with
the standard deviation and most of the scatter for Na, whereas the other two
groups tend to underestimate these values. In contrast to Na, Al does not
exhibit clear trends.

     * $\bullet$

In the case of $\alpha$ elements in our sample (Mg and Ca in our sample), we
found that Group A and Group B are able to reproduce most of the standard
deviation and scatter of Ca, while Group C tends to overestimate these values.
However, when it comes to the median of Ca, Group A and Group B are in good
agreement at all metallicities, while Group C only agrees with the median at
low metallicities. For Mg, both Group A and Group B tend to underestimate the
standard deviation and scatter, while Group C still overestimates them. The
trend for medians is similar for both Ca and Mg. In general, the median values
of both Ca and Mg can be accurately reproduced for any combination of groups,
although the majority of the standard deviation and scatter tend to be
overestimated.

     * $\bullet$

Our analysis of iron-peak elements shows that Group A is able to accurately
reproduce most of the median and standard deviation values of these elements
at all metallicities, while Group C tends to overestimate them, as shown in
Fig. 18. Group B is able to accurately reproduce the median values of these
elements, with the exception of Zn, but can be in agreement with the standard
deviation of Zn at -4.0 dex $\lesssim$ [Fe/H] $\lesssim$ -1.0 dex. When
combining the three groups, the median values of iron-peak elements can be
accurately reproduced at all metallicities, with the exception of a few data
points for Zn at low metallicities. However, all of the standard deviation
values for these elements tend to be overestimated by the combined groups, as
shown in Fig. 19. It’s worth noting that iron-peak elements can be reproduced
more accurately by the combination of the three groups than other elements,
which are primarily reproduced by Group A.

     * $\bullet$

Neutron-capture elements (Ba and Eu), exhibit the most significant tension
between observational data and theoretical models. Either single group or
groups combination underestimate most of the median of Eu and Ba. As for the
standard deviation, Group B and group combinations can be in agreement with
most of them at low metallicities. At -2.0 dex $\lesssim$ [Fe/H] $\lesssim$
0.0 dex, single group and groups combinations tend to overestimate Ba and
underestimate Eu. As for scatter, either single group or group combination
overestimate most of them.

This paper introduces statistical criteria that compare the median, standard
deviation, and scatter of observational data with different groups of models,
providing a measure of the ability of each yield combination to match
observational data. By analyzing the effectiveness of individual yields in
describing average Galactic patterns, we highlight the impact of various yield
modeling assumptions on theoretical tracks. Our work shows that the abundance
scatter patterns in the Milky Way cannot be explained solely by the
simultaneous application of the proposed yield prescriptions.

Our objective in this analysis was to investigate whether multiple yield
prescriptions and their respective scenarios could be simultaneously at play
in the evolution of our Galaxy and partially responsible for the scatter of
the data. We proposed statistical criteria that compares the median, standard
deviation, and scatter of observational data against groups of models to
evaluate the ability of each yield combination to match observational data.
However, our proposed method is inconclusive due to several factors. First, a
realistic mixing of yield prescriptions cannot be described by a simple
average of the models without taking into account weights or time/metallicity
dependence. Introducing additional modeling parameters and assumptions to
describe the variation of the occurrence of each yield prescription over time
would be necessary, but this goes beyond the scope of our analysis. Second,
the average of evolutionary tracks is different from a single evolutionary
track containing an average or mixing of the yields, with the latter being
more realistic.

One-zone GCE models are appropriate for describing global evolutionary trends
of elemental abundances. However, in order to analyze the spread in
observational data, it is insufficient to only vary yields on a fixed galactic
environment. The source of the spread in the data is likely due to varying
physical or dynamical properties of the local environment, rather than the
simultaneous involvement of multiple nucleosynthetic mechanisms.

While many yield prescriptions are generally effective at reproducing median
Galactic trends, the inconsistency of scatter and standard deviation leads us
to conclude that yield mixing may not be as significant a factor in the
observed scatter of the data compared to physical properties or dynamical
effects. At the same time, we cannot rule out that multiple yield
prescriptions could be occurring, because several of these prescriptions are
equally effective in describing average trends. To further investigate our
objectives, improvements can be achieved through the use of grids of GCE
models spanning a range of physical parameters, multi-zone GCE models, or
hydrodynamical simulations.

Out of all the elements analyzed in this study, neutron-capture elements are
the least accurately reproduced. Our analysis only accounts for merging
neutron stars as an r-process enrichment channel. We therefore highlight the
importance of exploring alternative production channels, such as collapsars
and magneto-hydrodynamic jets.

## Acknowledgements

We thank the anonymous referee for the meticulous feedback. We thank Benoit
Côté for providing the complete Milky Way NuPyCEE parametrization of his 2019
work (Côté et al., 2019) and for fruitful discussions. E.G. acknowledges the
support of the National Natural Science Foundation of China (NSFC) under
grants NOs. 12173016, 12041305. E.G. acknowledges the science research grants
from the China Manned Space Project with NOs. CMS-CSST-2021-A08, CMS-
CSST-2021-A07. E.G. acknowledges the Program for Innovative Talents,
Entrepreneur in Jiangsu. All the authors have been supported by the National
Natural Science Foundation of China under grant (No.11922303) and the
Fundamental Research Funds for the Central Universities (No.2042022kf1182).

## Data Availability

No new data were generated or analysed in support of this research.

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