# The Scale of Stellar Yields: Implications of the Measured Mean Iron Yield of
Core Collapse Supernovae

David H. Weinberg The Department of Astronomy and Center of Cosmology and
AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA
Emily J. Griffith NSF Astronomy and Astrophysics Postdoctoral Fellow  Center
for Astrophysics and Space Astronomy, Department of Astrophysical and
Planetary Sciences, University of Colorado, 389 UCB, Boulder, CO 80309-0389,
USA  James W. Johnson The Department of Astronomy and Center of Cosmology and
AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA  The
Observatories of the Carnegie Institution for Science, 813 Santa Barbara St.,
Pasadena, CA, 91101  Todd A. Thompson The Department of Astronomy and Center
of Cosmology and AstroParticle Physics, The Ohio State University, Columbus,
OH 43210, USA

###### Abstract

The scale of $\alpha$-element yields is difficult to predict from theory
because of uncertainties in massive star evolution, supernova physics, and
black hole formation, and it is difficult to constrain empirically because the
impact of higher yields can be compensated by greater metal loss in galactic
winds. We use a recent measurement of the mean iron yield of core collapse
supernovae (CCSN) by Rodriguez et al. (RMN23), $\bar{y}_{\rm Fe}^{\rm
cc}=0.058\pm 0.007M_{\odot}$, to infer the scale of $\alpha$-element yields by
assuming that the plateau of $[\alpha/{\rm Fe}]$ abundance ratios observed in
low metallicity stars represents the yield ratio of CCSN. For a Kroupa (2001)
initial mass function and a plateau at $[\alpha/{\rm Fe}]_{\rm cc}=0.45$, we
find that the population-averaged yields of O and Mg are about equal to the
solar abundance of these elements, $\log y_{\rm O}^{\rm cc}/Z_{{\rm
O},\odot}=\log y_{\rm Mg}^{\rm cc}/Z_{{\rm Mg},\odot}=-0.01\pm 0.1$, where
$y_{\rm X}^{\rm cc}$ is the mass of element X produced by massive stars per
unit mass of star formation. The inferred O and Fe yields agree with
predictions of the Sukhbold et al. (2016) CCSN models assuming their Z9.6+N20
neutrino-driven engine, a scenario in which many progenitors with
$M<40M_{\odot}$ implode to black holes rather than exploding. The yields are
lower than assumed in some models of galactic chemical evolution (GCE) and the
galaxy mass-metallicity relation, reducing the level of outflows needed to
match observed abundances. For straightforward assumptions, we find that one-
zone GCE models with $\eta=\dot{M}_{\rm out}/\dot{M}_{*}\approx 0.6$ evolve to
solar metallicity at late times. The ISM D/H ratio predicted by these models
is about 70% of the primordial D/H ratio, which is lower than observational
estimates but consistent at $2\sigma$. By requiring that models reach
$[\alpha/{\rm Fe}]\approx 0$ at late times, and assuming a mean Fe yield of
$0.7M_{\odot}$ per Type Ia supernova, we infer a Hubble-time integrated SNIa
rate of $1.1\times 10^{-3}M_{\odot}^{-1}$, compatible with estimates from
supernova surveys. The RMN23 measurement provides one of the few empirical
anchors for the absolute scale of nucleosynthetic yields, with wide-ranging
implications for stellar and galactic astrophysics.

##  1 Introduction

The nucleosynthetic yields of elements are the most basic ingredient of
galactic chemical evolution (GCE) models because they determine the rate at
which stars enrich their surroundings. The relative yields of different
elements can be constrained empirically through abundance ratios in stellar
populations, using time delays and theoretical models as a guide to separate
the contributions of different sources such as core collapse supernovae
(CCSN), Type Ia supernovae (SNIa), and asymptotic giant branch (AGB) stars
(e.g., Griffith et al. 2019, 2022; Weinberg et al. 2019, 2022; Johnson et al.
2023). However, the absolute scale of yields is difficult to constrain
empirically because it is largely degenerate with the effect of outflows
(a.k.a. galactic winds), which remove newly produced metals from the star-
forming interstellar medium (ISM). It is widely accepted that the mass-
metallicity relation (Tremonti et al., 2004; Andrews & Martini, 2013) is
primarily driven by an increasing efficiency of outflows in the shallower
potential wells of lower mass galaxies (Finlator & Davé, 2008; Peeples &
Shankar, 2011; Davé et al., 2012; Zahid et al., 2012; Lin & Zu, 2022). For the
Milky Way, on the other hand, some GCE models assume substantial metal
outflows (e.g., Schönrich & Binney 2009a; Johnson et al. 2021) while others do
not (e.g., Minchev et al. 2013; Spitoni et al. 2019), with both classes
reproducing observed chemical abundances because they adopt significantly
different population-averaged yields.111We use the term population-averaged to
encompass averaging over the stellar initial mass function (IMF) and other
properties such as rotation and binary fractions that may affect yields. The
absolute scale of yields is also crucial for assessing the heavy element
budget of galaxies and predicting the abundance of metals in the circumglactic
medium (e.g., Peeples et al. 2014).

Predicting nucleosynthetic yields of CCSN is a long-standing goal of supernova
modeling (e.g., Woosley & Weaver 1995; Chieffi & Limongi 2013; Nomoto et al.
2013; Sukhbold et al. 2016; Limongi & Chieffi 2018; Curtis et al. 2021). Given
yields as a function of progenitor mass and metallicity, there are two further
challenges in calculating the population-averaged yield. The first is the
choice of IMF; for example, models such as Minchev et al. (2013) and Spitoni
et al. (2019) have a low population-averaged $\alpha$-element yield because
they adopt a steep IMF (Scalo, 1986; Kroupa et al., 1993), while models such
as Schönrich & Binney (2009b) and Johnson et al. (2021) adopt a Kroupa (2001)
IMF with larger numbers of high mass stars. Vincenzo et al. (2016) demonstrate
the large impact that the choice of IMF can have on population-averaged
yields. The second and perhaps even thornier challenge is the uncertain
physics of black hole formation, because in many cases the massive star
progenitors that are most efficient in producing specific elements are also
those most susceptible to forming black holes and releasing no heavy elements
at all (e.g., Sukhbold et al. 2016, hereafter S16). Griffith et al. (2021,
hereafter G21) show that plausible variations in the degree of black hole
formation can produce a factor of three variation in the population-averaged
CCSN yield of O and Mg.

In a recent study, Rodriguez et al. (2022, hereafter RMN23) estimate the mean
Fe yield of stripped envelope CCSN (Type Ib, Ic), which they combine with a
similar estimate for Type II CCSN (Rodríguez et al., 2021) and the relative
frequency of CCSN types (Shivvers et al., 2017) to infer the mean Fe yield of
CCSN, $\bar{y}_{\rm Fe}^{\rm cc}=0.058\pm 0.007M_{\odot}$. The Fe yield of
individual supernovae can be estimated from the late-time light curve, which
is powered by the radioactive decay of 56Ni to 56Co, leading after a further
decay to 56Fe. Here we explore the implications of the RMN23 yield
determination for other population-averaged yields, for galactic outflows, for
black hole formation, and for the time-integrated rate of SNIa. In addition to
Fe, we focus on O and Mg, which are both $\alpha$-elements thought to come
almost entirely from CCSN (Andrews et al., 2017; Rybizki et al., 2017;
Johnson, 2019), and on the $\alpha$-element Si, which has a sub-dominant but
not negligible contribution from SNIa (Griffith et al., 2022; Weinberg et al.,
2022).

Our basic assumption — standard in GCE modeling — is that the plateau observed
in [O/Fe], [Mg/Fe], and [Si/Fe] at low metallicity reflects the population-
averaged yield ratios of CCSN, with the decline at higher metallicity caused
by the SNIa contribution to Fe. Combined with the RMN23 value of $\bar{y}_{\rm
Fe}^{\rm cc}$, the observed plateau level allows us to infer the mean yield
per supernova of O, Mg, and Si, which we denote $\bar{y}_{\rm O}^{\rm cc}$,
$\bar{y}_{\rm Mg}^{\rm cc}$, $\bar{y}_{\rm Si}^{\rm cc}$. What matters for GCE
models is the yield per unit mass of stars formed, which we denote with $y$
instead of $\bar{y}$. Going from $\bar{y}$ (with units of $M_{\odot}$) to $y$
(dimensionless) requires a choice of IMF and the fraction $F_{\rm exp}$ of
massive stars ($M>8M_{\odot}$) that explode as CCSN. The theoretical
uncertainty in $F_{\rm exp}$ is significant (see §2.2), but it is much smaller
than the uncertainty in the yield because the number of massive stars is
dominated by those of relatively low mass that are relatively easy to explode.

Our translations from $y_{\rm X}^{\rm cc}$ values to outflow constraints rely
on the analytic GCE models of Weinberg et al. (2017, hereafter WAF), with
arguments similar to those made in the mass-metallicity context by Peeples &
Shankar (2011), Davé et al. (2012), Zahid et al. (2012), and Lin & Zu (2022),
and in the dwarf galaxy context by Johnson et al. (2022b) and Sandford et al.
(2022). The drop between the plateau values of [X/Fe] and the solar ratios,
$[{\rm X}/{\rm Fe}]\approx 0$, characteristic of the present-day local disk
depends mainly on the population-averaged Fe yield of SNIa relative to that of
CCSN. Because the mean Fe yield per SNIa is reasonably well established at
$\bar{y}_{\rm Fe}^{\rm Ia}=0.6-0.7M_{\odot}$ (e.g., Howell et al. 2009),
$\bar{y}_{\rm Fe}^{\rm cc}$ can be used to normalize the SNIa delay time
distribution (DTD) and hence the number of SNIa produced per unit mass of star
formation. We also examine what the inferred Fe, O, Mg, and Si yields imply
for the CCSN models of S16/G21, in particular about which massive stars
explode.

Throughout this paper we assume a Kroupa (2001) IMF, with $dN/dM\propto
M^{-2.3}$ for $M=0.5-120M_{\odot}$ and $dN/dM\propto M^{-1.3}$ for
$M=0.08-0.5M_{\odot}$. Although the functional form is different, this IMF is
similar to that of Chabrier (2003), with both having a similar high mass slope
but fewer low mass stars than a Salpeter (1955) IMF ($dN/dM\propto
M^{-2.35}$). We refer to our choice as simply a Kroupa IMF, but we caution
that it is quite different from that of Kroupa et al. (1993), which has a
$-2.7$ high mass slope and thus leads to much smaller predicted yields
(Vincenzo et al. 2016; G21).

In Section 2 we combine the RMN23 value of $\bar{y}_{\rm Fe}^{\rm cc}$ with
the estimated level of the $[\alpha/{\rm Fe}]$ plateau to infer the overall
scale of CCSN element yields. In Section 3 we examine the implications of this
inferred scale for galactic outflows, for the SNIa rate, for theoretical
models of CCSN and black hole formation, and for the deuterium abundance of
the ISM. The logical flow of these arguments is summarized in Figure 5
(Section 4.1). In the remainder of Section 4 we discuss sources of uncertainty
in our results and broader implications for the mass-metallicity relation and
the chemical evolution of the Milky Way. Section 5 summarizes our conclusions.

##  2 Inferring the scale of yields

###  2.1 The $[\alpha/{\rm Fe}]$ plateau

In our analysis below we consider O, Mg, and Si as representative
$\alpha$-elements, using all three because each is affected by different
observational and theoretical uncertainties. Our goal is to combine the RMN23
estimate of $\bar{y}_{\rm Fe}^{\rm cc}$ with empirical estimates of $[{\rm
X}/{\rm Fe}]_{\rm cc}$ for these elements to infer their IMF-averaged CCSN
yields, where $[{\rm X}/{\rm Fe}]_{\rm cc}$ represents the element ratio that
would be produced by CCSN alone. In most observational studies, the values of
[O/Fe], [Mg/Fe], and [Si/Fe] show an approximately flat trend (with
significant scatter) in Milky Way halo stars with $-2\leq[{\rm Fe/H}]\leq-1$.
This plateau is usually taken to reflect the yield ratio of CCSN, on the
assumption that SNIa have not yet contributed significantly to the Fe
abundances of these stars. However, the value of the plateau varies noticeably
from study to study. For the most part these differences reflect the
systematic uncertainties in determining the absolute abundances from observed
stellar spectra, which are more acute in low metallicity stars because they
are further from solar calibration and may be more susceptible to the impact
of departures from local thermodynamic equilibrium on spectral synthesis.

The pioneering study of halo populations by Nissen & Schuster (2010) exhibits
plateaus at $[{\rm Mg/Fe}]\approx 0.35$ and $[{\rm Si/Fe}]\approx 0.3$ for
stars they identify with the in situ halo. Bensby et al. (2017) show trends
from microlensed bulge stars and the solar neighborhood sample of Bensby et
al. (2014), exhibiting a similar Si plateau and a slightly higher Mg plateau
at $[{\rm Mg/Fe}]\approx 0.4$. Kobayashi et al. (2020) present compilations of
$[{\rm X}/{\rm Fe}]$ measurements from many data sets. For Mg, the Zhao et al.
(2016) and Reggiani et al. (2017) data sets imply a plateau at $[{\rm
Mg/Fe}]\approx 0.3$, but the Andrievsky et al. (2010) data imply a higher
$[{\rm Mg/Fe}]\approx 0.6$ for very low metallicity stars with $-3.5\leq[{\rm
Fe/H}]\leq-2.5$. For Si, the Zhao et al. (2016) data set implies a plateau at
$[{\rm Si/Fe}]\approx 0.3$, while the values from Cayrel et al. (2004) and
Honda et al. (2004) are $\sim 0.2$ dex higher. For O, the Zhao et al. (2016)
and Amarsi et al. (2019) data sets imply a plateau at $[{\rm O/Fe}]\approx
0.6$, though it is not perfectly flat.

Figure 1: Distribution of APOGEE disk and halo stars in $[\alpha/{\rm
Fe}]$-$[{\rm Fe/H}]$ for the $\alpha$-elements O (left), Mg (middle), or Si
(right). Above $[{\rm Fe/H}]=-0.8$ stars are randomly downsampled by a factor
of 10; the transition to full sampling produces the edge at $[{\rm
Fe/H}]=-0.8$. Horizontal lines show $[\alpha/{\rm Fe}]=0.25$, 0.35, 0.45 for
visual reference.

Figure 1 shows [O/Fe], [Mg/Fe], and [Si/Fe] for stars from APOGEE (Majewski et
al., 2017) as reported in SDSS Data Release 17 (DR17; Abdurro’uf et al. 2022).
We selected stars with $0.5<\log(g)<3.5$, $3800<T_{\rm eff}<5200$, signal-to-
noise ratio ${\rm SNR}>200$ per pixel, Galactocentric radius $R_{\rm
gal}<15\rm\;kpc$, and midplane distance $|Z|<10\rm\;kpc$, using distances from
the DR17 AstroNN catalog (Leung & Bovy, 2019). As a rough cut to reject
accreted halo stars, we require ${\rm[Al/Fe]}>-0.05$ (Hawkins et al., 2015;
Belokurov & Kravtsov, 2022). Changes to these cuts do not change the
qualitative appearance of the plot, which shows all selected stars with $[{\rm
Fe/H}]<-0.8$ and a random 10% selection at higher $[{\rm Fe/H}]$. The
high-$\alpha$ stars with $-1.2<[{\rm Fe/H}]<-0.6$ suggest plateaus at
approximately $+0.35$ for O and Mg and $+0.25$ for Si. However, at lower
metallicity the scatter is large and there is no clear plateau. Furthermore,
motivated by data from the H3 survey, Conroy et al. (2022) present a model in
which the true CCSN plateau of the in situ halo lies at $[{\rm Mg/Fe}]\approx
0.6$ but is seen only at $[{\rm Fe/H}]\lesssim-2.5$, and the trend between
$-2.5<[{\rm Fe/H}]<-0.5$ is governed by a rapidly accelerating star formation
efficiency that maintains a roughly constant SNIa/CCSN ratio (see Figure 9
below). Maoz & Graur (2017), using chemical evolution models motivated by the
cosmic star formation history, also argue that $[{\rm Mg/Fe}]\approx 0.3$ is
an intermediate plateau arising from balanced CCSN and SNIa enrichment rather
than reflecting the CCSN ratio.

Since plausible observational values for the $[\alpha/{\rm Fe}]$ plateau range
from 0.3 to 0.6, we have decided to take 0.45 as a fiducial value for $[{\rm
O}/{\rm Fe}]_{\rm cc}$ and $[{\rm Mg}/{\rm Fe}]_{\rm cc}$. The 0.15-dex
systematic uncertainty in this choice is, unfortunately, a large (40%) source
of uncertainty in our eventual conclusions about the scale of yields. Based on
the separation between the [Si/Mg] ratios of the low-$\alpha$ and
high-$\alpha$ disk populations, Weinberg et al. (2022) infer that a fraction
0.81 of the solar Si abundance arises from CCSN, with the remainder from SNIa.
We therefore adopt $0.45+\log(0.81)=0.36$ as our fiducial value for $[{\rm
Si}/{\rm Fe}]_{\rm cc}$. This 0.09-dex difference is consistent with observed
differences between the [Si/Fe] and [Mg/Fe] plateaus in Figure 1 and in the
observational studies discussed previously. We sometimes use the notation
$[\alpha/{\rm Fe}]$ to refer generically to either $[{\rm O/Fe}]$ or $[{\rm
Mg/Fe}]$, or conceptually to an $\alpha$-element that is produced entirely by
CCSN.

Because observed abundance ratios are scaled to solar values, our choice of
solar abundances also matters for our results. We adopt solar photospheric
abundances from Magg et al. (2022), as this study uses state-of-the-art
atmospheric models and finds consistency between spectroscopically derived
abundances and helioseismic models. Relative to the photospheric abundances of
Asplund et al. (2009), these abundances are 0.08 dex higher for O and Si, 0.05
dex lower for Mg, and the same for Fe.

Table 5 of Magg et al. (2022) reports solar photospheric abundances on the
conventional scale $x=12+\log({\rm X}/{\rm H})$ of 8.77, 7.55, 7.59, and 7.50
for O, Mg, Si, and Fe, respectively. We add 0.04 dex to obtain proto-solar
abundances corrected for the impact of diffusion and gravitational settling
(Turcotte et al., 1998), the same correction adopted by Asplund et al. (2009).
Our calculations require a mass fraction $Z_{X}$ of these elements, which we
compute as

$\log Z_{\rm X}=(x-12)+\log 0.71+\log A~{},$ (1)

where 0.71 is the assumed solar hydrogen mass fraction and $A$ is the mean
atomic weight, which we take to be 16.0, 24.3, 28.09, and 55.85 for O, Mg, Si,
Fe. We therefore adopt

$\\{Z_{{\rm O},\odot},Z_{{\rm Mg},\odot},Z_{{\rm Si},\odot},Z_{{\rm
Fe},\odot}\\}=\\\ \\{73.3,6.71,8.51,13.7\\}\times 10^{-4}~{}.$ (2)

###  2.2 The explosion fraction

Parametric theoretical studies of the supernova mechanism tuned to produce SN
1987A generate a complicated landscape of successful explosions and
unsuccessful collapses that form black holes (Ugliano et al., 2012; Pejcha &
Thompson, 2015; Sukhbold et al., 2016; Ebinger et al., 2019). Such models
typically produce explosion fractions of 60-90% (see, e.g., Figure 2 below).
In general, lower mass progenitors explode relatively easily, but they do so
with low Ni yields, low energies, and low ejected O masses. Although
information on the metallicity dependence of $F_{\rm exp}$ is limited, the
Pejcha & Thompson (2015) and Ebinger et al. (2020) models show that black hole
formation should increase (and $F_{\rm exp}$ decrease) at lower metallicity.

On the observational side, Horiuchi et al. (2011) compare the number of
observed supernovae to that expected from the star formation rate, finding a
discrepancy at the factor of two level that indicates either $\sim 50$ % black
hole formation fraction or an undercounting of underluminous (and
underenergetic) supernovae in surveys. From the observation of a single
disappearing massive star and a sample of successful supernovae, Neustadt et
al. (2021, continuing the program of Kochanek et al. 2008; Gerke et al. 2015;
Adams et al. 2017) report a black hole formation fraction $F_{\rm bh}=1-F_{\rm
exp}$ of $\sim 20-40$ %. We will adopt $F_{\rm exp}=0.75$ as our fiducial
value in calculations below, but most of the range $0.5<F_{\rm exp}<1$ is
possible, and the value may be metallicity dependent.

###  2.3 The $\alpha$-element yield

By definition,

$[{\rm X}/{\rm Fe}]_{\rm cc}=\log\left({\bar{y}_{\rm X}^{\rm cc}/\bar{y}_{\rm
Fe}^{\rm cc}\over Z_{{\rm X},\odot}/Z_{{\rm Fe},\odot}}\right)~{}.$ (3)

We can rearrange this definition to find

$\displaystyle\bar{y}_{\rm X}^{\rm cc}=$ $\displaystyle\,\bar{y}_{\rm Fe}^{\rm
cc}\cdot 10^{[{\rm X}/{\rm Fe}]_{\rm cc}}\cdot{Z_{{\rm X},\odot}\over Z_{{\rm
Fe},\odot}}$ (4) $\displaystyle=$ $\displaystyle\,0.163\left({Z_{{\rm
X},\odot}\over Z_{{\rm Fe},\odot}}\right)\left({\bar{y}_{\rm Fe}^{\rm cc}\over
0.058M_{\odot}}\right)$ $\displaystyle\times 10^{({[{\rm X}/{\rm Fe}]_{\rm
cc}-0.45})}\,M_{\odot}~{}.$ (5)

Based on the discussion in §2.1, we adopt $[{\rm O}/{\rm Fe}]_{\rm cc}=[{\rm
Mg}/{\rm Fe}]_{\rm cc}=0.45$, $[{\rm Si}/{\rm Fe}]_{\rm cc}=0.36$, and the
Magg et al. (2022) solar abundances to obtain

$\\{\bar{y}_{\rm O}^{\rm cc},\bar{y}_{\rm Mg}^{\rm cc},\bar{y}_{\rm Si}^{\rm
cc},\bar{y}_{\rm Fe}^{\rm cc}\\}=\\\ \\{0.87,0.080,0.082,0.058\\}M_{\odot}~{}$
(6)

for the mean yield per CCSN.

For purposes of GCE modeling, we want to specify the population-averaged yield
in terms of $y_{\rm X}^{\rm cc}$, mass of element X produced per unit mass of
stars formed. For a Kroupa IMF, the number of massive stars per unit mass of
star formation is

${N_{\rm massive}\over M_{*}}=\frac{\int_{M_{\rm SN}}^{M_{\rm max}}{dN\over
dM}dM}{\int_{M_{\rm min}}^{M_{\rm max}}M{dN\over dM}dM}=0.0109M_{\odot}^{-1},$
(7)

where the numerical value assumes mass limits $M_{\rm min}=0.08M_{\odot}$ and
$M_{\rm max}=120M_{\odot}$ and a threshold mass $M_{\rm SN}=8M_{\odot}$ for
producing a CCSN. For a Salpeter (1955) IMF the ratio drops to 0.0068 because
of the larger number of low mass stars. For a Kroupa IMF with a higher
supernova threshold $M_{\rm SN}=11M_{\odot}$ it drops to 0.0071; for $M_{\rm
SN}=8M_{\odot}$ and $M_{\rm max}=60M_{\odot}$ it is 0.0104. To compute the
number of supernovae we also need to know the fraction $F_{\rm exp}$ of
massive stars that explode as CCSN rather than collapsing to black holes. We
thus have a core collapse supernova ratio

$R_{\rm cc}\equiv\frac{N_{\rm cc}}{M_{*}}=0.0109F_{\rm exp}F_{\rm
IMF}M_{\odot}^{-1}~{},$ (8)

where the $F_{\rm IMF}$ factor corrects for departures from a Kroupa IMF. The
$F_{\rm exp}$ factor is $\leq 1$ by definition, while the $F_{\rm IMF}$ factor
may be $<1$ (e.g., for Salpeter 1955 or Kroupa et al. 1993) or $>1$ for a “top
heavy” IMF. Based on the discussion in §2.2 and in §3.3 below, we adopt
$F_{\rm exp}=0.75$ and $F_{\rm IMF}=1$ as fiducial values.

The mean yield per CCSN can be converted to a mean yield per unit mass of star
formation with

$y_{\rm X}^{\rm cc}=R_{\rm cc}\bar{y}_{\rm X}^{\rm cc}~{}.$ (9)

Using Equation (5) and Equation (8) gives

$\displaystyle{y_{\rm X}^{\rm cc}\over Z_{{\rm X},\odot}}=$ $\displaystyle
0.973\left({\bar{y}_{\rm Fe}^{\rm cc}\over 0.058M_{\odot}}\right)\left({F_{\rm
exp}\over 0.75}\right)\left({F_{\rm IMF}\over 1.0}\right)$
$\displaystyle\times\left({0.00137\over Z_{{\rm Fe},\odot}}\right)\times
10^{([{\rm X}/{\rm Fe}]_{\rm cc}-0.45)}~{}.$ (10)

Because $[{\rm X}/{\rm Fe}]_{\rm cc}$ is expressed relative to the solar
abundance ratio, this formula for $y_{\rm X}^{\rm cc}$ in solar units does not
depend on the adopted solar abundance of element X. For Si we infer a lower
value of $y_{\rm X}^{\rm cc}/Z_{{\rm X},\odot}$ because we adopt $[{\rm
Si}/{\rm Fe}]_{\rm cc}=0.36$, which corresponds to $\approx 20\%$ of solar Si
arising from SNIa instead of CCSN.

Equation (10) is our first key result. Although the values of $F_{\rm exp}$,
$F_{\rm IMF}$, and $[{\rm X}/{\rm Fe}]_{\rm cc}$ are model dependent, and the
values of $\bar{y}_{\rm Fe}^{\rm cc}$ and $Z_{{\rm Fe},\odot}$ have
observational uncertainties, the equation itself follows directly from the
definition of these quantities, independent of supernova or GCE models. For
our fiducial parameter values, the RMN23 measurement of $\bar{y}_{\rm Fe}^{\rm
cc}$ implies that the population-averaged CCSN yields of $\alpha$ elements are
about equal to the solar abundance of those elements, a finding that will have
important implications for galactic outflows in chemical evolution models
(Section 3.1). This empirical conclusion does not rely on models of massive
stars and CCSN except through the choice of $F_{\rm exp}$.

Adopting the fiducial scalings of Equation (10), the $[{\rm X}/{\rm Fe}]_{\rm
cc}$ values of Section 2.1, and the solar abundances of Equation (2), we
obtain

$\\{y_{\rm O}^{\rm cc},y_{\rm Mg}^{\rm cc},y_{\rm Si}^{\rm cc},y_{\rm Fe}^{\rm
cc}\\}=\\\ \\{71.3,6.52,6.72,4.73\\}\times 10^{-4}$ (11)

for the population-averaged CCSN yields. Our calculations assume that these
yields are independent of metallicity. We discuss uncertainties associated
with this assumption in Section 4.2 below.

##  3 Implications of the inferred yield scale

###  3.1 Implications for outflows

In the one-zone GCE models described by WAF, the ISM mass fraction of an
element X produced by CCSN with a metallicity-independent yield evolves to an
equilibrium abundance

$Z_{\rm X,eq}={y_{\rm X}^{\rm cc}\over 1+\eta-r-\tau_{*}/\tau_{\rm SFH}}~{}.$
(12)

Here $\eta\equiv\dot{M}_{\rm out}/\dot{M}_{*}$ is the mass-loading factor of
outflows, $r$ is the recycling factor ($\approx 0.4$ for a Kroupa IMF),
$\tau_{*}=M_{\rm gas}/\dot{M}_{*}$ is the star formation efficiency (SFE)
timescale, and the star formation history (SFH) is assumed to be exponential
with $\dot{M}_{*}\propto e^{-t/\tau_{\rm SFH}}$. The analytic solution
approximates the recycling of material from stellar envelopes as
instantaneous, returning gas at the birth metallicity to the ISM at a rate
$r\dot{M}_{*}$, which (as shown by WAF) gives chemical evolution tracks nearly
identical to those of a numerical calculation with time-dependent recycling.
The full time evolution for this SFH is

$Z_{\rm X}(t)=Z_{\rm X,eq}\left(1-e^{-t/\bar{\tau}}\right)~{},$ (13)

where

$\bar{\tau}=\dfrac{\tau_{*}}{(1+\eta-r-\tau_{*}/\tau_{\rm SFH})}~{}.$ (14)

For a linear-exponential SFH, $\dot{M}_{*}\propto te^{-t/\tau_{\rm SFH}}$, the
late-time equilibrium is the same but the evolution to that equilibrium is
slower (WAF, Equation 56).

Observations of gas phase and Cepheid abundances imply that the ISM
metallicity is approximately solar at $R_{\rm gal}=8\rm\;kpc$ in the present-
day Galaxy (e.g., Lemasle et al. 2013; da Silva et al. 2022; Esteban et al.
2022). We can invert Equation (12) to find the value of $\eta$ that is
required to produce a solar metallicity ISM at equilibrium,

$\eta_{\odot}=y_{\rm X}^{\rm cc}/Z_{{\rm X},\odot}-1+r+\tau_{*}/\tau_{\rm
SFH}~{}.$ (15)

Johnson et al. (2021) present detailed GCE models of the Milky Way disk that
reproduce a wide range of observational constraints. They base $\tau_{\rm
SFH}$ on age profiles of Sa/Sb galaxies from Sánchez (2020), finding
$\tau_{\rm SFH}=15\,{\rm Gyr}$ at the solar annulus. In their models the value
of $\tau_{*}$ at the solar radius is slightly over $3\,{\rm Gyr}$ at the
present day, while a direct estimate of $\Sigma_{\rm SFR}/(\Sigma_{\rm
HI}+\Sigma_{{\rm H}_{2}})$ using values from Elia et al. (2022), Kalberla &
Kerp (2009), and Miville-Deschênes et al. (2017) gives $3.6\,{\rm Gyr}$. The
star formation efficiency may have been higher (shorter $\tau_{*}$) in the
past, when the galaxy was more gas rich, closer to the $2\,{\rm Gyr}$
timescale typical for molecular gas (Leroy et al., 2008; Sun et al., 2023).
These considerations suggest $0.13<\tau_{*}/\tau_{\rm SFH}<0.4$ as a plausible
range.

For $r=0.4$, $\tau_{*}/\tau_{\rm SFH}=0.2$, and $y_{\rm X}^{\rm cc}/Z_{{\rm
X},\odot}=0.973$ from Equation (10), we find $\eta_{\odot}=0.57$. Lowering
$[{\rm X}/{\rm Fe}]_{\rm cc}$ to 0.3 (implying $y_{\rm X}^{\rm cc}/Z_{{\rm
X},\odot}=0.689$) gives $\eta_{\odot}=0.29$, while raising $[{\rm X}/{\rm
Fe}]_{\rm cc}$ to 0.6 ($y_{\rm X}^{\rm cc}/Z_{{\rm X},\odot}=1.37$) gives
$\eta_{\odot}=0.97$. For our fiducial parameter choices, therefore, mild
outflows with $\eta\approx 0.6$ are required to reach a solar ISM at last
times. With a low value of $[{\rm X}/{\rm Fe}]_{\rm cc}$ and uncertainties in
$\bar{y}_{\rm Fe}^{\rm cc}$, $F_{\rm exp}$, and $\tau_{*}/\tau_{\rm SFH}$, a
solution with no outflows ($\eta_{\odot}=0$) is possible. With parameters
pushed in the other direction — $[{\rm X}/{\rm Fe}]_{\rm cc}=0.6$,
$\bar{y}_{\rm Fe}^{\rm cc}=0.065$, $F_{\rm exp}=0.85$, $\tau_{*}/\tau_{\rm
SFH}=0.4$ — Equation (15) implies $\eta_{\odot}=1.55$. This is still lower
than the value $\eta\approx 2.5$ adopted in the models of Andrews et al.
(2017) and WAF because their theoretically motivated oxygen yield $y_{\rm
O}^{\rm cc}=0.015$ corresponds to $y_{\rm O}^{\rm cc}/Z_{{\rm O},\odot}\approx
2$, twice the value suggested by the RMN23 determination of $\bar{y}_{\rm
Fe}^{\rm cc}$.

For the $\tau_{*}$ and $\tau_{\rm SFH}$ values advocated by Johnson et al.
(2021), the timescale to reach equilibrium is quite short, $\bar{\tau}\sim
3\,{\rm Gyr}$, so the departures from equilibrium are expected to be small.
However, with a longer value of $\tau_{*}$ these departures can become more
significant. In experimentation with the time-dependent solution (Equation
(13)) we find that the value of $\eta$ required to reach $Z_{\rm X}(t=13\,{\rm
Gyr})=Z_{{\rm X},\odot}$ is usually close to the value implied by Equation
(15) even accounting for these departures, but caution is required if the
denominators of Equations (12) or (14) approach zero. At fixed $\tau_{*}$, a
more sharply declining star formation history (shorter $\tau_{\rm SFH}$)
increases $Z_{\rm X}(t)$, so the required value of $\eta$ is higher even
though the abundance is further below equilibrium.

###  3.2 Implications for the SNIa rate

Once SNIa become an important enrichment channel, the value of $[\alpha/{\rm
Fe}]$ in the ISM and newly forming stars falls below $[\alpha/{\rm Fe}]_{\rm
cc}$ because Fe now includes the additional SNIa contribution. WAF
approximated the delay time distribution (DTD) of SNIa as an exponential,
$r_{\rm Ia}(t)\propto e^{-(t-t_{d})/\tau_{\rm Ia}}$, where $\tau_{\rm
Ia}\approx 1.5\,{\rm Gyr}$ and $t_{d}\approx 0.1\,{\rm Gyr}$ is the minimum
delay time required to produce a SNIa. The mass of iron produced by SNIa per
unit mass of star formation is

$y_{\rm Fe}^{\rm Ia}=R_{\rm Ia}\bar{y}_{\rm Fe}^{\rm Ia}~{},$ (16)

where $R_{\rm Ia}=N_{\rm Ia}/M_{*}$ is the time-integrated SNIa rate and
$\bar{y}_{\rm Fe}^{\rm Ia}$ is the mean iron yield per Type Ia supernova. We
take $\bar{y}_{\rm Fe}^{\rm Ia}=0.7M_{\odot}$, a typical value inferred from
the mass of radioactive 56Ni in the analysis of Howell et al. (2009), though
estimated 56Ni masses span a wide range from supernova to supernova (e.g.,
Childress et al. 2015). Like CCSN enrichment, SNIa enrichment also approaches
equilibrium at late times, and one can use the ratio of equilibrium abundances
(WAF Equations 29, 30) to show that

$[\alpha/{\rm Fe}]_{\rm eq}=[\alpha/{\rm Fe}]_{\rm
cc}-\log\left(1+\mu\frac{R_{\rm Ia}\bar{y}_{\rm Fe}^{\rm Ia}}{R_{\rm
cc}\bar{y}_{\rm Fe}^{\rm cc}}\right)~{},$ (17)

where

$\mu=\frac{\tau_{\rm SFH}}{\tau_{\rm SFH}-\tau_{\rm Ia}}e^{t_{d}/\tau_{\rm
SFH}}$ (18)

is a constant that approaches unity for $\tau_{\rm SFH}\gg\tau_{\rm Ia}$.
Equation (17) captures the expectation that the decrease of $[\alpha/{\rm
Fe}]$ at late times depends on the rate of SNIa enrichment relative to CCSN
enrichment, and the factor $\mu$ accounts for the fact that the SNIa rate is
tied to the past star formation rate while CCSN are tied to the current star
formation rate.

We can rearrange Equation (17) to solve for the SNIa rate,

$R_{\rm Ia}=\frac{R_{\rm cc}}{\mu}\frac{\bar{y}_{\rm Fe}^{\rm
cc}}{\bar{y}_{\rm Fe}^{\rm Ia}}\left(10^{\Delta[\alpha/{\rm
Fe}]}-1\right)~{},$ (19)

where $\Delta[\alpha/{\rm Fe}]=[\alpha/{\rm Fe}]_{\rm cc}-[\alpha/{\rm
Fe}]_{\rm eq}$ is the drop in $[\alpha/{\rm Fe}]$ between the CCSN plateau and
the late-time equilibrium. Taking $\mu=1.1$, $\bar{y}_{\rm Fe}^{\rm
cc}/\bar{y}_{\rm Fe}^{\rm Ia}=0.058/0.7=0.083$, $R_{\rm
cc}=0.008M_{\odot}^{-1}$, $[\alpha/{\rm Fe}]_{\rm eq}=0$, and $[\alpha/{\rm
Fe}]_{\rm cc}=0.45$ gives $R_{\rm Ia}=1.1\times 10^{-3}M_{\odot}^{-1}$. The
corresponding population-averaged yield is

$y_{\rm Fe}^{\rm Ia}=\left(7.7\times 10^{-4}\right)\times\left({F_{\rm
exp}\over 0.75}\right)\left({F_{\rm IMF}\over 1.0}\right)\left({\bar{y}_{\rm
Fe}^{\rm cc}\over 0.058M_{\odot}}\right)\\\
\times\left({1.1\over\mu}\right)\left({10^{\Delta[\alpha/{\rm Fe}]}-1\over
1.82}\right)~{},$ (20)

where we have restored the parameter dependencies to facilitate scaling to
other choices. Note that $\bar{y}_{\rm Fe}^{\rm Ia}$ cancels out of this
abundance-based inference of $y_{\rm Fe}^{\rm Ia}$, though the inferred
$R_{\rm Ia}$ itself scales as $(0.7M_{\odot}/\bar{y}_{\rm Fe}^{\rm Ia})$.

Maoz & Graur (2017) fit the observed cosmic star formation history and SNIa
history to infer a $t^{-1.1}$ DTD with a Hubble-time integrated normalization
$R_{\rm Ia}=(1.3\pm 0.1)\times 10^{-3}M_{\odot}^{-1}$ for a Kroupa IMF.
Compared to an exponential DTD, a $t^{-1.1}$ DTD leads to slightly lower
$[\alpha/{\rm Fe}]$ at late times, by roughly 0.05 dex (see Figure 11 of WAF).
However, the $[\alpha/{\rm Fe}]$ of recently formed stars in the solar
neighborhood may also be slightly sub-solar, compensating this difference. We
conclude that, within the uncertainties, the value of $R_{\rm Ia}$ implied by
RMN23’s value of $\bar{y}_{\rm Fe}^{\rm cc}$ is consistent with the value
found by Maoz & Graur (2017), assuming $[\alpha/{\rm Fe}]_{\rm cc}=0.45$. For
$[\alpha/{\rm Fe}]_{\rm cc}-[\alpha/{\rm Fe}]_{\rm eq}=0.3$ or 0.6, the value
of $R_{\rm Ia}$ implied by Equation (19) is lower by a factor of 1.8 or higher
by a factor of 1.6, respectively, so the uncertainty in the true level of the
$[\alpha/{\rm Fe}]$ plateau remains a substantial uncertainty in this
inference of $R_{\rm Ia}$.

###  3.3 Implications for black hole formation

For a given IMF, the population-averaged CCSN yield is sensitive to which
massive stars actually explode. We can therefore use the empirically inferred
mean element yields to constrain the degree of black hole formation. Here we
examine the solar metallicity models of S16 as extended by G21. S16 use the
KEPLER code (Weaver et al., 1978) to compute the evolution of a dense grid of
massive stars up to core collapse, then apply different models of neutrino-
driven central engines to compute the subsequent explosion or implosion. G21
extend this suite by forcing explosions at all masses, with explosion energies
and the boundary between ejected and fallback material calibrated by the
neutrino-driven engine results. With this grid of mass-dependent yields, one
can impose a black hole formation landscape a posteriori and compute the IMF-
averaged element yields from the CCSN and the pre-supernova stellar winds.

For the same models as S16, Ertl et al. (2016) find that exploding and non-
exploding massive star progenitors can be well separated by a critical curve
in the space of $\mu_{4}$, $M_{4}$ parameters linked to the mass infall rate
and neutrino luminosity at core collapse. Based on these results, G21 define
an explodability function

$E(M,e_{0})=\Theta\left(0.28M_{4}\mu_{4}-\mu_{4}+e_{0}\right)~{},$ (21)

where $\Theta$ is the Heaviside step function and progenitors with
$E(M,e_{0})=1$ explode and $E(M,e_{0})=0$ implode. The quantity $e_{0}$
encodes the power of the central engine. For $e_{0}\geq 0.065$ nearly all
massive stars explode, and for $e_{0}\leq 0.03$ only stars with $M\leq
14M_{\odot}$ and a handful of larger masses explode. At intermediate values of
$e_{0}$ the $E(M,e_{0})$ function traces a complex landscape of black hole
formation because the structure of the pre-supernova progenitor is a
sensitive, non-monotonic function of the initial mass $M$ (Pejcha & Thompson
2015; Ertl et al. 2016; S16; Ebinger et al. 2019). Figure 2, similar to Figure
5 of G21, illustrates the dependence of the explosion landscape and the
$F_{\rm exp}$ value on $e_{0}$. The “Z9.6+W18” neutrino-driven model that S16
adopt as representative corresponds to $e_{0}=0.043$.

Figure 2: Left: Continuous explosion landscapes for a range of $e_{0}$ values
($0.035-0.07$ and All Explode) as a function of progenitor ZAMS mass.
Horizontal lines indicate the masses where successful explosions occur. Lines
are colored by $e_{0}$, with low values of $e_{0}$ in yellow and high values
of $e_{0}$ in dark purple. While the $e_{0}=0.065$ and $e_{0}=0.07$ cases
appear to fully explode, there are small zones of black hole formation. Right:
$F_{\rm exp}$ for the $e_{0}$ landscapes shown in the left panel.

For each explosion landscape we calculate the IMF-averaged net yield,

$y_{\rm X}^{\rm cc}=\dfrac{\int_{M_{\rm SN}}^{M_{\rm
max}}\left[E(M,e_{0})m_{X,{\rm exp}}+m_{X,{\rm wind}}-b_{\rm X}\Delta
M\right]{\frac{dN}{dN}dM}}{\int_{M_{\rm min}}^{M_{\rm
max}}M\frac{dN}{dM}dM}~{}.$ (22)

Here $\Delta M=M-M_{\rm rem}$ is the difference between the star’s zero-age
main sequence (ZAMS) mass and that of its neutron star or black hole remnant,
and $m_{X,{\rm exp}}$ and $m_{X,{\rm wind}}$ are the mass of element X ejected
in the explosion and the pre-supernova wind, respectively. We subtract the
ejected mass $b_{X}\Delta M$ that was present in the star at birth to obtain
the net yield of newly produced material. We again adopt a Kroupa IMF with
$M_{\rm min}=0.08M_{\odot}$ and $M_{\rm max}=120M_{\odot}$, and we assume a
minimum CCSN progenitor mass $M_{\rm SN}=8M_{\odot}$. We only include the wind
component for O, as massive star winds do not carry newly produced Mg, Si, or
Fe.

We compute $M_{\rm rem}$ from the information tabulated by S16/G21 by
subtracting the sum of all element yields, including hydrogen and helium, from
the ZAMS mass:

$M_{\rm rem}=M-\Sigma m_{X,{\rm wind}}-E(M,e_{0})(\Sigma m_{X,{\rm exp}})~{}.$
(23)

For a non-exploding star, with $E(M,e_{0})=0$, the remnant black hole contains
all mass that was not ejected in winds. For exploding models our calculated
values of $M_{\rm rem}$ agree with the published values from S16.

Figure 3: Explosive (dark purple) and wind (dark orange) yields from S16 as a
function of progenitor mass for elements O (top left) Mg (top right), Si
(bottom left), and Fe (bottom right). We plot the progenitor birth abundances
in grey. The background colored lines indicate stellar models that explode
(light purple) and collapse (light orange) for a landscape with $e_{0}=0.046$.
For this $e_{0}$, which gives the best agreement with the RMN23 value of
$\bar{y}_{\rm Fe}^{\rm cc}=0.058M_{\odot}$, only the masses marked by purple
lines would explode and release newly synthesized Fe.

Figure 3 plots the S16/G21 yields of O, Mg, Si, and Fe as a function of
progenitor mass, similar to Figure 2 of G21. G21 force explosions for all
masses on this fine grid. Vertical orange lines indicate masses that would
implode (and thus produce no explosive yield) for a landscape with
explodability threshold $e_{0}=0.046$. For O, the predicted yield rises
steadily and steeply up to $M=60M_{\odot}$. The S16 models adopt an aggressive
mass loss prescription that strips the envelopes of stars with
$M>40M_{\odot}$. Above this mass the wind yields of O are significant, but
high mass progenitors have reduced explosive yields because of their early
mass loss. The sum of wind and explosive yields is roughly constant at about
$7M_{\odot}$ for $M>70M_{\odot}$. Because of IMF-weighting, stars with
$M>40M_{\odot}$ have only moderate impact ($\sim 20\%$) on the IMF-averaged O
yield, so although the uncertainties in mass loss are substantial, they do not
drastically affect the predicted $y_{\rm O}^{\rm cc}$.

The behavior for Mg is similar to O, but at intermediate masses there are
spikes in yield over narrow progenitor mass ranges that produce denser pre-
supernova cores, which create additional Mg during the explosion. The Si yield
is dominated by explosive nucleosynthesis rather than hydrostatic (pre-
explosion) nucleosynthesis, so this spikiness is more pronounced and the
overall mass trend is weaker. The sharp, non-monotonic variation is even
stronger for the Fe yield. Progenitor models near the threshold of
explodability tend to produce high Fe yields if they do explode because they
are characterized by dense cores that can approach nuclear statistical
equilibrium once explosive nucleosynthesis takes place. The properties of pre-
supernova cores can change sharply with small changes in ZAMS progenitor mass
because of shifts in the spatial position of nuclear fusion zones (S16;
Sukhbold et al. 2018).

Table 1 lists the explosion fraction $F_{\rm exp}$ and the population-averaged
net yields $y_{\rm Fe}^{\rm cc}$, $y_{\rm O}^{\rm cc}$, $y_{\rm Mg}^{\rm cc}$,
and $y_{\rm Si}^{\rm cc}$ computed from the S16/G21 models as a function of
the explodability threshold $e_{0}$, for a Kroupa IMF. These yields are
dimensionless, representing solar masses of element production per solar mass
of star formation. We also list the mean yield per supernova, which from
Equations (8) and (9) is

$\bar{y}_{\rm X}^{\rm cc}={y_{\rm X}^{\rm cc}\over R_{\rm cc}}=91.7F_{\rm
exp}^{-1}y_{\rm X,exp}^{\rm cc}\,M_{\odot}~{},$ (24)

where we assume $F_{\rm IMF}=1$. For Mg, Si, and Fe, $y_{\rm X,exp}^{\rm
cc}=y_{\rm X}^{\rm cc}$, but for O we omit the wind yield from non-exploding
stars so that $\bar{y}_{\rm O}^{\rm cc}$ represents the average yield of the
actual CCSN. Figure 4 plots these average yields as a function of $e_{0}$. As
illustrated in the lower right panel, the RMN23 value of $\bar{y}_{\rm
Fe}^{\rm cc}=0.058\pm 0.007M_{\odot}$ leads to a fairly tight constraint
$0.045<e_{0}<0.048$. The best-fit value $e_{0}=0.046$ is above the value
corresponding to the Z9.6+W18 model of S16 and very close to that of their
Z9.6+N20 model. The orange lines in Figure 3 mark the progenitor masses that
would implode to black holes without releasing any newly synthesized Fe for
this value of $e_{0}$.

Table 1: CCSN Yields From the S16/G21 Models

$e_{0}$ | $F_{\rm exp}$ | $y_{\rm O}^{\rm cc}$ | $y_{\rm Mg}^{\rm cc}$ | $y_{\rm Si}^{\rm cc}$ | $y_{\rm Fe}^{\rm cc}$ | $\bar{y}_{\rm O}^{\rm cc}$ | $\bar{y}_{\rm Mg}^{\rm cc}$ | $\bar{y}_{\rm Si}^{\rm cc}$ | $\bar{y}_{\rm Fe}^{\rm cc}$
---|---|---|---|---|---|---|---|---|---
0.035 | 0.472 | 3.51e-03 | 2.38e-05 | 7.15e-05 | 1.62e-04 | 6.82e-01 | 4.62e-03 | 1.39e-02 | 3.15e-02
0.040 | 0.546 | 4.01e-03 | 5.56e-05 | 1.13e-04 | 2.37e-04 | 6.73e-01 | 9.34e-03 | 1.90e-02 | 3.98e-02
0.045 | 0.658 | 5.54e-03 | 1.36e-04 | 2.20e-04 | 3.74e-04 | 7.72e-01 | 1.89e-02 | 3.06e-02 | 5.22e-02
0.050 | 0.785 | 9.23e-03 | 3.23e-04 | 3.82e-04 | 6.43e-04 | 1.08e+00 | 3.78e-02 | 4.46e-02 | 7.51e-02
0.055 | 0.851 | 1.20e-02 | 4.73e-04 | 4.69e-04 | 8.25e-04 | 1.29e+00 | 5.09e-02 | 5.05e-02 | 8.88e-02
0.060 | 0.933 | 1.51e-02 | 6.48e-04 | 6.23e-04 | 1.14e-03 | 1.48e+00 | 6.37e-02 | 6.12e-02 | 1.12e-01
0.065 | 0.981 | 1.80e-02 | 8.08e-04 | 7.15e-04 | 1.45e-03 | 1.68e+00 | 7.55e-02 | 6.68e-02 | 1.35e-01
0.070 | 0.983 | 1.80e-02 | 8.11e-04 | 7.18e-04 | 1.46e-03 | 1.68e+00 | 7.56e-02 | 6.69e-02 | 1.36e-01
All Exp | 1.000 | 1.83e-02 | 8.24e-04 | 7.29e-04 | 1.49e-03 | 1.68e+00 | 7.55e-02 | 6.68e-02 | 1.37e-01

Note. — For the $e_{0}=0.035-0.07$ and All Explode CCSN landscapes, columns
list $F_{\rm exp}$, net yield $y_{\rm X}^{\rm cc}$ in $M_{\odot}$ per
$M_{\odot}$ formed, and the average net explosive yield per supernova,
$\bar{y}_{\rm X}^{\rm cc}$. For O, the $y_{\rm X}^{\rm cc}$ calculation
includes wind contributions but the $\bar{y}_{\rm X}^{\rm cc}$ calculation
does not. Wind contributions to the net yields are negligible for Mg, Si, and
Fe.

Figure 4: Average yield per supernova $\bar{y}_{\rm X}^{\rm cc}$, in
$M_{\odot}$, as a function of explodability threshold for $e_{0}=0.035-0.07$
\+ All Explode for O (top left), Mg (top right), Si (bottom left), and Fe
(bottom right). In the bottom right panel for Fe we also plot the RMN23 value
of $\bar{y}_{\rm Fe}^{\rm cc}=0.058\pm 0.007M_{\odot}$, and in each panel we
shade the region between $e_{0}=0.045-0.048$, the $e_{0}$ values in agreement
with the RMN23 Fe yield.

At the best value of $e_{0}$, the predicted O, Mg, and Si yields are
$\bar{y}_{\rm O}^{\rm cc}=0.83M_{\odot}$, $\bar{y}_{\rm Mg}^{\rm
cc}=0.023M_{\odot}$, and $\bar{y}_{\rm Si}^{\rm cc}=0.033M_{\odot}$,
respectively. These can be compared to our empirically inferred values
(Equation (6)) of $0.87M_{\odot}$, $0.080M_{\odot}$, $0.082M_{\odot}$. For the
$e_{0}$ value implied by Fe, the predicted O yield is in good agreement with
our empirical inference, but the predicted Mg and Si yields are low by factors
of 3.5 and 2.5, respectively.

This inconsistency among O, Mg, and Si is already evident in the relative
yields of the models, as shown by G21 (see their Figure 9). For any choice of
explosion landscape, these models overpredict the solar O/Mg ratio by a factor
of 2.5-4, much larger than the observational uncertainty in this ratio. For a
landscape like Z9.6+N20 that gives agreement with the observed $\bar{y}_{\rm
Fe}^{\rm cc}$, the models also overpredict the solar Si/Mg ratio (after
accounting for the SNIa contribution to Si) by a factor $\sim 2$. Even if we
adopted a high value of $e_{0}$ for which essentially all massive stars
explode, the Mg and Si yields would remain (slightly) below our empirically
inferred values. We caution, however, that our empirical values rely on our
uncertain choice of $[\alpha/{\rm Fe}]_{\rm cc}=0.45$. If we lower
$[\alpha/{\rm Fe}]_{\rm cc}$ to 0.3, then the inferred $\bar{y}_{\rm Mg}^{\rm
cc}$ and $\bar{y}_{\rm Si}^{\rm cc}$ drop to $0.056M_{\odot}$ and
$0.058M_{\odot}$, respectively, which the S16/G21 models would produce for
$e_{0}\approx 0.057$. However, with this $e_{0}$ the models overproduce
$\bar{y}_{\rm Fe}^{\rm cc}$ and $\bar{y}_{\rm O}^{\rm cc}$ by factors of $\sim
1.5-2$.

It is encouraging that the S16 models with a physically plausible choice of
neutrino-driven engine can reproduce both the RMN23 Fe yield and our inferred
O yield, but the failure to reproduce Mg and Si tempers our assessment of this
success. In comparing the S16/G21 models to our empirically inferred yields,
we implicitly assume that these models represent all CCSN types including the
Ib and Ic supernovae that RMN23 include in their $\bar{y}_{\rm Fe}^{\rm cc}$
determination. It is possible that Ib and Ic supernovae have different
progenitor structure because of hydrogen envelope stripping and therefore
require separate treatment in yield calculations (Sukhbold & Adams, 2020;
Laplace et al., 2021).

###  3.4 Implications for ISM deuterium

Deuterium is the one isotope whose nucleosynthetic yield is securely and
precisely predicted by theory: stars destroy all the D they are born with
because in their fully convective proto-stellar phase they cycle their birth D
through layers hot enough to fuse it into 4He (Bodenheimer, 1966; Mazzitelli &
Moretti, 1980). All D present in the ISM must have originated in the big bang,
and it resides in fluid elements that were never processed through stars.
Weinberg (2017) shows that, in a variety of GCE models, the evolution of the
ISM D/H ratio $X_{\rm D}$ is accurately approximated by

$X_{\rm D}={X_{\rm D}^{\rm P}\over 1+rZ_{\rm X}/y_{\rm X}^{\rm cc}}~{},$ (25)

where $X_{\rm D}^{\rm P}$ is the primordial D/H and $Z_{\rm X}$ is the mass
fraction of a pure CCSN element with metallicity independent yield $y_{\rm
X}^{\rm cc}$. van de Voort et al. (2020) show that Equation (25) also
accurately describes the results of hydrodynamic cosmological simulations.
While the form of Equation (25) is motivated by analytic solutions at
equilibrium, the approximate relation follows from basic considerations. A
stellar population of mass $M_{*}$ produces a mass $y_{\rm X}^{\rm cc}M_{*}$
of element X and a mass $rM_{*}$ of gas, so its ejecta have a mean $Z_{\rm
X}=y_{\rm X}^{\rm cc}/r$. The ratio $X_{\rm D}/X_{\rm D}^{\rm P}$ tells what
fraction of the ISM is primordial (never processed through stars), and thus
the factor by which $Z_{\rm X}$ is diluted relative to the ratio in stellar
ejecta.

Taking $\bar{y}_{\rm X}^{\rm cc}/Z_{{\rm X},\odot}=0.973$ and $r=0.4$, we
predict $X_{\rm D}/X_{\rm D}^{\rm P}=0.71$ for solar metallicity. With
primordial D/H of 26 ppm (Cyburt et al., 2016) this implies ISM D/H of about
18.5 ppm. Linsky et al. (2006) measure D/H in absorption along many lines of
sight through the ISM finding values that span a factor of three. They
attribute this variation to depletion of D onto small dust grains along some
lines of sight, and based on the highest D/H (least depleted) sightlines they
advocate an ISM abundance $23.1\pm 2.4$ ppm. This is higher than the value
predicted at solar metallicity for our fiducial yield scale, but consistent at
$2\sigma$. Reducing the population-averaged $\alpha$-element yield
significantly below $Z_{\alpha,\odot}$ would make the high D/H values from
Linsky et al. (2006) difficult to reproduce within conventional GCE models.
For example, if we adopt the ratio $y_{\alpha}^{\rm
cc}/Z_{\alpha,\odot}=0.689$ implied by Equation (10) for $[\alpha/{\rm
Fe}]_{\rm cc}=0.3$, then the predicted ISM D/H falls to 16.4 ppm, a
$2.8\sigma$ conflict with Linsky et al.’s value.

##  4 Discussion

###  4.1 Overview

Figure 5: Overview of our results, indicating where parameter values and GCE
or CCSN models enter the chain of inference. The central column traces our
core results on the scale of $\alpha$-element yields and its implication for
outflows. The left column traces conclusions about Fe yields from CCSN and
SNIa. The right column traces implications for the S16/G21 models of CCSN and
black hole formation.

Figure 5 presents an overview of our findings, with an emphasis on where
different assumptions enter the logical flow. The center column traces the
route to our central result, the population-averaged yield of
$\alpha$-elements (Equation (10)). The assumed value of $[\alpha/{\rm
Fe}]_{\rm cc}$ is sufficient to translate the RMN23 measurement of the mean Fe
yield per CCSN to the corresponding mean yields of $\alpha$ elements (Equation
(6)). Converting to the population-averaged yields (solar masses produced per
solar mass of star formation) requires an assumed IMF and explosion fraction.
For our fiducial scalings, Equation (10) reduces to

$\log{y_{\alpha}^{\rm cc}\over Z_{\alpha,\odot}}=-0.46+[\alpha/{\rm Fe}]_{\rm
cc}~{}.$ (26)

Thus, for $[{\rm O}/{\rm Fe}]_{\rm cc}=[{\rm Mg}/{\rm Fe}]_{\rm cc}=0.45$, the
implied yields of O and Mg are almost exactly equal to their solar mass
fractions. Conclusions about outflows rely on a GCE model in combination with
these yields. For straightforward assumptions, a mass-loading factor
$\eta\approx 0.6$ is required to yield a solar metallicity ISM at late times
(Equation (15)).

The left column traces our conclusions about Fe yields. Deriving a population-
averaged CCSN Fe yield from the RMN23 measurement requires values of $F_{\rm
IMF}$ and $F_{\rm exp}$, but it does not depend on $[\alpha/{\rm Fe}]_{\rm
cc}$ because $[{\rm Fe}/{\rm Fe}]_{\rm cc}=0$ by definition. Inferring the
population-averaged SNIa Fe yield from the corresponding CCSN yield requires a
GCE model and a value for the gap $\Delta[\alpha/{\rm Fe}]$ between the CCSN
plateau and the late-time equilibrium (Equation (20)). For $[\alpha/{\rm
Fe}]_{\rm cc}=0.45$ and reasonable choices of star formation history and SNIa
DTD, a model with $y_{\rm Fe}^{\rm Ia}\approx 1.65y_{\rm Fe}^{\rm cc}$ evolves
to $[\alpha/{\rm Fe}]\approx 0$ at late times. With an empirically and
theoretically motivated choice of the mean Fe yield per SNIa, $\bar{y}_{\rm
Fe}^{\rm Ia}=0.7M_{\odot}$, one can infer the Hubble-time integrated SNIa rate
$R_{\rm Ia}\approx 1.1\times 10^{-3}M_{\odot}^{-1}$.

The right column traces the implications of the RMN23 measurement for the
S16/G21 models of CCSN and black hole formation. The measured $\bar{y}_{\rm
Fe}^{\rm cc}$ and the assumption of a Kroupa IMF leads directly to a
constraint on the explodability threshold in these models, $e_{0}\approx
0.046$, with an explosion landscape similar to the 3rd-from-bottom line in
Figure 2. In this model, the fraction of $M>8M_{\odot}$ stars that explode as
CCSN is $F_{\rm exp}\approx 0.7$. With $e_{0}$ fixed by matching the observed
$\bar{y}_{\rm Fe}^{\rm cc}$, the model predicts the SN-averaged yields of O,
Mg, and Si (Table 1). For O, the predicted yield of $\bar{y}_{\rm O}^{\rm
cc}=0.83M_{\odot}$ agrees well with the empirically inferred value for
$[\alpha/{\rm Fe}]_{\rm cc}=0.45$, but Mg and Si are underpredicted by a
factor $\sim 3$. Reproducing the inferred Mg and Si yields would require
nearly all $M>8M_{\odot}$ stars to explode, but the model would then
drastically overpredict O. This conflict among O, Mg, and Si is likely rooted
within the S16/G21 models themselves rather than choices we have made here,
perhaps reflecting inaccuracies in the adopted nuclear reaction rates (see G21
for further discussion).

###  4.2 Uncertainties

The single largest uncertainty in deriving $y_{\rm X}^{\rm cc}$ from the RMN23
measurement of $\bar{y}_{\rm Fe}^{\rm cc}$ is the choice of $[{\rm X}/{\rm
Fe}]_{\rm cc}$, the ratio of $\alpha$-element production to Fe production by
massive stars. We have chosen $[{\rm O}/{\rm Fe}]_{\rm cc}=[{\rm Mg}/{\rm
Fe}]_{\rm cc}=0.45$ as our fiducial value based on the plateau in
$[\alpha/{\rm Fe}]$ vs. $[{\rm Fe/H}]$ observed for stars with $[{\rm
Fe/H}]\leq-0.8$. However, the observed level of the plateau varies from study
to study and in some cases depends on the choice of $\alpha$-element
(Kobayashi et al., 2020). The RMN23 sample is likely dominated by CCSN
progenitors near solar metallicity because galaxies with $L\sim L_{*}$ and
$Z\sim Z_{\odot}$ contribute most to the global star formation rate. Taking
the low-metallicity plateau to represent $[\alpha/{\rm Fe}]_{\rm cc}$ for the
RMN23 supernovae implicitly assumes that this ratio does not change between
$[{\rm Fe/H}]\approx-1$ and $[{\rm Fe/H}]\approx 0$. This assumption is
reasonable because the predicted yields of these elements do not depend
strongly on metallicity (see Andrews et al. 2017, Figure 20). However, these
predictions could be incorrect if black hole formation or the stellar IMF
change systematically with metallicity in a way that favors Fe or
$\alpha$-element production.

RMN23 discuss a number of sources of uncertainty in their $\bar{y}_{\rm
Fe}^{\rm cc}$ measurement, and we assume that these are adequately reflected
in their $0.007M_{\odot}$ error bar. One specific concern is that the RMN23
sample might undercount low luminosity supernovae from low mass progenitors,
which have lower than average 56Ni production and thus lower than average Fe
yields. However, these faint supernovae also have lower than average
$\alpha$-element yields, so the ratio $\bar{y}_{\rm X}^{\rm cc}/\bar{y}_{\rm
Fe}^{\rm cc}$ may not change much by omitting them. To illustrate this point,
Figure 6 plots the fraction of O, Mg, Si, and Fe produced by CCSN with
progenitor mass $M=8-11M_{\odot}$, as a function of the explodability
threshold $e_{0}$, using the S16/G21 yields. For the $e_{0}=0.046$ favored by
our analysis in §3.3, these low mass progenitors produce only 15% of the Fe,
3-5% of the Mg and Si, and 1.5% of the O. These moderate fractions imply that
underrepresentation of these faint CCSN in the RMN23 sample would have limited
impact on our yield conclusions. The impact on the comparison to the S16/G21
models is more complex, because if the $8-11M_{\odot}$ mass range is omitted
then $e_{0}$ must be lowered (to $e_{0}\approx 0.4$) to reproduce
$\bar{y}_{\rm Fe}^{\rm cc}=0.058M_{\odot}$, which changes the predicted
$F_{\rm exp}$ and yield ratios.

Figure 6: Fraction of IMF-averaged yield contributed by stars with
$M=8-11M_{\odot}$, as a function of explodability threshold, for elements O
(top left), Mg (top right), Si (bottom left), and Fe (bottom right). We shade
the region between $e_{0}=0.045-0.048$, the $e_{0}$ values in agreement with
the RMN23 Fe yield. Note that the $y$-axis scale is different for each panel.

Equation (10) expresses the dependence of our inferred $\alpha$-element yield
scale on the quantities $F_{\rm exp}$, $F_{\rm IMF}$, and $[{\rm X}/{\rm
Fe}]_{\rm cc}$, so the equation itself is model-independent. To assign an
approximate error bar to our $y_{\rm X}^{\rm cc}$ values in Equation (11), we
take $[\alpha/{\rm Fe}]_{\rm cc}=0.45\pm 0.15$ to represent the “$\pm
2\sigma$” range of plausible values for this quantity, making the “$1\sigma$”
error bar 0.075 dex. We take $F_{\rm exp}=0.75\pm 0.1$. We then add the 0.075
dex uncertainty in $[\alpha/{\rm Fe}]_{\rm cc}$ in quadrature with the 0.05
dex uncertainty ($\approx\log[1+0.1/0.75]$) in $F_{\rm exp}$ and 0.05 dex
uncertainty ($=\log[1+0.007/0.058]$) of the RMN23 $\bar{y}_{\rm Fe}^{\rm cc}$
measurement to obtain

$\log y_{\alpha}^{\rm cc}/Z_{\alpha,\odot}=-0.01\pm 0.1$ (27)

for the $\alpha$-elements O and Mg. This error bar does not incorporate any
uncertainty in the IMF, and in general one should use Equation (10) to account
for specified changes in model assumptions.

###  4.3 Outflows and the mass-metallicity relation

For our fiducial choices of parameters, the inferred oxygen yield $y_{\rm
O}^{\rm cc}\approx Z_{{\rm O},\odot}\approx 0.007$ is substantially lower than
the value $y_{\rm O}^{\rm cc}\approx 0.015$ assumed in many earlier papers by
our group (e.g., Andrews et al. 2017; WAF; Johnson et al. 2021). The higher
$y_{\rm O}^{\rm cc}$, based on the Chieffi & Limongi massive star models
(Chieffi & Limongi, 2004, 2013; Limongi & Chieffi, 2006, 2018), a Kroupa
(2001) IMF, and minimal suppression of yield by black hole formation, is
similar to that adopted in much of the literature on the galaxy mass-
metallicity relation (e.g., Finlator & Davé 2008; Peeples & Shankar 2011; Davé
et al. 2012; Zahid et al. 2012). With the lower yield inferred here, the
efficiency of outflows required to reproduce observed galaxy metallicities
will be lower, by roughly a factor of two in the $\eta\gg 1$ regime that
applies at low halo masses. Similarly, the average fraction of metals ejected
by galaxies will be lower than the value of $\sim 75\%$ found by Peeples et
al. (2014) because that was based on comparing the total metals produced by a
galaxy’s stellar population to the total metals remaining in the stars and
ISM.

Figure 7: Outflow mass-loading required to reproduce the observed mass-
metallicity relation at equilibrium. Green points show the observed gas-phase
oxygen abundance as a function of stellar mass derived from stacked SDSS
spectra (Andrews & Martini, 2013), converted to $Z_{\rm O}/Z_{{\rm O},\odot}$
using the Magg et al. (2022) abundance scale. Curves show the values of $\eta$
that produce this abundance for a CCSN O yield $y_{\rm O}^{\rm cc}=0.015$
(black dashed) or for our empirical yield scaling with three different values
of $[\alpha/{\rm Fe}]_{\rm cc}$ (as labeled).

Figure 7 illustrates the impact of lower yields on inferred outflow mass-
loading, for the simplified assumptions of equilibrium abundances and constant
SFR. Open circles show the Andrews & Martini (2013) fit to the observed
relation between ISM oxygen abundance and galaxy stellar mass, which they
derive from auroral lines in stacked SDSS spectra. Over the galaxy stellar
mass range $M_{*}=10^{7.5}M_{\odot}-10^{10.5}M_{\odot}$ these abundances
increase from $\approx 0.1Z_{{\rm O},\odot}$ to $\approx Z_{{\rm O},\odot}$.
The solid black curve shows the value of $\eta$ required to produce the
observed oxygen abundance in equilibrium for our fiducial yield scale, $y_{\rm
O}^{\rm cc}=0.973Z_{{\rm O},\odot}$, and assuming a constant SFR (Equation
(12) with $r=0.4$ and $\tau_{*}/\tau_{\rm SFH}=0$). Red dotted and blue dashed
curves show the corresponding results for $[\alpha/{\rm Fe}]_{\rm cc}=0.6$ or
0.3, respectively. The black dashed curve shows the inferred $\eta$ using a
yield $y_{\rm O}^{\rm cc}=0.015=2.1Z_{{\rm O},\odot}$ characteristic of
studies based on the Limongi and Chieffi massive star yields with minimal
black hole formation.

The modeling assumptions adopted in Figure 7 are idealized, and the
observational relation has significant uncertainties, but the main
implications are robust: large outflow mass-loading is needed to explain the
low ISM metallicities of low mass galaxies, but the required $\eta(M_{*})$ is
sensitive to the adopted yield scale. For $M_{*}<10^{9}M_{\odot}$ the fiducial
curve in Figure 7 is well described by

$\eta\approx 4\left({M_{*}\over 10^{8}M_{\odot}}\right)^{-0.6}\approx
4\left({M_{h,{\rm peak}}\over 6\times 10^{10}M_{\odot}}\right)^{-1.2}~{},$
(28)

where the second $\approx$ adopts the $z=0.1$ relation between stellar mass
and peak halo mass from Figure 9 of Behroozi et al. (2019). The Andrews &
Martini (2013) gas-phase measurements cut off at $M_{*}\approx
10^{7.5}M_{\odot}$. Extrapolating to $M_{*}=10^{6}M_{\odot}$, Equations (28)
and (12) predict $[\alpha/{\rm H}]\approx-1.8$, in reasonable agreement with
the mean stellar metallicity found for galaxies in this mass range by Kirby et
al. (2013). However, this measurement is [Fe/H] rather than [$\alpha$/H], and
the relation $Z\propto M_{*}^{0.6}$ implied by Equation (28) at low masses is
steeper than the stellar mass-metallicity scaling $Z_{*}\propto M_{*}^{0.3}$
found by Kirby et al. (2013). Discussion of the outflows required to reproduce
the full stellar metallicity distributions of individual dwarf galaxies, and
the degeneracy between outflows and yields in this modeling, can be found in
Johnson et al. (2022b) and Sandford et al. (2022).

Returning to the Milky Way regime, we find in §3.1 that $\eta\approx 0.6$ is
required to produce a solar metallicity ISM at equilibrium, for an empirically
plausible choice of $\tau_{*}/\tau_{\rm SFH}$. Our GCE model assumes that
ejected material has the same metallicity as the ISM, and if winds were metal-
enhanced then the mass outflow would be lower while the metal outflow would be
similar. Even with $y_{\rm O}^{\rm cc}\approx Z_{{\rm O},\odot}$ as found
here, it is difficult to reproduce Milky Way disk abundances with no outflows,
though it is certainly easier than it would be for $y_{\rm O}^{\rm
cc}=2-3Z_{{\rm O},\odot}$. We will examine this issue more thoroughly in
future work that compares multi-zone GCE models to the Milky Way’s observed
gas and stellar abundance gradients, for a variety of assumptions about
outflows and radial gas flows (J.W. Johnson et al., in preparation).

###  4.4 Milky Way chemical evolution

Figure 8: Evolutionary tracks of one-zone GCE models compared to APOGEE
stellar abundance data from the solar annulus. Points show $[{\rm Mg/Fe}]$ vs.
$[{\rm Fe/H}]$ for APOGEE stars with $R_{\rm gal}=7-9\rm\;kpc$ and midplane
distance $|Z|<1\rm\;kpc$ (black, downsampled by a factor of four) or
$|Z|=1-2\rm\;kpc$ (red). Curves show models with the star formation histories
shown in the inset and with SFE and outflow parameters $\tau_{*},\eta=0.5,0$
(solid), $2,0.6$ (dashed), or $6,2$ (dotted).

In the $[\alpha/{\rm Fe}]-[{\rm Fe/H}]$ plane, stars in the Milky Way thick
disk and thin disk follow distinct “high-$\alpha$” and “low-$\alpha$”
sequences (Fuhrmann, 1998; Bensby et al., 2003). In the scenario proposed by
Schönrich & Binney (2009a), the low-$\alpha$ sequence is not an evolutionary
track itself but the superposition of endpoints of evolutionary tracks, with
radial migration mixing stellar populations that formed at different
Galactocentric radii. As a very simple illustration of this picture, Figure 8
superposes three one-zone evolutionary tracks on the $[\alpha/{\rm Fe}]-[{\rm
Fe/H}]$ distribution of APOGEE stars in the solar annulus. This Figure is
similar to Figure 16 of WAF (and Figures 15 and 17 of Nidever et al. 2014),
but with models adjusted to the lower yields of Equations (10) and (20). We
also adopt the 2-parameter star formation history of Johnson et al. (2021)
instead of the linear-exponential form used in the WAF figure, using a
calculational method described in Appendix A. We model the SNIa delay time
distribution with a sum of two exponentials that approximates a $t^{-1.1}$
power law (Maoz & Graur, 2017), with a minimum delay time $t_{d}=0.15\,{\rm
Gyr}$. For the APOGEE points we use the same sample cuts described in §2.1 but
restrict to $R_{\rm gal}=7-9\rm\;kpc$ and $|Z|<2\rm\;kpc$, and we downsample
the $|Z|<1\rm\;kpc$ stars by a factor of four for visual clarity. As expected,
the high-$\alpha$ population is more prominent at high $|Z|$.

For our “inner Galaxy” track we adopt $\eta=0$ (no outflow), a high SFE with
$\tau_{*}=0.6\,{\rm Gyr}$, and a star formation history that peaks at
$t=2\,{\rm Gyr}$ and declines exponentially ($\tau_{*}=6\,{\rm Gyr}$) at late
times. This track evolves to $[{\rm Fe/H}]\approx 0.3$ and slightly sub-solar
$[{\rm Mg/Fe}]$, and it roughly follows the upper envelope of the observed
high-$\alpha$ population. For the “solar radius” track we adopt $\eta=0.6$, a
lower SFE with $\tau_{*}=2\,{\rm Gyr}$, and a slower rise and shallower
decline of the star formation history. As expected from the discussion in
Sections 3.1-3.2, this track evolves to $[{\rm Mg/Fe}]\approx[{\rm
Fe/H}]\approx 0$. The “outer Galaxy” track adopts $\eta=2$, a low SFE with
$\tau_{*}=6\,{\rm Gyr}$, and a SFR that rises slowly and becomes approximately
constant. This track evolves to low $[{\rm Fe/H}]\approx-0.45$, principally
because of the strong outflow.

Although Figure 8 qualitatively resembles the corresponding WAF figure, with
lower $\eta$ values compensating for lower yields, the match between the
models and the APOGEE data is noticeably worse than in WAF. The principal
reason for this difference is that the lower values of $\eta$ lead to slower
growth of metallicity at early times (see Equations (13) and (14)), so that
the downturn of $[{\rm Mg/Fe}]$ from SNIa enrichment sets in at lower $[{\rm
Fe/H}]$. Only the inner Galaxy track has a knee in $[{\rm Mg/Fe}]$ at roughly
the location shown by the APOGEE data. The knee would shift to still lower
$[{\rm Fe/H}]$ if we adopted lower SFE or assumed a shorter minimum delay in
the $t^{-1.1}$ DTD. Additionally, the low-$\alpha$ population in this APOGEE
DR17 sample exhibits a kinked structure that was not evident in the DR12
sample shown in the WAF figure, and this kink is difficult to obtain with
smoothly changing star formation histories. One should not draw sharp
conclusions from a simple model overlay like Figure 8; for fully realized GCE
models with radial mixing see, e.g., Minchev et al. (2013, 2014), Loebman et
al. (2016), Johnson et al. (2021), and Chen et al. (2022). Nonetheless, Figure
8 suggests that the low-yield, low-outflow combination favored by the RMN23
measurement makes it more challenging to reproduce observed structure in the
$[\alpha/{\rm Fe}]-[{\rm Fe/H}]$ plane. Other groups have proposed very
different origins for the bimodality in $[\alpha/{\rm Fe}]$, such as the two-
infall model in which dilution resets the ISM metallicity before the
low-$\alpha$ sequence evolves (Chiappini et al., 1997; Spitoni et al., 2019),
and a clumpy burst scenario in which stochastic local self-enrichment takes
regions of low-$\alpha$ ISM temporarily to high-$\alpha$ (Clarke et al., 2019;
Garver et al., 2023).

Motivated by the $[{\rm Mg/Fe}]-[{\rm Fe/H}]$ trends of in situ halo stars in
the H3 survey, Conroy et al. (2022, hereafter C22) propose a different GCE
model in which the true $[{\rm Mg/Fe}]$ plateau lies at $+0.6$, and an
inflection to rising $[{\rm Mg/Fe}]$ between $[{\rm Fe/H}]\approx-1.5$ and
$[{\rm Fe/H}]\approx-0.5$ is caused by a rapidly growing SFE that drives an
accelerating star formation rate. Chen et al. (2023) suggest a variant of this
model in which the accelerating SFR is driven largely by gas inflow rather
than growing SFE.

To produce a high plateau and reach $[{\rm Mg/Fe}]\approx 0$ at late times,
the C22 model adopts $y_{\rm Mg}^{\rm cc}=3.45Z_{{\rm Mg},\odot}$ and $y_{\rm
Fe}^{\rm cc}=0.875Z_{{\rm Fe},\odot}$, and an SNIa Fe yield $y_{\rm Fe}^{\rm
Ia}=2.5y_{\rm Fe}^{\rm cc}$. Figure 9 shows $[\alpha/{\rm Fe}]-[{\rm Fe/H}]$
tracks and the SFE evolution for the original C22 model (solid curves) and a
revised version (dashed curves) in which all three yields are reduced by a
factor of 2.5, so that $y_{\rm Mg}^{\rm cc}=1.38Z_{{\rm Mg},\odot}$ matches
the value implied by Equation (10) for $[{\rm Mg}/{\rm Fe}]_{\rm cc}=0.6$. We
adopt $\eta=0.3$ so that the model evolves to the same metallicity as the
original high-yield model, which has $\eta=2$. We adjust the quantities in
Equation (3) of C22 to produce a nearly identical $[\alpha/{\rm Fe}]-[{\rm
Fe/H}]$ track, which C22 show to be a good match to the in situ population
measurements for H3 and APOGEE.

In C22, the $[{\rm Mg/Fe}]$ knee occurs at $[{\rm Fe/H}]\approx-2.75$, which
requires an extremely low initial SFE, with $\tau_{*}=50\,{\rm Gyr}$. Rapid
growth of SFE between 2.5 and $3.7\,{\rm Gyr}$ leads to the bump in
$[\alpha/{\rm Fe}]$, followed by another decline when the SFE becomes constant
at $t>3.7\,{\rm Gyr}$. In the revised yield model, the lower $y_{\rm Mg}^{\rm
cc}$ and $y_{\rm Fe}^{\rm cc}$ must be compensated by a higher initial SFE
(shorter $\tau_{*}$) so that $[{\rm Fe/H}]$ reaches $-2.75$ before SNIa
enrichment begins. In fact we find that simply increasing SFE at all times by
a factor of 2.5, the same factor by which the yields were reduced, leads to a
virtually identical model track once $\eta$ is reduced. The late-time SFE
required to put the second $[{\rm Mg/Fe}]$ downturn at $[{\rm
Fe/H}]\approx-0.6$ is fairly high, with $\tau_{*}=1\,{\rm Gyr}$, though not as
high as that of the inner Galaxy curve in Figure 8 because the yield scale of
this $[\alpha/{\rm Fe}]_{\rm cc}=0.6$ model is higher.

Based on the observed SNIa rate as a function of galaxy stellar mass (Brown et
al., 2019), Johnson et al. (2022a) argue that the SNIa rate increases with
decreasing metallicity, with a scaling similar to the $Z^{-0.5}$ dependence of
the close-binary fraction found in APOGEE (Moe et al., 2019). The dotted
curves in Figure 9 show a model in which we set $y_{\rm Fe}^{\rm Ia}=2.5y_{\rm
Fe}^{\rm cc}(Z_{\rm Mg}/Z_{{\rm Mg},\odot})^{-0.5}$ for $[{\rm Mg}/{\rm
H}]>-1$, saturating at $\sqrt{10}$ times the solar metallicity rate. The high
SNIa rate at low metallicity causes a steeper early drop in $[{\rm Mg/Fe}]$,
so it is not possible to reproduce the locus of the C22 model exactly.
Nonetheless, we are able to find a model with quite similar behavior by
adjusting the SFE history. This model requires a particularly high SFE at late
times so that the second downturn of $[{\rm Mg/Fe}]$ remains at $[{\rm
Fe/H}]\approx-0.6$ despite the elevated SNIa rate.

Figure 9: (Top) Evolutionary track of the original C22 model (solid curve), a
modified version of this model in which all yields are reduced by a factor of
2.5 and $\eta$ is reduced from 2.0 to 0.3 (dashed curve), and a third version
in which the SNIa yield scales with metallicity as $Z^{-0.5}$ for $[{\rm
Mg}/{\rm H}]>-1$ (dotted curve). In the modified versions we adjust the time-
dependence of SFE to approximately reproduce the $[\alpha/{\rm Fe}]-[{\rm
Fe/H}]$ track of the original model. (Bottom) The SFE ($=\tau_{*}^{-1}$) as a
function of $[{\rm Fe/H}]$ for the three models. All models are computed using
the Versatile Integrator for Chemical Evolution (VICE, Johnson & Weinberg
2020).

##  5 Conclusions

We use RMN23’s estimate of the mean CCSN Fe yield, $\bar{y}_{\rm Fe}^{\rm
cc}=0.058\pm 0.007M_{\odot}$, to anchor the scale of population-averaged
massive star yields. Taking $[\alpha/{\rm Fe}]_{\rm cc}=0.45$, a Kroupa IMF,
and a massive star explosion fraction $F_{\rm exp}=0.75$, we find that the
population-averaged dimensionless yields $y_{\rm O}^{\rm cc}$ and $y_{\rm
Mg}^{\rm cc}$ are nearly equal to their corresponding solar mass fractions
(Equation (10)). In other words, for every solar mass of stars formed, on
average, massive stars release a mass of freshly synthesized O and Mg equal to
the mass of these elements in the sun. For Si we estimate $y_{\rm Si}^{\rm
cc}\approx 0.8Z_{{\rm Si},\odot}$ based on the lower observed [Si/Fe] plateau
and the expectation that a fraction of solar Si comes from SNIa. Our estimated
“$1\sigma$” uncertainty in $y_{\alpha}^{\rm cc}/Z_{\alpha,{\odot}}$ is about
0.1 dex ($\sim$ 25%), contributed by the uncertainties in $[\alpha/{\rm
Fe}]_{\rm cc}$, in $F_{\rm exp}$, and in the RMN23 measurement itself.

Models with a Kroupa IMF in which all $M>8M_{\odot}$ stars (or even all
$M=8-40M_{\odot}$ stars) explode as CCSN predict a $y_{\rm O}^{\rm cc}$ that
is $2-3$ times higher than our empirically inferred value (see, e.g., Andrews
et al. 2017, G21). The lower yield scale reduces the need for outflows in GCE
models to reproduce observed abundances, though for our fiducial choices we
still find that $\eta=\dot{M}_{\rm out}/\dot{M}_{*}\approx 0.6$ is required to
produce a solar ISM abundance at late times. Many models of the galaxy mass-
metallicity relation have assumed a higher yield scale. Adopting our empirical
$y_{\rm O}^{\rm cc}$ would reduce the $\eta$ values inferred in these studies,
by about a factor of two in the low mass systems with $\eta\gg 1$. GCE models
with our empirical $y_{\rm O}^{\rm cc}$ predict an ISM D/H ratio that is 71%
of the primordial D/H (Equation (25)), which is lower than the ISM D/H
estimated by Linsky et al. (2006) but consistent at the $2\sigma$ level.
Improved determination of the D/H ratio in the local ISM — or even better, at
locations that probe a range of ISM metallicities — would provide a powerful
consistency test for our understanding of stellar yields and chemical
evolution.

Assuming a Kroupa IMF, the RMN23 value of $\bar{y}_{\rm Fe}^{\rm cc}$ and our
inferred $y_{\rm O}^{\rm cc}$ are well reproduced by the S16 CCSN models with
the Z9.6+N20 neutrino-driven engine, which corresponds to an explodability
threshold $e_{0}=0.046$ in the terminology of G21. Thus, these empirical
yields can be explained by massive star nucleosynthesis calculations with a
plausible level of black hole formation. However, with this black hole
formation landscape the S16 models underpredict the empirical yields of Mg and
Si by factors of 3.5 and 2.5, a discrepancy already evident in their
underprediction of the solar Mg/O and Si/O ratios.

By requiring that GCE models approach $[\alpha/{\rm Fe}]\approx 0$ at late
times, we can estimate the population-averaged SNIa Fe yield $y_{\rm Fe}^{\rm
Ia}$, again with a dependence on the assumed $[\alpha/{\rm Fe}]_{\rm cc}$
(Equation (20)). Assuming $[\alpha/{\rm Fe}]_{\rm cc}=0.45$ and a mean Fe
yield per SNIa of $0.7M_{\odot}$ (Howell et al., 2009), we infer a Hubble-time
integrated SNIa rate $R_{\rm Ia}=1.1\times 10^{-3}\,M_{\odot}^{-1}$,
consistent with the normalization $(1.3\pm 0.2)\times 10^{-3}\,M_{\odot}^{-1}$
that Maoz & Graur (2017) found by comparing the cosmic histories of star
formation and SNIa. Our estimate assumes that $R_{\rm Ia}$ and $y_{\rm
Fe}^{\rm Ia}$ are metallicity independent. If the SNIa rate depends on
metallicity (Johnson et al., 2022a) then more careful modeling is needed to
relate the CCSN and SNIa yield scales.

Theoretical uncertainties in massive star yields, especially those associated
with black hole formation, make ab initio predictions of the population-
averaged $\alpha$-element yield uncertain at the factor-of-three level. The
degeneracy between yields and outflows, on the other hand, makes it difficult
to infer the absolute scale of yields from GCE modeling of observed stellar
and ISM abundances. The measurement of $\bar{y}_{\rm Fe}^{\rm cc}$ by RMN23 is
arguably the best empirical anchor for the scale of stellar yields that is
currently available. Analogous measurements for other elements might be
possible with careful light-curve, spectroscopic, or supernova remnant
analyses of samples that span the full range of the CCSN population. The scale
of yields has wide-ranging implications for massive star models, galaxy
evolution, and the cosmic metal budget. The central role of yields highlights
the importance of stress-testing the $\bar{y}_{\rm Fe}^{\rm cc}$ measurement
and sharpening its precision, and of improving the determination of
$[\alpha/{\rm Fe}]_{\rm cc}$ through accurate measurements in low metallicity
stellar populations that span a range of galactic environments.

##  6 Acknowledgements

We thank Dan Maoz for helpful comments on a draft version of this manuscript.
We are grateful to him and to Brett Andrews, Charlie Conroy, Jennifer Johnson,
Chris Kochanek, Molly Peeples, Ralph Schoenrich, Krzysztof Stanek, Tuguldur
Sukhbold, and Michael Tucker for valuable conversations on these topics over
many years. This work is supported by NSF grants AST-1909841 and AST-2307621.
E.J.G. is supported by an NSF Astronomy and Astrophysics Postdoctoral
Fellowship under award AST-2202135.

##  Appendix A Analytic GCE for a 2-parameter SFH

WAF derive analytic solutions for the evolution of $\alpha$-element and Fe
abundances in a one-zone GCE model. These solutions assume time-independent
values of $\eta$ and $\tau_{*}$, metallicity-independent yields, instantaneous
CCSN enrichment, and SNIa enrichment with an exponential DTD. A sum of two
exponentials can be used to approximate the $t^{-1.1}$ power-law DTD advocated
by Maoz & Graur (2017). WAF present solutions for a constant SFR, an
exponential SFH with $\dot{M}_{*}\propto e^{-t/\tau_{\rm SFH}}$, and a linear-
exponential SFH with $\dot{M}_{*}\propto te^{-t/\tau_{\rm SFH}}$. The
$\alpha$-element (pure CCSN) solution for an exponential SFH is Equation (13)
of this paper.

The linear-exponential SFH (also known as “delayed tau”) has gained popularity
as a one-parameter model that captures the rise-and-fall behavior typical of
galaxies in semi-analytic calculations and cosmological simulations (e.g., Lee
et al. 2010; Simha et al. 2014; Carnall et al. 2019). However, in this model
the rise to the peak and the subsequent exponential decay are tied to each
other — one cannot have a fast rise and slow decline, or vice versa. The
2-parameter SFH used by Johnson et al. (2021),

$\dot{M}_{*}(t)=K\left(1-e^{-t/\tau_{1}}\right)e^{-t/\tau_{2}}~{},$ (A1)

is much more flexible. For $\tau_{1}\ll\tau_{2}$, the SFH rises linearly at
early times, reaches a maximum at $t\sim\tau_{1}$, then declines exponentially
with a timescale $\tau_{2}$. Limiting cases include a pure exponential SFH
($\tau_{1}=0$), a constant SFR ($\tau_{1}=0$, $\tau_{2}\rightarrow\infty$) and
a linearly rising SFH ($\tau_{1}\gg t$, $\tau_{2}\rightarrow\infty$). The
normalization constant $K$ scales the overall stellar mass of the galaxy but
cancels out in the evolution of chemical abundances. A GCE model with constant
SFE in which the gas supply starts at zero and the gas infall rate is $\propto
e^{-t/\tau_{\rm inf}}$ (e.g., Spitoni et al. 2017) follows the SFH of Equation
(A1) with $\tau_{2}=\tau_{\rm inf}$ and

$\tau_{1}={\tau_{*}\over 1+\eta-r-\tau_{*}/\tau_{\rm inf}}$ (A2)

(see Equation (129) of WAF).

Fortunately, the analytic solutions derived by WAF can be readily applied to
this 2-parameter “rise-fall” SFH. To see how, it is useful to write

$K\left(1-e^{-t/\tau_{1}}\right)e^{-t/\tau_{2}}=Ke^{-t/\tau_{2}}-Ke^{-t/\tau_{h}}$
(A3)

with

$\tau_{h}=\left(\frac{1}{\tau_{1}}+\frac{1}{\tau_{2}}\right)^{-1}~{}.$ (A4)

Equations (50) and (53) of WAF give solutions for the ISM mass fraction $Z(t)$
of, respectively, a pure CCSN element and the SNIa contribution to Fe, for an
exponential SFH. Equation (117) of WAF shows how to combine the solutions for
any two star formation histories to obtain a solution for the SFH that is the
sum of these histories. In this case, we simply take the solutions for
exponential histories with timescales $\tau_{2}$ and $\tau_{h}$ and combine
them using this equation. The second exponential SFH has a negative
normalization, but the overall SFH is positive-definite so this does not cause
unphysical results. We have confirmed that this procedure reproduces the
results of direct numerical integrations, as expected.

The [Mg/Fe] vs. [Fe/H] tracks in Figure 1 are computed using this analytic
method. The three SFHs shown correspond to $(\tau_{1},\tau_{2})=(1,6)$ for the
inner Galaxy track, (2,12) for the solar radius track, and (4,30) for the
outer Galaxy track, with all times in Gyr.

## References

  * Abdurro’uf et al. (2022) Abdurro’uf, Accetta, K., Aerts, C., et al. 2022, ApJS, 259, 35, doi: 10.3847/1538-4365/ac4414
  * Adams et al. (2017) Adams, S. M., Kochanek, C. S., Gerke, J. R., Stanek, K. Z., & Dai, X. 2017, MNRAS, 468, 4968, doi: 10.1093/mnras/stx816
  * Amarsi et al. (2019) Amarsi, A. M., Nissen, P. E., & Skúladóttir, Á. 2019, A&A, 630, A104, doi: 10.1051/0004-6361/201936265
  * Andrews & Martini (2013) Andrews, B. H., & Martini, P. 2013, ApJ, 765, 140, doi: 10.1088/0004-637X/765/2/140
  * Andrews et al. (2017) Andrews, B. H., Weinberg, D. H., Schönrich, R., & Johnson, J. A. 2017, ApJ, 835, 224, doi: 10.3847/1538-4357/835/2/224
  * Andrievsky et al. (2010) Andrievsky, S. M., Spite, M., Korotin, S. A., et al. 2010, A&A, 509, A88, doi: 10.1051/0004-6361/200913223
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481, doi: 10.1146/annurev.astro.46.060407.145222
  * Behroozi et al. (2019) Behroozi, P., Wechsler, R. H., Hearin, A. P., & Conroy, C. 2019, MNRAS, 488, 3143, doi: 10.1093/mnras/stz1182
  * Belokurov & Kravtsov (2022) Belokurov, V., & Kravtsov, A. 2022, MNRAS, 514, 689, doi: 10.1093/mnras/stac1267
  * Bensby et al. (2003) Bensby, T., Feltzing, S., & Lundström, I. 2003, A&A, 410, 527, doi: 10.1051/0004-6361:20031213
  * Bensby et al. (2014) Bensby, T., Feltzing, S., & Oey, M. S. 2014, A&A, 562, A71, doi: 10.1051/0004-6361/201322631
  * Bensby et al. (2017) Bensby, T., Feltzing, S., Gould, A., et al. 2017, A&A, 605, A89, doi: 10.1051/0004-6361/201730560
  * Bodenheimer (1966) Bodenheimer, P. 1966, ApJ, 144, 103, doi: 10.1086/148592
  * Brown et al. (2019) Brown, J. S., Stanek, K. Z., Holoien, T. W. S., et al. 2019, MNRAS, 484, 3785, doi: 10.1093/mnras/stz258
  * Carnall et al. (2019) Carnall, A. C., Leja, J., Johnson, B. D., et al. 2019, ApJ, 873, 44, doi: 10.3847/1538-4357/ab04a2
  * Cayrel et al. (2004) Cayrel, R., Depagne, E., Spite, M., et al. 2004, A&A, 416, 1117, doi: 10.1051/0004-6361:20034074
  * Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763, doi: 10.1086/376392
  * Chen et al. (2022) Chen, B., Hayden, M. R., Sharma, S., et al. 2022, arXiv e-prints, arXiv:2204.11413.  https://arxiv.org/abs/2204.11413
  * Chen et al. (2023) Chen, B., Ting, Y.-S., & Hayden, M. 2023, arXiv e-prints, arXiv:2308.15976, doi: 10.48550/arXiv.2308.15976
  * Chiappini et al. (1997) Chiappini, C., Matteucci, F., & Gratton, R. 1997, ApJ, 477, 765, doi: 10.1086/303726
  * Chieffi & Limongi (2004) Chieffi, A., & Limongi, M. 2004, ApJ, 608, 405, doi: 10.1086/392523
  * Chieffi & Limongi (2013) —. 2013, ApJ, 764, 21, doi: 10.1088/0004-637X/764/1/21
  * Childress et al. (2015) Childress, M. J., Hillier, D. J., Seitenzahl, I., et al. 2015, MNRAS, 454, 3816, doi: 10.1093/mnras/stv2173
  * Clarke et al. (2019) Clarke, A. J., Debattista, V. P., Nidever, D. L., et al. 2019, MNRAS, 484, 3476, doi: 10.1093/mnras/stz104
  * Conroy et al. (2022) Conroy, C., Weinberg, D. H., Naidu, R. P., et al. 2022, arXiv e-prints, arXiv:2204.02989.  https://arxiv.org/abs/2204.02989
  * Curtis et al. (2021) Curtis, S., Wolfe, N., Fröhlich, C., et al. 2021, ApJ, 921, 143, doi: 10.3847/1538-4357/ac0dc5
  * Cyburt et al. (2016) Cyburt, R. H., Fields, B. D., Olive, K. A., & Yeh, T.-H. 2016, Reviews of Modern Physics, 88, 015004, doi: 10.1103/RevModPhys.88.015004
  * da Silva et al. (2022) da Silva, R., Crestani, J., Bono, G., et al. 2022, A&A, 661, A104, doi: 10.1051/0004-6361/202142957
  * Davé et al. (2012) Davé, R., Finlator, K., & Oppenheimer, B. D. 2012, MNRAS, 421, 98, doi: 10.1111/j.1365-2966.2011.20148.x
  * Ebinger et al. (2019) Ebinger, K., Curtis, S., Fröhlich, C., et al. 2019, ApJ, 870, 1, doi: 10.3847/1538-4357/aae7c9
  * Ebinger et al. (2020) Ebinger, K., Curtis, S., Ghosh, S., et al. 2020, ApJ, 888, 91, doi: 10.3847/1538-4357/ab5dcb
  * Elia et al. (2022) Elia, D., Molinari, S., Schisano, E., et al. 2022, ApJ, 941, 162, doi: 10.3847/1538-4357/aca27d
  * Ertl et al. (2016) Ertl, T., Janka, H. T., Woosley, S. E., Sukhbold, T., & Ugliano, M. 2016, ApJ, 818, 124, doi: 10.3847/0004-637X/818/2/124
  * Esteban et al. (2022) Esteban, C., Méndez-Delgado, J. E., García-Rojas, J., & Arellano-Córdova, K. Z. 2022, ApJ, 931, 92, doi: 10.3847/1538-4357/ac6b38
  * Finlator & Davé (2008) Finlator, K., & Davé, R. 2008, MNRAS, 385, 2181, doi: 10.1111/j.1365-2966.2008.12991.x
  * Fuhrmann (1998) Fuhrmann, K. 1998, A&A, 338, 161
  * Garver et al. (2023) Garver, B. R., Nidever, D. L., Debattista, V. P., Beraldo e Silva, L., & Khachaturyants, T. 2023, ApJ, 953, 128, doi: 10.3847/1538-4357/acdfc6
  * Gerke et al. (2015) Gerke, J. R., Kochanek, C. S., & Stanek, K. Z. 2015, MNRAS, 450, 3289, doi: 10.1093/mnras/stv776
  * Griffith et al. (2019) Griffith, E., Johnson, J. A., & Weinberg, D. H. 2019, ApJ, 886, 84, doi: 10.3847/1538-4357/ab4b5d
  * Griffith et al. (2021) Griffith, E. J., Sukhbold, T., Weinberg, D. H., et al. 2021, ApJ, 921, 73, doi: 10.3847/1538-4357/ac1bac
  * Griffith et al. (2022) Griffith, E. J., Weinberg, D. H., Buder, S., et al. 2022, ApJ, 931, 23, doi: 10.3847/1538-4357/ac5826
  * Hawkins et al. (2015) Hawkins, K., Jofré, P., Masseron, T., & Gilmore, G. 2015, MNRAS, 453, 758, doi: 10.1093/mnras/stv1586
  * Honda et al. (2004) Honda, S., Aoki, W., Ando, H., et al. 2004, ApJS, 152, 113, doi: 10.1086/383201
  * Horiuchi et al. (2011) Horiuchi, S., Beacom, J. F., Kochanek, C. S., et al. 2011, ApJ, 738, 154, doi: 10.1088/0004-637X/738/2/154
  * Howell et al. (2009) Howell, D. A., Sullivan, M., Brown, E. F., et al. 2009, ApJ, 691, 661, doi: 10.1088/0004-637X/691/1/661
  * Johnson (2019) Johnson, J. A. 2019, Science, 363, 474, doi: 10.1126/science.aau9540
  * Johnson et al. (2022a) Johnson, J. W., Kochanek, C. S., & Stanek, K. Z. 2022a, arXiv e-prints, arXiv:2210.01818, doi: 10.48550/arXiv.2210.01818
  * Johnson & Weinberg (2020) Johnson, J. W., & Weinberg, D. H. 2020, MNRAS, 498, 1364, doi: 10.1093/mnras/staa2431
  * Johnson et al. (2023) Johnson, J. W., Weinberg, D. H., Vincenzo, F., Bird, J. C., & Griffith, E. J. 2023, MNRAS, 520, 782, doi: 10.1093/mnras/stad057
  * Johnson et al. (2021) Johnson, J. W., Weinberg, D. H., Vincenzo, F., et al. 2021, MNRAS, 508, 4484, doi: 10.1093/mnras/stab2718
  * Johnson et al. (2022b) Johnson, J. W., Conroy, C., Johnson, B. D., et al. 2022b, arXiv e-prints, arXiv:2210.01816.  https://arxiv.org/abs/2210.01816
  * Kalberla & Kerp (2009) Kalberla, P. M. W., & Kerp, J. 2009, ARA&A, 47, 27, doi: 10.1146/annurev-astro-082708-101823
  * Kirby et al. (2013) Kirby, E. N., Cohen, J. G., Guhathakurta, P., et al. 2013, ApJ, 779, 102, doi: 10.1088/0004-637X/779/2/102
  * Kobayashi et al. (2020) Kobayashi, C., Karakas, A. I., & Lugaro, M. 2020, ApJ, 900, 179, doi: 10.3847/1538-4357/abae65
  * Kochanek et al. (2008) Kochanek, C. S., Beacom, J. F., Kistler, M. D., et al. 2008, ApJ, 684, 1336, doi: 10.1086/590053
  * Kroupa (2001) Kroupa, P. 2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x
  * Kroupa et al. (1993) Kroupa, P., Tout, C. A., & Gilmore, G. 1993, MNRAS, 262, 545, doi: 10.1093/mnras/262.3.545
  * Laplace et al. (2021) Laplace, E., Justham, S., Renzo, M., et al. 2021, arXiv e-prints, arXiv:2102.05036.  https://arxiv.org/abs/2102.05036
  * Lee et al. (2010) Lee, S.-K., Ferguson, H. C., Somerville, R. S., Wiklind, T., & Giavalisco, M. 2010, ApJ, 725, 1644, doi: 10.1088/0004-637X/725/2/1644
  * Lemasle et al. (2013) Lemasle, B., François, P., Genovali, K., et al. 2013, A&A, 558, A31, doi: 10.1051/0004-6361/201322115
  * Leroy et al. (2008) Leroy, A. K., Walter, F., Brinks, E., et al. 2008, AJ, 136, 2782, doi: 10.1088/0004-6256/136/6/2782
  * Leung & Bovy (2019) Leung, H. W., & Bovy, J. 2019, MNRAS, 483, 3255, doi: 10.1093/mnras/sty3217
  * Limongi & Chieffi (2006) Limongi, M., & Chieffi, A. 2006, ApJ, 647, 483, doi: 10.1086/505164
  * Limongi & Chieffi (2018) —. 2018, ApJS, 237, 13, doi: 10.3847/1538-4365/aacb24
  * Lin & Zu (2022) Lin, Y., & Zu, Y. 2022, arXiv e-prints, arXiv:2212.01402.  https://arxiv.org/abs/2212.01402
  * Linsky et al. (2006) Linsky, J. L., Draine, B. T., Moos, H. W., et al. 2006, ApJ, 647, 1106, doi: 10.1086/505556
  * Loebman et al. (2016) Loebman, S. R., Debattista, V. P., Nidever, D. L., et al. 2016, ApJ, 818, L6, doi: 10.3847/2041-8205/818/1/L6
  * Magg et al. (2022) Magg, E., Bergemann, M., Serenelli, A., et al. 2022, A&A, 661, A140, doi: 10.1051/0004-6361/202142971
  * Majewski et al. (2017) Majewski, S. R., Schiavon, R. P., Frinchaboy, P. M., et al. 2017, AJ, 154, 94, doi: 10.3847/1538-3881/aa784d
  * Maoz & Graur (2017) Maoz, D., & Graur, O. 2017, ApJ, 848, 25, doi: 10.3847/1538-4357/aa8b6e
  * Mazzitelli & Moretti (1980) Mazzitelli, I., & Moretti, M. 1980, ApJ, 235, 955, doi: 10.1086/157700
  * Minchev et al. (2013) Minchev, I., Chiappini, C., & Martig, M. 2013, A&A, 558, A9, doi: 10.1051/0004-6361/201220189
  * Minchev et al. (2014) —. 2014, A&A, 572, A92, doi: 10.1051/0004-6361/201423487
  * Miville-Deschênes et al. (2017) Miville-Deschênes, M.-A., Murray, N., & Lee, E. J. 2017, ApJ, 834, 57, doi: 10.3847/1538-4357/834/1/57
  * Moe et al. (2019) Moe, M., Kratter, K. M., & Badenes, C. 2019, ApJ, 875, 61, doi: 10.3847/1538-4357/ab0d88
  * Neustadt et al. (2021) Neustadt, J. M. M., Kochanek, C. S., Stanek, K. Z., et al. 2021, MNRAS, 508, 516, doi: 10.1093/mnras/stab2605
  * Nidever et al. (2014) Nidever, D. L., Bovy, J., Bird, J. C., et al. 2014, ApJ, 796, 38, doi: 10.1088/0004-637X/796/1/38
  * Nissen & Schuster (2010) Nissen, P. E., & Schuster, W. J. 2010, A&A, 511, L10, doi: 10.1051/0004-6361/200913877
  * Nomoto et al. (2013) Nomoto, K., Kobayashi, C., & Tominaga, N. 2013, ARA&A, 51, 457, doi: 10.1146/annurev-astro-082812-140956
  * Peeples & Shankar (2011) Peeples, M. S., & Shankar, F. 2011, MNRAS, 417, 2962, doi: 10.1111/j.1365-2966.2011.19456.x
  * Peeples et al. (2014) Peeples, M. S., Werk, J. K., Tumlinson, J., et al. 2014, ApJ, 786, 54, doi: 10.1088/0004-637X/786/1/54
  * Pejcha & Thompson (2015) Pejcha, O., & Thompson, T. A. 2015, ApJ, 801, 90, doi: 10.1088/0004-637X/801/2/90
  * Reggiani et al. (2017) Reggiani, H., Meléndez, J., Kobayashi, C., Karakas, A., & Placco, V. 2017, A&A, 608, A46, doi: 10.1051/0004-6361/201730750
  * Rodríguez et al. (2022) Rodríguez, Ó., Maoz, D., & Nakar, E. 2022, arXiv e-prints, arXiv:2209.05552.  https://arxiv.org/abs/2209.05552
  * Rodríguez et al. (2021) Rodríguez, Ó., Meza, N., Pineda-García, J., & Ramirez, M. 2021, MNRAS, 505, 1742, doi: 10.1093/mnras/stab1335
  * Rybizki et al. (2017) Rybizki, J., Just, A., & Rix, H.-W. 2017, A&A, 605, A59, doi: 10.1051/0004-6361/201730522
  * Salpeter (1955) Salpeter, E. E. 1955, ApJ, 121, 161, doi: 10.1086/145971
  * Sánchez (2020) Sánchez, S. F. 2020, ARA&A, 58, 99, doi: 10.1146/annurev-astro-012120-013326
  * Sandford et al. (2022) Sandford, N. R., Weinberg, D. H., Weisz, D. R., & Fu, S. W. 2022, arXiv e-prints, arXiv:2210.17045.  https://arxiv.org/abs/2210.17045
  * Scalo (1986) Scalo, J. M. 1986, Fund. Cosmic Phys., 11, 1
  * Schönrich & Binney (2009a) Schönrich, R., & Binney, J. 2009a, MNRAS, 396, 203, doi: 10.1111/j.1365-2966.2009.14750.x
  * Schönrich & Binney (2009b) —. 2009b, MNRAS, 396, 203, doi: 10.1111/j.1365-2966.2009.14750.x
  * Shivvers et al. (2017) Shivvers, I., Modjaz, M., Zheng, W., et al. 2017, PASP, 129, 054201, doi: 10.1088/1538-3873/aa54a6
  * Simha et al. (2014) Simha, V., Weinberg, D. H., Conroy, C., et al. 2014, arXiv e-prints, arXiv:1404.0402, doi: 10.48550/arXiv.1404.0402
  * Spitoni et al. (2019) Spitoni, E., Silva Aguirre, V., Matteucci, F., Calura, F., & Grisoni, V. 2019, A&A, 623, A60, doi: 10.1051/0004-6361/201834188
  * Spitoni et al. (2017) Spitoni, E., Vincenzo, F., & Matteucci, F. 2017, A&A, 599, A6, doi: 10.1051/0004-6361/201629745
  * Sukhbold & Adams (2020) Sukhbold, T., & Adams, S. 2020, MNRAS, 492, 2578, doi: 10.1093/mnras/staa059
  * Sukhbold et al. (2016) Sukhbold, T., Ertl, T., Woosley, S. E., Brown, J. M., & Janka, H.-T. 2016, ApJ, 821, 38, doi: 10.3847/0004-637X/821/1/38
  * Sukhbold et al. (2018) Sukhbold, T., Woosley, S. E., & Heger, A. 2018, ApJ, 860, 93, doi: 10.3847/1538-4357/aac2da
  * Sun et al. (2023) Sun, J., Leroy, A. K., Ostriker, E. C., et al. 2023, ApJ, 945, L19, doi: 10.3847/2041-8213/acbd9c
  * Tremonti et al. (2004) Tremonti, C. A., Heckman, T. M., Kauffmann, G., et al. 2004, ApJ, 613, 898, doi: 10.1086/423264
  * Turcotte et al. (1998) Turcotte, S., Richer, J., Michaud, G., Iglesias, C. A., & Rogers, F. J. 1998, ApJ, 504, 539, doi: 10.1086/306055
  * Ugliano et al. (2012) Ugliano, M., Janka, H.-T., Marek, A., & Arcones, A. 2012, ApJ, 757, 69, doi: 10.1088/0004-637X/757/1/69
  * van de Voort et al. (2020) van de Voort, F., Pakmor, R., Grand, R. J. J., et al. 2020, MNRAS, 494, 4867, doi: 10.1093/mnras/staa754
  * Vincenzo et al. (2016) Vincenzo, F., Matteucci, F., Belfiore, F., & Maiolino, R. 2016, MNRAS, 455, 4183, doi: 10.1093/mnras/stv2598
  * Weaver et al. (1978) Weaver, T. A., Zimmerman, G. B., & Woosley, S. E. 1978, ApJ, 225, 1021, doi: 10.1086/156569
  * Weinberg (2017) Weinberg, D. H. 2017, ApJ, 851, 25, doi: 10.3847/1538-4357/aa96b2
  * Weinberg et al. (2017) Weinberg, D. H., Andrews, B. H., & Freudenburg, J. 2017, ApJ, 837, 183, doi: 10.3847/1538-4357/837/2/183
  * Weinberg et al. (2019) Weinberg, D. H., Holtzman, J. A., Hasselquist, S., et al. 2019, ApJ, 874, 102, doi: 10.3847/1538-4357/ab07c7
  * Weinberg et al. (2022) Weinberg, D. H., Holtzman, J. A., Johnson, J. A., et al. 2022, ApJS, 260, 32, doi: 10.3847/1538-4365/ac6028
  * Woosley & Weaver (1995) Woosley, S. E., & Weaver, T. A. 1995, ApJS, 101, 181, doi: 10.1086/192237
  * Zahid et al. (2012) Zahid, H. J., Dima, G. I., Kewley, L. J., Erb, D. K., & Davé, R. 2012, ApJ, 757, 54, doi: 10.1088/0004-637X/757/1/54
  * Zhao et al. (2016) Zhao, G., Mashonkina, L., Yan, H. L., et al. 2016, ApJ, 833, 225, doi: 10.3847/1538-4357/833/2/225