# The Chemical Evolution of Iron-Peak Elements with Hypernovae

J.J. Grimmett1, Amanda I. Karakas1,2, Alexander Heger1,3, Bernhard Müller1,
Christopher West4,5,6
1Monash Centre for Astrophysics, School of Physics and Astronomy, 19
Rainforest Walk, Monash University, VIC 3800, Australia
2ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
3Tsung-Dao Lee Institute, Shanghai 200240, P. R. China
4Department of Physics and Astronomy, Carleton College, Northfield, MN, 55057,
USA
5Joint Institute for Nuclear Astrophysics—Center for the Evolution of the
Elements, USA
6School of Physics and Astronomy, University of Minnesota, Minneapolis, MN
55455, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We calculate the mean evolution of the iron-peak abundance ratios
[(Cr,Mn,Co,Zn)/Fe] in the Galaxy, using modern supernova and hypernova
chemical yields and a Galactic Chemical Evolution code that assumes
homogeneous chemical evolution. We investigate a range of hypernova occurrence
rates and are able to produce a chemical composition that is a reasonable fit
to the observed values in metal-poor stars. This requires a hypernova
occurence rate that is large (50%) in the early Universe, decreasing
throughout evolution to a value that is within present day observational
constraints ($\lesssim 1\%$). A large hypernova occurence rate is beneficial
to matching the high [Zn/Fe] observed in the most metal-poor stars, although
including hypernovae with progenitor mass $\geq 60\,\mathrm{M}_{\odot}$ is
detrimental to matching the observed [(Mn,Co)/Fe] evolution at low [Fe/H]. A
significant contribution from HNe seems to be critical for producing
supersolar [(Co,Zn)/Fe] at low metallicity, though more work will need to be
done in order to match the most extreme values. We also emphasise the need to
update models for the enrichment sources at higher metallicity, as the
satisfactory recovery of the solar values of [(Cr,Mn,Co,Zn)/Fe] still presents
a challenge.

###### keywords:

supernovae: general – Galaxy: abundances, evolution

††pubyear: xxxx††pagerange: The Chemical Evolution of Iron-Peak Elements with
Hypernovae–A.1

##  1 Introduction

The chemical abundances in the upper layers of low-mass ($\lesssim
0.8\,\mathrm{M}_{\odot}$) stars remain relatively stable throughout their
evolution. With the exception of perhaps the CNO elements, the abundances
observed in the photosphere of low-mass stars reflect the chemical composition
of the gas from the time and place that the star formed. Assuming that this
gas is chemically enriched, a large part of this enrichment will have been
from massive stars ($\gtrsim 10\,\mathrm{M}_{\odot}$) ending their lives as
core-collapse supernovae. Ergo, a key metric in assessing our understanding of
galactic evolution, stellar evolution, and the relationship between the two is
given by how well we are able to reproduce the chemical abundances observed in
the photospheres of low-mass, long-lived stars with theoretical models for the
chemical end products of shorter lived, high-mass stars (e.g., Nomoto et al.,
2013; Frebel & Norris, 2015, for recent reviews). Although we can predict the
ejected abundances for individual supernova models to compare with individual
or small subsets of stellar observations, it is difficult to establish a "big
picture" understanding in this manner. Galactic Chemical Evolution (GCE)
models provide a useful framework to collectively integrate many different
stellar models into a representation of the Galactic population. This allows a
broad comparison to both the chemical abundances observed in large collections
of stellar observations, and in chronological trends (e.g., Timmes et al.,
1995; Kobayashi et al., 2006, 2011; Romano & Starkenburg, 2013; Côté et al.,
2016; Andrews et al., 2017). The evolutionary scales for galaxies and massive
stars are both spatially and temporally disparate, which poses a computational
challenge when considering the symbiotic evolution of the system. Furthermore,
both galactic and stellar evolution have many unknowns and hurdles to overcome
even as separate studies, and GCE is not simply a matter of combining two
completely understood phenomena. Nevertheless, the scientific benefits of
pursuing this goal are many, and several groups have made steady progress over
the past few decades (van den Bergh, 1957; Schmidt, 1959, 1963; Truran &
Cameron, 1971; Talbot & Arnett, 1971; Searle & Sargent, 1972; Pagel &
Patchett, 1975; Tinsley, 1980; Matteucci & Greggio, 1986; Kobayashi et al.,
2000; Kawata & Gibson, 2003; Nomoto et al., 2006; Kobayashi et al., 2011).

A persistent problem is the abundances of the iron-peak elements and in
particular [(Cr, Mn, Co, Zn)/Fe] observed in metal-poor stars, which have not
been accurately reproduced by GCE models to date. This disparity reflects a
significant shortcoming in our understanding of the chemical evolution in the
Galaxy. In general, for decreasing [Fe/H], and particularly for [Fe/H]
$\leq-3$, the ratios [(Cr,Mn)/Fe] are observed to decrease while [(Co,Zn)/Fe]
are seen to increase (McWilliam et al., 1995; Ryan et al., 1996; Cayrel et
al., 2004; Bonifacio et al., 2009; Yong et al., 2013; Roederer et al., 2014;
Reggiani et al., 2017). With the exception of perhaps [Mn/Fe], these chemical
ratios cannot be convincingly reconciled with the supernova yields available
(Chieffi & Limongi, 2002; Nomoto et al., 2006; Joggerst et al., 2010; Heger &
Woosley, 2010; Grimmett et al., 2018).

Stars with [Fe/H] $\leq-3$ are so iron-poor that they have most likely formed
at a very early time in the Universe, when only very little metal enrichment
had occurred. Though it is also possible that these stars may have formed in a
poorly mixed region of the Galaxy at a later time, the general consensus is
that these stars have been enriched by only one, or potentially a few, of the
most massive and shortest lived first stars, i.e. those which formed from the
primordial Universe (Audouze & Silk, 1995; Argast et al., 2000, 2002). Whereas
it has been shown that the iron-peak elemental abundances converge to the
solar value in up-to-date GCE calculations, there is no convincing fit for the
abundances of these elements in the most metal-poor stars (Kobayashi et al.,
2006, 2011). This suggests that the supernovae (SNe) of the first stars must
be in some way unusual.

Lacking robust multidimensional, neutrino-driven models for core-collapse
supernovae of massive stars (see, e.g., Müller 2016), and also in the interest
of computational cost, spherically symmetric, parametrised models are commonly
used to estimate the chemical yields of supernovae across a wide range of
progenitors (Chieffi & Limongi, 2002; Nomoto et al., 2006; Heger & Woosley,
2010; Fryer et al., 2018; Ebinger et al., 2020). Parameterisation of the
explosion leaves room to explore the effects of physically motivated
adjustments to the model parameters. The effectiveness and limitations of
modifications can be constrained by comparison to observation and the results
of state-of-the-art multidimensional models. This is a particularly useful
approach when assessing the quality of supernova models for metal free stars,
where the ejected abundances can be directly compared to those observed in the
most metal-poor stars. For example, the results of multidimensional supernova
models have shown that turbulent and convective mixing can redistribute the
inner material during the explosion, and that the presupernova composition
(particularly the neutron excess) of the innermost material can be
significantly altered by neutrino processes during collapse (e.g., Kifonidis
et al., 2000; Janka et al., 2003; Liebendörfer et al., 2003). This allows for
a range of reasonable values that can be set for the relevant parameters (e.g.
mixing or composition of the inner layers) in 1D models, and this parameter
space has been explored as a pathway to larger [(Co,Zn)/Fe] and smaller
[(Cr,Mn)/Fe] yields from 1D models (e.g., Nakamura et al., 1999; Umeda &
Nomoto, 2002, 2005; Heger & Woosley, 2010). 1D models that include neutrino
transport, and particularly those which mimic a neutrino-driven explosion, are
parameterised by the properties of the neutrino energy deposition, allowing
for a more self-consistent treatment of the mixing and chemical composition in
the innermost ejecta (Janka & Mueller, 1996; Perego et al., 2015). Recent
nucleosynthesis studies using these types of models have reaffirmed and
expanded on the idea that neutrino interaction with the innermost material can
ultimately result in an ejecta composition similar to that which is commonly
observed in metal-poor stars, in particular the enhanced [Zn/Fe] (Curtis et
al., 2019; Ebinger et al., 2020).

The most promising advance with regard to reproducing the observed Fe-peak
abundances in metal-poor stars, however, has been made by considering
unusually large explosion energies in models. By and large, most supernovae
are estimated to have explosion energies on the order of
$10^{51}\,\mathrm{erg}$ (e.g., Kasen & Woosley, 2009). Beginning with the
observation of the unusually energetic SN 1997bw, however, and of several
similarly energetic supernovae since, it has been realised that at least some
fraction of supernovae explode with energies an order of magnitude larger then
usual, i.e. $10^{52}\,\mathrm{erg}$, and have been commonly referred to as
hypernovae (HNe) (Galama et al., 1998; Iwamoto et al., 1998; Matheson et al.,
2003; Woosley & Bloom, 2006). The exact occurrence rate of HNe is not yet well
constrained, although recent estimates place it at less than 1 percent of the
currently observed SN rate, with speculation that it may have been $>$ 10
percent in the early universe (Podsiadlowski et al., 2004; Woosley & Bloom,
2006; Arcavi et al., 2010; Smith et al., 2011; Smidt et al., 2014). By
modifying existing 1D, parametrised supernova models to match the large
explosion energy observed in these energetic events, it has been shown that
spherical representations of hypernovae will typically heat larger regions of
the stellar envelope to temperatures required for complete silicon burning,
which results in more Fe, Co, Zn relative to Cr, Mn.

A growing library of hypernova observations have revealed a strong connection
between hypernovae and gamma-ray bursts (GRBs) which, alongside other emerging
evidence such as broad line features in hypernova spectra, indicate that
hypernovae may be intrinsically aspherical, and possibly driven by jets
(Woosley & Bloom, 2006; Maeda et al., 2008; Wang & Wheeler, 2008; Tanaka et
al., 2017). Preliminary studies into the chemical yields to be expected from
highly aspherical and energetic explosions indicate that they may also provide
a favourable match to the abundances observed in stars with [Fe/H]
$\lesssim-3$ (Maeda & Nomoto, 2003; Pruet et al., 2003, 2004; Tominaga et al.,
2007; Tominaga, 2009; Ezzeddine et al., 2019). Though, models of this type are
still in active development and will be refined as more observational
constraints become available. Jet-driven supernovae may also be a site of
r-process nucleosynthesis (Winteler et al., 2012; Nakamura et al., 2015;
Nishimura et al., 2017; Halevi & Mösta, 2018).

Several earlier studies have made GCE calculations including the chemical
yields from HNe, with promising results for the evolution of the iron-peak
elements in relation to the observed trends in the Galaxy (Kobayashi et al.,
2006, 2011; Komiya, 2011; Tsujimoto & Nishimura, 2018; Hirai et al., 2018).
These studies, however, have used SN and HN models which are now outdated
and/or implement HN occurrence rates (e.g., 50% through all time) which cannot
be reconciled with the observed rate. Meanwhile, many advances have been made
in our understanding of stellar evolution and nuclear physics, and updated
supernova and hypernova yields have become available (Heger & Woosley, 2010;
Côté et al., 2016; Grimmett et al., 2018). Additionally, we now have some
broad observational constraints on the HN occurrence rate (Podsiadlowski et
al., 2004; Woosley & Bloom, 2006; Arcavi et al., 2010; Smith et al., 2011;
Smidt et al., 2014). In this work, we seek to understand the range of chemical
evolution results that are possible with the most modern and comprehensive
chemical yield sets for SNe and HNe, and with realistic HN rates. For this
purpose, we use a GCE model that assumes homogeneous chemical evolution
throughout the galaxy, i.e., a one-zone model. One-zone models have proven to
be effective and accurate for modelling the chemical evolution of well-mixed
regions of the Galaxy, e.g., the thin disk, and are computationally
inexpensive (Matteucci & Greggio, 1986; Timmes et al., 1995; Kobayashi et al.,
2000, 2006; Nomoto et al., 2006; Romano & Starkenburg, 2013; Andrews et al.,
2017). They are therefore an excellent tool for calculating the mean trends of
chemical evolution in galaxies and assessing the viability of supernova yield
calculations. Given the uncertainties that are inherent to GCE modelling
(homogenous models in particular), and the broad observational constraints for
HN occurrence rates, we explore our results over a wide parameter space, to
determine both the strengths and shortcomings of our current SN, HN, and
thermonuclear Type Ia supernova (SN Ia) models. We aim to determine where we
currently stand on explaining the evolution of [(Cr,Mn,Co,Zn)/Fe] in the
galaxy, and in particular the role of HNe in this process.

##  2 Method

###  2.1 Stellar Yields

The sources of chemical enrichment that we include in our GCE model are SNe,
HNe, and SNe Ia. We have selected 6 HN models from the results of Grimmett et
al. (2018), with progenitor masses and explosion energies that are in
approximate agreement with the HN mass-energy relation found by Nomoto et al.
(2003). These are also models which produce relatively large [Zn/Fe] and low
[Cr/Fe], which is favourable to matching the observed abundances in stars with
[Fe/H] $\lesssim-3$. The model parameters are outlined in Table 1. These HN
models are used for all [Fe/H].

Table 1: The hypernova models that we use in the GCE calculation. HN models
---
mass / $\mathrm{M}_{\odot}$ | 15 | 20 | 30 | 40 | 60 | 80
explosion energy / $10^{51}$ erg | $7.0$ | $5.5$ | $9.5$ | 25 | 60 | 60

The SN models are taken from the results of Côté et al. (2016), with
progenitor masses to match our HNe models up to a maximum of
$30\,\mathrm{M}_{\odot}$. These SN models include four different progenitor
masses, each with 15 different progenitor metallicities, as presented in Table
2.

Table 2: The supernova models that we use in the GCE calculation. SN models
---
masses / $\mathrm{M}_{\odot}$ | 13, 15, 20, 30 |
metallicities ([Fe/H]) | $2.86\times 10^{-2}\,(+0.36)$ | $2.32\times 10^{-2}\,(+0.25)$
| $1.88\times 10^{-2}\,(+0.14)$ | $1.53\times 10^{-2}\,(+0.03)$
| $9.89\times 10^{-3}\,(-0.28)$ | $6.29\times 10^{-3}\,(-0.75)$
| $4.16\times 10^{-3}\,(-1.07)$ | $2.79\times 10^{-3}\,(-1.30)$
| $1.87\times 10^{-3}\,(-1.51)$ | $7.01\times 10^{-4}\,(-2.02)$
| $2.64\times 10^{-4}\,(-2.53)$ | $1.00\times 10^{-4}\,(-3.03)$
| $3.81\times 10^{-5}\,(-3.53)$ | $5.59\times 10^{-6}\,(-4.53)$
| $0.00$ |

We explore two implementations of the SN chemical yields in our GCE models. In
the first implementation, we assume that all progenitors explode successfully,
ejecting the majority of the envelope. We refer to this SN set as the standard
set. The second implementation that we investigate contains a portion of SN
progenitors that are presumed to not explode after core-collapse, and instead
collapse to black holes. The progenitor models are assessed for their
"explodability" with a prescription outlined by Ertl et al. (2016), wherein
the models are evaluated based on the progenitor structure and compactness. In
this set, which we refer to as the fallback set hereafter, the models that
fail to explode eject only the fraction of the envelope that was driven away
by stellar winds during evolution.

Of particular relevance in the context of GCE are the combined chemical yields
from a population of stars, weighted by the initial mass function (IMF). We
assume that stars form with a distribution of masses described by the Salpeter
IMF, $\xi(m)\propto m^{-1.35}$, normalised to unity between
$m_{\mathrm{u}}=100\,\mathrm{M}_{\odot}$ and
$m_{\mathrm{l}}=0.05\,\mathrm{M}_{\odot}$, i.e.
$\int^{m_{\mathrm{u}}}_{m_{\mathrm{l}}}N\xi(m)dm=1$, where $N$ is a
normalisation constant (Salpeter, 1955). We combine our SN/HN yields
accordingly, and the IMF-weighted chemical yields are shown in Figure 1.

SNe Ia are represented by the W7 model from Nomoto et al. (1984), with
chemical abundances taken from the yields table provided by Kobayashi et al.
(2006).

###  2.2 Observational Data

We compare the results of our GCE models with the observed chemical abundances
in a sample of Milky Way stars. The abundance data that we use is taken from
several high-quality observational studies in the literature. These studies
have obtained stellar abundance values using high quality spectroscopy, with
signal-to-noise ratios $>100$ and resolving power $>40,000$.
The observational data, shown in Figure 2, covers the range
$-4\lesssim\mathrm{[Fe/H]}\lesssim 0.5$ for each of the abundance ratios
[(Cr,Mn,Co,Zn)/Fe)]. The regime of lowest metallicity
($\mathrm{[Fe/H]}\lesssim-2.5$) is represented by data from the First Stars
Programme, including 35 giant stars (Cayrel et al., 2004) and 18 turnoff stars
(Bonifacio et al., 2009). The range $-2.5\lesssim\mathrm{[Fe/H]}\lesssim-1.5$
is covered by the abundances observed in 23 halo stars (Reggiani et al.,
2017). Observational data for the upper end of the metallicity distribution
($\mathrm{[Fe/H]}\gtrsim-1.5$) consists of 1111 FGK stars in the solar
neighbourhood (Adibekyan et al., 2012; Delgado Mena et al., 2017),
supplemented by the [Cr/Fe] value observed in 714 FG dwarf and subgiant stars
(Bensby et al., 2014).

All observational data has been normalised relative to the solar values of
Asplund et al. (2009), with the exception of the Bonifacio et al. (2009) data,
for which we use published data without modification, as we were unable to
obtain the adopted solar values for the original data.

###  2.3 Galactic Chemical Evolution Model

We use a basic one-zone chemical evolution model developed by the authors. The
code solves a set of equations that represent a simplified model for an
evolving galaxy, as introduced by Tinsley (1980). We emulate the formulation
of these equations as described by Kobayashi et al. (1998, 2000, 2006), with
minor changes to accommodate our implementation of HNe, described below.

Similar to Kobayashi et al. (2006), we assume two varieties of core-collapse
supernovae (CCSNe), those of typical supernova explosion energy of order
$10^{51}\,\mathrm{erg}$, and those of hypernova explosion energy, of order
$10^{52}\,\mathrm{erg}$ (specific values listed in Section 2.1). Rather than a
constant HN rate, however, we set the fraction of massive stars exploding as
hypernovae, $\epsilon_{\mathrm{hn}}$, with the following metallicity ($Z$)
dependent prescription:

$\epsilon_{\mathrm{hn}}=\max\left(\epsilon_{\mathrm{hn,}0}\exp\left(-\frac{Z}{0.001}\right),0.001\right)\,,$
(1)

where $\epsilon_{\mathrm{hn,}0}$ is the HN fraction at $t=0\,\mathrm{yr}$
(i.e. the beginning of the Universe). The evolution of
$\epsilon_{\mathrm{hn}}$ for each initial condition is shown in Figure 3. The
HN rate in the early universe is not well constrained but is predicted to be
much larger than the observed rate today (Woosley & Bloom, 2006; Smidt et al.,
2014). The current HN rate is loosely constrained to be $\lesssim 1\%$ of the
current SN rate (Podsiadlowski et al., 2004). This prescription allows us to
explore a range of HN rates in the early universe, while ensuring that the
rate drops rapidly to conform to the lower rate at higher metallicities. We
implement $\epsilon_{\mathrm{hn}}$ as a weighting factor to combine our SN and
HN yields. Additionally, to allow for more massive HN progenitors, we extend
the initial mass function (IMF) from a minimum mass of
$0.05\,\mathrm{M}_{\odot}$ to a maximum of $100\,\mathrm{M}_{\odot}$.

We do not consider the nucleosynthetic output from low and intermediate-mass
stars ($<8\,\mathrm{M}_{\odot}$), as the major contribution from this stellar
group are intermediate mass elements, particularly the CNO and s-process
elements (see, e.g., Kobayashi et al. 2011; Karakas & Lattanzio 2014), whereas
our investigation is focussed on the evolution of the Fe-peak elements. The
large amount of gas that low and intermediate-mass stars return to the
interstellar medium (ISM) via winds, however, is important to consider when
following the evolution of gas in the Galaxy. For this purpose, we adopt the
remnant mass ($m_{\mathrm{rem}}$) prescription given by Iben & Tutukov (1984);
Pagel (2009) for stars with mass $<10\,\mathrm{M}_{\odot}$:

$m_{\mathrm{rem}}=\begin{cases}m,&\text{if }m\leq 0.506\,\mathrm{M}_{\odot}\\\
0.11m+0.45,&\text{if }0.506\,\mathrm{M}_{\odot}<m\leq
9.5\,\mathrm{M}_{\odot}\\\ 1.5&\text{otherwise}\,.\end{cases}$

Here, $m$ is the main sequence mass of the star, and we leave the chemical
composition of the gas unchanged between star formation and wind ejection in
this mass range.

##  3 Results

In Figures 4 through 7 we show the results for the evolution of
[(Cr,Mn,Co,Zn)/Fe] as a function of [Fe/H]. For each GCE model we vary SN/HN
contributions, as described in Section 2.1. We compare our results to several
recent sets of observed stellar abundances, which can be more easily
differentiated in Figure 2, for reference. Due to the intrinsic uncertainties
associated with basic one-zone GCE models (Section 1), our main goal is not to
provide a perfect fit to the observed trends in chemical abundances. While
general improvements to the absolute fit are significant and are discussed
below, our central focus is to achieve a more comprehensive understanding of
the scope of possible results for various sets of SN yields and HN
contribution across a realistic parameter space. In the following, we discuss
our results for each of [(Cr,Mn,Co,Zn)/Fe] in turn. We first set about
reproducing the results of Kobayashi et al. (2006), with results shown in
Section A.1.

Figure 1: IMF-weighted Fe mass in ejecta (top row), and [(Cr,Mn,Co,Zn)/Fe]
IMF-weighted yields (bottom row) for individual model sets. (a,b); IMF-
weighted HN yields, as a function of maximum HN progenitor mass, the SN Ia
yields are plot as stars. (c,d); IMF-weighted standard SN set yields as a
function of [Fe/H]. (e,f); IMF-weighted SN yields from the fallback set as a
function of [Fe/H]. Models with primordial composition are plot at
$\mathrm{[Fe/H]}=-5$. Figure 2: The observed values of [(Cr,Mn,Co,Zn)/Fe] in
the set of Milky Way stars that we have selected for comparison to our GCE
results. The observational sources include metal-poor giant stars in the halo
(Cayrel et al., 2004, black plus signs), metal-poor turnoff stars (Bonifacio
et al., 2009, magenta squares), metal-poor halo stars (Reggiani et al., 2017,
gold stars), F and G dwarf stars in the solar neighbourhood (Bensby et al.,
2014, grey hexagons), FGK stars in the solar neighbourhood (Adibekyan et al.,
2012, green crosses) and (Delgado Mena et al., 2017, blue circles). Where
available, these abundances have been extracted from the STELLAB library
(Ritter & Côté, 2016). We have not included error bars for the Adibekyan et
al. (2012) data, as the original abundances and errors are given relative to
H, and without the errors for [Fe/H], we were unable to convert the
uncertainties. Figure 3: The HN fraction ($\epsilon_{\mathrm{hn}}$) as a
function of [Fe/H], for each value of initial HN fraction
($\epsilon_{\mathrm{hn,0}}$). The solid lines represent the models with the
standard SN (+HN/SN Ia) yields, and the dashed lines represent the models with
the fallback SN set (+HN/SN Ia). Here the HN mass upper limit is set as
$30\,\mathrm{M}_{\odot}$. For different values of HN mass upper limit, the
changes in $\epsilon_{\mathrm{hn}}$ evolution are negligible, and the
evolution shown here can be taken as representative for all values of HN mass
upper limit. Figure 4: The results of our GCE calculation for [Cr/Fe] as a
function of [Fe/H] (blue lines). The solid blue lines represent the models
with the standard SN (+HN/SN Ia) yields, and the dashed blue lines represent
the models with the fallback SN set (+HN/SN Ia). The HN mass upper limit in
each model increases down the rows (i.e. HN mass upper limit by row, top to
bottom; $30\,\mathrm{M}_{\odot}$, $40\,\mathrm{M}_{\odot}$,
$60\,\mathrm{M}_{\odot}$, and $80\,\mathrm{M}_{\odot}$), and the initial HN
fraction increases across the columns (i.e. initial HN rate by column, left to
right; 5%, 10%, 25%, and 50%). The grey symbols are the observed values of
[Cr/Fe] in stars, see Figure 2 for more detailed description of the
observations. Figure 5: Same as Figure 4, but for the evolution of [Mn/Fe].
Figure 6: Same as Figure 4, but for the evolution of [Co/Fe]. Figure 7: Same
as Figure 4, but for the evolution of [Zn/Fe].

###  3.1 Chromium

[Cr/Fe] is produced at a value close to zero in SNe of all [Fe/H] (Figure 1).
This is because Cr and Fe are synthesised in similar regimes of temperature
and neutron excess during explosive silicon burning, and therefore, it would
be rare to enhance or inhibit the creation of one without likewise affecting
the other. This is in contrast to the decreasing trend of [Cr/Fe] at low
metallicity. It has been found that one pathway to lower [Cr/Fe] is by
increased explosion energy in core collapse models, where Fe production (as
56Ni) is increased due to a larger volume of the envelope undergoing explosive
burning (Nakamura et al., 2001; Umeda & Nomoto, 2005; Nomoto et al., 2006;
Grimmett et al., 2018). This is the case in our HN models, which produce
slightly lower [Cr/Fe] than the SN models. The largest deviation in [Cr/Fe]
from zero occurs in the lower mass ($\leq 30\,\mathrm{M}_{\odot}$) HN models,
in which synthesis of Cr is suppressed as a result of extra heating from a
reverse shock due to the specifics of the progenitor structure in this mass
range (Grimmett et al., 2018). We have specifically selected HN models where
deviation in [Cr/Fe] is the largest to allow us to better investigate the full
range of possible [Cr/Fe] evolution histories. The W7 SN Ia model also
produces a relatively low [Cr/Fe] $\sim-0.2$.

The similarity in [Cr/Fe] between sources is reflected in our models of the
chemical evolution of the galaxy (Figure 4). In general, [Cr/Fe] is almost
constant with [Fe/H]. At $\mathrm{[Fe/H]}\gtrsim-1$, when SNe Ia begin to
contribute, there is a slight decrease in [Cr/Fe] in each GCE model. Due to
the similarity in [Cr/Fe] between most SNe and HNe, the chemical evolution
shows almost no change between models with different remnant mass
prescriptions for SNe, upper mass limit for HNe, or HN fraction. The only
(minor) exception to this result is for the models in which the contribution
from the lower mass HNe (with low [Cr/Fe]) is maximised. The results of these
models are shown in the upper right corner of Figure 4, particularly in panels
(c) and (d), which have a HN upper mass limit of $30\,\mathrm{M}_{\odot}$ and
the largest HN fraction. These models are still not able to match the low
$\mathrm{[Cr/Fe]}\sim-0.4$ observed in the most metal poor stars, but they are
the only models which show signs of a decreased [Cr/Fe] value for low [Fe/H].
There is an increasing sentiment, however, that the observed [Cr/Fe] values in
metal-poor stars may be underestimated when non-local thermodynamic
equilibrium effects are neglected in stellar atmosphere models (Cayrel et al.,
2004; Lai et al., 2008; Bergemann & Cescutti, 2010).
The evolution of the models with the fallback SN set maintain slightly lower
[Cr/Fe] relative to the models with the standard SNe, as the fallback SN set
contribute less ejecta overall, due to the stars which fail to explode, so the
HN yields dominate the [Cr/Fe] value. We are able to achieve [Cr/Fe] < 0 at
low metallicity, which is required for matching the observed values in stars
with $\mathrm{[Fe/H]}\lesssim-3$.

###  3.2 Manganese

[Mn/Fe] is produced by SNe in a ratio which is increasing with progenitor
metallicity (Figure 1). This is because Mn has only one stable isotope, 55Mn,
which is neutron-rich. Therefore, 55Mn is produced more abundantly in
environments with a supply of excess neutrons. An increasingly neutron-rich
environment is provided by SN progenitors with greater metallicity. In our HN
models, [Mn/Fe] is decreased due to the enhanced Fe production that
accompanies the large volume of explosively burned envelope during highly
energetic explosions. The W7 SN Ia model produces the largest [Mn/Fe] of our
enrichment sources, at approximately the solar value.

The trend of increasing [Mn/Fe] produced by higher metallicity SN models is
clearly evident in all of our chemical evolution calculations (Figure 5). The
decreasing contribution from HNe as the metallicity of the gas increases
serves to reinforce the positive relation between [Mn/Fe] and [Fe/H], as does
the increasing contribution from SNe Ia at [Fe/H] $\gtrsim-1$. Although the
[Mn/Fe] ratios from both SN sets are very similar, the models comprised of the
SNe from the fallback set are most strongly influenced by the HN and SN Ia
contributions. This is because the SNe in the fallback set collectively eject
less mass than the standard SN set, and therefore the HN and SN Ia abundances
are able to more strongly dominate. This is can be seen in the more extreme
values of low [Mn/Fe] at the lowest [Fe/H], and high values of [Mn/Fe] toward
solar [Fe/H] in the evolution of the models with the fallback SN set relative
to the models with the standard SN set. The HN sources provide the lowest
value of [Mn/Fe], and lower still from the most massive HNe, so the effect of
both a larger HN fraction, and a larger HN upper mass limit is to decrease the
[Mn/Fe] value in the evolution.
The best fit is provided by GCE models with a limited HN contribution, either
by low HN fraction, low HN upper mass limit, or a combination of both (e.g.,
M${}_{\mathrm{hn,u}}\leq 40\,\mathrm{M}_{\odot}$ or
$\epsilon_{\mathrm{hn,0}}\leq 0.1$), hence the lower right hand corner of the
grid of results in Figure 5 show the least favourable results. The increasing
[Mn/Fe] in the ejecta of SN progenitors with increasing metallicity, and the
large [Mn/Fe] contribution from SNe Ia ensures a fairly robust fit to the
solar value of [Mn/Fe] across all GCE models.
The models with the standard SN set typically provide a better fit to the
observed increasing trend in [Mn/Fe] with [Fe/H], although the flatter trend
in [Mn/Fe] in the models with the fallback SN set is likely attributed to the
contribution from only zero metallicity HNe. It is possible that there may be
a trend of higher [Mn/Fe] with progenitor [Fe/H] in HNe, as there is with SNe,
due to larger neutron excesses. This is the case in the HN models of Kobayashi
et al. (2006).

###  3.3 Cobalt

Similar to [Mn/Fe], [Co/Fe] is produced in an increasing value by SN
progenitors of larger [Fe/H] (Figure 1). Unlike [Cr/Fe] and [Mn/Fe], there are
significant differences between the values of [Co/Fe] produced by the fallback
SN set and the standard set. In Figure 1 we saw that for lower metallicities,
the IMF-weighted fallback SN set produce [Co/Fe] in a larger ratio than in the
standard SN set, though for higher metallicity the values converge. [Co/Fe] is
typically produced at a value $\mathrm{[Co/Fe]}\gtrsim 0.0$ by lower mass HNe,
and in decreasing values for increasing progenitor mass, hence Figure 1 shows
a decreasing IMF-weighted [Co/Fe] value for increasing HN upper mass limit.
The W7 SN Ia model produces $\mathrm{[Co/Fe]}\simeq-0.25$.

The impact of the difference in [Co/Fe] between the SN model sets can
immediately be seen in our results for the chemical evolution, shown in Figure
6. The models with the standard SN set typically have lower [Co/Fe] values,
though the results converge for higher HN fraction, higher upper mass limit
for HNe, or a combination of both. In this regime, the HN yields tend to
dominate the overall abundance ratios, and the differences between the SN
model sets have less impact. We find that a larger HN fraction will typically
increase the overall value of [Co/Fe]. This effect is more pronounced for the
models with the standard SN set, for two main reasons: (i) the HNe and
fallback SN set [Co/Fe] yields are similar, so an increased HN fraction has
little effect on the average between the two. On the other hand, there is a
large difference between the [Co/Fe] yields from the HNe and the standard SN
set, so a larger HN fraction has a stronger effect on the final value in this
case; and (ii) the fallback SN set eject less mass collectively, so the
averaged [Co/Fe] value is likely already dominated by HNe even for low HN
fraction, and increasing HN fraction makes little difference.

For $\mathrm{[Fe/H]}\gtrsim-1$, there is an increasing trend in the value of
[Co/Fe] for the models with the standard SN set, and a decreasing trend in the
same value for the models with the fallback SN set. This can be explained as
follows; At $\mathrm{[Fe/H]}\sim-1$ in each model, where HN contribution is
rapidly decreasing, the [Co/Fe] value at this point in the evolution is
essentially a weighted average between the SN and SN Ia models. For the models
with the fallback SN set, the increasing SN Ia contribution mostly dominates
the overall [Co/Fe] value, due to the smaller mass of SN ejecta from these
models, and the [Co/Fe] value in the ISM trends toward the value ejected from
SNe Ia, $\mathrm{[Co/Fe]}\simeq-0.25$. For the models with the standard SN
set, the SNe have a stronger contribution due to larger IMF-weighted ejecta
mass, and the final value resulting from the combined SN and SN Ia ejecta is
reflected in the ISM as $\mathrm{[Co/Fe]}\simeq 0$.

Overall, we find the GCE models with the fallback SN set produce a more robust
fit to the observed abundances of [Co/Fe] for $\mathrm{[Fe/H]}<-1$, whereas
models with the standard SNe provide a better fit for $\mathrm{[Fe/H]}\geq-1$.
Neither of the models can convincingly reproduce the large [Co/Fe] observed in
the lowest metallicity stars. The GCE models with the fallback SN set produce
a better fit when higher mass HNe do not contribute (e.g.,
M${}_{\mathrm{hn,u}}\leq 40\,\mathrm{M}_{\odot}$), though all of these models
underproduce [Co/Fe] at solar metallicity. For the GCE models with the
standard SNe, the best fit is produced with $40\,\mathrm{M}_{\odot}\leq$
M${}_{\mathrm{hn,u}}\leq 60\,\mathrm{M}_{\odot}$ and
$\epsilon_{\mathrm{hn,0}}\geq 0.25$, though the [Co/Fe] at the lowest
metallicities remains too low.

###  3.4 Zinc

There is a significant difference in the IMF-weighted [Zn/Fe] yields between
the SNe with different remnant mass prescriptions (Figure 1). On average, the
fallback SN set produces larger [Zn/Fe] relative to the standard SNe for
$\mathrm{[Fe/H]}\lesssim-1$, whereas the standard SN set produces larger
[Zn/Fe] relative to the fallback SN set for $\mathrm{[Fe/H]}\gtrsim-1$. The
reason for the difference in [Zn/Fe] between the two SN sets is not simple to
explain, and is essentially due to which particular stars are able to explode
at each given metallicity, depending on the progenitor structure, see Ertl et
al. (2016); Côté et al. (2016) for a more in-depth discussion on the
explodability of models. Figure 1 also shows that the IMF-weighted HN yields
consistently produce $\mathrm{[Zn/Fe]}\geq 0$, and $\mathrm{[Zn/Fe]}\geq 0.1$
for HN upper mass limit $\geq 40\,\mathrm{M}_{\odot}$. The W7 SN Ia model
produces very low $\mathrm{[Zn/Fe]}\simeq-1.5$.

The results for the evolution of [Zn/Fe] are shown Figure 7. We see that
typically, the GCE models with the fallback SN set produce a larger [Zn/Fe]
value for $\mathrm{[Fe/H]}\lesssim-1$, reflecting the high [Zn/Fe] from both
the low [Fe/H] fallback SN set, and dominant contribution from HN sources.
Each of the models with different SN sets converge for high HN fraction,
higher HN upper mass limit, or a combination of both, as the HN yields begin
to dominate the chemical abundances in this regime (lower right corner of
Figure 7). Variation of the HN upper mass limit and HN fraction has less
effect in the models with the fallback SN set source, relative to the effect
on those with the standard SNe. This is partly because the [Zn/Fe] yields from
the fallback SN set and HNe are similar, meaning that the variation of the HN
contribution has little effect on the averaged value of [Zn/Fe]. Additionally,
fallback SN set eject less mass than the standard SNe (Figure 1), so the
abundance ratio of [Zn/Fe] is strongly weighted toward the HN yields, even for
a low HN rate. Broadly speaking, each of the models with the fallback SN set
evolve with fairly constant $\mathrm{[Zn/Fe]}\sim 0.2$ up until
$\mathrm{[Fe/H]}\gtrsim-1$. At this point the contributions from SNe Ia and
higher metallicity SNe commence, and the value of [Zn/Fe] trends toward
$<-0.5$ for each model.

Both a higher HN fraction and a higher HN upper mass limit have a strong
effect on the [Zn/Fe] evolution in the models which contain the standard SNe.
When the HN contribution is minimal (upper left corner of Figure 7), we find
$\mathrm{[Zn/Fe]}\simeq-0.2$ for $\mathrm{[Fe/H]}\lesssim-1$. When the HN
contribution is increased, [Zn/Fe] is as high as $\sim 0.3$. This large value
for [Zn/Fe] requires either $\epsilon_{\mathrm{hn},0}\geq 0.1$ and HN upper
mass limit $\geq 60\,\mathrm{M}_{\odot}$, or $\epsilon_{\mathrm{hn},0}\geq
0.25$ and HN upper mass limit $\geq 40\,\mathrm{M}_{\odot}$. Whereas the SN Ia
contribution to [Zn/Fe] at $\mathrm{[Fe/H]}\gtrsim-1$ dominates the ISM
abundances in the models with the fallback SN set, the standard SNe have more
massive ejecta and therefore a stronger contribution to the [Zn/Fe] value. The
ISM [Zn/Fe] abundance in these models trends toward $\gtrsim-0.5$.

We are not able to match the high $\mathrm{[Zn/Fe]}\gtrsim 0.5$ at the lowest
metallicities with any of our models, nor are we able to match the mean solar
value of [Zn/Fe]. Whereas all of our models with the fallback SN set are able
to produce a more robust fit to the observed values for
$\mathrm{[Fe/H]}\leq-1$, the value of [Zn/Fe] toward solar metallicity is far
too low. Although the models with the standard SNe produce [Zn/Fe] that is too
low overall, these models do provide a better fit to the trend of [Zn/Fe] with
[Fe/H]. In particular, the [Zn/Fe] increase for low metallicity and plateau
for $\mathrm{[Fe/H]}\gtrsim-1$. The models with the standard SNe, with HN
upper mass limit $\geq 40\,\mathrm{M}_{\odot}$ and initial HN fraction
$\epsilon_{\mathrm{hn,0}}\geq 0.25$ provide the best fit, though the absolute
value is too low, most notably at solar metallicity.

###  3.5 Early Evolution and Extremely Metal Poor Stars

With the exception of perhaps [Mn/Fe], we have not been able to match the
observed abundances in stars with $\mathrm{[Fe/H]}\lesssim-3$. It is possible,
however, that the fit could be improved by varying some parameters that we
have not investigated here, such as the SFR or IMF. There are indications that
the IMF of the first stars may have been top-heavy (Bromm et al., 1999; Abel
et al., 2002; Susa et al., 2014; Hirano et al., 2014; Hosokawa et al., 2016).
It also has been predicted that the most metal-poor stars may have been
enriched by the chemical yields from a single, or perhaps just a few, SN/HN
events, whereas the one-zone GCE model assumes well-mixed material (Audouze &
Silk, 1995; Argast et al., 2000, 2002; Frebel et al., 2005; Keller et al.,
2014; Ji et al., 2015; Roederer et al., 2016; Frebel et al., 2019). In Figures
8 through 10, we show the yields of [(Cr,Mn,Co,Zn)/Fe] from each individual SN
and HN model available in the sets that we selected from, including models
that we did not use. In these figures it can be seen that we are in fact
unable to match all of the observed abundances of [(Cr,Co,Zn)/Fe] with any
single HN or SN model, and in turn, nor would any combination of the SN/HN
models that we have not already investigated be capable of doing so.
On the other hand, when considering the possibility that some low metallicity
stars may have been enriched by only one SN/HN, the lack of variation in
[Cr/Fe] across all of our HN/SN models is consistent with the small scatter in
the observed value (Cayrel et al., 2004). Furthermore, the range of [Mn/Fe]
values produced by our zero-metallicity models is consistent with the larger
scatter observed in [Mn/Fe]. The high energy ($\gtrsim 30\times 10^{51}$ erg)
explosions of the $40\,\mathrm{M}_{\odot}$ model can potentially provide a
reasonable fit to [(Mn,Co,Zn)/Fe], though the [Cr/Fe] becomes too large in
these models.

##  4 Discussion

We have calculated the mean evolution of the iron-peak ratios
[(Cr,Mn,Co,Zn)/Fe] in the Galaxy, using a GCE model that (i) applies the
instantaneous mixing approximation, (ii) includes the time delay between
formation and death of a star as a function of stellar mass and metallicity,
and (iii) allows for inflow of material from a galactic halo. We include the
chemical yields from SNe, HNe, and SNe Ia. Whereas this type of GCE model is
not novel, the SN and HN yields that we include are the most modern and
comprehensive sets available (Côté et al., 2016; Grimmett et al., 2018). Along
with an improved understanding of the relationship between nucleosynthesis and
explosion energy, there have been several developments in nuclear and stellar
physics, and in the general implementation of this knowledge in the
presupernova and supernova modelling (e.g., Rauscher et al., 2002; Heger &
Woosley, 2010). It is useful to periodically collate new nucleosynthetic
results with updated GCE models, in order to establish an understanding of
where we stand in explaining Galactic chemical trends. We have explored the
effect of altering the HN contribution by occurrence rate and maximum
progenitor mass, and also investigated SN models with two different remnant
mass prescriptions. By exploring our results across a wide parameter space,
our aim was to gain a thorough understanding of the strengths and shortcomings
of current SN/HN models to explain the chemical evolution in the Galaxy, and
in particular, to better understand the role of HNe in this process.

For the evolution of [Cr/Fe] and [Mn/Fe], we have found that a reasonable fit
to the observed relation between [(Cr,Mn)/Fe] and [Fe/H] can be made with
almost any combination of SN/HN models and HN rate. This statement is
especially true for [Cr/Fe], the evolution of which varies only slightly with
different SN sets and HN contributions. On one hand, the small variation in
[Cr/Fe] that is produced in our models is consistent with the particularly
small scatter of observed values of [Cr/Fe], as noted by Cayrel et al. (2004).
On the other hand, if our models are accurate, then the flat trend in [Cr/Fe]
that we see in our GCE results would support the suggestion that the
observational values of [Cr/Fe] reported for metal-poor stars suffer from
metallicity dependent corrections (Cayrel et al., 2004; Lai et al., 2008).
Bergemann & Cescutti (2010) find that neglect of non-local thermodynamic
equilibrium effects (NLTE) in the analysis of Cr spectral lines result in an
underestimate of the [Cr/Fe] value in the most metal-poor stars. In any case,
the consistency of the result for [Cr/Fe] across our range of model parameters
should be kept in mind when using [Cr/Fe] as a diagnostic for the quality of
any given GCE model.

Figure 8: The values of [(Cr,Mn,Co,Zn)/Fe] in the ejecta of the hypernova
models (Grimmett et al., 2018). The grey regions indicate models which undergo
significant fallback and eject negligible amounts of iron-peak elements.
Figure 9: The values of [(Cr,Mn,Co,Zn)/Fe] in the ejecta of the supernova
models from the standard SN set Figure 10: The values of [(Cr,Mn,Co,Zn)/Fe] in
the ejecta of the supernova models from the fallback set. The grey regions
indicate models which fail to explode and therefore eject negligible amounts
of iron-peak elements.

The evolution of [Mn/Fe] shows a larger variation, depending on the SN/HN
model parameters. Across the entire parameter space, however, we are still
able to achieve a reasonably good fit to the observed trend of low [Mn/Fe] at
early times, increasing to the solar value. The robustness of the fit can be
attributed to the low [Mn/Fe] ejected from HN models, which are the dominant
enrichment source at early times, shifting to the larger [Mn/Fe] in the SN Ia
ejecta, which becomes the dominant contribution for increasing metallicity.
This is the typical [Mn/Fe] evolution that has also been achieved in earlier
studies (e.g., Kobayashi et al., 2006). We find that if the HN contribution
becomes too large, i.e. when M${}_{\mathrm{hn,u}}\geq 60\,\mathrm{M}_{\odot}$
and $\epsilon_{\mathrm{hn,0}}\geq 0.25$, the [Mn/Fe] is driven too low to be
consistent with the observed values at lower metallicity, though we still
recover the solar value at later times in each of these models. This effect is
particularly pronounced for the models with the fallback SN set, where the HN
ejecta is able to more strongly dominate. Like many earlier studies our models
qualitatively explain the observed trend in [Mn/Fe] with [Fe/H]. It has been
argued, however, that the observed values of [Mn/Fe] at low metallicity are
significantly and systematically underestimated due to the neglected effects
of NLTE in spectral analyses (Bergemann & Gehren, 2008). If the large
corrections that have been suggested are proven true, this would have
substantial repercussions for the current understanding of the enrichment
sources and evolution of Mn abundances in the Galaxy.

We have not been able to achieve the super-solar values of [Mn/Fe] observed in
stars with $\mathrm{[Fe/H]}\gtrsim 0$. This is result is unavoidable, as none
of our individual SN or HN models produce super-solar [Mn/Fe] (Figure 1, and
Section 3.5). Seitenzahl et al. (2013) report that higher mass white dwarf
(WD) progenitors (i.e. near Chandrasekhar mass) are required to produce super-
solar [Mn/Fe], regardless of the exact pathway to explosion. The progenitor
systems of SNe Ia are yet to be identifed, although observational and
theoretical work in the near future will help to provide further constraints
on the progenitor properties (see Maoz et al., 2014, for a review). Our
results provide additional motivation to consider higher mass WD progenitors.

Both [Co/Fe] and [Zn/Fe] prove to be far more useful quantities for evaluating
our models and associated parameter space, as the evolution of these element
ratios show a large variation between GCE models with both different SN sets,
and HN contributions. We find a reasonable fit to the observed evolution when
implementing HN rates that are not irreconcilable with the observed rates.
Furthermore, our models are able to produce both [Co/Fe] and [Zn/Fe]
simultaneously in supersolar values at low metallicity. This is required to
make progress in explaining the high observed values of [(Co,Zn)/Fe] in stars
with $\mathrm{[Fe/H]}\lesssim-3$, which have proven to be difficult to
reconcile (Timmes et al., 1995; Kobayashi et al., 2006, 2011; Hirai et al.,
2018). We find that a strong contribution from HNe at low metallicity (e.g.,
$\gtrsim 25\%$ for $\mathrm{[Fe/H]}\lesssim-2$) is favourable to producing
large [(Co,Zn)/Fe] at low metallicity, though even the largest values that we
achieve are still too low to match the [Co/Fe] observed in stars with
$\mathrm{[Fe/H]}<-3$. We also find, however, that if some fraction of SNe fail
to explode as determined by the progenitor compactness criterion (i.e. the
fallback SN set), then the increasing contribution from SNe Ia dominates the
ISM abundance, and the evolution of [Zn/Fe] in our models becomes far too low
near solar [Fe/H] as a result. Although we have found that the models that
include the fallback SN set typically produce a more robust fit to the high
[(Co,Zn)/Fe] observed at low metallicity across the entire parameter space,
this is more likely attributed to the smaller IMF-weighted ejecta from these
SNe, which allows the HN contribution to more easily dominate. Moreover, the
models with the fallback SN set poorly fit the solar values of [(Co,Zn)/Fe].
The results of recent 3D supernova simulations performed by Burrows et al.
(2019) indicate that there is no correlation between progenitor compactness
and explodability, though, it certainly appears that not all progenitors in
the core-collapse mass range will successfully explode. It may be useful to
investigate other prescriptions for model explodability in future GCE studies.

Altogether, we find that the most favourable results are achieved with a
maximum HN mass of $40\,\mathrm{M}_{\odot}$, an initial HN occurrence rate of
50%, and the standard SN set i.e. the solid line in panel (h), Figures 4
through 7. In the GCE models with the standard SNe, we find that a combination
of both a large initial HN fraction and more massive HN progenitors can be
problematic, most notably for the fit to the observed trend in [Mn/Fe], which
is reasonably well fit with most other combinations of parameters.
Furthermore, this combination of parameters is also detrimental for producing
the slope in the observed values of [(Co,Zn)/Fe].

Ongoing investigations into the chemical yields from aspherical HN models will
provide a critical step forward in understanding the role of HNe in GCE.
Evidence has emerged that at least some HNe are accompanied by gamma-ray
bursts (GRBs) (Galama et al., 1998; van Paradijs et al., 2000; Stanek et al.,
2003; Malesani et al., 2004; Woosley & Bloom, 2006). The two most popular
models to explain the connection between the collapse of a massive star and a
GRB, the collapsar model of MacFadyen & Woosley (1999); MacFadyen et al.
(2001); Woosley & Heger (2003); Barkov & Komissarov (2008) and the magnetar
model (e.g., LeBlanc & Wilson, 1970; Wheeler et al., 2000; Akiyama et al.,
2003; Burrows et al., 2007; Komissarov & Barkov, 2007; Obergaulinger & Aloy,
2017), would also necessitate intrinsically aspherical HNe. A key ingredient
in both of these models is a rapidly rotating progenitor, so HNe from these
sources may also provide a natural explanation for larger HN rates in the
past, as rapid rotation in stars is predicted to be more common at low
metallicity (Woosley & Bloom, 2006; Woosley & Heger, 2006; Stacy et al., 2011;
Stacy & Bromm, 2013). Preliminary models for these types of jet-powered
explosions indicate that they may be able to provide ejecta with chemical
abundances favourable to matching the large [(Co,Zn)/Fe] observed in metal-
poor stars, and possess a total explosion energy which is of order
$10^{52}\,\mathrm{erg}$ (Maeda & Nomoto, 2003; Tominaga et al., 2007;
Tominaga, 2009; Nishimura et al., 2017). Moreover, the unique chemical yields
produced by aspherical HNe of the first stars may provide an explanation for
the large fraction of the most metal poor stars with enhanced values of
[C/Fe]. It has been suggested that the most metal poor stars may have been
chemically enriched through an "external enrichment scenario", whereby the
primordial gas in the mini-halo from which they form is polluted by the
chemicals ejected from nearby Population III SNe (Smith et al., 2015; Jeon et
al., 2017). Jet-powered HNe would be prime candidates for the type of
explosions that are likely to launch ejecta with sufficient velocities to
reach nearby mini-halos, and indeed, Ezzeddine et al. (2019) have recently
found indications that the unusual abundance patterns in stars with
$\mathrm{[Fe/H]}\lesssim-4$ may be well fit by aspherical HNe.

Along with the need for more realistic HN models, there are several other
sources of chemical enrichment and factors which could affect the GCE that we
have investigated here. As our intention was to investigate the SN/HN
contribution to GCE with modern nucleosynthetic results, we opted to keep
other contributions to a minimum to avoid obscuring the effect of HN
contribution. However, the following may have important implications in GCE
calculations and are worth further consideration;

  * •

The least massive ($\sim 9\,\mathrm{M}_{\odot}$) stars to undergo core-
collapse are thought to explode as electron-capture SNe (ECSNe) after forming
degenerate O-Ne-Mg cores (Nomoto, 1984, 1987; Poelarends et al., 2008). These
stars may make significant contributions to Galactic abundances of neutron-
rich elements including Zn (Wanajo et al., 2013, 2018; Hirai et al., 2018).
Although we do include CCSN yields down to $10\,\mathrm{M}_{\odot}$, the
unique yields provided by ECSNe may be significant in GCE.

  * •

There are indications that the IMF of the first stars may have been top-heavy,
and that stars might have formed with mass of the order
$100\,\mathrm{M}_{\odot}$ (Bromm et al., 1999; Abel et al., 2002; Susa et al.,
2014; Hirano et al., 2014; Hosokawa et al., 2016). Some fraction of these
stars are believed to explode as energetic pair-instability SNe (PISNe), with
unique nucleosynthetic yields (Heger & Woosley, 2002; Umeda & Nomoto, 2002).
If these stars existed, however, they would have been so short-lived that
their contribution to GCE may be obscured (Komiya, 2011).

  * •

In agreement with previous studies, we have found that the evolution of
Galactic chemical abundances towards solar metallicity depends critically on
the SN Ia rate and chemical yields. There is still no consensus on the
progenitor systems and explosion mechanisms for the thermonuclear events
observed as SNe Ia. Likewise. there is still uncertainty around the
nucleosynthetic end products of SNe Ia. This is an active area of research and
future constraints on SN Ia contribution to GCE will be very important to our
understanding of Galactic chemical abundances (Kobayashi et al., 1998, 2000;
Matteucci & Recchi, 2001; Kobayashi & Nomoto, 2009; Seitenzahl et al., 2013;
Seitenzahl & Townsley, 2017).

  * •

HN chemical yields are sensitive to the particular explosion energy chosen for
each model (Grimmett et al., 2018). We have opted to select HN models that
provide the most favourable fit to the abundances observed in EMP stars,
within reasonable agreement to the theoretical mass-energy relation (Nomoto et
al., 2003). Our aim was to test the limits of the most modern SN/HN models,
although we certainly could have achieved less favourable results with even a
small variation in the explosion energy of our chosen HN models. In finding
the best fit that we can achieve with currently available HN models, we hope
that that this can be built upon in the future with improved approximations
for the explosion mechanism in hypernova modelling.

  * •

Here we only use HN models from metal-free progenitors. The chemical end
products for higher metallicity HNe are likely to diverge from the zero
metallicity models, as we see for our SN models in Figure 1, and in the HN
models of Kobayashi et al. (2006). The changes in SN chemical yield with
metallicity, however, seem to be most significant towards solar [Fe/H], and
our HN rate quickly decreases as a function of metallicity, so the effect is
likely minimal.

##  5 Conclusion

Using a modern and comprehensive set of SN and HN yields, we have found that
we are able to achieve a reasonable fit to the observed Galactic trends in
[(Cr,Mn,Co,Zn)/Fe], with a hypernova rate that is within existing
observational constraints.
Our results indicate that the hypernova contribution to chemical enrichment is
made by HNe with an upper limit to the progenitor mass of
$40\,\mathrm{M}_{\odot}$, and an initial HN occurrence rate of 50%, decreasing
to $\lesssim 1\%$ at present day. This result is indicative of a moderate
contribution from HNe to the chemical enrichment of the Universe, but the
specific constraints may change when aspherical HN models and nucleosynthetic
results become available.
If some SNe fail to explode, as determined by the progenitor compactness
criterion, then some additional source of enrichment will be required to
reproduce the solar value of [(Co,Zn)/Fe].
Complementary to earlier investigations into the role of HNe in Galactic
chemical enrichment, our results demonstrate the crucial contribution that HNe
provide to understanding the observed Fe-peak ratios in the Galaxy.

Our aim was also to determine the areas where there is still significant
discrepancy between GCE modelling and observational data. Our findings
particularly make apparent the need for advancements in our understanding of
(i) [Zn/Fe] enrichment near solar metallicity, and (ii) in [(Co,Zn)/Fe] at the
lowest metallicities. In both cases we consistently achieve values which are
too low to be in agreement with observations. We suggest that (i) may be
improved with modern SN Ia chemical yields (which may also provide the
supersolar [Mn/Fe] observed in stars with high metallicity) and/or the
contribution from ECSNe, and (ii) indicates the need for developments in our
understanding of the HN explosion mechanism and resultant nucleosynethesis,
particularly with regards to the asphericity which seems to be intrinsic to
HNe. Finally, in order to continue to assess our progress in GCE modelling, we
emphasise the need for additional observational data sets, particularly those
which are homegenised, extend to low metallicities, and include the effects of
NLTE where necessary.

## Acknowledgements

We thank Chiaki Kobayashi for her helpful discussions and willingness to
assist us with the development of our GCE code. We also thank the anonymous
referee for their insightful comments and suggestions.
This work was supported by the Australian Research Council through ARC Future
Fellowship FT160100035 (BM) and Future Fellowship FT120100363 (AH). AIK
acknowledges financial support from the Australian Research Council
(DP170100521). AH has been supported, in part, by a grant from Science and
Technology Commission of Shanghai Municipality (Grants No.16DZ2260200) and
National Natural Science Foundation of China (Grants No.11655002). JG
acknowledges financial support from an Australian Government Research Training
Program (RTP) Scholarship. This material is based upon work supported by the
National Science Foundation under Grant No. PHY-1430152 (JINA Center for the
Evolution of the Elements). Parts of this research were supported by the
Australian Research Council Centre of Excellence for All Sky Astrophysics in 3
Dimensions (ASTRO 3D), through project number CE170100013. This research was
undertaken with the assistance of resources from the National Computational
Infrastructure (NCI), which is supported by the Australian Government and was
supported by resources provided by the Pawsey Supercomputing Centre with
funding from the Australian Government and the Government of Western
Australia.

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##  Appendix A Appendices

###  A.1 Reproducing the Results of Kobayashi et al. (2006)

In Figures 11 to 14, we show that we are able to reproduce the results of
Kobayashi et al. (2000, 2006) with our implementation of the chemical
evolution equations, using the chemical yields as provided by Kobayashi et al.
(2006). Small differences between our results may be due to different
treatments of the IMF discretisation, or in the interpolation between
metallicities for SN/HN models.
To reproduce these results we implement an IMF with upper limit
$m_{\mathrm{u}}=50\,\mathrm{M}_{\odot}$ and the solar values from Anders &
Grevesse (1989) as is used by Kobayashi et al. (2006). For the remainder of
our calculations, however, we use an upper mass limit of
$m_{\mathrm{u}}=100\,\mathrm{M}_{\odot}$ and the solar values provided by
Asplund et al. (2009).

Figure 11: Results for the evolution of [(Cr,Mn,Co,Zn)/Fe] from our model
(solid) and the model of Kobayashi et al. (2006) (dashed). Figure 12: Results
for the evolution of [Fe/H] from our model (solid) and the model of Kobayashi
et al. (2006) (dashed). Figure 13: Results for the evolution of the SFR from
our model (solid) and the model of Kobayashi et al. (2006) (dashed). Figure
14: The IMF weighted yields of [(Cr,Mn,Co,Zn)/Fe] from SNe+HNe in our
implementation (solid) and the implementation of Kobayashi et al. (2006)
(dashed).