# The relationship between age, metallicity, and abundances for disk stars in
a simulated Milky Way galaxy

Andreia Carrillo 11affiliation: Department of Astronomy, University of Texas
at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712-1205, USA
22affiliation: Center for Computational Astrophysics, Flatiron Institute,
Simons Foundation, 162 Fifth Avenue, New York, NY 10010, USA 33affiliation:
Institute for Computational Cosmology, Department of Physics, Durham
University, Durham DH1 3LE, UK Melissa K. Ness 22affiliation: Center for
Computational Astrophysics, Flatiron Institute, Simons Foundation, 162 Fifth
Avenue, New York, NY 10010, USA 44affiliation: Department of Astronomy,
Columbia University, Pupin Physics Laboratories, New York, NY 10027, USA Keith
Hawkins 11affiliation: Department of Astronomy, University of Texas at Austin,
2515 Speedway, Stop C1400, Austin, TX 78712-1205, USA Robyn Sanderson
22affiliation: Center for Computational Astrophysics, Flatiron Institute,
Simons Foundation, 162 Fifth Avenue, New York, NY 10010, USA 55affiliation:
Department of Physics and Astronomy, University of Pennsylvania, 209 S 33rd
St., Philadelphia, PA 19104, USA Kaile Wang 11affiliation: Department of
Astronomy, University of Texas at Austin, 2515 Speedway, Stop C1400, Austin,
TX 78712-1205, USA Andrew Wetzel 66affiliation: Department of Physics and
Astronomy, University of California, Davis, CA 95616 Matthew A. Bellardini
66affiliation: Department of Physics and Astronomy, University of California,
Davis, CA 95616<EMAIL_ADDRESS>
###### Abstract

Observations of the Milky Way’s low-$\alpha$ disk show that at fixed
metallicity, [Fe/H], several element abundances, [X/Fe], correlate with age,
with unique slopes and small scatters around the age-[X/Fe] relations. In this
study, we turn to simulations to explore the age-[X/Fe] relations for the
elements C, N, O, Mg, Si, S, and Ca that are traced in a FIRE-2 cosmological
zoom-in simulation of a Milky-Way like galaxy, m12i, and understand what
physical conditions give rise to the observed age-[X/Fe] trends. We first
explore the distributions of mono-age populations in their birth and current
locations, [Fe/H], and [X/Fe], and find evidence for inside-out radial growth
for stars with ages $<$ 7 Gyr. We then examine the age-[X/Fe] relations across
m12i’s disk and find that the direction of the trends agree with observations,
apart from C, O, and Ca, with remarkably small intrinsic scatters,
$\rm\sigma_{int}$ (0.01 $-$ 0.04 dex). This $\rm\sigma_{int}$ measured in the
simulations is also metallicity-dependent, with $\rm\sigma_{int}$ $\approx$
0.025 dex at [Fe/H] $=-0.25$ dex versus $\rm\sigma_{int}$ $\approx$ 0.015 dex
at [Fe/H] $=0$ dex, and a similar metallicity dependence is seen in the GALAH
survey for the elements in common. Additionally, we find that
$\rm\sigma_{int}$ is higher in the inner galaxy, where stars are older and
formed in less chemically-homogeneous environments. The age-[X/Fe] relations
and the small scatter around them indicate that simulations capture similar
chemical enrichment variance as observed in the Milky Way, arising from stars
sharing similar element abundances at a given birth place and time.

— methods: data analysis — methods: statistical — stars: evolution — stars:
fundamental parameters — techniques: spectroscopic

##  1 Introduction

Galaxy formation is a violent, chaotic process. Small galaxies are stripped of
their stars and accreted (Searle & Zinn, 1978) by their larger counterparts,
cold gas is accreted through filaments (e.g., Kereš et al. 2005), and
structures such as spiral arms and bars form and disrupt stellar orbits (e.g.,
Roškar et al. 2008). Yet, in some ways, galaxy formation is also very orderly.
Across populations, we observe that star formation proceeds from the central
regions to the outskirts (e.g., Bezanson et al. 2009; Carrillo et al. 2020) as
exhibited by abundance gradients with radius (Nidever et al., 2014; Hayden et
al., 2015; Kaplan et al., 2016; Belfiore et al., 2017; Weinberg et al., 2018),
and disk scale heights decrease as galaxies evolve (Wisnioski et al., 2015).
The combination of these orderly and disorderly processes gave rise to our
very own Milky Way.

With Galactic archaeology, and specifically through using stars as our rosetta
stone, we can begin to disentangle the detailed formation history of our
Galaxy. This is because stars exhibit atmospheric abundances that generally
reflect the chemistry of the gas from which they formed, barring dredge-up
processes that bring elements made in the core of stars onto the surface
(Iben, 1965). Although a star’s orbital and kinematic properties can change as
it migrates from where it was born, its chemical fingerprint or inventory is
an intrinsic property that remains relatively unchanged throughout its
lifetime. With every successive stellar birth and death, or mass loss from
stellar winds, the gas from which newer stars will be formed is further
enriched. Thereby, stars will have locked within them the chemical
fingerprints inherited from their environment, as expressed in the time and
place of their formation.

In the last decade, we have been able to peer into the detailed chemical
abundances of different stellar populations in the Milky Way with large
spectroscopic surveys. Large multi-object spectroscopic stellar surveys, such
as the Apache Point Observatory Galactic Evolution Experiment (apogee,
Majewski et al. 2017), the Galactic Archaeology with HERMES (galah, De Silva
et al. 2015), the Gaia–eso survey (Gilmore et al., 2012), and the Large Sky
Area Multi-Object Fibre Spectroscopic Telescope (LAMOST, Cui et al. 2012) have
expanded our stellar census of the Galaxy. These surveys have enabled an
extensive characterization of the Milky Way’s two-component disk, historically
called thin and thick disks due to their different spatial properties, and
later on understood in terms of the low-$\alpha$ and high-$\alpha$ disks,
respectively (Bland-Hawthorn et al., 2019). Earlier work (e.g., Chiappini et
al. 1997) have shown that these two populations have different formation
timescales, and with these large Milky Way surveys, it has been further
solidified that the two components are spatially, kinematically, and
chemically distinct, with varying contributions as a function of their
location in the Galaxy (Hayden et al., 2015). Many groups have posited
different origins for this chemical bimodality including clumpy star formation
(Clarke et al., 2019), radial migration (Sharma et al., 2020), or the presence
of a (gas-rich) merger (Grand et al., 2020; Lian et al., 2020; Buck, 2020;
Agertz et al., 2021). Following on from this example, chemical abundances,
especially for a large number of stars, e.g., N $>\rm 10^{5}$, are key to
understanding how the Milky Way built its disk. While chemistry informs which
stars formed together, age ultimately ties these processes to the temporal
evolution of the Galaxy.

There have been many advances in the realm of age derivation for individual
stars within large surveys. This has largely been driven by the availability
of a set of reference objects with precision ages. The Kepler mission, in
particular (Borucki et al., 2010), has provided asteroseismic ages for red
giant branch stars (RGB) (Pinsonneault et al., 2014, 2018). The Gaia satellite
(Gaia Collaboration, 2018) has also measured precision parallaxes, allowing
age derivation from isochrones around the main sequence turn off (MSTO).
Leveraging the mass-dependent, dredge-up-induced [C/N] ratio at the stellar
surface (Masseron & Gilmore, 2015; Martig et al., 2016), data-driven
approaches that use high precision reference ages have enabled ages to be
determined from large samples of spectra (e.g. Ness et al. 2016; Ho et al.
2017).

Ages have provided the means to understand nucleosynthesis over time, and link
the chemical evolution to the assembly history of a galaxy. There is a known
age-[Fe/H] relationship for the Milky Way disk, wherein stars decrease in
[Fe/H] with older ages (e.g., Twarog 1980; Soubiran et al. 2008). However,
whether there is a clear trend is debated in observational studies: previous
works (Haywood et al., 2013; Bergemann et al., 2014; Rebassa-Mansergas et al.,
2016) found that there is substantial scatter in the relation, with Nissen
(2015) finding this lack of correlation existing over a large age interval
(e.g., 8 Gyr). This same study, however, found that there is a tight
correlation between the age and [X/Fe] of stars. It is important to further
investigate these age-individual abundance, or age-[X/Fe], relations at a
given [Fe/H] in the disk because the tightness of these relations illustrate
that (1) by inverting the age-[X/Fe] relation, abundances can serve as
chemical clocks providing ages for stars, and that (2) [Fe/H] and age capture
majority of the crucial information about a star e.g., [X/Fe], orbits, and
birth location (Bedell et al., 2018; Ness et al., 2019; Jofré et al., 2020;
Hayden et al., 2020; Espinoza-Rojas et al., 2021; Casamiquela et al., 2021;
Sharma et al., 2022; Ratcliffe et al., 2022). Additionally, comparisons
between the slope in the [Fe/H]-[X/Fe] relation and the slope in the
age-[X/Fe] relation appear to group elements accordingly into three distinct
nucleosynthetic sites: core-collapse supernova (SNe II), white dwarf explosion
(SNIa), and stellar winds (Sharma et al., 2022). Furthermore, the slope in the
age-[X/Fe] relation is sensitive to the location across the [Mg/Fe] vs [Fe/H]
plane (Yuxi et al., 2021; Horta et al., 2021), but the scatter around these
relations is consistently small. Both the high resolution studies at [Fe/H] =
0 dex from Nissen (2015) and Bedell et al. (2018) as well as the larger
samples exploring a wider range of metallicities (Ness et al., 2019; Sharma et
al., 2022) have shown that the intrinsic scatter of stars around their
age-[X/Fe] relations is on the order of $<$ 0.01-0.04 dex. The small intrinsic
scatter in the age-[X/Fe] relations is also fairly insensitive to the current
location of the stars in the disk (Ness et al., 2019; Yuxi et al., 2021),
which is a natural consequence of radial redistribution or migration, whereby
stars at any given radius, R, are from a wide range of initial radii (Frankel
et al., 2018, 2019).

The growing literature on the age-[X/Fe] relations in observations could
constrain the nucleosynthetic processes that produce the elements in the
universe and the star formation history and interstellar medium conditions
that gave rise to the observed age-[X/Fe] relations and scatters in the Milky
Way. This is because stars not only remember their birth sites through their
detailed chemistry, but for an established disk, these birth sites, at a given
age, can be ordered in radius by sorting in metallicity. Likewise, for a given
metallicity, sorting in age probes systematically different birth radii. The
observed low intrinsic scatter around the age-[X/Fe] trends at fixed [Fe/H]
indeed corroborate this picture, wherein stars are formed with element
abundances governed by their birth radii and time of formation.

It is therefore timely that we understand what gives rise to this relationship
in a self-consistent manner. That is, investigating this relationship where we
know the galaxy formation history and we know the nucleosynthesic processes
that together bring about the resulting chemistry. Here, we utilize
cosmological simulations as a means to examine the age-[X/Fe] relations across
the disk. Within these simulations, we have access to the ages and chemistry
of stars with a direct knowledge of birth properties while unhindered by
observational uncertainties. Critically, recent cosmological zoom-in
simulations (e.g. Wetzel et al. 2016, Grand et al. 2016, Buck et al. 2019,
Wetzel et al. 2022) now implement physically motivated processes (e.g. sub-
grid turbulent metal diffusion, stellar feedback, chemical enrichment from SNe
II, SNIa, and stellar winds) at high resolutions that can distinguish
individual star forming regions.

Galactic archaeology studies with simulated Milky Way-like galaxies have
proven themselves instructive in demonstrating physical processes that can
give rise to observed trends. For example, cosmological simulations show that
gas-rich mergers can give rise to the low-$\alpha$ sequence seen in the Milky
Way disk (Buck, 2020; Agertz et al., 2021). Yu et al. (2021) used the Feedback
in Realistic Environments 2 (FIRE-2) simulations (Hopkins et al., 2018; Wetzel
et al., 2022), to demonstrate that the thick disk formation phase in these
Milky Way-like galaxies coincides with the star formation changing from a
bursty mode to a steady mode. Nikakhtar et al. (2021) similarly used FIRE-2
Milky Way-like galaxies and explored the different families of stars in the
solar neighborhood, determined through a Gaussian mixture modeling of
kinematics and [Fe/H]. They found simulated analogs to the real and observed
stellar population distinctions in the Milky Way solar neighborhood, and
subsequently related these to having different origins. Specifically, they
found a thin disk component that is young, a halo component consistent with
early and massive accretion events, and three thick disk components with
heated orbits due to satellite interactions. Bellardini et al. (2021, 2022)
also looked at [Fe/H] and abundance gradients in FIRE-2 Milky Way-like
galaxies, where they found that the [Fe/H] and abundances change from being
dominated by azimuthal variations to being dominated by radial variations at
lookback times 7–7.5 Gyr ago. This age broadly coincides with an earlier study
by Ma et al. (2017) as the transition from a chaotic, bursty mode of star
formation to a calmer, stable disk in the FIRE-1 galaxy m12i. As illustrated
by these studies, the star formation history, gas inflow and outflow,
nucleosynthetic processes, and environments that these galaxies live in are
known, and therefore aid in understanding the possible origins of observed
stellar population characteristics and interpret chemo-dynamical signatures in
the Milky Way.

Therefore, in this study, we similarly take advantage of a cosmological zoom-
in simulation of a Milky Way-like galaxy to explore and establish the
relationship of the age, metallicity, and individual element abundances of
stars. We use the Ananke Gaia synthetic survey (Sanderson et al., 2020) to
understand the observed tight age-[X/Fe] trend at a fixed [Fe/H].
Specifically, we aim to investigate this trend at different [Fe/H] and
locations in the disk, and to show where the simulations reproduce this
observed relationship, where it breaks down, and how this gap could be
bridged. Ultimately, we aim to be able to put into context the physical
processes in the simulated galaxy that give rise to the age-[X/Fe] trend it
exhibits. In Section 2 we describe the simulation data we use for this work
and the galaxy m12i. In Section 3 we discuss the observational data we use for
comparison. In Section 4 we explore the abundance and location distributions
of stars with similar ages. In Section 5 we show the age-[X/Fe] trends for the
various elements that are tracked in the simulations at different [Fe/H] and
galactocentric radii. In Section 6 we compare the intrinsic scatter in the
age-[X/Fe] trends between observations and simulations. Lastly, in Section 7
we discuss the implications of our results and provide a summary of this work.
We note that for the majority 111In Section 4, we analysed one aspect of birth
location of stars, i.e., their distributions with age, which is not public
data. of the analysis, we used the publicly available information in the
Ananke Gaia synthetic survey.

##  2 Simulation Data

###  2.1 Ananke: Gaia Synthetic Surveys

We take advantage of the Gaia synthetic surveys produced with the Ananke
framework (Sanderson et al., 2020) which generates synthetic phase-space
surveys from baryonic simulations. These synthetic surveys are based on the
Latte suite of zoom-in simulations of Milky Way-like galaxies (Wetzel et al.,
2016) utilizing FIRE-2 physics (Hopkins et al., 2018), that features state-of-
the-art implementations of hydrodynamics, radiative cooling and heating, star
formation, and stellar feedback.

In this simulation suite, stars are formed from self-gravitating molecular gas
that is self-shielding (following Krumholz & Gnedin 2011), dense (n $>$ 1000
$\rm cm^{-3}$), cold (T$<$ 10,000 K), and Jeans-unstable—prescriptions that
produce clustered stellar populations naturally. A star particle is
essentially a single stellar population with an initial mass of 7070 $\rm
M_{\odot}$ but with mass loss from stellar winds, it reduces to a typical mass
of $\sim$5000 $\rm M_{\odot}$ at z = 0.

The stellar evolution of star particles is produced from STARBURST99
(Leitherer et al., 1999) with a Kroupa (2001) initial mass function (IMF).
Chemical enrichment occurs through three main nucleosynthetic sources: SNe II,
SNIa, and OB/asymptotic giant branch (AGB) stellar winds. The yields for SNe
II are from Nomoto et al. (2006), for SNIa from Iwamoto et al. (1999), and
OB/AGB winds from Wiersma et al. (2009) compiling values from van den Hoek &
Groenewegen (1997), Marigo (2001), and Izzard et al. (2004). SNIa rates are
taken from Mannucci et al. (2006) including both prompt and delayed
populations. We note that the yields in the simulations are IMF-averaged
(Hopkins et al., 2018). These stellar and chemical evolution prescriptions
(listed in Appendix A) affect the present-day “observed” abundances in the
synthetic survey which is central to our analysis.

Sub-resolution gas metal diffusion is included, which leads to abundance
distributions that better match observations (Su et al., 2017; Hopkins et al.,
2018; Escala et al., 2018). For a more detailed exploration of how the
diffusion coefficient affects abundance scatters in Milky Way-like galaxies,
see Bellardini et al. (2021), Appendix C.

The simulations track the total mass of each element for each star particle,
but observationally, the abundance measurement is in terms of [X/Fe], where
[X/Fe] = [X/H] - [Fe/H] and the bracket,“[ ]”, notation denotes relative
abundance with respect to Solar abundance values. To put the abundances from
the simulations in the same scale, Asplund et al. (2009) Solar values were
used to calculate

$\rm[X/H]=\frac{m_{X}/m_{X,\odot}}{m_{H}/m_{H,\odot}},$ (1)

where X is the element, $\rm m_{X}$ is the mass of the element in the star
particle, and $\rm m_{X,\odot}$ is the Solar value for that element.

Three galaxies were taken from Latte, labeled m12f, m12i, and m12m, for the
generation of the Gaia synthetic surveys viewed from three different “Solar”
view points per galaxy (resulting in nine synthetic surveys in total), where a
“Solar” view point (Local Standard of Rest, LSR) is defined as 8.2 kpc from
the galactic center. These Milky Way-like galaxies were selected based only on
their total mass ($\rm M_{200m}$222$\rm M_{200m}$ is a proxy for virial mass
in simulations (white01) where $\rm 200m$ is 200 $\times$ the matter density =
1-2 $\rm\times 10^{12}M_{\odot}$) and environment (isolated, i.e., has no
nearby dark matter halo with similar mass up to 5 $\rm\times
R_{200}$333$R_{200}$ is a proxy for virial radius (white01)) and therefore
have a variety of morphologies, satellite populations, and star formation
histories. Each catalog has self-consistent dust extinction implemented, based
on scaling of the gas density of the simulated galaxies. For a proof of
concept in exploring age-[X/Fe] trends in simulations, and for simplicity, we
focus our analysis on one viewpoint (LSR 2) of the galaxy m12i which is also
publicly available
444https://girder.hub.yt/##collection/5b0427b2e9914800018237da. The Galactic
map of m12i is shown in Figure 1.

For the synthetic surveys, the single stellar populations are broken into
individual synthetic stars that are sampled similarly from a Kroupa (2001) IMF
and mapped to a grid of isochrones in age and [Fe/H] over a set of model
isochrones (PARSEC; Bressan et al. 2012,Marigo et al. 2013,Marigo et al.
2017). We note that the isochrones used for generating the synthetic surveys
are different from the isochrones used for the stellar evolution in the
simulations. Specifically, they differ in their treatment of high-mass star
evolution where the simulations use the Geneva tracks (Lejeune et al., 1997)
that have enhanced mass loss from rotation while PARSEC includes thermally-
pulsating AGB stars. However, Sanderson et al. (2020) prescribed mass-
independent mass-loss rate in sampling the number of stars from a star
particle, alleviating some of the differences introduced from the mismatch in
isochrones.

These individual stars then inherit the age, [Fe/H], and [X/H] of the parent
star particle, while their 6D phase-space information is sampled over a one-
dimensional kernel in each position and velocity axis centered on the parent
star particle. We use the true values for the “observed” properties (e.g.
distance, age, and chemistry) of stars in the synthetic survey because the
essence of our analysis is to understand the intrinsic scatter in the
age-[X/Fe] trend. However, we selected the stars from the synthetic survey to
mimic the constraints in observations which we ultimately draw comparisons to.

Figure 1: A picture of m12i. Aitoff projection of the star counts in m12i
viewed from LSR2. The galaxy shows structure similar to the Milky Way i.e. a
thin disk, a more diffuse thick disk, a central bulge-like concentration of
stars, and patchiness due to dust extinction.

###  2.2 Selection of stars for analysis

Figure 2: R and Z distributions at different [Fe/H] and ages. We select three
different samples of stars: one at solar metallicity ([Fe/H] = 0 dex, top),
and two at low metallicities ([Fe/H] = -0.25 dex, middle and [Fe/H] = -0.75
dex, bottom) and plot their current R (left) and Z (right) distributions. For
each [Fe/H] bin that has a size of $\pm$0.025 dex, we mark the distribution of
stars at different ages i.e. $<$1 Gyr (light blue), 1-3 Gyr (dark blue), 3-5
Gyr (green), 5-7 Gyr (olive), 7-10 Gyr (gold), and $>$10 Gyr (brick red). We
also mark the median R and Z at these metallicities (dashed purple line). The
R distributions move from the inner to the outer regions while the Z
distributions change from broad to narrow as we go from older to younger
stars.

In this work, we aim to compare the age-[X/Fe] relations for the thin disk
stars in the simulated Milky Way to those measured in the observed Milky Way
low-$\alpha$ disk (e.g. Bedell et al. 2018, Ness et al. 2019), that have in-
situ origins and have been the subject of most age-[X/Fe] studies. With this
in mind, we have constrained our analysis of stars in the simulations to those
that (1) have R$<$25 kpc, (2) [Fe/H] $\geq$ -0.25 dex, and (3) $\mid
Z\mid<500$ pc. We do this selection to obtain predominantly in-situ stars in
the current day spatial thin disk, though we note that there is no explicit
removal of ex-situ stars. We explain these criteria in detail below. We note
that for ease of processing, we utilize the subsampled independent survey that
is $1/100$ the size of the full synthetic survey for m12i at LSR2.

The first criterion (1) restricts to stars within 25 kpc of the galactic
center. This is motivated by Figure 3 of Sanderson et al. (2018), which shows
the current locations of stars in the FIRE simulated galaxies colored by their
formation locations. For m12i, these authors found that at R$\sim$20 kpc,
there is a sharp transition between stars that formed in-situ and stars that
formed at greater distances, i.e. R $>$30 kpc. Azimuthally, this transition
radius hovers around the R = 20 kpc value, though sometimes exceeds this
range. We therefore opted to use R = 25 kpc as our radius cut to take into
account the upper bound of this transition region. Interestingly, Sanderson et
al. (2018) also found that this transition region between mostly in-situ vs
mostly accreted stars is also traced by a transition in [Fe/H], which is a
promising diagnostic for observational studies. This is also similar to the
selection made in Bellardini et al. (2022) in defining the in-situ component
of Milky Way-like galaxies in FIRE-2.

As a sanity check for this selection, we investigate how the spatial
distributions, specifically in current (cylindrical) Radius, R, and current
height from the disk plane, Z, of our resulting sample change as a function of
age and metallicity. This is also motivated from observations (Ness et al.,
2019; Yuxi et al., 2021) that the tight relationship between [X/Fe] and age at
fixed [Fe/H] is due to stars reflecting the chemistry of their location;
therefore, we expect these distributions to be distinct at different ages and
[Fe/H]. In Figure 2, we show the R distributions on the left and Z
distributions on the right for stars at different [Fe/H]: solar at 0 dex
(top), -0.25 dex (middle), and -0.75 dex (bottom), with a bin size of [Fe/H]
$\pm$0.025 dex, and within R $<$ 25 kpc of the galactic center. We have
explored stellar populations in other [Fe/H] bins (from [Fe/H] = +0.25 dex to
-2.5 dex) as well, but focus on these three, which are sufficient to
succinctly portray the differences in the R and Z distributions at high and
low [Fe/H]. Each R and Z distribution is broken down into the separate
contributions from different stellar ages, i.e., $<$1 Gyr (light blue), 1-3
Gyr (dark blue), 3-5 Gyr (green), 5-7 Gyr (olive), 7-10 Gyr (gold), and $>$10
Gyr (brick red). We also mark the median R and Z values with the purple dashed
line. Though our first criterion makes a cut at R$<$25 kpc to select the
predominantly in-situ population, not many stars lie in that region. In order
to better compare the R trends at different ages, we therefore zoom in and
only show the distributions up to R = 20 kpc.

At solar metallicity, the R distribution peaks at R = 8 kpc, slightly
coinciding with the median R marked by the purple dashed line at R = 7.5 kpc.
Meanwhile the R distributions for the lower metallicity samples (i.e. [Fe/H] =
-0.25 dex and -0.75 dex) peak close to, but not quite at the center of the
galaxy at R = 2 kpc, in line with the results from El-Badry et al. (2018) for
the oldest stars in the FIRE-2 simulations. The median R for both low
metallicity bins is higher than the peaks in their distributions and lies at R
= 4 kpc, as both distributions have long tails towards larger R. For the Z
distributions, the trend at solar metallicity resembles a normal distribution
with mean at Z = 0. At lower metallicities, however, the distributions deviate
from a Gaussian and exhibit two peaks, below and above the plane of the disk.
The dispersion in Z also increases from high to low [Fe/H], as indicated on
the upper left corner of each subplot in Figure 2, from $\sigma_{Z}$ = 1.10
kpc at [Fe/H] = 0, to $\sigma_{Z}$ = 2.05 kpc at [Fe/H] = $-0.25$ dex, and to
$\sigma_{Z}$ = 3.27 kpc at [Fe/H] = $-0.75$ dex. This trend of increasing Z
dispersion with increasing age and decreasing metallicity exists in the Milky
Way as well (Meusinger et al., 1991), and could be explained by stars being
born in a thicker disk at early times (Wisnioski et al., 2015; Ma et al.,
2017; Bird et al., 2021; Yu et al., 2021), or by being heated kinematically
through interactions with giant molecular clouds, spiral arms, and bar (Aumer
et al., 2016) and with satellites (Hopkins et al., 2008; Villalobos & Helmi,
2008, 2009; Sales et al., 2009; Laporte et al., 2019). On the other hand, the
double-peak in the Z distribution and the asymmetry about Z = 0 is due to the
fact that it is harder to see stars through the disk because of the dust and
that older stars have larger scale heights.

In general, we find that the solar, [Fe/H] = 0 dex bin is dominated by stars
with ages $<$5 Gyr, at [Fe/H] = -0.25 dex by stars that are $5-7$ Gyr old, and
at [Fe/H] = -0.75 dex by stars that are $>$7 Gyr old. As expected, younger
stars dominate at high [Fe/H] and older stars dominate at lower [Fe/H]. These
stellar populations’ respective R distributions therefore also dominate the
total R distribution for each [Fe/H] bin. For example, although the [Fe/H] =
-0.25 dex bin is comprised of stars of different ages, the dominant stellar
age is 5-7 Gyr (olive green line); therefore, the total R distribution (gray),
across all ages, consequently follows the R distribution of the $5-7$ Gyr
stellar population. We note, however, that within the same [Fe/H] bin, these R
(and Z) distributions at different ages highly overlap.

At [Fe/H] = 0, the R distribution peak at higher R values for younger stars
than older stars. This is less evident for the lower [Fe/H] bins, where there
are fewer younger stars. As seen for the [Fe/H] = $-0.25$ dex sample, the peak
of the $5-7$ Gyr distribution has R value very close to that of the $7-10$ Gyr
distribution. At the lowest [Fe/H] bin, the R where the distributions peak for
stars with ages 5-7 Gyr, 7-10 Gyr, and $>$10 Gyr all coincide at $\sim$2 kpc,
showing no sequence in R with the dominant ages. Therefore, we primarily
observe evidence of an inside-out radial formation for younger and higher
metallicity stars, i.e. ages $<$ 7 Gyr and [Fe/H] $>$ -0.25 dex. That is, only
with these cuts do we isolate an inside-out forming disk with a clear current
day age gradient at a fixed [Fe/H]. This age is broadly consistent with the
transition age found by Yu et al. (2021) when FIRE-2 galaxies change from a
bursty to smooth star formation mode. This is also the transition age that
Bellardini et al. (2022) found for their sample of FIRE-2 Milky Way-like
galaxies, when the stellar abundances changed from being dominated by
azimuthal variations to radial variations. All Z distributions show increasing
dispersion with increasing age at a given [Fe/H]. For the [Fe/H] = 0 dex
sample, the Z distributions for stars with ages $<$1 Gyr and 1-3 Gyr are
Gaussian, but the distributions for older stars deviate from this behavior.
This is more obvious at [Fe/H] = -0.25 dex and [Fe/H] = $-0.75$ dex, which are
dominated by older stars that show this double peak more, as the effect of not
being able to “see” through the disk is more significantly felt. These R and Z
trends that we find in the simulations are in congruence with the inside-out
and upside-down scenario for the observed Milky Way stellar age-velocity
relation as shown by Bird et al. (2021), and is further support for the Milky
Way-like nature of this simulation and its earlier version in FIRE-1 (e.g., Ma
et al. 2017).

With this exploration and these trends in mind, we therefore apply a second
criterion in investigating the age-[X/Fe] relations (see Section 5) and (2)
only select stars with [Fe/H] $>$ -0.25 dex to ensure that we are probing the
regime in metallicity in which chemical evolution has been temporally smooth,
and age and [Fe/H] are a link to a star’s location.

Lastly, (3) we spatially constrain our analysis to the simulation’s thin disk,
which is the focus of most age-[X/Fe] trend studies in observations (e.g.
Nissen 2015, Bedell et al. 2018, Ness et al. 2019). We selected stars with Z
within $\pm$500 pc of the disk plane motivated by the measured scale height of
the thin disk for m12i (Sanderson et al., 2020) and by the Z distributions of
the radially-distinct mono-age, mono-abundance populations shown in Figure 2.

As a caveat, we note that this comparison is not entirely consistent because
in observational studies, the thin disk selection is usually done in the
[$\alpha$/Fe] vs [Fe/H] plane, while in most simulations and in this study,
the criteria for the thin disk is in approximation a spatial selection.
However, for the purpose of our analysis, this means we compare the stars in
the disk with small vertical oscillations around the plane, in both simulation
and data.

##  3 Observational Data

Figure 3: [$\alpha$/Fe] vs [Fe/H] from galah DR3. We focus our comparison to
the low-$\alpha$ disk, shown as the magma density plot. [Fe/H] = 0 dex is
marked with a solid vertical line and [Fe/H] = -0.25 dex is marked with a
dashed vertical line. We investigate these [Fe/H] regimes for the discussion
in Section 6

We briefly introduce the three sets of observational data that we compare to
regarding the [X/Fe] distributions, age-[X/Fe] relations, and the intrinsic
scatter, $\rm\sigma_{int}$, around these trends in the simulations.

First, we use the galah survey data release 3 (DR3, Buder et al. 2021). This
has an associated value added catalog from Sharma et al. (2022) that contains
ages from the BSTEP code (Sharma et al., 2018) which makes use of stellar
isochrones. In total, galah DR3 has abundances for 30 elements in 588,571
nearby stars in the Galactic disk. We derived a sample from this data set
consisting of 102,785 stars having applied the following quality cuts: signal-
to-noise ratio $>$40, flag_sp=0, flag_fe_h=0, and $4000<\mbox{$T_{\rm
eff}$}<7000$ K. We further apply a cut in [$\alpha$/Fe] vs [Fe/H] shown in
Figure 3 and as similarly done in previous observational studies (e.g., Ness
et al. 2019; Bland-Hawthorn et al. 2019), to focus on just the low-$\alpha$
disk, leaving us with 81,230 stars. We have the elements C, O, Mg, Si, and Ca
in common with the galah data and compare their [X/Fe] distributions for mono-
age populations with those from the simulations (see Section 4). The study by
Sharma et al. (2022) also provide comparison with age-[X/Fe] relations and the
$\rm\sigma_{int}$ around them at varying [Fe/H].

Second, we compare to the age-[X/Fe] relations from the set of 79 solar twin
stars from Bedell et al. (2018), where the ages were derived from the Yonsei-
Yale isochrones (Demarque et al., 2004). This sample of solar twin stars
generally have ages $<$8 Gyr. The authors report measurements for 30 element
abundances, where they have C, O, Mg, Si, S, and Ca in common with this study.
With regards to the [X/Fe]-age trends, the fits were derived through
minimizing the orthogonal distance between the data and the model, weighted by
the observational uncertainties (Hogg et al., 2010).

Lastly, we use the measurements from the set of $\approx$ 15,000 red clump
field stars from Ness et al. (2019) in the low-$\alpha$ disk, where the
authors utilize the apogee survey’s 14th data release (Majewski et al., 2017).
This work looked into the age-[X/Fe] relations in 19 different elements,
wherein C, N, O, Mg, Si, S, and Ca are similarly measured as in the
simulations. The ages were derived through data-driven modeling with The
Cannon (Ness et al., 2015) using the APOKASC catalogue (Pinsonneault et al.,
2018) as a training set which contains ages derived from asteroseismology.
This sample of giant stars have ages $<$10 Gyr. In deriving the intrinsic
scatter in the [X/Fe]-age trends in their study, the effect of the age
uncertainties were taken into account via Monte Carlo sampling. The error in
age can shift stars horizontally in the age-[X/Fe] trend. For flat trends,
accounting for the age error has minimal effect but for steep trends, this
essentially flattens the relationship. This comparison observational work
found that the effect of the age uncertainties changes the intrinsic scatter
by $<$0.01 dex and was therefore negligible. Likewise, the abundance error was
accounted for in deriving the $\rm\sigma_{int}$ as the measured scatter from
the best fit line will be a combination of the $\rm\sigma_{int}$ and the error
in the abundance measurement, $\rm\sigma_{abund}^{2}$. That is:
$\rm\sigma_{total}^{2}=\sigma_{abund}^{2}+\sigma_{int}^{2}$.

##  4 The spatial distribution of stellar chemical abundances at birth

Figure 4: Birth radius distributions for stars of different ages. Kernel
density estimations (KDE) for $\rm R_{birth}$ of stars within age bins
centered at 1 Gyr (purple) to 9 Gyr (yellow) in steps of 2 Gyr, prior to the
[Fe/H] cut. The KDEs are normalized to the number of stars per age group.
Stars of different ages trace distributions that, though highly overlapping,
show distinct birth annuli. However, the 9 Gyr stellar population does not
follow this trend due to a more chaotic mode of star formation at this point
in the m12i’s history.

A starting point to understanding the present day distributions of abundances
across the disk (see Section 5) is to look at the birth radius distributions
for different age population. In Figure 4, we show the kernel density
estimation (KDE) for the birth radius, $\rm R_{birth}$, of stars centered on
different age bins from ages = 1 to 9 Gyr. Because the number of stars within
each age bin is different, we normalize the KDEs such that the distributions
are on the same scale for easier comparison. We used the smallest possible bin
in age such that the measured dispersion across the [Fe/H] vs $\rm R_{birth}$
trend is not induced by the age binning, giving age bin sizes of (0.063,
0.125, 0.063, 0.125, 0.250) Gyr for the (1, 3, 5, 7, 9) Gyr samples,
respectively. For more details on this process, we refer to Appendix B. We
applied this same age binning in comparing the KDEs at different ages for
[Fe/H] and [X/Fe], where X=C, N, O, Mg, Si, S, Ca shown in Figure 5.

The $\rm R_{birth}$ distributions peak from the outskirts of the galaxy to the
central regions, from the youngest to the oldest stars, respectively. However,
this is not observed for the 9 Gyr stellar population, which is more spread
out across the disk. This is because at higher redshifts (i.e. lookback time
$>$ 8 Gyr), the star formation in m12i was violent and bursty until a final
minor merger finishes at lookback times of $\sim$6 Gyr, after which a more
stable and calm disk appears and persists (Ma et al., 2017).

Next, we focus on the distributions of element abundances as a function of age
as shown in Figure 5.

Metallicity: The [Fe/H] KDEs follow an increasing trend with decreasing age,
though the 1 Gyr and 3 Gyr age bins have similar [Fe/H] distributions, and the
width of the distribution also varies with different age bins. The similarity
in the [Fe/H] distributions for the younger stars is in line with the little
evolution in [Fe/H] and the similar radial abundance gradients at these ages
found by Bellardini et al. (2022) in their sample of Milky Way-like galaxies
in FIRE-2. Qualitatively, the [Fe/H] distribution widths seem to be related to
the $\rm R_{birth}$ distribution widths shown in Figure 4, although we caution
that this could be due to our age-binning, which was derived from the [Fe/H]
vs $\rm R_{birth}$ trend at different ages. For the 9 Gyr bin, there is star
formation everywhere in the galaxy, which induces a wide distribution in
[Fe/H]. There is also a large tail towards lower metallicities reflecting the
chemical enrichment at earlier times. Additionally, the large tail is affected
by the merger with another smaller system, seen as a slight bump in the $\rm
R_{birth}$ distribution at 20 kpc in Figure 4. For the 7 Gyr and 5 Gyr age
bins, their [Fe/H] distributions are narrower, following similarly narrow and
more-peaked distributions in $\rm R_{birth}$. Lastly, for the 3 Gyr and 1 Gyr
stellar populations, the [Fe/H] distributions are similar, peak above solar
values, and have broader distributions, reflecting the star formation that is
happening in a wider range of $\rm R_{birth}$ across the disk. In
observations, there is a known age-metallicity relationship in Milky Way disk
stars (Soubiran et al., 2008), that shows decreasing [Fe/H] with older ages.
On the other hand, many studies (Bergemann et al., 2014; Rebassa-Mansergas et
al., 2016) have also found that there is no clear trend in this relationship.
In a sense, the [Fe/H] distributions in this work agree with both scenarios,
with the important distinction being the age at which this holds true. In this
work, we find that the 1 and 3 Gyr age bins show little change in their
respective [Fe/H] distributions, while increasing in age from the 5 Gyr bin
shows progressively more metal-poor distributions. Bellardini et al. (2022)
shows this more clearly for their sample of Milky Way-like galaxies in FIRE-2
in looking at the age-metallicity relationship in the simulations (see their
Figure 1). The [Fe/H] shows no trend with age for the youngest stars up to
$\sim$4 Gyr, after which there is a slight decrease in [Fe/H] with increasing
age up to 7 Gyr, and a more drastic decrease in [Fe/H] at much older ages.

Figure 5: [X/Fe] distributions for stars at different ages. KDEs for the
[X/Fe] tracked in this study with legends similar to Figure 4 i.e., purple to
yellow go from 1 Gyr to 9 Gyr stellar populations. The KDEs are similarly
normalized to the number of stars per age group. All elements show distinct
[X/Fe] distributions for different ages, except for O, where the different
distributions show substantial overlap.

Looking at the trends in [X/Fe] distributions with age in Figure 5, we find
some elements follow expectations from observations while others diverge. We
discuss these [X/Fe] distribution trends in order of atomic mass, from the
light elements to the $\alpha$-elements. We also list the summary statistics
of abundance distributions in Table 1 and where available, we included summary
statistics from observations of the Milky Way low-$\alpha$ disk taken from
galah DR3 written in brackets. We note however that the age bins used for the
observations are much bigger (e.g., $\pm$0.8 Gyr) to take into account age
uncertainties. Due to a relatively smaller sample, we also did not include a 1
Gyr bin from the observations.

Carbon: The carbon abundance, [C/Fe], increases in peak [C/Fe] value with
younger populations in the simulation. In this work, we have shown that stars
in different age bins have roughly different [Fe/H] distributions in the -1
$<$ [Fe/H] $<$ 0 dex range; with this relation, we find that the [C/Fe]
distributions in the simulations are contrary to what is found in
observations, wherein [C/Fe] decreases with increasing metallicity at
[Fe/H]$>$-1 dex (e.g. Bensby & Feltzing 2006). This is also supported by our
comparison to the galah data as listed in Table 1, where the mean [C/Fe]
increases with older ages in the observations but decreases in the
simulations. C is mostly produced in massive stars, followed by low-mass AGB
stars (Kobayashi et al., 2020) in the Galaxy. In the simulations, there is a
larger contribution of C coming from stellar winds, which is one possible
source of discrepancy. In observations, the atmospheric abundance of C is also
affected by dredge-up in convective stars. Throughout the life of a star, its
C+N+O abundances remain relatively constant. However, the slowest reaction in
the CNO cycle produces a net increase in $\rm{}^{14}N$, decrease in
$\rm{}^{12}C$, and slight decrease in $\rm{}^{16}O$ (Masseron & Gilmore, 2015;
Martig et al., 2016). This is especially seen with RGB stars that have deep
convective zones which take materials formed in the core onto the surface of
the star, changing the atmospheric abundance. However, in the simulations, we
have yields (not necessarily atmospheric abundance) which are IMF-averaged and
all stars generated from a star particle inherit the same chemistry.
Therefore, although this process is prescribed on the population level in the
simulations (i.e., from van den Hoek & Groenewegen 1997, Marigo 2001, Izzard
et al. 2004), the abundances for individual stars are more nuanced than can be
achieved by this procedure.

Nitrogen: The nitrogen abundance, [N/Fe] distribution distinctly changes with
the stellar population age, from broader with lower [N/Fe] abundances at older
ages and narrower with higher [N/Fe] abundances at younger ages. The [N/Fe]
distributions at different ages also overlap the least among the elements
considered in this work, which is a result of its yield’s metallicity-
dependence from SNe II as implemented in the simulations (Hopkins et al.,
2018). In FIRE-2, only the elements N and O have yields from progenitors with
metallicity-dependence, which is linear with progenitor [Fe/H] up to
$Z/Z_{\odot}$ = 1.65. N is also intricately linked to C through the CNO cycle.
However, compared to C, N is mainly produced in intermediate-mass AGB stars
(Kobayashi et al., 2011). Like C, the effects of dredge-up on the N abundance
is applied in the simulation yields, although IMF-averaged. Works using large
spectroscopic survey data like apogee (e.g. Hayes et al. 2018) find a decrease
in [N/Fe] with increasing [Fe/H] up to [Fe/H]$\sim$-0.5 dex, after which
[N/Fe] increases with increasing [Fe/H], similar to what we find in the
simulations.

Oxygen: The oxygen abundance, [O/Fe], in Ananke shows the smallest evolution
across time among the elements in this study i.e., the [O/Fe] distributions
show the most insignificant differences for stars of different ages. The
[O/Fe] distributions are wide at older ages and become narrow at younger ages,
but, with each distribution overlapping substantially. In the simulations,
most of the O is produced via SNe II which is expected for this hydrostatic
$\alpha$-element. However, in FIRE-2, there is also non-negligible
contribution from stellar winds with a dependence on the progenitor
metallicity. The stellar winds contribute a comparable amount of O to the ISM
as SNe II at 3 $\times$ solar [Fe/H]. This trend in the simulations is
contrary to what is found in observations, where [O/Fe] is seen to decrease
with increasing [Fe/H] (e.g. Bensby et al. 2004), and loosely, age, as is
expected of an $\alpha$-element. This increase in abundance with older age is
indeed what we also find for the [O/Fe] distribution trends in galah. The
significant O production from stellar winds at higher metallicities is a
possible source of discrepancy.

$\alpha$-elements (Magnesium, Silicon, Sulfur, Calcium): Lastly, we consider
the other $\alpha$-element abundance distributions i.e. Mg, Si, S, and Ca.
Although these elements have distributions with different mean values, the
characteristics of the distributions for the different age bins are similar:
broad at older ages with higher abundances, evolving to narrow peak at younger
ages with lower abundances. This behavior we see for the mean [X/Fe] is
expected for $\alpha$-elements, which are mainly produced through SNe II with
shorter timescales, and diluted via SNIa that have delayed and longer
timescales, therefore decreasing the $\alpha$-abundance with younger
generations of stars. This is also what we observe in the galah data, aside
from Ca, which is intriguingly constant with age in observations (see also
discussion in Section 5), but increasing with older ages in the simulations
like the other $\alpha$-elements. Mg, like O, is a hydrostatic
$\alpha$-element produced during the shell burning phase in massive stars. Si,
S, and Ca are explosive $\alpha$-elements that also have non-negligible
contributions from SNIa. For the explosive $\alpha$-elements, the peaks in the
[X/Fe] distributions between the oldest and youngest stellar populations are
separated by only $<$0.1 dex in the simulations. On the other hand, the same
mono-age populations are more widely separated for Mg that has negligible
contributions from SNIa. In the simulations, Mg and Si are produced in similar
amounts from SNe II, with S yields being 1/3rd and Ca yields being 1/25th of
the Mg yields. Among the $\alpha$-elements, SNIa contributions are highest for
Si and S, and lower by a factor of 13 and 20 for Ca and Mg, respectively. We
also note a tail in the distributions of these elements towards lower [X/Fe],
at all age bins, and most pronounced at 1 Gyr.

Table 1: Summary Statistics of [X/Fe] Distributions. The mean ($\mu$) and standard deviation ($\sigma$) of each abundance distribution have been computed with standard numpy functions, while the skew and kurtosis have been calculated with scipy.stats. The $\mu$ and $\sigma$ have units of dex while the skew and kurtosis are unitless. The skewness is a measure of symmetry while kurtosis, though qualitatively a measure of how peaked a distribution is, is sensitive to outliers. Both skew and kurtosis are therefore affected by the tails in the distribution. For the skew, a value of zero is perfectly symmetric and resembles a Gaussian, negative values indicate more weight on the left tail of the distribution, and larger positive values indicate more weight on the right tail. For kurtosis, larger values indicate larger tails in the distribution or a narrower distribution, which is what we see in the simulations. We refrain from making conclusions based on these skew and kurtosis values, and mainly point out that the large values for the kurtosis and deviation from 0 for the skew, indicate the heavy tails in the [Fe/H] and [X/Fe] distributions inherited from the range of stellar $\rm R_{birth}$ at a given age. Where available, we included the values from observations from galah DR3, enclosed in brackets.  | 1 Gyr | 3 Gyr | 5 Gyr | 7 Gyr | 9 Gyr
---|---|---|---|---|---
[Fe/H], mean ($\mu$, dex) | 0.07 | 0.04 [-0.08] | -0.12 [-0.10] | -0.32 [-0.14] | -0.58 [-0.21]
scatter ($\sigma$, dex) | 0.17 | 0.15 [0.19] | 0.14 [0.23] | 0.17 [0.26] | 0.24 [0.31]
skewness ($S_{k}$) | -1.55 | -1.44 [-0.48] | 0.45 [-1.03] | 0.37 [-1.64] | -0.76 [-2.79]
kurtosis ($\kappa$) | 6.62 | 4.76 [2.94] | 5.69 [5.43] | 5.05 [8.56] | 3.58 [13.46]
[C/Fe], $\mu$, dex | 0.37 | 0.36 [0.02] | 0.32 [0.02] | 0.30 [0.04] | 0.25 [0.08]
$\sigma$, dex | 0.03 | 0.04 [0.12] | 0.05 [0.12] | 0.06 [0.13] | 0.06 [0.14]
$S_{k}$ | -2.12 | -2.23 [0.66] | -3.46 [0.59] | -2.31 [0.62] | -2.02 [0.91]
$\kappa$ | 18.4 | 25.08 [2.41] | 31.75 [3.22] | 27.50 [1.79] | 24.32 [2.43]
[N/Fe], $\mu$, dex | 0.35 | 0.32 | 0.22 | 0.11 | -0.04
$\sigma$, dex | 0.04 | 0.05 | 0.05 | 0.07 | 0.09
$S_{k}$ | -3.73 | -2.25 | -2.84 | -1.82 | -1.55
$\kappa$ | 35.90 | 22.62 | 26.69 | 18.37 | 14.19
[O/Fe], $\mu$, dex | 0.47 | 0.47 [0.05] | 0.47 [0.04] | 0.46 [0.09] | 0.47 [0.16]
$\sigma$, dex | 0.03 | 0.03 [0.17] | 0.04 [0.16] | 0.05 [0.17] | 0.06 [0.19]
$S_{k}$ | -7.33 | -7.55 [0.35] | -6.02 [0.25] | -5.36 [0.62] | -5.56 [0.89]
$\kappa$ | 96.98 | 137.38 [3.95] | 74.92 [6.68] | 62.39 [5.57] | 71.02 [5.25]
[Mg/Fe], $\mu$, dex | 0.22 | 0.25 [-0.04] | 0.27 [0.00] | 0.29 [0.03] | 0.31 [0.05]
$\sigma$, dex | 0.04 | 0.03 [0.08] | 0.04 [0.07] | 0.05 [0.08] | 0.05 [0.08]
$S_{k}$ | -3.64 | -4.91 [-0.14] | -5.90 [0.68] | -5.48 [0.68] | -5.78 [0.82]
$\kappa$ | 36.48 | 86.24 [4.21] | 73.79 [3.27] | 64.86 [5.73] | 75.11 [ 7.58]
[Si/Fe], $\mu$, dex | 0.26 | 0.28 [0.01] | 0.30 [0.02] | 0.31 [0.04] | 0.32 [0.07]
$\sigma$, dex | 0.03 | 0.02 [0.06] | 0.03 [0.06] | 0.04 [0.07] | 0.04 [0.09]
$S_{k}$ | -2.88 | -4.06 [1.32] | -5.26 [0.70] | -4.44 [1.50] | -4.38 [2.08]
$\kappa$ | 24.36 | 61.09 [36.74] | 60.45 [18.57] | 43.25 [16.77] | 42.38 [14.49]
[S/Fe], $\mu$, dex | 0.21 | 0.23 | 0.25 | 0.25 | 0.27
$\sigma$, dex | 0.02 | 0.02 | 0.03 | 0.04 | 0.04
$S_{k}$ | -2.66 | -3.80 | -5.05 | -4.16 | -4.05
$\kappa$ | 21.55 | 55.67 | 56.57 | 38.49 | 36.47
[Ca/Fe], $\mu$, dex | -0.06 | -0.05 [0.07] | -0.03 [0.06] | -0.02 [0.05] | -0.01 [0.06]
$\sigma$, dex | 0.02 | 0.02 [0.09] | 0.03 [0.08] | 0.03 [0.08] | 0.03 [0.09]
$S_{k}$ | -2.47 | -3.53 [-0.19] | -4.81 [0.22] | -3.92 [-0.03] | -3.78 [-0.24]
$\kappa$ | 19.20 | 50.59 [4.99] | 52.24 [4.22] | 34.74 [5.26] | 32.04 [19.88]

In this section, we found that mono-age populations in the thin disk of m12i
within $\rm R<25kpc$ trace birth locations (Figure 4) that overlap quite
broadly but peak at different $\rm R_{birth}$ and exhibit [Fe/H] and [X/Fe]
distributions that also show distinct trends (Figure 5). Interestingly, the
[Fe/H] trends mirror $\rm R_{birth}$ trends, where a wider $\rm R_{birth}$
distribution also corresponds to a wider [Fe/H] distribution; together, age
and [Fe/H] trace $\rm R_{birth}$ in the disk of m12i.

Irrespective of the [Fe/H] distributions, however, the width of the [X/Fe]
distributions increases at older stellar ages. The oldest stars (ages $>$9
Gyr) were born from less homogeneous gas, when star formation was happening
more chaotically in the galaxy as seen in previous works (e.g. Ma et al. 2017,
Agertz et al. 2021, Bellardini et al. 2021). This is further supported by the
larger azimuthal abundance scatter at higher lookback times for similar Milky
Way-like galaxies in FIRE-2 (Bellardini et al., 2022). This subsequently
produces wider abundance distributions. The increasing kurtosis in [X/Fe] with
younger stellar ages, best seen in the elements N and O, indicates that the
chemical enrichment in the disk becomes more homogeneous over time. At the
same time, the skew in the [X/Fe] distributions becomes more pronounced at
younger ages, which could be explained by where the stars were born. Focusing
on the 1 Gyr population, part of the sample was born at smaller radii, from
gas that has had further dilution from SNIa (i.e., greater overlap with older
populations as seen in Figure 4), therefore skewing the [X/Fe] distribution
more negatively. However, these populations can be disentangled with [Fe/H],
which would tell us which stars were born more centrally (i.e. higher [Fe/H])
or in the outskirts (i.e. lower [Fe/H]). Therefore, age is indeed intricately
linked to the [Fe/H], [X/Fe], and $\rm R_{birth}$ of a star.

In this section, we also showed that the trends in the [X/Fe] distributions
with age for the elements N, Mg, Si, and S in the simulations agree with
expectations from observations, while those for the elements of C, O, and Ca
do not. These similarities and differences have repercussions in the resulting
age-[X/Fe] trends that we discuss in the following section.

##  5 Age-abundance-present-day-location trends at fixed [Fe/H]

Figure 6: Current and birth radius for stars of different age ranges. Kernel
density estimations for $\rm R_{now}$ (solid) vs $\rm R_{birth}$ (dashed) of
stars within age bins centered at 1 Gyr (purple) to 9 Gyr (yellow) in steps of
2 Gyr (top to bottom). The KDEs are normalized to the number of stars per age
group. The location KDEs, both for $\rm R_{now}$ and $\rm R_{birth}$, exhibit
similar tracks for stellar populations of the same age. This shows that
although stars move from their birth location as shown by the changing R
distributions, the bulk expectation for an inside-out galaxy growth is still
observed, with old stars concentrated towards the center and young stars in
the outskirts.

We have seen in the prior section the spatial trends of stellar element
abundances, that are imprinted at birth. However, stars of different ages will
have since moved from their birth location, so any initial trends will not be
trivially maintained. As shown in Figure 6, there is a smoother distribution
of current stellar location, $\rm R_{now}$ compared to $\rm R_{birth}$, as
blurring and churning (i.e. radial migration) processes wash out the original
clumpiness that stars were born into (e.g., Frankel et al. 2020). However,
both $\rm R_{now}$ and $\rm R_{birth}$ still show distinct trends at different
age groups with younger stars in the outskirts and older stars more centrally-
located.

In this Section, we examine what the current distribution of element
abundances at different locations and [Fe/H] as a function of age looks like,
as a result of the combination of birth properties and migration. We note that
from within the cuts outlined in Section 2.2, we pivot to two fixed [Fe/H]
values, of [Fe/H] = 0 dex and [Fe/H] = $-0.25$ dex, for the purpose of
demonstrating the age–[X/Fe] trends for a high and a low [Fe/H] sample,
respectively.

###  5.1 Solar neighborhood at solar [Fe/H]

Figure 7: Age-individual element abundance relationships at [Fe/H] = 0 dex in
the solar neighborhood. [Fe/H] and [X/Fe] (y-axis) as a function of age in Gyr
(x-axis), for the abundances provided in the simulations. Included in the plot
are stars within 1 kpc of LSR2 in m12i with [Fe/H] = 0 $\pm$ 0.025 dex. We
plot the best fit line (solid purple) and the 1-$\sigma$ running dispersion
from the line (orange shaded region), which we mark on the legend of each
subplot. The $\alpha$ element abundances (Mg, Si, S, Ca) increase with age,
and C, N, O, and Fe decrease with age in the simulation. There is a very small
intrinsic scatter around these age-[X/Fe] relations, of $\rm\sigma_{int}$$<$
0.02 dex.

First, we investigate the age-[X/Fe] trends, at solar metallicity ([Fe/H] =
0$\pm$0.025 dex) in the solar neighborhood (1 kpc from LSR2), as shown in
Figure 7. We fit a linear function to the age-[X/Fe] trends and the best fit
is shown as the purple solid line with a running dispersion shown as the
orange shaded region. The quantities from these fits are included in each
figure sub-panel i.e., the median intrinsic dispersion ($\rm\sigma_{int}$),
the slope, m, and the intercept, b. Similar to the age binning derivation in
Section 4, we determined the best binning in [Fe/H] by progressively
calculating the $\rm\sigma_{int}$ at different bin values and choosing the bin
that gives the smallest value so that its measurement is the least sensitive
to the size of the [Fe/H] bin. We note that observational studies of the
age-[X/Fe] relations typically have fixed and larger [Fe/H] bins (e.g.,
0.05-0.10 dex) compared to this work, but we have added the step of
determining the best binning to remove the dependence of the $\rm\sigma_{int}$
in [X/Fe] on the scatter in [Fe/H]. Nonetheless, the discussion on the
$\rm\sigma_{int}$ from simulations and observations in Section 6 show that
even with different [Fe/H] binning, their comparison is still viable. We
excluded the running value of the intrinsic dispersion, $\rm\sigma_{int}$,
calculated for age-[X/Fe] bins with fewer than 500 stars, to avoid being
skewed to larger $\rm\sigma_{int}$ due to sparsity in certain regions of the
age-[X/Fe] parameter space. However, we note that sparsity still affects some
of our analysis.

The cuts we applied (see Section 3) allow for a more direct comparison of the
age-[X/Fe] trends between simulations and observations. However, there are
some discrepancies in the scaling of the [X/Fe] between observations and
simulations. Given this, we focus more on the similarities and differences in
the [X/Fe] trends with age. Absolute differences in the mean element
abundances between data and simulation are not unusual and could be caused by
many things in the simulations, like the different star formation history in
m12i compared to the Milky Way (Sanderson et al., 2020).

At [Fe/H] = $0\pm 0.025$ dex, the age-[Fe/H] relationship is essentially flat
(slope is -0.001 dex/Gyr), with a slight decrease at older ages, which is to
be expected as older stars generally have lower [Fe/H]. We measure a
$\rm\sigma_{int}$ of 0.014 dex for the age-[Fe/H]. Note that this is an
arbitrary $\rm\sigma_{int}$ because we fixed the [Fe/H] bin size at 0.05 dex;
it is merely a reflection of the [Fe/H] range at which we consider the other
age-element abundance trends at this [Fe/H].

The age-[C/Fe] relationship shows decreasing [C/Fe] as a function of age. This
is opposite to what is found in observations (e.g. Nissen 2015; Bedell et al.
2018; Ness et al. 2019). As discussed in Section 4, C is produced mainly in
SNe II in observations, and as a consequence, should have higher abundance for
older stars than younger stars. In fact, Nissen (2015) found that [C/Fe] has a
larger slope with age than $\alpha$-elements, and proposed that a top-heavy
IMF can possibly explain the strong trend in C abundance with age. There is a
small intrinsic dispersion around the age-[C/Fe] trend, of
$\rm\sigma_{int}$=0.019 dex.

The age-[N/Fe] trend shows a decreasing trend with older ages, similar to what
is found in observations of RGB stars from apogee (Ness et al., 2019). The
measured slopes from the Milky Way observations and the simulations agree,
with m=-0.019 $\rm dex~{}Gyr^{-1}$ from this study (and
$\rm\sigma_{int}$=0.019 dex) and m=-0.02 $\rm dex~{}Gyr^{-1}$ from Ness et al.
(2019). Unfortunately, N is hard to measure for stars in the optical so we are
unable to compare to more observational studies.

The [O/Fe] trend with age is slightly decreasing, but essentially flat, having
m=-0.001 $\rm dex~{}Gyr^{-1}$ and a $\rm\sigma_{int}$=0.014 dex around the
best fit line. This is contrary to what is observed for the Milky Way (Nissen,
2015; Bedell et al., 2018; Ness et al., 2019; Sharma et al., 2022). O is a
hydrostatic $\alpha$-element dispersed mostly through SNe II and is expected
to decrease with younger stars. As noted in Section 4, the metallicity-
dependence of O yields from stellar winds in the simulations significantly
contribute at higher [Fe/H], therefore driving the trend to be flat or even
decreasing with older age.

All the remaining $\alpha$-elements—Mg, Si, S, and Ca—show positive age-[X/Fe]
trends, in line with expectations from Galactic chemical evolution work
(Kobayashi et al., 2020) and from observations (Nissen, 2015; Bedell et al.,
2018; Ness et al., 2019; Sharma et al., 2022) with the exception of Ca, where
observations find a flat age-[Ca/Fe] trend. Ca is an explosive
$\alpha$-element, which is expected to have a similar slope as compared to
other explosive $\alpha$-elements like Si and S. We observe this in the
simulations where the slopes for Si, S, and Ca are 0.006, 0.006, and 0.005
$\rm dex~{}Gyr^{-1}$ respectively, as the SNIa contributions flatten the trend
with age, while in comparison, the hydrostatic $\alpha$-element Mg has m=0.008
$\rm dex~{}Gyr^{-1}$, a steeper trend because it is mostly produced in SNe II.
Ca, aside from showing a flat relationship with age in observations also shows
a flat relationship in stellar velocity dispersion with increasing abundance,
contrary to what is found for other $\alpha$-elements (Conroy et al., 2014).
This is especially intriguing because there is a known age-velocity dispersion
relation in observations wherein stellar populations with older ages have
higher stellar velocity dispersion. One way to explain this curious trend of
Ca with age is to include contributions from other sources, such as a subclass
of supernovae called “calcium-rich gap transients” that would flatten the
trend at later times (Mulchaey et al., 2014; Nissen, 2015).

For all the elements except Fe, the running $\rm\sigma_{int}$ gets larger at
older ages where there are (1) fewer stars in general at solar metallicity and
(2) the stars are born from less homogeneous gas.

Figure 8: Age-individual element abundance relationships at lower metallicity,
[Fe/H] = -0.25 dex, in the solar neighborhood. The legends and symbols are
similar to Figure 7. All the elements at this lower [Fe/H] follow a similar
trend to the solar [Fe/H] case with three main differences: (1) these
age-[X/Fe] relations extend to older ages, (2) the trends are much steeper for
all the elements (except for Fe), and (3) the scatter of the elements around
their age-[X/Fe] relations, $\rm\sigma_{int}$, are on average twice as large
as at [Fe/H] = 0 dex (although still very small) with $\rm\sigma_{int}$$<$
0.045 dex.

###  5.2 Solar neighborhood at [Fe/H]= -0.25 dex

We apply the same spatial selection and binning determination as per Figure 7,
but select stars at lower metallicity, with [Fe/H] = -0.25 $\pm$0.025 dex. We
compare our age-[X/Fe] trends with Ness et al. (2019) and Sharma et al.
(2022), that also investigated at lower [Fe/H] i.e. $-0.35$ dex for the former
and [Fe/H] $=-0.2$, $-0.4$, and $-0.5$ dex for the latter. We aim to compare
not only the trends for simulations vs observations but also the age-[X/Fe]
trend differences that are introduced at different [Fe/H]. As we noted in
Section 2.2 and Figure 2, for the simulated Milky Way-mass galaxy m12i,
exploring at much lower metallicities than [Fe/H] $=-0.25$ dex probes stars
that were born before the onset of a disk exhibiting distinct inside-out
growth.

Compared to the [Fe/H] = 0 dex sample, the stellar population at [Fe/H] =
-0.25 dex extends to older ages, as is expected from the [Fe/H] KDE in Figure
5. Interestingly, the age-[X/Fe] trends at [Fe/H] = -0.25 dex in Figure 8 show
more clumpiness and scatter than the [Fe/H] = 0 dex trend, especially at older
ages i.e., one clump at 5$<$age$<$7.5 Gyr and another at age $>$8 Gyr. For
each element, the age-[X/Fe] trend is steeper compared to the [Fe/H] = 0 dex
sample (see Figure 9 for comparison), even without the inclusion of stars in
the oldest clump. In fact, the age-[O/Fe] trend changes from slightly
decreasing (or flat) at [Fe/H] $=0$ dex to slightly increasing with a slope of
m=0.002 $\rm dex~{}Gyr^{-1}$, due to the less significant contribution from
stellar winds at this point of m12i’s chemical enrichment. The median
$\rm\sigma_{int}$ values are all higher for all the elements in this low-
metallicity regime, except for Fe, where we made a cut in the sample
selection.

Figure 9: Age-individual element abundance relationships at [Fe/H] = 0 dex and
[Fe/H] = $-0.25$ dex in the solar neighborhood. The running dispersions of the
data around the age-[X/Fe] trends are taken from Figures 7 and 8. The
age-[X/Fe] relations overlap and show flaring at older ages, especially for
the solar [Fe/H] stars. However, this is mostly due to a smaller sample of
stars at these ages.

Ness et al. (2019) find that their solar and low [Fe/H] samples have
comparable $\rm\sigma_{int}$ and slopes. They find offsets in some age-[X/Fe]
trends, especially for the $\alpha$-elements. In their observations, the
trends for O, Mg, Si, S, and Ca at [Fe/H] = 0 dex are all lower compared to
the trends at [Fe/H] = $-0.25$ dex. In the simulations, the trends for the
solar and low [Fe/H] samples occupy a very similar space but they intersect at
$\sim$5 Gyr for the $\alpha$-elements, as shown in Figure 9. However this
could be easily explained by stars with different ages dominating the
different [Fe/H] bins, where the median ages are 1 Gyr and 5 Gyr for the solar
and low [Fe/H] samples, respectively. At the same age, the lower [Fe/H] sample
is less chemically evolved and experienced less $\alpha$-element abundance
dilution, making their age-[X/Fe] relations for the $\alpha$ elements higher
compared to the solar sample. Additionally, a majority of the low [Fe/H]
sample is older compared to the solar [Fe/H] sample, and these stars skew the
age-[X/Fe] relations, resulting to the steepness in the trend (and therefore
the cross-over). Additionally, for stars with ages $>$ 4 Gyr, the
$\rm\sigma_{int}$ in the solar [Fe/H] is higher than the $\rm\sigma_{int}$ in
the low [Fe/H] bin at a given age; however, we note that this is most likely
due to the low number of stars at that age bin for the solar [Fe/H] sample
compared to the low [Fe/H] sample. The observational work by Sharma et al.
(2022) found that the age-[X/Fe] trends at different [Fe/H] overlap with each
other substantially at all ages, similar to what we find.

Figure 10: Age-individual element abundance relationships at [Fe/H] = 0 dex in
different distance slices. All legends and symbols are similar to Figure 7.
The top set of plots are stars in the inner disk at 1$\rm<R<$3 kpc, the center
set are for stars in the outer disk at 11$\rm<R<$13 kpc, and the bottom set
are for all stars within R = 25 kpc and Z$\pm$500 pc. The scatter around the
age-[X/Fe] relations is higher for the inner disk compared to the outer
disk.This is a consequence of older stars (of which there are more in the
inner disk) exhibit wider abundance distributions. Interestingly, when all
distance slices are included, the median $\rm\sigma_{int}$ is still quite
small at $\rm\sigma_{int}$$<$0.025 dex.

###  5.3 Different disk locations at solar metallicity, [Fe/H] = 0 dex

In this section, we explore the age-[X/Fe] trends in the simulations for stars
at [Fe/H] = 0 $\pm$ 0.025 dex, Z $<$500 pc, and at different distance slices
to represent the inner disk (1-3 kpc), outer disk (13-15 kpc), and the whole
disk ($<$25 kpc) as shown in Figure 10. We particularly want to understand if
different locations in the galaxy exhibit distinct age-[X/Fe] trends, and how
this ties to the m12i’s history.

Compared to the solar neighborhood trend in Figure 7, the inner disk region
(top set of subplots) shows higher $\rm\sigma_{int}$ around the age-[X/Fe]
relations. There is also a more pronounced increase in $\rm\sigma_{int}$ for
ages $>$ 4 Gyr. In fact, this behavior extends to much older ages where the
number of stars is below the threshold that we have set. These old stars, as
we have shown in Section 4, and Figures 4 and 5, are born with a larger spread
in their abundances when the star formation in the galaxy was more chaotic and
clumpier. The stellar populations and the general age-[X/Fe] relations in the
inner disk are both more complex, a reflection of the inner disk region’s
intricate history where stars of different ages all contribute. Nonetheless,
we adopt a linear fit to the age-[X/Fe] distributions so as to enable more
direct and interpretable comparison to observations and to the age-[X/Fe]
trends at other locations. The increase that we see in $\rm\sigma_{int}$,
which signals the difference between steady and bursty star formation, is
supported by the recent study by Yu et al. (2021), which finds a similar
transition at $\sim$3 Gyr for m12i. This is only seen closer to the inner
disk, but is slightly noticeable in the solar neighborhood sample as well.

At the same [Fe/H] (i.e. [Fe/H] = 0 dex), the inner disk shows age-[X/Fe]
relations in the $\alpha$-elements that are higher at older ages and lower at
younger ages compared to the solar neighborhood (Figure 7) and outer disk
(middle set of subplots in Figure 10). The higher $\alpha$-element abundance
in the inner disk is in agreement with the rapid star formation and enrichment
scenario in the center of galaxies, before the onset of SNIa. On the other
hand, the lower $\alpha$-element abundance in the same region is in line with
the longer history of star formation in the inner regions of the galaxy where
the [$\alpha$/Fe] abundances have been more diluted by SNIa compared to the
outskirts of the galaxy. Compared to the age-[X/Fe] trends in the inner disk,
the stars in the outer disk are mostly young with ages $<$5 Gyr and therefore
have a more well-behaved age-[X/Fe] trend across the board that is well-fit by
a linear trend.

Lastly, we look at the age-[X/Fe] trends for the entire disk (bottom set of
subplots in Figure 10). Even with the inclusion of stars from various
locations in the galaxy—from the inner disk that experienced different modes
of star formation to the outer disk that has experienced later and more steady
star formation—the median $\rm\sigma_{int}$ remains small with values of
$\rm\sigma_{int}$$<$0.025 dex. We note that the gray data points are meant to
show the total extent of the age-[X/Fe] trends, but not the density. This
could be a misleading representation for the whole disk, where at ages $<$5
Gyr, the running $\rm\sigma_{int}$ is small, but the extent in [X/Fe] at a
given age is large. Nonetheless, this provides a direct comparison to the
age-[X/Fe] relations discussed earlier. We see an increase in the running
$\rm\sigma_{int}$ towards older stars, bearing similarity to the inner disk.
The small $\rm\sigma_{int}$ measured in m12i, even after including all the
stars “observed” across the full disk sample points to a case where this small
$\rm\sigma_{int}$ is representative of a population whereby knowing the age
and [Fe/H] of a star can predict its abundance, with increasing scatter at
older ages. Ultimately, the ability to predict the abundance from the age and
[Fe/H], highly depends on when the galaxy changed to a steady star formation
from a more bursty mode as this determines the $\rm\sigma_{int}$ and
precision.

##  6 Intrinsic Dispersion in the age-[X/Fe] -metallicity relation across the
Galaxy’s disk

Figure 11: Intrinsic dispersions around the age-individual element abundance
relations in simulations vs observations in the solar neighborhood. We compare
the $\rm\sigma_{int}$ in the simulation (diamond) with those in observations:
Solar twins from Bedell et al. 2018 (circle) and red clump stars from Ness et
al. 2019 (cross), both at [Fe/H] = 0 dex. The symbols are also colored by the
age-[X/Fe] slopes, with blue corresponding to increasing abundance with age
and red to decreasing. $\rm\sigma_{int}$ = 0.02 dex is marked to guide the
eye. In general, the $\rm\sigma_{int}$ from observations and simulations
agree, barring C,N,O and S that have higher $\rm\sigma_{int}$ for the red
giant stars. For the most part, the $\rm\sigma_{int}$ tracked in the
simulations is $<$0.02 dex. Aside from C,O, and Ca, the rest of the elements
show similar trends in the simulations and observations.

We take a deeper look at the $\rm\sigma_{int}$ in the simulation around the
age-[X/Fe] relations, compared to Milky Way, as well as at different [Fe/H]
and locations. We aim to see if this measurement of the scatter around the
age-[X/Fe] relations can be used as a diagnostic and/or as a metric to capture
the chemical enrichment in the simulated Milky Way galaxy m12i, and to relate
this quantity to what is happening in the galaxy at that point in time.

###  6.1 $\rm\sigma_{int}$ in simulation vs observations

The intrinsic scatter around the age-[X/Fe] trends, $\rm\sigma_{int}$, as
measured for stars in the solar neighborhood at solar [Fe/H], from this work
and other observational studies, is shown in Figure 11. The diamonds are stars
from the simulations (Ananke), while the circles are solar twins (Bedell et
al., 2018), and crosses are RGB stars (Ness et al., 2019) from observations.
As noted in Section 3, these observational data take into account the effects
of the [X/Fe] and age uncertainties on the resulting $\rm\sigma_{int}$. The
symbols are colored by the age-[X/Fe] slope where red is negative and
decreasing with age and blue is positive and increasing with age.

The $\rm\sigma_{int}$ for the elements produced in the simulated Milky Way are
similar to the $\rm\sigma_{int}$ from Milky Way observations, both for the
solar twins and RGB stars, and are generally $\rm\sigma_{int}$$\leq$0.02 dex,
marked by the gray dashed line. This is especially true for the
$\alpha$-elements Mg and Si. This small scatter is similarly found by
Bellardini et al. (2022) for [Mg/Fe] at different age bins in their sample of
FIRE-2 Milky Way-like galaxies. The general similarity in the
$\rm\sigma_{int}$ means that the level of homogeneity of star forming gas in
the Milky Way at a given metallicity is well-reproduced in the simulated
galaxy m12i.

As discussed in Sections 4 and 5, C, O, and Ca exhibit dissimilar trends with
age compared to observations (and again note we refer to age-[X/Fe] relations
in narrow bins of [Fe/H]). The element C shows a negative slope with age in
the simulation, while it has a positive slope measured both from dwarf and
giant stars in the Milky Way. The element Ca, on the other hand, has a
positive slope, typical of $\alpha$-elements in the simulations. However, Ca
has an observed negative slope with age. The element O is essentially flat
with age in the simulations, in contrast to the increasing trend with age seen
in observations. In a simple exploration of the age-[X/Fe] relations in the
other galaxies in Ananke (m12m and m12f) with different star formation
histories, we find that they similarly exhibit these discrepancies in the
direction of the trends. This highlights the need for updates to the chemical
enrichment sources, yields, and rates prescribed in the star formation in
cosmological simulations in order to further their utility for Galactic
archaeology purposes.

There have been many recent efforts to look at the age-[X/Fe] trends in
different Milky Way stellar populations or spectral types. These recent
efforts are a reflection of how powerful a tool chemical abundances could be,
especially for deriving ages (e.g. Hayden et al. 2020). Yuxi et al. (2021)
compared the age-[X/Fe] trends for the high- and low-$\alpha$ disk using
apogee red clump stars and found that the median $\rm\sigma_{int}$ for both
stellar populations are similar at $\rm\sigma_{int}$$\sim$0.04 dex; even with
stellar populations with different chemical enrichment histories, the
abundance scatter is comparable. Although there is no $\alpha$-element
dichotomy in m12i in the [$\alpha$/Fe]-[Fe/H] plane, we do find that the
$\rm\sigma_{int}$ measured at different radial parts of the disk are small and
generally $\rm\sigma_{int}$$<$0.04 dex, even if they experienced different
formation histories. In this sense, our work agrees with the findings from
Yuxi et al. (2021). However, a further exploration of the age-[X/Fe] relations
for different galaxy components, as Nikakhtar et al. (2021) recently
determined for the Ananke galaxies, would be worth exploring in detail to
compare to observations. Using galah DR3 data, Sharma et al. (2022) found that
for ages $<$10 Gyr, a disk star’s abundance can be predicted from its age and
[Fe/H] with a $\rm\sigma_{int}$ of 0.03 dex. They also found that comparing
the slopes in the age-[X/Fe] trends to those from the [Fe/H]-[X/Fe] relations
split elements into three groups, corresponding to their main channel of
production: SNe II, SNIa, and stellar winds; these are the same sources used
for the chemical enrichment in our simulated Milky Way.

Figure 12: Intrinsic dispersion of individual element abundances [X/Fe] around
the age-[X/Fe] trends at different disk locations and metallicities, for
simulations and observations. The $\rm\sigma_{int}$ values are reported across
different bins in galactocentric radii (1-3 kpc, 4-6 kpc, 7-9 kpc, and 10-12
kpc) for the elements in this study (solid symbols) as well as comparison with
observations from GALAH DR3 (marked as a pentagon at solar [Fe/H] and star at
low [Fe/H] in the 7-9 kpc bin for C, O, Mg, Si, and Ca). At each radius bin,
we show the $\rm\sigma_{int}$ for a solar metallicity sample (circles) and
low-metallicity sample ([Fe/H] = -0.25 dex, triangles). The symbols are
colored by the median stellar population age. Across all the elements for both
simulations and observations, the stars with lower [Fe/H] show higher
$\rm\sigma_{int}$ as well as higher median ages. There is also a trend of
decreasing $\rm\sigma_{int}$ with radius for the solar [Fe/H] sample which is
not present in the low [Fe/H] sample.

###  6.2 $\rm\sigma_{int}$ at different disk locations and [Fe/H] in the
simulations

We now explore the $\rm\sigma_{int}$ of each element around their age-[X/Fe]
relations for the seven elements included in this work (excluding [Fe/H]) at
different radial locations in the disk (e.g. 1-3 kpc, 4-6 kpc, 7-9 kpc, and
10-12 kpc) and at different metallicities (solar at [Fe/H] $=$ 0 dex and low
at [Fe/H] $=$ -0.25 dex) as shown in Figure 12. We color our markers by the
median age of that stellar population in a given location and [Fe/H] bin. The
2kpc radial bin is consistent with the radial range over which the stars are
similar in chemistry (Bellardini et al., 2021).

The $\rm\sigma_{int}$ around the age-[X/Fe] trends differ for the two
metallicity bins we examine, and this difference is larger than what is seen
for the samples at different median ages or varying locations for stars at
fixed [Fe/H]. The low-[Fe/H] sample, across all elements and radial bins,
aside from the innermost region, exhibits higher $\rm\sigma_{int}$ than the
solar [Fe/H] sample. This is especially pronounced for the element N, which
has metallicity-dependent yields from SNe II, thereby causing N to have the
largest difference in $\rm\sigma_{int}$ between the high and low [Fe/H]
samples. In fact, only N shows a noticeable difference in the
$\rm\sigma_{int}$ at different [Fe/H] in the innermost radial bin. The element
C shows the smallest $\rm\sigma_{int}$ difference between the high and low
[Fe/H] stars, meaning that a change in [Fe/H] produces minimal change in
$\rm\sigma_{int}$. The $\alpha$-elements i.e. O, Mg, Si, S, and Ca all
generally show similar $\rm\sigma_{int}$ trends with [Fe/H]: The solar [Fe/H]
sample shows an exponential decline in $\rm\sigma_{int}$ as a function of
radial bin, while the low [Fe/H] sample does not exhibit a clear trend, and is
mostly flat.

As expected from how chemical enrichment proceeds and from Section 4, the
solar [Fe/H] stars have a younger median age compared to the low [Fe/H] stars
at all radial bins. The median ages at a given [Fe/H] also vary depending on
the location. For the solar [Fe/H] sample, the median age in the central
region is $\sim$4 Gyr old, but for the outer radial bins, the median ages are
$<$2 Gyr old. There is a larger range in median ages for the lower [Fe/H] bin,
from 6.5 Gyr old in the central region of the disk, to 2.5 Gyr old in the disk
outskirts. At the same radial bin, a larger difference in the median age
between the two [Fe/H] bins does not correspond to a larger difference in
$\rm\sigma_{int}$.

In summary, we show that in comparing the variables of galactocentric radius,
median age, and [Fe/H], the scatter of the elements, $\rm\sigma_{int}$, around
their age-[X/Fe] relations, changes the most with [Fe/H]. All the elements
show differences in $\rm\sigma_{int}$ between the low and high [Fe/H] bins,
with a mean difference of $\sim$0.010 dex or a 60% increase in
$\rm\sigma_{int}$ at low [Fe/H]. On the other hand, only for the solar [Fe/H]
sample does the $\rm\sigma_{int}$ vary with location, and at the same radius
bin, the difference in median ages is not proportional to the difference in
$\rm\sigma_{int}$(for example, comparing between the 5 kpc and 11 kpc bin).
Ultimately, there is the caveat that the $\rm\sigma_{int}$ trends as a
function of [Fe/H] and location in this work are unique to m12i.

###  6.3 $\rm\sigma_{int}$ in the real solar neighborhood at different [Fe/H]

In addition to the results from the simulation data, we derive the scatter
around age-[X/Fe] relations, $\rm\sigma_{int}$, from galah DR3 observations of
real solar neighborhood stars at different metallicities, shown in Figure 12.
The following cuts were made to ensure that the stars from galah data
(described in Section 3) are comparable to our bins of simulated stars and
that the abundances are of good quality:

  * •

7$<$R$<$9 kpc,

  * •

signal-to-noise ratio $>$40,

  * •

[Fe/H]=0$\pm$0.05 dex for the solar [Fe/H] bin,

  * •

[Fe/H]=-0.25$\pm$0.05 dex for the low [Fe/H] bin,

  * •

flag_sp=0,

  * •

flag_X_fe=0,

  * •

5000$<$$T_{\rm eff}$$<$6100 K, and

  * •

3.2$<$$\log g$$<$4.1 dex,

where X = C, O, Mg, Si, and Ca—the elements this study has in common with
galah. Note that our $T_{\rm eff}$ and $\log g$ cuts are adopted from the MSTO
stars cut from Sharma et al. (2022). With these selections, we end up with
2,882 stars for the solar [Fe/H] sample and 1,546 stars for the low [Fe/H]
sample. In general, we performed a similar exercise as was done for the
simulations, and calculated the median running dispersion in the best fit
linear age-[X/Fe] trend for these elements.

Like the other comparison observational data (see Section 3), the
uncertainties in [X/Fe] and age have to be taken into account to calculate the
$\rm\sigma_{int}$. To account for the abundance uncertainties, we subtracted
the mean $\rm\sigma_{abund}^{2}$ from the measured $\rm\sigma_{total}^{2}$ to
get the $\rm\sigma_{int}^{2}$. The typical value for $\rm\sigma_{abund}$ is
0.05 dex for Mg, Si, and Ca and 0.10 dex for C and O. Next, to understand how
the age uncertainty affects the measured $\rm\sigma_{int}$, we drew a new age
from a normal distribution centered on the age given by BSTEP with the age
uncertainty as standard deviation. The mean age error for our sample is 0.7
Gyr. We do this 100 times and find that the change in $\rm\sigma_{int}$ is
$\ll$0.01 dex and therefore negligible.

We show the $\rm\sigma_{int}$ in galah for stars in the solar neighborhood at
[Fe/H] = 0 dex and [Fe/H] = $-0.25$ dex, together with the $\rm\sigma_{int}$
derived in the simulations in Figure 12. The [Fe/H] = 0 dex sample from galah
has a median age of 5 Gyr, while the [Fe/H] = -0.25 dex sample has a median
age of 6 Gyr. The $\rm\sigma_{int}$ for the low [Fe/H] bin is higher than the
solar [Fe/H] bin for all the elements with a mean offset of $\rm\sigma_{int}$
0.014 dex, similar to what we find in the simulations. All measured
$\rm\sigma_{int}$ from the observations are lower than 0.07 dex, and the
difference in $\rm\sigma_{int}$ between the two [Fe/H] bins ranges from
$<$0.01 dex (e.g. C, Ca) to $\sim$0.03 dex (e.g. Mg). The $\rm\sigma_{int}$
trend for Si in observations resembles the $\rm\sigma_{int}$ in the
simulations the most, while the $\rm\sigma_{int}$ for Ca in galah resembles
the simulations the least, where the $\rm\sigma_{int}$ from the observations
are larger by $\sim$0.03 dex compared to their simulated counterpart. Even
with these differences, it is noteworthy that the $\rm\sigma_{int}$ from the
observations and simulations are very similar and of the same magnitude. This
comparison highlights that even if the absolute values and trends for [X/Fe]
in the simulations are dependent on many assumptions (i.e., on yields, rates,
sources) and the star formation history, the general abundance scatter in
observations is reproduced well (Hopkins et al., 2018; Escala et al., 2018;
Bellardini et al., 2021, 2022). We expect that this agreement holds true even
if realistic observational uncertainties were modeled for m12i as (1) the
effect of the abundance error is removed when deconvolving the sources of the
scatter and (2) the slopes in the age-[X/Fe] relations in the simulations are
shallower than what the observations measure. Applying age uncertainties in
the simulations will therefore have negligible effects on the resulting
$\rm\sigma_{int}$.

##  7 Discussion and Summary

The stars in the Milky Way have been shown to exhibit tight age-[X/Fe]
relations at a given [Fe/H], such that by knowing a star’s age and [Fe/H], one
can predict its abundance to a precision of $\sim$0.02 dex. Indeed, this has
been studied in greater detail and in many axes e.g., high vs low-$\alpha$
disk, high vs low [Fe/H], MSTO vs RGB stars, thanks to large spectroscopic
survey data (Ness et al., 2019; Yuxi et al., 2021; Sharma et al., 2022).

The main goal of this work is to explore this observed age-[X/Fe] relations in
cosmological zoom-in simulations that have been shown to reproduce Milky Way
results from Gaia DR2 (Sanderson et al., 2020). One advantage to doing this is
that we have absolute knowledge of when and where stars formed in the
simulations. In contrast, ages in observations are harder to derive, the
approach for deriving them is dependent on the type of star, and birth
locations are unknown. Our main focus has been to determine if the current
state-of-the-art simulations that trace the chemical enrichment of galaxies,
can successfully reproduce the observed age-[X/Fe] trends in the Milky Way,
and where they fall short and disagree. However, we can go beyond this and
even explore what is physically happening in the simulated galaxy, and
investigate where stars of a given age or chemistry are born.

Therefore, in understanding where the age-[X/Fe] trends in the simulations
come from, we have also come to establish the intricate relationship of age,
metallicity, individual element abundances, and location of stars in this
simulated Milky Way galaxy, m12i. We have shown that aside from the chemical
enrichment in the galaxy being a product of the prescribed nucleosynthesis
(see Section 4), the individual element abundances, and specifically their
scatter around the age-[X/Fe] relations, $\rm\sigma_{int}$, generally reflect
m12i’s star formation history. Focusing on the solar [Fe/H] sample, at all
distances, we find that the running $\rm\sigma_{int}$ across age is constant
at young ages, up to about $\sim$4 Gyr, and then increases to higher
$\rm\sigma_{int}$ values at older ages. The detailed study of m12i by Ma et
al. (2017) find that the disk changes from a bursty and clumpy star formation
mode to a stable star formation mode at look back times of 6 Gyr. This galaxy
was also included in the more recent analysis by Yu et al. (2021) where they
found that m12i changes to a near constant state of star formation at $\sim$3
Gyr. Indeed, for stars that are $<$3 Gyr, the $\rm\sigma_{int}$ is constant,
for 3$<$age$<$6 Gyr, the $\rm\sigma_{int}$ increases, and at $>$6 Gyr, the
$\rm\sigma_{int}$ is the largest. We would like to point out however, that we
mainly see this behavior for the solar [Fe/H] sample.

To make sense of the observed age-[X/Fe] trends, we have explored their
distributions at different ages in Section 4. Importantly, we also explored
where these stars are born and where they are currently as shown in Figures 4
and 6. Here we find that stars in the oldest age bin (9 Gyr) are spread all
across the disk within R $<$ 10 kpc, extending farther than stars that are
younger in the 5 Gyr and 7 Gyr age bins. This might seem counter-intuitive in
the inside-out formation scenario, where star formation starts in the central
regions and proceeds outwards. However, for these oldest stars, the galaxy
that eventually forms into m12i is composed of multiple smaller galaxies (see
Figure 1 in Ma et al. 2017), which gives rise to the more extended and spread
out $\rm R_{birth}$ distribution that we see. Indeed, Santistevan et al.
(2020) found that the Milky Way/M31-mass galaxies in the FIRE simulations are
made up of a hundred distinct dwarf galaxies before z$\sim$2\. Therefore, the
stars at the oldest age bin show the largest spread in the abundance
distributions (as similarly seen in the analysis from Bellardini et al. 2022
of the stellar abundances of Milky Way/M31-mass galaxies), which reflects the
chemistry of this clumpier and more chaotic star formation, and with the
[Fe/H] KDE showing a long tail towards lower values. Curiously, the stars in
the 3 Gyr bin have a similar $\rm R_{birth}$ (as well as current R)
distribution to the 9 Gyr age bin, but its abundance distributions are
narrower. This could be explained however by the existence of a rotationally-
supported and stable disk that at 3 Gyr, has stars with abundances that are
more azimuthally homogeneous at a given radius (Bellardini et al., 2022).

We also note that stars with different ages have significant overlap in their
distributions of [X/Fe], $\rm R_{birth}$, and current R, but that the peak of
those distributions are distinct from each other. For younger stars, the peak
of the radial distributions of stars is at larger R, and the abundance
distributions exhibit progressive enrichment. There is an overlap in the
[X/Fe] KDEs, which is manifested in the gentle slopes in the age-[X/Fe]
trends. The element N, whose yields from SNe II have a progenitor metallicity-
dependence, has the most distinct [N/Fe] distributions as a function of age
(i.e., [N/Fe] distributions overlap the least at varying age bins), and
therefore also the steepest age-[X/Fe] trend. On the other hand, O, whose
yields from stellar winds have a progenitor metallicity-dependence, has
abundance KDEs that overlap for different ages, and therefore exhibits the
flattest trend with age. We therefore find that for the simulated Milky Way-
like galaxy m12i, the direction of the age-[X/Fe] trends is a reflection of
the chemical evolution prescriptions in the simulations, and the scatter
around these relations is a reflection of the mode of star formation, with
higher scatter at earlier times when star formation was more bursty and
chaotic, and lower scatter at later times when the disk is more stable.
Meanwhile, the magnitude of the slopes in the age-[X/Fe] relations is a
convolution of the effects from the chemical evolution prescription as well as
the star formation history. With the next generation of spectroscopic surveys
dedicated to densely mapping stars in the Milky Way such as SDSS-V (Kollmeier
et al., 2017), WEAVE (Bonifacio et al., 2016), and 4MOST (de Jong et al.,
2019), as well as novel ways to derive ages for these stars, we can use the
$\rm\sigma_{int}$ in the age-[X/Fe] trend as another diagnostic in
understanding the evolution of the Milky Way.

In Section 6 we compared the slopes of and $\rm\sigma_{int}$ around the
age-[X/Fe] relations from this work to those derived in observations.
Specifically, in Figure 11 we show the $\rm\sigma_{int}$ in the solar
neighborhood at solar metallicity for the stars in Ananke, solar twins (Bedell
et al., 2018), and RGB stars (Ness et al., 2019). We find that the
$\rm\sigma_{int}$ for m12i and the Milky Way are similar, and in fact, some
elements such as Mg and Si have near-identical $\rm\sigma_{int}$ across all
three samples. The elements C, O, and Ca, however, show the opposite trends in
their slopes with age in the simulations compared to observations. The
age-[C/Fe] relation increases with older ages in observations and has been
noted to even have a steeper slope with age than $\alpha$ elements Mg and Si
(Nissen, 2015). In contrast, we find that in m12i, the age-[C/Fe] relations
have a negative slope; decreasing with increasing age. We expect that this is
because of the substantial contribution from stellar winds at later times
prescribed in the simulations. The age-[O/Fe] relations show a positive slope
in observations, as is expected for an $\alpha$-element. However, we find that
the O abundance is flat, or decreasing with older ages in the simulations.
This is due to the large O yields from stellar winds at higher metallicities
as prescribed in the simulations. Lastly, observations find that the [Ca/Fe]
trend with age is flat or decreasing with older ages, contrary to the trend
for the other $\alpha$ elements (Nissen, 2015; Bedell et al., 2018; Ness et
al., 2019; Sharma et al., 2022) and to what we find in this study, though this
discrepancy could potentially be alleviated by the inclusion of other
astrophysical sources (e.g., Mulchaey et al. 2014). The chemical enrichment in
this cosmological zoom-in simulation of a Milky Way-like galaxy and
observational comparisons highlight that our knowledge of how elements form
and evolve through time is incomplete. Continued efforts for Galactic chemical
evolution modeling (e.g. Kobayashi et al. 2020) in tandem with a more flexible
way of propagating chemical enrichment are therefore important for these
simulated galaxies to be an even more realistic laboratory for Galactic
archaeology. Nonetheless, it is truly noteworthy that m12i, which was evolved
in a large cosmological volume (later on zoomed-in which the Ananke synthetic
survey was made from), and was matched as a Milky Way galaxy based only on its
mass and isolation, reproduce comparable age-[X/Fe] trends to the Milky Way.

Moving forward, there are many avenues to delve deeper into Galactic
archaeology in simulations, especially in the Ananke catalog, to link observed
Milky Way properties to a plausible galaxy formation history. For example, it
is worth looking more into the $\rm\sigma_{int}$ for the different Milky Way
galaxies in Ananke that have masses 1-2 $\times 10^{12}\rm M_{\odot}$ but have
varying morphologies and star formation histories. This would be important in
contextualizing what brings about the observed $\rm\sigma_{int}$ in the Milky
Way by comparing to the derived $\rm\sigma_{int}$ in simulated Milky Way
galaxies with different properties. In line with this, we have shown in
Section 6 how the $\rm\sigma_{int}$ for observations and simulations compare
in the solar neighborhood, with the former calculated from galah abundances of
Milky Way stars and the latter based on this analysis. With future surveys
that will observe stars in the Milky Way more extensively (e.g. Kollmeier et
al. 2017), we will be able to map the $\rm\sigma_{int}$ in the Galaxy in its
different regions, that have, in turn, their distinct stellar populations to
aid in a more holistic picture of its formation history. Bellardini et al.
(2021) used simulated Milky Way galaxies to show that the abundance scatter in
gas is dominated by radial trends at lookback times $<$6.9 Gyr, and follow-up
work show that this transition lookback time is $<$7.5 Gyr for stars
(Bellardini et al., 2022), beyond which the abundance scatter is dominated by
azimuthal inhomogeneity. It is therefore worthwhile to also explore the
age-[X/Fe] trends from different solar viewpoints in the same galaxy to
ultimately understand if the age-[X/Fe] relationship at a given metallicity is
dominated by radial or azimuthal trends.
The future of the field is bright with large spectroscopic surveys in tandem
with novel ways of deriving ages, as well as updates on hydrodynamical
simulations to produce more realistic star formation in galaxies. This will
allow the community to tackle how our Galaxy formed more expansively and from
different perspectives, bridging our knowledge and the gap between
observations and simulations, and with stars, specifically their chemical
fingerprint, as the main tool. In this work, we give but a glimpse of this
future, outlining how the abundance $\rm\sigma_{int}$ can be a diagnostic for
unraveling the evolution of the Milky Way. In the end, we are still left with
many questions to investigate (e.g. galactic chemical evolution modeling,
azimuthal vs radial abundance trends, flexible chemical prescriptions) both
from the observational and simulation point of view, and we look forward to
answering them in the future, in this era of big data in Galactic archaeology.

##  8 acknowledgments

AC is grateful to the Flatiron Institute Center for Computational Astrophysics
Pre-Doctoral Program through which this project was made possible. AC also
acknowledges support from the Science and Technology Facilities Council (STFC)
[grant number ST/T000244/1] and thanks the Large Synoptic Survey Telescope
Corporation (LSSTC) Data Science Fellowship Program, which is funded by LSSTC,
NSF Cybertraining Grant 1829740, the Brinson Foundation, and the Moore
Foundation. MKN is in part supported by a Sloan Research Fellowship. AW
received support from: NSF via CAREER award AST-2045928 and grant AST-2107772;
NASA ATP grant 80NSSC20K0513; HST grants AR-15809, GO-15902, GO-16273 from
STScI. KH and AC acknowledge support from the National Science Foundation
grant AST-1907417 and AST-2108736 and from the Wootton Center for
Astrophysical Plasma Properties funded under the United States Department of
Energy collaborative agreement DE-NA0003843. This work was performed in part
at Aspen Center for Physics, which is supported by National Science Foundation
grant PHY-1607611. This work was performed in part at the Simons Foundation
Flatiron Institute’s Center for Computational Astrophysics during KH’s tenure
as an IDEA Fellow. RES acknowledges support from the Research Corporation
through the Scialog Fellows program on Time Domain Astronomy, from NSF grant
AST-2007232, from NASA grant 19-ATP19-0068, and from HST-AR-15809 from the
Space Telescope Science Institute (STScI), which is operated by AURA, Inc.,
under NASA contract NAS5-26555.

Simulations in this project and the mock catalogs generated from them were run
on the Caltech compute cluster “Wheeler,” allocations from XSEDE TG-AST130039
and PRAC NSF.1713353 supported by the NSF, NASA HEC SMD-16-7592, and the High
Performance Computing at Los Alamos National Lab., and analyzed using
computing resources supported by the Scientific Computing Core at the Flatiron
Institute. Further simulations were run using Early Science Allocation 1923870
on Frontera, which is made possible by National Science Foundation award
OAC-1818253. This work used additional computational resources of the
University of Texas at Austin and TACC, the NASA Advanced Supercomputing (NAS)
Division and the NASA Center for Climate Simulation (NCCS), and the Extreme
Science and Engineering Discovery Environment (XSEDE), which is supported by
National Science Foundation grant number OCI-1053575. Last but not the least,
huge thanks to Jason Hunt who generously hosted the data as we were closing in
on the finish line.

##  Appendix A Yield prescriptions in the simulation

As discussed in Section 2, the stellar evolution in FIRE-2 Hopkins et al.
(2008) is produced from STARBURST99 (Leitherer et al., 1999) with a Kroupa
(2001) IMF. The three sources of nucleosynthesis included are SNe II, SNIa,
and stellar winds from OB/AGB stars. In the simulations, SNe II rates are a
function of time (i.e. age of a star particle) dictated by the following:

$\rm
R_{SNeII}~{}(SNe~{}Myr^{-1}~{}M_{\odot}^{-1})=\left\\{\begin{array}[]{ll}0&\quad\rm
age<3.4~{}Myr\\\ 5.408\times 10^{-4}&\quad\rm 3.4~{}Myr<age<10.37~{}Myr\\\
2.516\times 10^{-4}&\quad\rm 10.37~{}Myr<age<37.53~{}Myr\\\ 0&\quad\rm
age>37.53~{}Myr,\end{array}\right.$

with IMF-averaged ejecta mass of $\rm 10.5~{}M_{\odot}$ per explosion, most of
which are in H. The SNe II yields for the elements in this study are listed on
the first column of Table 2. On the other hand, SNIa rates have the following
prescriptions:

$\rm
R_{SNIa}~{}(SNe~{}Myr^{-1}~{}M_{\odot}^{-1})=\left\\{\begin{array}[]{ll}0&\quad\rm
age<37.53~{}Myr\\\ \rm 5.3\times 10^{-8}+1.6\times
10^{-5}e^{-((age-50)/10)^{2}/2}&\quad\rm age>37.53~{}Myr,\end{array}\right.$

with ejecta mass of $\rm 1.4~{}M_{\odot}$ per SNIa explosion. The element
yields per SNIa explosion are tabulated in the second column of Table 2.
Lastly, mass-loss from OB/AGB stellar winds were included whose yields are
listed on the third column of Table 2. For more details on the stellar
evolution in FIRE-2, we refer to Appendix A in Hopkins et al. (2018).

Table 2: Element yields from different sources. SNe II yields are from Nomoto et al. (2006), SNIa from Iwamoto et al. (1999), and stellar winds from the compilation by Wiersma et al. (2009) of van den Hoek & Groenewegen (1997), Marigo (2001), and Izzard et al. (2004). Yields are in terms of $\rm M_{\odot}$. We note that the N abundance has a metallicity dependence and is multiplied by N where N=max($Z/Z_{\odot}$,1.65). Elements | SNe II | SNIa | OB/AGB winds
---|---|---|---
Fe | 0.0741 | 0.743 | -
C | 0.133 | 0.049 | 0.016
N | 0.0479N | 1.2 $\times 10^{-6}$ | 0.0041
O | 1.17 | 0.143 | 0.0118N
Mg | 0.0987 | 0.0086 | -
Si | 0.0933 | 0.156 | -
S | 0.0397 | 0.087 | -
Ca | 0.00458 | 0.012 | -

##  Appendix B Optimal age-binning

In Section 4, we determined the optimal binning in age to look at the $\rm
R_{birth}$, current R, [Fe/H] and [X/Fe] distributions as a function of age.
We started with a bin size of $\pm$ 0.5 Gyr and scrutinized the [Fe/H] vs $\rm
R_{birth}$ of the stars encompassed by this selection. We reduce the bin size
by half, and calculate how the dispersion across the linear fit changes. A
decrease in dispersion warrants a further decrease in the age bin; an increase
in dispersion or a sample size $<$50,000 stars stops the iteration. In
general, we define the age bin as sufficient when it is (1) not too wide such
that the measured dispersion in the [Fe/H] vs $\rm R_{birth}$ trend is
independent of the binning and (2) not too narrow to be affected by small
number statistics.

Figure 13 illustrates this process for 5 Gyr old stars, wherein we show their
2D histogram, with the best fit line for [Fe/H] vs $\rm R_{birth}$, and the
running dispersion around this trend. We also note the number of stars
included in the bin in the bottom left corner and the median running
dispersion in the top right corner.

As we reduced the age bin size from 0.5 Gyr, the dispersion in [Fe/H] vs $\rm
R_{birth}$ also continue to decrease until we reach 0.03 Gyr. At this point,
the nth iteration has a small sample size that drives up the dispersion (e.g.
0.147 dex). We therefore choose the age binning from the n-1 iteration which
gives us the smallest dispersion in the [Fe/H] vs $\rm R_{birth}$ diagram. We
perform a similar exercise for the other ages included in the analysis.

Figure 13: [Fe/H] vs $R_{birth}$ for different age binning. Each panel shows
this trend for different age bins of 5 Gyr old stars. We mark the best fit
line (cyan) and the running dispersion around this trend (gray shaded region),
and also note the number of stars included in the selection in the bottom left
corner and the median of the running dispersion on the top right corner.

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