# Universal properties of the high- and low$-\alpha$ disk: small intrinsic
abundance scatter and migrating stars

Yuxi(Lucy) Lu Department of Astronomy, Columbia University, 550 West 120th
Street, New York, NY, USA  American Museum of Natural History, Central Park
West, Manhattan, NY, USA  Melissa Ness  Department of Astronomy, Columbia
University, 550 West 120th Street, New York, NY, USA  Tobias Buck  Leibniz
Institute for Astrophysics Potsdam, An der Sternwarte 16, 14482 Potsdam,
Germany  Joel Zinn  NSF Astronomy and Astrophysics Postdoctoral Fellow
American Museum of Natural History, Central Park West, Manhattan, NY, USA

###### Abstract

The detailed age-chemical abundance relations of stars measures time-dependent
chemical evolution. These trends offer strong empirical constraints on
nucleosynthetic processes, as well as the homogeneity of star-forming gas.
Characterizing chemical abundances of stars across the Milky Way over time has
been made possible very recently, thanks to surveys like Gaia, APOGEE and
Kepler. Studies of the low-$\alpha$ disk have shown that individual elements
have unique age-abundance trends and the intrinsic dispersion around these
relations is small. In this study, we examine and compare the age distribution
of stars across both the high and low-$\alpha$ disk and quantify the intrinsic
dispersion of 16 elements around their age-abundance relations at [Fe/H] = 0
using APOGEE DR16. We find the high-$\alpha$ disk has shallower age-abundance
relations compared to the low-$\alpha$ disk, but similar median intrinsic
dispersions of $\approx$ 0.04 dex, suggesting universal element production
mechanisms for the high and low-$\alpha$ disks, despite differences in
formation history. We visualize the temporal and spatial distribution of disk
stars in small chemical cells, revealing signatures of upside-down and inside-
out formation. Further, the metallicity skew and the [Fe/H]-age relations
across radius indicates different initial metallicity gradients and evidence
for radial migration. Our study is accompanied by an age catalogue for 64,317
stars in APOGEE derived using The Cannon with $\approx$ 1.9 Gyr uncertainty
across all ages (APO-CAN stars) as well as a red clump catalogue of 22,031
stars with a contamination rate of 2.7%.

Giant stars; Stellar nucleosynthesis; Milky Way Galaxy; Stellar abundances;
Stellar ages; Galactic archaeology

††journal: The Astronomical Journal††facilities: Gaia, Kepler,
APOGEE††software: Astropy (Astropy Collaboration et al., 2013; Price-Whelan et
al., 2018), Numpy (Oliphant, 2006), sklearn (Pedregosa et al., 2011), The
Cannon (Ness et al., 2015), Astraea (Lu et al., 2020)

##  1 Introduction

Large spectroscopic surveys such as Apache Point Observatory Galactic
Evolution Experiment (APOGEE) (Majewski et al., 2017), Large Sky Area Multi-
Object Fibre Spectroscopic Telescope (LAMOST) (Cui et al., 2012), GALactic
Archaeology with HERMES (GALAH) (De Silva et al., 2015; Buder et al., 2019)
and time-domain surveys such as Kepler (Borucki et al., 2010) and TESS (Ricker
et al., 2015) are observing hundreds of thousands of stars. These surveys
provide the data that gives us insight as to the formation and evolution of
the Galaxy as well as the nucleosynthetic channels of chemical enrichment.

The APOGEE survey (Majewski et al., 2017) is an IR survey at R=22,500 that
primarily targets the disk, where the majority of the baryonic matter of the
Milky Way resides (e.g. Bland-Hawthorn & Gerhard, 2016). From the APOGEE
R=22,500 spectra, more than 20 precision element abundances [X/Fe] (García
Pérez et al., 2016a; Ahumada et al., 2020), imprecise spectroscopic ages (e.g.
Leung & Bovy, 2019; Ness et al., 2016; Martig et al., 2015), and precision
distances (e.g. Leung & Bovy, 2019; Hogg et al., 2019) can be determined.
Time-domain missions, most notably to date the Kepler survey (Borucki et al.,
2010), are enabling precision ages to be determined via asteroseismology by
examining internal oscillation frequencies of stars. A population of Kepler
red giants with both asteroseismic data and APOGEE spectra are collated in the
APOKASC catalogue (Pinsonneault et al., 2018). This provides 2,616 precise
asteroseismic ages. This catalogue has proven to be a useful benchmark for
building larger catalogues of stellar ages using machine learning (e.g., Ness
et al., 2016, 2019; Mackereth et al., 2019).

Using both 1) small local benchmark samples of stars with high precision
abundances and ages from stellar spectra and astroseismology and 2) large
samples of stars with high precision abundances and imprecise ages across the
Galactic disk allows for testing the chemical enrichment of the disk over wide
ranges in time and spatial position. With these data, we can examine global
age distributions of stars across the Milky Way as well as the temporal and
spatial properties of stars with different chemical compositions. Globally,
this can link the star formation history to the galaxy formation history and
reveal evolutionary processes at work like radial migration (e.g. Roškar et
al., 2008).

The spectroscopic age distributions built using large surveys have shown the
detailed mapping from the old populations in the inner Galaxy, to the young
populations in the outer disk (e.g. Ness et al., 2016; Martig et al., 2016;
Bovy et al., 2019; Bensby et al., 2017). Further, younger stars are clearly
concentrated to the plane of the disk and old stars at larger heights with
flaring across radius (e.g. Mackereth et al., 2019; Martig et al., 2016). The
age gradient indicates an inside-out formation for the Galaxy. Although stars
also evolve from their birth sites over time, stellar ages have been used to
model the so called radial migration across part of the disk, which has been
determined to be strong (Frankel et al., 2018, 2019). To connect the star
formation environment and history to formation and evolutionary processes like
radial migration, we need to explore age-individual chemical abundance
relations at different locations of the disk.

Age-individual chemical abundance trends at fixed metallicity find utility as
chemical clocks, via which we can understand:

  * •

Nucleosynthesis processes: Different elements are believed to be produced in
different processes on different timescales. For example, light elements such
as C and N are produced in large part during the phase of asymptotic giant
branch (AGB) stars; iron-peak elements (e.g. V, Cr, Mn, Ni, Co) are produced
mostly by type Ia supernovae; $\alpha$-elements (e.g. O, Mg, Si, S, and Ca)
derive from core-collapse supernovae. Many elements are produced by multiple
channels are have both both mass and metallicity dependent yields (more
detailed description and references see Kobayashi et al., 2020). These
complicated nucleosynthesis processes and stellar yields are in detail based
on many approximations and estimates. By studying the age-chemical abundance
trends, one can learn the chemical yields as informed by the data, and
constrain the theoretical models (Rybizki et al., 2017).

  * •

Formation processes in the Milky Way: By combining the insights from chemical
clocks with spatial and kinematic properties of stars across the Galaxy, we
can study how the disk has formed and evolved subsequently through radial
migration (e.g. Frankel et al., 2018, 2019). Ultimately we can use this
information in combination with simulations to link the current day properties
of stars to their birth location and environments.

Chemically, the disk is broadly characterised by the presence of a high and
low-$\alpha$ sequence of stars. The [$\alpha$/Fe]-[Fe/H] bi-modality was
discovered by Fuhrmann (1998). The stars in the high-$\alpha$ disk are
predominantly old and those in the low alpha disk are predominantly young
(e.g. Bensby et al., 2014). This bi-modality has been linked to the structural
“thin” and “thick” disk Gilmore & Reid (1983), and certainly, the
[$\alpha$-Fe] versus [Fe/H] plane changes dramatically with spatial position
over the Galaxy (Nidever et al., 2014; Hayden et al., 2015). However, as
pointed out in Bland-Hawthorn et al. (2019), a star’s kinematic changes
throughout its lifetime but not its chemistry. As a result, if it is
advantageous to break up the disk into constituents to study it, it is often
desirable to divide it in the chemical rather than the dynamical plane. The
high and low-$\alpha$ disks have different element abundance ratios in a
multitude of elements, indicative of their different star formation histories
(e.g. Bensby et al., 2014; Masseron & Gilmore, 2015). Recent work leveraging
large data shows that the high and low-$\alpha$ sequence appear to have
different dynamical properties, at all ages (Mackereth et al., 2019; Gandhi &
Ness, 2019).

Different hypothesis have been proposed for the formation of the
$\alpha$-bimodality. Vertical disk heating driven by an encounter between the
Milky Way and satellite galaxies Quinn et al. (1993) and the accretion of
satellite stars Abadi et al. (2003) has been invoked as potential culprits.
More recent simulations demonstrate other scenarios such as clumpy formation
to form the high-$\alpha$ disk (e.g. Clarke et al., 2019; Debattista et al.,
2019) or gas accretion to form the low-$\alpha$ sequence (Agertz et al., 2020;
Buck, 2020). Regardless of the mechanisms via which the high and low-$\alpha$
disk were respectively formed, and whether or not they represent a shared or
separate star formation history, the empirical differences in chemical
composition and dynamical properties lead us to study these two “populations”
separately.

In this paper, we adopt an ad-hoc separation of the high- and low-$\alpha$
disk for part of our analysis. However, we go beyond this dichotomy and
explore the characteristics of the disk across a grid of chemical cells in the
[Fe/H]-[$\alpha/Fe]$ plane (see section 3). This is perhaps are far more
powerful approach to study the disk. Indeed it is now readily enabled with
large samples of stars from surveys like APOGEE. A similar line of analysis
was already suggested by (Bovy et al., 2011). Under this decomposition
approach they reported a a continuous and monotonic distribution of disk
thicknesses rather than a bi-modal disk. Nevertheless, the visual appearance
of an high-$\alpha$/low-$\alpha$ bimodality in [Fe/H]-[$\alpha$/Fe] space is a
prediction of several models. In this sense, the bimodality is broadly
indicative of different formation mechanisms of the two populations, even if
an exact, simple division between the apparent populations in chemical space
may be undesirable for a number of analyses.

We first examine the global properties of the high- and low-$\alpha$ disk. We
investigate how their mean age distributions change spatially and chemically
(how the mean age distribution changes at different location in the
[$\alpha$/Fe]-[Fe/H] plane). We then investigate the overall age-metallicity
relation (where by metallicity we refer to [Fe/H]), which can place broad
constraints on galactic and chemical evolution of the Galaxy (e.g. Edvardsson
et al., 1993; Casagrande et al., 2011). Studies have revealed a large range of
stellar ages at any fixed metallicity throughout the disk (e.g. Feuillet et
al., 2019; Jönsson et al., 2020), showcasing that [Fe/H] itself is not a
chemical clock. Finally, we quantify the relationship between ages and
individual chemical element abundances using red clump stars. We identify the
red clump stars in the APOGEE catalogue from their spectra, using data-driven
modeling of the correlation between flux variability and evoluionary state
(Hawkins et al., 2018). Red clump stars provide a narrow region in
evolutionary state, thus mitigating systematic imprints of abundance variation
in the data (Jofré et al., 2019). Furthermore, they enable precision distance
estimates across a large radial extent.

Specifically, we examine the age-abundance properties for 16 elements for
stars of the low- compared to the high$-\alpha$ disk. Studies of detailed age-
individual chemical abundance relations - at fixed metallicity - have
previously found low intrinsic dispersion for stars around these relations
(e.g. Ness et al., 2019; Bedell et al., 2018; Sharma et al., 2020; Hayden et
al., 2020). However, these studies have focused on all the stars in the disk
or only stars in the low-$\alpha$ disk. The small intrinsic dispersion around
the individual age-abundance relations ($\approx$ 0.02 dex on average for the
low$-\alpha$ disk) implies we can use these element abundance trends to age-
date stars.

The age-individual chemical abundance results set out strong constraints on
nucleosynthetic channels and the initial composition of the star-forming gas.
Examining these separately for the high and low-$\alpha$ disk gives us insight
as to which properties are shared and which are distinct across this chemical
plane. This gets toward understanding the relationship between these sequences
and if the high-$\alpha$ disk could be an ancestor of the low, or if its
formation channel must be entirely distinct. In detail, we compare and
contrast the age-abundance relations and the intrinsic dispersions around
these relations. We highlight which elements are most similar between the two
disks and which are least similar. We examine the mean age distributions, both
spatially and in the chemical-plane, and showcase the signatures of radial
migration. Using the age variable in concert with metallicity directly
demonstrates how the formation and evolutionary signatures of the disk are
imprinted in the data. We also provide reader an age catalog for 64,317 stars
from APOGEE DR16 (Ahumada et al., 2020) with an mean age error of 0.25 dex
(APO-CAN stars) as well as a red clump catalog with 22,031 stars with a
contamination rate of 2.7%.

Section 2.1 describes the data used in this project. Section 2.2 details how
we determined ages and red clump membership, and how we separated the high-
and low-$\alpha$ disk. In Section 3.1, we look at the overall age distribution
of the APO-CAN stars from APOGEE DR16 across the Galaxy. In Section 3.2, we
investigate the age distribution of the stars in the chemical plane at
different locations in the Galaxy. Then, we examine the temporal and spatial
distributions of the high- and low-$\alpha$ disk in a grid of chemical cells
across [$\alpha$/Fe]-[Fe/H]. In Section 3.4, we explore the detailed age-
element abundance trends for 16 different elements and the age-metallicity
relation for the high- and low-$\alpha$ disk. Finally, in Section 4, we
compare our results to simulations.

##  2 Data & Methods

We used two data sets for this work – the APOGEE survey DR16 spectra and
abundances data (Ahumada et al., 2020), and the APOKASC catalogue that
contains ages and asteroseismic parameters (Pinsonneault et al., 2018). The
APOGEE spectrograph has a resolution of $R=22,500$ and is mounted on the 2.5-m
telescope of the Sloan Digital Sky Survey (Wilson et al., 2019; Gunn et al.,
2006). For details on the data reduction process, see Nidever et al. (2015).
APOGEE spectroscopic analysis is performed using the APOGEE Stellar Parameter
and Chemical Abundance Pipeline (ASPCAP; García Pérez et al., 2016b), with
temperatures calibrated to the infrared flux method scale of González
Hernández & Bonifacio (2009) (Holtzman et al., 2015).

Figure 1: Histograms of the parameters in our training set of 2,616 stars for
The Cannon.

We combined both data sets to examine the abundance-age relations. The APOKASC
catalogue serves not only as a benchmark data set of stars with precision
ages, but also as a training set for the data driven approach of The Cannon to
estimate ages for the rest of the red giant stars in APOGEE from their spectra
and to identify red clump stars.

We worked in a narrow region of $T_{\mbox{\scriptsize eff}}$-$\log g$ for our
analysis to circumvent systematics (e.g. Jofré et al., 2019) and we selected
the red clump stars for our endeavour as we can determine precise distances
for them to use in our follow on dynamical analyses of these stars and their
age-abundance relations.

In order to create the age and the red clump catalog, we used The Cannon (Ness
et al., 2015)111Available at https://annayqho.github.io/TheCannon/intro.html.
The Cannon is a data driven approach to derive stellar parameters from stellar
spectra. Here we use a quadratic combination of the labels to predict each
pixel of the spectrum, as is consisent with previous implementations (e.g.
Ness et al., 2015; Ho et al., 2017; Casey et al., 2017; Wheeler et al., 2020).

###  2.1 Data

In order to use The Cannon to determine stellar ages and identify the red
clump stars, we needed a high fidelity set of reference objects for training
the model. The Cannon is a tool that determines the relationship between flux
and labels that describe the variability of the flux. Therefore, it is
important to include the labels that describe most of the variability in the
flux, hence we included the set of labels of metallicity,
$T_{\mbox{\scriptsize eff}}$, $\log g$, and [Mg/Fe] in addition to the
asteroseismic parameter and ages that we wished to infer for APOGEE DR16
spectra.

We used measurements of frequency spacing between $p$-modes, $\Delta\nu$, and
period spacing of the mixed $g$ and $p$ modes, $\Delta$P, from Vrard et al.
(2016) for 6,111 Kepler stars. We obtained estimates of ages from the second
APOKASC catalog (Pinsonneault et al., 2018) and parameters of
$T_{\mbox{\scriptsize eff}}$, $\log g$, [Fe/H], and [Mg/Fe] from APOGEE’s DR16
data release (Jönsson et al., 2020). We included $\Delta\nu$ and $\Delta$P
since these astroseismic parameters can be used to better separate the red
clump stars with the red giant branch stars (Bedding et al., 2011; Ting et
al., 2018). After cross-matching these two catalogs, we were able to find
2,616 common stars to construct the training set. Figure 1 shows the parameter
space occupied by the training set.

The distances that we use in our analysis are from StarHorse (Queiroz et al.,
2018), which is a bayesian tool for determining stellar masses, ages,
distances, and extinctions for field stars. To study the detailed age-element
abundance relations, we also included 16 individual element abundances, C, N,
O, Mg, Al, Si, S,K, Ca, Ti, V, Mn, Ni, P, Cr, Co from the APOGEE DR16 catalog,
inferred using ASPCAP 222Available at
https://www.sdss.org/dr16/irspec/spectro_data/.. We removed Na as this showed
anomalous behaviour and indeed in Ness et al. (2019) was unable to be
recovered in cross-validation with The Cannon.

We downloaded APOGEE DR16 spectra from the SDSS-IV Science Archive Server
(SAS)333Avaliable at https://data.sdss.org/sas/.

###  2.2 Methods

####  2.2.1 Creating the age/red clump catalog with The Cannon

For our implementation of The Cannon we use a second-order polynomial to fit
the spectra flux ($F$) for each star, $n$, with labels at each wavelength,
$\lambda$. The labels for each star used in this project, in vector form, is
$l_{n}$ = [$T_{\mbox{\scriptsize eff}}$, $\log g$, [Mg/Fe], [Fe/H], $\Delta$P,
$\Delta\nu$, and $Log_{10}$(age)]. As a result, the model can be described as:

$\begin{split}F_{n\lambda}&=\theta^{0}_{\lambda}\\\
&+\theta^{T_{\mbox{\scriptsize eff}}}_{\lambda}T_{\mbox{\scriptsize
eff}}+...+\theta^{\log_{10}(age)}_{\lambda}\log_{10}(age)\\\
&+\theta^{T_{\mbox{\scriptsize eff}}^{2}}_{\lambda}T_{\mbox{\scriptsize
eff}}^{2}+...+\theta^{\log_{10}(age)^{2}}_{\lambda}\log_{10}(age)^{2}\\\
&+\theta^{T_{\mbox{\scriptsize eff}}\log g}_{\lambda}T_{\mbox{\scriptsize
eff}}\log
g+...+\theta^{\log_{10}(age)\Delta\nu}_{\lambda}\log_{10}(age)\Delta\nu\\\
&+error\end{split}$

In order to infer the stellar parameters, we have to train The Cannon on a set
of reference stars in order to fit for the coefficients, $\theta_{\lambda}$.
To train The Cannon, we first excluded stars that were flagged as “bad” (stars
with ASPCAPFLAG flag 23), and/or had signal-to-noise ratio less than 100. This
left us with 2,480 stars with 7 labels in our training set of stars (parameter
range shown in Figure 1).

To test the performance of The Cannon, we performed a 10-fold cross-validation
test, in which we left 10% of the data untouched and trained the model on the
rest of the 90% and predicted the labels for stars in the rest of that 10%. We
then repeated the same test 10 times with a different 10% of the data left out
(hence called 10-fold). The cross-validation result for stellar age is shown
in Figure 2. We added the cross-validation root mean squared (rms) scatter to
the error estimated from The Cannon to obtain the final systematic age
uncertainty, which yield a median uncertainty of around 1.9 Gyr across all
ages. The rms scatter for other labels is — 0.017 dex for [Fe/H], 26.8 K for
$T_{\mbox{\scriptsize eff}}$, 0.056 dex for $\log g$, 0.028 dex for [Mg/Fe],
40.1 s for $\Delta$P, and 0.6 $\mu$Hz for $\Delta\nu$.

Figure 2: 10-fold cross-validation result for stellar ages estimated with The
Cannon for all 2,480 stars. The ages are in unit of Myr.

After training The Cannon, we applied the trained model to the rest of the
APOGEE DR16 spectra. One caveat of using data-driven method is that we were
not able to (reliably) infer stellar parameters for stars outside of the range
of those values of the training set since the The Cannon extrapolates beyond
the training sample regime. Therefore, we first discarded stars outside of the
training parameter range. We only included stars with $T_{\mbox{\scriptsize
eff}}$ between 4,400 K - 5,200 K, $\log g$ between 2.2 dex - 3.5 dex, and
[Fe/H] between -0.8 dex - 0.5 dex. We also excluded stars with abnormal
element abundances (absolute abundance values $>$ 1 dex) for the 16 elements
we are interested in. This left us with 64,317 stars.

To create the red clump catalog, we follow the method described in Ting et al.
(2018) to select stars with $\Delta$P $>$ 230 s as red clump stars. Figure 3
shows our results. The black dots show all the stars in $\Delta$P-$\Delta\nu$
space and the red clump stars are shown in red. It is clear that $\Delta$P
separates the two types of stars.

Figure 3: $\Delta$P vs $\Delta\nu$ for the APO-CAN stars (black). The red dots
show the 22,031 red clump stars ($\Delta$P $>$ 230 s) in this parameter space.
It is clear that the $\Delta$P separates the red clump stars and the red giant
branch stars.

We calculate the contamination rate following their method, where we measured
the false positive rate by taking the ratio of stars that have predicted
$\Delta$P $>$ 230 s but true $\Delta$P $<230$ s. This yield a contamination
rate of 2.7%, which is extremely similar to that from Ting et al. (2018).
However, we might be underestimating the contamination rate with this method
since by assuming their catalog is the ground truth, we misclassify 13% of the
stars.

####  2.2.2 Separation of the high- and low-$\alpha$ disk

For the purpose of examining the different properties of the high and
low$-\alpha$ stars, we separated the high- and low-$\alpha$ disk with an ad-
hoc line, (0.1$\times$[Fe/H]+0.063) We also explored separating the two disks
with a clustering algorithm described in Ratcliffe et al. (2020) in the
Appendix. Figure 4 shows the APO-CAN stars in the [$\alpha$/Fe]-[Fe/H] plane.
The left plot shows the stars colored by age and the right plot shows the
high- (red) and the low-$\alpha$ (blue) disk. This color code will be used
throughout the paper to distinguish between these two disks. It is clear that
the stars in the high-$\alpha$ disk are on average older than those in the
low-$\alpha$ disk, which is what we expected. Within the low-$\alpha$ disk,
stellar ages increase with [$\alpha$/Fe].

Figure 4: 64,317 APO-CAN stars plotted in [$\alpha$/Fe]-[Fe/H] space. The
black dotted line separates the high- (red) and low-$\alpha$ (blue) disk. The
bi-modal distribution of these two disks and the positive $\alpha$-age
gradient are clearly visible.

##  3 Results

In this section, we first examine the global age and metallicity trends across
the disk. We then explore the age distribution in the chemical plane; first,
across the disk spatially, and then across small cells in [$\alpha$/Fe]-[Fe/H]
(section 3.2). In section 3.4, we report the age-chemical abundance trends for
16 elements and calculate the intrinsic dispersions around these relations. We
examine the high- and low-$\alpha$ disk separately, comparing and contrasting
the relations and their intrinsic dispersions.

###  3.1 The global age/metallicity skew distribution across the Milky Way:
two episodes of star formation

In Figure 5, the top left plot shows the age distribution of the APO-CAN stars
with inferred ages from The Cannon across the APOGEE footprint (mean age: 6.3
Gyr; standard deviation: 3.3 Gyr). These stars range from $R$ = 0 $\sim$ 18
kpc in radius and $|z|$ $<$ 5 kpc in galactic height from the plane. It is
worth pointing out that this range does not reflect the underlying density
distribution of the disk, but only the observing strategy and selection
function of APOGEE. However, even without correcting for the selection
function, we are still able to see clear mean trends across the disk, which
are signatures of its formation. The middle left plot shows the mean ages of
the low-$\alpha$ disk stars (45,983 stars; mean age: 5.4 Gyr; standard
deviation: 2.9 Gyr); and the bottom left plot shows the mean age distribution
of the high-$\alpha$ disk (16,416 stars; mean age: 8.8 Gyr; standard
deviation: 3.1 Gyr). We found that 7% of the $\alpha$-enriched stars are
younger than 5 Gyr, which are also observed by Martig et al. (2015); Chiappini
et al. (2015); Feuillet et al. (2018). Within the low-$\alpha$ disk stars, 15%
are older than 8 Gyr.

Looking first at the age distribution of the low-$\alpha$ disk: Young stars
are concentrated to the mid-plane and stars with larger ages are seen higher
above the mid-plane, as expected (see also Ness et al., 2016; Mackereth et
al., 2017). Looking at the middle left figure, the low-$\alpha$ disk shows
flaring in the young population (Mackereth et al., 2018; Bovy et al., 2016)
and at a given height from the plane, $|z|$, the mean age decreases with
radius, R. The concentration of old stars in the inner region and young in the
outer is indicative of inside-out formation of the Milky Way disk. Looking,
second, at the age distribution of the high-$\alpha$ disk: There are no age
gradients across either R or $|z|$. The high-$\alpha$ disk stars are old even
along the mid-plane and extend to larger heights compared to the low-$\alpha$
stars.

There are a range of ages in the high-$\alpha$ disk. Yet, the absence of any
age gradient suggests a rapid formation history for the high-$\alpha$ disk
where all the stars were formed early, before the stars of the low-$\alpha$
disk.

Next, we examine the spatial distribution of the metallicity skew across the
disk, where the skewness is used as a measure of how far the distribution
deviates from a Gaussian. A positive age skew means there is an of stars with
ages older than the average stellar age in that bin, and visa-versa. The top
panel shows the skew in all stars, the middle in the low-$\alpha$ disk and the
bottom for the high-$\alpha$disk.

Hayden et al. (2015) and also Loebman et al. (2016) highlighted the change in
the direction of the metallicity skewness across radius in the disk as a
possible signature of radial migration. In the presence of migration and an
initial disk metallicity gradient, the distribution can skew in opposite ways
in the inner and outer region, respectively, as stars migrate in and out,
across the disk.

We calculated the skewness of the metallicity distribution in spatial bins
with a bin size of (R, z) = (0.4 kpc, 0.4 kpc) using the Python package
scipy.stats.skew, excluding any bins with fewer than 20 stars.

From the middle right plot, we can see that there is a clear trend from
negative skewness to positive skewness as we move from the inner Galactic disk
to the outer disk in the low-$\alpha$ disk (see also Kochukhov, 2021). This
supports the idea that the disk formed with a negative metallicity gradient,
and radial migration has been significant in the low-$\alpha$ disk.

We note there is a strong negative skewness for the group of stars around R =
17.5 kpc ($\sim$ 140 stars). We further investigated these stars and we did
not find any specific APOGEE programs associated with these stars. This group
of (on average) young stars has an excess number of metal poor stars. We note
that the mean metallicity of these stars is on average higher than that of
stars around R= 12 kpc and is in fact similar to that of stars that are around
R= 5 kpc.

For the high-$\alpha$ disk (bottom right plot), there is a weak positive
metallicity skew across (R,z), but no gradient in the skew across R as seen
for the low-$\alpha$ disk above it. However, this does not indicate that
radial migration is not significant. Rather, this is consistent with the
high-$\alpha$ disk having no initial metallicity gradient, at formation;
subsequently any migration would not affect the metallicity skewness across
radius, in each spatial bin. The lack of metallicity gradient in the
high-$\alpha$ disk is also seen in cosmological simulations (e.g. Agertz et
al., 2020). In the high-$\alpha$ disk there is an overall positive skewness on
the order of 0.19 dex in metallicity. Presumably this places constraints on
the enrichment rate and formation history of the high-$\alpha$ disk.

Figure 5:  The top plots show the overall age distribution/metallcity
skewness, the middle plots shows the age distribution/metallcity skewness of
the low-$\alpha$ disk, and the bottom plots shows the age
distribution/metallcity skewness of the high-$\alpha$ disk. The overall age
distribution of the stars and flaring of the low-$\alpha$ disk stars suggests
an inside-out formation history and the negative skew in metallicity suggests
radial migration (for more explanation, see section 3.1). The lack of any age
gradient across the Galaxy in the high-$\alpha$ disk places constraints on the
enrichment rate and formation history of the high-$\alpha$ disk.

###  3.2 The age distribution of the chemical plane across the Galaxy:
signatures of radial migration and two modes of star formation

We dissect the APO-CAN stars into spatial (figure 6) and chemical (figure 7)
bins in order to piece-wise examine the structure of these two disks.

Figure 6 shows the age distribution of stars in [$\alpha$/Fe]-[Fe/H] plane at
different locations of the Galaxy’s disk. From left to right, the Galactic
radius increases, from $R$ = 0 to 13 kpc, and from bottom to top, the absolute
vertical height increases, from $|z|$ = 0 to 2 kpc, similar to the figure in
Nidever et al. (2014); Hayden et al. (2015) but coloured here by age. Moving
away from the galactic plane, the mean stellar age clearly increases, and in
the inner disk at small Galactic radius, the (older) high-$\alpha$ disk stars
dominate. On the other hand, the (younger) low-$\alpha$ disk dominates nearer
to the mid- plane and in the outer Galactic disk, where low-$\alpha$ stars
with large galactic heights in the outer disk indicate signatures of flaring.
It is clear that a significant mean age gradient is imputed across (R,z)
merely by the changing ratio of the number of (younger) low-$\alpha$ and
(older) high-$\alpha$ stars, respectively.

We qualitatively compare this result to the expectation from simulations
(Figure A1 and B1 top two plots) from Buck (2020). These simulations suggest
that the $\alpha$-bimodality is a generic consequence of a low-$\alpha$ disk
forming gas-rich merger after the high-$\alpha$ disk is in place. The
simulations presented in Buck (2020) reveal a similar age gradient along the
[$\alpha$/Fe] axis in the low-$\alpha$ disk at any spatial bin whilst almost
no age gradient is found for the high-$\alpha$ disk. Overall, in both
simulation and data, there is a strong age gradient with [$\alpha$/Fe] (at
fixed [Fe/H]) and a weak gradient with [Fe/H] (at fixed [$\alpha$/Fe]. The
shape of the distribution of stars with similar ages in each spatial bin in
Figure 6 is inclined with respect to the [Fe/H] axis. Interestingly, the
inclination in the age distribution of the low-$\alpha$ disk as shown across
(R,z) in this figure is only seen in the simulation with a strong bar (Fig. A1
in Buck, 2020).

The simulations further show a similar trend of increasing [Fe/H] with
decreasing radius. Note, this comparison is only of qualitative nature since
no APOGEE selection function nor age uncertainties were taken into account in
their work.

From Figure 6, we see that there is no age gradient for the high-$\alpha$ disk
along either [Fe/H] or [$\alpha$/Fe], at any given location that has been
observed. The absence of any age gradient in the high-$\alpha$ disk across
[Fe/H] or [$\alpha$/Fe] is consistent with the results from Agertz et al.
(2020), in which, similarly to Buck (2020) they connect the last gas rich
merger as a low-$\alpha$ disk formation mechanism around the in-place
high-$\alpha$ disk (Renaud et al., 2020)

Figure 6: The age distribution of the APO-CAN stars in APOGEE DR16 in mono
spatial bins inspired by Figure 4 in Hayden et al. (2015). The radius
increases moving to the right, and the Galactic height increases moving
towards the top. The high-$\alpha$ disk dominates as the Galactic height
increases, and the low-$\alpha$ disk dominates at larger Galactic radius. The
low-$\alpha$ stars at large Galactic height in the outer disk is the signature
of the flaring young disk.

We have examined the mean spatial trends of age across the overall chemical,
[$\alpha$/Fe]-[Fe/H], plane. We now look deeper into the conditions of the
star-forming gas by examining the age distribution across small cells in
[$\alpha$/Fe]-[Fe/H] across the disk. This is related to the analysis route of
Bovy et al. (2011), who examined the scale height and lengths of mono-
abundance populations.

Figure 7 shows the age distribution of the same APO-CAN stars as shown in
Figure 6, in a grid of [$\alpha$/Fe]-[Fe/H] bins. We call these chemical cells
(they are not strictly ‘mono’ abundance populations; the cells are larger than
the errors on the [Fe/H] and [$\alpha$/Fe]). The metallicity, [Fe/H] increases
toward the right and [$\alpha$/Fe] increases upwards. The bin sizes are 0.04
dex in [$\alpha$/Fe] and 0.14 dex in [Fe/H]. Each individual cell shows the
spatial distribution of stars in the $R$-$z$ plane. The blue dashed lines show
the $z$=0 plane and the radius location of the sun.

Across the matrix of chemical cells, we see different portions of the stellar
disk, that together, comprise the full disk stellar distribution. Globally,
the chemical cell projection shows a chemical population of spatially and
temporally distinct disks, that looks to be consistent with an inside-out and
upside-down formation process (Bird et al., 2013). Note, we do not take into
account the selection function. Nonetheless, the age gradient of older to
younger stars from the inner to outer region and spatial flattening across
cells of mean decreasing age is indicative of these processes.

Figure 7 shows that the transition from the young to old stars is marked and
rapid across the chemical plane, in that the transformation happens within a
small range of [Fe/H]-[$\alpha$/Fe], across chemical cells. Outside of these
clear and most dramatic changes across chemical cells, there are subtle
spatial and temporal variations along both rows of [Fe/H] and columns of
[$\alpha$/Fe]. We see using the chemical cells, that away from where there are
strong age gradients across cells, most cells do not show age gradients within
them.

Along fixed rows in ([Fe/H]) the disks are far more similar than across
columns of [$\alpha$/Fe]. Most notably, moving along rows reveals subtle age
gradients and shifting radial distribution of the stars. Along fixed columns
in [$\alpha$/Fe], there are strong changes in the mean age of populations and
more dramatic spatial changes. Looking along say the forth column from left,
the bottom row shows young stars concentrated to the outer disk and
distributed around the mid-plane, and old stars concentrated to the inner
Galaxy and diffusely distributed around the plane at top.

The matrix of the chemical cells reveals an inversion of the age gradient
along [Fe/H] moving from highest to lowest [$\alpha$/Fe]. Looking along the
top five rows, for the chemical cells with highest $\alpha$-enhancement, there
is a mean decrease in age as [Fe/H] increases. Correspondingly, along these
rows, the stars are less diffusely distributed around the plane spatially, and
as metallicity increases the mean radius of the stars increases. Note that in
chemical cell projection, along the bottom rows, the age gradient inverts; the
stars become older moving to increasing [Fe/H]. However, they also are on
average nearer to the Galactic center, which is the reverse spatial trend to
high [$\alpha$/Fe] stars. This is indicative of the initial negative [Fe/H]
gradient in the low-alpha disk. The flattening distribution of stars around
the mid-plane as they become younger across the chemical cells, and at fixed
[$\alpha$/Fe], is indicative of an upside-down formation of an ensemble of
disks represented within the chemical cells. The oldest stars in the disk at
the high [Fe/H] and low [$\alpha$/Fe] have presumably migrated from the inner
disk, which would explain this age gradient inversion and change in spatial
trend compared to the high-$\alpha$ rows of stars.

We repeated this analysis using the ages included in the APOGEE DR16 release,
derived using a neural network (Mackereth et al., 2019). We found a small but
important difference in doing this: the most metal-poor high-$\alpha$ stars
are younger than the more metal-rich high-$\alpha$ stars using the ages
provided in Mackereth et al. (2019). This is the opposite to what we find
using our age catalogue and this instead suggests a outside-in formation of
the high-$\alpha$ disk. The differences in the results from our inference
using The Cannon and the neural network approach are likely a consequence of
sparse training data in this chemical realm. This small discrepancy has
significant implications for the formation history inferred under the
different catalogues for the high-$\alpha$ stars.

We also investigated the skew and standard deviation of ages in this plane and
found the high-$\alpha$ disk has the strongest negative age skew, and the
low-$\alpha$ disk has the strongest positive age skew (see Figure C.1 in
Appendix). The largest age dispersion is seen for the low $\alpha$ and metal
rich stars (see Figure C.2 in Appendix). This large dispersion could support
the two in-fall analytic model described in Spitoni et al. (2021). Overall,
Figure 7 suggests the disk can be well considered as a continuum of
populations, rather than two distinct populations. The nature of the changing
age distribution in chemical cells resonates with the scale height and scale
length analysis of Bovy et al. (2012).

Figure 7: Spatial age distribution for all APO-CAN stars in mono
[$\alpha$/Fe]-[Fe/H] bins. The mean values of [$\alpha$/Fe] and metallcity for
each subplot are displayed along the axis, and the bin sizes are 0.039 dex and
0.14 for [$\alpha$/Fe] and metallcity, respectively. The metallicity increases
towards the right, and stars are more $\alpha$ enhanced towards the top. The
blue dashed line mark the $z$=0 and solar radius. Every column is at a fixed
[Fe/H], and every row is at a fixed [$\alpha$/Fe]. The transition from low- to
high-$\alpha$ disk as the $\alpha$ element increases is gradual in the
chemical cells, which means in order to understand the formation of these two
disks in the chemical plane, one should consider a smooth transition function
instead of separating them into two disks. The skew and dispersion of ages in
each bin are shown in Figure C.1 and Figure C.2.

Now that we have examined the age and spatial distribution of our stars in
chemical cells, we turn to the age-metallicity relations of stars and their
detailed age-abundance trends for 16 individual elements. In doing so we seek
to quantify the star formation environment in chemical enrichment over time
across the disk.

###  3.3 Age-metallicity trends in the high- and low-$\alpha$ disk

We now explore the age-metallicity relations for our sample across its spatial
extent (R, z). This is similar to the analysis of Feuillet et al. (2019), who
examined the age-metallicty relation for stars in their sample at different
Galactic radius and heights (Figure 3 in their paper). They subsequently
argued the trends to be signature of radial migration (see Minchev et al.,
2013; Buck, 2020). We employ this same analysis but we separate the stars into
high and low-$\alpha$ populations. Since high and low-$\alpha$ stars have
different initial conditions, spatial and temporal distributions, it is a
natural next step to examine the age-[Fe/H] relations conditioned on narrower
regions of chemical space (including beyond our bi-modal population model
here). This should place stronger empirical constraints on the formation and
evolution mechanisms of the disk - and here reveals interesting differences
between the high and low-$\alpha$ sequences.

Figure 8 shows the age-metallicity relation in (four) different spatial bins,
separated by the high- (red) and low-$\alpha$ (blue) disk for all the 64,339
stars. This figure spans a radial range from $R$ = 5 kpc to $R$ = 13 kpc, and
3 different Galactic height bins, from $|z|$ = 0 kpc to $|z|$ = 2 kpc. In each
spatial bin, we calculated the mean age in each small metallicity bin. These
50 metallicity bins range from -0.8 to 0.5 dex so that each bin spans 0.027
dex. We then calculated the standard error on the mean by
$\sigma_{age}$/$\sqrt{N-1}$, in which $\sigma_{age}$ is the standard deviation
of the ages in each bin, and $N$ is the number of stars in that bin, to be the
uncertainty on the mean stellar ages. The lines connecting the mean age bins
vertically are generated by running a 1D Gaussian filter with a kernel size of
3 using the scipy.ndimage.gaussian_filter function from scipy (Virtanen et
al., 2020).

The black dots and dashed line show the overall trends for the entire sample.
The average age for stars in each spatial bin is shown in the legend.

Figure 8: Age-metallicity relations from all APO-CAN stars (black),
high-$\alpha$ disk stars (red), and low-$\alpha$ disk stars (blue) in
different spatial bins. Each point is the mean age in that metallicity bin
(width of 0.027 dex), and the errorbars are calculated as
$\sigma_{age}$/$\sqrt{N-1}$, where $\sigma_{age}$ is the standard deviation of
the ages in each bin, and $N$ is the number of stars in that bin. The dashed
lines are calculated using a 1D Gaussian filter
(scipy.ndimage.gaussian_filter, 2020SciPy) with a kernel size of 3. There are
significant differences between the age-metallicity relations for the high-
and low-$\alpha$ disks.

Compared to Figure 3 from Feuillet et al. (2019), in which they measured the
age-metallicity trends for stars in the APOGEE DR14, our overall trends show
very similar results throughout the Galaxy. One of the main features in these
relations is the primary turnover point in the age-metallicity trends, which
is a presumed marker of radial migration (see Frankel et al., 2018, 2019).
These turnover points locate the oldest/youngest stars in the trends where the
age-metallicity relations change. For stars with $|z|<$ 0.5 kpc, the turnover
point is at $\sim$ 0.2 dex for 5 $<$ R $<$ 7\. The location of this point
gradually moves towards lower metallicity as the radius increases ($\sim$ 0.1
dex for 7 $<$ R $<$ 9, $\sim$ -0.2 dex for 9 $<$ R $<$ 11, and $\sim$ -0.4 dex
for 11 $<$ R $<$ 13). We found the change in the turning point metallicity
seen in Feuillet et al. (2019) comes from the behaviour of the low-$\alpha$
stars.

To aide interpretation of this figure consider the following: The age-[Fe/H]
relationship in the solar neighbourhood is fairly flat (Nordström et al.,
2004; Ibukiyama & Arimoto, 2002, e.g.). Thus, [Fe/H] is not an age indicator,
and presumably stars in the solar neighbourhood have a large range of initial
birth radii. Any given spatial location will comprise stars born across the
disk and at different times. Nucleosynthetic enrichment processes increase
overall metallicity over time. Assuming an initial radial metallicity gradient
in the Galaxy and self-enriching disk, this means that the first born, oldest
stars would have the lowest metallicities compared to the younger stars at a
fixed birth radius (assuming the [Fe/H] is a birth property and stays
relatively constant throughout the stars’ lifetime). As a result, without
stars migrating throughout the Galaxy, the metallicity should monotonically
decrease with increasing stellar age, which is expected from temporal gas
enrichment.

However, we see turning points where the age-metallicity relation has changed,
away from a presumed fiducial monotonic decrease. Under this scenario, the
most metal rich old stars in the sample that are present across all radii must
have migrated from the inner region where the gas was more enriched than the
stars formed at the same time at larger radii in the disk.

The most striking result in this figure is the differences in the low- and
high-$\alpha$ age-metallicity relations. The high-$\alpha$ and low-$\alpha$
disk have different mean ages, but also show different locations of the
age-[Fe/H] turning points across the galactic disk. The difference in the
turning point location between the low- and high-$\alpha$ disk suggests that
the high-$\alpha$ disk has had different mean migration directions and/or
initial age-metallicity gradient in the star-forming gas, compared to the
low-$\alpha$ disk. Unlike the low-$\alpha$ disk where the turning points are
at lower metallicity moving away from the Galactic center, suggesting an
initial negative metallicity gradient in the gas, the high-$\alpha$ disk has
age-[Fe/H] turning points roughly at the same metallicity at all locations
across the Galaxy. This suggests the absence of any initial metallicity
gradient in this population.

The age-[Fe/H] turning point in the high-$\alpha$ disk does suggest that
radial migration is a relevant evolutionary process in the high-$\alpha$ disk
too. The empirical role and details of radial migration have been
characterised to date only for the low-$\alpha$ disk (e.g. Frankel et al.,
2018, 2019). However, the role of radial migration in the high-$\alpha$ disk
has been recognised really only within simulations (eg Roškar et al., 2008;
Minchev et al., 2012; Buck, 2020; Khoperskov et al., 2020).

One peculiar feature of Figure 8 is the reversal of the turning point in the
high-$\alpha$ disk for 7 kpc $<$ R $<$ 9 kpc and $|z|<$ 0.5 kpc as well as the
reversal of that of the low-$\alpha$ disk at large $|z|$. One explanation is
that the method of using a line to distinguish the high- and low-$\alpha$ disk
is inappropriate. Ratcliffe et al. (2020) suggested a clustering algorithm to
separate the high- and low-$\alpha$ disk which resulted in a very different
separation compared to that of a line. We will discuss evidence that this
formalised method of assigning stars to groups using their chemical similarity
may be preferable than a by-eye division in the Appendix.

###  3.4 The intrinsic dispersion of elements around their age-abundance
trends

Figure 9: The age-abundance distributions and best-fit second-order polynomial
relations for 16 elements as well as metallicity and [C/N] for the 1,651
low-$\alpha$ and 224 high-$\alpha$ disk red clump stars, with ages determined
from The Cannon. The shaded area shows the 1-$\sigma$ dispersion of the
trends, and the typical error bars are shown in the bottom right corner. Note
that the $y$-axis scales are different to show the details in the age-
abundance relations. It is clear that the high- and low-$\alpha$ disks have
different age-abundance trends for most elements besides [C/N], [N/Fe],
[V/Fe], and [Cr/Fe] ([Na/Fe] is excluded due to high systematic uncertainty).
The age-metallicity relations are similar between the two disks ensure the
differences in their relations are not caused by the metallicity.

In previous sections, we looked at the overall age distribution for different
spatial and chemical bins as well as the age-metallicity relation separated by
the two disks. We now examine the enrichment channels for the high-$\alpha$
and low-$\alpha$ disk. Specifically we look to see if they are similar or show
marked differences.

Since the metallicity of a star can significantly impact the element
abundances (see Jofré et al., 2019), we constrain our analysis to a narrow
range in [Fe/H]. We look into detailed age-element abundance relations at a
single reference solar metallicity, [Fe/H] = 0. We examine these relations for
16 elements (C, N, O, Mg, Al, Si, S,K, Ca, Ti, V, Mn, Ni, P, Cr, Co) observed
by APOGEE. They mostly belong to the following nucleosynthetic families —
Iron-peak: V, Mn, Cr, Ni, Co, Sc; $\alpha$-element: O, Mg, Si, Ti, Ca; Odd-z:
Al, K, P and light: C, N

We measure the age-abundance trends for each element at solar metallicity and
calculate the intrinsic dispersions ($\sigma_{int}$) around these relations
(Ness et al., 2019). That is, the scatter around the age-individual abundance
trends not accounted for by the measurement errors.

Previous work has done this for the low-$\alpha$ disk (eg Bedell et al., 2018;
Ness et al., 2019) and a small $\sigma_{int}$ has been found for almost all
the elements. Recent work using GALAH has found small $\sigma_{int}$ combining
both high- and low-$\alpha$ populations together (Hayden et al., 2020; Sharma
et al., 2020) In this section, we will investigate and compare the trends in
the age-individual element relations and $\sigma_{int}$ for the low- and
high-$\alpha$ disk.

We use two sets of stars for this (i) the benchmark set of asteroseismic stars
identified as red clump stars with 1,261 high-$\alpha$ and 3,173 low-$\alpha$
stars, with typical age errors of 0.88 Gyr and (ii) the red clump stars
identified as per section 2, with 5,290 high-$\alpha$ and 16,784 low alpha
stars respectively, with typical age errors of 3.4 Gyr.

To do so, we selected stars with asteroseismic ages from Pinsonneault et al.
(2018) as well as red clump stars with solar metallicity ($|$[Fe/H]$|<$ 0.05
dex). We also restricted to stars with metallicity error $<$ 0.03 dex. We
excluded 11 high-$\alpha$ and 12 low-$\alpha$ stars from the asteroseismic
sample and 46 high-$\alpha$ and 114 low-$\alpha$ red clump stars with age $>$
10 Gyr. The relations for stars $>$ 10 Gyr are fairly flat (also in part
presumably due to large age errors) and here we characterise the gradient
where a trend is present (see Figure 9 and Ness et al. 2019). We also excluded
stars that have $\chi^{2}>10,000$ (with $\bar{\chi}^{2}>1.17$) between the
spectra model by The Cannon and the real spectra. This leaves us with 642
low-$\alpha$ disk stars, 53 high-$\alpha$ disk stars with asteroseismic ages
and 1,650 low-$\alpha$ disk stars, 224 high-$\alpha$ disks stars with ages
determined from The Cannon.

To make sure there are no systematic temperature dependencies for the
abundances, we examined the abundances of these stars versus stellar age, as a
function of temperature. We see no temperature gradients along the abundance
axis, with the exception of the element V. This could indicate bias in the
measurement of V, as a result, we excluded this element in analyzing the
intrinsic dispersion and the slope of the age-abundance relations. For each
element we determine the age-abundance relations, using both samples of stars.

The results for the solar metallicity red clump stars are shown in Figure 9.
We show lines fit using a second order polynomial, to quantify these age-
abundance relations. The blue and red lines show the fit to the low- and
high-$\alpha$ stars, respectively. The shaded region represent the total
dispersion around the relations, and the typical error bar for each element is
shown in the bottom right corner. In general, the average abundances for the
high-$\alpha$ disk are higher than those of the low-$\alpha$ disk and the
relations between the two disks are different apart from [C/N], [N/Fe], [V/Fe]
and [Cr/Fe]. The differences in the age-[Mg/Fe] trends for the high- and
low-$\alpha$ disk is also observed in Kochukhov (2021). The [Fe/H]-age
relations are similar between the two disks ensure the differences in the
other abundance relations are not caused by the difference in metallicity.

We also calculated the slopes of these trends by fitting straight lines
through the age-abundance relations for the red clump stars. However, the
relations in Figure 9 suggests the linear relation is in log(age)-abundance
space ([X/Fe]=a log(age)+b), but we calculated the slope in linear age space
to compare our results with literature values from Bedell et al. (2018); Ness
et al. (2019).

The results are shown in Figure 10, in which the red dots represent the slopes
for the trends in the high-$\alpha$ disk, the blue dots represent those for
the low-$\alpha$ disk, the black circles show the results from Bedell et al.
(2018), and the black squares show the results from Ness et al. (2019). The
elements are arranged so that the average absolute slope of the two sequence
decreases towards the right of the $x$-axis. We excluded Na and V from our
calculation as we suspect systematic bias in the measurements as previously
discussed.

The uncertainties (shown as the shaded area) were measured by perturbing each
point within its uncertainties both in age and abundance, fitting a new line
each time. We recalculated the slope 500 times weighting in the uncertainties
of the data, and the uncertainty was then determined by the standard deviation
of all these 500 measurements.

We also tested how the slope varies across Galactic radius by calculating the
age-abundance relation slope in bins of 1 kpc width, between R = 8 kpc to 13
kpc. In doing this, we found no significant spatial variation of the age-
abundance slopes. Thus, we conclude the age-individual abundance relations are
global across the disk, conditioned on [Fe/H]; note that they change across
[Fe/H] so would presumably change spatially if not examined at this fixed
metallicity (see Ness et al., 2019).

Figure 10: Slopes of the detailed age-element abundance relations for solar
metallicity stars [Fe/H] = 0 $\pm$ 0.05 dex, ordered by the average absolute
slope of the high- and low-$\alpha$ disk (from largest to smallest). The
shaded area represents the $1-\sigma$ confidence of the measurement of the
slope, determined by calculating the $1-\sigma$ dispersion in the best-fit
slope from 500 re-draws of the measurements from their age and individual
abundance errors, for each element. The slope is calculated using a straight
line. Results from the main sequence study of Bedell et al. (2018) and Ness et
al. (2019) are included for comparison. Note these two studies are for the
low-$\alpha$ stars only.

The slopes between the high- and low-$\alpha$ disk stars are similar for some
elements (e.g. N, V, Cr, Mn) and quite different for others (e.g. Al, Mg, O,
C). We expected the slopes for the low-$\alpha$ disk to be slightly dissimilar
from the results shown in Bedell et al. (2018) since they used dwarf stars
around solar temperature as opposed to giant stars in this work. Differences
may hint at the contribution of stellar versus galactic chemical evolution.
Similarly, the slopes are very similar to those reported in Ness et al.
(2019), comparing the low-$\alpha$ disk results.

We then calculated the $\sigma_{int}$ using the method described in Ness et
al. (2019),

$\sigma_{int}^{2}=\sigma_{tot}^{2}-\sigma_{measure}^{2},$ (1)

where $\sigma_{tot}$ is the total dispersion, measured by calculating the
dispersion around the best-fit 2nd order polynomial, and $\sigma_{measure}$ is
the measurement dispersion. The measurement dispersion is estimated by
perturbing the abundance and age of each point within their uncertainties and
then measure the dispersion around the original determined 2nd order
polynomial. We perform this 100 times and $\sigma_{measure}$ was then taken to
be the standard deviation of the 100 dispersion measurements.

Figure 11 shows the intrinsic dispersion for the stars with asteroseismic ages
from Pinsonneault et al. (2018) (circles) and ages determined from The Cannon
(squares), separated by the high- (red) and low-$\alpha$ (blue) disk. The
dashed lines show the median dispersion for the high- (red) and low-$\alpha$
(blue) disk and the bars are the mean abundance error for the high- (red) and
low-$\alpha$ (blue) red clump stars. Our intrinsic dispersion measurements are
very similar to those of Bedell et al. (2018) (and Ness et al. (2019) who also
found consistent results), with the exception of V, Na, and Co. The slopes of
the age-abundance relations for the high- and low-$\alpha$ disk are very
similar, and are not correlated with the intrinsic dispersion.

Figure 11: Intrinsic dispersion at solar metallicity, [Fe/H] = 0 $\pm$ 0.05
dex, for each element in the high- (red) and low-$\alpha$ (blue) disk for both
stars with asteroseismic ages from Pinsonneault et al. (2018) (circles) and
ages determined from The Cannon (squares). The black dots are results from
Bedell et al. (2018); the bars show the mean abundance error for the high-
(red) and low-$\alpha$ (blue) red clump stars; and the dashed lines show the
median intrinsic dispersion of the elements, which is 0.039 and 0.035 for the
high- and low-$\alpha$ sequence respectively. These indicate that there is no
obvious correlation between measurement error on the elements and the
calculated intrinsic dispersion. Elements are ordered by the average
dispersion of the asterosismic low-$\alpha$ stars and the red clump
low-$\alpha$ stars.

We found very similar intrinsic dispersion between the high- and the
low-$\alpha$ disk, with a median of 0.039 dex and 0.035 dex, respectively.
Note Vincenzo et al. (2021) also calculated the intrinsic dispersion for
[Mg/Fe] and found a value of $\sim$ 0.04 dex for both the high- and
low-$\alpha$ sequence. The slightly higher dispersion in the high-$\alpha$
disk might simply indicate a faster rate of enrichment which leads to more
variability in the range of each element’s abundance at any given age, at
fixed metallicity [Fe/H].

The low intrinsic dispersion that we report here suggests we should be able to
determine ages using the detailed age-element abundance relations for both the
low- and the high-$\alpha$ disk (see also Hayden et al., 2020; Sharma et al.,
2020). Further, our result hints at universal chemical enrichment processes
that have given rise to the abundance distributions of the high- and
low-$\alpha$ disk. The small variation in the intrinsic dispersions indicate
subtle differences in the nucleosynthesis processes that are taking place.

Figure 12 shows the absolute difference of $\sigma_{int}$ between the high-
and low-$\alpha$ disk for the red clump stars. Elements with small
$\Delta$$\sigma_{int}$ are mostly iron-peak elements, and those with large
$\Delta$$\sigma_{int}$ are mostly odd-z elements.

Figure 12: Absolute difference of the intrinsic dispersion around the age-
abundance relations $\sigma_{int}$ between the high- and low-$\alpha$ disk for
the red clump stars at solar [Fe/H] = 0 $\pm$ 0.05 dex. These are ordered by
increasing difference in the intrinsic dispersion.

##  4 Discussion & future work

Large spectroscopic surveys such as Apache Point Observatory Galactic
Evolution Experiment (APOGEE) (Majewski et al., 2017), Large Sky Area Multi-
Object Fibre Spectroscopic Telescope (LAMOST) (Cui et al., 2012), GALactic
Archaeology with HERMES (GALAH) (De Silva et al., 2015; Buder et al., 2019)
and time-domain surveys such as Kepler (Borucki et al., 2010) and TESS (Ricker
et al., 2015) are observing hundreds of thousands of stars. These surveys
enable us to test galaxy formation and evolution mechanisms as well as
channels of element production.

By using the APOGEE DR16 spectra, measurements of frequency spacing between
$p$-modes, $\Delta\nu$, and period spacing of the mixed $g$ and $p$ modes,
$\Delta$P, from Vrard et al. (2016), as well as estimation of ages from the
second APOKASC catalog (Pinsonneault et al., 2018) and $T_{\mbox{\scriptsize
eff}}$, $\log g$, metallicity [Fe/H], and [Mg/Fe] from APOGEE’s DR16, we
constructed an age catalogue for 64,317 stars in APOGEE derived using The
Cannon with $\sim$ 1.9 Gyr uncertainty across all ages (APO-CAN stars) as well
as a red clump catalogue of 22,031 stars with a contamination rate of 2.7%.

Combining our catalogs and 16 element abundances (C, N, O, Mg, Al, Si, S, K,
Ca, Ti, V, Mn, Ni, P, Cr, Co) from Ahumada et al. (2020), we concluded several
similarities and differences between the high- and low-$\alpha$ disk:

  1. 1.

Similarities:

     * •

Both disks show evidence for radial migration and upside-down formation
(Figures 5 & 8).

     * •

Intrinsic dispersions around the age-abundance relations are small,
highlighting the universality of temporal chemical enrichment pathways (Figure
11).

     * •

Given large numbers of stars as in our study, the high and low-$\alpha$ disk
can be described within a single framework as an ensemble of temporal and
spatial populations across chemical space that underlie a global disk
distribution. Presumably the gradients in the mean age, dynamics and spatial
extent (and higher order moments of these distributions) afford very strong
constraints on the Galaxy’s formation and evolution (Figure 7, 11).

  2. 2.

Differences:

     * •

Our analysis suggests that the high-$\alpha$ disk had no initial metallicity
gradient, where as the low-$\alpha$ disk formed with a gradient in place
(Figure 5).

     * •

The high-$\alpha$ disk is old and has a larger vertical reach and narrower
radial expanse than the young low-$\alpha$ disk, which has a shorter vertical
reach and extends over a larger radial range (Figure 6, 7).

     * •

There are differences in the age-[Fe/H] relations for the high- and
low-$\alpha$ sequences across (R,z). These suggest distinct rates and/or
directions of radial migration and/or different initial metallicity gradients
with Galactic radius (Figure 8).

     * •

Most of the age-individual abundance relations for the high- and the
low-$\alpha$ disk (Figure 9) show differences, with some exceptions. Some
elements, such as N, show near-identical relations. Other elements, like Mg
and Al, have quite different slopes (see Figure 10).

     * •

Although some elements have identical intrinsic dispersions around the age-
individual abundance relations for the high and low-$\alpha$ disk, on average
this is abour 10% larger in the high-$\alpha$ disk. This hints at either
different star formation efficiencies/rates or the level of mixing of chemical
elements in the star-forming gas (Figure 11, 12).

###  4.1 Comparing with simulations/analytic models

In this paper, we provide a population study of the similarities and
differences between the high- and low-$\alpha$ disks using APOGEE DR16 and
Gaia data. In this section, we will briefly summarise our results in the
context of formation mechanisms seen in simulations and in analytic models.

Figure 5 examines the global age distribution and the metallicity skewness of
the two disks. The skewness gradient for all stars is seen in the simulations
of Agertz et al. (2020), suggesting radial migration.

Figure 6 revealed the age distribution of stars in small spatial bins. The
lack of age gradient in the high-$\alpha$ disk is seen in the simulations of
Agertz et al. (2020), and the strong age gradient in the low-$\alpha$ disk
resembles that of the simulations in Buck (2020). Furthermore, the flat age
gradient in the high-$\alpha$ disk points towards a fast formation mechanism
which is in line with the recent findings of Di Matteo et al. (2020).

Figure 7 showcased the age distribution of stars in different chemical cells
together with their spatial distribution across the disk. In general, older
stars have shorter scale length and extend to larger heights above the plane
compared to younger stars. This is in line with early star formation happening
in a compact turbulent gas disk (e.g. Buck et al., 2020) while the
low-$\alpha$ disk forms later (e.g. Lian et al., 2020). We found the largest
age dispersion appears at the most metal rich and $\alpha$ poor region (see
Figure C.2), supporting the analytic two in-fall model described in Spitoni et
al. (2021). Furthermore, the different age-abundance relations for low- and
high-$\alpha$ stars shown in Fig. 9 might hint towards different formation
mechanisms/time scales or star formation efficiencies and/or the level of
mixing of chemical elements in the star-forming gas. Similar conclusions have
also been made by Nissen et al. (2020) using HARPS data.

Figure 8 shows the age-metallicity relations at different locations in the
disk in (R, z). These results suggest that radial migration has been
significant in both disks. This result is supported by Buck (2020); Khoperskov
et al. (2020) but are in tension with recent formation scenarios without
strong radial migration (Khoperskov et al., 2021).

###  4.2 Limitations

We did not take into account the selection function, this means we are not
able to calculate the scale height or scale length of the two disks. However,
we argue that the major results will not change because of the simplicity of
the APOGEE selection function.

We only examined stars with solar metallicity, [Fe/H] = 0 for the detailed
age-abundance trends. However, by testing the intrinsic dispersion for stars
with metallicity centered around -0.3 dex, we concluded the intrinsic
dispersion does not vary much, although the age-abundance relations are
different for stars with different overall metallicity, [Fe/H].

##  5 Conclusions

With such a large and detailed benchmark comparison of the age-abundance
relations across the chemically defined high- and low-$\alpha$ disk, these
results can potentially constrain nucleosynthesis channels across broadly
different chemical regimes. While the similarity across chemical space in the
intrinsic dispersions in the age-abundance relation indicates the universality
of chemical enrichment, the small element-dependent differences we see are
relevant in informing metallicity dependent yields, and the impact of
different star formation rates and environment (Buck et al. in prep.).

Using the distributions of ages of stars across chemical and spatial cells, we
hope to provide the empirical data to distinguish between different formation
scenarios for the Galaxy, by e.g., comparing to cosmological simulations.
Moving forward, we expect a powerful analysis tool to constrain the origin of
the bi-modality in the disk will be to examine the age, dynamical and spatial
properties of stars as a function of their chemical distributions. To first
order, simulations that reproduce a bi-modality in the [$\alpha$/Fe] plane
must also reproduce the results we see in Figure 7, given equivalent sampling,
to reflect the formation channel(s) that underlie the Milky Way’s current set
of observed properties. To second order, Figure 8 is demonstrative of the
strength of evolutionary processes like radial migration across different
chemical (and correspondingly, initial spatial) spaces. Next generation
surveys (e.g. Kollmeier et al., 2017) will enable this analysis to be taken to
the next level by completing this exploration over a vast expanse of the disk
into the bulge and to sample the disk finely across its temporal and spatial
variables, within true mono-abundance chemical populations.

###  A Predicting stellar ages with Astraea

We also tried to predict stellar ages estimated using The Cannon with Astraea
(Lu et al., 2020)444Avaliable at https://astraea.readthedocs.io/en/latest/..
Astraea uses Random Forest, a machine learning algorithm, to predict label
from features.

We performed a simple cross-match between APOGEE and the GaiaKepler cross-
match catalog555Avaliable at https://gaia-kepler.fun. and found 3,417 stars
with the measurements we needed. The features we trained on are all the 17
element abundances from APOGEE, metallicity, and all the Gaia parameters. The
important features are listed in table A.1.

Feature name |  Description |  source
---|---|---
N_FE |  [N/Fe] |  APOGEE
C_FE |  [C/Fe] |  APOGEE
MG_FE |  [Mg/Fe] |  APOGEE
TI_FE |  [Ti/Fe] |  APOGEE
AL_FE |  [Al/Fe] |  APOGEE
ALPHA_M |  [$\alpha$/Fe] |  APOGEE
TEFF_SPEC |  spectroscopic temperature |  APOGEE
RV_CCFWHM |  radial velocity |  APOGEE
radius_val |  radius value |  Gaia
pmdec |  proper motion in dec |  Gaia
pmra |  proper motion in ra |  Gaia
parallax |  parallax |  Gaia
b |  Galactic latitude |  Gaia
l |  Galactic longtitude |  Gaia
Log(g) |  $\log g$ |  The Cannon
r_est |  estimated distances |  Bailer-Jones et al. (2018)
Table A.1: Stellar parameters used to train Astraea to get stellar ages.

We trained on 80% of the data and predicted the stellar ages for the rest of
the 20%. We were able to predict these ages from The Cannon with a median
relative error of 15%. Figure A.1 shows the “gini” importance (ranges from 0
to 1, where 1 being the most important) of the 18 most important feature in
predicting these ages. This importance can be determined by calculating the
mean decrease in impurity (MDI), which indicates whether a single feature
alone can predict the outcome. For example, if one can predict the stellar
ages of a star just by the effective temperature, then the gini importance for
the effective temperature will be 1.

Figure A.1: The “gini” importance of the features used to predict stellar ages
using Astraea. This indicates that we are able to predict stellar ages with
only a few stellar parameters.

It is not surprising [N/Fe] is the most important feature in determining the
stellar ages, as age-[N/Fe] relation has one of the steepest slopes (see
Figure 10) and smallest intrinsic dispersion (see Figure 11). The fact that
[Na/Fe] is not important in determining the ages despite the fact that it has
the steepest slope further indicates that there is anomaly in this abundance
measurement. The importance of [$\alpha$/Fe] is relatively low, suggesting the
two disks experience similar enrichment process. The distribution of
importance is relatively spread-out (in that several features are relatively
important to predict ages) suggesting determining stellar ages is complicated.
It also suggests stellar ages are indeed related to a sum of complicated
stellar processes such as nucleosynthesis, gravity, kinematics.

Even with a spread in importance, it is still striking that we were able to
predict stellar ages within 20% uncertainty with only a handful of stellar
parameters. This means we should be able to estimate stellar ages for a large
number of field stars fairly straightforwardly with large spectroscopic survey
such as LAMOST and GALAH.

###  B Separating the high- and low-$\alpha$ disk with a clustering algorithm

In this paper, we separated the high- and low-$\alpha$ disk with an ad-hoc
straight line. However, it is not clear whether this is the most appropriate
way to separate the two disks. Particularly since the [$\alpha$/Fe] and
metallicity, [Fe/H] only represent two of the chemical dimensions of a larger
chemical space. Ratcliffe et al. (2020) suggested a clustering algorithm
approach to deconstruct stars of the disk, using the 19 dimensions of
available APOGEE abundances. In their work, they reported that a small group
of the high-$\alpha$ stars was more stronly associated to the low-$\alpha$
disk than the high. We performed the same ward hierarchical clustering as per
Ratcliffe et al. (2020), using sklearn (Pedregosa et al., 2011) with the same
17 elements and [Fe/H] and found the same two-cluster projection in the
[$\alpha$/Fe]-[Fe/H] as in that work. The clustering result is shown in Figure
B.1. The green points are stars that are considered high-$\alpha$ stars with
the line separation used in this paper, but are classified as low-$\alpha$
stars in Ratcliffe et al. (2020).

Figure B.1: The high- (red) and low-$\alpha$ (blue) disk separated using ward
hierarchical clustering. We were able to reproduce the result in Ratcliffe et
al. (2020). The green points are stars that were considered high-$\alpha$ disk
stars with our separation but classified as low-$\alpha$ disk stars with the
clustering algorithm. The separation of the two disk in the age-[$\alpha$/Fe]
plain (right) favors the clustering algorithm.

One line of independent evidence that these nominallly high-$\alpha$ disk
stars could be associated with the low-$\alpha$ disk more than the high, is
from the age-metallicity relations. As pointed out in the section 3.3, we were
not able to explain some of the features in the age-metallicity relations for
the high-$\alpha$ disks. For example, the reversal of the turning point in age
for the spatial bin of 7 $<$ R $<$ 9 and $|z|<$ 0.5. Using the high and
low-$\alpha$ disks defined by hierarchical clustering, we are able to extract
what look to be clearer and more distinctly different relations for the
high-$\alpha$ disk stars across the galaxy.

Figure B.2: Same as figure 8 but the high- and low-$\alpha$ disks are
determined by the clustering algorithm.

We also tried using more than 2 clusters and we see that there only exists two
distinct modes, meaning the age-metallicity relation of the third (and above)
cluster overlaps with that of the low-$\alpha$ disk.

However, note that clustering algorithms, similarly to by-eye designations, do
not offer a “best” solution. They are subject to algorithmic choices. These
results should be interpreted with caution and used to inform what varies
under different analysis choices.

###  C Additional graphs

Figure C.1 and C.2 show the skew and standard deviation of the ages in each
bin using the matplotlib.axes.Axes.hexbin function.

Figure C.1: Same as Figure 7 but plotting the skew in age. Only included bins
with $>$ 10 stars. Figure C.2: Same as Figure 7 but plotting the standard
deviation in age. Only included bins with $>$ 10 stars.

We thank Adrian Price-Whelan and Kate Daniel for helpful discussions. JCZ is
supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under
award AST-2001869. Melissa Ness is supported in part by a Sloan Fellowship.
This work made use of the gaia-kepler.fun crossmatch database created by Megan
Bedell. This paper includes data collected by the Kepler mission. Funding for
the Kepler mission is provided by the NASA Science Mission directorate. This
work has made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. Funding for the Sloan
Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the
U.S. Department of Energy Office of Science, and the Participating
Institutions. SDSS-IV acknowledges support and resources from the Center for
High Performance Computing at the University of Utah. The SDSS website is
www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for
the Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science, Carnegie
Mellon University, Center for Astrophysics — Harvard & Smithsonian, the
Chilean Participation Group, the French Participation Group, Instituto de
Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the
Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the
Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz
Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie
(MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-
Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatário Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber
Spectroscopic Telescope LAMOST) is a National Major Scientific Project built
by the Chinese Academy of Sciences. Funding for the project has been provided
by the National Development and Reform Commission. LAMOST is operated and
managed by the National Astronomical Observatories, Chinese Academy of
Sciences. This research made use of Astropy,666http://www.astropy.org a
community-developed core Python package for Astronomy (Astropy Collaboration
et al., 2013; Price-Whelan et al., 2018).

## References

  * Abadi et al. (2003) Abadi, M. G., Navarro, J. F., Steinmetz, M., & Eke, V. R. 2003, ApJ, 591, 499, doi: 10.1086/375512
  * Agertz et al. (2020) Agertz, O., Renaud, F., Feltzing, S., et al. 2020, arXiv e-prints, arXiv:2006.06008.  https://arxiv.org/abs/2006.06008
  * Ahumada et al. (2020) Ahumada, R., Allende Prieto, C., Almeida, A., et al. 2020, ApJS, 249, 3, doi: 10.3847/1538-4365/ab929e
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Bailer-Jones et al. (2018) Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Mantelet, G., & Andrae, R. 2018, AJ, 156, 58, doi: 10.3847/1538-3881/aacb21
  * Bedding et al. (2011) Bedding, T. R., Mosser, B., Huber, D., et al. 2011, Nature, 471, 608, doi: 10.1038/nature09935
  * Bedell et al. (2018) Bedell, M., Bean, J. L., Meléndez, J., et al. 2018, ApJ, 865, 68, doi: 10.3847/1538-4357/aad908
  * Bensby et al. (2014) Bensby, T., Feltzing, S., & Oey, M. S. 2014, A&A, 562, A71, doi: 10.1051/0004-6361/201322631
  * Bensby et al. (2017) Bensby, T., Feltzing, S., Gould, A., et al. 2017, A&A, 605, A89, doi: 10.1051/0004-6361/201730560
  * Bird et al. (2013) Bird, J. C., Kazantzidis, S., Weinberg, D. H., et al. 2013, ApJ, 773, 43, doi: 10.1088/0004-637X/773/1/43
  * Bland-Hawthorn & Gerhard (2016) Bland-Hawthorn, J., & Gerhard, O. 2016, ARA&A, 54, 529, doi: 10.1146/annurev-astro-081915-023441
  * Bland-Hawthorn et al. (2019) Bland-Hawthorn, J., Sharma, S., Tepper-Garcia, T., et al. 2019, MNRAS, 486, 1167, doi: 10.1093/mnras/stz217
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977, doi: 10.1126/science.1185402
  * Bovy et al. (2011) Bovy, J., Hogg, D. W., & Roweis, S. T. 2011, Annals of Applied Statistics, 5, 1657, doi: 10.1214/10-AOAS439
  * Bovy et al. (2019) Bovy, J., Leung, H. W., Hunt, J. A. S., et al. 2019, MNRAS, 490, 4740, doi: 10.1093/mnras/stz2891
  * Bovy et al. (2012) Bovy, J., Rix, H.-W., Liu, C., et al. 2012, ApJ, 753, 148, doi: 10.1088/0004-637X/753/2/148
  * Bovy et al. (2016) Bovy, J., Rix, H.-W., Schlafly, E. F., et al. 2016, ApJ, 823, 30, doi: 10.3847/0004-637X/823/1/30
  * Buck (2020) Buck, T. 2020, MNRAS, 491, 5435, doi: 10.1093/mnras/stz3289
  * Buck et al. (2020) Buck, T., Obreja, A., Macciò, A. V., et al. 2020, MNRAS, 491, 3461, doi: 10.1093/mnras/stz3241
  * Buder et al. (2019) Buder, S., Lind, K., Ness, M. K., et al. 2019, A&A, 624, A19, doi: 10.1051/0004-6361/201833218
  * Casagrande et al. (2011) Casagrande, L., Schönrich, R., Asplund, M., et al. 2011, A&A, 530, A138, doi: 10.1051/0004-6361/201016276
  * Casey et al. (2017) Casey, A. R., Hawkins, K., Hogg, D. W., et al. 2017, ApJ, 840, 59, doi: 10.3847/1538-4357/aa69c2
  * Chiappini et al. (2015) Chiappini, C., Anders, F., Rodrigues, T. S., et al. 2015, A&A, 576, L12, doi: 10.1051/0004-6361/201525865
  * Clarke et al. (2019) Clarke, A. J., Debattista, V. P., Nidever, D. L., et al. 2019, MNRAS, 484, 3476, doi: 10.1093/mnras/stz104
  * Cui et al. (2012) Cui, X.-Q., Zhao, Y.-H., Chu, Y.-Q., et al. 2012, Research in Astronomy and Astrophysics, 12, 1197, doi: 10.1088/1674-4527/12/9/003
  * De Silva et al. (2015) De Silva, G. M., Freeman, K. C., Bland-Hawthorn, J., et al. 2015, MNRAS, 449, 2604, doi: 10.1093/mnras/stv327
  * Debattista et al. (2019) Debattista, V. P., Gonzalez, O. A., Sanderson, R. E., et al. 2019, MNRAS, 485, 5073, doi: 10.1093/mnras/stz746
  * Di Matteo et al. (2020) Di Matteo, P., Spite, M., Haywood, M., et al. 2020, A&A, 636, A115, doi: 10.1051/0004-6361/201937016
  * Edvardsson et al. (1993) Edvardsson, B., Andersen, J., Gustafsson, B., et al. 1993, A&A, 500, 391
  * Feuillet et al. (2019) Feuillet, D. K., Frankel, N., Lind, K., et al. 2019, MNRAS, 489, 1742, doi: 10.1093/mnras/stz2221
  * Feuillet et al. (2018) Feuillet, D. K., Bovy, J., Holtzman, J., et al. 2018, MNRAS, 477, 2326, doi: 10.1093/mnras/sty779
  * Frankel et al. (2018) Frankel, N., Rix, H.-W., Ting, Y.-S., Ness, M., & Hogg, D. W. 2018, ApJ, 865, 96, doi: 10.3847/1538-4357/aadba5
  * Frankel et al. (2019) Frankel, N., Sanders, J., Rix, H.-W., Ting, Y.-S., & Ness, M. 2019, ApJ, 884, 99, doi: 10.3847/1538-4357/ab4254
  * Fuhrmann (1998) Fuhrmann, K. 1998, A&A, 338, 161
  * Gandhi & Ness (2019) Gandhi, S. S., & Ness, M. K. 2019, ApJ, 880, 134, doi: 10.3847/1538-4357/ab2981
  * García Pérez et al. (2016a) García Pérez, A. E., Allende Prieto, C., Holtzman, J. A., et al. 2016a, AJ, 151, 144, doi: 10.3847/0004-6256/151/6/144
  * García Pérez et al. (2016b) —. 2016b, AJ, 151, 144, doi: 10.3847/0004-6256/151/6/144
  * Gilmore & Reid (1983) Gilmore, G., & Reid, N. 1983, MNRAS, 202, 1025, doi: 10.1093/mnras/202.4.1025
  * González Hernández & Bonifacio (2009) González Hernández, J. I., & Bonifacio, P. 2009, A&A, 497, 497, doi: 10.1051/0004-6361/200810904
  * Gunn et al. (2006) Gunn, J. E., Siegmund, W. A., Mannery, E. J., et al. 2006, AJ, 131, 2332
  * Hawkins et al. (2018) Hawkins, K., Ting, Y.-S., & Walter-Rix, H. 2018, ApJ, 853, 20, doi: 10.3847/1538-4357/aaa08a
  * Hayden et al. (2015) Hayden, M. R., Bovy, J., Holtzman, J. A., et al. 2015, ApJ, 808, 132, doi: 10.1088/0004-637X/808/2/132
  * Hayden et al. (2020) Hayden, M. R., Bland-Hawthorn, J., Sharma, S., et al. 2020, MNRAS, 493, 2952, doi: 10.1093/mnras/staa335
  * Ho et al. (2017) Ho, A. Y. Q., Ness, M. K., Hogg, D. W., et al. 2017, ApJ, 836, 5, doi: 10.3847/1538-4357/836/1/5
  * Hogg et al. (2019) Hogg, D. W., Eilers, A.-C., & Rix, H.-W. 2019, AJ, 158, 147, doi: 10.3847/1538-3881/ab398c
  * Holtzman et al. (2015) Holtzman, J. A., Shetrone, M., Johnson, J. A., et al. 2015, AJ, 150, 148
  * Ibukiyama & Arimoto (2002) Ibukiyama, A., & Arimoto, N. 2002, A&A, 394, 927, doi: 10.1051/0004-6361:20021157
  * Jofré et al. (2019) Jofré, P., Heiter, U., & Soubiran, C. 2019, ARA&A, 57, 571, doi: 10.1146/annurev-astro-091918-104509
  * Jönsson et al. (2020) Jönsson, H., Holtzman, J. A., Allende Prieto, C., et al. 2020, AJ, 160, 120, doi: 10.3847/1538-3881/aba592
  * Khoperskov et al. (2020) Khoperskov, S., Haywood, M., Snaith, O., et al. 2020, arXiv e-prints, arXiv:2006.10195.  https://arxiv.org/abs/2006.10195
  * Khoperskov et al. (2021) —. 2021, MNRAS, 501, 5176, doi: 10.1093/mnras/staa3996
  * Kobayashi et al. (2020) Kobayashi, C., Karakas, A. I., & Lugaro, M. 2020, ApJ, 900, 179, doi: 10.3847/1538-4357/abae65
  * Kochukhov (2021) Kochukhov, O. 2021, A&A Rev., 29, 1, doi: 10.1007/s00159-020-00130-3
  * Kollmeier et al. (2017) Kollmeier, J. A., Zasowski, G., Rix, H.-W., et al. 2017, arXiv e-prints, arXiv:1711.03234.  https://arxiv.org/abs/1711.03234
  * Leung & Bovy (2019) Leung, H. W., & Bovy, J. 2019, MNRAS, 483, 3255, doi: 10.1093/mnras/sty3217
  * Lian et al. (2020) Lian, J., Thomas, D., Maraston, C., et al. 2020, MNRAS, 497, 2371, doi: 10.1093/mnras/staa2078
  * Loebman et al. (2016) Loebman, S. R., Debattista, V. P., Nidever, D. L., et al. 2016, ApJ, 818, L6, doi: 10.3847/2041-8205/818/1/L6
  * Lu et al. (2020) Lu, Y. L., Angus, R., Agüeros, M. A., et al. 2020, AJ, 160, 168, doi: 10.3847/1538-3881/abada4
  * Mackereth et al. (2018) Mackereth, J. T., Bovy, J., Schiavon, R. P., & SDSS-IV/APOGEE Collaboration. 2018, in Rediscovering Our Galaxy, ed. C. Chiappini, I. Minchev, E. Starkenburg, & M. Valentini, Vol. 334, 265–268, doi: 10.1017/S1743921317006627
  * Mackereth et al. (2017) Mackereth, J. T., Bovy, J., Schiavon, R. P., et al. 2017, MNRAS, 471, 3057, doi: 10.1093/mnras/stx1774
  * Mackereth et al. (2019) Mackereth, J. T., Bovy, J., Leung, H. W., et al. 2019, MNRAS, 489, 176, doi: 10.1093/mnras/stz1521
  * Majewski et al. (2017) Majewski, S. R., Schiavon, R. P., Frinchaboy, P. M., et al. 2017, AJ, 154, 94, doi: 10.3847/1538-3881/aa784d
  * Martig et al. (2016) Martig, M., Minchev, I., Ness, M., Fouesneau, M., & Rix, H.-W. 2016, ApJ, 831, 139, doi: 10.3847/0004-637X/831/2/139
  * Martig et al. (2015) Martig, M., Rix, H.-W., Silva Aguirre, V., et al. 2015, MNRAS, 451, 2230, doi: 10.1093/mnras/stv1071
  * Masseron & Gilmore (2015) Masseron, T., & Gilmore, G. 2015, MNRAS, 453, 1855, doi: 10.1093/mnras/stv1731
  * Minchev et al. (2013) Minchev, I., Chiappini, C., & Martig, M. 2013, A&A, 558, A9, doi: 10.1051/0004-6361/201220189
  * Minchev et al. (2012) Minchev, I., Famaey, B., Quillen, A. C., et al. 2012, A&A, 548, A126, doi: 10.1051/0004-6361/201219198
  * Ness et al. (2015) Ness, M., Hogg, D. W., Rix, H. W., Ho, A. Y. Q., & Zasowski, G. 2015, ApJ, 808, 16, doi: 10.1088/0004-637X/808/1/16
  * Ness et al. (2016) Ness, M., Hogg, D. W., Rix, H. W., et al. 2016, ApJ, 823, 114, doi: 10.3847/0004-637X/823/2/114
  * Ness et al. (2019) Ness, M. K., Johnston, K. V., Blancato, K., et al. 2019, ApJ, 883, 177, doi: 10.3847/1538-4357/ab3e3c
  * Nidever et al. (2014) Nidever, D. L., Bovy, J., Bird, J. C., et al. 2014, ApJ, 796, 38, doi: 10.1088/0004-637X/796/1/38
  * Nidever et al. (2015) Nidever, D. L., Holtzman, J. A., Allende Prieto, C., et al. 2015, AJ, 150, 173
  * Nissen et al. (2020) Nissen, P. E., Christensen-Dalsgaard, J., Mosumgaard, J. R., et al. 2020, A&A, 640, A81, doi: 10.1051/0004-6361/202038300
  * Nordström et al. (2004) Nordström, B., Mayor, M., Andersen, J., et al. 2004, A&A, 418, 989, doi: 10.1051/0004-6361:20035959
  * Oliphant (2006) Oliphant, T. E. 2006, A guide to NumPy, Vol. 1 (Trelgol Publishing USA)
  * Pedregosa et al. (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., et al. 2011, Journal of Machine Learning Research, 12, 2825
  * Pinsonneault et al. (2018) Pinsonneault, M. H., Elsworth, Y. P., Tayar, J., et al. 2018, ApJS, 239, 32, doi: 10.3847/1538-4365/aaebfd
  * Price-Whelan et al. (2018) Price-Whelan, A. M., Sipőcz, B. M., Günther, H. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Queiroz et al. (2018) Queiroz, A. B. A., Anders, F., Santiago, B. X., et al. 2018, MNRAS, 476, 2556, doi: 10.1093/mnras/sty330
  * Quinn et al. (1993) Quinn, P. J., Hernquist, L., & Fullagar, D. P. 1993, ApJ, 403, 74, doi: 10.1086/172184
  * Ratcliffe et al. (2020) Ratcliffe, B. L., Ness, M. K., Johnston, K. V., & Sen, B. 2020, ApJ, 900, 165, doi: 10.3847/1538-4357/abac61
  * Renaud et al. (2020) Renaud, F., Agertz, O., Andersson, E. P., et al. 2020, arXiv e-prints, arXiv:2006.06012.  https://arxiv.org/abs/2006.06012
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003, doi: 10.1117/1.JATIS.1.1.014003
  * Roškar et al. (2008) Roškar, R., Debattista, V. P., Quinn, T. R., Stinson, G. S., & Wadsley, J. 2008, ApJ, 684, L79, doi: 10.1086/592231
  * Rybizki et al. (2017) Rybizki, J., Just, A., & Rix, H.-W. 2017, A&A, 605, A59, doi: 10.1051/0004-6361/201730522
  * Sharma et al. (2020) Sharma, S., Hayden, M. R., Bland-Hawthorn, J., et al. 2020, arXiv e-prints, arXiv:2011.13818.  https://arxiv.org/abs/2011.13818
  * Spitoni et al. (2021) Spitoni, E., Verma, K., Silva Aguirre, V., et al. 2021, arXiv e-prints, arXiv:2101.08803.  https://arxiv.org/abs/2101.08803
  * Ting et al. (2018) Ting, Y.-S., Hawkins, K., & Rix, H.-W. 2018, ApJ, 858, L7, doi: 10.3847/2041-8213/aabf8e
  * Vincenzo et al. (2021) Vincenzo, F., Weinberg, D. H., Miglio, A., Lane, R. R., & Roman-Lopes, A. 2021, arXiv e-prints, arXiv:2101.04488.  https://arxiv.org/abs/2101.04488
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods
  * Vrard et al. (2016) Vrard, M., Mosser, B., & Samadi, R. 2016, A&A, 588, A87, doi: 10.1051/0004-6361/201527259
  * Wheeler et al. (2020) Wheeler, A., Ness, M., Buder, S., et al. 2020, ApJ, 898, 58, doi: 10.3847/1538-4357/ab9a46
  * Wilson et al. (2019) Wilson, J. C., Hearty, F. R., Skrutskie, M. F., et al. 2019, PASP, 131, 055001, doi: 10.1088/1538-3873/ab0075