# The GALAH Survey: Chemical Clocks

Michael R. Hayden1,2, Sanjib Sharma1,2, Joss Bland-Hawthorn1,2, Lorenzo
Spina3,2,4,Sven Buder5,2, Martin Asplund6, Andrew R. Casey3,2, Gayandhi M. De
Silva7,8,Valentina D’Orazi4,3, Ken C. Freeman5,2, Janez Kos9, Geraint F.
Lewis1, Jane Lin5,2,Karin Lind10, Sarah L. Martell11,2, Katharine J.
Schlesinger5, Jeffrey D. Simpson11,2,Daniel B. Zucker12,8,2, Tomaž Zwitter9,
Boquan Chen1,2, Klemen Čotar9, Diane Feuillet13,Jonti Horner14, Meridith
Joyce5,2, Thomas Nordlander5,2, Dennis Stello11,2,Thor Tepper-Garcia1,2,15,
Yuan-sen Ting (丁源森) 16,17,18,5, Purmortal Wang (王梓先)1,2, Rob Wittenmyer14
1Sydney Institute for Astronomy, School of Physics, A28, The University of
Sydney, NSW 2006, Australia
2Centre of Excellence for Astrophysics in Three Dimensions (ASTRO-3D),
Australia
3School of Physics and Astronomy, Monash University, Australia
4Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Padova, vicolo
dell’Osservatorio 5, 35122, Padova, Italy
5Research School of Astronomy & Astrophysics, Australian National University,
ACT 2611, Australia
6Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85741
Garching, Germany
7Australian Astronomical Optics, Faculty of Science and Engineering, Macquarie
University, Macquarie Park, NSW 2113, Australia
8Macquarie University Research Centre for Astronomy, Astrophysics &
Astrophotonics, Sydney, NSW 2109, Australia
9Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19,
1000 Ljubljana, Slovenia
10Department of Astronomy, Stockholm University, AlbaNova University Centre,
SE-106 91 Stockholm, Sweden
11School of Physics, UNSW, Sydney, NSW 2052, Australia
12Department of Physics and Astronomy, Macquarie University, Sydney
13Lund Observatory, Department of Astronomy and Theoretical Physics, Box 43,
SE-221 00 Lund, Sweden
14Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD
4350, Australia
15Centre for Integrated Sustainability Analysis, School of Physics, The
University of Sydney, NSW 2006, Australia
16Institute for Advanced Study, Princeton, NJ 08540, USA
17Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08544, USA
18Observatories of the Carnegie Institution of Washington, 813 Santa Barbara
Street, Pasadena, CA 91101, USA
E-mail<EMAIL_ADDRESS>
###### Abstract

Previous studies have found that the elemental abundances of a star correlate
directly with its age and metallicity. Using this knowledge, we derive ages
for nearly 250,000 stars of the GALAH DR3 sample using only their overall
metallicity and chemical abundances. Stellar ages are estimated via the
machine learning algorithm $XGBoost$, using main sequence turnoff stars with
precise ages as our input training set. We find that the stellar ages for the
bulk of the GALAH DR3 sample are accurate to 1-2 Gyr using this method. With
these ages, we replicate many recent results on the age-kinematic trends of
the nearby disk, including the age-velocity dispersion relationship of the
solar neighborhood and the larger global velocity dispersion relations of the
disk found using Gaia and GALAH. The fact that chemical abundances alone can
be used to determine a reliable age for a star have profound implications for
the future study of the Galaxy as well as upcoming spectroscopic surveys.
These results show that the chemical abundance variation at a given birth
radius is quite small, and imply that strong chemical tagging of stars
directly to birth clusters may prove difficult with our current elemental
abundance precision. Our results highlight the need of spectroscopic surveys
to deliver precision abundances for as many nucleosynthetic production sites
as possible in order to estimate reliable ages for stars directly from their
chemical abundances. Applying the methods outlined in this paper opens a new
door into studies of the kinematic structure and evolution of the disk, as
ages may potentially be estimated for a large fraction of stars in existing
spectroscopic surveys. This would yield a sample of millions of stars with
reliable age determinations, and allow precise constraints to be put on
various kinematic processes in the disk, such as the efficiency and timescales
of radial migration.

###### keywords:

Galaxy: abundances – Galaxy: structure – Galaxy: stellar content – Galaxy:
kinematics and dynamics

††pubyear: 2020††pagerange: The GALAH Survey: Chemical Clocks–LABEL:lastpage

##  1 Introduction

How the Milky Way formed and its evolution through time is one of the critical
questions facing astrophysics today. Galactic Archaeology (Freeman & Bland-
Hawthorn, 2002) is the midst of a revolution in its attempt to answer these
questions with the advent of Gaia (Gaia Collaboration et al., 2016), enabling
a plethora of new discoveries about the structure, formation, and evolution of
the Galaxy. Even now, however, the origin of various Galactic substructures
and the relative importance of secular processes such as blurring and
migration is still a matter of great debate (e.g., Rix & Bovy 2013; Bland-
Hawthorn & Gerhard 2016). Stars and stellar populations are one of the primary
tracers by which the Galaxy can be studied, as stars contain the chemical
imprint of the gas from which they formed, allowing the evolutionary history
of the Galaxy to be traced through time (Freeman & Bland-Hawthorn, 2002).
However, our knowledge and understanding of the Milky Way has been hampered by
the lack of large samples of stars for which reliable age estimates are
available.

Traditionally, stellar ages can be determined only for small subsets of the
Hertzspring-Russell (H-R) diagram. Isochrone matching can be used to determine
ages for stars along the main sequence turn off (MSTO) and sub-giant branch,
where there is a large separation between stars of different ages in the Teff
vs. luminosity plane. Ages can also be determined from high quality studies of
solar twins, where stellar parameters can be determined with much higher
precision than other stellar types. Asteroseismology provides an additional
avenue for age determination, particularly for giant stars, by providing
accurate mass and radius determinations. However, this requires extremely
accurate photometry and long baseline observations, and is generally
restricted to specific areas of the sky like the Kepler (Borucki et al., 2010)
fields. In giants, the C/N ratio can also be used as a proxy for age, as the
first dredge up along the giant branch is mass dependant (e.g., Masseron &
Gilmore 2015; Martig et al. 2016; Ness et al. 2016; Casali et al. 2019). This
relation can be calibrated using asteroseismic observations. However, each of
these methods is only able to provide ages for a relatively small number of
stars, and require significant effort to provide those ages. None of these
methods are therefore able to estimate ages for the full sample of stars that
would be observed by a large-scale spectroscopic survey such as GALAH.

Studies of the chemical abundances of nearby stars have found that there are
clear age-[$\alpha$/Fe] relations (e.g., Haywood et al. 2013; Bensby et al.
2014; Hayden et al. 2017), and that the [$\alpha$/Fe] ratio is a good proxy
for the age of a star in [$\alpha$/Fe]-enhanced populations. For younger
stellar populations belonging to the thin disk this is no longer the case, as
most stars have similar [$\alpha$/Fe] abundances; therefore [$\alpha$/Fe]
ceases to be a direct measurement of stellar age for thin disk stellar
populations. However, for the thin disk slow neutron capture process
(s-process) abundances have been found to have a strong correlation with a
stars age. In particular, the [Y/Mg] or [Y/Al] ratio has been observed to have
a tight correlation with stellar age in studies of local MSTO stars (Nissen,
2015; Spina et al., 2016; Nissen, 2016; Feltzing et al., 2017; Bedell et al.,
2018; Spina et al., 2018; Lin et al., 2020). Additionally, there have been
differences observed between the age-abundance trends of the thin and thick
disks, or as a function of metallicity (e.g., Feltzing et al. 2017; Titarenko
et al. 2019; Nissen et al. 2020; Casali et al. 2020; Lin et al. 2020), Still,
these studies highlight the potential of using chemical abundances such as
yttrium to directly determine the age of a star, and potentially estimate a
birth radius. Several studies have attempted to estimate the birth radii of
stars by comparing their age and metallicities with models of Galaxy evolution
(e.g., Minchev et al. 2018; Feltzing et al. 2019), placing a star at a
specific place and time in the Galaxy when it was born.

Table 1: Data selection criteria used to select stars in this paper. Parameter | Value
---|---
sp_flag | 0
abundance_flag $\mathrm{[X/Fe]}$ | 0
SNR | $>20$
Teff |  $<6200$ K
$\sigma_{\mathrm{T_{eff}}}$ |  $<150$ K
Spectral Fit ${\chi}^{2}$ | $<4$
$\mathrm{[Fe/H]}$ |  $-1.0<\mathrm{[Fe/H]}<0.5$ dex
Figure 1: The H-R diagram for the sample presented in this paper. The MSTO
selection criteria is shown by the black box.

Tagging a star to a particular birth location is known as chemical tagging
(Freeman & Bland-Hawthorn, 2002). In strong chemical tagging, as outlined by
Bland-Hawthorn et al. (2010), a star can be assigned to an individual birth
cluster by measuring many abundances to high precision. Open clusters have
been found to have very uniform chemical abundances, with scatter lower than
0.03 dex, which is the typical measurement uncertainty in such studies (De
Silva et al., 2006; Bovy, 2016). However, it is unclear to what degree each
cluster has a unique chemical signature, or how this changes with time. For
example, if the thick disk formed from many massive clumps (Clarke et al.
2019, but see Ting et al. 2016 also), these clumps may have unique chemical
signatures enabling strong chemical tagging, while the smaller clusters
forming in the thin disk today may not be very chemically distinct from other
nearby star forming regions. If the ISM is in general well-mixed and the
mixing timescale is short, it is possible that tagging to a specific cluster
may be very difficult with the abundance precision in current spectroscopic
surveys. Even in the case where strong chemical tagging is not possible,
however, weak chemical tagging can still be extremely useful for Galactic
Archaeology. Tagging a star to a general location in a Galaxy (i.e., a birth
radius), rather than a specific star cluster, is known as weak chemical
tagging. Weak chemical tagging allows the study of the spatial and kinematic
structure of the Galaxy through time, and can aid in determining the
effectiveness of secular processes such as blurring or migration (Sellwood &
Binney, 2002; Schönrich & Binney, 2009).

Figure 2: The age-[$\alpha$/Fe] relation for the MSTO training set. Table 2: Data cuts used on the GALAH DR3 sample to select training set of MSTO stars. Parameter | Value
---|---
$\log{g}$ | $3.5<\log{g}<4.1$
SNR | $>45$
$\tau$ (Gyr) |  $>1.75$ Gyr
$\frac{\sigma_{\tau}}{\tau}$ | $<0.2$

Ness et al. (2019) used APOGEE (Majewski et al., 2017) data and found that the
age and metallicity of a star alone could be used to estimate the initial
orbits and abundances for stars, and that deviations from calculated age-
abundance relations were quite small, within the measurement errors. These
authors argue that this makes strong chemical tagging unrealistic for much of
the disk, at least with current measurement precision. Casali et al. (2020)
use high resolution HARPS observations, and argue that the age-abundance
relations, particularly those involving yttrium, are not universal, as they
find that there are variations between the [Y/Mg]-age relation as a function
of metallicity. These s-process elements were not observed by APOGEE. Casey et
al. (2019) use GALAH (De Silva et al., 2015) to group stars by their chemical
abundances, in particular by latent factors linked to the relative
contribution of different nucleosynthetic production sites (e.g., Kobayashi et
al. 2020); these authors identify six different latent factors linked to
elemental production sites are required to explain the abundance trends
observed in GALAH. Sharma et al. (2020c) also use GALAH data to measure age-
abundance relations for nearly 30 elements and find that almost all of the
elements studied show significant trends with age. They find, similar to Ness
et al. (2019), that to first order age and metallicity alone can be used to
estimate abundances across the disk, but do find minor deviations from these
relations for some elements often related to the SNR or Teff.

The arguments laid out in Ness et al. (2019) and Sharma et al. (2020c) have
profound implications for the study of the kinematic structure and evolution
of the disk. These authors argue that the age and metallicity alone can be
used to predict chemical abundances. If correct, these relations can also be
inverted: the age of a star can be inferred based on its overall metallicity
and chemical abundances. In essence, the chemical abundances of a star act as
a clock for the chemical evolution of the Galaxy, from which we can date the
age of a star based on its abundances. This means that if abundances can be
measured to high precision, ages can potentially be estimated for every star
in a stellar survey, expanding sample sizes with reliable ages by orders of
magnitude compared to existing studies. In the ideal case, a survey would
measure abundances for as many different metal production sites as possible
(Type II Supernovae, Type Ia Supernovae, s-process, and rapid neutron capture
process). In practice, not all surveys are able to observe elements from all
production sites given their resolution or wavelength coverage. Additionally,
some elements may be difficult to measure for different stellar types or due
to non-LTE effects. Compromises must be made when selecting which chemical
abundances to use between having more abundances that make an age estimate
more precise, or using fewer abundances that are well measured for a larger
fraction of the stars in a stellar survey.

In this paper, we estimate the ages of several hundred thousand stars directly
from their chemical abundances as measured in GALAH DR33 (Buder et al., 2020).
We use MSTO stars as a training set for Bayesian and machine learning models
for age estimation. With these ages, we reproduce recent age-kinematic
observations of the disk with a sample size an order of magnitude larger than
was previously available. GALAH DR3 provides abundance determinations for
nearly 30 elements, allowing us to be selective and choose elements that are
well estimated for a large fraction of the sample while also covering the
different nucleosynthetic production sites in the Milky Way.

This paper is organized as follows: in Section 2, we describe the sample
selection, data cuts, and the MSTO training selection used in our analysis. In
Section 3, we outline the methods we use to determine stellar ages for a large
fraction of the GALAH DR3 sample, using Bayesian analysis as well as $XGBoost$
(Chen & Guestrin, 2016) to generate a model for the age of a star based on its
chemical abundances. In Section 4, we demonstrate the reliability of our age
determinations and reproduce many of the recent kinematic studies of the
Galaxy. In Section 5, we discuss our findings in the context of strong and
weak chemical tagging, implications for future kinematic studies of the disk,
the chemical evolution of the Galaxy, and future survey design.

Figure 3: Comparison of chemical ages with those based on isochrones and
stellar parameters.

##  2 Data

Spectroscopic data are taken from GALAH DR3 (Buder et al., 2020), and along
with additional fields from the K2-HERMES (Wittenmyer et al., 2018) and TESS-
HERMES surveys (Sharma et al., 2018). GALAH uses the High Efficiency and
Resolution Multi-Element Spectrograph (HERMES, Sheinis et al. 2015)
instrument, which is a high resolution ($R\sim 28000$) multi-fibre
spectrograph mounted on the 3.9 meter Anglo Australian Telescope (AAT). HERMES
covers four wavelength ranges ($4713-4903$Å, $5648-5873$Å, $6478-6737$Å, and
$7585-7887$Å), carefully selected to maximise the number of elemental
abundances that are able to be measured. Observations are reduced through a
standardised pipeline developed for the GALAH survey as described in Kos et
al. (2017). Stellar atmospheric parameters and individual abundances are
derived using SME (Valenti & Piskunov, 1996; Piskunov & Valenti, 2017). The
precision of individual abundances [X/Fe] is typically $\sim 0.05$ dex, while
the random errors in radial velocities are $\sim 100\ \mathrm{m\ s^{-1}}$
(Zwitter et al., 2018). In this analysis, we use a selection of 13 well
measured elemental abundances from GALAH DR3: [Fe/H], [Mg/Fe], [Ca/Fe], [Ti
I/Fe], [Si/Fe], [O/Fe], [Mn/Fe], [Cr/Fe], [Na/Fe], [K/Fe], [Y/Fe], [Ba/Fe],
and [Sc/Fe] which span a range of nucleosynthetic production sites.

Figure 4: The chemical age versus age determined from isochrone matching for the training set of MSTO stars using $XGBoost$. The black line denotes a 1:1 correlation, while the grey lines denote $\pm 1$ Gyr. The orange line denotes the median and one sigma scatter about the relation. Table 3: $XGBoost$ Parameters Used In Model Generation Parameter | Value
---|---
${\Gamma}$ | 5
Max Depth of Tree | 7
Subsample Ratio | 0.7
Learning Rate $\eta$ | 0.05
Minimum Child Weight | 5
Column Sampled by Tree | 0.8

Because of slight differences in both abundance precision and zeropoint shifts
between giants ($\log{g}<3.5$) and dwarfs ($\log{g}>3.5$), we report results
for these subsamples as well as all of GALAH DR3 separately in this paper;
however the same model trained on MSTO stars is applied to both subsamples. We
implement several quality cuts on the signal-to-noise ratio of the spectra,
stellar parameter and abundance flags, as well as a temperature cutoff of
Teff$<6200$ K, as outlined in Table 1. We require a SNR$>20$, a quality
flag$=0$ for both the stellar parameters and the individual abundances (see
the GALAH DR3 release paper, Buder et al. 2020). The reason for the Teff cut
at 6200 K is we have found that there are significant issues for abundances
measured for the hottest stars. We restrict our parameter space to that
covered by the disk, $-1<$[Fe/H]$<0.5$, as the age-abundance relations of any
accreted objects, e.g., the halo, follow a different enrichment history. These
cuts give us a sample of 155,519 dwarf stars and 92,645 giant stars for which
we determine ages for in this paper.

Figure 5: The chemical age versus age determined from isochrone matching for
the test set of MSTO stars using $XGBoost$. The black line denotes a 1:1
correlation, while the grey lines denote $\pm 1$ Gyr. The orange line denotes
the median and one sigma scatter about the relation. Note that the test set
performs worse than the training set as $XGBoost$ is slightly overfitting.
However, even here the scatter is relatively small, and even the largest
systematic age error is $<1$ Gyr for the worst performing cases (age between
$8-10$ Gyr.

For the training set we use main sequence turn off stars (MSTO), with ages
computed with the code BSTEP (Sharma et al., 2018) with isochrone matching
using PARSEC-COLIBRI isochrones (Marigo et al., 2017). BSTEP provides a
Bayesian estimate of intrinsic stellar parameters by comparing observed
parameters to those of the PARSEC-COLIBRI isochrones. For determining the age
of an MSTO star through BSTEP, the relevant stellar and astrometric parameters
are Teff, $\log{g}$, [Fe/H], [$\alpha$/Fe], $J$, $Ks$, and parallax.
Astrometric parameters are taken from Gaia DR2 (Gaia Collaboration et al.,
2018; Lindegren et al., 2018). The stellar distances and velocities used in
this analysis are taken from the GALAH DR3 dynamics value added catalog, see
Buder et al. (2020) for details.

The relative accuracy for ages derived for MSTO stars in BSTEP is on the order
of 10-15%. With these high precision ages, MSTO stars make an ideal training
set for age-abundance relations. In addition to the abundance and stellar
parameter quality conditions applied to the rest of the GALAH sample, we apply
additional criteria for the MSTO training set as outlined in Table 2. To
isolate the MSTO stars, we use a cut in $\log{g}$ space with
$3.5<\log{g}<4.1$, as well as stricter criteria on the SNR, with SNR$>45$ for
the training set. See Fig. 1 for the H-R diagram of the sample which meets our
selection criteria. The black box denotes the MSTO sample for which accurate
ages are able to be estimated from isochrones. There are issues in determining
the abundances for the youngest stars, $\tau<1.75$ Gyr, and we elect to remove
these stars from the training set. For stars belonging to the thick disk
([$\alpha$/Fe]$>0.15$), we require these stars to have an age $\tau>8$ Gyr
(i.e., we remove the young $\alpha$-rich stars from the training set, see
Chiappini et al. 2015), see Fig. 2 for the age-$\alpha$ relation found in our
training set. We also require the fractional uncertainty in the age
determination from the isochrone matching to be better than 20%, i.e.,
$\frac{\sigma_{\tau}}{\tau}<0.2$. These restrictions, along with the quality
cuts from Table 1, leave a sample of 15,424 MSTO stars belonging to the
training set.

##  3 Methods

Ages can be estimated from chemical abundances in several ways. Initial
studies of the [Y/Mg] ratio used linear fits (e.g., Nissen 2016; Bedell et al.
2018; Spina et al. 2018). However, variations have been found with metallcity
in the age-abundance trends (e.g., Feltzing et al. 2017; Titarenko et al.
2019; Casali et al. 2020; Lin et al. 2020), and as our sample covers a large
range in Galactocentric radii, more sophisticated methods are required. Sharma
et al. (2020c) determine age-abundance relations for various elements measured
in GALAH DR3, under the assumption that an abundance can be determined given
its age and metallicity. For elements which follow this assumption, this
formalism can be inverted, and an age determined given a metallicity and a set
of abundances, using Bayes Theorem. Let $X_{i}$ denote the observed abundance
[X/Fe] of the $i$-th element with measurement uncertainty $\sigma_{Xi}$. Let
$F$ be the observed metallicity [Fe/H] and $\sigma_{F}$ its uncertainty. Given
age $\tau$ and metallicity $F^{\prime}$ we can predict the abundance
$X_{i,{\rm Model}}(\tau,F^{\prime})$ using a model with some intrinsic
dispersion $\epsilon_{Xi}$. Using Bayes theorem the probability distribution
of age of a star given its metallicity and elemental abundances can then be
written as:

$\displaystyle
p(\tau|X_{i},\sigma_{Xi},F,\sigma_{F})=\int{\mathrm{d}}F^{\prime}p(\tau|F^{\prime})p(F^{\prime}|F,\sigma_{F})\times$
(1) $\displaystyle\prod_{i}p(X_{i}|\tau,F^{\prime},\sigma_{Xi}).$ (2)

where

$p(X_{i}|\tau,F^{\prime},\sigma_{Xi})=\mathcal{N}\left(X_{i}|X_{i,{\mathrm{Model}}(\tau,F^{\prime})},\sigma_{Xi}^{2}+\epsilon_{Xi}^{2}\right)$
(3)

and

$p(F^{\prime}|F,\sigma_{F})=\mathcal{N}\left(F|F^{\prime},\sigma_{F}^{2}\right)$
(4)

We apply this methodology with the age-abundance trends measured in Sharma et
al. (2020c), and results for the age determination using Bayesian analysis are
shown in Fig. 3. There is in general a good agreement between the age
determined from isochrone matching and the age determined using Bayes theorem,
although there are significant number of outliers. The primary benefit of the
Bayesian approach is that the errors in abundances are directly incorporated
into the age estimation, along with the fact that missing or flagged
abundances still allow an age to easily to be determined.

Figure 6: Top: The relative gain of different chemical abundances in
determining the age of a star. Bottom: The relative weight of different
chemical abundances in determining the age of a star.

Figure 7: Top: The random age errors due to abundance uncertainties as a
function of SNR. The giants and dwarfs have similar median trends, but the
giants have significantly more scatter as a function of SNR than dwarfs.
Bottom: The total uncertainty in the age estimates, combining the random and
systematic errors.

In addition to Bayesian analysis, ages can also be estimated using machine
learning tools such as Neural Networks or gradient boosting algorithms. For
this paper, we use the gradient boosting algorithm $XGBoost$ (Chen & Guestrin,
2016). We split our initial sample of 15,424 MSTO stars into a training set
(70% of the original MSTO sample) and test set (30% of the MSTO sample). The
training set uses 13 chemical abundances described in the data section as
input, along with the desired output parameter of age $\tau$ determined from
MSTO isochrone fitting. We ran a 5-fold cross validation grid of several
hundred thousand $XGBoost$ hyper parameters to obtain the model that best
reproduced the trends of the test set while trying to minimize the over-
fitting of the training set. The parameters used for $XGBoost$ are given below
in Table 3.

The chemical age estimated from the best fit model relative to the input
isochrone age determination is shown in Fig. 4 and 5. The training set is
slightly overfit relative to the test set, with a scatter $\sim 1.1$ Gyr in
the training set compared to the test set ($\sigma_{\tau}\sim 1.35$ Gyr).
However, this overfitting problem is less severe than in other methods we
attempted (e.g., neural networks), while still giving age determinations for
the test set that are more robust than those determined using Bayesian
analysis. This model is able to reproduce the general age trend found in the
test set, with a slight overestimate of ages for stars $\tau<6$ Gyr and a
slight underestimate in ages for stars $\tau>6$ Gyr, i.e., the slope of the
chemical age vs isochrone age relation is generally shallower at intermediate
ages relative to the 1:1 line, as shown by the running median (orange line) in
Fig. 5. Most importantly, the number of critical failures/outliers is
significantly reduced compared to the Bayesian age estimates.

Figure 8: Left: The age-metallicity relation for MSTO stars determined from
isochrone ages. Right: The age-metallicity relation determined from chemical
ages.  Figure 9: The age-abundance trends for MSTO using isochrone ages.

The relative importance of different abundances in determining the age using
$XGBoost$ is shown in Fig. 6. The feature gain, which measures how useful each
element is in improving the accuracy of the age estimates, is shown in the top
panel. Perhaps unsurprisingly, the [$\alpha$/Fe] elements [Mg/Fe] and [Si/Fe]
are at the top, followed by the s-process element [Y/Fe]. This makes sense
from a qualitative perspective: we know the oldest stars are enhanced in
$\alpha$ elements, so we have an immediate age indicator just from the [Mg/Fe]
and [Si/Fe] abundance if a star is (relatively) young or old. For younger and
intermediate aged stars with lower [Mg/Fe] or [Si/Fe], [$\alpha$/Fe] is less
able to discriminate age. This is where the s-process elements become
important in distinguishing young from intermediate aged populations. The
bottom panel of Fig. 6 is the relative weight of each element in $XGBoost$.
The weight describes how often an element occurs in different decision trees
in $XGBoost$. For this metric, the $\alpha$ elements are much lower down in
the metric, with s-process, odd-Z, and iron peak elements being dominate, as
they help distinguish between different locations within the Galaxy for a
given age (i.e, for a given age there can be a range of metallicities which
each correspond to a different birth radius).

Uncertainties in the age estimates are calculated by doing 1,000 Monte Carlo
runs of the input abundances through the model for each star, with the
standard deviation of the age distribution being reported. The age
distribution generated from the Monte Carlo run is generally a single peaked
Gaussian. However, there is an exception for stars with ages between 8-10 Gyr.
These stars typically have an age distribution that is bimodal, and the ages
are quite sensitive to errors in [$\alpha$/Fe]. Stars in this age range
generally have thin disk/solar-[$\alpha$/Fe] measurements, but still have very
low s-process abundances, similar to that of the thick disk population. This
means that $\alpha$ elements are the dominant age indicator distinguishing
them from the thick disk. For these older thin disk stars, an error in
[$\alpha$/Fe] of 0.1 dex, for example, might shift a star towards the thick
disk as the other most age-sensitive abundances are not yet reliable for age
determination.

Figure 10: The age-abundance trends for the various GALAH DR3 samples analyzed
in this paper. Note the large discrepancy for the s-process elements [Y/Fe]
and [Ba/Fe] for the giants.

The median age uncertainty due to random errors as a function of SNR is shown
in the top panel of Fig. 7. The median random errors for dwarfs and giants as
a function of SNR are comparable, and between 0.5 and 1.5 Gyr, with the error
in the age decreasing as signal to noise increases. The giants have a larger
scatter as a function of SNR, however. We estimate the systematic errors by
using the results of the MSTO test set. As we have calculated the random
errors due to abundance uncertainties for the test set stars, the systematic
errors inherent in the method can be estimated using the scatter about the
relation found in Fig. 5; this systematic uncertainty is ${\sim}1$ Gyr. The
total uncertainty, taking both the random and systematic errors into account,
is shown in the bottom panel of Fig. 7; typical age errors for each star are
$\sim 1-2$ Gyr. These results are more robust than those determined using
Bayesian statistics, and we elect to use the ages derived from $XGBoost$ going
forward.

The age-metallicity relation for MSTO stars is shown in Fig. 8. The trends
between the input training set from the isochrones is quite similar to the
trends found using chemical ages; the age determined from chemistry having
slightly less scatter, as expected given that the ages are determined directly
from the chemistry in this case. The global features of the relations are
present and in the same locations: the oldest stars are [$\alpha$/Fe]
enhanced, and there is a trend of increasing metallicity with decreasing age
for these thick disk stars. Once $\tau\lesssim 9$ Gyr, however, there is a
distinct lack of an obvious age-metallicity relation in the thin disk, which
mirrors results from previous studies (e.g., Casagrande et al. 2011; Haywood
et al. 2013; Bergemann et al. 2014).

###  3.1 The Impact of Atomic Diffusion and Convective Mixing Length Variation

The photospheric chemical abundances (i.e., those we estimate) can be
different than the bulk chemical composition of a star due to atomic
diffusion. Atomic diffusion is a catch all term to describe the various
transport processes that are most efficient in the radiative zones of stars,
driven by gradients in pressure, temperature, and concentration. Differences
between the surface and bulk composition can lead to age errors on the order
of 20% for MSTO stars (see e.g., Thoul et al. 1994; Dotter et al. 2017; Liu et
al. 2019), becoming preferentially larger for older stars. Additionally, for
MSTO stars in particular, atomic diffusion can cause abundance underestimates
of $\sim 0.1$ dex relative to the bulk composition. This means that atomic
diffusion, if not taken into account, can cause an error in our calibration
sample both in the age determination and in the chemical abundance relations
used to derive the ages for the rest of the sample. The version of the PARSEC
isochrones used in this analysis do take atomic diffusion into account, and we
use the bulk composition in our age estimates, so the effect on age
determination of MSTO stars is mitigated. Still, there remains the potential
for diffusion to effect the abundances of our training set. The overall impact
of this is likely small, given that the magnitude of the effect is only
slightly larger than our abundance uncertainties, and most importantly because
the abundance variation of [X/H] varies roughly in lock step with [Fe/H]. This
means that the [X/Fe] ratio used in our analysis will be relatively
unaffected, and the primary error in a chemical age determination will then be
due to any [Fe/H] offset between the surface and bulk composition. This [Fe/H]
offset could then yield a systematic error in age of up to 20% in the chemical
age of a star, but is in general much smaller than this value as [Fe/H] is not
as important as $\alpha$ or s-process elemental abundances in the age
estimations.

Variations have been found in the convective mixing length both with
metallicity (e.g., Tayar et al. 2017) and there are also hints that there
could be variations due to mass as well (Joyce & Chaboyer, 2018b). The primary
impact of this variation in the convective mixing length is to cause an offset
between the observed effective temperature of a star and the temperature of a
star of that mass and metallicity in model isochrones, if the variation in
mixing length is not accounted for in the models. As temperature is a critical
component to the age estimation of MSTO stars, any offsets between
observations and isochrones will therefore cause an error in the age
determination. The impact on MSTO stars is largest for older (lower mass) and
more metal-poor stars; Viani et al. (2018) find for example a 0.3 difference
in the mixing length parameter for an old metal-poor star relative to the
solar mixing length, this causes a 200K shift towards cooler temperatures of
the models on some portions of the turn off region. Similarly, in their study
of metal-poor stars with [Fe/H]$\sim-2$, Joyce & Chaboyer (2018a) find
variations of the mixing length of 0.3, which also causes a shift of $sim150$K
cooler along the MSTO region. This means that the ages would be underestimated
compared to reality. However, the areas of the turnoff region which are most
effected by this are relatively small (the turnoff itself only, see their Fig.
16), and the majority of the stars in our training set lie further along the
MSTO or subgiant branch where the effective temperature difference between
models is small. For the majority of our training set, our age determinations
will be unaffected. For the older, more metal-poor stars, we might
underestimate the age of an MSTO star by up to $\sim 15$% based on a
temperature shift of 200K in the isochrones.

As a whole, these effects are relatively minor compared to our global
uncertainties, but do add to the systematic error budget. Particularly, these
effects could be another possible explanation for the slight discrepancy
between the chemical and isochrone ages for stars with $8<\tau<10$ Gyr.

##  4 Results

###  4.1 Age-Abundance Relations

Fig. 9 shows the age-abundance trends for the input training set. These are
the same trends found in Sharma et al. (2020c). In the ideal case, the trends
should match the training set sample exactly, as the chemical abundances
themselves are being used to determine the ages. For the most part, we find
that this is true, and the same abundance trends found in the training set are
well matched to those for the entire sample derived with chemical ages, as
shown in Fig. 10. These results closely match the age-abundance trends
measured in Sharma et al. (2020c). However, there is a particular issue with
the [Y/Fe] and [Ba/Fe] abundances for giants, where there is a strong
discrepancy between the expected output and what is observed, with the [Y/Fe]
and [Ba/Fe] abundances having large scatter for the oldest stellar populations
which is not present in the MSTO or dwarf subsamples. The large scatter in
[Y/Fe] and [Ba/Fe] for the giant subsample is shown in Fig. 11.

Figure 11: The age-abundance trends of [Y/Fe] and [Ba/Fe] for giant stars.
Note the large scatter in these s-process abundances for old stellar
populations. This highlights a potential issue in the s-process abundance
estimates for the old, metal-poor thick disk populations.

The expectation for [Y/Fe] and [Ba/Fe], as s-process elements, is for low
abundances at old ages, and high abundances at young ages, as is measured in
the MSTO and dwarf samples; this trend is not reproduced in the giants
highlighting an issue either with the abundance of [Y/Fe] and [Ba/Fe] or with
the chemical age determination. We perform several tests to validate the
reliability of the giant ages in light of the issue surrounding the s-process
abundances for old stars. The [$\alpha$/Fe] vs. [Fe/H] plane for the giant
sample is shown in Fig. 12, with the color code being the [Y/Fe] abundance.
The problem with the [Y/Fe] measurements for the oldest and most metal-poor
stars is immediately apparent: there are a large fraction of stars with
[$\alpha$/Fe] and [Fe/H] belonging to the thick disk which have high [Y/Fe]
measurements. These stars are likely some of the oldest in the Galaxy, based
purely on their [$\alpha$/Fe] and metallicity: this is the metal-poor end of
the thick disk, so their [Y/Fe] should therefore be low. This hints at a
problem in the [Y/Fe] and [Ba/Fe] abundances for metal-poor high-[$\alpha$/Fe]
stars. Luckily, for stars belonging to the thick disk, the [Y/Fe] and [Ba/Fe]
are not useful tracers for age, as AGB stars have not produced large amounts
of s-process elements, and the age can be derived almost entirely from the
[$\alpha$/Fe] and [Fe/H] abundances alone (see e.g., Haywood et al. 2013;
Hayden et al. 2017).

The question of reliability for the [Y/Fe] and [Ba/Fe] and their impact on the
age determination is then primarily for the thin disk, i.e., stars with age
$<9$ Gyr. For these stars [$\alpha$/Fe] is no longer a useful age tracer, and
the [Y/Fe] and [Ba/Fe] become critically important for reliable age
determination. For younger stars, the giant age-abundance trends match closely
to those of the dwarfs and MSTO stars, which is an encouraging sign and hints
that the [Y/Fe] and [Ba/Fe] are reliable where they are important for the age
measurement. Further, we have additional abundance tracers that were not used
in the age determinations available for giants for which to test the age
reliability. As mentioned in the data section, [Eu/Fe] is not well measured in
the MSTO or dwarf samples, but is measured in the majority of giants. As an
r-process element, [Eu/Fe] is expected to have a strong age-abundance trend
(e.g., Sharma et al. 2020c). This makes [Eu/Fe] a useful diagnostic in our
chemical age determinations, as we can measure the [Eu/Fe] vs. chemical age
trend and compare this to expected trends from other studies. We find a strong
trend of increasing [Eu/Fe] as a function of chemical age as shown in Fig. 13,
as expected for an r-process element. Our result closely matches the
age-[Eu/Fe] trends measured in other studies (e.g., Sharma et al. 2020c). This
highlights that the ages derived for giants from chemical abundances are
likely trustworthy, and the issue with the [Y/Fe] and [Ba/Fe] abundances only
affects thick disk stars for which these abundances are not critically
important in the age determination.

###  4.2 Kinematic Trends

We demonstrate both the utility and accuracy of ages derived from chemical
abundances by reproducing some of the recent results on the kinematic
structure of the disk (e.g., Mackereth et al. 2019; Sharma et al. 2020a).
These results also highlight that the potential issue with the [Y/Fe] and
[Ba/Fe] abundances for giants belonging to the metal-poor thick disk does not
dramatically impact the age determination, as the recovered kinematic
relations generally agree for both giants and dwarfs. As shown in Fig. 14, we
reproduce the age-velocity dispersion relation for the solar neighborhood
($7.1<R<9.1$ kpc, $|z|<0.5$ kpc), and find that all of the subsamples agree
well to within the errors in both the measured ages and the velocities. This
result also closely matches previous studies of the vertical velocity
dispersion in the solar neighborhood, who find a low velocity dispersion of
$\sim 10-15$ km s-1 for younger stellar populations that increases with age up
to values between 40-50 km s-1 for the oldest stars in the disk (Nordström et
al., 2004; Haywood et al., 2013; Feuillet et al., 2016; Hayden et al., 2017;
Yu & Liu, 2018; Sharma et al., 2020a). Some authors have found a step function
in the velocity dispersion between low- and high-[$\alpha$/Fe] populations
(e.g., Silva Aguirre et al. 2018; Miglio et al. 2020), but this is likely due
to the definition of low- and high-[$\alpha$/Fe] and imposing a hard
[$\alpha$/Fe] cut on the boundary between thin and thick disks, when the
chemical distributions are actually rather continuous (see Sharma et al.
2020b). We find that the velocity dispersion increases smoothly with age and
that there is no step function in the estimated dispersion, although there is
a clear inflection point around $\sim 8-9$ Gyr, with a steeper increase in
dispersion with age for older stellar populations.

Figure 12: The [$\alpha$/Fe] vs. [Fe/H] plane for the giants. It is expected
that young stars have higher [Y/Fe] than older stars, as [Y/Fe] is an
s-process element. We identify a potential issue with the [Y/Fe] abundance for
the metal-poor thick disk in the giant subsample ([Fe/H$<-0.5$,
[$\alpha$/Fe]$>0.15$), where many stars have enhanced [Y/Fe]. These stars
should be among the oldest in the disk based on their belonging to the metal-
poor thick disk. Figure 13: The age-[Eu/Fe] trend for giants. [Eu/Fe] is an
r-process element and should have a strong trend with age, which we recover in
our analysis. [Eu/Fe] was not one of the elements used in the age generation.
This means that the age determinations for giants are robust, despite the
issue with the s-process abundances.

We also apply the data set to cover a larger fraction of the Galaxy, and
reproduce the results shown in Sharma et al. (2020a). Sharma et al. (2020a)
measure how the velocity dispersion of the disk varies with age, angular
momentum, [Fe/H], and height about the plane ($|z|$). We apply the same
formalism used in their paper to disentangle how the velocity dispersion
depends on these parameters, as outlined below. The dispersion
$\sigma_{\varv}$ of velocity $\varv$ (for either $\varv_{R}$ or $\varv_{z}$),
is assumed to depend on the stellar age $\tau$, angular momentum $L_{z}$,
metallicity [Fe/H], and vertical height from the disc midplane $z$, via the
following multiplicatively separable functional form:

$\sigma_{\varv}(X,\theta_{\varv})=\sigma_{\varv}(\tau,L_{z},{\rm[Fe/H]},z,\theta_{\varv})=\sigma_{0,\varv}f_{\tau}f_{L_{z}}f_{{\rm[Fe/H]}}f_{z}.$
(5)

Here, $X=\\{\tau,L_{z},{\rm[Fe/H]},z\\}$ is a set of observables that are
independent variables and

$f_{\tau}=\left(\frac{\tau/{\rm Gyr}+0.1}{10+0.1}\right)^{\beta_{\varv}},$ (6)
$f_{L_{z}}=\frac{\alpha_{L,\varv}(L_{z}/L_{z,\odot})^{2}+\exp[-(L_{z}-L_{z,\odot})/\lambda_{L,\varv}]}{1+\alpha_{L,\varv}},$
(7) $f_{\rm[Fe/H]}=1+\gamma_{{\rm[Fe/H]},\varv}{\rm[Fe/H]},$ (8)
$f_{z}=1+\gamma_{z,\varv}|z|,$ (9)

and
$\theta_{\varv}=\\{\sigma_{0,\varv},\beta_{\varv},\lambda_{L,\varv},\alpha_{L,\varv},\gamma_{{\rm[Fe/H]},\varv},\gamma_{z,\varv}\\}$
is a set of free parameters. We assume the same values used in their paper
outlined in their Table 2 for these free parameters.

Figure 14: The vertical velocity dispersion for different samples as a
function of age in the solar neighborhood ($7.1<R<9.1$ kpc, $|z|<0.5$ kpc)
Figure 15: The vertical velocity dispersion for different samples as a
function of age, angular momentum, [Fe/H], and height above the plane. Note
that the test set relation is omitted in this figure due to insufficient
sampling.

Our results, shown in Fig. 15, match closely with the results of Sharma et al.
(2020a) (Fig. 1 in their paper) for all of our samples. Briefly, we find that
the velocity dispersion increases with age following a shallow power law
slope. The velocity dispersion as a function of the angular momentum, or
guiding radius, is high in the inner Galaxy, while being lower in the solar
neighborhood with a flattening, or perhaps hints of an increase, in the outer
disk. As GALAH is a southern hemisphere survey, it is difficult to probe the
outer Galaxy as compared to the analysis in Sharma et al. (2020a), who also
utilized observations from LAMOST (Deng et al., 2012) to determine the
relation in the outer disk. The dependence on metallicity is roughly linear,
although there is an S shape present in the dispersions for the dwarf stars
relative to the model; this is still similar to what is observed by Sharma et
al. (2020a). The dependence on metallicity is the only panel that shows any
difference on the velocity dispersion relation between the various subsamples
characterized in this paper, with the giants following the linear trend more
closely, but even here the difference is relatively minor between dwarfs and
giants. The only major difference in our results compared to Sharma et al.
(2020a) is the vastly increased coverage of the thick disk in the sample
presented in this paper, showing that while a linear dependence on $|z|$ is a
good fit for stars up to $|z|<1.5$ kpc, it may not hold for stars high above
the plane. The vastly increased sample size in our analysis, in particular for
the thick disk, is made possible by the use of chemical ages, rather than
being forced to rely on MSTO or Kepler asteroseismic values for age estimates
which have more limited Galactic coverage.

###  4.3 Catalog and Access

This catalog will be available online through MNRAS, as well as locally at the
University of Sydney here:
http://physics.usyd.edu.au/~mhay5097/chemicalclocks/ once the paper is
accepted. The catalog will contain the sobject_id for a given star, the age
estimate for that star, and the total uncertainty in the age estimate.

##  5 Discussion

We calculate the ages of stars in GALAH DR3 using their measured chemical
abundances. With these chemical ages, we are able to reproduce the global
kinematic properties of the disk, such as the age-angular momentum-velocity
dispersion relation found by Sharma et al. (2020a). We have demonstrated that
ages can be accurately predicted for stars with a large range of birth radii,
metallicity, and chemical abundances. These results, as well as those in Ness
et al. (2019) and Sharma et al. (2020c), demonstrate that the age,
metallicity, and overall chemical abundance do not vary dramatically at a
given birth radius.

The results presented here, as well as those in Ness et al. (2019) and Sharma
et al. (2020c) highlight that the variation in abundances at a given place in
time in the Galaxy is quite small. This means that strong chemical tagging,
particularly for stars belonging to the thin disk, may prove to be difficult
with our current abundance precision. The deviations from age-abundance trends
measured by Ness et al. (2019) are roughly the same size as the abundance
measurement uncertainties, and Sharma et al. (2020c) find similar results
using GALAH observations covering a wider range of chemical abundances and
Galactic spatial coverage. To first order, this means that the dominant factor
in the abundance of a star is its age and birth radius. Early strong chemical
tagging efforts find many groups in chemical space, but have been unable to
recover open clusters observed in major spectroscopic surveys relative to the
background disk stars (e.g., Price-Jones et al. 2020). Verification of these
groups is difficult, as our results demonstrate that by having similar
chemical abundances, stars grouped in chemical space will by necessity have
similar age. This means that having a similar age or falling along an
isochrone cannot be used as validation for strong chemical tagging, as stars
will by definition meet this criteria simply by having similar chemical
abundances. Until it is demonstrated that open clusters can be chemically
tagged reliably, claims of strong chemical tagging should be viewed with some
scepticism, given our results and the arguments laid out in Ness et al. (2019)
and Sharma et al. (2020c).

However, it is possible that future improvements in the precision of abundance
determination, as well as including additional elements, could make strong
chemical tagging more tangible going forward. For instance, Bland-Hawthorn et
al. (2010) estimate that abundance precision of 0.01 to 0.02 dex is needed to
enable reliable strong chemical tagging. This is still likely lower than our
current uncertainties, but we are approaching this value, particularly with
data driven machine learning methods such as the Cannon (Ness et al., 2015) or
Astro-NN (Leung & Bovy, 2019). Continued improvements in abundance
determination, such as improved corrections for non-local thermal equilibrium
effects (see e.g., Amarsi et al. 2020) as well as better reduction techniques
to eliminate issues such as fiber cross-talk (Kos et al., 2018), could mean
that we are close to the required abundance precision necessary to enable
strong chemical tagging.

For the Milky Way, the fact that abundances can be determined merely by a
stars age and metallicity (Ness et al., 2019; Sharma et al., 2020c), or
conversely accurate ages directly from abundances, likely means that the gas
azimuthal mixing timescales are quite short. Strong azimuthal abundance
variation would yield a much larger scatter in these relations and hints that
the Milky Way must be fairly uniform with azimuth. However, recent
observations of HII regions in the Milky Way make claims of azimuthal
metallicity variation across the Galaxy (Wenger et al., 2019). The results
from the HII regions are difficult to mesh with the conclusions of Ness et al.
(2019), Sharma et al. (2020c), and this paper; it is possible that the results
coming from HII regions is due to large uncertainties in the distances to
Galactic HII regions. Observations of azimuthal abundance variation in
external spiral galaxies are inconclusive: some galaxies have azimuthal
variation, often correlating with metallicity peaks along the spiral arms or
due to interactions with the bar, while other galaxies with seemingly similar
properties do not (e.g., Kreckel et al. 2019). This presents a somewhat
muddled view of azimuthal abundance variations and it is unclear how the Milky
Way fits into this picture, other than that at least for most of its history
the Milky Way seems to have been fairly well mixed azimuthally, as otherwise
the age-abundance relations and the chemical age determinations would not work
as well as they do.

While strong chemical tagging may prove difficult with current spectroscopic
studies, this is not necessarily bad news for Galactic Archaeology. The
results presented in this paper highlight the capability of weak chemical
tagging which enable a host of new opportunities and pathways for studying the
Milky Way, and in particular the kinematic structure and evolution of the
Galaxy. The fact that we can use chemistry alone to estimate a robust age for
a star has important implications for future studies of the Milky Way.
Previously, kinematic studies were often limited to MSTO stars or
asteroseismic targets (Sharma et al., 2020a), or use chemistry where ages were
not available (e.g., Hayden et al. 2018, 2020). The ability to use abundances
to estimate a reliable age provides not only a greatly increased sample size,
but also vastly increases the Galactic coverage. MSTOs probe a limited volume
given the combination of their absolute magnitudes and the magnitude limits
imposed by spectroscopic surveys; for example, in GALAH MSTO stars probe a
volume of $\sim 1.5$ kpc around the solar position. Relying on asteroseismic
targets for ages restricts you to the Kepler and K2 fields, or the bright
magnitude limits of the TESS mission. Additionally, asteroseismic ages also
have much larger uncertainties than those in MSTO stars, and potentially the
ages presented in this paper. For example, the age-[$\alpha$/Fe] relation
found in studies of asteroseismic giants has significantly more scatter than
those of MSTO stars (see e.g., Silva Aguirre et al. 2018; Miglio et al. 2020).

Applying the methods outlined in this paper is also possible for other
completed, ongoing, and future spectroscopic surveys. In particular, the
existing datasets of LAMOST and APOGEE and the upcoming surveys 4-MOST (de
Jong et al., 2014) and WEAVE (Dalton et al., 2014) are prime targets for such
an analysis. In the ideal case, putting these surveys on the same abundance
scale (and, by the methods outlined in this paper, the same age scale) would
enable them to be used together, providing a sample of several million stars
with reliable ages determined directly from chemical abundances. A data set of
this magnitude would allow for precise constraints on important kinematic
properties that govern the secular evolution of the disk such as the
efficiency and timescales of blurring and migration. It would also allow, for
example, a detailed study of the age structure of the ridges observed in
$V_{\phi}$-$R$ space (Antoja et al., 2018; Khanna et al., 2019), the phase-
spiral (Antoja et al., 2018; Bland-Hawthorn et al., 2019), or
galactoseismology (Widrow et al., 2012; Bland-Hawthorn & Tepper-Garcia, 2020).

The results presented here also highlight the need for future spectroscopic
surveys to cover as many nucleosynthetic channels as possible, while also
providing high quality and high signal to noise ratio spectra. As clearly seen
in Fig. 7, the age precision derived from chemical abundances is a direct
function of the signal to noise ratio of the spectra. Higher quality
observations will lead to a direct improvement in the age determinations. The
need for precision abundances is also highlighted in Sharma et al. (2020c):
while the age-abundance trend for every element is dominated by the age and
metallicity, there are still smaller second order effects with signal to
noise, Teff, and $\log{g}$. There is also the potential for Galactic evolution
effects on smaller scales, such as the offsets in the [Y/Mg] relations as a
function of metallicity identified by Casali et al. (2020); these effects will
be difficult to detect if other sources of uncertainty in the age-abundance
relation dominate due to low quality observations. Additional elements,
particularly those coming from the s-process, will also provide tighter
constraints on the age determination as well.

##  6 Conclusions

In this paper we have determined ages for a large fraction of the stars in the
GALAH survey with well-measured chemical abundances, using the $XGBoost$
algorithm. The training set consists of high SNR MSTO stars with precision
ages determined from isochrone matching. The catalog is publicly available and
ideal for kinematic studies of the Galaxy. We demonstrate that the stellar
ages derived from chemical abundances are reliable and can reproduce many of
the age-kinematic results that have been observed since the advent of Gaia. In
particular, we reproduce the age-velocity dispersion of the solar
neighbourhood, and of the global relation found by Sharma et al. (2020c), but
with a greatly increased GALAH sample and much improved coverage of the thick
disk. This form of weak chemical tagging, which enables an order of magnitude
increase in the sample size as well as increasing the sample volume of stars
which have reliable age estimates, enables more detailed studies of the
kinematic structure of the disk.

When combined with the results from Ness et al. (2019) and Sharma et al.
(2020c), strong chemical tagging seems unlikely with our current abundance
precision of $\sim 0.05$ dex. The ability to measure precise ages and recreate
the age-kinematic trends across the disk, as well as find very small
dispersions in age-abundance relations, requires that the chemical abundance
variation at a given radius and time is small. This is a boon for weak
chemical tagging and the study of the kinematic structure of the Galaxy, but
means that our current abundance precision is likely not high enough to engage
in strong chemical tagging.

Going forward, the methods outlined in this paper can be applied to several of
the existing and upcoming Galactic Archaeology surveys. If these surveys can
be put on the same metallicity scale, reliable ages can be determined for
millions of stars while also having excellent coverage of the Galaxy
spatially. This dataset would be truly unprecedented, and when combined with
Gaia DR3, our understanding of the formation and evolution of the Milky Way
could undergo another gigantic leap forward.

##  7 Acknowledgements

This work is also based on data acquired from the Australian Astronomical
Telescope. We acknowledge the traditional owners of the land on which the AAT
stands, the Gamilaraay people, and pay our respects to elders past and
present. This research was supported by the Australian Research Council Centre
of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through
project number CE170100013. This work has made use of data from the European
Space Agency (ESA) mission Gaia(https://www.cosmos.esa. int/gaia), processed
by the GaiaData Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa. int/web/gaia/dpac/consortium).

In addition to ASTRO3D, MRH received support from ARC DP grant DP160103747. LS
acknowledges financial support from the Australian Research Council (discovery
Project 170100521). JK and TZ acknowledge financial support of the Slovenian
Research Agency (research core funding No. P1-0188) and the European Space
Agency (PRODEX Experiment Arrangement No. C4000127986). KL acknowledges funds
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (Grant agreement No. 852977). SLM and
JDS acknowledge support from the UNSW Scientia Fellowship program and the
Australian Research Council through grant DP180101791. YST is grateful to be
supported by the NASA Hubble Fellowship grant HST-HF2-51425.001 awarded by the
Space Telescope Science Institute.

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