# The APO-K2 Catalog. II. Accurate Stellar Ages for Red Giant Branch Stars
Across the Milky Way

Jack T. Warfield Department of Astronomy, The University of Virginia, 530
McCormick Road, Charlottesville, VA 22904, USA  Joel C. Zinn Department of
Physics and Astronomy, California State University Long Beach, Long Beach, CA
90840, USA  Jessica Schonhut-Stasik Neurodiversity Inspired Science and
Engineering Graduate Fellow  Department of Physics & Astronomy, Vanderbilt
University 6301 Stevenson Center Ln, Nashville, TN 37235, USA  James W.
Johnson The Observatories of the Carnegie Institution for Science, 813 Santa
Barbara St., Pasadena, CA, 91101, USA  Department of Astronomy, The Ohio State
University, 140 W 18th Ave, Columbus, OH 43210, USA  Center for Cosmology and
AstroParticle Physics, The Ohio State University, 191 W Woodruff Ave,
Columbus, OH 43210, USA  Marc H. Pinsonneault Department of Astronomy, The
Ohio State University, 140 W 18th Ave, Columbus, OH 43210, USA  Center for
Cosmology and AstroParticle Physics, The Ohio State University, 191 W Woodruff
Ave, Columbus, OH 43210, USA  Jennifer A. Johnson Department of Astronomy, The
Ohio State University, 140 W 18th Ave, Columbus, OH 43210, USA  Center for
Cosmology and AstroParticle Physics, The Ohio State University, 191 W Woodruff
Ave, Columbus, OH 43210, USA  Dennis Stello School of Physics, University of
New South Wales, NSW 2052, Australia  Sydney Institute for Astronomy (SIfA),
School of Physics, University of Sydney, NSW 2006, Australia  Stellar
Astrophysics Centre, Department of Physics and Astronomy, Aarhus University,
DK-8000 Aarhus C, Denmark  Rachael L. Beaton Space Telescope Science
Institute, Baltimore, MD 21218, USA  Department of Physics and Astronomy,
Johns Hopkins University, Baltimore, MD 21218, USA  Yvonne Elsworth  Stellar
Astrophysics Centre, Department of Physics and Astronomy, Aarhus University,
Ny Munkegade 120, DK-8000 Aarhus C, Denmark  School of Physics and Astronomy,
University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK  Rafael A. García
Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-
sur-Yvette, France  Savita Mathur Instituto de Astrofísica de Canarias (IAC),
E-38205 La Laguna, Tenerife, Spain  Universidad de La Laguna (ULL),
Departamento de Astrofísica, E-38206 La Laguna, Tenerife, Spain  Benoît Mosser
LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université,
Université de Paris, 92195 Meudon, France  Aldo Serenelli Institute of Space
Sciences (ICE, CSIC), Carrer de Can Magrans S/N, Campus UAB, E-08193
Bellaterra, Spain  Institut d’Estudis Espacials de Catalunya, Carrer Gran
Capitá 2, E-08034, Barcelona, Spain  Jamie Tayar Department of Astronomy,
University of Florida, Bryant Space Science Center, Stadium Road, Gainesville,
FL 32611, USA

(Received December 21, 2023; Revised March 9, 2021; Accepted March 10, 2024)

###### Abstract

We present stellar age determinations for 4,661 red giant branch (RGB) stars
in the APO-K2 Catalog, derived using mass estimates from K2 asteroseismology
from the K2 Galactic Archaeology Program and elemental abundances from the
Apache Point Galactic Evolution Experiment (APOGEE) survey. Our sample
includes 17 of the 19 fields observed by K2, making it one of the most
comprehensive catalogs of accurate stellar ages across the Galaxy in terms of
the wide range of populations spanned by its stars, enabling rigorous tests of
Galactic chemical evolution models. Taking into account the selection
functions of the K2 sample, the data appear to support the age-chemistry
morphology of stellar populations predicted by both inside-out and late-burst
scenarios. We also investigate trends in age versus stellar chemistry and
Galactic position, which are consistent with previous findings. Comparisons
against APOKASC-3 asteroseismic ages show agreement to within $\sim$3%. We
also discuss offsets between our ages and spectroscopic ages. Finally, we note
that ignoring the effects of $\alpha$-enhancement on stellar opacity (either
directly or with the Salaris metallicity correction) results in an $\sim$10%
offset in age estimates for the most $\alpha$-enhanced stars, which is an
important consideration for continued tests of Galactic models with this and
other asteroseismic age samples.

Stellar ages (1581) — Asteroseismology (73) — Stellar abundances (1577) —
Milky Way evolution (1052) — Milky Way formation (1053) — Galaxy stellar
content (621) — Red giant stars (1372) — Catalogs (205) — Stellar evolutionary
models (2046)

##  1 Introduction

The complex formation history of the Milky Way (MW) is both important to
understand and difficult to decode. Stellar ages are a crucial clue,
complementing studies of stellar positions, dynamics, and composition.
However, due to the fact that the observed properties of stars are mostly
insensitive to age, precise and accurate ages of stars are difficult to infer.

With the rise of ensemble asteroseismology over the last two decades—thanks to
large-scale, space-based, time-domain surveys such as CoRoT (Baglin, 2003),
Kepler (Borucki et al., 2010), K2 (Howell et al., 2014), and TESS (Ricker et
al., 2015)—it is possible to measure solar-like oscillation patterns in many
stars. These oscillations are due to near-surface turbulent convection motions
that generate sound waves. These sound waves, when at distinct resonant
frequencies in the stellar interior, create standing waves that form a
frequency pattern of overtone modes of differing spherical degree and radial
order. The characteristic spacing between these frequencies, known as the
large frequency-spacing ($\Delta\nu$), is related to the mean density of the
star (Tassoul, 1980; Kjeldsen & Bedding, 1995). The frequency of maximum
acoustic power ($\nu_{\rm max}$) is related to the acoustic cutoff frequency,
and therefore the density scale height and surface gravity of the star (Brown
et al., 1991; Kjeldsen & Bedding, 1995). These global parameters allow us to
derive the masses of the stars through well-understood scaling relations as
long as these two parameters and the stars’ effective surface temperatures
($T_{\mathrm{eff}}$) are known. These mass measurements, along with
composition information, allow model-based age determinations.

Analyses from asteroseismic data have been significantly furthered due to
support from similarly large, ground-based, spectroscopic surveys such as the
Apache Point Galactic Evolution Experiment (APOGEE; Majewski et al. 2017), The
Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST) survey (Cui
et al., 2012), and the GALactic Archaeology with HERMES (GALAH) survey (De
Silva et al., 2015). These surveys, in select cases, have intentionally large
overlaps with targets from space-based asteroseismic missions. The resulting
ages due to the combination of spectroscopic compositions and temperatures
with asteroseismic masses have allowed significant work in Galactic
archaeology, revealing the Galaxy’s evolution history by linking stellar
chemistry and age at Galaxy-wide scales (e.g., Anders et al. 2017, Silva
Aguirre et al. 2018, Rendle et al. 2019, Miglio et al. 2021, Willett et al.
2023, Imig et al. 2023, Stokholm et al. 2023).

One avenue through which the formation of the MW can be examined with stellar
compositions is by comparing populations of $\alpha$-rich versus $\alpha$-poor
stars (as discussed by, e.g., Aller & Greenstein 1960 and Wallerstein 1962).
These stars are rich (or poor) in $\alpha$-capture elements (e.g., Mg, O)
compared to the Sun. The mix of heavy elements in stars is not universal;
instead, it arises from distinct sources contributing on different timescales.
As a result, regions with rapid star formation will have a different mix of
elements versus regions with more gradual, or episodic, star formation.
$\alpha$ elements are primarily produced in core-collapse supernova (SNe II),
which—owing to their massive, short-lived progenitors—have rates that closely
track the Galaxy’s (at least the local) star formation history (SFH). SNe Ia,
a significant source of iron-peak elements, only enrich the interstellar
medium at later times due a combination of the longer lifetime of
intermediate-mass SNe Ia progenitors and delay time distributions from
Chandrasekhar mass overflow (Timmes et al., 1995; Kobayashi et al., 1998;
Ruiter et al., 2009). In models of Galactic star formation, as the Galaxy
evolves and the rate of SNe Ia increases, the predicted
${\rm[\alpha/Fe]}$111${\rm[\alpha/Fe]}=\log_{10}{(N_{\alpha}/N_{\rm
Fe})}-\log_{10}{(N_{\alpha}/N_{\rm Fe})_{\odot}}$. of new stars simultaneously
decreases as the Fe-peak elements become more abundant. Eventually, an
equilibrium ratio is reached (e.g., Weinberg et al. 2017).

Regardless, this expected trend is not observed in the solar neighbourhood
(Prochaska et al., 2000; Bensby et al., 2003); rather, stars with
$-1<{\rm[Fe/H]}<0$ present a discontinuous range of [$\alpha$/Fe] values. This
has become known as the $\alpha$ bi-modality, due to its bi-modal distribution
and a distinct ridge-line in [$\alpha$/Fe] vs. [Fe/H] space. Investigations
have led to further discoveries related to the $\alpha$ bi-modality, such as
the relationship between $\alpha$-richness and the geometrically and
kinematically defined thin and thick disks (e.g., Gilmore & Reid, 1983; Bovy
et al., 2012; Hayden et al., 2015).

The literature presents several Galactic chemical evolution models that
attempt to explain the observed spatial, chemical, and age trends associated
with the $\alpha$ bi-modality. One class of models explains the $\alpha$ bi-
modality with an initial, rapid star formation episode that forms the
$\alpha$-rich population, with a subsequent lull in star formation that is
pierced by an infall of pristine gas. This resets the metallicity of the disk,
and quiescent formation proceeds to form the $\alpha$-poor population (“two-
infall”; e.g., Chiappini et al. 1997; Spitoni et al. 2019). Another class of
models describes two separate star formation episodes for the $\alpha$-rich
disk and the $\alpha$-poor disk: the inner disk forms an $\alpha$-rich
population at early times, followed by a smooth transition to an $\alpha$-poor
population, with the inner part of the $\alpha$-poor population having higher
metallicity than the outer part due to forming from the gas enriched by the
inner disk (Haywood et al., 2013; Ciucă et al., 2021). An alternate scenario
described by Schönrich & Binney (2009) and expanded by Sharma et al. (2021b)
envisions a disk whereby stars occupying a wide range in chemical space are
born simultaneously, though at higher rates in the inner disk than the outer
disk. This causes $\alpha$-poor stars born in the slow chemical enrichment
environment of the outer disk to have low metallicities that are otherwise
associated with $\alpha$-rich stars. Radial migration (e.g., Sellwood &
Binney, 2002) then brings populations of different chemistry into the solar
neighborhood, causing the observed $\alpha$ bi-modality. A more recent model
(Clarke et al., 2019) predicts overlap in ages between $\alpha$-rich and
$\alpha$-poor populations thanks to a clumpy star formation scenario, where
star formation proceeds at different rates simultaneously throughout the disk
in small clumps.

Although models have been shown to reproduce observed abundance and spatial
trends, there has been little direct comparison between these model
predictions and observed age-abundance patterns beyond generic predictions
that the $\alpha$-rich population should be generally older than the
$\alpha$-poor population. The consideration of the age trends in
[Fe/H]-$\alpha$ space is therefore a potentially crucial test of these models,
which we investigate here using comparisons between asteroseismic ages and the
Galactic chemical evolution model of Johnson et al. (2021).

In this paper, we expand on the work presented by Warfield et al. (2021),
where asteroseismic-based ages were derived for 735 RGB stars across three K2
campaigns. Here, we present accurate age measurements for 4,661 RGB stars from
the APO-K2 Catalog (Schonhut-Stasik et al. 2024, hereafter JSS24), a cross-
match between the asteroseismic K2 Galactic Archaeology Program (K2 GAP)
catalog (Stello et al., 2015, 2017) and APOGEE DR17. Thanks to the observing
strategy forced upon the K2 mission, this catalog provides asteroseismic and
spectroscopic parameters along 17 lines of sight in the MW, making it one of
the most comprehensive asteroseismic-spectroscopic surveys of the diverse
populations of our Galaxy to date.

In §2, we discuss our data selection, showcasing the spectroscopic data in
§2.1, the asteroseismic data in §2.2, and the selection function in §2.3. In
§2.4, we discuss the cuts we have made to the APO-K2 data set. In §3, we
detail our methods (§3.1), our comparison with the APOKASC-3 methodology
(§3.2), and the effects of $\alpha$-abundance on age determination (§3.3). We
compare our ages with spectroscopic ages from AstroNN (Mackereth et al., 2019)
in §3.4. We move on to analysing our population in §4, with a discussion of
the K2 fields in §4.1, similarities between the Kepler field and the K2 fields
in §4.2, stellar age and chemistry as a function of Galactic position in §4.3,
and a comparison to modeled populations from Johnson et al. (2021) in §4.4. We
conclude and discuss our results in-context in §5. In addition to the online
journal, age data is publicly available for the APO-K2 catalog at
https://github.com/jesstella/apo-k2 and
https://github.com/jackwarfield/apo-k2.

##  2 Data

Our base data set is the APO-K2 Catalog
(JSS24),222https://github.com/Jesstella/APO-K2 a cross-match between data from
the K2 Galactic Archaeology Program Data Release 3 (K2 GAP DR3; Zinn et al.
2022), the Apache Point Observatory Galactic Evolution Experiment Data Release
17 (APOGEE DR17; Majewski et al., 2017; Abdurro’uf et al., 2022), and Gaia DR3
(Gaia Collaboration et al., 2016, 2023). In addition to this summary of the
data, we detail the entire data pipeline, from K2 light curves to ages, in
Appendix A.

###  2.1 Spectroscopic Data

APOGEE DR17 is a part of the final data release of the Sloan Digital Sky
Survey Phase IV (SDSS-IV; Blanton et al., 2017) and provides high-resolution
near-infrared spectra (using twin, $R\sim 22{,}500$ $H$-band spectrographs;
Wilson et al. 2019) for 657,000 unique targets (with targeting described by
Beaton et al. 2021), encompassing observations dating back to the first
edition of the APOGEE survey during SDSS Phase III in 2011. Spectra were
collected using the 2.5-meter Sloan Foundation Telescope (Gunn et al., 2006)
at the Apache Point Observatory in New Mexico, USA (APOGEE-North) from
2011-2020 and the 2.5-meter du Pont Telescope (Bowen & Vaughan, 1973) at Las
Campanas Observatory in Chile (APOGEE-South) from 2017-2020.

In addition to providing the raw data, APOGEE spectra have been put through a
reduction pipeline that, in the end, provides spectrally-derived and
calibrated estimates for stellar parameters such as effective temperature
($T_{\rm eff}$), log surface gravity ($\log{g}$), and chemical abundances
(including ${\rm[Fe/H]}$ and ${\rm[\alpha/M]}$333${\rm[\alpha/M]}$ is
conceptually equivalent to ${\rm[\alpha/Fe]}$, but instead measuring the ratio
of $\alpha$-elements to the total metallicity (M) rather than just to Fe.).
Nidever et al. (2015; with updates from Holtzman et al. 2018 and Jönsson et
al. 2020) describe the schema for extracting the spectra and performing
wavelength calibrations, flat-fielding, and measuring radial velocities.

###  2.2 Asteroseismic Data

K2 GAP (Stello et al., 2015) is the source for the asteroseismic data that we
use in this work. The targeting for the program used simple color and
magnitude cuts to select a sample of red giant solar-like oscillators,
prioritizing bright and red targets. Dwarfs, which would not oscillate at low
enough frequencies to be detected with K2 long-cadence data, were additionally
selected against using a reduced proper motion selection cut. Details of the
selection function can be found in Sharma et al. (2022). The result is a well-
understood sample ideal for Galactic archaeology applications.

Zinn et al. (2022) derived values for the asteroseismic parameters $\nu_{\rm
max}$ and $\Delta\nu$. These derived values for each star are made from the
amalgamation of values from six independent pipelines, each based on an
independent analysis of K2 light-curves, corrected for instrumental
systematics (Luger et al., 2018). We have further corrected $\Delta\nu$ using
the prescription from Sharma et al. 2016 in combination with APOGEE DR17
abundances (see JSS24, and/or Appendix A.3, for details). Stellar surface
gravities ($g$, or $\log{g}$), masses, and radii are derived using
asteroseismic scaling relations, which are calibrated to be on the Gaia DR2
radius scale (see Zinn et al. 2022 and/or Appendix A.3).

###  2.3 Selection Function

Because the targeting strategy used for selecting potential solar-like
oscillators for K2 GAP is distinct from that used for targeting the same
fields for APOGEE (which is described by Beaton et al. 2021), the composite
age of a given stellar population is potentially vulnerable to bias. The
selection function between these two targeting strategies has been worked out,
in part, by JSS24 (see also Appendix A.5) as functions of magnitude, color,
mass, radius, $\nu_{\rm max}$, and metallicity. Additionally, JSS24 presents a
selection function for translating from K2 GAP’s stellar distribution to the
true Galactic stellar populations of detectable asteroseismic giants as
indicated by Galaxia (Sharma et al., 2011).

In the analysis of our results in §4.2, we present the composite ages of our
sample’s populations both un-scaled as well as scaled to the selection
function in mass and metallicity that exists between the APO-K2 sample and the
Galaxia model; i.e., this selection function allows us to re-scale the APO-K2
age distributions to how they should approximately have appeared if the sample
truly represented an unbiased sample of the Galaxy’s asteroseismically
detectable red giant population. The wider implications of accounting for
selection functions for population age-dating is discussed in that section.

###  2.4 Cuts to the APO-K2 data set

The APO-K2 Catalog provides asteroseismically-derived masses and spectroscopic
measurements of metallicity ([Fe/H]) and $\alpha$-element abundances
([$\alpha$/M]) for 7,672 red giant branch (RGB) and red clump (RC) stars.
Stellar evolutionary states are assigned using a spectroscopic classification
that has been calibrated using stars from the APOKASC-3 sample (M.
Pinsonneault et al. 2024, in preparation), for which evolutionary states have
been determined asteroseismically (our process is described in War21 using
APOKASC-2 data, with updated parameters using APOKASC-3 provided in §2.3 of
JSS24 as well as Appendix A.5). In this work, we derive stellar ages from
stellar evolutionary tracks, using mass as a fundamental proxy for age.
Therefore, we only consider stars for our analysis that are classified as
being on the RGB. Though it is known that stars lose mass transitioning
between the RGB and RC, without a detailed prescription of this change ages
derived from the masses of RC stars will tend to be systematically biased to
older ages (e.g., as shown by Casagrande et al. 2016; we discuss this further
in Appendix B). We also limit the sample to stars with masses between 0.6 and
2.6 M⊙, [$\alpha$/M] values between 0.0 and 0.4 dex, and [Fe/H] values between
-1.0 and 0.6 dex, which is the parameter space encompassed by our evolutionary
tracks and for which asteroseismic scaling relations are well
behaved.444Though our tracks reach down to ${\rm[Fe/H]}=-2$, the behavior of
the asteroseismic scaling relations where ${\rm[Fe/H]}\lesssim-1$ is still
precarious (e.g., Epstein et al. 2014 and Valentini et al. 2019). We look at
calculating the ages for these stars in Appendix C.

Figure 1: [$\alpha$/M] versus [Fe/H] for all stars in the K2 GAP sample. The
black, dashed line represents our by-eye ridge-line to separate the
$\alpha$-poor (cyan) from the $\alpha$-rich (orange) populations. The black
unfilled markers represent stars that cannot be classified as either
$\alpha$-poor or $\alpha$-rich with $>$95% confidence, due to the
uncertainties in their chemical abundances.

In addition to our cuts, we have also defined two independent flags to
categorize stars. The first is $\alpha$-rich versus $\alpha$-poor, which is
made by drawing a ridge-line by-eye along the lower over-density of stars in
the bi-modal [$\alpha$/M] vs. [Fe/H] distribution, with stars $\geq 2\sigma$
below the line being considered $\alpha$-poor (value of $0$ in the
ALPHA_RICH_FLAG column) and above the line as $\alpha$-rich (ALPHA_RICH_FLAG =
$1$). Stars falling within $2\sigma$ of this ridge-line are given neither
classification (ALPHA_RICH_FLAG = $-1$). Our $\alpha$-rich versus
$\alpha$-poor classifications are shown in Figure 1. The second classification
is between the luminous and low-luminosity RGB (LL-RGB). Apropos
asteroseismology, luminous giants are subject to measurement systematics due
to the lower frequencies of their oscillations (versus their lower luminosity
counterparts; Mosser et al. 2013, Pinsonneault et al. 2018, Zinn et al. 2019).
War21 show that the larger uncertainties of the luminous giants are able to
moderately diffuse the distribution of ages for a given population of stars
beyond its intrinsic spread. We explore this again in §4, examining age
distributions separately for stars with $\log{g}>2.5$, which we classify as
low-luminosity.

##  3 Age Determination and Methodology Comparisons

###  3.1 Method

We have calculated underlying per-star age distributions in a manner identical
to §3 of War21. Briefly, stellar evolutionary tracks generated with the Yale
Rotating Evolution Code (YREC; Pinsonneault et al. 1989, with updates from van
Saders & Pinsonneault 2012, and generated as described by Tayar et al. 2017)
were used to create a regular grid (i.e., equally spaced along each axis),
with axes for log(mass), [Fe/H], [$\alpha$/Fe], and log(age). Monte-Carlo
sampling is used to draw sets of mass, [Fe/H], and [$\alpha$/M] values from
each star’s distributions (which are assumed to be Gaussian), and the implied
age for each draw is then estimated from the grid of evolutionary tracks via
multi-dimensional four-point Lagrange interpolation. (See also Appendix A.6).

Table 1: The partial data table for the APO-K2 RGB sample, including our ages. The complete table is available in CSV format in the online journal, as well as with the APO-K2 catalog online at https://github.com/jesstella/apo-k2 and https://github.com/jackwarfield/apo-k2. The main identifier for each star is its ID in the Ecliptic Plane Input Catalog (EPIC). In addition to the columns shown here, our table contains Galactic longitude and latitudes; Galactocentric $R$ and $Z$ positions; the uncalibrated values for APOGEE $T_{\rm eff}$, $\log{g}$, [Fe/H], [$\alpha$/M], and [O/Fe], as well as the associated errors for all relevant columns. The $\alpha$-rich Flag column has a value of 1 for $\alpha$-rich stars, 0 for $\alpha$-poor stars, and -1 for unclassified stars. The S.F. Weight column is the weighting for that star given by the Galaxia selection function, and have been normalized so that the maximum weight is 100.  EPIC ID | APOGEE ID | Gaia EDR3 Source ID | RA | DEC | $T_{\rm eff}$ | $\log(g)_{\rm APO}$ | [Fe/H] | [$\alpha$/M] | [O/Fe] |  $\alpha$-rich Flag | Mass | Radius | $\log(g)_{\rm seis}$ | $\nu_{\rm max}$ | $\Delta\nu$ | Age | Modal Age | S.F. Weight
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  | (Deg.) | (Deg.) | (K) | ($\log({\rm cm/s}^{2})$) | (dex) | (dex) | (dex) |  | (M⊙) | (R⊙) | ($\log({\rm cm/s}^{2})$) | ($\mu$Hz) | ($\mu$Hz) | (Gyr) | (Gyr) |
220648976 | 2M01161528$+$1009159 | 2580092098586391168 | 19.0637 | 10.1544 | 4947 | 3.13 | -0.251 | 0.064 | 0.006 |  | 1.18 | 4.87 | 3.13 | 165.1 | 13.66 | 5.4 | 5.4 | 6.07
212123262 | 2M08302828$+$2228487 | 665901492034489600 | 127.6179 | 22.4802 | 4509 | 2.46 | 0.203 | 0.025 | 0.048 |  | 1.35 | 10.32 | 2.54 | 44.0 | 4.73 | 4.6 | 4.6 | 55.68
203757434 | 2M16095435$-$2502223 | 6049759992483427072 | 242.4765 | -25.0395 | 4487 | 1.91 | -0.663 | 0.312 | 0.371 |  | 0.60 | 12.47 | 2.03 | 13.5 | 2.38 | 45.7 | 30.0 | 0.51
212570575 | 2M13283736$-$1113561 | 3611427412665830784 | 202.1557 | -11.2323 | 4797 | 3.06 | -0.270 | 0.226 | 0.260 |  | 1.00 | 4.65 | 3.10 | 156.8 | 13.51 | 10.6 | 10.6 | 8.25
212458977 | 2M13271006$-$1336353 | 3609924895666745216 | 201.7919 | -13.6098 | 4783 | 2.90 | -0.378 | 0.265 | 0.357 |  | 0.97 | 5.46 | 2.95 | 110.1 | 10.44 | 11.3 | 11.3 | 5.94
206005182 | 2M22072683$-$1440432 | 6827450163146087168 | 331.8618 | -14.6787 | 4702 | 3.19 | 0.289 | 0.060 | 0.077 |  | 1.16 | 8.86 | 2.61 | 50.4 | 5.52 | 8.5 | 8.0 | 70.08
212562020 | 2M13490785$-$1124552 | 3613482601761697024 | 207.2827 | -11.4153 | 4874 | 3.09 | -0.364 | 0.274 | 0.295 |  | 1.08 | 4.47 | 3.17 | 180.3 | 14.82 | 7.7 | 7.1 | 5.16
212396190 | 2M13564344$-$1500050 | 6301760184888854400 | 209.1810 | -15.0014 | 4787 | 2.85 | -0.439 | 0.290 | 0.324 |  | 0.92 | 5.64 | 2.90 | 98.2 | 9.70 | 13.1 | 12.7 | 4.70
205976299 | 2M22254038$-$1531593 | 2596147343468963840 | 336.4183 | -15.5332 | 4989 | 3.14 | -0.661 | 0.283 | 0.247 |  | 1.39 | 5.42 | 3.11 | 157.1 | 12.65 | 2.5 | 2.5 | 0.18
… | … | … | … | … | … | … | … | … | … |  | … | … | … | … | … | … | … | …

In War21, the median and $\pm 1\sigma$ percentiles of 500 log(age) values were
reported for each star in the sample, and these values were used to construct
the histograms and accompanying kernel-density estimations (KDEs) to discuss
the overall characteristics of the populations. We repeat this process for our
sample in this work, but have increased the number of runs from 500 to 5000
per star, made possible by optimizing the script from War21. In addition to
median ages, we have also calculated ages defined by the mode of a KDE that we
have fit over the 5000 log(age) values for each star individually.555KDEs were
fit using SciPy (https://scipy.org; Virtanen et al. 2020). This is done with
the intention of spotting potential biases on the median from extended tails
in the age distributions to unphysically old ages ($\gtrapprox$14 Gyrs).
However, we use the median age estimates for all of our analysis in this work.
Our ages for the APO-K2 Catalog sample are available in Table 1 (as well as
online at https://github.com/jesstella/apo-k2 and
https://github.com/jackwarfield/apo-k2).

###  3.2 Ages in the Kepler Field and Comparison with an APOKASC-3 Methodology

Table 2: The partial data table for the re-calibrated APOKASC-2 RGB sample, including our ages. The complete table is available in CSV format in the online journal. The main identifier for each star is its ID in the Kepler Input Catalog (KIC), in the column KEPLER_ID. In addition to the columns shown here, our table contains Galactic longitude and latitudes; Galactocentric $R$ and $Z$ positions; the uncalibrated values for APOGEE $T_{\rm eff}$, $\log{g}$, [Fe/H] and [$\alpha$/M], as well as the associated errors for all relevant columns. The $\alpha$-rich Flag column has a value of 1 for $\alpha$-rich stars, 0 for $\alpha$-poor stars, and -1 for unclassified stars.  Kepler ID | APOGEE ID | Gaia EDR3 Source ID | RA | DEC | $T_{\rm eff}$ | $\log(g)_{\rm APO}$ | [Fe/H] | [$\alpha$/M] |  $\alpha$-rich Flag | Mass | Radius | $\log(g)_{\rm seis}$ | $\nu_{\rm max}$ | $\Delta\nu$ | Age | Modal Age
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  | (Deg.) | (Deg.) | (K) | ($\log({\rm cm/s}^{2})$) | (dex) | (dex) |  | (M⊙) | (R⊙) | ($\log({\rm cm/s}^{2})$) | ($\mu$Hz) | ($\mu$Hz) | (Gyr) | (Gyr)
8176543 | 2M19414369$+$4405382 | 2079615472445407744 | 295.4321 | 44.0939 | 4366 | 2.06 |  | 0.060 | 0 | 1.10 | 17.08 | 2.02 | 13.4 | 1.93 | 7.8 | 7.3
8277362 | 2M18440905$+$4417307 | 2117361186928151936 | 281.0377 | 44.2919 | 4507 | 2.40 |  | 0.260 | 1 | 0.96 | 10.64 | 2.37 | 29.4 | 3.65 | 12.6 | 12.7
2161831 | 2M19270967$+$3731187 | 2051785390040144640 | 291.7903 | 37.5219 | 4902 | 3.05 |  | 0.122 | -1 | 1.06 | 4.94 | 3.08 | 145.0 | 12.41 | 7.6 | 7.4
6664533 | 2M18452413$+$4209269 | 2104693683403661952 | 281.3506 | 42.1575 | 4567 | 2.70 |  | 0.058 | 1 | 1.16 | 7.98 | 2.70 | 62.8 | 6.25 | 8.1 | 8.0
11723893 | 2M19473766$+$4949200 | 2087238180400724736 | 296.9069 | 49.8222 | 4651 | 2.80 |  | 0.035 | 0 | 1.15 | 6.83 | 2.83 | 84.1 | 7.88 | 8.0 | 7.9
10517437 | 2M18524613$+$4745349 | 2107664151504239104 | 283.1922 | 47.7597 | 3895 | 1.48 |  | 0.019 | 0 | 1.28 | 36.22 | 1.43 | 3.7 | 0.68 | 6.5 | 6.0
3441473 | 2M19232529$+$3833418 | 2052842158148906880 | 290.8554 | 38.5616 | 4643 | 2.57 |  | 0.295 | 1 | 0.99 | 7.84 | 2.65 | 55.3 | 5.90 | 11.0 | 10.9
6501676 | 2M18552344$+$4159111 | 2104874759224452096 | 283.8477 | 41.9864 | 4609 | 2.56 |  | 0.117 | 1 | 1.04 | 9.14 | 2.53 | 43.0 | 4.81 | 9.1 | 8.9
10272641 | 2M19250623$+$4718546 | 2129158263800815232 | 291.2760 | 47.3152 | 4859 | 2.86 |  | 0.063 | 0 | 1.29 | 6.95 | 2.87 | 89.8 | 8.17 | 3.9 | 3.9
… | … | … | … | … | … | … |  | … | … | … | … | … | … | … | … | …

Ages in the APOKASC-2 asteroseismic sample of stars in the Kepler field (as
provided by Pin18, but also by, e.g., Silva Aguirre et al. 2018 for
APOKASC-1), being one of the largest homogeneous sets of accurate age
estimates to-date, have become a pillar for investigating the evolutionary
history of the MW (e.g., Spitoni et al. 2019, Mackereth et al. 2019, Sharma et
al. 2021a, Sharma et al. 2021b). Expecting that the upcoming update to this
sample will play a similar role (APOKASC-3; M. Pinsonneault et al. 2024, in
preparation), it is crucial to understand how the Kepler and K2 samples
compare, both from methodological and astrophysical standpoints. Therefore, in
addition to ages for the complete APO-K2 data set, we have calculated ages for
the stars in the Kepler field, having recalculated the masses from Pin18 with
new corrections to $\Delta\nu$ ($f_{\Delta\nu}$) calculated in the same manner
as we have for APO-K2. As was done in Pin18, our masses for stars in the
Kepler field have been calibrated from an open cluster contained in the data
set, which carries a $2.1\%$ systematic uncertainty due to uncertainties in
eclipsing binary mass measurements. Our data for the Kepler Field can be found
in Table 2.

As noted in JSS24, K2 GAP DR3 $\nu_{\rm max}$ values were calibrated to a
different scale than the APOKASC sample: the Gaia DR2-based astrometric scale
(Gaia Collaboration et al., 2018), with a careful correction for parallax bias
that took into account the different selection functions of RGB and RC stars
(Schönrich & Aumer, 2017; Schönrich et al., 2019; Zinn et al., 2022). If we
instead used parallaxes as corrected by the Gaia DR3 team without selection
function corrections (Gaia Collaboration et al., 2018; Lindegren et al., 2021)
to perform the calibration, this would result in an $\sim 4.5\%$ downward
revision of the mass scale. We acknowledge potential variations in the Gaia
calibration by noting a $2\%$ systematic uncertainty in the $\nu_{\rm max}$
calibration scale and $6\%$ in mass (Zinn et al., 2019) in the K2 data used in
this work. Therefore, mass comparisons between K2 and Kepler could carry
relative shifts with respect to each other up to a potential 6% in mass, or
20% in age. As we will see, it appears that the K2 and Kepler ages agree much
better than this conservative estimate. We also note that whatever offsets
there might be in the native K2 and Kepler $\nu_{\rm max}$ scales (e.g., due
to differences in the time baselines of the datasets; Sharma et al. 2019; Zinn
et al. 2022), are removed by this Gaia calibration.

War21 observed an offset of up to 2 Gyr between K2 stellar ages and those
reported by Pin18. The authors postulated that this could have potentially
been due to the lack of $\alpha$-enhanced interior opacities in the models
used by Pin18, instead relying on the metallicity correction from Salaris et
al. (1993); we explore this hypothesis more fully in the following subsection
(§3.3).

One of the APOKASC-3 age determination methods will use mass (with no
assumption for RGB mass-loss), $\log{g}$, [Fe/H], and [$\alpha$/M] as look-up
parameters in a YREC stellar model grid (also from Tayar et al. 2017). For the
sake of comparison, we have generated ages also using this APOKASC-3
methodology for all of the RGB stars in the APO-K2 sample. As our
methodologies rely on the same axes within a common set of tracks, but utilize
slightly different look-up procedures, this comparison should provide a
valuable sense of our method’s systematic uncertainty.

Figure 2: A comparison between the asteroseismic age determination method
described in this work (and Warfield et al. 2021) and a method to be used for
the APOKASC-3 catalog (M. Pinsonneault et al. 2024, in preparation). The
x-axis is ages from this work, in Gyr, and the y-axis is the percent-offset of
the ages found using the APOKASC-3 methodology. The gray, dashed line
represents the one-to-one line and the blue, solid line tracks the measured
median offset between the two ages.

The comparison between these ages is shown in Figure 2. In general, we see
sufficient agreement between these ages, only with the APOKASC-3 methodology
producing ages that are a median of 3.2% older than ours, and consistently
$\lesssim$6% older than ours. Though not totally negligible on a star-by-star
basis, this systematic offset is much smaller than the random uncertainties of
our measurements.

One possible explanation for this zero-point offset is that it is due to how
$\log{g}$ is taken into account by our different methods. In this work, we
take the amount of time that a star spends on the main sequence (the main-
sequence lifetime, or MSLT) as the age of a star. The time for a star to move
up the RGB and onto the asymmetric giant branch is $\lesssim 10\%$ of its
MSLT, and how far along on the RGB a star is will be associated with its
$\log{g}$. Therefore, because this age-dating method from APOKASC-3 takes
$\log{g}$ into account directly as a lookup parameter, it is possible to
expect an average offset in ages at this level. Overall, this reminds us that
age estimates from evolutionary tracks can be noticeably sensitive to
methodology, even when working with the same sets of observational and
theoretical data. This is especially relevant when taking into account that
RGB ages derived from different stellar evolution codes also differ at the
2-5% level, even for similar physical inputs (Silva Aguirre et al., 2020).

###  3.3 Effects of Alpha-Abundance on Age Determination

Figure 3: The left column of panels compares the ages for stars given in this
work to the ages that would be obtained if interior opacity variations due to
non-solar [$\alpha$/M] are not considered in the underlying evolutionary
tracks, but rather each star’s metallicity is corrected using the formula from
Salaris et al. (1993). The right column has the same x-axis, but the y-axis
shows the ages inferred if the metallicities of stars are given no correction.
The bottom panels compare these ages one-to-one, and the top panels show the
fractional offset of these ages versus ours. Each individual star in our data
set is represented by an orange point, and we plot rolling medians with 1.5
Gyr bin-widths for stars with $0.0<{\rm[\alpha/M]}<0.2$ (blue, dashed line)
and for $0.2<{\rm[\alpha/M]}<0.4$ (blue, solid line). The excellent agreement
in this left column indicates that the Salaris et al. (1993) approximation
captures the effect of non-solar $\alpha$ abundances on age even using updated
microphysics prescriptions. The large age disagreement shown in this right
column demonstrates that the ages of $\alpha$-rich stars are strongly
systematically biased by assuming solar $\alpha$ abundance.

In order to more fully explore the importance of $\alpha$-abundance in age
calculations, we have estimated ages for each star in the APO-K2 sample with
two alternative treatments of [$\alpha$/M]. These are:

  * i.

${\rm[Fe/H]}$ values are corrected using the prescription
${\rm[Fe/H]}=\log_{10}{(0.638\cdot 10^{\rm[\alpha/M]}+0.362)}$ from Salaris et
al. (1993), and then ${\rm[\alpha/M]}$ is set to a value of 0 when
interpolating through the evolutionary tracks; and

  * ii.

${\rm[\alpha/M]}$ is set to 0, with no correction applied to ${\rm[Fe/H]}$.

Calculating ages in the manner of (i) invites a test of the Salaris et al.
(1993) correction against ages using models with non-solar $\alpha$ interior
opacities taken into account, but with updated microphysics compared to those
used in formulating the original correction (e.g., Grevesse & Sauval 1998
abundance mixture; OPAL equation of state [Rogers et al. 1996; Rogers &
Nayfonov 2002]; and OPAL opacity tables [Iglesias & Rogers 1996]). Calculating
ages in the manner of (ii) quantifies the systematic age uncertainty incurred
when not using the Salaris et al. (1993) correction at all.666We note that
although $\alpha$-enhanced opacities are used in our models and in the Salaris
et al. (1993) models, neither the equation of state tables used in our models
nor those of the Salaris et al. (1993) models are calculated using
$\alpha$-enhanced mixtures (Chieffi & Straniero, 1989).

The comparison between these two alternative treatments and our fiducial ages
for the APO-K2 sample is shown in Figure 3. Along the left-hand column of this
figure, we see the comparison between our catalog ages and the ages derived
using (i). We can see that, as a median function of age, ages generated using
the [Fe/H] correction from Salaris et al. (1993) are tightly consistent (no
observed systematic offset) with ages generated when using
[$\alpha$/M]-enhanced opacity tables in modern stellar structure calculations,
with a scatter much tighter than the random uncertainty on these ages. In the
right-hand column, we compare between our catalog ages and those generated
using (ii). Here, we see a systematic offset to younger ages, at about the
2-5% level for ${\rm[\alpha/M]}\leq 0.2$ and 10% for ${\rm[\alpha/M]}>0.2$.
This tells us two things:

  1. 1.

the prescription from Salaris et al. (1993) to take [$\alpha$/M] into account
via a correction on [Fe/H] yields ages that are consistent to within $\sim
5\%$ with those from using $\alpha$-enhanced stellar models with updated
microphysics;

  2. 2.

age is a non-negligible function of [$\alpha$/M] at fixed [Fe/H], and failing
to properly account for $\alpha$-enrichment may lead to offsets of up to 2 Gyr
for samples of old stars.

These results somewhat muddy the hypothesis made in War21, where it was
assumed that the offset between their ages and the ages provided in the
APOKASC-2 catalog at old ages was mainly due to the opacity effects of the
different treatments of $\alpha$ by the evolutionary tracks used to calculate
the respective ages. From our comparison above, it seems that, alternatively,
a similar offset could be realized if the correction from Salaris et al.
(1993) was never applied to the metallicities. However, as we discussed in
§3.2, ages calculated with different sets of evolutionary tracks, or even
merely differing look-up procedures, can lead to offsets at the $\sim$5%
level, and so it is still very possible that these various effects added
together produced this offset.

###  3.4 Comparison with spectroscopic ages from AstroNN

AstroNN is an APOGEE value-added catalog that provides stellar abundances,
distances (Leung & Bovy, 2019), and ages (Mackereth et al., 2019) for stars
through implementations of Bayesian convolutional neural networks. In
particular, hoping to exploit the relation between surface C and N abundances
and stellar mass/age in red giants (e.g., as presented by Salaris et al. 2015,
and also explored in the APOGEE data by Martig et al. 2016), Mackereth et al.
(2019) derives ages for red giants in the APOGEE catalog using ages from
APOKASC-2 and the associated APOGEE elemental abundances (since updated using
APOGEE DR17) as the training set.

Figure 4: A comparison between age estimates for RGB stars in the APO-K2
sample produced using the method described in this work and Warfield et al.
(2021) versus those produced by Mackereth et al. (2019) with AstroNN. The
bottom panel compares these ages directly and the top panel shows the age
offsets, $\Delta{\rm age}=\tau_{\rm NN}-\tau$. The gray, dashed line
represents the one-to-one line and the blue, solid curve tracks the rolling
inverse-variance weighted-mean of the data with a bin width of 2 Gyr. Orange,
circular markers are stars classified as $\alpha$-poor and green, triangular
markers are for $\alpha$-rich stars. In order to not overcrowd the plots,
error bars are shown for a random 1% of the sample.

The comparison between our ages and the ages for the same RGB stars from
Mackereth et al. (2019) is shown in Figure 4. For our sample, age versus age
trends are similar to what is shown between astroNN and APOKASC-2 in Appendix
A of Mackereth et al. (2019). We find that, below $\sim$4 Gyr, astroNN tends
to predict ages up to $\sim$2 Gyr larger than ours, and above $\sim$8 Gyr,
astroNN tends to predict lower ages, with an offset of up to 5 Gyr for the
oldest stars. The astroNN ages also demonstrate a clear upper limit of
approximately 10 Gyr. Similar trends have been found in other works
independently deriving spectroscopic masses and ages (Martig et al. 2016; Das
& Sanders 2019; Ting & Rix 2019; Anders et al. 2023; Wang et al. 2023; Stone-
Martinez et al. 2023), though encouraging efforts have shown progress in
addressing bias in spectroscopic ages (e.g., Ciucă et al. 2021; Leung et al.
2023).

Regarding the mismatch at old ages, the $\alpha$-rich population is known to
have a very strongly peaked age distribution, which is located at $\sim$9 Gyr
for the APOKASC sample when using the ages provided by Pin18. [C/N] is not
strongly mass-dependent at low masses (see, e.g., Martig et al. 2016, Figure
3; Roberts et al. 2024). Therefore, algorithms relying on [C/N] are likely to
assign any low-mass star to approximately the typical mass/age of an
$\alpha$-rich star within random variation.

However, in general, this limitation for spectroscopic methods at old ages may
not be as problematic as it initially appears. As it stands, astroNN is quite
effective at tagging low-mass/old stars. War21 (as well as, e.g., Silva
Aguirre et al. 2018) show that the spread in ages around the median for
$\alpha$-rich stars in the Kepler field is consistent with those stars’ random
age uncertainties (which we show to be the case for the K2 fields in §4.2).
Similarly, the standard deviation for the $\alpha$-rich astroNN ages in Figure
4 is comparable to the spread in the APOKASC-2 $\alpha$-rich ages. The
mismatch in the Figure 4 $\alpha$-rich age spread and that of K2 is driven by
the larger K2 asteroseimic age uncertainties compared to APOKASC-2 (§4.2).
Therefore, if it is true that the $\alpha$-rich population is approximately
coeval across the Galaxy, and the APOKASC sample is the most precise
measurement of this population’s age, then astroNN would be accurately
predicting the age of these stars at the composite level. However, this does
still rely heavily on the assumption of a coeval $\alpha$-rich population. If
there are genuine, intrinsic, astrophysical trends at old ages, or a
significant position–age relation at old ages, for instance, then a
spectroscopic method, assigning a median $\alpha$-rich age, would be unable to
uncover them.

At young ages, astroNN and other spectropic age determinations face the same
limitations as the asteroseismic RGB data. That is, though the trends found
from studying RGB ages may still remain (adjusted by a multiplicative zero-
point, and perhaps with more random scatter than the underlying asteroseismic
data set), there are still important questions to be answered regarding how
well the age trends derived from RGB stars actually are reflective of, and can
be applied to, the general underlying stellar population. For instance,
because a star’s RGB lifetime is proportional to its MSLT, we would expect to
find few young RGB stars. RC stars, with their longer lifetimes, may be a
valuable tracer for the true density of younger stellar populations. Roberts
et al. (2024) does show RC stars to have a more reliable [C/N]–mass
relationship at high mass, potentially offering a promising avenue for
tackling this question. However, today, RC stars still have limited utility,
as it is difficult to put these stars on an absolute age scale due to their
yet-fully understood mass loss (see Appendix B).

##  4 Population Analysis

###  4.1 K2 age distributions

Figure 5: Density distributions for the ages of stars in the $\alpha$-poor
(blue) and $\alpha$-rich (orange) RGB populations across all K2 campaigns.
Densities are calculated per chemical population, such that the areas under
the $\alpha$-poor and $\alpha$-rich curves histograms both, individually, sum
to one, and so the relative heights of these distributions are not reflective
of the relative stellar counts between each population. Along the top row, we
plot these distributions for our unaltered sample and, on the bottom, for our
re-scaled sample, accounting for the selection function from JSS24. The
vertical, colored dashed lines mark the modal peak of each distribution. In
the top right of the top panels, we show the representative Poisson
uncertainties at a density of 0.1 for both the $\alpha$-poor and $\alpha$-rich
sample’s histograms. The plots in the left column include stars within the
full range of $\log{g}$, which includes 2176 $\alpha$-poor and 2467
$\alpha$-rich stars, where the right column is limited to the low-luminosity
RGB ($\log{g}>2.5$), with 1463 $\alpha$-poor and 1402 $\alpha$-rich stars. In
all panels, only stars with ${\rm[Fe/H]}<0.35$ are included, restricting the
sample to the range within which our selection function can be interpolated.

For Figure 5, we have plotted one-dimensional age histograms for the
$\alpha$-poor and $\alpha$-rich populations across all K2 campaigns within our
sample. In the top panel, we have included all stars for which
$-1.0{\rm\,dex}\leq{\rm[Fe/H]}\leq 0.35$ and $0.60{\rm\,M_{\odot}}\leq M\leq
2.6{\rm\,M_{\odot}}$,777These limits in metallicity and mass were chosen
because it is the range in which the bins of the selection function are well
sampled. with no weighting, and in the bottom panel we have re-weighted these
same stars using the Galaxia-to-K2 metallicity and mass selection function
(described in §2.3). In the left column of this plot we include stars with all
values of $\log{g}$, where in the right column we only include stars on the
LL-RGB (§2.4).

We see, in comparing the widths of the distributions in the first row of
Figure 5, that the smaller age uncertainties for the LL-RGB stars (with median
age uncertainties of $\pm_{1.7}^{2.2}$ Gyrs, versus $\pm_{2.1}^{2.9}$ Gyrs for
the full sample) result in correspondingly smaller spreads in the age
distributions. This is perhaps more notably the case for the $\alpha$-rich
population, where the standard deviation of ages for for the full sample is
5.6 Gyrs versus 4.2 Gyrs for the LL-RGB, and which is a population with
considerably less intrinsic age spread versus the $\alpha$-poor population.
Additionally, the percentage of very young $\alpha$-rich stars ($<4$ Gyr) in
the LL-RGB sample is considerably lower (4.5% versus 8.5%). In the full
catalog, there is a comparable number of stars at $<4$ Gyr as there are $>25$
Gyr. In the LL-RGB sample, there are more exceptionally young than
exceptionally old stars, suggesting that the LL-RGB population of young
$\alpha$-rich stars may be a purer sample of truly young or otherwise high-
mass sources. However, despite these changes, the median ages of the
$\alpha$-rich populations from these two samples are essentially the same
(11.5 Gyrs for the full sample, 11.6 Gyrs for the LL-RGB). We suggest that
this indicates that the LL-RGB represents a valid, precise subset of our
sample with no biasing effect on the distributions of our populations defined
by $\alpha$ abundance, and therefore we may expect the distributions from this
subset to be more indicative of the intrinsic spread in ages for these
populations. It also suggests that the population of young $\alpha$-rich stars
may partially, but not entirely, be a product of age uncertainty (e.g., as
discussed by Anders et al. 2017).

Along the bottom row of Figure 5, we see that the selection function weighted
distributions are very similar to the unweighted distributions. Particular
differences are that the Galaxia models seem to predict slight shifts to older
ages in both populations, mainly predicting fewer very young $\alpha$-poor
stars, with a larger overlap between the two populations at intermediate ages.
We also see more pronounced secondary modes in both of these populations. In
particular, the double-peaked profile of the $\alpha$-poor population with the
selection function applied now more more closely the profile of the
$\alpha$-poor population in the Kepler field (§4.2). Otherwise, the median
ages of both populations agree within uncertainties.

###  4.2 Kepler age distributions and similarities with the K2 fields

Figure 6: Density distributions for the ages of stars in the $\alpha$-poor
(blue; 1557 stars) and $\alpha$-rich (orange; 452 stars) LL-RGB populations
within the Kepler field. Densities are calculated per chemical population,
such that the areas under the $\alpha$-poor and $\alpha$-rich histograms both,
individually, sum to one, and so the relative heights of these distributions
are not reflective of the relative stellar counts between each population. On
the left, we plot the age distribution as it would be calculated using the
methodology described in §3.1. On the right, we show the ages for the same
stars calculated after those stars’ masses have been randomly perturbed
randomly by the typical mass uncertainties for similar stars in the K2 fields.
The dashed curves on each plot represent the KDE fits for the K2 LL-RGB
sample, which is also shown in Figure 5. In the top right of each panel, we
show the representative Poisson uncertainties at a density of 0.1 for both the
$\alpha$-poor and $\alpha$-rich sample’s histograms.

In Figure 6, we present the age distribution for LL-RGB stars in the Kepler
field. On the left, we show the ages for these stars, calculated as described
in §3.1 (with masses re-calibrated from APOKASC-2, as described in §3.2).
Overall, these age distributions show good agreement with those for the same
sample of stars made by Silva Aguirre et al. (2018), with the primary peak of
the $\alpha$-poor distribution coming in at $\sim 3$ Gyr, and the
$\alpha$-rich distribution being strongly peaked at $\sim 10$ Gyr.

In comparison to the K2 sample, however, though the median ages of the
$\alpha$-rich and $\alpha$-poor populations are consistent between the
samples, there is still a qualitative difference in the shapes of these
distributions. On the right side of Figure 6, we have plotted the age
distribution of the Kepler field as it would appear if the masses of those
stars had the same uncertainties as similar stars in the APO-K2 data. We have
also, on both sides of the figure, plotted the distributions of the APO-K2 LL-
RGB $\alpha$-poor and $\alpha$-rich samples as dashed lines. Comparing these
scenarios, it seems that, by accounting for the differences in the
asteroseismic mass (and, therefore, age) uncertainties between the samples, we
can infer that the Kepler field’s stronger $\alpha$-rich peak and the K2
fields’ lack of a potentially double-peaked $\alpha$-poor age distribution
(both seen by, e.g., Silva Aguirre et al. 2018 and Miglio et al. 2021 in the
Kepler field) may be due—at least in part—to the difference in data quality
between the samples (as is also noted by, e.g., Rendle et al. 2019).
Inversely, we also might infer that the $\alpha$-rich stars of K2 are
consistent with the $\alpha$-rich population being approximately coeval as,
with K2-like uncertainties, the Kepler field’s $\alpha$-rich distribution
seems to match the qualitative features of the K2 distribution, including the
appearance of a potential second peak.

Despite some differences, the existing agreement between the K2 and Kepler age
distributions shows the reassuring progress made with these data since War21.
In War21, the authors found a median age of $\sim 8.6$ Gyrs for the
$\alpha$-rich populations from Campaigns 4, 6, and 7 (K2 GAP DR2; Zinn et al.
2020), with this population having a modal age about 2 Gyrs lower than the age
they found for the Kepler field. The non-astrophysical explanation given by
the authors was that this was related to the overall $\nu_{\rm max}$ scaling
used for K2 asteroseismic data. In K2 GAP DR2 and War21, the scaling was not
tied to Gaia, and so an $\sim$2% systematic remained between the asteroseismic
and astrometric radii. Using the K2 GAP DR3 data (which is calibrated to the
Gaia radius scale), we now find the median age for the same set of
$\alpha$-rich stars from War21 to be $\sim$20% larger, at $\sim$10.4 Gyr,
which is more consistent with the age of similar stars in the Kepler field.

As we will explore in the next section, remaining discrepancies between the K2
and Kepler populations may due to real differences between stellar
populations, tied to their spatial distributions across the Galaxy. Stokholm
et al. (2023) similarly support this idea with age data, showing the evolution
of the ages of stellar populations as a function of Galactocentric radius.
This idea is also supported by Sharma et al. (2022). They compare the
asteroseismic masses of the Kepler and K2 samples, both observationally and
theoretically (based on Galactic models), to show that there is potentially a
real difference in the modes of stellar masses across different fields
independent of spectroscopic data.

###  4.3 Age versus Galactic position in the Kepler field and K2 fields

As we have stated above, the K2 fields are unique compared to the Kepler field
because—apart from observing strategies and time baselines—they cover a much
wider positional sample of the MW, both in terms of radial distance from the
Galactic center ($R$) and vertical distance from the plane of the Galaxy’s
disk ($Z$).

Previous studies, such as Hayden et al. (2015) using APOGEE DR12 (Holtzman et
al., 2015), have already shown that both the relative and absolute
distributions of stars in chemical phase-space are functions of both $R$ and
$|Z|$, with, for instance, more $\alpha$-rich stars appearing at larger $|Z|$,
and mostly within $R\lesssim 11$ kpc (representing the older, “thick” disk)
and $\alpha$-poor stars being present at all values of $R$, but with
$|Z|\lesssim 1$ kpc. In addition to these trends in the number of stars in
each population, Hayden et al. (2015) observed that $\alpha$-poor stars in the
range $3{\rm\,kpc}<R<5{\rm\,kpc}$ have typical [Fe/H] values of about $0.2$
dex, where stars on the outskirts of the Galaxy
($13{\rm\,kpc}<R<15{\rm\,kpc}$) have a typical [Fe/H] of about $-0.4$ dex.

Figure 7: Density distributions for the ages of stars in the $\alpha$-poor
(blue) and $\alpha$-rich (orange) RGB populations within the K2 fields (left)
and Kepler field (right), in different bins of height above or below the
Galactic plane, $|Z|$. Densities are calculated per chemical population, such
that the areas under the $\alpha$-poor and $\alpha$-rich curves both,
individually, sum to one, and so the relative heights of these distributions
are not reflective of the relative stellar counts in each population. In the
top right corner of each panel, we show the representative Poisson
uncertainties at a density of 0.1 for both the $\alpha$-poor and $\alpha$-rich
samples histograms.

Looking at both the K2 and Kepler fields, we do not see obvious trends in the
age of the $\alpha$-poor population with $R$ alone. However, we do see a clear
dependence on $|Z|$, which is shown in Figure 7.888Each star’s Galactocentric
$R$ and $Z$ position was calculated with Astropy (http://www.astropy.org;
Astropy Collaboration et al. 2013; Astropy Collaboration et al. 2018; Astropy
Collaboration et al. 2022) and using Gaia eDR3-based distances calculated
using the methodology described by Bailer-Jones (2015). Similar to what was
shown by War21, we see that that the peak age of the $\alpha$-poor population
shifts older with increasing $|Z|$. In the K2 fields, this age is just above 4
Gyrs under 0.5 kpc, versus $\sim$6 Gyrs above 2 kpc. This seems to be
connected to the bi-modal age distribution of the $\alpha$-poor population.
This bi-modal distribution is most obvious in the Kepler data, where the
majority of stars are at $|Z|<0.5$ kpc, and where we observe peaks in the
$\alpha$-poor distribution both around 3 Gyrs and 6 Gyrs. In fact, when we
look at K2 stars with $|Z|<0.5$ kpc (Figure 7, first panel), we see the
younger peak emerge in this bi-modal distribution, which is not clearly
present—nor dominant—when all of the data is combined. Instead, we see that,
in the composite of Figure 5, this shelf in the K2 distribution is smoothed
over by intermediate-age $\alpha$-poor stars, which we see (moving down the
columns of Figure 7) are dominant at $|Z|>0.5$ kpc. Identically, when we look
at stars in the Kepler field in the range $1{\rm\,kpc}<|Z|<2{\rm\,kpc}$, we
see that the proportion of younger (versus intermediate-age) $\alpha$-poor
stars has decreased, and the overall distributions in these bins more closely
resemble those same bins for the K2 fields.

In Lu et al. (2022), it is shown that the $\alpha$-rich population (which we
know from, e.g., Hayden et al. 2015, among others, to be more prominent at
higher $|Z|$ due to being an older population that has experienced more
dynamic heating) lacks a clear age-metallicity relation. Over time, stars at a
given Galactic birth radius tend to be born with increasingly lower
[$\alpha$/M] and higher [Fe/H] than the initial $\alpha$-rich population born
at that radius. In a gas-rich environment with efficient star formation, this
happens quickly, creating a radial trend in stellar metallicity distribution
such that metallicities are higher in the center of the Galaxy than the
exterior after the same amount of time. Then, though $\alpha$-poor stellar
abundances are a function of Galactic birth radius and age, radial migration
(e.g., as described by Sellwood & Binney 2002, Bland-Hawthorn et al. 2010) is
sufficient to mostly wipe out the present-day radial dependence, giving rise
to the $\alpha$ bi-modality (this is also discussed by, e.g., Sharma et al.
2021a). An age-$|Z|$ trend in the $\alpha$-poor population would be expected
from either vertical heating mechanisms or imprinted from upside-down disc
formation (e.g., Bird et al., 2021, and references therein). For our data, we
seem to be seeing this same phenomenon, with a lack of strong age trends with
$R$, but trends in $|Z|$ reflecting the evolution of the $\alpha$-poor
sequence over time. We stress that while the age distributions of the
$\alpha$-rich and $\alpha$-poor populations are typically older and younger,
respectively, the $\alpha$-poor population age distribution varies with
Galactic height in K2 and Kepler data. What correlations there may be in the
ages of the $\alpha$-rich population with position or metallicity are not
obvious with the precision afforded by either K2 or Kepler data. Ultimately,
the $\alpha$ bi-modality is a multi-variate problem, which requires models
that take into account metallicity- and position-dependent SFHs to begin to
understand. We explore the K2 $\alpha$ bi-modality in this context in what
follows.

###  4.4 Comparison to modeled populations

In order to test whether the predictions of various Galaxy formation theories
are consistent with our data, we have compared our results to Galactic
evolution and chemical enrichment models from Johnson et al. (2021). These
Galactic chemical evolution models predict elemental abundances of stellar
populations across the disk given two different models of the Galactic SFH.
The inside-out SFH model has a generic “rise-fall” shape, while the late-burst
model follows the same formalism, but with a slow Gaussian-shaped bump in the
rate $\sim$2 Gyr ago (motivated by the observations of Isern 2019 and Mor et
al. 2019). Stellar populations are subject to radial migration and vertical
heating over their lifetimes under a prescription based on the h277
hydrodynamic simulation (e.g., Christensen et al. 2012). We refer to §2.2 and
§2.5 of Johnson et al. (2021) for further details.

The product of these models is a table of stellar populations along with each
population’s age, stellar mass, birth and present-day Galactic radius and
height above (or below) the Galactic plane, [Fe/H], and [O/Fe].999Oxygen,
being an $\alpha$ element, behaves similarly in tracking the Galactic $\alpha$
bi-modality. In order to convert these data into a selection that roughly
mimics our data from K2 GAP, we assigned every population of stars from the
model a weight corresponding to the fraction of stars in that population that
are in the mass range that would be occupied by stars on the RGB at that
population’s present-day age, given the initial-mass function from Salpeter
(1955). The RGB mass range was calculated assuming that a star spends
$\sim$10% of its main-sequence lifetime on the RGB,101010Though this may be a
slight overestimate of the RGB lifetime, adjusting this estimate by $\pm 5\%$
negligibly affects our results. and given the mass-luminosity relationship of
$M\propto L^{3.8}$ (for the data from Torres et al. 2010, fit in Pinsonneault
& Ryden 2023). In order to match the radial distribution of stars from the
APO-K2 sample, we created bins 2 kpc wide from $R=0{\rm\,kpc}$ to
$R=16{\rm\,kpc}$ and recorded the number of stars in the APO-K2 sample that
fall within each of these bins, $N_{\rm bin}$. We then cut the models down to
sub-samples to match the bounds of each bin in $R$, then draw $N_{\rm bin}$
populations from the sub-sample using the weights defined above. Finally, in
order to produce model data sets that roughly mimic our observed sample, the
abundances and ages of these populations have been randomly perturbed using
the mean uncertainties for these parameters in our catalog.

Figure 8: Top row: Number density distributions for RGB stars in the APO-K2
Catalog within 0.5 kpc of the Galactic plane compared to two models from
Johnson et al. (2021): a stellar population drawn from inside-out formation
models and a stellar population drawn from “late-burst” formation models in
[Fe/H] vs. [O/Fe] space. The scaling is logarithmic. The black line drawn
across the distributions represents the ridge-line separating the
$\alpha$-rich and $\alpha$-poor populations in the K2 data. Middle row: The
median age of stars within each bin in the corresponding plot on the top row.
Bottom row: Age density distribution KDE curves for the $\alpha$-poor (blue)
and the $\alpha$-rich (orange) populations from each respective sample. The
area under the $\alpha$-poor KDE is normalized independently from the
$\alpha$-rich. For the model populations, we plot the actual age distributions
(i.e., ages from the models without APO-K2-like uncertainties) as dashed
curves, which are normalized to fit under the solid curves. Vertical lines
mark each chemical population’s median age—this does not appear for the
$\alpha$-rich actual age distributions as the median is about identical to
that with uncertainties. Within each panel, we give the value for the ratio of
the number of $\alpha$-poor stars to the number of $\alpha$-rich stars,
$N_{\rm poor}/N_{\rm rich}$. The total number of stars drawn from each model
matches the number of stars in the plotted subset of APO-K2.

Histograms of the densities of the resulting populations, in [Fe/H] vs. [O/Fe]
space, are shown in the top row of Figure 8 for the APO-K2 sample (left
column) and the inside-out (middle column) and late-burst (right column) star
formation scenarios. Here, we have only considered stars from the APO-K2
sample that are within 0.5 kpc of the Galactic plane.

One obvious difference between the models and the data from these plots is the
existence of the $\alpha$ bi-modality in this chemical space; though there are
arguable over-densities of $\alpha$-rich and $\alpha$-poor stars in these
plots, the distinct ridge between these populations is less pronounced than it
is observationally. The failure to reproduce the separation between the two
sequences is a known problem with the Johnson et al. (2021) models, whereby
more intermediate [O/Fe] stars are predicted than are observed (see Johnson et
al. 2021, Figure 12). Despite this shortcoming, however, the models do
accurately predict the observed variations in the abundance distributions
between different Galactic regions. In addition to the ridge-line in the bi-
modality, the $\alpha$-poor loci of these models are less constrained to a
clear sequence versus the APO-K2 sample, and include additional over-densities
of $\alpha$-poor stars with ${\rm[Fe/H]}\lesssim-0.5$ and $\alpha$-rich stars
with ${\rm[Fe/H]}\gtrsim 0$.

Regardless of these differences, both the inside-out and late-burst models
seem to do qualitatively well at matching the age distribution of stars within
this chemical space (Figure 8, middle row), including an old $\alpha$-rich
population with a scattered age distribution and an $\alpha$-poor population
that transitions from intermediate to young ages as a function of decreasing
$\alpha$ abundance. Translated to 1D histograms for the age distributions for
the $\alpha$-rich and -poor populations (bottom row), we also see quantitative
agreement. Considering the observational uncertainties associated with APO-K2,
the median ages of these model populations are in excellent agreement with our
data, and—though models do predict different proportions of stars in each of
the respective chemical populations—the KDE curves seem to qualitatively
resemble those of the data. Therefore, despite some remaining discrepancies
between the data and models, we are unable to rule out the plausibility of
either of these scenarios.

Comparing the actual, pre-perturbation age distributions from the models
(plotted as dashed curves under the inside-out and late-burst data in the
bottom row of Figure 8) to the ages produced when including K2-like
uncertainties (solid curves in Figure 8), we see significant differences not
only in the relative width of the distributions, but also in the shapes of the
distributions. The age uncertainty convolution process, of course, will widen
the distributions, washing out any substructure that may (or may not) be
present. It will also tend to skew the distributions due to the fact that the
stars have roughly constant fractional age uncertainties, not absolute age
uncertainties. Nonetheless, the median age of each chemical population remains
relatively unchanged after convolution, and should allow for meaningful
inferences on true, population-level median ages.

##  5 Discussion and Conclusions

In §4.2 and §4.3, we compared the distributions of ages in the K2 and Kepler
fields, and found differences in the ages between these populations are
related to differences in $|Z|$ and $R$, which may be related to the generally
lower mean metallicity of the Kepler sample. In fact, when we re-sample the K2
stars to reproduce the ${\rm[Fe/H]}$ distribution of the Kepler stars, we see
a significant decrease in the offsets between the median ages of the
populations. Re-sampling by $(R,|Z|)$ yields similar results.

Considering the relationship between the mean ${\rm[Fe/H]}$ of a given
population and $(R,|Z|)$ (as shown by, e.g., Hayden et al. 2015, Imig et al.
2023), it is reasonable to assume that these two avenues of re-weighting are
achieving the same ends. This relationship between chemistry and position—and
the extended relationship that we have confirmed with age—demonstrates the
importance of accounting for all properties of a given stellar population
before applying these ages more broadly, particularly in the case of the ages
of the Kepler field stars. I.e., otherwise similar stars found in different
locations of the Galaxy may have different ages, which is related specifically
to gradients in the SFH and the migration of stars across the Galaxy.

In §4.4, we compared the age distributions of APO-K2 stars to Johnson et al.’s
(2021) Galactic chemical evolution models. Their late-burst SFH differs from
their inside-out SFH only in that it includes a recent, slow burst of star
formation (observationally motivated by Isern 2019 and Mor et al. 2019).
Broadly, the predictions of both models are consistent with our APO-K2 sample.
However, due to the substantial age uncertainties for our data, we are unable
to distinguish between these models in terms of the Galaxy’s recent SFH, and,
therefore, neither model can be preferred nor ruled out.

However, it is also important to note that these inside-out and late-burst
models from Johnson et al. (2021) fail to fully recover the MW’s $\alpha$ bi-
modality. Much has been said in the recent decades for possible histories able
to produce both the kinematic and the chemical thin- and thick-disks. Clarke
et al. (2019) shows that bursty episodes of brief, high star formation in
simulated galaxies are able to account for both the chemical bi-modality and
the associated age bi-modality of the MW. This finding seems especially
promising, considering that these clumpy star-forming regions are also
observed in disk galaxies at high redshift (e.g., as first identified by Cowie
et al. 1995, van den Bergh et al. 1996). However, the fine-grained age
predictions from this model do not agree with the data (see Clarke et al.
2019, Figure 12). On the other hand, two- and three-infall models—where the
bi-modality is primarily produced by the separation of two epochs of star
formation driven by distinct infall events of pristine gas into the
Galaxy—have been shown capable of recovering the general age-$\alpha$ relation
(Chiappini et al., 2015; Spitoni et al., 2019) and age-$Z_{\rm max}$ relation
(Spitoni et al., 2022), at least as they are observed in the Kepler field/the
Solar circle (using data from, e.g., Silva Aguirre et al. 2018, Ting & Rix
2019, Leung & Bovy 2019). As it stands, in-depth comparisons of the
predictions of these models to those of parallel evolution models (such as
those of Johnson et al. 2021) have not been done. These comparisons, done via
tests of predicted age distributions as a function of chemical space with
fine-grained age binning (as presented here) would place interesting
constraints on the relative strengths of each class of models versus observed
trends.

In this paper, we have presented precise, asteroseismically-derived ages for a
large and comprehensive sample of stars in the K2 and Kepler fields. We have
shown that, despite being rather ubiquitous across the Galaxy, the
$\alpha$-rich and $\alpha$-poor populations show small variations in their
ages that are at least partial functions of their locations in
$(R,|Z|,{\rm[Fe/H]})$ phase-space. In-line with this, we conclude that nearly
all of the differences that we find between the ages of $\alpha$-rich
populations of the Kepler field versus the K2 fields are due to the larger
uncertainties of the K2 data, and offsets in the age distribution of the
$\alpha$-poor population are attributable to the differences in stellar
Galactic position and metallicity between the samples. Although K2 affords a
wider selection of stellar populations than Kepler, comparisons of the K2 ages
against models as a function of ${\rm[\alpha/Fe]}$-${\rm[Fe/H]}$ space are not
precise enough to distinguish between scenarios with versus without recent
star formation. Combining Gaia distance information with TESS asteroseismology
is promising for achieving more precise ages across a wide range of Galactic
environment (e.g., Aguirre et al. 2020, Stello et al. 2022), which will allow
for further investigations of Galactic chemical evolution models beyond the
Kepler field.

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. S.M. acknowledges support by the Spanish Ministry of Science and Innovation with the Ramon y Cajal fellowship number RYC-2015-17697 and the grant number PID2019-107187GB-I00, and through AEI under the Severo Ochoa Centres of Excellence Programme 2020–2023 (CEX2019-000920-S). D.S. is supported by the Australian Research Council (DP190100666). R.A.G acknowledges the support from the PLATO Centre National D’Études Spatiales grant.

##  Appendix A Asteroseismic Data and Catalog Pipeline

Throughout this paper, we have described our use and modifications of various
published data sets, namely the APO-K2 Catalog as presented by JSS24, which
itself is constructed primarily using the K2 GAP DR3 catalog (Zinn et al.,
2022) and APOGEE DR17 (Abdurro’uf et al., 2022). From their origins as K2
imaging to the ages we present in this paper, these data have come together
through the collective efforts of many different individuals and teams. In
this appendix, we summarize the process used to build and calibrate the
catalogs presented here and by JSS24.

###  A.1 Data processing

The K2 Galactic Archaeology Program selected red giant asteroseismic targets
for K2 observations based on simple color-magnitude selection criteria, with
typical K2 campaign selection criteria being (J-$\mathrm{K_{s}}$) > 0.5 and (9
< V < 15). Additional targets were selected based on previous spectroscopic
identification from surveys, including APOGEE. Ultimately, more than 110,000
targets were observed from these target lists in campaigns C1-C8 and C10-C18.
The selection process is discussed in more detail by Sharma et al. (2022).

Light curves were generated using EVEREST (Luger et al., 2018), which uses
pixel-level data to remove systematics associated with K2’s degraded pointing
compared to that of Kepler. The K2 GAP light curves were reduced and
calibrated in a manner appropriate for asteroseismology, including high-pass
filtering and inpainting missing data (Pires et al., 2015). Members of the
Kepler Asteroseismic Science Consortium (KASC)111111https://kasoc.phys.au.dk
analyzed each star, resulting in a set of asteroseismic parameters from up to
six independent pipelines. The final adopted values in the K2 GAP DR3 catalog
are averages of the frequency of maximum oscillation power
($\nu_{\mathrm{max}}$) and the large frequency separation ($\Delta\nu$) values
across the pipelines (and campaigns, if a star was observed multiple times),
with discrepant pipeline results rejected with sigma clipping (Zinn et al.,
2022).

###  A.2 Crossmatches

To build our catalog, we started with Table 6 from Zinn et al. (2022), which
has a row for each star identified by its Ecliptic Plane Input Catalog (EPIC)
ID and the K2 campaign it was observed during, providing the asteroseismic
$\nu_{\rm max}$ and $\Delta\nu$ values from each of the six pipelines. Each
unique EPIC ID was matched to 2MASS IDs (Skrutskie et al., 2006) using the
kepler.k2_epic table in the MAST
Query/CasJobs121212https://mastweb.stsci.edu/mcasjobs/ database, which gives a
2MASS match for every target with an EPIC ID. These 2MASS IDs were then used
to match to the APOGEE DR17 catalog, where stars are given APOGEE IDs that are
equivalent to 2MASS IDs. As-is, the APOGEE catalog will have more than one
entry for some 2MASS IDs due to the same source being targeted in distinct
observing programs. In order to assure a one-to-one crossmatch, we first
sorted the APOGEE table by SNR, then dropped the duplicate 2MASS entries with
the lower SNR value. From this crossmatched table, we removed all objects for
which there was no measured $\nu_{\rm max}$, $\Delta\nu$, or [Fe/H].

###  A.3 $\Delta\nu$ and $\nu_{\rm max}$ Calibration

Asteroseismic masses and radii are calculated according to scaling relations
following the notation of Sharma et al. (2016):

$\frac{M}{M_{\odot}}\approx\left(\frac{\nu_{\rm max}}{f_{\nu_{\rm
max}}\nu_{\rm
max,\odot}}\right)^{3}\left(\frac{\Delta\nu}{f_{\Delta\nu}\Delta\nu_{\odot}}\right)^{-4}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{3/2}\equiv\kappa_{M}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{3/2},$ (A1)
$\frac{R}{R_{\odot}}\approx\left(\frac{\nu_{\rm max}}{f_{\nu_{\rm
max}}\nu_{\rm
max,\odot}}\right)\left(\frac{\Delta\nu}{f_{\Delta\nu}\Delta\nu_{\odot}}\right)^{-2}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{1/2}\equiv\kappa_{R}\left(\frac{T_{\rm
eff}}{T_{\rm eff,\odot}}\right)^{1/2}.$ (A2)

These scaling relations are approximations linked to solar values and require
calibration factors, $f_{\Delta\nu}$ and $f_{\nu_{\mathrm{max}}}$, because
$\Delta\nu$ does not scale exactly as stellar mean density (White et al.,
2011) and $\nu_{\rm max}$ does not scale exactly as g/$T_{\rm eff}$ (Brown et
al., 1991; Kjeldsen & Bedding, 1995). These calibration factors vary by star
and depend on, in part, the star’s temperature and metallicity.

For APO-K2, we used APOGEE DR17 temperatures and metallicities to update
$f_{\Delta\nu}$ using Asfgrid131313Asfgrid is publicly available at
http://www.physics.usyd.edu.au/k2gap/Asfgrid/ (Sharma et al., 2016) with the
low-mass, low-metallicity extension (Stello & Sharma, 2022). Specifically, we
used Asfgrid to perform a multi-dimensional interpolation of $f_{\Delta\nu}$
using evolutionary state (see below), a Salaris-corrected metallicity using
APOGEE [$\alpha$/M] (Salaris et al., 1993), APOGEE temperature, $\Delta\nu$,
and $\nu_{\rm max}$.

The K2 GAP DR3 asteroseismic data that feed into the APO-K2 catalog were
additionally calibrated using Gaia parallaxes inferred from bulk stellar
motions from Gaia DR2 (Gaia Collaboration et al., 2018) according to the
methodology detailed in Schönrich et al. (2019) and accounting for selection
functions according to Schönrich & Aumer (2017). This technique corrects for
parallax bias, and indicates $\sim 10$ $\mu$as positional variations in the
Gaia parallax zero-point across K2 campaigns. The resulting Gaia parallaxes
were then compared to asteroseismic parallaxes, which can be computed from
asteroseismic radii in combination with the Stefan-Boltzmann law.
$f_{\nu_{\mathrm{max}}}$ was defined to bring the asteroseismic parallaxes
into agreement with the Gaia parallaxes. A separate value was computed for RGB
and RC stars, in recognition of their different stellar structure (and
therefore potentially different asteroseismic systematics) as well as their
potentially different selection functions (and therefore potentially different
Gaia parallax systematics).

###  A.4 Evolutionary States

With a long enough time baseline, the evolutionary state of stars can be
determined directly through an asteroseismic analysis of light curves (e.g.,
Bedding et al. 2011). Though this is possible for the original Kepler/APOKASC
data (e.g.; Pin18; Elsworth et al. 2019; M. Pinsonneault et al. 2024, in
preparation 2024), and some work has been done on doing the same light curve
analysis for shorter-baseline data using neural networks (e.g., Hon et al.
2018), these methods are still far from ideal for the relatively noisy K2
data.

Figure 9: A random sample of asteroseismically-classified RGB (filled) and RC
(unfilled) stars from the APOKASC-3 catalog, plotted in the space of the
spectroscopic fit. The $y$-axis is the uncalibrated APOGEE [C/N] and the
$x$-axis is an empirical surface gravity- and metallicity-dependent parameter
(see text for details). Stars with low surface gravities and therefore assumed
to be RGB/AGB stars spectroscopically are also shown. The dashed line is one-
to-one, under which stars would be spectroscopically classified as RGB and
above RC.

An alternative is to determine the evolutionary states of stars
spectroscopically using cuts in temperature, surface gravity, and element
abundances (specifically, [Fe/H] and [C/N]; e.g., Bovy et al. 2014, Elsworth
et al. 2019). To do this, we first define a “reference” temperature, which is
defined as the typical effective temperature we would expect for an RGB of a
given metallicity and surface gravity:141414The uncalibrated metallicity
([Fe/H]RAW) and surface gravity ($\log(g)_{\rm SPEC}$) from APOGEE DR17 are
used as the calibrated versions are dependent on APOGEE’s own evolutionary
state determinations.

$T_{\rm ref}=\alpha+\beta\,{\rm[Fe/H]_{\rm RAW}}+\gamma\,(\log(g)_{\rm
SPEC}-2.5).$ (A3)

Spectroscopic parameters are taken from APOGEE DR17, and $\alpha$, $\beta$,
and $\gamma$ are fit parameters that are determined through a nonlinear least-
squares fit for stars classified as RGB in the APOKASC-3 catalog. After
determining these values ($\alpha=4427.1779$, $\beta=-399.5105$,
$\gamma=553.1705$), a Monte Carlo optimization is used to find values for $a$,
$b$, and $c$ in a second expression:

$a-b\,(c\,{\rm[Fe/H]_{\rm RAW}}+T_{\rm eff}^{\rm SPEC}-T_{\rm
ref})-{\rm[C/N]_{\rm RAW}},$ (A4)

for which at least 98% of stars with a value less than 0 have an asteroseismic
RGB classification (Figure 9). We found best-fit values of $a=0.05915$,
$b=0.003455$, and $c=155.1$. This expression is then applied to our APO-K2
data set, assigning all stars for which A4 $<0$ as RGB and $>0$ as RC. The
exception is for stars on the upper-RGB (U-RGB), which we have conservatively
defined as $\log(g)<2.3$. These stars fall in a region of a Kiel diagram
beyond what is possible for the RC, and so have all been classified as RGB,
regardless of their value for A4.

###  A.5 Selection Function

Due to being the product of a cross-match between K2 GAP and APOGEE DR17, the
APO-K2 catalog is subject to selection and targeting choices made by both of
these surveys. By understanding the selection function of the underlying
samples, JSS24 assessed the completeness of the sample and existing effects on
the distributions of stellar parameters by comparing the selection function
for the whole APO-K2 sample to that of K2 GAP. This selection function is
defined in bins of mass and stellar metallicity, giving the ratio of the
number of stars in the APO-K2 sample versus the K2 GAP sample for each bin.
These ratios can then be used as weights when analyzing the APO-K2 sample.

This same comparison is also done between the APO-K2 sample and a set of
simulated stars drawn from a parent mock Galactic asteroseismic red giant
population generated with Galaxia (Sharma et al., 2011). These simulated stars
are drawn in color and magnitude space in accordance with the original K2 GAP
targeting selection function, and then have an expected asteroseismic
selection function applied in order to only retain stars with a $>90\%$
probability of having detectable asteroseismic signals (Sharma et al., 2022).

###  A.6 Ages

Our ages in this paper use the methodology and code that some of these authors
first presented in War21. Each star is defined by its values for $\log({\rm
Mass})$ (from asteroseismology), [Fe/H] and [$\alpha$/M] (from APOGEE DR17),
and the 1$\sigma$ uncertainties for each of these, which are then used as
look-up parameters in a regular four-dimensional grid—including $\log({\rm
age})$—constructed from sets of stellar evolutionary tracks.

Our stellar evolutionary tracks were generated originally by/for Tayar et al.
(2017) using the Yale Rotating Evolution Code (Pinsonneault et al., 1989; van
Saders & Pinsonneault, 2012). Given tabulated values for stellar
mass,151515These tracks do not apply any mass loss, and so treat the birth
mass as the present-day mass. present-day surface element abundances, surface
gravity, and temperature, these tracks provide the age of a star when leaving
the main sequence/joining the RGB (the MSLT). From these tracks, we created
three sets of grids at fixed $\log(g)$ values of 3.30, 2.50, and 1.74 (chosen
as discrete values in the tracks that approximately separate and bracket the
low-luminosity giants from the luminous giants), and assuming a solar He
abundance of 0.272683 and solar mixing length of 1.72. These grids are
tabulated at mass values between 0.6 M⊙ and 2.6 M⊙, with 0.1 M⊙ steps; [Fe/H]
values between -2.0 and 0.6, with 0.2 dex steps; and [$\alpha$/M] values of
0.0, 0.2, and 0.4.

We chose this approach because the MSLT is well-defined as a function of mass.
The age of a star at a specific location on the RGB requires at least
accounting for $T_{\rm eff}$, which is strongly correlated to age in this
regime. However, this also leads to small uncertainties in temperature (or
systematic offsets between model and data zero-points) being associated with
large swings in age, with a 50 K uncertainty in $T_{\rm eff}$ (typical of the
APOGEE data) corresponding to an $\sim 70\%$ uncertainty in age (Tayar et al.
2017, War21). For these reasons, and considering also that the amount of time
spent on the RGB ($\sim 10\%$ of the MSLT) is smaller than our stellar age
uncertainties (even for the APOKASC sample), the MSLT, as determined via mass,
serves as a very consistent age measurement across both asteroseismic samples.

Given values of $(x,y,z)\equiv(\log{M},{\rm[Fe/H]},{\rm[\alpha/M]})$, we can
then find an associated value for $w\equiv\log({\rm age})$ from our model grid
via four-point, four-dimensional Lagrange interpolation. In two dimensions,
four-point Lagrange interpolation works by constructing a Lagrange
interpolation polynomial of the form:

$L(x)=\sum_{j=0}^{3}w_{j}l_{j}(x),$ (A5)

where

$l_{j}(x)=\prod_{0\leq m\leq 3\atop m\neq j}\frac{x-x_{m}}{x_{j}-x_{m}}$ (A6)

are the Lagrange basis polynomials. For a set of four ordered pairs
$(x_{0},w_{0}),...,(x_{3},w_{3})$, this equation gives an estimate for values
of $w=L(x)$ for any given $x\in(x_{0},x_{3})$ while keeping that
$L(x_{j})=w_{j}$. Expanded to four-dimensions, this then becomes:

$L(x,y,z)=\sum_{j=0}^{3}\sum_{k=0}^{3}\sum_{m=0}^{2}w_{j,k,m}l_{j}(x)l_{k}(y)l_{m}(z).$
(A7)

Using a Monte Carlo method to produce 5,000 $(x,y,z)$ tuples for each star
(assuming a Gaussian error distribution for each parameter), we then used
Equation A7 to make two $\log({\rm age})$ estimates for each tuple, one with
each of the tracks for the $\log{g}$ values that bracket the input star’s
$\log{g}$. For this calculation, four nearest-neighbor points were chosen from
the model grid so that, for each parameter, $x_{0}<x_{1}<x<x_{2}<x_{3}$. The
exception is for $z\equiv{\rm[\alpha/M]}$ where, since the tracks from Tayar
et al. (2017) only sample three values of [$\alpha$/M],
$(z_{0},z_{1},z_{3})=(0.0,0.2,0.4)$ always, and so the associated basis
polynomial, $l_{m}(z)$, is always constructed using these points.

After obtaining 5,000 $\log({\rm age})$ estimates, the median and $\pm
1\sigma$ were calculated in $\log$ space for each grid, and then converted to
linear space for tabulation. The mean of the low- and high-$\log{g}$ grid
values is used, with the difference between them added as a systematic error.
However, in all cases, there is no offset between these two values.

##  Appendix B Ages on the Red Clump

In §2.4, we explain our reasoning for only considering RGB stars as being due
to the mass loss in the RC not being fully understood, leading to biases in
the ages of these stars (e.g., Casagrande et al., 2016). This issue could
theoretically be resolved with a prescription for this mass loss. However, it
is not clear how—if at all—mass loss on the RGB is related to stellar age or
abundance (An et al., 2019). Absent a physics-driven understanding of mass
loss that is applicable to stars on an individual basis, it may instead be
possible to characterize this bias empirically for a given population. For
instance, we might assume that the mean mass (or age) of $\alpha$-rich RC
stars in the APO-K2 sample should be the same as that of the $\alpha$-rich RGB
stars, and therefore apply the multiplicative offset between the means to each
star’s mass (or age) ex post facto. However, this would grant these stars very
limited utility: there would be little reason to trust the “corrected” age of
individual stars, and it would be difficult to separate genuine features of
the age profile from a mis-accounting of mass loss. Regardless, this may still
be an intriguing avenue for generally characterizing the expected magnitude of
this change.

Figure 10: Age distributions for the 2298 $\alpha$-poor and 1357 $\alpha$-rich RC stars in the APO-K2 sample. These ages have been estimated using present day asteroseismic masses, uncorrected for potential mass loss. Table 3: The partial data table for the APO-K2 RC sample, including our ages. The complete table is available in CSV format in the online journal. The main identifier for each star is its ID in the Ecliptic Plane Input Catalog (EPIC). In addition to the columns shown here, our table contains Galactic longitude and latitudes; Galactocentric $R$ and $Z$ positions; the uncalibrated values for APOGEE $T_{\rm eff}$, $\log{g}$, [Fe/H], [$\alpha$/M], and [O/Fe], as well as the associated errors for all relevant columns. The $\alpha$-rich Flag column has a value of 1 for $\alpha$-rich stars, 0 for $\alpha$-poor stars, and -1 for unclassified stars.  EPIC ID | APOGEE ID | Gaia EDR3 Source ID | RA | DEC | $T_{\rm eff}$ | $\log(g)_{\rm APO}$ | [Fe/H] | [$\alpha$/M] | [O/Fe] |  $\alpha$-rich Flag | Mass | Radius | $\log(g)_{\rm seis}$ | $\nu_{\rm max}$ | $\Delta\nu$ | Age | Modal Age
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  | (Deg.) | (Deg.) | (K) | ($\log({\rm cm/s}^{2})$) | (dex) | (dex) | (dex) |  | (M⊙) | (R⊙) | ($\log({\rm cm/s}^{2})$) | ($\mu$Hz) | ($\mu$Hz) | (Gyr) | (Gyr)
212532733 | 2M13122093$-$1202115 | 3621787595338351872 | 198.0872 | -12.0366 | 4644 | 2.25 | -0.296 | 0.220 | 0.307 |  | 0.77 | 11.33 | 2.21 | 20.4 | 3.10 | 28.2 | 27.5
211163406 | 2M03582817$+$2541126 | 67156795835480448 | 59.6174 | 25.6869 | 4616 | 2.45 | -0.060 | 0.037 | 0.065 |  | 1.22 | 10.70 | 2.46 | 36.6 | 4.26 | 5.7 | 5.1
212451149 | 2M13180826$-$1346410 | 3609158841003939968 | 199.5344 | -13.7781 | 4925 | 2.55 | -0.239 | 0.043 | 0.060 |  | 1.15 | 9.83 | 2.51 | 39.5 | 4.69 | 5.9 | 5.4
210618287 | 2M04235747$+$1713227 | 3313965708686969728 | 65.9895 | 17.2230 | 4438 | 2.21 | -0.059 | 0.073 | 0.108 |  | 1.10 | 12.27 | 2.30 | 25.7 | 3.30 | 8.4 | 7.9
212750556 | 2M13191487$-$0657577 | 3634414902267654400 | 199.8120 | -6.9660 | 4264 | 1.93 | -0.059 | 0.107 | 0.147 |  | 1.24 | 19.06 | 1.97 | 12.2 | 1.81 | 5.5 | 3.7
204987895 | 2M16111595$-$1956386 | 6245469248293963648 | 242.8165 | -19.9441 | 4556 | 2.12 | -0.411 | 0.154 | 0.199 |  | 0.89 | 12.54 | 2.19 | 19.7 | 2.88 | 13.4 | 10.2
205170082 | 2M16384287$-$1903121 | 4131201704833326592 | 249.6786 | -19.0534 | 4817 | 2.52 | -0.018 | 0.020 | 0.025 |  | 1.13 | 9.85 | 2.51 | 39.4 | 4.65 | 7.3 | 7.2
220637427 | 2M00504808$+$0952289 | 2582124996801990528 | 12.7003 | 9.8747 | 4618 | 2.52 | -0.304 | 0.235 | 0.275 |  | 0.99 | 9.31 | 2.49 | 39.1 | 4.72 | 11.0 | 11.1
246144695 | 2M23021866$-$0610311 | 2634924193707214080 | 345.5778 | -6.1753 | 4586 | 2.08 | -0.383 | 0.109 | 0.134 |  | 0.96 | 13.59 | 2.15 | 18.0 | 2.65 | 10.3 | 10.5
… | … | … | … | … | … | … | … | … | … |  | … | … | … | … | … | … | …

In Figure 10, we have plotted the age distributions for the $\alpha$-poor and
$\alpha$-rich populations of RC stars in the APO-K2 catalog, calculated
without any correction for mass loss. In one sense, these ages may still be
useful. We see a slight peak in $\alpha$-rich ages at $\sim$14 Gyrs, and a
gradual peak of younger $\alpha$-poor stars. However, the main difference
between these ages and those for the RGB stars is that there are many more
$\alpha$-rich RC stars at intermediate and young ages. Still, it is possible
that this is telling us something real about these populations. For instance,
we would expect more RC stars to go through mergers, and one possible
explanation for the young $\alpha$-rich population (that is also observed
among the RGB stars) is that these are merger products that have larger
present-day masses than they had at birth, and so they are interpreted as
being younger than they actually are (Chiappini et al., 2015; Martig et al.,
2015). These ages can be found in Table 3, though we strongly encourage
readers to pay heed to the limitations we have highlighted above and to
exercise caution when using these data.

##  Appendix C Ages for Metal Poor Stars

Thus far, we have excluded metal poor stars from our analysis due to
significant uncertainty in the calibration of the asteroseismic scaling
relations at ${\rm[Fe/H]}\lesssim-1$, which, in turn, leads to systematic
overestimates in the masses of these stars. Epstein et al. (2014) notably
presented this issue using APOKASC-1 data, showing an $\sim$10% overestimate
for the asteroseismic masses of halo stars as compared to model predictions.
Subsequent studies have shown mixed levels of systematics (see JSS24 and
references therein).

Table 4: The partial data table for the stars in the APO-K2 sample with ${\rm[Fe/H]}<-1$, including our ages, which have their masses calibrated and calculated using the uncalibrated APOGEE $T_{\rm eff}$ values. The complete table is available in CSV format in the online journal. The main identifier for each star is its ID in the Ecliptic Plane Input Catalog (EPIC). In addition to the columns shown here, our table contains Galactic longitude and latitudes; Galactocentric $R$ and $Z$ positions; the uncalibrated values for APOGEE [Fe/H], [$\alpha$/M], and [O/Fe], as well as the associated errors for all relevant columns. The $\alpha$-rich Flag column is not populated for these stars, as the $\alpha$-rich and $\alpha$-poor populations are not well defined at these metallicities, though it may be assumed that these stars are more similar to the $\alpha$-rich population.  EPIC ID | APOGEE ID | Gaia EDR3 Source ID | RA | DEC | $T_{\rm eff}^{\rm uncal.}$ | $\log(g)_{\rm APO}^{\rm uncal.}$ | [Fe/H] | [$\alpha$/M] | [O/Fe] | Mass | Radius | $\log(g)_{\rm seis}$ | $\nu_{\rm max}$ | $\Delta\nu$ | Age | Modal Age
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  | (Deg.) | (Deg.) | (K) | ($\log({\rm cm/s}^{2})$) | (dex) | (dex) | (dex) | (M⊙) | (R⊙) | ($\log({\rm cm/s}^{2})$) | ($\mu$Hz) | ($\mu$Hz) | (Gyr) | (Gyr)
212498924 | 2M13181597$-$1245237 | 3609527147335455744 | 199.5666 | -12.7566 | 4729 | 1.55 | -1.908 | 0.199 | 0.680 |  | 13.72 | 2.07 | 14.4 | 2.38 | 11.3 | 8.6
201226802 | 2M12005757$-$0333311 | 3600860727964887680 | 180.2399 | -3.5587 | 4567 | 1.70 | -1.172 | 0.242 | 0.323 |  | 15.73 | 1.94 | 11.0 | 1.92 | 13.4 | 9.9
228946727 | 2M12390889$-$0227021 | 3682958718591170560 | 189.7870 | -2.4506 | 4849 | 2.09 | -1.814 | 0.218 | 0.186 |  | 7.17 | 2.64 | 53.1 | 6.35 | 11.9 | 11.0
251545861 | 2M13223901$-$0228562 | 3638105756643622912 | 200.6626 | -2.4823 | 4766 | 2.17 | -1.075 | 0.207 | 0.574 |  | 10.01 | 2.33 | 26.4 | 3.77 | 14.4 | 13.0
211480777 | 2M08441675$+$1251409 | 602285256783119616 | 131.0698 | 12.8614 | 4598 | 1.81 | -1.038 | 0.159 | 0.409 |  | 18.97 | 1.92 | 10.4 | 1.71 | 4.4 | 4.2
216439618 | 2M18540793$-$2220424 | 4078727070039971968 | 283.5331 | -22.3451 | 4894 | 2.16 | -1.401 | 0.344 | 0.018 |  | 9.99 | 2.23 | 20.7 | 3.36 | 31.9 | 29.8
212510240 | 2M13171736$-$1230546 | 3609593083673337472 | 199.3223 | -12.5152 | 4517 | 1.77 | -1.031 | 0.274 | 0.383 |  | 15.33 | 1.92 | 10.5 | 1.90 | 21.4 | 18.8
203520011 | 2M16110245$-$2551498 | 6043468754447485696 | 242.7602 | -25.8638 | 4706 | 2.11 | -1.144 | 0.315 | 0.500 |  | 9.12 | 2.35 | 27.7 | 4.04 | 24.5 | 22.0
211326502 | 2M08550079$+$1022483 | 597826737133200256 | 133.7533 | 10.3801 | 5033 | 2.73 | -1.200 | 0.352 | 0.483 |  | 5.83 | 2.86 | 88.0 | 9.15 | 9.4 | 9.1
… | … | … | … | … | … | … | … | … | … |  | … | … | … | … | … | …

Table 4 contains all stars (230) from the APO-K2 catalog with
${\rm[Fe/H]}<-1$. All of these stars are assumed to be RGB stars, as no RC
stars should exist in this metallicity range. The masses (and, thereby, ages)
of these stars have been calculated using the uncalibrated temperatures from
APOGEE, which are those inferred from fitting to a grid of synthetic spectra
generated by the spectral synthesis code Turbospectrum (Alvarez & Plez, 1998;
Plez, 2012) under 1D local thermodynamic equilibrium. These temperatures are
shown by JSS24 to produce masses more in-line with what is expected for halo
populations from stellar isochrones, though a mean offset of up to $\approx
10\%$ may remain.

##  Appendix D Per-Campaign Age Distributions

Figure 11: Density distributions for the ages of stars in the $\alpha$-poor
(blue) and $\alpha$-rich (orange) RGB populations, split up by K2 campaign.
Excluded is Campaign 15, for which we were only able to recover ages for 3 RGB
stars. Densities are calculated per chemical population, such that the areas
under the $\alpha$-poor and $\alpha$-rich curves both, individually, sum to
one, and so the relative heights of these distributions are not reflective of
the relative stellar counts in each field. Embedded in the plot, we have also
included the number of stars per chemical population per campaign. Figure 12:
Map of the K2 Campaigns by Galactic Longitude and Latitude. Each point
represents an individual star, with each point being colored by age. The
background image is modified from ESA/Gaia/DPAC, and is applied using the
mwplot Python module and the MWSkyProjection map “equirectangular”.

In Figure 11, we show histograms for the age distributions of the
$\alpha$-rich and $\alpha$-poor populations, separated by campaign. In Figure
12, we map these campaigns on the sky, colored by age.

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