# The APO-K2 Catalog. I. 7,673 Red Giants with Fundamental Stellar Parameters
from APOGEE DR17 Spectroscopy and K2-GAP Asteroseismology

Jessica Schonhut-Stasik Neurodiversity Inspired Science and Engineering
Graduate Fellow.  Vanderbilt University, Department of Physics & Astronomy,
6301 Stevenson Center Ln., Nashville, TN 37235, USA  Joel C. Zinn NSF
Astronomy and Astrophysics Postdoctoral Fellow.  Department of Astrophysics,
American Museum of Natural History, Central Park West at 79th Street, New
York, NY 10024, USA  Keivan G. Stassun Vanderbilt University, Department of
Physics & Astronomy, 6301 Stevenson Center Ln., Nashville, TN 37235, USA  Marc
Pinsonneault  Department of Astronomy, The Ohio State University, McPherson
Laboratory, 140 W 18th Ave, Columbus, OH 43210, USA  Jennifer A. Johnson
Department of Astronomy, The Ohio State University, McPherson Laboratory, 140
W 18th Ave, Columbus, OH 43210, USA  Center for Cosmology and AstroParticle
Physics, Physics Research Building, 190 West Woodruff Ave, Columbus, OH 43210,
USA  Jack T. Warfield Department of Astronomy, The University of Virginia, 530
McCormick Road, Charlottesville, VA 22904, 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  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  Benoit Mosser
LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université,
Université de Paris, 92195 Meudon, France  Jamie Tayar Department of
Astronomy, University of Florida, Bryant Space Science Center, Stadium Road,
Gainesville, FL 32611, USA  Guy S. Stringfellow Center for Astrophysics and
Space Astronomy, University of Colorado at Boulder, 389 UCB, Boulder, CO
80309-0389, USA  Rachael L. Beaton Space Telescope Science Institute,
Baltimore, MD, 21218, USA  Department of Physics and Astronomy, Johns Hopkins
University, Baltimore, MD 21218, USA  Henrik Jönsson Materials Science and
Applied Mathematics, Malmö University, SE-205 06 Malmö, Sweden  Dante Minniti
Instituto de Astrofísica, Facultad de Ciencias Exactas, Universidad Andres
Bello, Av. Fernández Concha 700, Santiago, Chile e-mail<EMAIL_ADDRESS>Vatican Observatory, V00120 Vatican City State, Italy

###### Abstract

We present a catalog of fundamental stellar properties for 7,673 evolved
stars, including stellar radii and masses, determined from the combination of
spectroscopic observations from the Apache Point Observatory Galactic
Evolution Experiment (APOGEE), part of the Sloan Digital Sky Survey IV (SDSS),
and asteroseismology from K2. The resulting APO-K2 catalog provides
spectroscopically derived temperatures and metallicities, asteroseismic global
parameters, evolutionary states, and asteroseismically-derived masses and
radii. Additionally, we include kinematic information from Gaia. We
investigate the multi-dimensional space of abundance, stellar mass, and
velocity with an eye toward applications in Galactic archaeology. The APO-K2
sample has a large population of low metallicity stars ($\sim$288 at [M/H]
$\leq$ $-$1), and their asteroseismic masses are larger than astrophysical
estimates. We argue that this may reflect offsets in the adopted fundamental
temperature scale for metal-poor stars rather than metallicity-dependent
issues with interpreting asteroseismic data. We characterize the kinematic
properties of the population as a function of $\alpha$-enhancement and
position in the disk and identify those stars in the sample that are candidate
components of the Gaia-Enceladus merger. Importantly, we characterize the
selection function for the APO-K2 sample as a function of metallicity, radius,
mass, $\nu_{\mathrm{max}}$, color, and magnitude referencing Galactic
simulations and target selection criteria to enable robust statistical
inferences with the catalog.

††software: Python 3 (Van Rossum & Drake, 2009), numpy (Harris et al., 2020),
matplotlib Hunter (2007), pandas (Wes McKinney, 2010), Astropy (Astropy
Collaboration et al., 2013, 2018), mw_plot (https://milkyway-
plot.readthedocs.io/en/latest/)††facilities: Du Pont (APOGEE), Sloan (APOGEE),
2MASS, Kepler, K2

##  1 Introduction

Galactic archaeology probes the stars in our Galaxy as a fossil record,
investigating their histories to determine the formation and evolution of the
Milky Way. Studying the Milky Way’s fossil record is best achieved using
precise stellar ages, abundances, and kinematics; this data reveals the _when_
, _what_ , and _where_ of the Milky Way’s formation.

Galactic archaeology gains the most from extensive data sets, with large
numbers of stars over a broad parameter space, as ideally, these data sets
represent the Galaxy’s stellar population. Galactic stellar population
research has experienced considerable growth in the last decade, with an
abundance of space- and ground-based telescopes providing asteroseismic,
spectroscopic, and kinematic measurements with which to determine ages,
compositions, and positions for hundreds of thousands of stars; such data
allows for a cogent description of the Milky Way’s stellar building blocks for
the first time.

In terms of stellar targets, red giant stars are particularly beneficial for
Galactic archaeology (e.g., Stello et al., 2015). Due to their high
luminosities, red giant stars can be seen to greater distances than less
evolved stars, providing a better understanding of the edge of our Galaxy;
furthermore, their solar-like oscillations are observationally possible in a
longer cadence than that which is necessary to observe similar oscillations in
dwarfs and subgiants. Asteroseismic, spectroscopic, and kinematic catalogs of
giant stars already exist for various samples.

CoRoT (Baglin et al., 2006), Kepler (Borucki et al., 2010), K2 (Howell et al.,
2014) and TESS (Ricker et al., 2014) have been transformational for taking the
once-boutique field of asteroseismology into the era of large data sets. There
are catalogs of comprehensive asteroseismic mass data for red giants available
from Kepler ($\sim$16,000 Yu et al., 2018, see also Pinsonneault et al.
(2018)) and TESS ($\sim$158,000; Hon et al. (2021) and $\sim$1,700; Mackereth
et al. (2021)). Other asteroseismic data sets have published stellar
parameters but without spectroscopic cross-matching; for example, in TESS
($\sim$1,700 Hon et al., 2022) and K2 ($\sim$19,000, K2-GAP (DR3) Zinn et al.,
2022), the latter of which we use in this work.

Spectroscopic data provide temperatures, gravities, and detailed chemical
abundances. These data, for millions of stars, are being accumulated by APOGEE
(Majewski et al., 2017), GALAH (Buder et al., 2021), and LAMOST (Cui et al.,
2012).

Finally, Gaia (Gaia Collaboration et al., 2016) has also been ground-breaking
in producing precise astrometric and kinematic information for millions of
stars.

We can learn _when_ stellar populations formed in our Galaxy using precise
asteroseismic stellar ages (e.g., Miglio et al., 2009; Sharma et al., 2016;
Silva Aguirre et al., 2018; Rendle et al., 2019; Mackereth et al., 2020). Such
age estimates rely on asteroseismic masses in combination with stellar models,
metallicities, and temperatures. When performing population asteroseismology
on large samples of solar oscillators, it is common to depict the oscillation
pattern with two global seismic parameters for faster processing instead of
performing detailed mode-by-mode analysis on all targets. These asteroseismic
global parameters are the large frequency separation ($\Delta\nu$; related to
the mean density) and the frequency of maximum power ($\nu_{\mathrm{max}}$;
related to $\log$(g) and Teff). These global parameters can, in turn, provide
the input to calculate high-precision stellar masses and radii using scaling
relations (See Section 2.2 and Kjeldsen & Bedding, 1995; Brown et al., 1991).
These scaling relations, and resulting ages, depend explicitly on temperature
measurements; therefore, spectroscopic surveys provide a powerful complement
to time-domain asteroseismology.

Spectroscopic data measures stellar abundances powerfully describing _what_
the Galaxy’s stellar composition is. For example, Zinn et al. (2022) explored
the origin of neutron-capture elements for the K2 red giant sample, combining
asteroseismology with abundances from GALAH, and used them to test their
assumed production mechanisms by comparing them to models from Kobayashi et
al. (2020). An avenue where asteroseismic and spectroscopic measurements show
symbiosis is the research of $\alpha$-capture abundances. The consensus states
that as stars formed in our Galaxy, the proportion of metals increased through
a series of supernovae, providing an increasingly metal-rich medium for star
formation and altering the proportions of $\alpha$ elements to other elements
(hereafter, [$\alpha$/Fe]) (e.g., Burbidge et al., 1957; Timmes et al., 1995).
Therefore, the most metal-poor old stars present an excellent probe of the
nucleosynthesis pathways in the early Galaxy. Studies of the evolution of
$\alpha$ element abundances can reveal much about the past epochs of star
formation (Kobayashi et al., 2020). A major prediction from early models is a
single-valued function of $\alpha$ elements as a function of [Fe/H] (Haywood
et al., 2016). However, with the support of spectroscopic measurements, data
have shown a double-valued function (Fuhrmann, 1998; Prochaska et al., 2000;
Bensby et al., 2003). There are many proposed mechanisms for this
[$\alpha$/Fe]-[Fe/H] bimodality, such as radial migration (Sellwood & Binney,
2002; Schönrich & Binney, 2009; Nidever et al., 2014; Weinberg et al., 2017;
Sharma et al., 2021), two separate episodes of star formation (Chiappini et
al., 1997; Spitoni et al., 2019; Lian et al., 2020), and stars forming
throughout the Galaxy in clumpy bursts (Clarke et al., 2019). Ages from
asteroseismology are unique tools allowing us to test the models of
[$\alpha$/Fe]-[Fe/H] bimodality thanks to their precision, large sample sizes,
and a relatively large range of distances compared to isochronal aging
techniques. Samples from Kepler, K2, and TESS are promising for continued work
to constrain these models (e.g., Silva Aguirre et al., 2018; Rendle et al.,
2019; Mackereth et al., 2019a; Warfield et al., 2021).

Because stars differ in their formation histories in the Galaxy, kinematic
information is crucial to understand _where_ stars formed. With stellar
position and kinematic energy values derived from astrometry, we can determine
where stars formed and to which Milky Way component they belong (the halo,
thick disk, thin disk, or bulge); we can also determine if the stars
originated as part of an accretion event by considering the Galactic halo. As
the Milky Way developed its structure, it accreted much smaller galaxies, and
in some cases, they are still actively accreting. For example, the tidal
disruption of the Sagittarius dwarf galaxy is ongoing (Ibata et al., 1994),
the Magellanic Clouds are on their first infall (Besla et al., 2007), and
there are dozens of globular cluster streams currently encircling our Galaxy
(Bonaca et al., 2020). These accreted bodies come with unique kinematic
profiles, and they share integrals of motion (Helmi & de Zeeuw, 2000; Font et
al., 2011; Simpson et al., 2019) even several billion years later, allowing
us, in some cases, to identify their stellar components. These stars also
share chemical abundance values, aiding in their identification (Freeman &
Bland-Hawthorn, 2002; Venn et al., 2004; Lee et al., 2015; Grunblatt et al.,
2021).

Combining the abundances, masses, radii, velocities, and positions of evolved
stars allows us to assemble a catalog containing the essential data to address
the significant questions in Galactic archaeology. Furthermore, combining
these parameters can improve the precision of other diagnostic criteria. For
example, evolutionary states can be inferred to higher precision by
correlating spectroscopic properties with asteroseismic evolutionary (Jönsson
et al., 2020), as it is otherwise difficult to separate shell H-burning (RGB)
stars from core He-burning (RC) stars. It is also challenging to infer
parameters for red giant branch (RGB) stars, such as ages, with spectroscopic
information alone (e.g., Soderblom, 2010).

Another problem our extensive data set can explore is the conflict between
astrophysical priors and asteroseismic masses for low-metallicity stars in the
halo (Epstein et al., 2014). Asteroseismic masses for main sequence and RGB
stars are both accurate and precise to around $\sim$5% (e.g., Huber et al.,
2017; Zinn et al., 2019; Li et al., 2020) in the near–solar metallicity
regime. However, at low metallicities ([Fe/H] $=$ $-$1), asteroseismic masses
of RGB stars are consistently larger than the values expected for old halo
stars. Overestimates masses at low metallicities has been explored in Kepler
data (Pinsonneault et al., 2014); however, the sample of such stars was small,
and more data is needed to understand the origin of this effect.

Our catalog introduces the first APOGEE and K2 combination and includes the
incorporation of Gaia data. Because the K2 fields sample very different
populations from those seen in the original Kepler field, we can gain powerful
insights into the formation history of the Milky Way. This paper presents the
results from the dedicated targeting efforts of SDSS-IV (Beaton et al., 2021)
to observe the K2 fields. The resulting APO-K2 catalog contains the combined
data sets for 7,673 evolved Milky Way stars, combining spectroscopic (APOGEE
DR17, Abdurro’uf et al., 2021), asteroseismic (K2-GAP, Stello et al., 2015),
and astrometric (Gaia EDR3, Gaia Collaboration et al., 2021) data, with a
well-understood selection function (Sharma et al., 2022, hereafter S22). In
Section 2, we discuss the data and construction of the catalog. Section 3
presents the final sample, including a discussion of metallicity, the
selection function, and a comparison between the APO-K2 and APOKASC2
(Pinsonneault et al., 2018) samples. In Section 4, we investigate stellar
masses in the low metallicity regime, the kinematic properties, and the
[$\alpha$/Fe]-[Fe/H] bimodality as seen in the data set; we also identify halo
stars and potential GES members (Helmi et al., 2018; Mackereth et al., 2019b).
Section 5 presents the conclusions. The publication of asteroseismic ages for
this sample will follow in a companion paper (Warfield et al. in prep.), as
will a detailed analysis of sample abundances and multiplicity (Schonhut-
Stasik et al. in prep.).

##  2 Sample Data

Our catalog uses time domain data from the K2 Galactic Archaeology Program
(K2-GAP DR3, Zinn et al., 2022), spectroscopic data from APOGEE DR17
(Abdurro’uf et al., 2021), and astrometric data from Gaia EDR3 (included in
the APOGEE DR17 release (Abdurro’uf et al., 2021)). Our initial sample is
compiled by cross-matching K2-GAP DR3 with APOGEE DR17, keeping any source
that appears in both (Section 2.1). Having combined these sources we
calculated fundamental asteroseismic parameters (Section 2.2), incorporated
spectroscopic parameters (Section 2.3), and collated astrometric values
(Section 2.4), resulting in the APO-K2 catalog.

###  2.1 Cross-match

The K2-GAP DR3 catalog is the base of our cross-match, which has been analyzed
in various works (Stello et al., 2017; Zinn et al., 2020, 2022). The complete
list of K2-GAP targets observed by K2 contains 121,715 stars. From here, a
cross match with APOGEE DR17 was performed. Around 32% of the K2-GAP stars
were also observed in APOGEE (39,319 stars); by then cross-matching those
stars with accurate asteroseismology, we have a sample of 8,581 stars (7% of
those observed for K2-GAP).

The primary axis for our cross-match between the K2-GAP and APOGEE DR17
catalogs is their associated 2MASS IDs, which come from the Ecliptic Plane
Input Catalog (EPIC, Huber et al., 2016), compiled to support target selection
and management for the K2 mission. Prior to the match, we sort the APOGEE DR17
targets by signal-to-noise ratio (SNR), dropping the rows with lower SNR where
multiple entries for the same target are available. Dropping low SNR entries
ensures that each entry in the K2-GAP catalog receives a single spectroscopic
match, with the highest available SNR observation. To show which stars were
dropped, we created a flag for the cross-match (apo_crossmatch_flag) and set
the value to 1 where no drop was made and 0 otherwise. A minority of stars
were observed in multiple K2 campaigns (10% of the sample), and are noted in
the final catalog as having observations in multiple campaigns. Gaia EDR3 data
is already in the APOGEE DR17 table, and so no stars are dropped by adding
this information.

###  2.2 Asteroseismic Data

Asteroseismic detections for thousands of stars became possible with missions
such as CoRoT (Auvergne et al., 2009; De Ridder et al., 2009) and Kepler
(Huber et al., 2009; Stello et al., 2013). However, these missions had limited
sky coverage, selecting stars primarily for planet-hunting (Sharma et al.,
2016, 2017). In Kepler, this resulted in only a fraction of the available
oscillating targets observed, at a given combination of age, metallicity, and
distance (Silva Aguirre et al., 2018).

The Kepler mission targeted a field in the Cygnus and Lyra constellations,
with a four-year nominal mission, to find Earth-like planets around Sun-like
stars. Conveniently, Kepler’s precise photometry also allowed for the study of
asteroseismic variability in thousands of cool stars; the long cadence of
Kepler being particularly advantageous, with 29.4-minute observations ideal
for studying the oscillation periods of red giants (e.g., Bedding et al.,
2010).

Following the failure of the second of Kepler’s four reaction wheels in May
2013, the Kepler spacecraft began on a second mission, K2. The spacecraft was
re-purposed using solar wind for partial stabilization, allowing the telescope
to continue to observe, although the acquisition of Kepler’s original field
was no longer possible due to the lack of stable pointing. Therefore, unlike
the fixed original field, K2 observed 18 regions across a 360∘. ecliptic field
of view for $\sim$80 days at a time (Howell et al., 2014). Each K2 campaign
covers 115.64 sq. deg. — over 21 CCD modules, each made of 1024 $\times$ 2200
pixel CCDs (2$\farcs$98 pixel scale); there are slight gaps between the CCDs
and between each of the 21 modules.

The K2 mission had several scientific objectives proposed through guest
observer programs. The international collaboration of K2-GAP (Stello et al.,
2015) created a dedicated program to target red giants far beyond the solar
neighborhood, which provided a trove of new asteroseismic data by detecting
oscillations, with an aim to study the formation and evolution of the Milky
Way. Roughly 25% of the observed K2 targets were allocated to K2-GAP.

K2 performed 18 full observing campaigns before a lack of fuel forced
retirement in 2018. In Zinn et al. (2022, hereafter Z22), the authors present
K2-GAP for campaigns (C) C1-C8 and C10-C18111Some differences between
campaigns are worth noting: C3 had a slightly shifted field of view due to a
late change in roll angle; therefore, some of the proposed targets were
unobservable. During C10, a permanent failure of one of the CCD modules
occurred, resulting in stars in this, and all subsequent campaigns, being
observed partially, or not at all. C18 has very few seismic detections. C9 was
not used as it was a dense field chosen for microlensing, nor C19, which had
very few asteroseismic detections due to its short duration.. The Z22 catalog
represents the largest asteroseismic sample of red giants, with both
$\nu_{max}$ and $\Delta\nu$, in the literature to date.

Our catalog includes K2-GAP asteroseismic data from C1-C8 and C10-C18, and was
analysed by members of the Kepler Asteroseismic Science Consortium
(KASC)222https://kasoc.phys.au.dk. The K2-GAP lightcurves were reduced and
calibrated in a manner appropriate for asteroseismology (see Z22, where the
interested reader can find a full explanation). Briefly, each star in the
sample received analysis from six independent pipelines, which returned
asteroseismic parameters. The final asteroseismic values are the average of
the pipeline values, with an outlier rejection algorithm applied. For the
stars observed in multiple campaigns, a weighted average of the averaged
pipeline values from each campaign serves as the final set of parameters.

The final two asteroseismic parameters in the K2-GAP catalog are the frequency
of maximum oscillation power ($\nu_{\mathrm{max}}$) and the large frequency
separation ($\Delta\nu$); we only adopt stars that have a measurement of both
these values in K2-GAP. We determine asteroseismic masses and radii by
combining these parameters with scaling relations that use both spectroscopic
temperatures (APOGEE DR17) and correction factors defined in Z22, referred to
as the radius and mass coefficients ($\kappa_{\mathrm{M}}$ and
$\kappa_{\mathrm{R}}$, respectively). Full details are in Z22, but we include
the equations here for completeness:

$\begin{split}\frac{R}{R_{\odot}}\approx\bigg{(}\frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{max},\odot}}\bigg{)}\bigg{(}\frac{\Delta\nu}{f_{\Delta\nu}\Delta\nu_{\odot}}\bigg{)}^{-2}\bigg{(}\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\odot}}\bigg{)}^{1/2}\\\
\equiv\kappa_{R}\bigg{(}\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\odot}}\bigg{)}^{1/2}\end{split}$
(1)

$\begin{split}\frac{M}{M_{\odot}}\approx\bigg{(}\frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{max},\odot}}\bigg{)}^{3}\bigg{(}\frac{\Delta\nu}{f_{\Delta\nu}\Delta\nu_{\odot}}\bigg{)}^{-4}\bigg{(}\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\odot}}\bigg{)}^{3/2}\\\
\equiv\kappa_{M}\bigg{(}\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\odot}}\bigg{)}^{3/2}\end{split}$
(2)

In the above, $f_{\Delta\nu}$ represents a model-dependent correction to the
observed $\Delta\nu$ values. Its value depends on, in part, the temperature
and metallicity of the star. In K2-GAP DR3 (Z22), the EPIC served as the
source for this temperature and metallicity. With APOGEE DR17 temperatures and
metallicities in hand, we update $f_{\Delta\nu}$ using Asfgrid (Sharma et al.,
2016) with its low-mass low-metallicity extension (Stello & Sharma, 2022), in
accordance with the treatment described in Z22, including alpha-dependent
corrections to the metallicities. In the APOKASC publications, the so far
published K2-GAP work, and in this work, our asteroseismic parameters adopt
the method found in APOKASC2, where we adopt an empirical calibration against
fundamental data to set the $\nu_{\mathrm{max}}$ zero-point (see Z22 for more
details). Our catalog includes masses and radii using $f_{\Delta\nu}$ and
$\nu_{\mathrm{max}}$ values that have been updated since their initial
calculation in Z22. For $\nu_{\mathrm{max}}$, the empirical evolutionary-state
corrections from Z22 are removed and re-applied using the spectroscopic
evolutionary states used here; the solar reference value,
$\nu_{\mathrm{max,}\odot}$, for both red giant branch and red clump stars, is
taken to be 3076$\mu\mathrm{Hz}$. $f_{\Delta\nu}$ values have similarly been
recalculated with our new spectroscopic evolutionary states and updated APOGEE
DR17 temperatures and metallicities. We include the updated $f_{\Delta\nu}$
and an associated flag, mass and radii coefficients, stellar masses and radii,
and their associated errors in Table 1.

###  2.3 Spectroscopic Data

Through detailed studies of the stars in the Milky Way, APOGEE (Majewski et
al., 2017) is unravelling the compositional form of our Galaxy. APOGEE uses a
high-resolution (R $\sim$ 22,500), infrared spectrograph (Wilson et al., 2019)
operating in the H-band. For the Northern Hemisphere observations, APOGEE is
mounted on the Sloan Foundation 2.5m telescope (Gunn et al., 2006) at Apache
Point Observatory, New Mexico. In the southern hemisphere the 40-inch Irénée
DuPont telescope (Bowen & Vaughan, 1973) at Las Campanas Observatory, Chile,
houses the instrument.

APOGEE data are reduced, wavelength-calibrated, and co-added according to
Nidever et al. (2015). Spectroscopic parameters are calculated using the
APOGEE Stellar Parameters and Chemical Abundances Pipeline (ASPCAP; Holtzman
et al., 2015; García Pérez et al., 2016) and calibrated according to Holtzman
et al. (2018), with the model grids and the interpolation method described by
Jönsson et al. (2020). APOGEE primarily targeted K2 evolved stars (RGB and Red
Clump (RC)) as they are intrinsically luminous with significant flux in the
infrared; allowing high SNR observations at large distances. K2-GAP stars that
were not already observed in the GALAH survey (Buder et al., 2021) were
prioritized, although overlapping stars at lower priority were also observed.
About half of the targets observed by APOGEE reside in the disk of the Milky
Way ($b$ $\leq$ 16∘), with the remaining targets split between the bulge and
halo. Information on K2 object targetting can be found in Zasowski et al.
(2017); Beaton et al. (2021) and Santana et al. (2021); the latter two papers
include information on the relative weighting of the different classes of
targets.

In this work, we use APOGEE data obtained during the fourth phase of the Sloan
Digital Sky Survey (hereafter SDSS-IV) (Blanton et al., 2017) and analyzed in
the seventeenth (and final) data release (Abdurro’uf et al., 2021) of SDSS-IV
(hereafter DR17). DR17 contains 675,000 APOGEE targets over an additional two
years of SDSS observations, in both hemispheres, compared to DR16 (Jönsson et
al., 2020). The cumulative nature of SDSS data sets means that DR17 contains a
reprocessing of all data obtained, processed, and released in previous data
releases. Although, many beneficial, and significant, updates appear in the
DR17 release we only detail changes to the data release and reduction process
relevant to this work. These changes primarily affect the abundances, because
APOGEE uses the infrared flux method (IRFM) as an absolute standard for
effective temperature, and asteroseismology for the absolute standard for
$\log(g)$. For instance, ASCPAP was updated for DR17, with new sets of
spectral synthesis grids including NTLE effects for Na, Mg, K, and Ca, which
will be considered more in the upcoming APO-K2 abundance paper. Furthermore,
Gaia EDR3 data is incorporated into the DR17 data products, allowing us to
include astrometry in the catalog. DR17 also includes eight value-added
catalogs, including the “SB2” catalog of spectroscopic binaries (Kounkel et
al., 2021) and the “RedClump” catalog of RC stars (derived using a spectro-
photometric selection).

In the APOKASC samples (Pinsonneault et al., 2014, 2018), spectroscopic
evolutionary states were employed to differentiate between the RC and RGB
stars. For the purposes of this work, we adopt spectro-asteroseismic
evolutionary states. These are calculated by a similar method as described in
Warfield et al. (2021), and they depend on recalculating the temperature-,
surface gravity-, and abundance-dependent cut performed in Warfield et al.
(2021). Therefore, these spectroscopic evolutionary states rely on values of
$T_{\rm eff}$, $\log{(\mathrm{g})}$, and elemental abundances from APOGEE
DR17, and are trained on the asteroseismically-derived RGB and RC
classifications from APOKASC3 (Pinsonneault et al., in prep). Ultimately, we
assign stars as being on the RGB if their uncalibrated surface gravity333In
the APOGEE catalog, the “SPEC” subscript marks the uncalibrated version of the
parameter. $\log{(g)}_{\rm SPEC}<2.30$, or, where $\log{(g)}_{\rm SPEC}\geq
2.30$, if

${\rm[C/N]}\times 10^{3}<59.15\ -\ 3.455\ (155.1{\rm[Fe/H]_{SPEC}}\ +\ \Delta
T)$ (3)

In this equation, $\Delta T=T_{\rm eff}^{\rm(SPEC)}-T_{\rm ref}$ is the
difference between a star’s uncalibrated effective temperature and its
‘reference’ temperature, calculated from a fit to the ridge-line of known RGB
stars in the APOKASC3 data-set, which we define as

$T_{\rm ref}=4427.18\ -\ 399.5{\rm[Fe/H]_{\rm SPEC}}\ +\ 553.17(\log{(g)}\ -\
2.5)$ (4)

Although these new evolutionary states show promise in that they are reliable
in classifying the states accurately, there is a harder boundary in the
$\log(g)$ space than we see when using spectroscopy-only derived evolutionary
states. For example, there are few RC stars above a $\log(g)$ of 2.2 and this
should be considered when dealing with stars in this domain.

break

###  2.4 Astrometric Data

The largest ever source of precise astrometric data comes from the Gaia
mission (Gaia Collaboration et al., 2016), which was launched in 2013 and
measured the six-dimensional spatial and velocity distribution of nearly two
billion stars in the Milky Way. The astrometric information in this work comes
from the early-release of the third data release of Gaia (hereafter Gaia EDR3
or EDR3 Gaia Collaboration et al. (2021)), included in the APOGEE DR17 data
release.

##  3 Results

In this section we begin by exploring the selection function (Section 3.1). We
then detail the catalog provided in this work (Section 3.2). In Section 3.3 we
discuss the overall metallicity as a function of campaign, before finishing
with an exploration of the sample in terms of evolutionary states (Sections
3.4 and 3.5).

###  3.1 Selection Function

Figure 1: A collection of plots describing the completeness of our sample
through an analysis of the selection function, by first comparing the
simulated and K2-GAP samples, then the K2-GAP and APO-K2 samples. _Top Left:_
A 10$\times$10 grid showing the fractional density of K2-GAP stars compared to
the simulated sample in the mass [$M_{\odot}$] vs. radius [$R_{\odot}$] space.
The title displays the number of stars in each of the samples overall; these
values correspond to the scaling relation in the second term of Equation 7.
The color-bar shows the varying density in each bin from an over abundance of
simulated stars in red to an over abundance of K2-GAP stars in blue, compared
to expectations. Because $\hat{D}$ contains an asinh scale, the color-bar is
centered around 0, which corresponds to an equal number of stars in both bins.
The grey area, bordered in white, delineates between bins with real numbers
allocated and those that are calculated to have ‘nan’ or ‘inf’ values, due to
the bin being populated by no star in either samples. _Top Right:_ The same as
the top left but for the density of APO-K2 stars over K2-GAP stars, with an
abundance of APO-K2 stars shown in green and an abundance of K2-GAP stars,
again shown in blue. _Bottom Left:_ Three histograms showing the distribution
in radius between 2 $<$ R [R⊙] $<$ 30 of each sample (when cut on radius and
mass); the simulated sample (red), the K2-GAP sample (blue), and the APO-K2
sample (green). The simulated stars have been multiplied by a scaling factor
of 0.1, in all examples, to account for the over-sampling discussed in S22.
_Bottom Right:_ The same as the bottom left but showing the distribution of
masses over 0.25 $<$ M [M⊙] $<$ 2\.  Figure 2: Same as Figure 1 but in mass
[$M_{\odot}$] vs metallicity [dex] space. The scaling values can be determined
from the plot titles. The metallicity histogram (bottom left) ranges from -2.5
$<$ [M/H] [dex] $<$ 0.5, and the mass histogram (bottom right) ranges from 0
$<$ Mass [M⊙] $<$ 2\. See Figure 1 for more general information.  Figure 3:
Same as Figure 1 but in color (J - K) vs magnitude ($V_{mag}$). Again, the
scaling value can be determined from the plot titles. The color histogram
(bottom left) ranges from 0.4 $<$ J - K $<$ 1.3, and the magnitude histogram
(bottom right) ranges from 8 $<$ Vmag $<$ 16\. See Figure 1 for more general
information.  Figure 4: Same as Figure 1 but in magnitude ($V_{mag}$) vs
$\nu_{\mathrm{max}}$ [$\mu$Hz]. The scaling values can be determined from the
plot titles. The $\nu_{\mathrm{max}}$ histogram (bottom left) ranges from 0
$<$ $\nu_{\mathrm{max}}$ [$\mu$Hz] $<$ 240, and the magnitude histogram
(bottom right) ranges from 8 $<$ Vmag $<$ 16\. See Figure 1 for more general
information.

This section investigates the multiple layers of targeting and selection that
lead to the APO-K2 sample, so that interested readers may consider
completeness when using our catalog. The APO-K2 sample is fundamentally
comprised of K2-GAP DR3, cross-matched with APOGEE DR17, which therefore
includes selection and targeting choices made by both APOGEE DR17 and K2-GAP.
In order to understand how our sample maps onto a distribution of the Galaxy’s
red giant population, we must first understand how these selection choices
affect our data set. The K2-GAP data benefits from a simple and well-
understood selection function (S22). The advantages of K2-GAP is the clear and
reproducible target selection criteria, which can be used by us to map from
K2-GAP to APO-K2.

Although the purpose here is to understand the completeness of the APO-K2
sample, we stop short of comparing it to any suggested Galactic distribution;
instead we compare it to a theoretical target sample that was initially used
in targeting for K2-GAP. Further comparison to underlying Galactic populations
lie in the comparison between the Galaxia models (Sharma et al., 2011) used to
simulated the sample and the accuracy of their underlying assumptions.

Some important assumptions are made in the creation of this model that could
effect the way we perceive the completeness of our sample. For example, the
assumed metallicity of the thin and thick disks (discussed further in our
interpretation of Figure 2), and the choices related to star formation history
(which directly impact the distribution of $\kappa_{\mathrm{M}}$, and as a
consequence, mass). All choices made in the creation of the simulated sample
and it’s initial comparison to the K2-GAP targets are explored in Sharma et
al. (2021).

####  3.1.1 K2-GAP Selection

The targets for the K2-GAP sample were chosen with a color cut. Broadly, the
cut removed mostly dwarfs, with (J - Ks) $<$ 0.5, corresponding to dwarfs with
$M_{\mathrm{K}\mathrm{s}}$ $>$ 1\. This cut did exclude some giants, such as
those on the horizontal branch (the blue extension of the RC); however, these
are mostly rare, metal-poor stars, and are too hot to support solar-like
oscillations.

For C1, C2, and C3, the 2MASS H-band magnitude was used to select in
brightness. For later campaigns, an approximation of the V-band magnitude
measured from 2MASS J and K bands was used, as per Eq. 5 below. This cut was
chosen for later campaigns because K2 collects data in the $K_{\mathrm{p}}$
band, which is significantly bluer than the H-band, and more consistent with
the V-band (see Sharma et al. (2022) for more details):

$V_{JK}=K_{s}+2.0(J-K_{s}+0.14)+0.382\exp{[(J-K_{s}-0.2)/0.50]}$ (5)

####  3.1.2 APOGEE Selection

By cross-matching with APOGEE we insert elements of the APOGEE selection
function into our catalog. Beaton et al. (2021) discusses the prioritization
scheme for the targeting in each field of the K2 program. This provides a
meaningful comparison between targeted and observed stellar populations, and
can be used to judge the completeness of our sample, in parameter spaces that
may have been affected by our cross-match.

####  3.1.3 Comparing Targeting and Selection Functions

We compare our sample of APO-K2 stars to the observed K2-GAP stars and the
simulated stars from S22444github.com/sanjibs/k2gap and
http://www.physics.usyd.edu.au/k2gap/ — using data_name =
‘Galaxia-K2-sydai2-mrtd5’ for K2 observed stars and data_name = ‘k2-sydai2’
for the simulated stars.. We note that the values of asteroseismic mass and
radius that we report are the average over a number of pipelines from Z22,
whereas the K2 data only use the SYD pipeline (Huber et al., 2009), which may
cause a small difference in the inferred selection function when mapping from
K2 data to APO-K2.

From here we investigate four parameter spaces: mass vs. radius (Figure 1),
mass vs. metallicity (Figure 2), color vs. magnitude (Figure 3), and magnitude
vs. $\nu_{\mathrm{max}}$ (4). In each space, depending on the relative density
being computed, we cut down the larger sample to match the limits of the
smaller sample555For example, when calculating the number density between the
K2 and simulated data, we cut the simulated data to match the K2-GAP data
limits.. We created two 10 $\times$ 10 grids, in the investigated parameter
space; for example, in the mass vs. radius regime we bin in a 10 $\times$ 10
grid of mass and radius. We populated these grids with the relative density of
stars; one corresponding to the observed K2 sample/simulated sample, and one
corresponding to the observed K2 sample/APO-K2 sample. This allows us to
determine which stars are lost during the K2 observation phase and which from
the APOGEE selection function. When considering the simulated sample we
multiplied the amount of stars in each bin by 0.1, to compensate for how the
simulated stars were over-sampled by a factor of 10 to reduce Poisson noise
(see explanation in Sharma et al. (2022)). In all the density plots (Figures
1, 2, 3, and 4) grey areas cover bins where ‘nan’ or ‘inf’ values were
calculated; this occurs when one of the samples has no stars in that bin. The
resulting relative densities (D) were calculated and multiplied by a scaling
factor (the second term in Equations 6 and 7) that is specific to the
parameter space. These scalings allow the reader to plainly see the
differences in the bins. These equations are shown below, for the density of
the K2 sample relative to the simulated sample:

$\mathrm{D}=\frac{N_{i(\mathrm{K2})}}{N_{i(\mathrm{sim})}}\times\frac{N_{\mathrm{sim}}}{N_{\mathrm{K2}}}$
(6)

and for the density of the APO-K2 sample relative to the K2 sample,

$\mathrm{D}=\frac{N_{i(\mathrm{APO-K2})}}{N_{i(\mathrm{K2})}}\times\frac{N_{\mathrm{K2}}}{N_{\mathrm{APO-K2}}},$
(7)

where the first term represents the relative density between the samples for
that bin. Another scaling was applied to all bins, to more easily see
structure in the selection function (Equation 8). Using the value of $x$ = 0.6
(for the mass-radius space) and $x$ = 1.0 (for the other parameter spaces) in
the denominator will tend the scaling toward a log-scale at $<-$0.6 and $>$
0.6, and $<-$1.0 and $>$ 1.0, respectively.

$\mathrm{\hat{D}}=\mathrm{arcsinh}\Bigg{[}\frac{D-1}{x}\Bigg{]}$ (8)

One of the key distributions, when considering Galactic archaeology, is mass.
The mass distribution we would expect to see would be indicative of the most
recent burst of star formation and the lifetime of the giant branch as a
function of mass and metallicity (Wu et al., 2017); these factors conspire to
produce the observed mass distribution. In terms of individual mass limits we
expect a couple of clear boundaries. For example, no massive stars above 5 M⊙.
This is because high-mass red giants do not show solar-like oscillations and
therefore are not observed in our sample, while their evolved counterparts
have lower surface gravities than we probe. The lowest mass stars we expect
would correspond to the age of the Galaxy, discussed further in Section 4.1.
Higher or lower masses than these bounds likely reflect mergers, binary mass
transfer, or mass loss.

Figure 1 shows density plots in mass [$M_{\odot}$] vs. radius [$R_{\odot}$]
and histograms for these parameters. The color-bar on the density plot
indicates the scaled fractional density of stars. The scaling factor described
in the second term of Equation 6 for the left hand plot is (538414/95503)
$\sim$ 5.6 (Number of K2-GAP stars/Number of simulated stars) and (13722/8460)
$\sim$ 1.6 (Number of APO-K2 stars/Number of K2-GAP stars) on the right
(Equation 7). In the top left density plot, for K2/simulated stars, we see few
simulated stars in the low mass regime. This is likely due to the
uncertainties on mass adopted by the simulation, as they correspond to the
median uncertainties for the data (Sharma et al., 2011). We used the
temperatures, $\kappa_{\mathrm{M}}$ values, a 3% uncertainty on temperature
(in alignment with EPIC temperature), and SYD $\kappa_{\mathrm{M}}$ errors to
compute the fractional error on mass for the simulated stars. Over the whole
sample, the fractional mass error for the simulated stars is around 24%, but
51% when the sample is cut to M $<$ 1M⊙ stars. Therefore, as the simulation
adopts median uncertainties from the data, the simulation errors for mass are
underestimated. This likely causes the deficit in mass we see in the low-mass
regime for the simulated sample; this is also seen at the low mass end of the
mass histogram.

Looking to the top right plot, of the relative APO-K2 stars/observed K2 stars,
we see more APO-K2 stars at lower masses and radii (this is represented by the
green bump on the left of the plot). This may be the result of underestimated
masses, caused by different asteroseismic coefficients (and therefore
temperatures) being used to calculate the masses and radii in these samples.
To investigate the extent to which temperatures are the cause, we calculate
the mean raw APOGEE temperatures and mean EPIC temperatures (used for the K2
sample). The resulting temperature difference corresponds to a difference of
$\sim$6% in mass. One could argue that the abundance of low-mass stars seen in
the sample, that are absent in the simulated data, may be the result of mass
loss (including mass transfer and binary interaction that the simulation does
not account for), as discussed in works that investigate inferred mass loss
from asteroseismic data (Miglio et al., 2012, 2021; Tailo et al., 2022; Howell
et al., 2022; Kallinger et al., 2018, and Roberts et al., (in prep.)), however
due to the level of uncertainty in the simulated sample at low masses (as
discussed above), we do not wish to draw this conclusion here. The same
argument could be made for the K2-observed (green) histogram appearing to rise
above the (APO-K2) blue histogram in the bottom right plot; although we would
expect that in all cases the APO-K2 histogram should fall below the
K2-observed histogram as cuts are being made to the K2-GAP sample when
observed by APOGEE, the masses in these samples were calculated using
different temperatures and will result in different masses, likely creating
this effect. We discuss further the importance of the temperature scale for
asteroseismic masses in Section 4.1.

On the bottom left, in the radii histogram, we see a bump at 10R⊙ that does
not appear in the observed samples. This likely represents clump stars, which
are underrepresented in the data, as it can be difficult to detect
oscillations in RC (Mosser et al., 2018) with their lower oscillation
amplitudes (Yu et al., 2018). The extent of this bump in the histogram is due
to the relatively long lifetime the stars spend in the clump relative to the
RGB. RC stars share a similar core mass, which dictates their similar radii to
one another (however they do not share the same total mass).

Figure 2 shows the density plots for mass [M⊙] and metallicity [dex]. In this
case, we used [M/H] values666When discussing abundances in this work we use
the standard notation:
$[\mathrm{X/Fe}]\equiv\log_{10}\bigg{(}\frac{\mathrm{X}}{\mathrm{Fe}}\bigg{)}-\log_{10}\bigg{(}\frac{\mathrm{X}_{\odot}}{\mathrm{Fe}_{\odot}}\bigg{)}$.
for the simulated stars, and [Fe/H] for APO-K2 and the observed K2 sample, as
[Fe/H] was unavailable for the simulated sample777[M/H] is the average bulk
metallicity in APOGEE, while [Fe/H] involves a selected subset of iron lines.
In practice, the two agree closely for most APOGEE stars, since optical
metallicity values track iron abundances (See
https://www.sdss3.org/dr10/irspec/aspcap.php for more information.). For the
relative density of the K2/simulated sample we see a bimodal distribution
separated by an area where the relative amount of observed stars is higher.
The bimodal distribution is likely due to the assumed metallicity of the thin
and thick disks in the Galaxia model used to create the simulated stars. For
example, the metallicity adopted in the model of the thick disk (-0.162)
corresponds to the peak of the simulated histogram (Sharma et al., 2019).

On the right hand side, showing the relative density of observed K2
stars/APO-K2, we see more colored bins, indicating the presence of more
populated bins. This demonstrates that although these stars were observed in
K2 they did not appear in the simulated sample, resulting in the grey area on
the left hand density plot.

In the mass histogram, again we see an wealth of lower mass stars ($\approx$
0.6 $-$ 1 M⊙) in the APO-K2 sample. In the right hand histogram we see that
more stars were observed at lower masses in K2 than simulated. Note the
difference in mass range in Figure 1 compared to Figure 2. The mass range in
Figure 1 is purposefully more condensed to show the detail toward smaller
masses.

In Figure 3, we show the color-magnitude space. These plots clearly show color
cuts enforced by the K2-GAP target selection, where C1 and C2 have a maximum
magnitude of 7 in H-band, and all other campaigns have a maximum magnitude of
$V_{\mathrm{JK}}$ $=$ 9 (see Table 1 of S22). In the top left density plot for
the K2/simulated sample we see an abundance of K2 observed stars towards the
bluer end of the color axis, and more simulated stars toward the red end. This
is due to a number of factors: firstly, we would expect that as K2 reaches its
magnitude limit it will be less able to see faint, red, stars that would be
found in the simulation. Furthermore, the Galaxia simulation does not include
extinction, which is a process left to the user, so this blue edge may be due
to the cut off in J-K observation; therefore, the simulated sample is not
seeing some stars that would be reddened and pass the J-K cut. This lack of
extinction is also evident in the histograms, which show the simulated stars
blueward of the K2 observed stars. Finally, the simulations assume a color
dependence to the amplitude of oscillation, which may cause a mis-match
between the assumed and actual color dependence of the amplitudes.

Figure 4 shows the $\nu_{\mathrm{max}}$ vs. magnitude space. In both the
density plots we see fewer K2 and APO-K2 stars at low magnitudes (brighter
stars), likely due to targeting based on color-magnitude cuts (J – Ks $\geq$
0.5 and 7 $\lesssim$ H $\lesssim$ 14) across the sky. We also see more K2
stars towards fainter stars with high $\nu_{\mathrm{max}}$ values. S22 plot
their selection function in this same parameter space (bottom right hand plot
of Figure 10 in S22), and the top left density plot in this figure shows
similar results, confirming consistency in our work. In the density plot on
the right, we see a shift in the distribution, imposed by the APOGEE
selection. In the upper right corner of this plot, we see less APO-K2 stars,
corresponding to less of the seismically low SNR stars.

The histograms between the K2 stars and the simulated stars match well in this
set of plots because the selection of stars for the K2 sample were based on
the ability to determine pulsations, and the simulated sample were selection-
function-matched to the catalog (provided in S22) and designed to determine
the completeness of the K2 observed stars. In the right hand histogram we see
very good agreement between the simulated stars and the observed K2 stars.
This is due to the selection function for targeting taking place in V-band,
and these cuts being easily replicated in the simulation. The histograms also
show an overabundance of stars at lower $\nu_{\mathrm{max}}$ values
corresponding to the RC.

The selection function plots for each of the campaigns can be found on the
companion GitHub. It is important to consider the selection function over each
campaign due to potential differences (discussed in S22), such as light curve
duration, pointing accuracy, and the variation in crowding. For those
interested in studying completeness in the sample, the supplemental material
on the GitHub also includes tabulated values of the 2D histograms in XXX vs.
YYY vs. ZZZ etc., where X is the bin edges for the x-axis, Y is the bin edges
for the y-axis and Z is the relative density of stars in the bin.

###  3.2 The APO-K2 Catalog

The public catalog distributed with this publication contains a row for each
star, including the EPIC ID, K2 campaign number, and spectroscopic
evolutionary state888Seismic evolutionary states for the K2-GAP sample can be
found in] Zinn et al. (2022). From APOGEE, each row contains
$T_{\mathrm{eff}}$ and $\log(g)$, with associated errors and [M/H], [Fe/H],
[$\alpha$/M]. For K2, we include asteroseismic $\nu_{\mathrm{max}}$,
$f_{\Delta\nu}$, and $\Delta\nu$, mass, radius, their asteroseismic errors,
along with the mass and radius coefficients. We derived kinematic information
from Gaia positions, parallax, proper motions, and radial velocities. These
kinematics include Galactic eccentricity, angular momentum, and total energy,
derived using the Python module Gala (Price-Whelan, 2017; Price-Whelan et al.,
2022). We also provide two flags, one for determining whether a source is
high-[$\alpha$/M] or low-[$\alpha$/M], with an extra condition for stars that
are close to the dividing line. The second flag pertains to the
$f_{\Delta\nu}$, where 0.0 corresponds $f_{\Delta\nu}$ value within the bounds
of the grid and with a complete [Fe/H], the value is 1.0 if 75% of the Monte
Carlo chains go outside the bounds of the $f_{\Delta\nu}$ grid, and 2.0 if
there is incomplete [Fe/H], Teff, $\nu_{\mathrm{max}}$, or $\Delta\nu$
information to compute $f_{\Delta\nu}$.

Access to the APO-K2 catalog can be found as an electronic table with this
paper and on the companion GitHub (https://github.com/Jesstella/APO-K2). The
public catalog contains all information needed to re-create the plots in this
paper. The GitHub and paper website (https://www.jessicastasik.com/apo-k2)
also contain supplemental plots, including the selection function density
plots for the individual K2 campaigns, and their density matrices as .csv
files. Furthermore, in the interest of accessibility, alternative text999For
those who may be blind or visually impaired. for plots can also be found at
these sources, as well as author information, and relevant conference
presentations. The APOKASC2 data used in this paper can be found directly at
http://vizier.u-strasbg.fr/viz-bin/VizieR-3?-source=J/ApJS/215/19, or by way
of Pinsonneault et al. (2018).

###  3.3 Sample Metallicity

Figure 5: Each histogram shows the APOGEE [M/H] distribution for the K2
campaigns. The bottom right-hand plot, with hatched markings, is the
metallicity distribution of the Kepler field taken from APOKASC2 (Pinsonneault
et al., 2018). Color indicates the mean metallicity, with a darker blue
corresponding to a higher mean metallicity. The title of each plot gives the
campaign number, the number of stars in the campaign, the average metallicity,
and the average Galactic latitude of the campaign field in.

K2 samples multiple Galactic lines of sight providing a broad overview of the
metallicity distribution of the Milky Way. Overall, our catalog provides a
range of metallicities for evolved stars, particularly in campaigns of high
and low Galactic latitudes.

Figure 5 shows a histogram of [M/H] distribution for each K2 campaign. Each
histogram contains all stars observed in the campaign, including those that
appear in multiple campaigns. The bottom right histogram shows the metallicity
distribution for the APOKASC2 sample. C10 boasts the lowest mean metallicity
with $\langle$[M/H]$\rangle=-$0.50 [dex] at b = 59.6∘101010Where b is the
average Galactic latitude for the field.. The highest average metallicity is
in C11 ($\langle$[M/H]$\rangle=-$0.05 [dex] at b = 9.0∘). This is in contrast
to the APOKASC2 histogram with a mean metallicity of $-$0.02 [dex] (at b =
13.5∘); all of the K2 histograms have lower mean metallicity than the Kepler
field.

Kepler’s objective of observing near-by dwarf stars probably resulted in a
dearth of metal-poor giants; therefore, displaying a striking difference to
our sample. One factor in the difference in average metallicity is that many
Kepler giants were selected as planet candidate targets in a (generally)
magnitude limited sample, strongly disfavouring luminous giants. This was
then, in turn, biased against the metal-poor targets because they are (on
average) further away, thus more likely to have a truly high luminosity at
fixed magnitude. Reinforcing the dearth of distant red giants, Wolniewicz et
al. (2021) found a strong selection bias against cool, low-luminosity, red
giant stars in Kepler, where the observed red giants decrease from $\approx$
80% at $K_{\mathrm{p}}$ = 14 mag to $\approx$ 50% at $K_{\mathrm{p}}$ = 15
mag; with only $40\%$ of red giants at $K_{\mathrm{p}}$ = 15 being observed
for more than 8 quarters. They also note that the scarcity of observed red
giants could be because the goal of Kepler was to observe solar-type stars;
therefore, many identified red giants were removed from the target list after
one quarter. The K2-GAP sample was selected with completeness of the evolved
stars in mind, a sample ideal for Galactic archaeology. Furthermore, the K2
sample is, on average, farther away from the Galactic plane than the stars in
the Kepler field. This is evident in Figures 6 and 5, as the stars at higher
and lower Galactic latitude are generally more metal-poor.

###  3.4 Sample Overview

Figure 6: Footprint of each of the K2 campaigns in this sample, on a backdrop
of the Milky Way, with axes representing the coordinates [degrees]. Each point
represents a star in our catalog and color indicates its metallicity. The
position of the Kepler mission field is shown for reference, also colored by
metallicity [Fe/H], as taken from APOKASC2. Gaps seen in the telescopes field
of view correspond to CCD modules 3 and 7, which failed prior to the K2
mission. Each campaign is labelled with the campaign number in the format
‘C[number]’. The metallicity color-bar has been scaled between -1.0 and 0.5 to
show the resulting metallicity distribution within each campaign. C9 and C19
are not used and therefore not shown. Background image modified from
ESA/Gaia/DPAC, and is applied using the mw_plot Python module and the
MWSkyProjection map ‘equirectangular.’ An accessible black and white plot with
alternative text can be found at the companion GitHub and paper website.

Figure 7: Two H-R diagrams showing the APO-K2 sample. Left: RC stars. Right:
RGB stars. The title of each plot includes the number of stars in the plot.
Only one observation of each star is plotted for stars that were observed in
multiple campaigns. Stars with R $>$ 30$R_{\odot}$ are removed. The
‘ASPCAPFLAG’∗ must be 0 for the stars to be included in this plot. This
removes 1,174 stars from the total sample. The marker size, for filled
circles, corresponds to the asteroseismic stellar mass [$M_{\odot}$] (See
Section 2.2) and the color scale corresponds to [M/H]] [dex] from APOGEE. A
representative error bar is given in the lower right of each plot for the
stars shown as filled circles. The points are ordered by metallicity in
descending order so that low metallicity stars are plotted on the top of the
scatter. Luminosity is calculated using the asteroseismic radius and APOGEE
temperature. Accessible versions of this plot can be found at the GitHub and
website.

*‘ASPCAPFLAG’ corresponds to a bit mask of warnings produced by the ASPCAP pipeline. The APOGEE bit masks are used to provide information associated with individual pixels in a one-dimensional spectrum (For more information, see https://www.sdss4.org/dr17/irspec/apogee-bitmasks/.) When a star has a value of a 0 integer from the bit mask, no warnings have been flagged for the observation.

Figure 8: Two Kiel diagrams showing the APO-K2 sample. All cuts, color, and
placement are the same as in Figure 7 with the exception of the circle size
for the RGB plot (right), which is indicative of asteroseismic stellar radius,
as opposed to stellar mass.  Figure 9: Four scatter plots showing the RC
stars, in various parameter spaces. Each plot has Teff on the x-axis and the
y-axis depicting stellar Luminosity [L⊙] (top left), $\log(g)$ [dex] (top
right), Radius [R⊙] (bottom left), and Mass [M⊙] (bottom right). The entire
sample of RC stars is shown by grey circles. Stars with L $>$ 100L⊙ are
represented by purple triangles and stars with L $<$ 100L⊙ and M $>$ 1.8M⊙ are
shown by green diamonds. Error bars are shown for the radii and masses in the
bottom left and right panels, respectively.

Our final APO-K2 sample contains 7,673 unique stars with spectroscopic,
asteroseismic, and astrometric data. The sample includes a total of 8,460
observations, occurring across multiple campaigns, sometimes observing the
same star in multiple campaigns. When separated into evolutionary states, we
have 2,465 unique RC stars and 5,208 unique RGB stars. The extensive overlap
(See Section 2.1) between the APOGEE catalog and K2-GAP program is due to the
priority ordering for target selection (See Table 3 of S22). Targets were
chosen for observation in K2-GAP according to criteria such as 2MASS and SDSS
color and membership in existing spectroscopic survey catalogues, with each
criterion given a priority ranking. This selection method gave the highest
priority to APOGEE targets; APOGEE, in turn, prioritized observing spectra of
K2-GAP stars resulting in the large overlap we see.

The sample reaches $\sim$60∘ above and below the plane of the Milky Way, and
explores the Galactic Center. The position of these stars relative to the
plane of the Milky Way are shown in Figure 6. Color represents [Fe/H] for each
star in our sample (from APOGEE), with the color-bar scaled between -1.0 and
0.5 [dex]; the actual maximum and minimum values of [Fe/H] are -2.45 and 0.51
[dex], respectively. This scaling shows the relatively metal-rich state of the
Kepler field in comparison to K2 and the metal-rich state of C2, C7, C11, and
C15 relative to the other Campaigns; we discussed campaign-specific
metallicities in Section 3.3. We do not include C9, nor C19 (see Section 2.2).
Throughout this work we investigate, in particular, the low-metallicity stars
specifically those below $-$1.0 [dex], which have not been available in large
asteroseismic catalogs until now. For comparison, there are $\sim$ 288 stars
in this catalog with [Fe/H] $\leq$ $-$1.0, and only $\sim$ 35 in the APOKASC2
catalog.

###  3.5 Identifying Stellar Sub-Populations

We separated the APO-K2 sample into two sub-samples (RC and RGB) using their
evolutionary states (as described in Section 2.3). Figure 7 illustrates that
combining asteroseismology and spectroscopy enables us to decipher areas on
the H-R diagram containing interesting details like the RGB bump and secondary
red clump stars (Tayar et al., 2019). Figure 7 represents the stars on a H-R
diagram. The left plot presents the RC stars prominently, with the RGB stars
indicated by grey crosses in the background, and the right-hand side presents
the RGB stars prominently, with the RC stars indicated by the grey crosses. By
separating these two groups of stars, we hope to make the different samples
clear to the reader. Figure 8 displays the same stars as in Figure 7 but in
$\log$(g) $-$ Teff space, also known as a Kiel diagram. The size of the points
indicates the asteroseismically derived mass for Figure 7 and the left hand
plot of Figure 8. The right hand side of Figure 8 has asteroseismically
derived radius represented by point size.

The size of our sample is clear in Figures 7 and 8, and coneys a large
collection of RC and RGB stars. These figures show a rich array of features
that are qualitatively in agreement with the physics of RGB and RC phases of
stellar evolution (Cox & Giuli, 1968; Kippenhahn et al., 2013; Girardi, 2016).
We mention some of these patterns here, and ongoing investigation explores to
what extent the RC observations are consistent with models as a function of
mass and metallicity. These plots indicate the importance of precisely derived
evolutionary states (see Section 2.3); there is overlap between higher mass
RGB stars and RC stars, which are more precisely distinguished
asteroseismically that by using classical H-R distinction.

The combination of spectroscopic temperatures with well constrained
asteroseismic radii (used to calculate luminosity) allows us to investigate
small sub-samples of secondary RC and RGB bump stars. The RGB bump stars are
clearly shown by an over density of grey crosses on the left-hand side of
Figure 7, at a temperature of $\sim$4650K and 30$L_{\odot}$.

When considering these plots as one sample, the RC stars are generally at
higher temperatures than their RGB counterparts. The RC stars show an increase
in mass with temperature and luminosity. Using mass as a proxy for age, this
gradient implies youth at higher temperature and luminosity. A clear
metallicity gradient can be seen with temperature, with lower metallicities
corresponding to hotter stars. The existence of the secondary red clump is
seen on the left-hand side of Figure 8, with a collection of stars around
$\log(g)$ $\sim$2.75 dex and $T_{\mathrm{eff}}$ $\sim$5000 K. We explore
possible secondary red clump stars further in Figure 9, as they are not seen
clearly in the left hand plot of Figure 7.

The RGB stars also show a gradient in metallicity towards higher temperature
and luminosity, most clearly in Figure 8, where metallicity decreases toward
the left of the branch. This gradient also corresponds to increased radii (see
marker size in Figure 8). These relationships and the slight dispersion of
points at higher luminosities (representing a wider range of temperatures for
the more luminous stars) may be representative of AGB stars, although it is
difficult to fully determine stars belonging to the AGB. Using MIST models we
were able to rule out AGB stars at $\log(g)$ $>$ 2.0 with reasonable
certainty, for low metallicity stars ([Fe/H] $<$ -1.5), and with relative
confidence at solar metallicities. However, we found a $\log(g)$ cut alone
does not rule out all AGB stars.

Using this sample we attempt to make a distinction between secondary RC stars
in different stages of evolution. In Figure 9 we further investigate possible
members of the secondary red clump. This plot shows RC stars plotted using
four different parameters as a function of temperature. Each of these plots
uses grey circles to show the RC sample as a whole, purple triangles to
indicate stars with a luminosity $>$ 100L⊙, and green diamonds to indicate
stars with a mass $>$ 1.8M⊙ and a luminosity $<$ 100L⊙. The 1.8M⊙ cut was used
to select for secondary clump stars, which do not undergo a helium flash and
therefore show a range of luminosities due to their range of core masses
(Girardi, 2016). In Teff vs. luminosity we see a number of high luminosity
stars above the RC, whilst the more massive stars are hidden within the RC.
However, when plotted using $\log$(g), the high luminosity stars are inside
the RC, and the high mass stars fall below it, indicating that these two
samples are separate and distinct.

The high luminosity clump stars are likely secondary red clump stars due to
their high mass, but may be later in their evolution compared to the other
high-mass clump stars. We would expect their radii to expand with evolution as
the core of the star contracts and heats up, consistent with the majority of
them having radii $>$ 12.5 R⊙.

On the other hand, the stars that may be earlier in their evolution (green
diamonds) appear on the Kiel diagram as secondary red clump stars, falling
below the main RC. These stars also have higher masses than the main clump, as
seen in the bottom-right figure, but their radii are more consistent with the
RC than the purple triangles, as they have not yet started to expand.

###  3.6 Comparison to APOKASC2

Pinsonneault et al. (2014) presented the initial combination of asteroseismic
(Kepler) and spectroscopic (APOGEE) data for 1,916 evolved stars in the first
APOKASC catalog (hereafter, APOKASC1). They used the asteroseismic data to
calibrate the relationships between parameters such as mass and age with
spectroscopic observables. The second APOKASC release (Pinsonneault et al.,
2018, hereafter APOKASC2) looked at an additional 4,760 evolved stars (6,676
in total) with an empirical approach, combining asteroseismic measurements
across different methodologies to calculate averaged values and reduce
systematic errors. APOKASC1 used SDSS DR10 (Mészáros et al., 2013) parameters,
and the APOKASC2 used SDSS DR14 (Holtzman et al., 2015).

The K2-GAP asteroseismic parameters used here follow a similar averaging
approach (Zinn et al., 2022) to APOKASC2. The main difference between the
approachs is in target selection. The K2 stars were chosen =as a function of
magnitude and color with the intention of creating a clean and easy to
reproduce sample, unlike the Kepler stars.

K2 is better suited to Galactic archaeology as compared to Kepler. Due to its
multiple Galactic lines of sight, it allows wider coverage of the Galaxy,
observing multiple stellar populations at greater distances both radially and
above and below the Galactic plane, thus broadening our understanding of the
Galaxy’s stellar composition as a whole. However, the downside to K2’s wide
coverage is the shorter length of lightcurves in comparison to Kepler,
resulting in lower SNR, meaning that oscillation spectra are harder to
analyze.

APO-K2 and APOKASC2 also differ to APOKASC1 in the addition of stars in the
low-metallicity regime (See Figure 5). Furthermore, in APO-K2 we also observe
stars across a wide range of Galactic latitude and longitude and to farther
distances. These broader parameter spaces subsequently extend our
understanding of related parameters, i.e. the high-mass vs. low-metallicity
space (See Section 4.1) and the [$\alpha$/Fe]-bimodality (See Section 4.2).

##  4 Discussion: Example Applications of the APO-K2 Catalog

###  4.1 Asteroseismic Mass vs. Metallicity

Figure 10: Asteroseismic mass [M⊙] as a function of APOGEE [Fe/H] [dex],
calculated using asteroseismic scaling relations and mass and radius
coefficients (as discussed in Section 2.2). Dartmouth isochrones are shown by
grey lines and labelled with their ages to the right of the plots (8 Gyr, 10
Gyr, and 14 Gyr). The black points in the top plot represent masses derived
using calibrated temperatures from APOGEE and the existence of so many black
points above the isochrone lines demonstrates the existence of potential
overestimates in asteroseismic mass for the low-metallicity regime. Red
triangles in the bottom plot represent masses calculated using the
uncalibrated temperatures from APOGEE. The title shows cuts made to this plot
(e.g., halo and RGB stars only). Binned medians of both samples have been
added using a black (red) line on the top (bottom) plot with 1$\sigma$ errors
shown by the shaded regions. Any stars with the APOGEE flag ‘STAR_BAD’ have
been removed and those with and ASPCAPFLAG $=$ 0.0 (corresponding to no
warnings) are surrounding by blue.  Figure 11: A Toomre diagram of our sample,
showing stellar velocity relative to the local standard of rest. The color-bar
in this plot represents [Fe/H] from APOGEE and a dotted black line represents
a velocity of 220 [km/s]. This line is used to delineate between halo stars
and the rest of the sample, for use in Figure .

Epstein et al. (2014) showed that the asteroseismic masses determined through
scaling relations in the APOKASC1 sample are higher than expected for the halo
stars. Given that the halo stars are old, the median true mass in this regime
should be lower. Sharma et al. (2016) applied the model-motivated $\Delta\nu$
corrections, f${}_{\Delta}\nu$, to the scaling relations, which largely
removed discrepancy reported by Epstein et al. (2014).

Other authors have investigated low-metallicity asteroseismic masses and
compared them against Gaia-derived masses (Zinn et al., 2019) and M4 cluster
masses (Miglio et al., 2016; Valentini et al., 2019; Matsuno et al., 2021;
Tailo et al., 2022; Howell et al., 2022). The latter studies, which also use
the (Sharma et al., 2016) $\Delta\nu$ corrections also find no evidence for
the overmassive asteroseismic masses, in agreement with Sharma et al. (2016).
Another so far unconsidered aspect is a $\nu_{\mathrm{max}}$ metallicity
dependence, as discussed by Viani et al. (2017), who conclude that although
the effects on main sequence and subgiant stars are minimal the effects on
giant stars are much larger. With our relatively large sample of low
metallicity stars, we therefore revisit the issue of seismic scaling-based
mass for the low metallicity regime.

In Figure 10 we show a selection of the low metallicity ($<$ -1.0 dex) halo
stars (V $\geq$ 220 km$s^{-1}$) in our sample (See Figure 11 for the graphic
depiction of our halo star cut). To ensure that the stars with masses higher
than expected are not the result of explainable factors we make the following
cuts: all stars with the evolutionary type ‘RC’ are removed as it is likely
that the evolutionary states in this regime are wrong, because hot RC stars
would not show oscillations. All stars with a mass $>$ 1.6 M⊙ are removed to
reduce the likelihood of merger products. There is also the recognition that
there are asteroseismic biases in the highly luminous stars, so we remove
stars with R $>$ 30R⊙ (Mosser et al., 2013; Stello et al., 2014; Kallinger et
al., 2018; Zinn et al., 2019).

We use Dartmouth isochrones (Dotter et al., 2008), with $\alpha$-enhancement
(of $\alpha$ = 0.4), to demonstrate where we would expect our stars to
fall111111In Section 3.5 we use MIST models (Dotter, 2016; Choi et al., 2016)
instead of the Dartmouth isochrones used here. The use of Dartmouth at this
point was to allow for an $\alpha$-enhancement factor that was not necessary
for the AGB cut.. We also removed stars with a ‘STAR_BAD’ flag.

Initially, we plotted our stars using the calibrated temperatures from APOGEE
(Top plot in Figure 10). The calibrated spectroscopic temperatures available
in APOGEE are calibrated to González Hernández & Bonifacio (2009). Full
details are available in Holtzman et al. (2018). Using the standard
(f${}_{\Delta}\nu$ = 1) scaling relation we find, similarly to Epstein et al.
(2014), that the asteroseismic masses of our metal-poor halo stars are much
higher than Galactic models would suggest given the age of the Universe.

Given the uncertain temperature calibration in the low-metallicity regime, we
then considered the effect of using uncalibrated, ionization balance
temperatures from APOGEE. Plotting the stars using the uncalibrated
temperatures (Bottom plot in Figure 10) we found much better agreement with
the isochrones; this indicates that temperature calibration may be a key
factor in the resulting mass values at low metallicities. The 2009 IRFM
relationship (González Hernández & Bonifacio, 2009) was anchored on a small
sample of metal-poor stars, and a modest zero-point adjustment is reasonable.
The mean mass of stars with the calibrated temperatures (black dots in Figure
10) is 1.02 $\pm$ 0.25 M⊙, and the mean mass using the uncalibrated
temperatures (red triangles in Figure 10) is 0.88 $\pm$ 0.24 M⊙, indicating an
overestimate of masses (using calibrated temperatures) of $\sim$14% around the
mean. This is similar to the value reported by Epstein et al. (2014),
describing an overestimate of 11% $\pm$ 4%.

Two mass limits on this plot are of particular interest considering the
precision of our asteroseismic masses. The first are the few RGB stars at M
$<$ 0.8 M⊙, which corresponds to the approximate minimum mass of an RGB star
at the current age of the Galaxy (see mass for 14 Gyr isochrone with
Dartmouth). These stars may have undergone mass loss throughout their
evolution and represent interesting future studies. Conversely, 1.6 M⊙
corresponds to the maximum possible mass (the maximum merger mass of two 0.8
M⊙ stars). The RGB stars above this limit warrant further follow-up as they
could represent interaction products.

###  4.2 [$\alpha$/Fe]-Bimodality

Figure 12: [$\alpha$/Fe]-bimodality plot with the metallicity [Fe/H]
([$\alpha$/Fe]) on the x-axis (y-axis). Marker color represent the
asteroseismic mass [M⊙], which have been truncated for clarity, with the full
range of masses show in the title. The darker blue colors correspond to lower
masses, with the data sorted by mass so that higher mass stars appear on the
top of the scatter. The size of each marker represents the Galactic
eccentricity from Gala. A dividing line is drawn to separate the low- and
high-[$\alpha$/Fe] stars, and this separation results in the flag contained in
the APO-K2 catalog. Representative error bars are shown for both the
relatively high and relatively low metallicity stars, where the separation
occurs at [Fe/H] = -1.0.  Figure 13: $\alpha$-bimodality plot for both
APOKASC2 (grey) and APO-K2 (blue) overplotted to highlight the increase of
sample size in this parameter space.

An important abundance relation for exploring chemical enrichment in the Milky
Way is the [$\alpha$/Fe]-bimodality. This association compares $\alpha$
process elements (e.g., O, Mg, Ca, and Si) to Fe abundance in stellar
populations, which results in two groups, clearly separable on a plot of
[Fe/H] vs. [$\alpha$/Fe], called the high- and low-[$\alpha$/Fe] stars.

There are multiple theories about the origin of this double sequence. It has
been suggested that the stars with larger $\alpha$ abundance form under
different circumstances than the low-[$\alpha$/Fe] stars (Mackereth et al.,
2018). An enhanced $\alpha$ with high Fe suggests that the majority of the
heavy elements come from core-collapse SNe, whilst a low-[$\alpha$/Fe] mixture
arises from a combination of SNe Ia and core-collapse SNe. The
low-[$\alpha$/Fe] stars tend to be young, reside in the thin disk, and form
their own sequence. One possibility is that they result from decreased star
formation efficiency as the Galaxy ages (Nidever et al., 2014); they also show
different birth radii and an anti-correlation between angular momentum and
[Fe/H], which suggests the existence of radial migration could be needed to
form the sequence (Sharma et al., 2021). The old, high-[$\alpha$/Fe] stars
generally reside in the thick disk; they display enhanced $\alpha$ abundance,
sometimes low metal abundance, and are relatively kinematically hot (e.g.,
Haywood et al., 2013). These stars can form from intense episodes of star
formation, mainly from core-collapse supernova, or from gas polluted by core-
collapse supernova. The [$\alpha$/Fe]-bimodality has been seen both in the
solar neighbourhood and beyond (Hayden et al., 2017); recent studies using
asteroseismic ages find the age distribution for high- and low-[$\alpha$/Fe]
stars converge with increasing distance from the Galactic plane, support the
[$\alpha$/Fe]-bimodality (Warfield et al., 2021), but questions still remain
about whether the high-[$\alpha$/Fe] population is old.

Our K2 data allows investigation of the [$\alpha$/Fe]-bimodality. Figure 12
shows the $\alpha$-bimodality plot for our sample. Asteroseismic masses are
defined as in Section 2.2 and abundance information is taken from APOGEE. The
eccentricity is defined in Section 4.3.

Our data extends the abundance ranges to lower metallicities, higher $\alpha$
abundances, and farther distances than Kepler. The APO-K2 sample offers a
sample with asteroseismic masses including a known selection function to build
on existing asteroseismic catalogs (e.g., Rendle et al. (2019); Mackereth et
al. (2019b); Imig et al. (2022).

Using the asteroseismic mass as a proxy for age, we can see a clump of higher
mass stars in the low-[$\alpha$/Fe] regime, suggesting these stars are
generally younger. They are also in a more condensed mass range than the
high-[$\alpha$/Fe] stars, where the high-[$\alpha$/Fe] regime includes some
stars with considerably lower mass, suggesting that they are older. We use the
size of the marker to represent eccentricity and see that the highest
eccentricity stars generally have high [$\alpha$/Fe]-abundance and low
metallicity.

Our sample adds over 1,000 stars to the [$\alpha$/Fe]-bimodality plot as
compared to the APOKASC2 sample ($\sim$6,000 stars), extending to lower
metallicities and higher [$\alpha$/Fe] abundance, allowing us to more clearly
separate the high- and low-[$\alpha$/Fe] samples. Figure 13 shows both the
APOKASC2 and APO-K2 catalogs over-plotted in metallicity-[$\alpha$/Fe],
illustrating the extent to which the APO-K2 catalog has expanded on the Kepler
field, and confirming the convergence of the bimodality into a single
distribution at higher metallicities.

Warfield et al. (2021) explored this space in K2 C4, C6, and C7, and
discovered overlap between high- and low-$\alpha$ populations with stars of
similar age. This discussion will continue in the companion paper of ages for
this sample (Warfield et al. in prep.).

###  4.3 Kinematics

Figure 14: Kinematic plots for the APO-K2 sample. The plot on the left (right)
corresponds to RC stars (RGB stars). Only stars with positive Gaia parallaxes
are plotted. The grey area of the plots correspond to the area likely to host
GES stars, and are defined by the lines drawn (and stated in the text). In
both of these plots, the black (white) stars correspond to the
high-[$\alpha$/Fe] (low-[$\alpha$/Fe]) stars, as defined by Figure 12.

The Gaia-Encaledus (GES) structure is thought to represent the remnants of a
dwarf galaxy that merged with the Milky Way (Belokurov et al., 2018; Helmi et
al., 2018; Montalbán et al., 2021) in its early history. Though initially
identified by kinematics, GES can also be identified by its combination of low
metallicity and particular abundance pattern (Haywood et al., 2018; Mackereth
et al., 2019b).

To define the dynamical information for our sample we use Gala, an Astropy-
affiliated (Astropy Collaboration et al., 2013, 2018) Python package. Gala
uses the Astropy Galactocentric frame parameters adopted in Astropy v4.0.
These are defined with a solar position of $R_{\odot}$ = 8.122 kpc and
$z_{\odot}$ = 20.8 pc. The velocity of the Sun in the Galactocentric frame is
(U, V, W)⊙ = (12.9, 245.6, 7.78) km/s. For our analysis we adopt the Milky Way
Potential available with Gala (Bovy, 2015), using the default parameters of
Milky Way mass, virial radius etc. In Section 4.2 we used eccentricities ($e$)
from Gala defined as

$e=\frac{r_{apo}-r_{peri}}{r_{apo}+r_{peri}},$ (9)

where $r_{apo}$ is the orbital apocentre and $r_{peri}$ is the orbital
pericentre.

Figure 14 shows orbital angular momentum (Lz) as a function of orbital energy
(E). This space is most often used to identify merger debris from past
accretion events. Colors correspond to the high-[$\alpha$/Fe] (green) and
low-[$\alpha$/Fe] (purple) samples. We display the sample broken down by
evolutionary state (RGB and RC), with boundaries for the GES overlaid. The
vertical lines denote the GES limits in angular momentum from Helmi et al.
(2018), between $-$150 $<$ Lz [kms-1kpc] $<$ 1500\. The GES distinction in
energy is taken from Koppelman et al. (2019) and placed between $-$1.1$\times
10^{5}$ km2s-2 and $-$1.5$\times 10^{5}$ km2s-2. Inside the grey box lie a few
dozen GES substructure candidates.

Koppelman et al. (2019) discuss the necessity of a chemical tagging analysis
to determine whether substructures are related to accretion events. They
studied the distribution of nearby thick disk and halo stars using 6D phase-
space data from Gaia DR2 and found that not all substructure is due to
accretion, nor is it due to the settling of the gravitational potential after
major activity (Haywood et al., 2018). The range of kinematic parameter space
probed in this sample may prove useful for future analysis, in combination
with the selection functions presented in S22.

Figure 14, shows the low-[$\alpha$/Fe] stars mainly occupy the disk, and hence
are seen in fairly circular orbits that sit close to the hyperbola that
defines the minimum energy curve in this space. By contrast, the
high-[$\alpha$/Fe] stars, residing mainly in the thick disk and halo are
expected to be kinematically hot (possessing eccentric orbits) and also occupy
regions above the minimum energy hyperbola, but with a larger spread.

##  5 Conclusions

In this paper we summarize the APO-K2 evolved star sample, with corresponding
spectroscopic (APOGEE DR17), asteroseismic (K2-GAP), and astrometric (Gaia
EDR3) parameters. Our sample of 7,673 unique stars contains RGB, RC, secondary
red clump, and RGB bump evolutionary states, from various areas of the Galaxy.
Our work provides precise asteroseismic radii and masses as well as
evolutionary states and metallicities, explored in multiple parameter spaces.

Throughout this work, we overviewed many parameter spaces that our catalog
extends, some for the first time. We investigate the completeness of our
sample by comparing it to the S22 selection function, in the mass-radius,
mass-metallicity, color-magnitude, and magnitude-$\nu_{\mathrm{max}}$ regimes.
We examine our astereosismic masses in the low-metallicity regime, resulting
in higher masses than expected for the low-metallicity stars, even when taking
corrections to the $\Delta\nu$ sclaing relation into account. We find that
using raw APOGEE temperatures to derive stellar masses results in a better
agreement with astrophysical estimates for very metal-poor stars. As specific
science applications, we also identify possible stellar merger products.
Further, we use this low-metallicity sample to extend our understanding of the
$\alpha$-bimodality in new regions of parameter space. Finally, we looked at
our sample in energy/kinematic space, with Gaia EDR3, and identified potential
GES stars.

The overview presented in our paper only scratches the surface of the rich
data sample; some of the spaces explored in this paper will be further
investigated in follow-up papers. Further work will be undertaken to explore
the multiplicity (Schonhut-Stasik et al. in prep.), and abundance space
provided by APOGEE (Schonhut-Stasik et al. in prep.), for example, to
investigate the carbon-enhanced stars known to exist at low metallicity
([Fe/H] $<$ 2.0 in Beers & Christlieb (2005) and [Fe/H] $\leq$ -2.5 [dex] in
Suda et al. (2008)). Furthermore, a deeper analysis of interesting individual
objects, such as high-mass stars at low metallicities and potential GE/merger
candidates will be explored. A companion paper will release age information
(Warfield et al. in prep.).

The amount of stars accessible to Galactic archaeology using asteroseismology
will only grow with future missions. For example, the NASA planet-finding
mission TESS (Ricker et al., 2014; Hon et al., 2022) and the Nancy Grace Roman
Telescope (Gould et al., 2015; Spergel et al., 2015), as well as the ESA
missions Euclid (Laureijs et al., 2011; Gould et al., 2016) and PLATO (Rauer
et al., 2014; Miglio et al., 2017), will yield enormous harvests of
asteroseismic detections. In terms of spectroscopic measurements, the upcoming
projects WEAVE (Dalton et al., 2020), MOONS (Cirasuolo et al., 2014), and
4MOST (de Jong et al., 2012) will increase chemical abundance yields. Finally
with future releases from Gaia, our astronometry and kinematic data will only
increase in precision.

EPIC | APOGEE ID | Campaign | Gal Lat | Gal Long | Evol | $T_{\mathrm{eff}}$ | $T_{\mathrm{eff}}$ | log(g) | log(g)
---|---|---|---|---|---|---|---|---|---
|  |  | [Deg] | [Deg] | State | [K] | Error [K] | [dex] | Error [dex]
211705076 | 2M08374542+1558546 | 5,16,18 | 30.556 | 209.677 | RGB | 4664 | 7.37 | 2.87 | 0.021
212873074 | 2M13305142-0302596 | 17 | 58.359 | 321.945 | RGB | 4789 | 11.35 | 2.78 | 0.029
204651310 | 2M16593864-2127129 | 11 | 12.794 | 0.400 | RGB | 4667 | 7.05 | 3.03 | 0.020
212386819 | 2M13364485-1513074 | 18,16 | 46.249 | 318.838 | RC | 4761 | 8.04 | 2.47 | 0.022
228774437 | 2M12441419-0751315 | 10 | 54.969 | 299.824 | RC | 5006 | 9.48 | 2.85 | 0.021
212349887 | 2M13394793-1608035 | 1 | 45.174 | 319.515 | RC | 4828 | 8.84 | 2.46 | 0.024
204970434 | 2M16494197-2001358 | 12 | 15.510 | 0.145 | RGB | 4827 | 13.78 | 2.42 | 0.035
228909546 | 2M12254205-0306337 | 3 | 59.131 | 290.334 | RC | 4725 | 7.78 | 2.38 | 0.0217
… | … | … | … | … | … | … | … | … | …

[M/H] | [Fe/H] | $[\alpha/M]$ |  $\alpha$ flags | $V_{\mathrm{mag}}$ | J-K | $\nu_{max}$ | $\Delta\nu$ | $f\Delta\nu$ | $\kappa_{M}$
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  $\mu$Hz |  $\mu$Hz |  | [$M_{\odot}$]
0.00 | -0.01 | 0.07 | 1 | 9.25 | 0.64 | 98.0 | 9.27 | 1.024 | 1.463
-0.27 | -0.28 | 0.06 | 1 | 6.87 | 0.66 | 49.9 | 5.20 | 1.026 | 1.955
0.19 | 0.17 | 0.03 | 1 | 9.60 | 0.734 | 148.6 | 12.95 | 1.016 | 1.338
-0.05 | -0.06 | 0.01 | 1 | 9.34 | 0.64 | 33.0 | 3.95 | 0.998 | 1.686
0.04 | 0.03 | -0.02 | 1 | 9.05 | 0.57 | 89.2 | 7.56 | 0.993 | 2.495
-0.24 | -0.25 | 0.05 | 1 | 9.59 | 0.69 | 31.6 | 3.97 | 0.998 | 1.457
-0.45 | -0.46 | 0.27 | 2 | 13.19 | 0.70 | 26.0 | 2.98 | 1.021 | 2.554
-0.03 | -0.03 | -0.00 | 1 | 9.35 | 0.68 | 31.1 | 3.69 | 0.997 | 1.857
… | … | … | … | … | … | … | … | … | …

Mass | Mass | $\kappa_{R}$ | Radius | Radius | Galactic | Angular Momentum | Total Energy |  $f\Delta\nu$ Flag
---|---|---|---|---|---|---|---|---
M⊙ | Error [$M_{\odot}$] | [$R_{\odot}$] | [$R_{\odot}$] | Error [$R_{\odot}$] | Eccentricity | [kpc km s-1] | [km2 s-2] |
1.06 | 0.069 | 6.776 | 6.09 | 0.125 | 0.84 | -6.5 | -2711.8 | 0
1.48 | 0.050 | 10.974 | 9.99 | 0.111 | 0.90 | -58.3 | -2890.9 | 0
0.97 | 0.033 | 5.262 | 4.73 | 0.053 | 0.99 | -346.9 | -2783.3 | 0
1.26 | 0.196 | 12.540 | 11.38 | 0.575 | 0.97 | 253.4 | -2594.4 | 0
2.01 | 0.189 | 9.273 | 8.63 | 0.261 | 0.95 | -1662.3 | -2632.9 | 0
1.11 | 0.097 | 11.908 | 10.89 | 0.309 | 0.60 | -3263.7 | -2551.1 | 0
1.95 | 0.217 | 17.386 | 15.89 | 0.576 | 0.85 | -2641.1 | -1960.7 | 0
1.37 | 0.370 | 13.557 | 12.26 | 1.048 | 0.79 | -1210.8 | -2580.7 | 0
… | … | … | … | … | … | … | … | …

Table 1: APO-K2 catalog. EPIC column gives the EPIC ID number for the star,
and corresponds to the same row in each of the smaller tables. All observed
campaigns are listed in the second column, followed by the Galactic Latitude
and Longitude [degrees]. Next, we show the evolutionary state, as defined in
the text as either red clump (RC) or red giant branch (RGB). The remaining
columns in the top section of the table represent Teff [K], and log(g) [dex]
with their associated errors. The second section of the table gives
spectroscopic abundance values from APOGEE: [M/H], [Fe/H], and [$\alpha$/M].
Next the alpha flag is given, with 1 corresponding to high [$\alpha$/Fe], 0 to
low [$\alpha$/Fe], and -1 to the middle of the [$\alpha$/Fe] bimodality, with
a 95% confidence interval around the dividing line. Next, we provide the
V-band magnitude and the j-k colors used in the selection function plots The
final four columns of this section are the frequency of maximum oscillation
power [$\mu Hz$] and large frequency separation [$\mu Hz$], followed by the
f$\Delta\nu$ value. The final column of this section of the table shows the
coefficient of mass [M⊙]. The stars mass, with the coefficient used, and the
appropriate error come first in the bottom section of the table, followed by
the radius coefficient, stellar radius, and radius error (all [R⊙]). Finally
we provide the kinematic information from Gala with values of Galactic
eccentricity, angular momentum and total energy. The final column is the
f$\Delta\nu$ flag defined in the text. Full version available electronically,
in one table.

## Acknowledgements

We would like to thank the community that supports APO-K2. J.S.S. is supported
a Neurodiversity Inspired Science and Engineering (NISE) Fellowship under
award. J.C.Z. is supported by an NSF Astronomy and Astrophysics Postdoctoral
Fellowship under award AST-2001869. D.M. gratefully acknowledges support by
the ANID BASAL projects ACE210002 and FB210003 and by Fondecyt Project No.
1220724. S.M. acknowledges support from the Spanish Ministry of Science and
Innovation (MICINN) with the Ramón y Cajal fellowship no.
$\sim$RYC-2015-17697, grant no. PID2019-107187GB-I00 and PID2019-107061GB-C66
for PLATO, and through AEI under the Severo Ochoa Centres of Excellence
Programme 2020–2023 (CEX2019-000920-S). D.S. acknowledges support from the
Australian Research Council Discover Project DP190100666. R.A.G. and B.M.
acknowledge the support from the PLATO Centre National D’Études Spatiales
grant.

This paper includes data collected by the Kepler mission and the K2 mission.
Funding for the Kepler mission and K2 mission are provided by the NASA Science
Mission directorate.

This work has made use of data from the European Space Agency (ESA) mission
_Gaia_ (https://www.cosmos.esa.int/gaia), processed by the _Gaia_ Data
Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the _Gaia_ Multilateral Agreement. This research has made use
of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI:
10.26093/cds/vizier). The original description of the VizieR service was
published Ochsenbein et al. (2000).

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
web site 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, the Chilean Participation Group, the French Participation Group,
Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de
Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, 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. Land Acknowledgement?
This paper includes data collected by the Kepler mission and obtained from the
MAST data archive at the Space Telescope Science Institute (STScI). Funding
for the Kepler mission is provided by the NASA Science Mission Directorate.
STScI is operated by the Association of Universities for Research in
Astronomy, Inc., under NASA contract NAS 5–26555.

## References

  * Abdurro’uf et al. (2021) Abdurro’uf, Accetta, K., Aerts, C., et al. 2021, arXiv e-prints, arXiv:2112.02026.  https://arxiv.org/abs/2112.02026
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Auvergne et al. (2009) Auvergne, M., Bodin, P., Boisnard, L., et al. 2009, A&A, 506, 411, doi: 10.1051/0004-6361/200810860
  * Baglin et al. (2006) Baglin, A., Auvergne, M., Boisnard, L., et al. 2006, in 36th COSPAR Scientific Assembly, Vol. 36, 3749
  * Beaton et al. (2021) Beaton, R. L., Oelkers, R. J., Hayes, C. R., et al. 2021, AJ, 162, 302, doi: 10.3847/1538-3881/ac260c
  * Bedding et al. (2010) Bedding, T. R., Huber, D., Stello, D., et al. 2010, ApJ, 713, L176, doi: 10.1088/2041-8205/713/2/L176
  * Beers & Christlieb (2005) Beers, T. C., & Christlieb, N. 2005, ARA&A, 43, 531, doi: 10.1146/annurev.astro.42.053102.134057
  * Belokurov et al. (2018) Belokurov, V., Erkal, D., Evans, N. W., Koposov, S. E., & Deason, A. J. 2018, Monthly Notices of the Royal Astronomical Society, 478, 611
  * Bensby et al. (2003) Bensby, T., Feltzing, S., & Lundström, I. 2003, A&A, 410, 527, doi: 10.1051/0004-6361:20031213
  * Besla et al. (2007) Besla, G., Kallivayalil, N., Hernquist, L., et al. 2007, ApJ, 668, 949, doi: 10.1086/521385
  * Blanton et al. (2017) Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, AJ, 154, 28, doi: 10.3847/1538-3881/aa7567
  * Bonaca et al. (2020) Bonaca, A., Conroy, C., Cargile, P. A., et al. 2020, ApJ, 897, L18, doi: 10.3847/2041-8213/ab9caa
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977, doi: 10.1126/science.1185402
  * Bovy (2015) Bovy, J. 2015, ApJS, 216, 29, doi: 10.1088/0067-0049/216/2/29
  * Bowen & Vaughan (1973) Bowen, I. S., & Vaughan, A. H., J. 1973, Appl. Opt., 12, 1430, doi: 10.1364/AO.12.001430
  * Brown et al. (1991) Brown, T. M., Gilliland, R. L., Noyes, R. W., & Ramsey, L. W. 1991, ApJ, 368, 599
  * Buder et al. (2021) Buder, S., Sharma, S., Kos, J., et al. 2021, MNRAS, 506, 150
  * Burbidge et al. (1957) Burbidge, E. M., Burbidge, G. R., Fowler, W. A., & Hoyle, F. 1957, Rev. Mod. Phys., 29, 547
  * Chiappini et al. (1997) Chiappini, C., Matteucci, F., & Gratton, R. 1997, ApJ, 477, 765, doi: 10.1086/303726
  * Choi et al. (2016) Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102, doi: 10.3847/0004-637X/823/2/102
  * Cirasuolo et al. (2014) Cirasuolo, M., Afonso, J., Carollo, M., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9147, Ground-based and Airborne Instrumentation for Astronomy V, ed. S. K. Ramsay, I. S. McLean, & H. Takami, 91470N
  * Clarke et al. (2019) Clarke, A. J., Debattista, V. P., Nidever, D. L., et al. 2019, MNRAS, 484, 3476, doi: 10.1093/mnras/stz104
  * Cox & Giuli (1968) Cox, J. P., & Giuli, R. T. 1968, Principles of stellar structure (Gordon and Breach)
  * Cui et al. (2012) Cui, X.-Q., Zhao, Y.-H., Chu, Y.-Q., et al. 2012, Research in Astronomy and Astrophysics, 12, 1197, doi: 10.1088/1674-4527/12/9/003
  * Dalton et al. (2020) Dalton, G., Trager, S., Abrams, D. C., et al. 2020, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11447, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 1144714
  * de Jong et al. (2012) de Jong, R. S., Bellido-Tirado, O., Chiappini, C., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8446, Ground-based and Airborne Instrumentation for Astronomy IV, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 84460T
  * De Ridder et al. (2009) De Ridder, J., Barban, C., Baudin, F., et al. 2009, Nature, 459, 398, doi: 10.1038/nature08022
  * Dotter (2016) Dotter, A. 2016, ApJS, 222, 8, doi: 10.3847/0067-0049/222/1/8
  * Dotter et al. (2008) Dotter, A., Chaboyer, B., Jevremović, D., et al. 2008, ApJS, 178, 89, doi: 10.1086/589654
  * Epstein et al. (2014) Epstein, C. R., Elsworth, Y. P., Johnson, J. A., et al. 2014, ApJ, 785, L28, doi: 10.1088/2041-8205/785/2/L28
  * Font et al. (2011) Font, A. S., McCarthy, I. G., Crain, R. A., et al. 2011, MNRAS, 416, 2802, doi: 10.1111/j.1365-2966.2011.19227.x
  * Freeman & Bland-Hawthorn (2002) Freeman, K., & Bland-Hawthorn, J. 2002, ARA&A, 40, 487, doi: 10.1146/annurev.astro.40.060401.093840
  * Fuhrmann (1998) Fuhrmann, K. 1998, A&A, 338, 161
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1, doi: 10.1051/0004-6361/202039657
  * García Pérez et al. (2016) García Pérez, A. E., Allende Prieto, C., Holtzman, J. A., et al. 2016, AJ, 151, 144
  * Girardi (2016) Girardi, L. 2016, ARA&A, 54, 95, doi: 10.1146/annurev-astro-081915-023354
  * González Hernández & Bonifacio (2009) González Hernández, J. I., & Bonifacio, P. 2009, A&A, 497, 497, doi: 10.1051/0004-6361/200810904
  * Gould et al. (2015) Gould, A., Huber, D., Penny, M., & Stello, D. 2015, Journal of Korean Astronomical Society, 48, 93, doi: 10.5303/JKAS.2015.48.2.93
  * Gould et al. (2016) Gould, A., Huber, D., & Stello, D. 2016, Journal of Korean Astronomical Society, 49, 9, doi: 10.5303/JKAS.2016.49.1.9
  * Grunblatt et al. (2021) Grunblatt, S. K., Zinn, J. C., Price-Whelan, A. M., et al. 2021, ApJ, 916, 88, doi: 10.3847/1538-4357/ac0532
  * Gunn et al. (2006) Gunn, J. E., Siegmund, W. A., Mannery, E. J., et al. 2006, AJ, 131, 2332
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * Hayden et al. (2017) Hayden, M. R., Recio-Blanco, A., de Laverny, P., Mikolaitis, S., & Worley, C. C. 2017, A&A, 608, L1, doi: 10.1051/0004-6361/201731494
  * Haywood et al. (2018) Haywood, M., Di Matteo, P., Lehnert, M., et al. 2018, A&A, 618, A78
  * Haywood et al. (2013) Haywood, M., Di Matteo, P., Lehnert, M. D., Katz, D., & Gómez, A. 2013, A&A, 560, A109, doi: 10.1051/0004-6361/201321397
  * Haywood et al. (2016) Haywood, M., Lehnert, M. D., Di Matteo, P., et al. 2016, A&A, 589, A66
  * Helmi et al. (2018) Helmi, A., Babusiaux, C., Koppelman, H. H., et al. 2018, Nature, 563, 85, doi: 10.1038/s41586-018-0625-x
  * Helmi & de Zeeuw (2000) Helmi, A., & de Zeeuw, P. T. 2000, MNRAS, 319, 657, doi: 10.1046/j.1365-8711.2000.03895.x
  * Holtzman et al. (2015) Holtzman, J. A., Shetrone, M., Johnson, J. A., et al. 2015, AJ, 150, 148, doi: 10.1088/0004-6256/150/5/148
  * Holtzman et al. (2018) Holtzman, J. A., Hasselquist, S., Shetrone, M., et al. 2018, AJ, 156, 125, doi: 10.3847/1538-3881/aad4f9
  * Hon et al. (2022) Hon, M., Kuszlewicz, J. S., Huber, D., Stello, D., & Reyes, C. 2022, arXiv e-prints, arXiv:2208.06478
  * Hon et al. (2021) Hon, M., Huber, D., Kuszlewicz, J. S., et al. 2021, ApJ, 919, 131
  * Howell et al. (2022) Howell, M., Campbell, S. W., Stello, D., & De Silva, G. M. 2022, MNRAS, 515, 3184
  * Howell et al. (2014) Howell, S. B., Sobeck, C., Haas, M., et al. 2014, PASP, 126, 398, doi: 10.1086/676406
  * Huber et al. (2009) Huber, D., Stello, D., Bedding, T. R., et al. 2009, Communications in Asteroseismology, 160, 74, doi: 10.48550/arXiv.0910.2764
  * Huber et al. (2016) Huber, D., Bryson, S. T., Haas, M. R., et al. 2016, ApJS, 224, 2, doi: 10.3847/0067-0049/224/1/2
  * Huber et al. (2017) Huber, D., Zinn, J., Bojsen-Hansen, M., et al. 2017, ApJ, 844, 102
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Ibata et al. (1994) Ibata, R. A., Gilmore, G., & Irwin, M. J. 1994, Nature, 370, 194, doi: 10.1038/370194a0
  * Imig et al. (2022) Imig, J., Holtzman, J. A., Yan, R., et al. 2022, AJ, 163, 56, doi: 10.3847/1538-3881/ac3ca7
  * Jönsson et al. (2020) Jönsson, H., Holtzman, J. A., Allende Prieto, C., et al. 2020, AJ, 160, 120, doi: 10.3847/1538-3881/aba592
  * Kallinger et al. (2018) Kallinger, T., Beck, P. G., Stello, D., & Garcia, R. A. 2018, A&A, 616, A104, doi: 10.1051/0004-6361/201832831
  * Kippenhahn et al. (2013) Kippenhahn, R., Weigert, A., & Weiss, A. 2013, Stellar Structure and Evolution (Springer), doi: 10.1007/978-3-642-30304-3
  * Kjeldsen & Bedding (1995) Kjeldsen, H., & Bedding, T. R. 1995, A&A, 293, 87.  https://arxiv.org/abs/astro-ph/9403015
  * Kobayashi et al. (2020) Kobayashi, C., Karakas, A. I., & Lugaro, M. 2020, ApJ, 900, 179
  * Koppelman et al. (2019) Koppelman, H. H., Helmi, A., Massari, D., Price-Whelan, A. M., & Starkenburg, T. K. 2019, A&A, 631, L9, doi: 10.1051/0004-6361/201936738
  * Kounkel et al. (2021) Kounkel, M., Covey, K. R., Stassun, K. G., et al. 2021, AJ, 162, 184, doi: 10.3847/1538-3881/ac1798
  * Laureijs et al. (2011) Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, arXiv e-prints, arXiv:1110.3193.  https://arxiv.org/abs/1110.3193
  * Lee et al. (2015) Lee, D. M., Johnston, K. V., Sen, B., & Jessop, W. 2015, ApJ, 802, 48, doi: 10.1088/0004-637X/802/1/48
  * Li et al. (2020) Li, Y., Bedding, T. R., Stello, D., et al. 2020, MNRAS
  * Lian et al. (2020) Lian, J., Thomas, D., Maraston, C., et al. 2020, MNRAS, 494, 2561, doi: 10.1093/mnras/staa867
  * Mackereth et al. (2018) Mackereth, J. T., Crain, R. A., Schiavon, R. P., et al. 2018, MNRAS, 477, 5072, doi: 10.1093/mnras/sty972
  * Mackereth et al. (2019a) Mackereth, J. T., Bovy, J., Leung, H. W., et al. 2019a, MNRAS, 489, 176, doi: 10.1093/mnras/stz1521
  * Mackereth et al. (2019b) Mackereth, J. T., Schiavon, R. P., Pfeffer, J., et al. 2019b, MNRAS, 482, 3426
  * Mackereth et al. (2020) Mackereth, J. T., Miglio, A., Elsworth, Y., et al. 2020, arXiv e-prints, arXiv:2012.00140
  * Mackereth et al. (2021) —. 2021, MNRAS, 502, 1947, doi: 10.1093/mnras/stab098
  * Majewski et al. (2017) Majewski, S. R., Schiavon, R. P., Frinchaboy, P. M., et al. 2017, AJ, 154, 94, doi: 10.3847/1538-3881/aa784d
  * Matsuno et al. (2021) Matsuno, T., Aoki, W., Casagrande, L., et al. 2021, ApJ, 912, 72, doi: 10.3847/1538-4357/abeab2
  * Mészáros et al. (2013) Mészáros, S., Holtzman, J., García Pérez, A. E., et al. 2013, AJ, 146, 133, doi: 10.1088/0004-6256/146/5/133
  * Miglio et al. (2009) Miglio, A., Montalbàn, J., Baudin, F., et al. 2009, A&A, 503, L21, doi: 10.1051/0004-6361/200912822
  * Miglio et al. (2012) Miglio, A., Brogaard, K., Stello, D., et al. 2012, MNRAS, 419, 2077
  * Miglio et al. (2016) Miglio, A., Chaplin, W. J., Brogaard, K., et al. 2016, MNRAS, 461, 760
  * Miglio et al. (2017) Miglio, A., Chiappini, C., Mosser, B., et al. 2017, Astronomische Nachrichten, 338, 644, doi: 10.1002/asna.201713385
  * Miglio et al. (2021) Miglio, A., Chiappini, C., Mackereth, J. T., et al. 2021, A&A, 645, A85
  * Montalbán et al. (2021) Montalbán, J., Mackereth, J. T., Miglio, A., et al. 2021, Nature Astronomy, 5, 640, doi: 10.1038/s41550-021-01347-7
  * Mosser et al. (2018) Mosser, B., Gehan, C., Belkacem, K., et al. 2018, A&A, 618, A109, doi: 10.1051/0004-6361/201832777
  * Mosser et al. (2013) Mosser, B., Dziembowski, W. A., Belkacem, K., et al. 2013, A&A, 559, A137, doi: 10.1051/0004-6361/201322243
  * Nidever et al. (2014) Nidever, D. L., Bovy, J., Bird, J. C., et al. 2014, ApJ, 796, 38, doi: 10.1088/0004-637X/796/1/38
  * Nidever et al. (2015) Nidever, D. L., Holtzman, J. A., Allende Prieto, C., et al. 2015, AJ, 150, 173, doi: 10.1088/0004-6256/150/6/173
  * Ochsenbein et al. (2000) Ochsenbein, F., Bauer, P., & Marcout, J. 2000, A&AS, 143, 23, doi: 10.1051/aas:2000169
  * Pinsonneault et al. (2014) Pinsonneault, M. H., Elsworth, Y., Epstein, C., et al. 2014, ApJS, 215, 19, doi: 10.1088/0067-0049/215/2/19
  * Pinsonneault et al. (2018) Pinsonneault, M. H., Elsworth, Y. P., Tayar, J., et al. 2018, ApJS, 239, 32, doi: 10.3847/1538-4365/aaebfd
  * Price-Whelan et al. (2022) Price-Whelan, A., Sipőcz, B., Wagg, T., et al. 2022, adrn/gala: v1.6.1, v1.6.1, Zenodo, doi: 10.5281/zenodo.7299506
  * Price-Whelan (2017) Price-Whelan, A. M. 2017, The Journal of Open Source Software, 2
  * Prochaska et al. (2000) Prochaska, J. X., Naumov, S. O., Carney, B. W., McWilliam, A., & Wolfe, A. M. 2000, AJ, 120, 2513, doi: 10.1086/316818
  * Rauer et al. (2014) Rauer, H., Catala, C., Aerts, C., et al. 2014, Experimental Astronomy, 38, 249, doi: 10.1007/s10686-014-9383-4
  * Rendle et al. (2019) Rendle, B. M., Miglio, A., Chiappini, C., et al. 2019, MNRAS, 490, 4465
  * Ricker et al. (2014) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9143, Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave, ed. J. Oschmann, Jacobus M., M. Clampin, G. G. Fazio, & H. A. MacEwen, 914320, doi: 10.1117/12.2063489
  * Santana et al. (2021) Santana, F. A., Beaton, R. L., Covey, K. R., et al. 2021, AJ, 162, 303, doi: 10.3847/1538-3881/ac2cbc
  * Schönrich & Binney (2009) Schönrich, R., & Binney, J. 2009, MNRAS, 396, 203, doi: 10.1111/j.1365-2966.2009.14750.x
  * Sellwood & Binney (2002) Sellwood, J. A., & Binney, J. J. 2002, MNRAS, 336, 785, doi: 10.1046/j.1365-8711.2002.05806.x
  * Sharma et al. (2011) Sharma, S., Bland-Hawthorn, J., Johnston, K. V., & Binney, J. 2011, ApJ, 730, 3, doi: 10.1088/0004-637X/730/1/3
  * Sharma et al. (2021) Sharma, S., Hayden, M. R., & Bland-Hawthorn, J. 2021, MNRAS, 507, 5882, doi: 10.1093/mnras/stab2015
  * Sharma et al. (2016) Sharma, S., Stello, D., Bland-Hawthorn, J., Huber, D., & Bedding, T. R. 2016, ApJ, 822, 15, doi: 10.3847/0004-637X/822/1/15
  * Sharma et al. (2017) Sharma, S., Stello, D., Huber, D., Bland-Hawthorn, J., & Bedding, T. R. 2017, ApJ, 835, 163, doi: 10.3847/1538-4357/835/2/163
  * Sharma et al. (2022) Sharma, S., Stello, D., Zinn, J. C., et al. 2022, MNRAS, doi: 10.1093/mnras/stac2031
  * Sharma et al. (2019) Sharma, S., Stello, D., Bland-Hawthorn, J., et al. 2019, Monthly Notices of the Royal Astronomical Society, 490, 5335, doi: 10.1093/mnras/stz2861
  * Silva Aguirre et al. (2018) Silva Aguirre, V., Bojsen-Hansen, M., Slumstrup, D., et al. 2018, MNRAS, 475, 5487
  * Simpson et al. (2019) Simpson, C. M., Gargiulo, I., Gómez, F. A., et al. 2019, MNRAS, 490, L32, doi: 10.1093/mnrasl/slz142
  * Soderblom (2010) Soderblom, D. R. 2010, ARA&A, 48, 581
  * Spergel et al. (2015) Spergel, D., Gehrels, N., Baltay, C., et al. 2015, arXiv e-prints, arXiv:1503.03757.  https://arxiv.org/abs/1503.03757
  * Spitoni et al. (2019) Spitoni, E., Silva Aguirre, V., Matteucci, F., Calura, F., & Grisoni, V. 2019, A&A, 623, A60, doi: 10.1051/0004-6361/201834188
  * Stello & Sharma (2022) Stello, D., & Sharma, S. 2022, Research Notes of the American Astronomical Society, 6, 168, doi: 10.3847/2515-5172/ac8b12
  * Stello et al. (2013) Stello, D., Huber, D., Bedding, T. R., et al. 2013, ApJ, 765, L41, doi: 10.1088/2041-8205/765/2/L41
  * Stello et al. (2014) Stello, D., Compton, D. L., Bedding, T. R., et al. 2014, ApJ, 788, L10, doi: 10.1088/2041-8205/788/1/L10
  * Stello et al. (2015) Stello, D., Huber, D., Sharma, S., et al. 2015, ApJ, 809, L3, doi: 10.1088/2041-8205/809/1/L3
  * Stello et al. (2017) Stello, D., Zinn, J., Elsworth, Y., et al. 2017, VizieR Online Data Catalog, J/ApJ/835/83
  * Suda et al. (2008) Suda, T., Katsuta, Y., Yamada, S., et al. 2008, PASJ, 60, 1159, doi: 10.1093/pasj/60.5.1159
  * Tailo et al. (2022) Tailo, M., Corsaro, E., Miglio, A., et al. 2022, A&A, 662, L7
  * Tayar et al. (2019) Tayar, J., Beck, P. G., Pinsonneault, M. H., García, R. A., & Mathur, S. 2019, ApJ, 887, 203, doi: 10.3847/1538-4357/ab558a
  * Timmes et al. (1995) Timmes, F. X., Woosley, S. E., & Weaver, T. A. 1995, ApJS, 98, 617
  * Valentini et al. (2019) Valentini, M., Chiappini, C., Bossini, D., et al. 2019, A&A, 627, A173
  * Van Rossum & Drake (2009) Van Rossum, G., & Drake, F. L. 2009, Python 3 Reference Manual (Scotts Valley, CA: CreateSpace)
  * Venn et al. (2004) Venn, K. A., Irwin, M., Shetrone, M. D., et al. 2004, AJ, 128, 1177, doi: 10.1086/422734
  * Viani et al. (2017) Viani, L. S., Basu, S., Chaplin, W. J., Davies, G. R., & Elsworth, Y. 2017, ApJ, 843, 11
  * Warfield et al. (2021) Warfield, J. T., Zinn, J. C., Pinsonneault, M. H., et al. 2021, AJ, 161, 100, doi: 10.3847/1538-3881/abd39d
  * Weinberg et al. (2017) Weinberg, D. H., Andrews, B. H., & Freudenburg, J. 2017, ApJ, 837, 183, doi: 10.3847/1538-4357/837/2/183
  * Wes McKinney (2010) Wes McKinney. 2010, in Proceedings of the 9th Python in Science Conference, ed. Stéfan van der Walt & Jarrod Millman, 56 – 61, doi: 10.25080/Majora-92bf1922-00a
  * Wilson et al. (2019) Wilson, J. C., Hearty, F. R., Skrutskie, M. F., et al. 2019, PASP, 131, 055001
  * Wolniewicz et al. (2021) Wolniewicz, L. M., Berger, T. A., & Huber, D. 2021, AJ, 161, 231, doi: 10.3847/1538-3881/abee1d
  * Wu et al. (2017) Wu, Y., Xiang, M., Bi, S., et al. 2017, Monthly Notices of the Royal Astronomical Society, 475, 3633, doi: 10.1093/mnras/stx3296
  * Yu et al. (2018) Yu, J., Huber, D., Bedding, T. R., et al. 2018, ApJS, 236, 42
  * Zasowski et al. (2017) Zasowski, G., Cohen, R. E., Chojnowski, S. D., et al. 2017, AJ, 154, 198
  * Zinn et al. (2019) Zinn, J. C., Pinsonneault, M. H., Huber, D., et al. 2019, ApJ, 885, 166, doi: 10.3847/1538-4357/ab44a9
  * Zinn et al. (2020) Zinn, J. C., Stello, D., Elsworth, Y., et al. 2020, ApJS, 251, 23, doi: 10.3847/1538-4365/abbee3
  * Zinn et al. (2022) —. 2022, ApJ, 926, 191, doi: 10.3847/1538-4357/ac2c83