# Spectroscopic identification of rapidly rotating red giant stars in
APOKASC-3 and APOGEE DR16

Rachel A. Patton,1,2 Marc H. Pinsonneault,1,2 Lyra Cao,1,2 Mathieu Vrard,1,2
Savita Mathur,3,4 Rafael A. García,5 Jamie Tayar,6 Christine Mazzola Daher,2
and Paul G. Beck3,4,7
1Department of Astronomy, The Ohio State University, 140 West 18th Ave,
Columbus, OH 43210, USA
2Center for Cosmology and AstroParticle Physics, The Ohio State University,
191 West Woodruff Avenue, Columbus, OH 43210
3Instituto de Astrofísica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain
4Universidad de La Laguna (ULL), Departamento de Astrofísica, 38206 La Laguna,
Tenerife, Spain
5AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, Sorbonne Paris
Cité, 91191 Gif-sur-Yvette, France
6Department of Astronomy, University of Florida, Bryant Space Science Center,
Stadium Road, Gainesville, FL 32611, USA
7 Institut für Physik, Karl-Franzens Universität Graz, Universitätsplatz 5/II,
NAWI Graz, 8010 Graz, Austria E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Rapidly rotating red giant stars are astrophysically interesting but rare. In
this paper we present a catalog of 3217 active red giant candidates in the
APOGEE DR16 survey. We use a control sample in the well-studied Kepler fields
to demonstrate a strong relationship between rotation and anomalies in the
spectroscopic solution relative to typical giants. Stars in the full survey
with similar solutions are identified as candidates. We use vsini measurements
to confirm 50+/- 1.2 % of our candidates as definite rapid rotators, compared
to 4.9+/-0.2% in the Kepler control sample. In both the Kepler control sample
and a control sample from DR16, we find that there are 3-4 times as many
giants rotating with 5 < vsini < 10 km s-1 compared to vsini > 10 km s-1, the
traditional threshold for anomalous rotation for red giants. The vast majority
of intermediate rotators are not spectroscopically anomalous. We use binary
diagnostics from APOGEE and Gaia to infer a binary fraction of 73+/-2.4 %. We
identify a significant bias in the reported metallicity for candidates with
complete spectroscopic solutions, with median offsets of 0.37 dex in [M/H]
from a control sample. As such, up to 10% of stars with reported [M/H]<-1 are
not truly metal poor. Finally, we use Gaia data to identify a sub-population
of main sequence photometric binaries erroneously classified as giants.

###### keywords:

stars: rotation – stars: binaries: general – stars: low-mass

††pubyear: 2023††pagerange: Spectroscopic identification of rapidly rotating
red giant stars in APOKASC-3 and APOGEE DR16–References

##  1 Introduction

Rapid rotation in low-mass giant stars is a sign of a star with an unusual
history. For stars born below the Kraft break (Teff < 6250 K; M < 1.3 M⊙
Kraft, 1967), the convective envelope is deep enough that magnetized winds
form, which carry away angular momentum. These low mass stars rotate slowly,
even on the main sequence. Even higher mass stars that leave the main sequence
rotating more rapidly experience magnetic braking when they become giants with
deep surface convection. Furthermore, all giants expand and slow down through
angular momentum conservation. The net effect is that the bulk of red giants
are very slow rotators, typically with undetected rotation rates (de Medeiros
et al., 1996).

Yet we know rapidly rotating red giants exist, as there are several ways in
which rapid rotation manifests itself. Rotational broadening of spectral lines
corresponding to vsini > 10 km s-1 is the typical threshold adopted in the
literature to designate rapid rotation in red giants (e.g. Tayar et al.,
2015). With the advent of large time-domain surveys, star spot modulation can
also be used to infer rotation periods for giants (e.g., García et al.,
2014c). Magnetic activity, caused by a rotationally-driven, convective dynamo
in the giant’s envelope results in star spots (e.g., Yadav et al., 2015), thus
rotational modulation in its light curve (e.g., Mathur et al., 2014). Position
in an HR diagram can also indicate binary interaction and current rapid
rotation. For example, there is a field turnoff related to the finite age of
the Galaxy, and a corresponding minimum effective temperature in the red giant
branch. Stars cooler than this locus, sometimes referred to as sub-subgiants,
are preferentially active stars, and frequently interacting binaries (Geller
et al., 2017a; Leiner et al., 2017; Geller et al., 2017b; Leiner et al.,
2021b). Stars with typical evolutionary histories are either completely absent
or nearly so in these domains.

Still, each of these detection methods have their shortfalls. In principle,
rotation in the form of vsini can be measured directly from spectra. However,
the large majority of giants are slow rotators, and adding rotation as an
additional degree of freedom in spectroscopic fitting pipelines is
computationally expensive for a quantity that is an upper limit for 98% or
more of the sample. Furthermore, rapidly rotating giants have spectra that can
be difficult to fit with conventional methods. Time-domain surveys are biased
towards higher amplitude spot modulation and can miss lower amplitude
modulation from targets with moderate rotational enhancement. Although
detected from rotational line broadening, Tayar et al. (2015) found a
significant population of moderately rotating (5 < vsini < 10 km s-1) red
clump stars. Such low-amplitude rotators should exist but are likely to be
missed in photometric surveys. Lastly, it is not guaranteed that all or most
rapidly rotating red giants are photometrically distinct from their non-
rotating counterparts. New diagnostics are needed to extract the underlying
population of rapidly rotating red giants.

As such, the census of rapidly rotating red giants is largely incomplete, and
their expected occurrence rates remain uncertain due to the broad range of
values from observations. Massarotti et al. (2008) found a 0.3% rotation
fraction from 761 Hipparcos giants. Carlberg et al. (2011) found that 2.2% of
their sample of roughly 1300 K giants was rapidly rotating. Daher et al.
(2022) found that 0.8 - 3.5%, depending on the threshold, of 79 308 field
giants rapidly rotate. In the Kepler field, where the stars are well
characterized, Tayar et al. (2015) identified 10 rapidly rotating stars (0.5%)
in a sample of 1950 red giants with asteroseismic detections in the APOKASC-1
survey (Pinsonneault et al., 2014), Ceillier et al. (2017) measured surface
rotation periods for 361 (2%) rapidly rotating and active red giants from a
sample of 17 377 red giants, and, most recently, Gaulme et al. (2020)
identified 370 (out of 4500; 8%) red giants, some single, some in binaries,
displaying rotational modulation in their light curves. Selection effects in
samples, differences in the definition of what constitutes a rapid rotator,
and true differences in distinct stellar populations, likely account for this
dispersion. Furthermore, the bulk of the above studies on rapidly rotating red
giants have been done on targets in the Kepler field, which we cannot assume
is a representative sample of the entire Galaxy.

Knowing the true underlying population of rapidly rotating red giants is
desirable for a variety of reasons. First, binary interaction is the dominant
cause of rapid rotation in red giants. Main sequence mergers can produce blue
stragglers, which tend to rotate more rapidly than old field stars (see, for
example, Peterson et al., 1984, for blue stragglers in M67). Subgiant mergers
with stellar or planetary companions will also produce fast spinning stars
(e.g. Leiner et al., 2017). Both are a natural source for rapidly rotating
single giants. On the lower giant branch, the timescale for tidal
synchronization is shorter than the evolutionary time scale, so close binaries
will be tidally synchronized (Zahn, 1977; Verbunt & Phinney, 1995). This
produces much more rapid rotation than expected from single star expansion and
angular momentum conservation. Even on the main sequence, tidal
synchronization with a binary companion is a significant contributor to rapid
rotation (Zahn & Bouchet, 1989). Not only are these interesting systems in
their own right, enabling the study of, for example, synchronization
timescales and formation channels of white dwarf-white dwarf
binaries/supernova type Ia progenitors, but they also allow us to calibrate
uncertain binary interaction and population synthesis models, such as merger
rates and the physics of mass transfer (e.g. Leiner & Geller, 2021).
Additionally, a large sample of rapidly rotating giants, the majority of which
we expect to be in binaries, allows us to probe activity metrics in binaries,
the demographics of active giants (e.g., Gaulme et al., 2020), and the types
of evolutionary tracks which populate non-standard parts of the HR diagram
(e.g. Leiner et al., 2021b).

The trouble lies in finding large samples of rapidly rotating, red giants.
Typical studies of rapidly rotating red giants only yield tens to hundreds of
stars. This makes sense given the initial sample sizes. But in larger stellar
samples, on the order of hundreds of thousands of stars, we want to be able to
easily identify the thousands of expected rapidly rotating red giants without
searching for rotation signatures giant by giant. Furthermore, rapid rotation,
and the activity caused by it, is known to degrade data quality. It can
suppress asteroseismic signals (e.g., García et al., 2010; Tayar et al., 2015;
Mathur et al., 2019; Gaulme et al., 2020), add extra modulation to light
curves (e.g., Mathur et al., 2014), and prevent good fits to spectra due to
the dramatically different temperatures of spotted and non-spotted surfaces
(Cao & Pinsonneault, 2022). Rapid rotators produce data that may not survive
quality cuts, leading to an underestimate of their true numbers. To get a true
representative sample from a large stellar population, we need to be able to
find rapid rotators even among stars with poor quality data.

Data Release 16 (DR16) from the Apache Point Observatory Galactic Evolution
Experiment (APOGEE; Majewski et al., 2017; Wilson et al., 2019; Ahumada et
al., 2020) contains high-resolution (R of 22,500) infrared spectra for over
437,445 nearby stars. This survey is ideal for locating a large data set of
active, evolved stars. The manner in which stellar spectra are fit in this
survey results in natural bins by which we can sort stars and search for
spectroscopic anomalies indicating rotation.

In this study, we explore whether we can use anomalies in spectroscopic
solutions to preferentially identify rapidly rotating giants. We use the well-
studied Kepler field to infer a census of rapid rotators, use asteroseismic
data to distinguish between different evolutionary states, and demonstrate
that our technique recovers the majority of these targets in that sample. We
describe the catalogs from which we draw our data in section §2. We then
evaluate the efficacy of each spectroscopic criterion at identifying rapid
rotators in §3. Having defined new spectroscopic criteria to quickly identify
rapidly rotating red giants, we construct a DR16 catalog of several thousand
rapidly rotating red giant candidates and validate them with vsini
measurements, which we discuss in section §4. §5 discusses the implications of
a large population of rapidly rotating red giants, biases in our selection
criteria, and how adaptable the criteria are for other data releases and other
spectroscopic surveys. Finally, we summarize our results in §6.

##  2 Data

For photometric and spectroscopic identification of red giants as well as
independent confirmation of rotation, we drew data from several sources. We
describe each catalog and its utility below.

###  2.1 APOGEE DR16

APOGEE uses an automated pipeline to infer stellar parameters from a target’s
spectra. The APOGEE Stellar Parameter and Chemical Abundances Pipeline
(ASPCAP; Nidever et al., 2015; García Pérez et al., 2016) extracts stellar
parameters in two parts. First, the code FERRE (Allende Prieto et al., 2006)
matches the input spectrum to the best-fit template from a grid of dwarf and
giant synthetic spectra, by minimizing $\chi^{2}$ and returning estimates for
stellar parameters. This is an 8-D fit to get log $g$, $T_{\mathrm{eff}}$,
carbon abundance, nitrogen abundance, overall metal abundance,
$\alpha$-element abundance, microturbulent velocity, and either vsini or
macroturbulent velocity for dwarfs and giants respectively. In the former
case, mactroturbulent velocity is assumed to be negligible and in the latter,
vsini is fixed at 1.5 km s-1.

Synthetic spectral grids cover G-M type giants and F-M dwarfs, with the giant
grids spanning -0.5< log $g$ < 4.5 cm s-2, depending on spectral type, and the
dwarf grids spanning 2.5 < log g < 5.5 cm s-2 (Jönsson et al., 2020). Stars
can be fit using multiple grids, where the model yielding the lowest
$\chi^{2}$ gets adopted. Due to the overlap in log g between the dwarf and
giant grids, some stars will be fit with both giant and dwarf template
spectra. In these instances, the goodness-of-fit is not only important for
determining stellar parameters but also evolutionary state. Spectra that
cannot be fit at this stage trigger the appropriate quality control flag and
the pipeline returns no solution.

Figure 1: Power spectra for two stars that have complete spectroscopic
solutions but are anomalously cool. The top panel shows the power spectrum for
KIC12253381, a confirmed slowly rotating giant with solar-like oscillations.
Note the characteristic cluster of oscillation modes centered near 20 $\mu$Hz.
The bottom panel shows the power spectrum for KIC10929752, a confirmed rapidly
rotating giant with a strong rotation signature around 0.5 $\mu$Hz and its
harmonics around 1.5 and 2 $\mu$Hz.

Log $g$ and $T_{\mathrm{eff}}$ estimates from the FERRE fit are then
calibrated against independent measures of the same parameters. For giants,
the calibrations come from asteroseismology for log $g$ and trends in
photometric estimates for $T_{\mathrm{eff}}$ for stars with minimal reddening.
Spectra without the calibrated stellar parameters we dub partial solutions,
whereas spectra that have both the initial FERRE fits and the calibrated
stellar parameters we deem complete solutions.

We might expect rapid rotators to preferentially fall into the first two
categories because rapid rotation can cause a non-uniform surface temperature,
especially in active, cool stars with large spot filling factors, making a
single-temperature solution difficult (Cao & Pinsonneault, 2022). However,
rapid rotation still affects data quality even among stars with complete
solutions. ASPCAP does not include rotation as a free parameter when fitting
the spectra of giants. The derived stellar parameters are then strongly
perturbed by excess line broadening, which can be artificially produced by
correlated changes in effective temperature, surface gravity, and elemental
abundances. Rapidly rotating giants whose spectra are fit without rotation as
a free parameter have artificially high surface gravity (Thompson et al.,
2019) and artificially low effective temperatures (Dixon et al., 2020).

Among the stars with complete solutions, we still need a tool to
preferentially identify rapidly rotating red giants. To construct this tool,
two pieces are essential: a definition for characteristic properties of
typical red giants and a method for identifying outliers. Conveniently, DR16
gives us the definition of a typical red giant due to the inclusion of a means
to spectroscopically separate first-ascent red giant branch (RGB) stars from
red clump (RC) stars. This is possible because there are systematic surface
gravity and temperature offsets between the two populations.

From stellar theory, we expect there to be a mapping between mass,
metallicity, $\alpha$-enrichment, and the location of the RGB on an HR
diagram. DR16 defined the mean locus of the RGB using a reference temperature,
$T_{\mathrm{ref}}$, which represents the expected effective temperature for an
average red giant of a given mass and metallicity (Jönsson et al., 2020). The
carbon-to-nitrogen ratio can be used as a proxy for mass. Holtzman et al.
(2018) found this reference temperature by fitting the average
$T_{\mathrm{eff}}$ of red giants in the APOKASC-2 catalog (Pinsonneault et
al., 2018). The difference between the star’s uncalibrated spectroscopic
$T_{\mathrm{eff}}$ and $T_{\mathrm{ref}}$ is then used to calculate a
reference number based on the temperature difference and a star’s total,
uncalibrated metallicity [M/H]. If a star’s uncalibrated carbon-to-nitrogen
ratio [C/N] is less than the reference number, then it is classified as an RGB
star.

Combining the [C/N] dependence with the reference temperature, we can write
the condition for separating RGB stars from RC stars as

$\Delta T=T_{\mathrm{eff,spec}}-T^{{}^{\prime}}_{\mathrm{ref}}$ (1)

where $T_{\mathrm{eff,spec}}$ is the uncalibrated spectroscopic effective
temperature and

$T^{{}^{\prime}}_{\mathrm{ref}}=3032.786-357.13\mathrm{[C/N]}+552.6\mathrm{log}g-488.886\mathrm{[M/H]}$
(2)

for uncalibrated values of log $g$, [C/N], and [M/H]. If $\Delta T$ is
negative, the star is on the RGB. This evolutionary state separation only is
calibrated to the lower giant branch, for 2.38 < log $g$ < 3.5 cm s2. However,
to capture the sub-subgiants, which are interacting red giants, we must extend
the range of log g to 3.85 cm s2. We do not have to worry about contamination
from true subgiants, however, because they tend to be hotter and will not
scatter into the sub-subgiant domain.

To understand the origin of the rapid rotation we must assess binarity. Radial
velocity (RV) variations suggest the presence of a companion. Since we expect
most of the rapid rotators to be close, interacting binaries, the RV
amplitudes should be high, so we can adopt a more lax criterion, requiring at
least two visits instead of more visits. We define as binaries those stars
which have a scatter around the average heliocentric velocity, VSCATTER,
greater than 3*VERR_MED, the average measurement error of the heliocentric
velocity. This criterion does not exclusively identify close binaries,
however. We deem close binaries those stars with at least two APOGEE visits
where VSCATTER > 10*VERR_MED. This choice is comparable to a $\Delta
RV_{\mathrm{max}}$ of 1 km s-1, the threshold adopted in Moe et al. (2019) and
Daher et al. (2022).

Lastly, ASPCAP provides a lengthy list of quality control flags to indicate
any issues with the fit. in the instance of rapid rotation, ASPCAP will return
SUSPECT_BROAD_LINES,ROTATION_WARN, or ROTATION_BAD. We include these flags in
our final catalog of rapid rotators.

###  2.2 APOKASC-3 and rapidly rotating red giant catalogs from Kepler

We need a control sample containing well-studied, well-characterized stars in
which we can test various rotation diagnostics. Any of the three criteria
which is particularly good at identifying rapid rotators in the control sample
can then be applied to the entire DR16 catalog, picking out stars which are
most likely to be rapid rotators instead of searching for rotation giant by
giant.

The APOKASC-3 catalog (Pinsonneault et al., in prep) is an ideal sample to use
as our control because it combines the IR spectroscopy of APOGEE with high
cadence photometry from Kepler for 23 363 stars observed by both surveys. For
our purposes, the key advantages of this over other public catalogs
(Pinsonneault et al., 2018; Yu et al., 2018) is that the full set of giants
has been studied for both seismology and rotation, and all of the giants in
the Kepler fields targeted by APOGEE are included. The APOGEE survey selected
giants in the field based on color and magnitude cuts, and as a result the
sample is not biased toward asteroseismic detections. This is important,
because rapid rotators are selected against in purely asteroseismic samples
(Tayar et al., 2015; Gaulme et al., 2020).

Giants in APOKASC-3 are selected for having log g < 3.6 cm s-2 and
$T_{\mathrm{eff}}$ < 6000 K. We verify the spectroscopic identification of
giants with Gaia photometry. Additionally, for any target which did not have a
complete solution, giants can only be selected from Gaia photometry. Giants
are defined as having a dereddened Gaia BP \- RP color > 0.9 and dereddened,
absolute Gaia G-band magnitude < 0.3 + 3.4 * Gaia color. This is a strict
photometric cut and it does remove the blue end of the clump, but we are not
losing many targets. In total, there are 15 220 giants.

We use these giants as our control sample, testing the efficacy of each
spectroscopic criterion at picking out rapid rotators. To do this, we need a
uniform measure of rapid rotation. In short, we measure vsini for all giants
in APOKASC-3, the details of which are in §3.4. The method to measure vsini we
use only works for stars with a best-fit template spectrum, meaning stars with
no spectroscopic solution are not included in the sample of rapid rotators. In
this instance, we crossmatch the 32 giants with no solution to various
catalogs of rapidly rotating red giants in Kepler as well as Gaia. We check
the Gaulme et al. (2020) binary catalog and extract orbital periods and/or
rotation velocities at the target’s equator, where measured, for stars which
appear in that catalog. If a star only has a measured rotation period and no
other indicators of rotation, we calculate rotation speed using the radii from
Berger et al. (2020). Any speed > 10 km s-1 is considered rapid. Then we check
the Ceillier et al. (2017) catalog of red giants with measured surface
rotation periods and extract a rotation period for the one star, KIC 7531136,
in our sample which appears in this catalog. From this rotation period and a
radius estimate (Berger et al., 2020), we infer a rotation speed of 16 km/s.
Stars which appeared in any of these catalogs, and had measured orbital
periods or v sini > 10 km s-1 where applicable, were designated as rapid
rotators. We maintain the 10 km s-1 threshold for these measurements of
equatorial rotation speed to be conservative.

We also inspect their light curves. Using the Kepler
KEPSEISMIC111https://archive.stsci.edu/prepds/kepseismic/ lighcurves and power
spectrum densities calibrated by the KADACS pipeline (García et al., 2011,
2014b), we inspect the stars’ power spectra for rotation signatures. While the
presence of a high amplitude peak at low frequency along with its harmonics in
some cases can provide an estimate of the surface rotation period of a star
(e.g., Santos et al., 2019, 2021), the modulation can come from a nearby star
polluting the Kepler light curve as also explained in Ceillier et al. (2017)
and Colman et al. (2017). The cross-check with the APOGEE data should allow us
to discard those cases. If the analysis from both instruments yield a fast
rotation, we can consider that this star has a high surface rotation rate.
However we should be cautious that cases of pollution by a nearby star can
occur in Kepler data. An example of a high-amplitude, low-frequency peak is
shown in the lower panel of Fig. 1 compared to a giant exhibiting normal
oscillation modes shown in the top panel. Any star which displays a rotation
peak in its power spectrum we deem a rapid rotator.

Finally, we check SIMBAD for signs of variability and any other information on
the nature of the star. Stars which were labeled as rotational or ellipsoidal
variables, as well as eclipsing binaries, were designated as rapid rotators.

###  2.3 Gaia

For stars without complete spectral solutions and as an additional check for
stars with complete spectroscopic solutions, we photometrically identified the
giants using Gaia. All stars in our sample are found in Gaia Data Release 2
(DR2) (Gaia Collaboration et al., 2016, 2018b). We choose to use DR2 to remain
consistent with the Bailer-Jones et al. (2018) distance estimates. Since the
photometry is used only for a coarse sorting of evolutionary states, APOGEE
giants are bright and 2.9 Kpc away on average, so zero-point issues are not as
large, and DR3 results agree well on average with DR2, we find DR2 to be
sufficient. We de-redden our stars using the same methods as Godoy-Rivera et
al. (2021) based on the process described in Gaia Collaboration et al. (2018),
where extinction coefficients depend on both the Gaia BP \- RP color and AV.
We obtain $A_{V}$ for both the APOKASC-3 and for the entire APOGEE DR16 sample
by integrating over 3D dust maps using the Python package mwdust (Bovy et al.,
2016). We use the combined dust maps of Marshall et al. (2006) and Green et
al. (2019). When converting from $A_{V}$ to Gaia-band extinctions, we do not
adopt any of the singular values from the literature because the conversion
factor for each band depends on the temperature of the star. Instead, we
iteratively solve for the conversion factors following the methods in Godoy-
Rivera et al. (2021). Once stars are placed on the Gaia color-magnitude
diagram (CMD), we then make a cut on the upper bound of the main sequence,
separating the dwarfs below from the red giants above. The exact slope of this
line is not important; the line simply must exclude the main sequence. As
such, the cuts made for the APOKASC-3 data differ from those made for the
APOGEE data due to the larger color range spanned by the main sequence in
APOGEE. Examples of our cuts are shown in Fig. 3 and Fig. 11.

With the recently released spectra from Gaia DR3 we can also verify rotation
and binarity for a subset of APOKASC-3 and DR16 giants bright enough to be in
the sample. The parameter $v_{\mathrm{Broad}}$ captures the total line
broadening due to rotation (v sin$i$), macroturbulence (vmacro), and other
sources. Macroturbulence in a cool giant is on the order of 2 - 3.5 km s-1, so
in principle, in the instance of a rapid rotator, rotation is the dominant
source of broadening (Doyle et al., 2014; Frémat et al., 2022). If we treat
v$\mathrm{Broad}$ as a proxy for vsini, however, we significantly overestimate
the number of rapid rotators. At a 10 km s-1 threshold in
$v_{\mathrm{Broad}}$, which Frémat et al. (2022) claims is comparable to vsini
for cool giants with 10.5 < GRVS magnitude < 11.5, we find that $\sim$ 40% of
APOKASC-3 giants and DR16 giants with the appropriate magnitude are “rapidly
rotating." As such, we do not use $v_{\mathrm{Broad}}$ to draw conclusions
about the rates of rapid rotation in giants. Phillips et al. (in prep) sees
that there is good agreement between vsini and $v_{\mathrm{Broad}}$ above 20
km s-1, so we adopt this as our threshold for rapid rotation just for giants
without a spectroscopic solution, where we have limited additional constraints
on rapid rotation.

Gaia also lets us probe binarity. High values (> 1.4) of the renormalized unit
weight error (RUWE) often indicate the presence of a wide binary companion,
which impacts the astrometric solution (Belokurov et al., 2020; Stassun &
Torres, 2021; Kervella et al., 2022). Adopting a less conservative cut, we
label as binaries all targets with RUWE > 1.2. DR3 also contains the non-
single star catalog, which contains targets with complete binary
classifications. More broadly we can also find radial velocity variable stars
using the criteria laid out in Katz et al. (2022). While binarity alone does
not imply rapid rotation, binarity can suggest the origins of rotation. Binary
rapid rotators are likely tidally synchronized, whereas single rapid rotators
are likely mergers.

##  3 Spectroscopic selection criteria

The aim of this work is to develop a set of spectroscopic criteria which could
identify, with high fidelity, rapidly rotating red giants in the full APOGEE
DR16 catalog. Natural spectroscopic categories to search for rapid rotators
arise from the steps ASPCAP takes to process spectra. ASPCAP assesses the
quality of the spectra as they travel through the pipeline, rejecting spectra
outright and returning no solution, performing an initial fit and rejecting
spectra whose fits do not pass quality control, and finally doing a calibrated
fit to spectra which do not trigger any quality control flags. Below, we
assess how well each spectroscopic category, those with no, partial, and
complete solutions, preferentially picks out rapidly rotating giants.

###  3.1 Anomalously cool stars with complete spectroscopic solutions

Figure 2: Top: A Gaia CMD of APOKASC-3 (grey) and the rapidly rotating,
anomalously cool stars shows that rapid rotators cannot be distinguished from
regular red giants. Photometric binaries and hot stars are excluded from
subsequent plots. Middle: Calibrated log g versus $T_{\mathrm{eff}}$ for
APOKASC-3 (grey) and the rapid rotators, which are more distinguishable from
the RGB, but not completely. Bottom: Calibrated log g versus $\Delta T$ for
all of APOKASC-3 (grey) and the anomalously cool sample. The $\Delta T$
calculation is only valid for stars with 2.3 < log g < 3.5 cm s-2, though we
extrapolate to higher log g. $\Delta T$ calculations outside this log g range
are illustrative only. Black dashed lines denote the log g limits and the
purple dotted line marks the 3$\sigma$ threshold. The anomalously cool stars
are classified as rapid rotators (blue points), slow rotators (green crosses),
or having not enough info (red stars).

The evolutionary state separator (equations 1 and 2), besides being able to
isolate the RGB from the red clump, provides a novel way to quickly identify
stars with stellar properties which significantly deviate from those of
standard red giants. A temperature offset relative to the mean locus is a
useful figure of merit. The position of a red giant in the HR diagram is
predicted to be a nearly vertical line close to the Hayashi track. The zero-
point is a function of the mass and heavy element content, and the difference
between clump and giant is nearly constant. As a result, correcting for these
zero point differences produces two distinct temperature domains - for clump
and giant - that are a useful evolutionary state diagnostic. The abundances
and log g are observables in APOGEE, and the mass is correlated with [C/N]. As
a result, we can measure the temperature offset using a star’s [C/N] ratio,
metallicity, and surface gravity. It is valid only for giants with 2.3 < log g
< 3.5 cm s-2 (Jönsson et al., 2020) due to the narrow log g spanned by red
clump stars (see discussion in, e.g., Girardi, 2016). Because of the evidence
that the log g measurements for rapidly rotating giants are inflated (Thompson
et al., 2019), we examine this figure of merit for the full sample down to log
g = 4.5 cm s-2. The vast majority of anomalously cool stars have 2.3 < log g <
3.85 cm s-2 (bottom panel of Fig. 2). This higher upper limit introduces
minimal contamination from dwarfs. Stars with spectroscopic temperatures at
least 3$\sigma$ cooler ($\Delta T$ < -287 K) than the mean locus of red giants
of equivalent mass and metallicity we deem anomalously cool. For the few stars
in our sample with total metallicity [M/H] < -1, we only deem them anomalously
cool if they satisfy the metallicity-dependent criterion described in §4.
Ultimately we identify 71 anomalously cool stars.

Table 1: Column names for the table containing our APOKASC-3 anomalously cool rapid rotator candidate sample Kepler ID | Brown et al. (2011)
---|---
APOGEE ID | APOGEE DR16
log g (cm s-2) | APOGEE DR16
$T_{\mathrm{eff}}$ (K) | APOGEE DR16
[M/H] | APOGEE DR16
$\Delta T$ (K) | this work
mG |  Gaia DR2
m${}_{\mathrm{B}_{\mathrm{P}}}$ |  Gaia DR2
m${}_{\mathrm{R}_{\mathrm{P}}}$ |  Gaia DR2
$A_{V}$ | this work
distance (pc) |  Gaia DR3
vsini (km s-1) | this work
$v_{\mathrm{Broad}}$ (km s-1) |  Gaia DR2
period (d) | Ceillier et al. (2017); Gaulme et al. (2020)
rotation peak | this work
RUWE |  Gaia DR3
Gaia RV variable | this work
APOGEE RV variable | this work
notes | this work
Table 2: Four different measures of binary fractions of the anomalously cool giants |  vsini > 10 km s-1 | 5 < vsini < 10 km s-1
---|---|---
RUWE fbin | 16$\pm$8% (5/31) | 23$\pm$13% (3/13)
Gaia RV fbin | 88$\pm$18% (23/26) | 92$\pm$27% (12/13)
APOGEE fbin | 84$\pm$21% (16/19) | 100$\pm$33% (9/9)
APOGEE fbin,close | 74$\pm$20% (14/19) | 100$\pm$33% (9/9)

The top panel of Fig. 2 shows where the rapid rotators from the anomalously
cool sample fall in a Gaia CMD. The plotted stars are indistinguishable from
standard red giants despite their spectroscopic anomalies. Anomalously cool
stars are not actually cooler than a typical red giant, however; they are not
particularly red. Their cooler spectroscopic temperatures are a byproduct of
how ASPCAP handles rapidly rotating, potentially active stars.

Thirteen of our targets that were spectroscopically classified as giants are
actually dwarfs, as inferred from Gaia astrometry and photometry (Fig. 2). The
cause is a rare failure mode in the APOGEE pipeline, which considers both
dwarf and giant solutions for a subset of the data. In some cases, both fits
were poor, but the giant solution had a slightly better goodness of fit. For
both the APOKASC-3 sample and the full DR16 sample we use a Gaia CMD cut to
remove such stars from our sample. There are an additional 8 stars which are
too blue to be giants. We make a cut at BP \- RP of 0.9 to remove these
targets. This leaves us with 50 targets.

While these remaining stars appear as normal red giants in a Gaia CMD, when we
examine them in log g \- $T_{\mathrm{eff}}$ space, we begin to see some
differentiation from typical red giants. The middle panel of Fig. 2 shows that
the majority of anomalously cool stars are cooler than the red edge of the
RGB.

The separation between typical red giants and rapid rotators is best seen if
instead of $T_{\mathrm{eff}}$ we plot $\Delta T$ versus log g. The bottom
panel of Fig. 2 shows the two distinct populations. As expected, red clump
stars fall to the left of the $\Delta T$ = 0 line while the RGB stars are
clustered to the right, occupying a narrow range between 0 and -200 K.
Notably, the anomalously cool stars span a range of roughly 800 K.

The majority of anomalously cool giants are rapidly rotating. Of 50
anomalously cool stars, 45 of them (90%) are confirmed rapid rotators, with 31
having vsini > 10 km s-1 and 14 having 5 < vsini < 10 km s-1. Only 4 stars
(8%) can be confirmed as true slow rotators due to their low vsini
measurements. The remaining 1 star (2%) does not have enough information to
classify it as true slow rotators or rapid rotators. All data used to select
and verify rotation in these targets are compiled in a table released with
this paper. Table 1 lists the column names.

Table 2 lists the binary fraction of this sample from four separate measures.
We expect the majority of these targets to be tidally synchronized binaries.
While just about 75% of the most rapidly rotating giants do have a close
binary companion, the Poisson errors are large. For the intermediate rotators,
the close binary fraction is 100%, which is surprising since the tidally
synchronized systems should have the largest spinup. However, the Poisson
errors are large and the close binary fraction for rapid and intermediate
rotators agrees within 1$\sigma$. It is possible that the inclination of the
rapidly rotating systems is low. We also see that about 10% of giants have a
wide binary companion, as determined from Gaia RUWE. Even when the RUWE binary
detection limit is set at 1.4, the wide binary fraction is still non-zero.
This is surprising since a companion at a distance cannot spin up a star.
While most of the targets caught by this criterion are binaries, we see that
they are not exclusively close, tidally synchronized systems like we expected.

###  3.2 Stars with partial spectroscopic solutions

Table 3: Column names for the table containing our APOKASC-3 rapid rotator candidate sample of giants with partial or no spectroscopic solutions. Binarity is not assessed for giants with no spectroscopic solution. Kepler ID | Brown et al. (2011)
---|---
APOGEE ID | APOGEE DR16
mG |  Gaia DR2
m${}_{\mathrm{B}_{\mathrm{P}}}$ |  Gaia DR2
m${}_{\mathrm{R}_{\mathrm{P}}}$ |  Gaia DR2
$A_{V}$ | this work
distance (pc) |  Gaia DR2
vsini (km s-1) | this work
$v_{\mathrm{Broad}}$ (km s-1) |  Gaia DR3
period (d) | Ceillier et al. (2017); Berger et al. (2020)
rotation peak | this work
RUWE |  Gaia DR3
Gaia RV variable | this work
APOGEE RV variable | this work
Flags | APOGEE DR16
notes | this work
no solution | APOGEE DR16
Figure 3: A Gaia CMD of stars with partial spectroscopic solutions compared to the entire APOKASC-3 sample (grey). All targets which fall above and to the right of the dashed line we consider giants. Blue points represent confirmed rapid rotators, whereas the red stars are all stars that have partial solutions. Table 4: Four different measures of binary fractions of the giants with partial spectroscopic solutions. |  vsini > 10 km s-1 | 5 < vsini < 10 km s-1
---|---|---
RUWE fbin | 27$\pm$7% (14/52) | 27$\pm$16% (3/11)
Gaia RV fbin | 70$\pm$15% (23/33) | 25$\pm$18% (2/8)
APOGEE fbin | 63$\pm$15% (17/27) | 71$\pm$31% (5/7)
APOGEE fbin,close | 59$\pm$15% (16/27) | 57$\pm$29% (4/7)

These spectra were high enough quality that the FERRE code could generate an
initial fit but not high enough quality to support the subsequent calibrated
line fitting. Quality is assessed in several ways. Stars with too low signal-
to-noise (< 50), too high of a $\chi^{2}$ ($\chi^{2}$/(signal-to-noise/100)2 >
50), too much variation between photometric and spectroscopic temperatures (>
1000 K discrepancy) from the FERRE fit, or with spectral lines that are more
than twice the width of the template spectrum due to rotational broadening or
line blending from double-lined spectroscopic binaries, all get flagged as
bad.

Rotation can affect these spectra in two ways, first causing line broadening
captured by the rotation flag mentioned above. Second, it can trigger the
color-temperature flag either due to activity-induced temperature gradients on
a star’s surface, which are difficult to capture with a single spectroscopic
temperature, or due to the presence of a binary companion, where, again, a
single temperature cannot accurately reflect the combined light of the binary.
While rotation is not the sole cause of poor fits, we want to quantify how
significant of a role it plays in triggering the quality flags.

We identify 684 stars in APOKASC-3 with partial spectroscopic solutions.
Though we have an initial guess at the stars’ spectral parameters, we choose
to identify red giants by their position on a Gaia CMD for greater accuracy.
Fig. 3 shows the CMD of this sample compared to the entire APOKASC-3 catalog.
The vast majority of these stars are dwarfs, so we impose a cut at a Gaia BP
\- RP of 0.9 mag ($T_{\mathrm{eff}}\approx 5450$ K) to eliminate hot stars and
a cut near the base of the RGB to eliminate dwarfs and subgiants. All stars
which fall above the dividing line are selected as red giants, a population of
95 in total. Table 3 lists the properties of these stars included in the table
released with this paper. Note that these stars are not photometric outliers,
instead lying well within the red giant branch.

Of our 95 giants, 63 (66%) are rapid rotators, with 52 having vsini > 10 km
s-1 and 11 with 5 < vsini < 10 km s-1. Some of the remaining giants are indeed
rapid rotators - they have triggered the ROTATION_BAD flag - but we do not
measure a vsini, so they do not contribute to the count. Though this criterion
requires more work in separating out the red giants, the high fraction of
rapid rotators it identifies, even greater than the anomalously cool stars,
makes it a good tool to target rapidly rotating stars.

The binary fraction of this sample is listed in Table 4. We did not expect a
single binary configuration to explain the rapid rotation of these systems
like we did for the anomalously cool giants. Giants with partial solutions
span the RGB and clump and a star’s position on the giant branch changes its
likelihood of being in a close, tidally synchronized system. The higher wide
binary fraction is surprising, though giants at the tip of the giant branch
can interact with stars at a larger separation. Even when the RUWE binary
detection limit is set at 1.4, the wide binary fraction is still around 10%.

###  3.3 Stars with no spectroscopic solution

Figure 4: A Gaia CMD of stars with no spectroscopic solution compared to the
entire APOKASC-3 sample (grey). All points above and to the right of the
dashed line we consider giants. Red stars are all targets with no solution.
Red squares with a blue outline represent the stars which have other
indicators of rapid rotation, such as rotational variability, but no vsini
measurement. While these giants are rapid rotators, they do not count toward
our uniform assessment of rapid rotation.

ASPCAP was unable to obtain solutions for some targets. Rotation could
contribute to the poor quality for the same reasons outlined above, noisy
spectra from activity, or combined spectral features from two stars in a
binary.

The lack of spectroscopic information about these stars makes them the most
difficult to study but points to the usefulness of the APOKASC-3 catalog. We
still have the light curves of these stars to search for rotation signatures.
However, we are limited by which of these targets show up in other surveys
because we need external information in order to classify each star first as a
red giant and then as a rapid rotator.

There are 232 stars in APOKASC-3 without a spectroscopic solution. Table 3
lists the properties of these stars included in the table released with this
paper. Cross matching with Gaia to photometrically separate the red giants
from main sequence stars, we end up with 32 giants. Their locations on a Gaia
CMD is shown in Fig. 4. These stars do not appear to be photometric outliers,
lying well within the red giant branch. Since these stars lack spectroscopic
solutions, thus lack a best-fit template spectrum, we cannot measure their
vsini. However, Gaia reports a $v_{\mathrm{Broad}}$ measurement for 18
targets, none of which are rapid rotators at the 20 km s-1 threshold. 6
targets do show signs of rotational variability, so we do not think that
giants in this category are devoid of rapid rotators. However, a uniform
assessment of rapid rotation for this sample is difficult; these giants
require significant follow-up.

These stars are difficult to characterize because they lack APOGEE stellar
parameters and radial velocity data. There are also very few of them. As such,
we do not report binary fractions for giants in this category. While these
stars may still contribute significantly to the total population of rapid
rotators, a proper characterization, including an assessment of binarity, is
needed.

###  3.4 Rapid rotation in the complete set of APOKASC-3 giants

Figure 5: Top: A Gaia CMD of the complete set of APOKASC-3 rotators with vsini
> 10 km s-1(black) compared to stars with no spectroscopic solution compared
to those extracted by our spectroscopic criteria (red). Bottom: a comparison
of the spectroscopic stellar parameters for the complete set of APOKASC-3
rotators (black) with vsini > 10 km s-1 with those picked up in the
anomalously cool criterion (red)

Figure 6: Top: A Gaia CMD of the complete set of APOKASC-3 rotators with 5 < vsini < 10 km s-1(black) compared to stars with no spectroscopic solution compared to those extracted by our spectroscopic criteria (red). Bottom: a comparison of the spectroscopic stellar parameters for the complete set of APOKASC-3 rotators (black) with 5 < vsini < 10 km s-1 with those picked up in the anomalously cool criterion (red) Table 5: The counts of rapid rotators at the 5 and 10 km s-1 threshold and their corresponding binary fractions broken down by evolutionary state. We lump together giants on the lower RGB where we could not separate the true first-ascent giants from the RC stars. | Upper RGB | Lower RGB | RC | RGB/RC
---|---|---|---|---
vsini > 10 km s-1 | 28 | 41 | 34 | 48
RUWE fbin | 18$\pm$8% (5/28) | 24$\pm$8% (10/41) | 3$\pm$3% (1/34) | 17$\pm$6% (8/48)
Gaia RV fbin | 48$\pm$14% (11/23) | 69$\pm$15% (22/32) | 14$\pm$7% (4/28) | 86$\pm$17% (24/28)
APOGEE fbin | 85$\pm$26% (11/13) | 71$\pm$16% (20/28) | 73$\pm$26% (8/11) | 62$\pm$15%(18/29)
APOGEE fbin,close | 69$\pm$23% (9/13) | 64$\pm$15% (18/28) | 9$\pm$9% (1/11) | 59$\pm$14%(17/29)
5 < vsini < 10 km s-1 | 207 | 71 | 304 | 13
RUWE fbin | 17$\pm$3% (36/207) | 23$\pm$6% (16/70) | 13$\pm$2% (40/304) | 23$\pm$13% (3/13)
Gaia RV fbin | 9$\pm$2% (16/172) | 42$\pm$9% (23/55) | 14$\pm$2% (35/247) | 23$\pm$13% (3/13)
APOGEE fbin | 74$\pm$9% (69/93) | 66$\pm$13% (25/38) | 42$\pm$6% (59/139) | 30$\pm$17% (3/10)
APOGEE fbin,close | 58$\pm$8% (54/93) | 50$\pm$11% (19/38) | 11$\pm$3% (15/139) | 10$\pm$10% (1/10)

To assess the yields and completeness of the criteria, we must know what the
true underlying population of rapid rotators is in the APOKASC-3 sample.
Measurements of vsini can automatically confirm whether a star is rapidly
rotating. Using the same methods as in Tayar et al. (2015), Dixon et al.
(2020), and Daher et al. (2022), we determine vsini by cross-correlating the
combined DR16 spectra for 15 220 giants with their respective best-fit
template and an artificially broadened version of that template. This only
works for stars with a best-fit template spectrum. We cannot estimate vsini
for stars with no spectroscopic solution. While Tayar et al. (2015) adopts 10
km s-1 as their threshold for rapid rotation, they find that stars with 5 <
vsini < 10 km s-1 are rotationally enhanced, likely from binary interaction.
The same is seen more broadly in the field in Daher et al. (2022). As such, we
adopt 5 km s-1 as the threshold for rapid rotation but maintain a distinction
between stars with v > 10 km s-1 and those with 5 < vsini < 10 km s-1. We
employ the same photometric cuts as for stars identified by our spectroscopic
selection criteria: Gaia BP \- RP color > 0.9 mag and absolute Gaia G-band
magnitude > 0.3 + 3.4*(dereddened BP \- RP).

Because the APOKASC-3 giants span the giant branch and the clump, a constant
vsini threshold for rapid rotation corresponds to vastly different rotation
periods depending on the star’s radius. As such, we break down the rapid
rotators by their evolutionary state. Upper giant branch stars are defined as
giants with log g < 2.3 cm s-2 or, for giants without complete solutions, Gaia
BP \- RP color > 1.5 mag. For giants on the lower giant branch we separate the
first-ascent giants from the clump using seismically-determined evolutionary
states. For stars lacking seismology we adopt the spectroscopic evolutionary
state determined in DR16. Table 5 shows the breakdown of rapid rotators by
evolutionary state as well as their respective binary fractions from Gaia and
APOGEE data.

We demonstrated that our spectroscopic selection criteria have a high purity
but we still need to assess completeness. Our method selects for stars with
rotation, or activity, substantial enough to produce anomalies in the
spectroscopic solution. Furthermore we focus on the lower RGB, where these
anomalies are the largest. Slower rotators, or those in other regimes, will
not be detected with our technique.

In APOKASC-3, we detect 151 stars with vsini > 10 km s-1, and our
spectroscopic criteria combined capture 85 (56%) of them. However, on the
lower RGB, where our anomalously cool and partial solution criteria are most
sensitive, we capture 33 (80%) of the 41 confirmed lower RGB rapid rotators.
Still, we see rapid rotators outside the evolutionary states where we are
sensitive. Of the 67 stars we do not recover, 19 are confirmed on the upper
RGB, 8 are on the lower RGB, and the remaining 40 are either in the clump or
were a combination of first-ascent and clump giants. Fig. 5 shows that our
spectroscopic selection criteria are efficient in recapturing the rapid
rotators with v > 10 km s-1.

We observe 595 giants with 5 < vsini < 10 km s-1. Our results are consistent
with Tayar et al. (2015), who found that stars with vsini in this range
significantly outnumbered the traditional rapid rotators. Even when binaries
are targeted first and then rapid rotation is assessed, the trend holds (Daher
et al., 2022). The vast majority of these targets are found in the clump or
the upper RGB, seen most clearly in the bottom panel of Fig. 6. As such, we
reach the same conclusions as Tayar et al. (2015), namely that these stars are
interacting, and likely merging, on the upper RGB and retaining enhanced
rotation in the clump as a result. This is supported by the low binary
fraction of these intermediate rotators in the clump. Because the majority of
these targets do not live on the lower RGB, where our criteria are most
sensitive, we only recover 26 of these rotators.

Two things are apparent. The first is that our spectroscopic selection
criteria preferentially pick out the most rapidly rotating giants, suggesting
that higher rotation speeds are needed to induce spectroscopic anomalies. The
second is that population of rotationally enhanced giants is far larger than
we expected. Ultimately, 4.9% of the APOKASC-3 catalog is rapidly rotating at
the 5 km s-1 threshold, implying a previously unidentified population of
giants in the Kepler field which have experienced binary interaction.

As an aside, we call attention to the population of rotators to the left of
the giant branch in Fig. 5 and Fig. 6. These stars occupy a well-defined
region of log g \- $T_{\mathrm{eff}}$ space, namely 2.3 < log g < 3.85 cm
S-2and 5300 - 6000 K. There are only a few stars in this region because we
impose a photometric cut at Gaia BP \- RP = 0.9. Without this cut, this region
is much more populated with rapid rotators. These stars are likely crossing
the Hertzsprung gap. We suggest the above spectroscopic criteria for
identifying these targets in APOGEE.

###  3.5 Binary Origins of Rotation in APOKASC-3

The binary fraction of giants should change as a function of evolutionary
state (Badenes et al., 2018; Price-Whelan et al., 2020). As giants expand up
the RGB, systems which were tidally synchronized on the lower RGB should merge
on the upper RGB. This then leads to a lower binary fraction in the clump.
This is mostly reflected in the binary fractions in the entire set of
APOKASC-3 rotators; the binary fraction are comparable between the lower and
upper RGB but overall lower in the clump.

The giants with partial solutions reflect this distribution of binaries, which
makes sense since these giants span the RGB and clump. The anomalously cool
giants have higher binary and close binary fractions, suggesting that most of
these stars are in close binaries on the lower RGB.

##  4 Rapidly rotating red giants in APOGEE DR16

###  4.1 Assembling the catalog

Figure 7: Calculated $\Delta T$ for all DR16 stars (grey) with 2.3 < log g < 3.85 versus metallicity. Photometric binaries are highlighted in pink. They do not contribute to our final counts. The solid lines represent the fits to the binned data (black squares), with blue corresponding to the high metallicity population and red corresponding to the low metallicity population. Dashed lines represent the 3$\sigma$ threshold, below which stars are deemed anomalously cool. We limit the y-axis to focus on the main group of anomalously cool stars, though some targets have much lower $\Delta T$. Figure 8: A Gaia CMD of a subset of DR16 stars (grey) with the anomalously cool stars highlighted in red. The blue distribution represents the location of a sample of RS CVn-like variables, which preferentially occupy the subgiant and lower giant branch. The black line represents color-magntiude cuts we make to remove dwarfs from our sample. Figure 9: Calibrated log g versus $T_{\mathrm{eff}}$ for the entire DR16 sample (grey) with the anomalously cool stars highlighted in red. YSOs are visible above the main sequence with temperatures below 4000 K, our adopted cutoff (dashed line). Figure 10: Log g versus $\Delta T$ for log g-restricted set DR16 stars (grey) and the anomalously cool stars (red). The anomalously cool stars are distinct from the rest of DR16. Figure 11: A Gaia DR2 CMD for a subset of the APOGEE DR16 sample (grey) and stars with partial spectroscopic solutions (red). Black lines denote the photometric cuts we impose to isolate the giants. Table 6: Column names for the table containing our APOGEE DR16 anomalously cool rapid rotator candidate sample APOGEE ID | APOGEE DR16
---|---
Gaia ID |  Gaia DR3
log g (cm s-2) | APOGEE DR16
$T_{\mathrm{eff}}$ (K) | APOGEE DR16
[M/H] | APOGEE DR16
$\Delta T$ (K) | this work
mG |  Gaia DR2
m${}_{\mathrm{B}_{\mathrm{P}}}$ |  Gaia DR2
m${}_{\mathrm{R}_{\mathrm{P}}}$ |  Gaia DR2
$A_{V}$ | this work
distance (pc) |  Gaia DR3
vsini (km s-1) | this work
$v_{\mathrm{Broad}}$ (km s-1) |  Gaia DR2
RUWE |  Gaia DR3
Gaia RV variable | this work
APOGEE RV variable | this work
Table 7: Column names for the table containing our APOGEE DR16 rapid rotator candidate sample of giants with partial solutions APOGEE ID | APOGEE DR16
---|---
Gaia ID |  Gaia DR3
mG |  Gaia DR2
m${}_{\mathrm{B}_{\mathrm{P}}}$ |  Gaia DR2
m${}_{\mathrm{R}_{\mathrm{P}}}$ |  Gaia DR2
$A_{V}$ | this work
distance (pc) |  Gaia DR3
vsini (km s-1) | this work
$v_{\mathrm{Broad}}$ (km s-1) |  Gaia DR2
RUWE |  Gaia DR3
Gaia RV variable | this work
APOGEE RV variable | this work
Flags | APOGEE DR16

Our total candidate list contains 6933 targets, including 1484 anomalously
cool giants, 1733 giants with partial solutions (plus 340 luminous giants with
partial solutions), and 3376 giants with no solution. We discuss each category
below and follow up with vsini confirmation in §4.2.

To apply the anomalously cool criterion to the entire APOGEE DR16 sample, we
must impose an additional cut that was not necessary in the APOKASC-3 sample
due to it being an older population. Some young stellar objects (YSOs) fall in
the same log g range as the lower giant branch but have lower effective
temperatures. The YSO population is very cleanly separated from the red
giants, as shown in Fig. 9, and we can remove them with a simple
$T_{\mathrm{eff}}$ cut at 4,000 K and below.

We must also make corrections to account for the different metallicity ranges
covered by DR16 and APOKASC-3. The evolutionary state classifier, which
provides the basis for finding anomalously cool stars, was calibrated to the
Kepler field, where metallicity extends down to [M/H] of -2.46, with the vast
majority of stars having metallicity above -0.5. There is therefore no
guarantee that the Kepler relationships will extrapolate to the low
metallicity regime.

We see that they do not. Fig. 7 shows the distribution of all stars in DR16
with 2.3 < log g < 3.85 cm s-2 in $\Delta T$ \- metallicity space. By
definition, all RGB stars have $\Delta T<0$. Down to [M/H] of about -1 RGB
stars tend to lie in an even distribution centered on $\Delta T=-83.6$K. At
lower metallicity, the RGB stars trend downward in $\Delta T$.

To identify anomalously cool stars at low metallicity, we first bin the stars
with metallicity between -2.4 and -1 in increments of 0.1 dex and iteratively
reject stars which are more than 3$\sigma$ from the median $\Delta T$ in each
bin. Here $\sigma$ represents the absolute median deviation. Red clump stars
have $\Delta T$ values above zero, and could in principle bias the results.
This contamination is mitigated by the usage of median statistics and by
stellar population effects. The majority of core He burning stars in this
metallicity domain are hotter than 6000 K and therefore outside of the domain
considered here. We then fit a second-order polynomial to the medians of each
bin, the solid red line in Fig. 7. To align the median lines of the red giants
above and below [M/H] = -1, we apply an offset of +3.24 K to the polynomial.
Finally, we define a boundary at 3$\sigma$ to isolate the anomalously cool
stars. The dashed lines in Fig. 7 represent the 3$\sigma$ boundary, below
which all stars are flagged as anomalously cool. Above [M/H] of -1, the line
is fixed at -287K, and below it is metallicity dependent. We discuss the
metallicity dependence of our yields in Section 4.2.

We then remove 310 candidates that are not giants by excluding all targets
with absolute Gaia G-band magnitude MG > 1 + 3*Gaia BP \- RP and Gaia BP \- RP
> 0.77 mag. As in the APOKASC-3 sample, these stars lie along the photometric
binary sequence, and we believe that they represent a failure mode where the
combined light of two stars contributing significant flux yields inaccurate
stellar parameters. In total, our anomalously cool sample consists of 1,484
giants. Like the APOKASC-3 sample, these stars are not actually redder than
typical giants. A Gaia CMD of the anomalously cool stars and a subset of the
entire DR16 sample, Fig. 8, shows that most stars fall comfortably on the
lower RGB. We note that this list includes 82 targets that we could not verify
as giants because of either very large formal $A_{V}$ values or unreliable
distances. Table 6 lists the column names of the table containing the
properties of these giants.

Active field giants are preferentially found on the lower RGB. Many active
giants are in synchronized binaries, and while synchronization timescales
increase steeply with orbital period, evolutionary timescales decrease with
luminosity. On the upper RGB, stars do not have enough time to synchronize
with their companion before coming into contact. Observations of active giants
confirm this. RS CVn type variables, a binary where the primary is an active
giant or subgiant, live predominantly on the lower RGB with orbital periods
mostly below the synchronization limit of 20 days (Fig 7 in Leiner & Geller
(2021)). Since we expect many of our targets to be active giants in binaries,
RS CVn candidates, they should fall in the same CMD location as the RS CVns.
We can see from the overlap of our targets with the blue distribution in Fig.
8, the CMD location of a sample of about 550 known RS CVns, that they do.

In spectroscopic parameter space, our sample looks like the APOKASC-3 sample.
Fig. 9 shows that these stars, mostly, but not completely, stand out from
typical red giants. In log g \- $\Delta T$ space, these stars are completely
isolated from the rest of the sample, shown in Fig. 10. Like the APOKASC-3
sample, the majority of stars have temperature offsets between -1000 and -287
K, though there are some which extend to higher offsets. The value of $\Delta
T$ does not correlate with the rotation speed, so it remains unclear as to why
some stars have such large $T_{\mathrm{eff}}$ offsets from a typical red
giant.

Stars with partial spectroscopic solutions are more difficult to identify
because they are not isolated to one evolutionary state. There are 31821 stars
spread out across a Gaia CMD (Fig. 11). To isolate the stars along the RGB, we
start with the stars with the best characterized photometry, excluding any
target with an $A_{\mathrm{V}}>1$. We then impose 3 cuts: a cut at a Gaia BP
\- RP of 0.77 to exclude hot main sequence stars, a diagonal cut at the
turnoff, the same as in the anomalously cool sample, to exclude the main
sequence, and a cut along the red edge of the RGB to exclude YSOs. In total,
we identify 1733 rapidly rotating giant candidates. Table 7 lists the
properties of these targets we include in the catalog released with the paper.
Combining the targets identified by both spectroscopic criteria, our main
catalog ultimately contains 3217 rapid rotator candidates.

In the top-right section of Fig. 11, there is a population of 340 luminous
giants excluded by our photometric cuts. Our independent spectroscopic fitting
method is not sensitive to these very red stars, so we cannot assess rotation.
Furthermore, these giants require a more careful analysis due to more sources
of variability, including Mira-like variables and AGB thermal pulses, which
could contribute to their poorer spectral quality. We also see in Fig. 3 and
Fig. 5 that in the APOKASC-3 catalog these stars are mostly devoid of rapid
rotators, which our criteria are most sensitive to. As such, we include the
APOGEE IDs of these stars in a separate table but do not include them in our
main catalog.

There are 3376 stars with no spectroscopic solution, selected via the same
photometric cuts as the stars with partial solutions. In APOKASC-3, we did not
have a uniform, definitive means to assess the fraction of these stars which
exhibit rapid rotation. Quality control is difficult for stars with no
spectroscopic solution, so we do not include these in our main sample.
However, we do include the APOGEE IDs of targets in a separate table, as these
targets likely contain rapid rotators.

###  4.2 Confirming rapid rotation

Figure 12: The distribution of vsini measurements from our independent
spectral fits for our rapid rotator candidates (blue) and a subset of typical
red giants (yellow) in APOGEE. The vertical dashed line indicates the uptick
in rotation starting at log vsini of 0.7, corresponding to a distinct
population of rapid rotators.

Figure 13: The distribution in a Gaia CMD (top), log g versus
$T_{\mathrm{eff}}$ (middle), and log g versus $\Delta T$ (bottom) of the
confirmed anomalously cool rapid rotators. The grey histogram represents a
subset of the entire DR16 sample. Figure 14: A Gaia CMD of the confirmed rapid
rotators with partial spectroscopic solutions. The grey histogram represents a
subset of the entire DR16 sample. Figure 15: The fraction of giants in a bin
of 0.1 dex in [M/H] that are rapidly rotating. The errors bars represent
Poisson noise. There is an apparent metallicity dependence, with a break at
[M/H] = -1, where we see that few of the low metallicity stars are rapid
rotators.

Because the field stars in DR16 are less well-characterized than those in the
Kepler field, we take a more careful approach when identifying the rapid
rotators by extracting vsini. The ASPCAP (García Pérez et al., 2016) pipeline
has been applied successfully in the characterization of large samples of
giants; however, it does not currently have a prescription to measure
rotational velocity in spectra simultaneously with other stellar parameters.
To obtain rotational velocity estimates from APOGEE H-band spectra we modify
the existing ASPCAP pipeline, which includes seven dimensions including
microturbulence, [C/M], [N/M], [alpha/M], [M/H], surface gravity, and
effective temperature, with a fit for macroturbulence. We use a fiducial value
of 2 km/s in microturbulence and replace this dimension with surface rotation
(sampled at $v\,\sin\,i$ of 1.5, 3.0, 6.0, 12.0, 24.0, 48.0, and 96.0 $km/s$)
in our modified 7-D fit. We use stellar atmospheres from MARCS with the same
extent in metallicity, chemical abundances, and effective temperatures as in
the ASPCAP pipeline; we kept the grid spacing similar in most dimensions but
reduced it from 0.25 to 0.5 dex in others for computational purposes, which
does not significantly impact stars with S/N > 70 (García Pérez et al., 2016).
Combined spectra from APOGEE DR17 were continuum normalized simultaneously
with the synthetic spectra and fit with ferre222ferre is publicly available
from http://hebe.as.utexas.edu/ferre., a least-squares fitting spectra
analysis code. Stellar parameters were obtained through the unconstrained
optimization by quadratic approximation algorithm and errors were estimated
through inversion of the curvature matrix (García Pérez et al., 2016).

We compare our measured log vsini from our sample to a subset of 1021 regular
APOGEE red giants on the lower RGB in Fig. 12. Two things are apparent. The
first is that the regular stars rarely exceed a log vsini of 0.7 km s-1; they
are true slow rotators as expected. The second is that there is a break in the
distribution of log vsini at 0.7 instead of 1, as we would expect if the
threshold for rapid rotation were 10 km s-1. Instead, this gap suggests that
the distinct population of rapid rotators starts at 5 km s-1, consistent with
Tayar et al. (2015). As such, we adopt log vsini of 0.7 km $s^{-1}$ as our
threshold for rapid rotation in the vsini we measure ourselves. For the values
reported by APOGEE, we maintain the 10 km s-1 threshold.

Using vsini from APOGEE, our measured vsini, and Gaia $v_{\mathrm{Broad}}$, we
confirm rotation in 1598 (50%) stars, with 1188 having vsini > 10 km s-1 and
410 having 5 < vsini < 10 km s-1. Given that we have no constraint on the
inclinations of these systems, some stars will be true rapid rotators with low
vsini. For example, 15% of stars with random inclinations have sini < 0.5.
Rapid rotation could still induce spectroscopic anomalies even if the star is
viewed pole-on - a large star spot filling fraction from stellar activity, for
example. Correcting for random inclination effects, our data corresponds to a
true rapid rotator fraction of 65%.

Like our test sample from the APOKASC-3 catalog, the vast majority of these
stars have vsini > 10 km s-1, further supporting that, at least in DR16,
significant rotation is needed in order to induce spectroscopic anomalies.
Based on the yield from the anomalously cool stars in APOKASC-3, we expect 979
anomalously cool giants to have vsini > 10 km s-1. In DR16, we find 541 (55%).
Furthermore, in APOKASC-3, both criteria recover 56% of the total number of
giants rotating with at least 10 km s-1. Based on this yield, our catalog
captures 1188 (66%) of the expected 1802 giants with vsini > 10 km s-1.

Fig. 13 shows the observed properties of the rapid rotators. The anomalously
cool rotators do not favor a particular range in log g nor do they cluster
around a particular $\Delta T_{\mathrm{eff}}$. Like the APOKASC-3 sample, most
rotators are cooler than the giants, but not entirely separated from the giant
branch. The same is true in the Gaia CMD. The rapid rotators fall on the lower
giant branch, though their median color is 0.13 magnitudes redder than the
non-rotating, metal-rich stars. This could be due to spots. The rapidly
rotating giants with partial solutions also fall predominantly on the lower
giant branch, as shown in Fig. 14, completely indistinguishable from typical
giants.

We know from the APOKASC-3 sample that our spectroscopic criteria
preferentially identify rotators with vsini > 10 km s-1 and that the number of
rotators with 5 < vsini < 10 km s-1 far outnumbers those with vsini > 10 km
s-1. While this could be a population effect, it is important to see if the
same pattern is observed in the field. We use the same control sample as
above, randomly selected from lower giant branch stars (with 2.3 < log g < 3.6
cm s-2) that we photometrically confirm as giants. Among these 1021 giants, we
recover the same general trends of rapid rotation as the APOKASC-3 sample,
even when the APOKASC-3 sample is restricted to the same log g, though at half
the same absolute rate. 2% (18/1021) of the DR16 control sample rotates with 5
< vsini < 10 km -1, compared to 4% in the APOKASC-3 sample, and 0.4% (4/1021)
of the DR16 control rotatoes with vsini > 10 km s-1, compared to 1% in the
APOKASC-3 sample.

There is also a striking trend with metallicity, but not in the direction we
might expect. Fig. 15 shows that the fraction of anomalously cool giants where
we confirm rapid rotation is half to a quarter of the rates for higher
metallicity stars. There are several things at play here. The first is that
there is a metallicity offset of -0.37 dex between the median [M/H] of all
anomalously cool giants and a random sample of “regular" giants. This suggests
that in addition to a lower $T_{\mathrm{eff}}$ and a higher log g, ASPCAP also
returns lower [M/H].

The second is that the anomalously cool criterion catches more metal-poor
stars than metal-rich stars. Fig. 7 shows that the scatter in $\Delta T$ below
[M/H] of -1 is much larger. As such, the anomalously cool criterion captures
18% of all metal-poor giants compared to just 1.5% of metal-rich giants. If we
correct for the -0.37 dex metallicity offset, the fraction of anomalously
cool, metal-poor giants drops to 9%. This implies two things. The first is
that 10% of metal-poor giants are false positives; it is a result of a poor
fit that the metallicity is low. The second is that the temperature errors on
the low metallicity stars are larger than for those with high metallicity.
This is reflected in the distribution of $\Delta T$ values shown in Fig. 7.
The anomalously cool criterion is defined to be a 3$\sigma$ deviation from the
median values of each metallicity bin. However, when we fit a Gaussian
distribution to the $\Delta T$ values of giants in each metallicity bin below
-1, we see non-Gaussian tails and large scatter such that the 3$\sigma$
boundary to separate the anomalously cool giants is on average only 1.14
standard deviations from the peak. As such, if there are large temperature
errors in for these metal-poor giants, $DeltaT$ is larger and we mistakenly
flag more of the population as anomalously cool. Furthermore, since there is
such a broad scatter in $\Delta T$ among the rapid rotators and the false
positives, there is no way we could have defined a boundary where rapid
rotators are preferentially identified.

The third is that there are also just fewer rapid rotators at lower
metallicity. The drop in the rapid rotator fraction at low metallicity results
from the combination of a fewer rapid rotators and a higher fraction of the
overall metal-poor giant population being deemed anomalously cool. The
numerator is small and the denominator is large compared to the metal-rich
giants.

###  4.3 The origin of rapid rotation

Table 8: Four different measures of binary fractions of the confirmed rapid rotators in DR16. |  vsini > 10 km s-1 | 5 < vsini < 10 km s-1
---|---|---
Anomalously cool | 541 | 281
RUWE fbin | 16$\pm$2% (86/537) | 21$\pm$3% (59/277)
Gaia RV fbin | 78$\pm$7% (138/178) | 70$\pm$9% (66/94)
APOGEE fbin | 88$\pm$4% (381/434) | 88$\pm$6% (207/234)
APOGEE fbin,close | 74$\pm$4% (322/434) | 79$\pm$6% (185/234)
Partial solution | 647 | 129
RUWE fbin | 19$\pm$2% (124/644) | 26$\pm$4% (33/128)
Gaia RV fbin | 74$\pm$5% (206/279) | 52$\pm$14% (13/25)
APOGEE fbin | 70$\pm$4% (352/506) | 50$\pm$8% (38/76)
APOGEE fbin,close | 54$\pm$3% (274/506) | 28$\pm$6% (21/76)

To begin to understand the origin of rapid rotation in our sample, we search
for binary companions among our rapid rotators. We see the same general trends
as in the APOKASC-3 sample. The anomalously cool giants have a higher binary
fraction than the giants with partial solutions and the majority of the
anomalously cool binaries are close. However, the giants with partial
solutions show lower binary fractions across all measures. We also find that
about 20% of stars have a wide binary companion, comparable to what we find in
APOKASC-3 but still higher than anticipated. We discuss this further in §5.
Table 8 lists the binary fractions of this sample.

We suspect that the remaining giants which had enough data to assess binarity
but had no indication of a companion are merger products. They occupy the same
space as the binary rapid rotators in a Gaia CMD, and those with complete
spectroscopic solutions are indistinguishable from binaries in log g \-
$T_{\mathrm{eff}}$ space. Among the stars with complete spectroscopic
solutions the fraction of single rapid rotators is strongly metallicity-
dependent. Below [M/H] of -1, 36% of rapid rotators are single, compared to 7%
at high metallicity, after correcting for the metallicity bias. This is not a
byproduct of ASPCAP, as Carlberg et al. (2011) also found that their
population of single rapidly rotating K giants had lower [Fe/H] metallicity
(0.6 dex on average) than their slowly rotating counterparts. Several things
could lead to this. These giants could be merger products, either with a
stellar or substellar companion. Since the close binary fraction is higher at
low metallicity and there i evidence that the period distribution of those
bianries is also metallicity-dependent (Moe et al., 2019), a merger scenario
makes sense. Low metallicity stars are also subject to less mass loss, which
would weaken magnetic breaking on the main sequence and keep the star spinning
more rapidly as it evolved to the giant branch. Also, we are subject to small
number statistics here as there are only 11 single rapid rotators with [M/H] <
-1.

##  5 Discussion

Individual samples of rapidly rotating red giants typically contain on the
order of tens to hundreds of stars (e.g., Tayar et al., 2015; Ceillier et al.,
2017; Gaulme et al., 2020). With APOGEE, we can extract samples an order of
magnitude larger (e.g., Daher et al., 2022). With our new spectroscopic
criteria, we have created an sample of 1598 rapid rotators. We also provide a
comparably-sized cohort of stars (3376 targets) without spectroscopic
solutions. With independent validation, this group would yield a significant
increase in the rapidly rotating population. While the stars in this catalog
are interesting in their own right, we have also uncovered many unforeseen but
interesting results during the catalog assembly process.

The first is that the population of rapidly rotating red giants is far larger
than we previously thought. With approximately 10% of giants expected to
interact at some point on the RGB (Carlberg et al., 2011; Tayar et al., 2015;
Badenes et al., 2018), finding that 4.9% of APOKASC-3 giants and 2.5% of DR16
giants rotate with at least 5 km s-1 implies that the number of true rapid
rotators in APOGEE is in the tens of thousands. This is seen in Daher et al.
(2022) as well. No one criterion will capture all of these rapid rotators,
especially since we are not sensitive to the majority of them. By definition,
the anomalously cool criterion is only sensitive to the lower RGB. While
giants with partial solutions span the giant branch, they make up a minority
of giants in all of DR16. Furthermore, the bulk of these rapid rotators,
especially those at the 5 km s-1 threshold do not even appear to be
spectroscopically anomalous, and there are $\sim$ 4x as many stars rotating at
the 5 km s-1 threshold compared to the 10. Without measuring their vsini, we
would not know that they are rapid rotators.

These stars can only be identified if rotation is included as a free parameter
in the giant grids used in ASPCAP. This is a computationally expensive task,
however, there are two strong reasons to do this. The first, is that there are
tens of thousands of post-interaction red giants in the field that are
currently undetected. Finding these giants completes the census of interacting
binaries, allowing us to better constrain the relative rates and outcomes of
different types of binary interaction. The second is that including rotation
in our independent spectroscopic fits allowed us to obtain fits for giants
where ASPCAP could not find solutions, and it improved the fits for stars
where ASPCAP could find solutions. We saw shifts in the stellar parameters
returned by ASPCAP for the anomalously cool stars to more realistic values,
such as increase in [M/H] and decreases in log g. Including rotation for the
giants not only enables more accurate fits, it also enables more fits in the
first place.

Furthermore, measuring vsini for all giants ensures a uniform assessment of
rotation. Aggregating rapid rotators from different surveys, or searching for
rotators using a variety of techniques, leads to complicated selection
functions and biases that are difficult to correct for. For example,
identification of rapid rotators from rotational modulation is biased towards
higher spot amplitudes, and the telescope with which a star is observed leads
to a bias in rotation periods; TESS is biased to short periods whereas Kepler
can detect longer periods. A lack of uniform measures of rotation make it
difficult to assess completeness.

The biases might also not be what one might think. This leads to the second
surprising result. We thought that our spectroscopic selection criteria,
especially the anomalously cool stars, would almost exclusively pick out
tidally-synchronized binaries. They instead yielded a mix of systems. While we
found that the majority of giants in our sample are rapid rotators in
binaries, the close binary fraction for these giants was lower overall than
anticipated. This is the opposite of what one would expect. The fraction of
longer period binaries, probed by RUWE and the less strict APOGEE RV
criterion, are also non-negligible. Rapid rotation in a giant with a distant
companion is difficult to explain, since there is no interaction to cause
spin-up. However, three-body interactions could explain these systems if
instead, the distant companion was actually a tertiary and the rapid rotator
resulted from a merger of the inner binary. It is outside the scope of this
paper to investigate the feasibility and expected rates of three-body
interactions producing these systems. We present it here as a possible
solution.

While our criteria are effective at finding rapid rotators, we have learned
that we cannot use these criteria to preferentially identify rapid rotators of
a singular origin. To pick out just the tidally synchronized systems, the
contact binaries, or the merger products, for example, one would need a vsini
measurement as well as a careful selection function to isolate stars in
particular regions of the HR diagram or spectroscopic stellar parameter space.

However, using vsini to identify rapid rotators has its shortcomings too.
Fixed values of vsini do not correspond to the same rotation periods due to
the factor of 10 change in radius between the lower and upper RGB. The
convective overturn timescales also increase as a star moves up the giant
branch meaning that comparable vsini values in stars do not necessarily imply
comparable levels of activity. Furthermore, ASPCAP can return spuriously high
vsini measurements for more slowly rotating but equal mass (comparable
luminosity) binary stars due to the confusion between orbital and rotation
period (Simonian et al., 2019). While we do not expect this to be a
significant source of contamination, we still must be aware of it.

Nevertheless, this catalog is a unique grouping of rapid rotators whose rapid
rotation originates from a variety of sources. There are RS CVn variables,
merger products, and synchronized binaries. The catalog could be searched for
cataclysmic variables or non-interacting compact object companions. The merger
and binary populations taken individually can probe a variety of interesting
astrophysics. For example, a comparison between the number of mergers seen
versus how many we would expect to see based on the time it takes to spin down
post-merger can calibrate spin-down models. The properties of the binary
systems, such as their mass and period distributions, can be compared to known
binary distributions, like the ones in Moe & Di Stefano (2017). For stars with
seismology, the internal structure of interacting or post-interaction giants
can be assessed, or, masses derived from binary systems can calibrate
asteroseismic scaling relations (e.g., Beck et al., in prep). While assembling
this catalog has revealed more about the underlying population of rapid
rotators, the catalog itself is a useful resource for studying en masse giants
with unique evolutionary histories.

Beyond what can be done with this catalog, is the question of whether the same
or similar criteria can be applied to subsequent APOGEE data releases or other
large spectroscopic surveys. In principle they can, but how well the criteria
perform depends on the quality of the spectra and the performance of the
processing pipelines.

In APOGEE DR17 (Abdurro’uf et al., 2022, Holtzman et al. in prep), the
anomalously cool criterion is still valid, but it is not as good at capturing
rapidly rotating red giants. The $T_{\mathrm{ref}}$ calculation for DR17 is

$T^{{}^{\prime}}_{\mathrm{ref}}=3082.884-533.781\mathrm{[Fe/H]}+553.171\mathrm{log}g-282.087\mathrm{[C/N]},$
(3)

using the stellar parameters derived in DR17. When we take our sample of 178
APOKASC-3 targets and see if we can identify them as spectroscopic outliers
using DR17 parameters, we cannot recapture the sample. 37 stars have log g and
$T_{\mathrm{eff}}$ outside the range where the anomalously cool criterion is
valid. Of the 95 stars with stellar parameters in the range where we can
calculate temperature offsets, about half appear anomalously cool. The last 46
stars have no spectroscopic solution, but 35 of them were confirmed rapid
rotators. In DR17, selecting giants with no spectroscopic solution could yield
a high-fidelity sample of rapid rotators. We cannot study stars with partial
solutions in DR17 because these types of stars do not exist in DR17. All stars
that had initial FERRE fits were pushed through the pipeline; they had
complete fits.A combination of giants with no spectroscopic solution and
anomalously cool giants should recover roughly one third of the expected
population of rapid rotators in DR17. In DR17 ASPCAP did a better job of
getting more accurate stellar parameters for rapidly rotating stars.

Regardless of the criterion adopted to find rapidly rotating red giants in a
particular spectroscopic survey, it is promising that this analysis is
possible at all. Being able to exploit lower quality data and systematic
errors in a pipeline to find astrophysically interesting stars helps maximize
the utility of the survey.

The existence of lower quality data also means that there are gains to be made
in processing the spectra. We are able to fit all but 29 of them. Allowing
rotation to be a free parameter changes the other stellar parameters as well.
The median log g in our sample is higher by 0.13 dex and the median [Fe/H] is
lower by 0.25 dex compared to our independent spectral fits. The independent
fits shifted the median $T_{\mathrm{eff}}$ colder by about 150 K, opposite of
the direction we expect. However, this may be explained by the presence of
star spots. While a spot analysis (Cao & Pinsonneault, 2022) is not included
in this work, we expect many of these stars to be active, and the presence of
spots could further bring down $T_{\mathrm{eff}}$.

Moving forward, spectroscopic identification through vsini measurements or
photometric detection through spot modulation seem to be the most promising
means for identifying rapid rotators. The disagreement between
$v_{\mathrm{Broad}}$ and vsini makes Gaia $v_{\mathrm{Broad}}$ difficult to
use. Furthermore, less than 500 giants in our sample had Gaia
$v_{\mathrm{Broad}}$ measurements. Rotation periods from TESS offer a
promising means to find rapid rotators, especially with a new pipeline which
can extract rotation periods longer than the 13.7 day limit imposed by TESS’s
27 day viewing window (Claytor et al., 2022).

##  6 Conclusions

For giants born below the Kraft break, rapid rotation is a direct consequence
of binary interaction. These products of interaction produce sets of
astrophysically interesting systems which follow evolutionary tracks often not
accessible in single star evolution, including evolved blue stragglers, merger
products, active giants, and contact binaries, to name a few. In this paper,
we have identified a large cohort of rapid rotator candidates selected from
evolved, cool giants in APOGEE.

We find that the three spectroscopic categories of giants with no solution,
giants with partial solutions, and giants with complete solutions, created as
APOGEE assesses the quality of its spectra, make natural and efficient groups
for identifying rapid rotators. Anomalously cool giants with complete
solutions and giants with partial solutions are particularly good at
preferentially identifying rapid rotators, with rotator fractions of 90% and
78% respectively in our APOKASC-3 control sample (Fig. 2, Fig. 3). The
majority of these stars rotate with vsini > 10 km s-1. This makes sense since
we are targeting the biggest spectroscopic outliers and we know that rapid
rotation can change reported stellar parameters like log g (Thompson et al.,
2019) and $T_{\mathrm{eff}}$ (Dixon et al., 2020). The rotator fraction at the
>5 km s-1 threshold drops to 55% and 45% in our DR16 catalog, which, while
still high, could result from population differences between the Kepler field,
which is old and above the plane, and general field stars. The difference
could also result from the different ways we measure vsini in the two
populations, though we detect no systematic biases in our vsini measurements
in the APOKASC-3 field.

We also discovered two failure modes of ASPCAP. In both the APOKASC-3 and DR16
samples 15% of stars identified as anomalously cool giants were actually main
sequence photometric binaries based on their Gaia photometry (Fig. 2 and Fig.
9). For stars where both dwarf and giant template spectra were fit, the giant
templates fit these spectra slightly better. For stars at the base of the
giant branch we recommend photometric verification of stars selected to be
spectroscopic giants.

The second failure mode is a metallicity bias. We found two issues with this
low metallicity population. The first is that the median [M/H] for the
anomalously cool giants is 0.37 dex lower than the median for “normal" red
giants. As a result, intrinsically higher metallicity stars, which are more
common, scatter into the poorly populated metal-poor domain, artificially
inflating the apparent fraction of metal-poor rotators. Correcting for this
offset resulted in a 10% reduction in the total number of low-metallicity
giants. The second is that large random errors in temperature for metal-poor
stars make our spectroscopic temperature anomaly an inffective diagnostic of
rapid rotation there, resulting in a large number of false positives. The net
result is that our method does not yield strong results about rapid rotation
in the metal-poor domain.

In assessing the completeness of our spectroscopic selection criteria we
measure vsini for all of the giants in APOKASC-3. This revealed a population
of giants rotating with 5 < vsini < 10 km s-1 that was almost 4x larger than
those with vsini > 10 km s-1. This is not a population effect. In a control
sample of “normal" DR16 giants we find the same relative rates of rapid
rotation at the 5 and 10 km s-1 threshold. Our criteria are largely
insensitive to these stars (Fig. 6), meaning that giants which are
rotationally enhanced fly under the radar due to their lack of spectroscopic
anomalies.

We therefore recommend that current and future spectroscopic surveys include
rotation as a free parameter in their solutions for cool evolved stars. It is
a computationally expensive task, but the payoff is a census of interacting
and post-interaction binaries on the order of ten thousand. vsini is the only
way to have a uniform measure of rotation, as we cannot guarantee rotation
periods from TESS for all of these targets. Gaia is not helpful here either
because there are $v_{\mathrm{Broad}}$ measurements for only some DR16 giants
and it is only sensitive down to 10 km s-1. Furthermore, our criteria are
mostly sensitive to rapid rotation on the lower RGB. It is only with vsini
measurements for all giants that we can find the rotators on the upper RGB and
in the clump.

Lastly, vsini measurements for all giants lets us first identify the rapid
rotators and then assess the binary origins of their rotation. This is
important because our spectroscopic selection criteria do not actually isolate
the systems we thought they would. We expected to almost exclusively capture
tidally-synchronized binaries on the lower RGB, but the close binary fraction
was only about two-thirds in the APOKASC-3 sample. This means that our
criteria identify giants with a variety of binary origins. To select rapid
rotators that have particular features, like those which are just tidally-
synchronized, just RS CVns, or just merger products, more careful
consideration must be given to where those systems would cluster in a CMD or
in spectroscopic parameter space.

Rapid rotation in giants is more common than we expected. We have shown this
by producing the largest catalog of both single and binary rapidly rotating
red giants that are not exclusively variable to date. We have shown how to
exploit pipeline failure modes to find astrophysically interesting systems
quickly and with high fidelity, and we have shown that the prevalence of
binary interaction in low-mass giants is much greater than previously thought.
Measurements of vsini are good ways to find interacting and post-interaction
binaries. With subsequent APOGEE data releases and the wealth of stellar
spectra from SDSS-V, we are poised to fill in gaps in the census of rapidly
rotating red giants if rotation is included for all giants when fitting
spectra. An accurate census and careful characterization of these systems are
critical for concretely testing predictions from binary population synthesis,
probing mass transfer physics and common envelope evolution, and generally
exploring the effects of binary interaction on evolved stars.

## Acknowledgements

RAP would like to thank Diego Godoy-Rivera for providing the de-reddening
scripts and for guidance on how to process the Gaia photometry. S.M.
acknowledges support from the Spanish Ministry of Science and Innovation
(MICINN) with the Ramón y Cajal fellowship no. RYC-2015-17697, grant no.
PID2019-107187GB-I00 and PID2019-107061GB-C66, and through AEI under the
Severo Ochoa Centres of Excellence Programme 2020–2023 (CEX2019-000920-S).

## Data Availability

The catalog of rapidly rotating stars we identify in APOKASC-3 and APOGEE
DR16, as well as the stellar parameters we use to identify them, are all
available upon request. Upon acceptance to the journal, the data will be made
publicly available in tables released with this paper.

## References

  * Abdurro’uf et al. (2022) Abdurro’uf, Accetta, K., Aerts, C., et al. 2022, ApJS, 259, 35. doi:10.3847/1538-4365/ac4414
  * Ahumada et al. (2020) Ahumada, R., Prieto, C. A., Almeida, A., et al. 2020, ApJS, 249, 3. doi:10.3847/1538-4365/ab929e
  * Allende Prieto et al. (2006) Allende Prieto, C., Beers, T. C., Wilhelm, R., et al. 2006, ApJ, 636, 804. doi:10.1086/498131
  * Andronov et al. (2006) Andronov, N., Pinsonneault, M. H., & Terndrup, D. M. 2006, ApJ, 646, 1160. doi:10.1086/505127
  * Badenes et al. (2018) Badenes, C., Mazzola, C., Thompson, T. A., et al. 2018, ApJ, 854, 147. doi:10.3847/1538-4357/aaa765
  * Bailer-Jones et al. (2018) Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., et al. 2018, AJ, 156, 58. doi:10.3847/1538-3881/aacb21
  * Belokurov et al. (2020) Belokurov, V., Penoyre, Z., Oh, S., et al. 2020, MNRAS, 496, 1922. doi:10.1093/mnras/staa1522
  * Berger et al. (2020) Berger, T. A., Huber, D., van Saders, J. L., et al. 2020, AJ, 159, 280. doi:10.3847/1538-3881/159/6/280
  * Bovy et al. (2016) Bovy, J., Rix, H.-W., Green, G. M., et al. 2016, ApJ, 818, 130. doi:10.3847/0004-637X/818/2/130
  * Brown et al. (2011) Brown, T. M., Latham, D. W., Everett, M. E., et al. 2011, AJ, 142, 112. doi:10.1088/0004-6256/142/4/112
  * Cao & Pinsonneault (2022) Cao, L. & Pinsonneault, M. H. 2022, arXiv:2209.10549
  * Carlberg et al. (2011) Carlberg, J. K., Majewski, S. R., Patterson, R. J., et al. 2011, ApJ, 732, 39. doi:10.1088/0004-637X/732/1/39
  * Ceillier et al. (2017) Ceillier, T., Tayar, J., Mathur, S., et al. 2017, A&A, 605, A111. doi:10.1051/0004-6361/201629884
  * Colman et al. (2017) Colman, I. L., Huber, D., Bedding, T. R., et al. 2017, MNRAS, 469, 3802. doi:10.1093/mnras/stx1056
  * Claytor et al. (2022) Claytor, Z. R., van Saders, J. L., Llama, J., et al. 2022, ApJ, 927, 219. doi:10.3847/1538-4357/ac498f
  * Daher et al. (2022) Daher, C. M., Badenes, C., Tayar, J., et al. 2022, MNRAS, 512, 2051. doi:10.1093/mnras/stac590
  * de Medeiros et al. (1996) de Medeiros, J. R., Da Rocha, C., & Mayor, M. 1996, A&A, 314, 499
  * de Medeiros & Mayor (1999) de Medeiros, J. R. & Mayor, M. 1999, A&AS, 139, 433. doi:10.1051/aas:1999401
  * Dixon et al. (2020) Dixon, D., Tayar, J., & Stassun, K. G. 2020, AJ, 160, 12. doi:10.3847/1538-3881/ab9080
  * Doyle et al. (2014) Doyle, A. P., Davies, G. R., Smalley, B., et al. 2014, MNRAS, 444, 3592. doi:10.1093/mnras/stu1692
  * Fekel (1997) Fekel, F. C. 1997, PASP, 109, 514. doi:10.1086/133908
  * Frémat et al. (2022) Frémat, Y., Royer, F., Marchal, O., et al. 2022, arXiv:2206.10986
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Babusiaux, C., van Leeuwen, F., et al. 2018, A&A, 616, A10. doi:10.1051/0004-6361/201832843
  * Gaia Collaboration et al. (2018b) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1. doi:10.1051/0004-6361/201833051
  * 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
  * García Pérez et al. (2016) García Pérez, A. E., Allende Prieto, C., Holtzman, J. A., et al. 2016, AJ, 151, 144. doi:10.3847/0004-6256/151/6/144
  * García et al. (2014c) García, R. A., Ceillier, T., Salabert, D., et al. 2014, A&A, 572, A34. doi:10.1051/0004-6361/201423888
  * García et al. (2011) García, R. A., Hekker, S., Stello, D., et al. 2011, MNRAS, 414, L6. doi:10.1111/j.1745-3933.2011.01042.x
  * García et al. (2014b) García, R. A., Mathur, S., Pires, S., et al. 2014, A&A, 568, A10. doi:10.1051/0004-6361/201323326
  * García et al. (2010) García, R. A., Mathur, S., Salabert, D., et al. 2010, Science, 329, 1032. doi:10.1126/science.1191064
  * Gaulme et al. (2020) Gaulme, P., Jackiewicz, J., Spada, F., et al. 2020, A&A, 639, A63. doi:10.1051/0004-6361/202037781
  * Geller et al. (2017a) Geller, A. M., Leiner, E. M., Bellini, A., et al. 2017, ApJ, 840, 66. doi:10.3847/1538-4357/aa6af3
  * Geller et al. (2017b) Geller, A. M., Leiner, E. M., Chatterjee, S., et al. 2017, ApJ, 842, 1. doi:10.3847/1538-4357/aa72ef
  * Girardi (2016) Girardi, L. 2016, ARA&A, 54, 95. doi:10.1146/annurev-astro-081915-023354
  * Godoy-Rivera et al. (2021) Godoy-Rivera, D., Pinsonneault, M. H., & Rebull, L. M. 2021, ApJS, 257, 46. doi:10.3847/1538-4365/ac2058
  * Green et al. (2019) Green, G. M., Schlafly, E., Zucker, C., et al. 2019, ApJ, 887, 93. doi:10.3847/1538-4357/ab5362
  * Holtzman et al. (2018) Holtzman, J. A., Hasselquist, S., Shetrone, M., et al. 2018, AJ, 156, 125. doi:10.3847/1538-3881/aad4f9
  * Jönsson et al. (2020) Jönsson, H., Holtzman, J. A., Allende Prieto, C., et al. 2020, AJ, 160, 120. doi:10.3847/1538-3881/aba592
  * Katz et al. (2022) Katz, D., Sartoretti, P., Guerrier, A., et al. 2022, arXiv:2206.05902
  * Kervella et al. (2022) Kervella, P., Arenou, F., & Thévenin, F. 2022, A&A, 657, A7. doi:10.1051/0004-6361/202142146
  * Kraft (1967) Kraft, R. P. 1967, ApJ, 150, 551. doi:10.1086/149359
  * Leiner & Geller (2021) Leiner, E. M. & Geller, A. 2021, ApJ, 908, 229. doi:10.3847/1538-4357/
  * Leiner et al. (2021b) Leiner, E. M., Geller, A. M., Gully-Santiago, M. A., et al. 2021, arXiv:2112.09100
  * Leiner et al. (2017) Leiner, E., Mathieu, R. D., & Geller, A. M. 2017, ApJ, 840, 67. doi:10.3847/1538-4357/aa6aff
  * Leiner et al. (2019) Leiner, E., Mathieu, R. D., Vanderburg, A., et al. 2019, ApJ, 881, 47. doi:10.3847/1538-4357/ab2bf8
  * 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
  * Marshall et al. (2006) Marshall, D. J., Robin, A. C., Reylé, C., et al. 2006, A&A, 453, 635. doi:10.1051/0004-6361:20053842
  * Massarotti et al. (2008) Massarotti, A., Latham, D. W., Stefanik, R. P., et al. 2008, AJ, 135, 209. doi:10.1088/0004-6256/135/1/209
  * Mathur et al. (2014) Mathur, S., García, R. A., Ballot, J., et al. 2014, A&A, 562, A124. doi:10.1051/0004-6361/201322707
  * Mathur et al. (2019) Mathur, S., García, R. A., Bugnet, L., et al. 2019, Frontiers in Astronomy and Space Sciences, 6, 46. doi:10.3389/fspas.2019.00046
  * Moe & Di Stefano (2017) Moe, M. & Di Stefano, R. 2017, ApJS, 230, 15. doi:10.3847/1538-4365/aa6fb6
  * Moe et al. (2019) Moe, M., Kratter, K. M., & Badenes, C. 2019, ApJ, 875, 61. doi:10.3847/1538-4357/ab0d88
  * 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
  * Peterson et al. (1984) Peterson, R. C., Carney, B. W., & Latham, D. W. 1984, ApJ, 279, 237. doi:10.1086/161887
  * 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. (2020) Price-Whelan, A. M., Hogg, D. W., Rix, H.-W., et al. 2020, ApJ, 895, 2. doi:10.3847/1538-4357/ab8acc
  * Santos et al. (2019) Santos, A. R. G., García, R. A., Mathur, S., et al. 2019, ApJS, 244, 21. doi:10.3847/1538-4365/ab3b56
  * Santos et al. (2021) Santos, A. R. G., Mathur, S., García, R. A., et al. 2021, MNRAS, 508, 267. doi:10.1093/mnras/stab2402
  * Simonian et al. (2019) Simonian, G. V. A., Pinsonneault, M. H., & Terndrup, D. M. 2019, ApJ, 871, 174. doi:10.3847/1538-4357/aaf97c
  * Simonian et al. (2020) Simonian, G. V. A., Pinsonneault, M. H., Terndrup, D. M., et al. 2020, ApJ, 898, 76. doi:10.3847/1538-4357/ab9a43
  * Stassun & Torres (2021) Stassun, K. G. & Torres, G. 2021, ApJ, 907, L33. doi:10.3847/2041-8213/abdaad
  * Tayar et al. (2015) Tayar, J., Ceillier, T., García-Hernández, D. A., et al. 2015, ApJ, 807, 82. doi:10.1088/0004-637X/807/1/82
  * Thompson et al. (2019) Thompson, T. A., Kochanek, C. S., Stanek, K. Z., et al. 2019, Science, 366, 637. doi:10.1126/science.aau4005
  * Verbunt & Phinney (1995) Verbunt, F. & Phinney, E. S. 1995, A&A, 296, 709
  * Wilson et al. (2019) Wilson, J. C., Hearty, F. R., Skrutskie, M. F., et al. 2019, PASP, 131, 055001. doi:10.1088/1538-3873/ab0075
  * Yadav et al. (2015) Yadav, R. K., Gastine, T., Christensen, U. R., et al. 2015, A&A, 573, A68. doi:10.1051/0004-6361/201424589
  * Yu et al. (2018) Yu, J., Huber, D., Bedding, T. R., et al. 2018, ApJS, 236, 42. doi:10.3847/1538-4365/aaaf74
  * Zahn (1977) Zahn, J.-P. 1977, A&A, 57, 383
  * Zahn & Bouchet (1989) Zahn, J.-P. & Bouchet, L. 1989, A&A, 223, 112