# Rotation Distributions around the Kraft Break with TESS and Kepler: The
Influences of Age, Metallicity, and Binarity

Ellis A. Avallone Institute for Astronomy, University of Hawai‘i at Mānoa,
2680 Woodlawn Drive, Honolulu, HI 96822, USA  Jamie N. Tayar NASA Hubble
Fellow  Institute for Astronomy, University of Hawai‘i at Mānoa, 2680 Woodlawn
Drive, Honolulu, HI 96822, USA  Department of Astronomy, University of
Florida, Bryant Space Science Center, Stadium Road, Gainesville, FL 32611, USA
Jennifer L. van Saders Institute for Astronomy, University of Hawai‘i at
Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Travis A. Berger NASA
Postdoctoral Program Fellow  Exoplanets and Stellar Astrophysics Laboratory,
Code 667, NASA Goddard Space Flight Center, Greenbelt, MD, 20771, USA
Institute for Astronomy, University of Hawai‘i at Mānoa, 2680 Woodlawn Drive,
Honolulu, HI 96822, USA  Zachary R. Claytor Institute for Astronomy,
University of Hawai‘i at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
Rachael L. Beaton Much of this work was completed while this author was a NASA
Hubble Fellow at Princeton University.  Carnegie-Princeton Fellow  Department
of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ
08544  The Observatories of the Carnegie Institution for Science, 813 Santa
Barbara St., Pasadena, CA 91101  Johanna Teske  Earth and Planets Laboratory,
Carnegie Institution for Science, 5241 Broad Branch Road, NW, Washington, DC
20015, USA  Diego Godoy-Rivera Department of Astronomy, The Ohio State
University, 140 West 18th Avenue, Columbus, OH 43210, USA  Kaike Pan  Apache
Point Observatory and New Mexico State University, P.O. Box 59, Sunspot, NM,
88349-0059, USA

(Received 26 August 2021; Revised 21 March 2022; Accepted 22 March 2022)

###### Abstract

Stellar rotation is a complex function of mass, metallicity, and age and can
be altered by binarity. To understand the importance of these parameters in
main sequence stars, we have assembled a sample of observations that spans a
range of these parameters using a combination of observations from The
Transiting Exoplanet Survey Satellite (TESS) and the Kepler Space Telescope.
We find that while we can measure rotation periods and identify other classes
of stellar variability (e.g., pulsations) from TESS lightcurves, instrument
systematics prevent the detection of rotation signals longer than the TESS
orbital period of 13.7 days. Due to this detection limit, we also utilize
rotation periods constrained using rotational velocities measured by the
APOGEE spectroscopic survey and radii estimated using the Gaia mission for
both TESS and Kepler stars. From these rotation periods, we 1) find we can
track rotational evolution along discrete mass tracks as a function of stellar
age, 2) find we are unable to recover trends between rotation and metallicity
that were observed by previous studies, and 3) note that our sample reveals
that wide binary companions do not affect rotation, while close binary
companions cause stars to exhibit more rapid rotation than single stars.

stars: rotation — stars: evolution — stars: binaries

††software: Astropy (Astropy Collaboration et al., 2013, 2018), isoclassify
(Huber, 2017; Berger et al., 2020), Kiauhoku (Claytor et al., 2020),
Lightkurve (Lightkurve Collaboration et al., 2018), Matplotlib (Hunter, 2007),
NumPy (van der Walt et al., 2011), SciPy (Virtanen et al., 2020), and Starspot
(Angus, 2021).††facilities: Du Pont (APOGEE), Sloan (APOGEE), 2MASS, _Gaia_ ,
TESS, _Kepler_

##  1 Introduction

Stars in nature rotate, and rotation is an important ingredient in stellar
evolution for the most rapid rotators. Stellar models, however, often do not
take rotation into account or simplify rotation when they do (Dantona &
Mazzitelli, 1984; Maeder & Meynet, 1989; Pinsonneault et al., 1989a; Ekström
et al., 2012; van Saders & Pinsonneault, 2013; Choi et al., 2016, 2017;
Ostrowski et al., 2017). This is primarily because the physics governing
angular momentum transport and loss are complex and uncertain.

Angular momentum loss occurs as a consequence of magnetized stellar winds,
which carry angular momentum away from stars and cause stellar spin-down as
stars evolve (Weber & Davis, 1967; Skumanich, 1972). Most braking laws predict
angular momentum loss that goes as $dJ/dt\propto\omega^{3}$, where the angular
momentum loss strongly depends on the rotational velocity of the star
(Kawaler, 1988). This has the effect of producing tight rotational sequences
in intermediate-age open clusters, despite the fact that stars begin their
lives with a wide range of rotation periods (e.g. Meibom et al., 2011, 2015;
Barnes et al., 2016; Douglas et al., 2017; Rebull et al., 2017; Curtis et al.,
2020; Fritzewski et al., 2020; Godoy-Rivera et al., 2021).

The primary focus of previous observational studies of rotation as it relates
to stellar evolution has been on main sequence stars that experience
substantial spin-down, with some focus on cool evolved stars (e.g. Mamajek &
Hillenbrand, 2008; Meibom et al., 2009, 2011; Nielsen et al., 2013; van Saders
& Pinsonneault, 2013; Tayar & Pinsonneault, 2018). The Kepler Space Telescope
enabled many of these studies of rotation across the main sequence (e.g
McQuillan et al., 2013a, b, 2014; Amard et al., 2020; Simonian et al., 2020).
Its successor, the Transiting Exoplanet Survey Satellite (TESS; Ricker et al.,
2015), has enabled further studies of rotation across stellar populations
(e.g. Lu et al., 2020; Martins et al., 2020).

As with Kepler, rotation studies with TESS rely on measurements from starspot
modulation in lightcurves. These rotation rates are often preferred to
rotation rates measured via other methods due to their independence from
stellar inclination. Previous work has shown that rotation periods from
starspot modulation agree with rotation periods derived from spectroscopic
$v\sin i$ (Nielsen et al., 2013; Simonian et al., 2020) and asteroseismically
inferred rotation periods (Reinhold & Gizon, 2015; Beck et al., 2017; Hall et
al., 2021).

Attempts at studying rotation with TESS lightcurves, however, have been met
with several challenges (e.g. Lu et al., 2020; Martins et al., 2020). The
$27.4$-day observing sectors in TESS limit the measurement of rotation signals
beyond tens of days for much of the survey area. Stars observed in the
southern continuous viewing zone (SCVZ), however, have a combined observing
baseline of $\sim$ 2 years from the 13 southern hemisphere observing sectors
and extended mission observations. Although the combined baseline of these
observations should enable studies of rotation signals that span multiple TESS
observing sectors, attempts at measuring rotation periods longer than the
length of a single sector ($27.4$ days) have been unsuccessful with standard
rotation period measurement methods (e.g. Günther et al., 2020; Martins et
al., 2020), although there has been some success with machine learning-based
methods of extracting rotation signals (e.g. with neural networks Lu et al.,
2020; Claytor et al., 2021). These prior studies using traditional methods of
measuring rotation periods from lightcurves have suggested that challenges in
measuring rotation signals across multiple sectors in TESS are likely due to
scattered light at the start and end of observing sectors and signals from the
orbital motion of the spacecraft around Earth. These timescale constraints
have limited the types of stars where we can effectively probe rotation.

We therefore focus our efforts here on main sequence stars near the Kraft
Break, which tend to be rapid rotators. The Kraft Break is a transition that
is seen across stellar effective temperatures ($T_{\mathrm{eff}}$). Hot stars
($T_{\mathrm{eff}}>6250$ K) typically experience little to no angular momentum
loss due to the presence of vanishingly thin convective zones that cannot
sufficiently support a magnetized wind. Cool stars ($T_{\mathrm{eff}}<6250$ K)
typically experience substantial angular momentum loss due to the presence of
large convection zones that can sustain a magnetized wind (Kraft, 1967). Main
sequence stars above the Kraft Break were the first stars to have reliable
rotation measurements and lend themselves well to spectroscopic investigations
due to their rapid rotation (Kraft, 1967).

Our focus on stars near the Kraft Break can provide a look into the
distribution of rotation periods at the transition between stars which
experience Solar-like spin-down and stars which experience little to no
angular momentum loss. Some work has already been done to characterize the
main sequence distribution of rotation periods as a function of other stellar
parameters with the Kepler Space Telescope (e.g McQuillan et al., 2014; Amard
et al., 2020; Simonian et al., 2020).

Here, we measure rotation periods from starspot modulation for a sample of
stars observed by TESS and augment this sample with stars whose rotation
periods were measured from Kepler lightcurves by Nielsen et al. (2013),
Reinhold et al. (2013), and McQuillan et al. (2014). We use spectroscopic
parameters measured by the Apache Point Observatory Galactic Evolution
Experiment (APOGEE; Majewski et al., 2017; Ahumada et al., 2020; Jönsson et
al., 2020) and photometric radii estimated using the Gaia mission to carefully
characterize each star in our sample and derive alternate rotation diagnostics
from spectroscopic $v\sin i$ and Gaia radii. The combination of these data
enables us to obtain a more complete picture of rotation in a sample with
significant detection bias in spots.

##  2 Data & Methods

###  2.1 Observational Data

We use Cycle 1 data from the TESS (Ricker et al., 2015) SCVZ and spectroscopic
parameters from an APOGEE (Majewski et al., 2017; Ahumada et al., 2020;
Jönsson et al., 2020) External Program through Carnegie Observatories. The
37,000 stars in the TESS-APOGEE SCVZ sample have been observed near the
ecliptic poles, where stars could be observed in a maximum of 13 TESS sectors,
although the number of sectors with observations depends on the exact location
of the star relative to the observing CCDs. These stars additionally have
homogeneous spectroscopic characterization, making this the ideal sample to
search for rotation periods.

To select our sample, we constrain the APOGEE-TESS SCVZ sample to stars with
$\log g$ values between $3$ and $5$ dex and $T_{\mathrm{eff}}$ values between
$5600$ K and $8000$ K. These constraints on $\log g$ and $T_{\mathrm{eff}}$
allow us to encompass stars on the main sequence without risking contamination
from giant-branch stars or stars with masses far below 1$M_{\bigodot}$. These
constraints additionally exclude M type stars, where APOGEE $\log g$ values
have high uncertainties (Ahumada et al., 2020; Sarmento et al., 2021). We
restrict our sample to only include stars that have been observed with TESS’s
short cadence observing mode, as these stars have available lightcurves
generated with the TESS Science Processing Operations Center pipeline (SPOC;
Jenkins et al., 2016). We finally only select stars that have been observed
across at least 9 TESS observing sectors to optimize our chance of detecting
rotation signals that span multiple sectors. Of the 37,000 stars observed in
the APOGEE-TESS SCVZ sample, our final TESS sample contains 2115 stars. The
top panel in Figure 1 shows a Kiel diagram of all stars in the APOGEE-TESS
SCVZ sample along with our final TESS sample.

APOGEE data are collected with the $2.5$ meter Sloan Digital Sky Survey (SDSS)
telescope (Gunn et al., 2006) at Apache Point Observatory and the $2.5$ meter
du Pont telescope (Bowen & Vaughan, 1973) at Las Campanas Observatory using
near-twin H-band spectrographs (Wilson et al., 2019) as part of the fourth
campaign of SDSS (SDSS-IV; Blanton et al., 2017). The APOGEE data reduction,
Spectral Parameters, and Chemical Abundances Pipeline (ASPCAP; Nidever et al.,
2015; García Pérez et al., 2016; Shetrone et al., 2015; Smith et al., 2021)
provides stellar spectra and computes several stellar parameters including
$T_{\mathrm{eff}}$, surface gravity ($\log g$), metallicity ($[Fe/H]$), and
surface rotational velocity ($v\sin i$). We specifically use propriety
parameters from APOGEE Data Release 16 plus (Jönsson et al., 2020; Ahumada et
al., 2020; Smith et al., 2021, DR16) for stars in the TESS SCVZ and public
data from the APOGEE DR16 standard release for stars in the Kepler field.
APOGEE DR16 plus includes observations up to March 2020 (MJD=58932) and APOGEE
DR16 includes observations up to August 2018 (MJD=58358).

Figure 1: Kiel diagrams showing our selected sample of stars near the Kraft
Break observed by both APOGEE and TESS (top) and by both APOGEE and Kepler
(bottom). The full APOGEE-TESS SCVZ (top) and APOGEE-Kepler (bottom) samples
are shown in the background, and our selected sample in both TESS (purple) and
Kepler (green) are shown.

We augment our TESS-APOGEE SCVZ sample with stars observed by APOGEE and the
Kepler Space Telescope. We specifically select stars in the APOGEE-Kepler
sample that have values for rotation periods measured by Nielsen et al.
(2013), Reinhold et al. (2013), or McQuillan et al. (2014) or are listed as
nondetections in McQuillan et al. (2014). For stars with rotation periods from
more than one of these samples, we adopt the rotation period from McQuillan et
al. (2014). When compared to the other studies via a hare-and-hounds exercise
(Aigrain et al., 2015), the McQuillan et al. (2014) rotation periods were
found to be more reliable than those found by Nielsen et al. (2013) and
Reinhold et al. (2013). We note that although other studies were found to have
more reliable rotation periods than McQuillan et al. (2014), Nielsen et al.
(2013), and Reinhold et al. (2013) by Aigrain et al. (2015) (e.g. García et
al., 2014), their samples contain far fewer stars. We limit the APOGEE-Kepler
sample using the same $\log g$ and $T_{\mathrm{eff}}$ constraints we used on
the APOGEE-TESS SCVZ sample, $3<\log g<5$ and $5600K<T_{\mathrm{eff}}<8000K$.
Of the 23,000 stars observed in the APOGEE-Kepler sample, our final Kepler
sample contains 2101 stars. The bottom panel in Figure 1 shows a Kiel diagram
of all stars in the APOGEE-Kepler sample along with our final Kepler sample.

We finally use stellar parallaxes from Gaia data release 2 (Gaia Collaboration
et al., 2018, DR2), K-band magnitudes from 2MASS (Skrutskie et al., 2006), and
$T_{\mathrm{eff}}$ and $[Fe/H]$ from APOGEE to improve our estimates of the
radii of stars in our final sample. This is done using the direct method of
isoclassify (Huber, 2017; Berger et al., 2020). There were no parallax
measurements for 68 stars in our sample, therefore we exclude them from our
analysis. isoclassify is currently being adapted to analyze Gaia Early Data
Release 3 (Gaia Collaboration et al., 2021) photometry. This update to
isoclassify was not complete at the time of writing.

###  2.2 Processing TESS Lightcurves

The TESS SPOC pipeline currently provides lightcurves generated via Simple
Aperture Photometry (SAP) and Pre-search Data Conditioning SAP (PDCSAP)
(Jenkins et al., 2016). PDCSAP lightcurves use singular value decomposition to
remove trends caused by the spacecraft. However, some of these trends are on
timescales similar to the rotation signals we want to recover. Because of the
possibility of suppressing rotation signals, we choose to use SAP lightcurves
for our analysis.

Figure 2: Creating short-period rotator-optimized lightcurves. The top panel
shows the uncorrected SAP lightcurve for TIC 141482802. The second from top
panel shows the sigma-clipped, normalized lightcurve with NaN values and
values around discontinuous regions removed along with our identified
discontinuity points. The third from top panel shows the flattened lightcurve
with segment-length trends removed. The bottom panel shows the final
lightcurve with all jumps corrected using the first segment as a reference. We
use short-period rotator-optimized lightcurves in the remainder of our
analyses.

To generate SAP lightcurves, the TESS pipeline sums the flux values of pixels
within an aperture that optimizes the target star’s signal-to-noise (Morris et
al., 2017). However, this process retains significant instrument systematics
caused by spacecraft pointing, safe modes, and scattered light. Additionally,
our stars have been observed across at least nine sectors, and these
systematics are observing sector-dependent. To correct and stitch together our
lightcurves, we initially followed the methods performed in García et al.
(2011) on the Kepler sample. However, upon implementing this method on several
simulated TESS lightcurves from Claytor et al. (2021), we found that these
methods preserved too many TESS-specific systematics for us to accurately
measure rotation periods. In light of this, we devise our own method to
correct for TESS systematics while preserving any possible rotation signals.

We start by removing all NaN values from each lightcurve, sigma clipping any
values greater than $4\sigma$ from the mean in each individual sector, and
removing any data points with a nonzero quality flag. We then clip the first
and last two days of each observing sector where scattered light effects are
most likely to be present. We also clip regions around TESS safe modes. We
identify and clip regions around discontinuities and jumps that are found in
all lightcurves in our sample. We then identify timestamps where we expect the
lightcurve to be discontinuous (e.g. at the start and end of each sector and
at safe mode times), and median-normalize the flux in each part of the
lightcurve. Portions of the lightcurve between discontinuities and the start
and end of observing sectors are hereafter referred to as "segments".

Figure 3: Creating long period rotator-optimized lightcurves. The top panel
shows the uncorrected SAP lightcurve for TIC 141482802. The second from top
panel shows the sigma-clipped, normalized lightcurve with NaN values and
values around discontinuous regions removed along with our identified
discontinuity points. The bottom panel shows the final lightcurve with all
jumps corrected using each segment as the reference for the next segment. This
method preserves systematic trends that are inconsistent with expectations
from $v\sin i$ more than the short rotation period-optimized method, as seen
by the $\sim 50$ day linear increase in brightness. Because of this, we do not
use long period rotator-optimized lightcurves in the remainder of our
analyses.

To correct discontinuous regions and long-term trends, we take a two-tiered
approach optimized for the range of rotation periods we expect to find. We
devise a method for stars with rotation periods less than one observing sector
(hereafter referred to as short-period rotators) and for stars with rotation
periods greater than one observing sector (hereafter referred to as long-
period rotators).

For creating lightcurves optimized for short-period rotators, we do a linear
fit to each segment. We then subtract this line and re-normalize each segment
so the median value is 1. This yields flattened lightcurve segments that only
differ by a scalar factor. We then compute the median flux of the first
observed sector and use that as a reference. We correct the rest of the
lightcurve using the difference between each segment’s median flux and the
reference flux. The steps for creating lightcurves optimized for short-period
rotators are shown in Figure 2.

Figure 4: All of our rotation period measurement methods run on TIC 370115640.
The top left panel shows a 30 day segment of the full lightcurve. The bottom
left panel shows the autocorrelation function and its measured rotation
period. The middle panels show the full Lomb-Scargle periodogram (top) and the
zoomed-in periodogram (bottom). The right panels show the full wavelet power
spectrum and global wavelet power spectrum (top) and the zoomed-in wavelet
power spectrum and global wavelet power spectrum (bottom). Red lines in all
plots correspond to periods associated with signals from the TESS instrument.

We utilize a different approach for creating lightcurves optimized for long
period rotators. We do not subtract any trends from the existing lightcurve to
preserve any rotation periods that are longer than one sector. Instead of
using the first observed segment as a reference, the segment directly before a
discontinuity is used as a reference for the segment directly after the
discontinuity. We correct each segment of the lightcurve using the difference
between the median fluxes of each segment. The steps for creating lightcurves
optimized for long period rotators are shown in Figure 3. After correcting all
jumps in both sets of lightcurves, we smooth the lightcurve using a boxcar
kernel of 0.5 days to remove any high-frequency noise while also preserving
any possible short period rotation signals.

We run both the long period rotator and short-period rotator-optimized methods
on our full sample of stars. After doing this, we run our rotation period
measurement methods (see Section 2.3) and test the validity of the recovered
periods on both sets of lightcurves (see Section 3). From visual inspection
comparing both sets of lightcurves with the rotation periods we recover, we
find that the long period rotator-optimized method still retains substantial
systematic signals that both prevent the recovery of rotation signals longer
than 13.7 days and suppress real rotation signals shorter than 13.7 days.
Therefore our long period rotator-optimized method is not a viable method for
processing lightcurves for stars observed in multiple TESS observing sectors.
Because of this, we only use the short-period rotator-optimized lightcurves in
this paper.

###  2.3 Measuring Rotation Periods in TESS

After generating short-period rotator-optimized lightcurves for our full
sample of stars, we compute rotation periods using a combination of Lomb-
Scargle periodograms, wavelet analysis, and autocorrelation function (ACF)
analysis. We apply these methods to all the stars in our sample. Previous work
has shown that a combination of these methods optimizes the recovery of
correct stellar rotation periods in injection and recovery exercises (Aigrain
et al., 2015; Ceillier et al., 2017).

The Lomb-Scargle periodogram uses Fourier transforms to recover any periodic
trends found in a dataset and their associated periods and shows which
recovered periods have the strongest power using methods optimized for
unevenly time-sampled data (Scargle, 1982; Reinhold et al., 2013). We utilize
the Lomb-Scargle periodogram method within the lightcurve processing package
lightkurve (Lightkurve Collaboration et al., 2018). We ignore peaks in the
periodogram that are within 10% of periods related to the orbital period and
observing cadence of the TESS instrument and their aliases at 13.7, 27.4, 90,
and 120 days. We then recompute the periodogram to search the region 10 days
around the highest peak in the initial periodogram with finer sampling. Our
final rotation period from Lomb-Scargle periodograms is defined as the highest
peak in the recomputed periodogram that does not correspond to a TESS alias.
An example of our computed periodograms for TIC 370115640 is shown in the
middle panel of Figure 4.

Table 1: Properties of stars with recovered rotation periods, potential binaries, pulsating stars, and stars with no detected variability in our TESS and Kepker samples. $T_{\mathrm{eff}}$, $\log g$, $v\sin i$, metallicity, and $\alpha$ abundance are from APOGEE spectra. Stellar radii and masses are estimated using isoclassify (Berger et al., 2020, similar to). This table is available in its entirety in machine-readable format. A portion is shown here for guidance regarding its form and content. ID | Flag |  $T_{\mathrm{eff}}$ (K) |  $\log g$ (dex) |  $v\sin i$ ($kms^{-1}$) | [$Fe/H$] (dex) | [$\alpha/M$] (dex) |  $R$ ($R_{\odot}$) |  $M$ ($M_{\odot}$)
---|---|---|---|---|---|---|---|---
TIC 30631031 | Rotator | $7136\pm 183$ | $3.89\pm 0.07$ | 58.09 | $-0.02\pm 0.03$ | $0.12\pm 0.03$ | $2.21^{+0.05}_{-0.05}$ | $1.63^{+0.05}_{-0.05}$
TIC 55480220 | RUWE Binary | $6947\pm 170$ | $3.98\pm 0.07$ | $13.49$ | $0.13\pm 0.02$ | $0.03\pm 0.01$ | $1.98^{+0.05}_{-0.04}$ | $1.58^{+0.05}_{-0.05}$
TIC 150393064 | Spectroscopic Binary | $5868\pm 121$ | $4.47\pm 0.08$ | $20.06$ | $-0.02\pm 0.01$ | $-0.04\pm 0.01$ | $1.32^{+0.07}_{-0.07}$ | $1.11^{+0.05}_{-0.06}$
TIC 272127517 | Pulsator | $6774\pm 163$ | $3.95\pm 0.07$ | $80.05$ | $0.05\pm 0.02$ | $0.04\pm 0.02$ | $1.98^{+0.04}_{-0.04}$ | $1.50^{+0.04}_{-0.05}$

The Lomb-Scargle periodogram, however, falls short in that it is optimized for
periodic phenomena. Often, a rotation signal will be a mixture of periodic and
aperiodic signals, as starspots are not permanent features on a star’s
surface. Wavelet analysis overcomes perturbations in the lightcurve caused by
the evolution of starspots by determining the periodic signals present in a
slice of timeseries data and stepping through the rest of the time series to
test the strength of those periods (Torrence & Compo, 1998). We use the Morlet
wavelet transform in the SciPy library (Virtanen et al., 2020) with the power
spectral density correction of Liu et al. (2007) to perform these
measurements. We first compute the wavelet power spectrum and global wavelet
power spectrum (GWPS), which is equal to the wavelet power spectrum summed
over the full timeseries, with coarse spacing in frequency. As with the Lomb-
Scargle periodogram, we ignore any peaks in the wavelet power spectrum and
GWPS that correspond to TESS instrument signals. We then recompute the wavelet
power spectrum and the GWPS with finer sampling in the region within 10 days
of the highest peak in the GWPS. Our final rotation period from wavelet
analysis is defined as the highest peak in the zoomed-in GWPS that doesn’t
correspond to a TESS alias. An example of our computed wavelet power spectra
and GWPS for TIC 370115640 is shown in the right panel of Figure 4.

ACF analysis searches for periodic signals by correlating time series data
with itself (McQuillan et al., 2013a). This allows for confirmation of signals
found through the Lomb-Scargle periodogram and wavelet analysis while also
searching for signals that are not perfectly periodic, sinusoidally shaped, or
present throughout the full dataset. We use the methods outlined in McQuillan
et al. (2013a) and the Python package starspot (Angus, 2021) to compute
rotation periods with ACF analysis. As with the Lomb-Scargle periodogram and
wavelet analysis, we ignore any peaks in the ACF that correspond to TESS
instrument signals and their aliases. Our final rotation period from ACF
analysis is defined as the first peak that does not correspond to a TESS alias
or the second peak in the ACF if the second peak is larger than the first peak
(McQuillan et al., 2013a). An example of our computed ACF for TIC 370115640 is
shown in the bottom left panel of Figure 4.

We then compute the total uncertainty on our measured rotation periods. The
statistical uncertainty for each of our methods is defined as the full-width
at half maximum of the peak corresponding to the rotation period in the
recomputed Lomb-Scargle periodogram, recomputed GWPS, and ACF. Additionally,
Epstein & Pinsonneault (2014) argue that one should adopt a limit on the
precision of any measured rotation period of 10% due to stellar differential
rotation. We propagate the statistical uncertainty from each of our rotation
period measurement methods to obtain the final uncertainty on the photometric
rotation period for each star. If that uncertainty is less than the assumed
systematic floor from Epstein & Pinsonneault (2014), we adopt 10% as the final
uncertainty. We list the properties of all stars in our sample, including
those with detected variability and no detected variability, in Table 1.

###  2.4 Modeling

Recent observations reveal that angular momentum loss is likely more
complicated for stars unlike the Sun (e.g. van Saders & Pinsonneault, 2013;
Kurtz et al., 2014; Douglas et al., 2017; Tayar & Pinsonneault, 2018; Curtis
et al., 2019, 2020). A recent parameterization, was devised by Pinsonneault,
Matt, and MacGregor and was derived from magnetohydrodynamic simulations in
Matt et al. (2012) coupled to mass loss and magnetic field scalings (see van
Saders & Pinsonneault, 2013). This law includes a stronger dependence on
stellar radius and an explicit dependence on a star’s convective overturn
timescale, and is better able to match observed trends in hot ($>6000\,K$) and
more evolved stars (van Saders & Pinsonneault, 2013; Tayar & Pinsonneault,
2018). Discussions of this law can be found in Matt et al. (2012), van Saders
& Pinsonneault (2013), and Amard et al. (2020).

To analyze how well our measured rotation periods follow model predictions and
to determine additional stellar parameters, we use the Python package kiauhoku
(Claytor et al., 2020). kiauhoku uses model grids computed using the Yale
Rotating stellar Evolution Code (Pinsonneault et al., 1989b; Demarque et al.,
2008; Tayar et al., 2020, YREC). A summary of the input physics used in the
YREC models can be found in van Saders & Pinsonneault (2013) and Claytor et
al. (2020). We specifically use model grids that compute a star’s rotational
evolution using the angular momentum loss law described Matt et al. (2012);
van Saders & Pinsonneault (2013). kiauhoku has the ability to interpolate
between model grids computed at multiple masses and metallicities to provide
parameters of interest (e.g. $P_{\mathrm{rot}}$, radius, mass, and zero-age
main sequence $T_{\mathrm{eff}}$) given observed values for
$T_{\mathrm{eff}}$, $\log g$, $[Fe/H]$, and $[\alpha/M]$ from APOGEE.

##  3 Rotation Period Validation

After collecting our final samples in Kepler and TESS and running the rotation
period measurement methods described in Section 2.3 on our TESS sample, we
validate the rotation periods in both samples.

###  3.1 Pulsators in TESS

Figure 5: Identifying pulsators in our TESS sample. We show the Lomb-scargle
periodograms in frequency space for a $\delta$ Scuti pulsator in TESS
identified by Antoci et al. (2019) (top) and a candidate pulsator identified
in this work (bottom, purple). We additionally plot an example rotator
periodogram (grey) for TIC 370115640 in the bottom panel. We observe that the
spacings between peaks in the periodograms are the same for the $\delta$ Scuti
pulsator identified by Antoci et al. (2019) and our candidate pulsator,
indicating that the star in our sample is a likely pulsator.

Previous work has shown that pulsational and rotational modulation has been
detected in lightcurves of stars in the $T_{\mathrm{eff}}$ range probed by our
sample (Murphy et al., 2019; Antoci et al., 2019). We visually inspect all
TESS lightcurves to determine whether the period found by our pipeline is more
likely to be caused by pulsations than rotation. Pulsating stars are
distinguished from rotators by multiple, regularly-spaced peaks in their Lomb-
Scargle periodograms. Common pulsating contaminants in the region of the Kiel
diagram occupied by our sample are $\delta$ Scuti and $\gamma$ Doradus type
pulsators. $\delta$ Scuti type pulsators display equally spaced peaks in
frequency from $5d^{-1}$ to $50d^{-1}$ (see e.g. Murphy et al., 2019), while
$\gamma$ Doradus type pulsators display peaks that are equally spaced in
period with frequencies below $5d^{-1}$ (see Figure 5 for an example of a
$\delta$ Scuti type pulsator). We find that our rotation period measurement
pipeline incorrectly identifies these potential pulsation periods as rotation
periods. We remove these stars from our final rotation sample and flag them
for future users.

Figure 6: Kiel diagram comparing the pulsators identified in our TESS sample
(purple) to the $\delta$ Scuti pulsators identified by Murphy et al. (2019)
(green) in their search of stars in the Kepler field (gray). We also show
$\gamma$ Doradus pulsators identified by Van Reeth et al. (2015) (orange) in
the Kepler sample. The pulsators in our sample follow the distribution of
pulsators in the Kepler sample and lie near the $\delta$ Scuti instability
strip (red line) identified in Kepler by Murphy et al. (2019), thus supporting
our identification of them as pulsators.

Our visual inspection yields 28 potential pulsating stars in our TESS sample,
which have been identified with the flag "Pulsator" in Table 1. Figure 6 shows
a Kiel diagram of our identified TESS pulsators along with $\delta$ Scuti and
$\gamma$ Doradus pulsators found in the Kepler sample by Murphy et al. (2019)
and Van Reeth et al. (2015). Our identified TESS pulsators clearly follow the
$\delta$ Scuti instability strip identified in the Kepler sample (Murphy et
al., 2019). Additionally, this initial investigation suggests that several of
our pulsators also populate the $\delta$ Scuti - $\gamma$ Doradus hybrid
regime, where stars are particularly useful testbeds for stellar physics (Van
Reeth et al., 2015). However, we leave confirmation and direct
characterization of the pulsation modes of these stars to future, more
targeted studies.

Figure 7: Kiel diagram comparing the pulsators and rotators identified in our
TESS sample (purple) to the pulsators and rotators identified by Murphy et al.
(2019), Van Reeth et al. (2015), Nielsen et al. (2013), Reinhold et al.
(2013), and McQuillan et al. (2014) (green) in the Kepler sample. The
pulsators and rotators in both samples overlap in $T_{\mathrm{eff}}$ and$\log
g$.

We compare our identified pulsators and rotators in TESS to those identified
in Kepler in Figure 7. We find that pulsators and rotators overlap in the
region between $6500\,K\,<T_{\mathrm{eff}}<7000\,K$. The McQuillan et al.
(2014) study applied a cut at $T_{\mathrm{eff}}=6500\,K$ when selecting their
sample to avoid any pulsators when recovering rotation periods. However,
previous studies (e.g. Antoci et al., 2019) and our results in TESS show that
pulsators exist at effective temperatures less than $6500\,K$ and rotational
variables with apparent starspot modulation exist at effective temperatures
greater than $6500\,K$.

###  3.2 Binaries in TESS and Kepler

We use the re-normalized unit-weight error (RUWE) statistic from Gaia DR2 to
identify any stars with likely wide (within 4", $\sim$1000 AU) binary
companions in both the TESS and Kepler samples (see Berger et al., 2020, for
further discussion of the RUWE statistic). Following Berger et al. (2020), we
flag any stars with $RUWE>1.2$ as potential binaries. We find that 509 stars
in our TESS sample (24.1%) and 333 stars in our Kepler sample (15.8%) are
flagged as potential wide binaries based on Gaia RUWE. We identify these stars
for future users with the flag "RUWE Binary" in Table 1.

We visually inspect APOGEE spectra for all stars in both our Kepler and TESS
samples to identify any likely double-lined spectroscopic (within 2", $\sim
500$ AU) binaries (e.g. Fernandez et al., 2017; Kounkel et al., 2021). We
compare each spectrum to a model spectrum computed by ASPCAP, which assumes a
spectrum for a single star. We flag stars that exhibit multiple spectral lines
in regions where the ASPCAP model spectrum expects a single line as suspect.
We find that 33 stars in our TESS sample and 20 stars in our Kepler sample are
likely spectroscopic binaries. We identify these stars for future users with
the flag "Spectroscopic Binary" in Table 1. Stars which were not identified as
spectroscopic or RUWE binaries are referred to as "undetected" binaries in our
following discussion since they may be single stars, but we cannot exclude the
possibility of the undetected companion.

###  3.3 Spectroscopic Validation

We use $v\sin i$ from APOGEE and stellar radii estimated using isoclassify
with Gaia parallaxes and 2MASS photometry (similar to Berger et al., 2018) to
compute $P_{\mathrm{rot}}/\sin i$ for all stars in both our Kepler and TESS
samples. $P_{\mathrm{rot}}/\sin i$ is an upper limit on the rotation period we
expect to observe from starspot modulation and is given by:

$P_{\mathrm{rot}}/\sin i=\frac{2\pi R_{Gaia}}{v\sin i}$ (1)

where $R_{Gaia}$ is the stellar radius constrained using Gaia parallaxes.
Simonian et al. (2020) validated APOGEE $v\sin i$ values with rotation periods
from Kepler lightcurves and found a detection threshold of 10 $kms^{-1}$,
where $v\sin i$ values less than 10 $kms^{-1}$ are considered suspect. Tayar
et al. (2015) adopted a more generous threshold of 5 $kms^{-1}$ based off of
comparisons between their measured $v\sin i$ values to previously published
values. We adopt the same threshold as Tayar et al. (2015) and flag stars with
($v\sin i<$5 $kms^{-1}$) as suspect. Although we calculate
$P_{\mathrm{rot}}/\sin i$ values for all stars, including those with
unreliable $v\sin i$ values, we ignore these stars in our analysis of rotation
periods derived with $v\sin i$.

We cross-validate our estimated stellar radii computed with the direct method
in isoclassify to model radii predicted by YREC models with kiauhoku. We find
that $3\%$ of our Kepler sample and $3\%$ of our TESS sample have Gaia radii
that differ from YREC radii by more than $1.3$ solar radii, which is the
median radius of stars in our sample and the radius difference we expect for
equal-mass binaries. We find that of these stars with inconsistent Gaia and
YREC radii, $21\%$ in our Kepler sample are classified as RUWE or
spectroscopic binaries and $51\%$ in our TESS sample are classified as RUWE or
spectroscopic binaries (see 3.2 for further discussion of our binary
classification techniques). Given that an estimated $40\%$ of astrometric
binaries in Gaia are expected to not be detected and Gaia is unable to detect
binaries with angular separations below 20 milliarcseconds (Gaia Collaboration
et al., 2018), we attribute inconsistent Gaia and YREC radii to undetected
binaries. We additionally find that these inconsistent stars do not lie on the
apparent main sequence when plotted on a Gaia radius v. APOGEE
$T_{\mathrm{eff}}$ diagram, supporting our suspicion that these are likely
undetected binaries in Gaia.

Given our confidence in our APOGEE $v\sin i$ and Gaia radii values, we
validate our photometric rotation periods from TESS and Kepler with
$P_{\mathrm{rot}}/\sin i$. We consider any photometric rotation periods longer
than the expected maximum of $P_{\mathrm{rot}}/\sin i$ as suspect, as there
exists no inclination $i$ where the photometric rotation period can be greater
than $P_{\mathrm{rot}}/\sin i$. Additionally, any rotation periods where the
implied inclination angle is less than $20$ degrees are also flagged, as
starspot modulation is typically not possible to detect in the lightcurves of
these stars (Santos et al., 2017). We do this for both TESS rotation periods
and our adopted Kepler rotation periods computed by Nielsen et al. (2013),
Reinhold et al. (2013), or McQuillan et al. (2014).

$P_{\mathrm{rot}}$ v. $P_{\mathrm{rot}}/\sin i$ and our inferred distributions
of $\sin i$ for both our Kepler and TESS samples are shown in Figure 8. We
find that for stars with reliable $v\sin i$ measurements, 536 stars in our
TESS sample (41.3%) and 957 stars in our Kepler sample (66.9%) have rotation
periods that are inconsistent with $P_{\mathrm{rot}}/\sin i$ measured with
$v\sin i$ from APOGEE spectra and stellar radii estimated using Gaia, where
stars are considered suspect detections when their inferred $\sin i$ values
are greater than 1 or less than $\sin(20^{\circ})$. We additionally find that
removing RUWE binaries does not significantly change our fraction of stars
flagged as having reliable rotation periods, and therefore include them in
Figure 8. We identify stars with inconsistent $P_{\mathrm{rot}}$ and
$P_{\mathrm{rot}}/\sin i$ for future users with the flag "Suspect Detection"
in Table 1 and exclude them from our photometric rotation period analysis in
Section 4.

Figure 8: Validation of photometric rotation periods with $v\sin i$ from
APOGEE and stellar radii estimated using Gaia. The top panel shows the
comparison between $P_{\mathrm{rot}}$ and $P_{\mathrm{rot}}/\sin i$ for our
full TESS sample (left) and our Kepler detection sample (right), including
stars flagged as RUWE binaries. The different lines on both panels indicate
where a star is expected to fall for a given inclination angle $i$, as shown
in the right legend. The hatched regions indicate where $P_{\mathrm{rot}}$ and
$P_{\mathrm{rot}}/\sin i$ are inconsistent (i.e. no real star can live in
these regions). Stars flagged as suspect detections in our TESS sample are
shown in grey, while stars in our TESS (Kepler) sample with reasonable
detections are shown in purple (green). The shade of points on the top panels
indicate the trustworthiness of $v\sin i$ values used to calculate
$P_{\mathrm{rot}}/\sin i$; the lightest color indicates that $v\sin i$ is far
below the APOGEE detection threshold of $10\,kms^{-1}$, while the darkest
color is above the APOGEE detection threshold. The bottom panel shows the
distribution of $\sin i$ for the same TESS (left) and Kepler (right) samples
shown in the top panels, in the same color scheme as the top panels. The solid
light blue line shows the expected distribution for a random sample of
inclination angles. While the distribution of $\sin i$ values in our TESS
detection sample does follow an expected distribution for a random sample of
inclination angles, the Kepler detection sample extends far above this
distribution, as shown by the existence of many $\sin i$ values greater than
1, suggesting a problem with some of the Kepler measurements.

The bottom right panel of Figure 8 shows that the distribution of $\sin i$
values in our Kepler sample contains many stars with apparent $\sin i>1$
(23.4% of stars with reliable $v\sin i$ values in our Kepler sample). The
inferred Kepler $\sin i$ distribution has many measured periods that cannot
match the measured $v\sin i$ at any inclination angle, even after removing
stars with suspect $v\sin i$ values as shown by the differently shaded
distributions in the bottom right panel of Figure 8. Additionally, the
distribution of Kepler $\sin i$ values is strikingly different from the
expected distribution for a random sample of inclination angles (blue line in
bottom right panel of Figure 8). While we cannot conclusively identify the
source of this inconsistency, previous work has suggested that many incorrect
measurements of rotation periods in Kepler lightcurves are caused by
detections of rotation period aliases at twice the real rotation period
(Simonian et al., 2019). This hypothesis is additionally consistent with
previous studies which found that 20-30% of stars in the Kepler sample will
yield incorrect rotation periods even if they display starspot modulation in
their lightcurves (Aigrain et al., 2015; Santos et al., 2019).

##  4 Photometric Rotation Periods

###  4.1 Rotation in TESS

Table 2: Rotation periods for our TESS sample measured with Lomb-Scargle periodograms, wavelet analysis, and autocorrelation function analysis, the mean photometric rotation period, and the rotation period derived from APOGEE spectra and Gaia radii. ID | Binary Flag | Lomb-Scargle (d) | Wavelet (d) | ACF (d) |  $P_{\mathrm{rot}}$ (d) |  $P_{\mathrm{rot}}/\sin i$ (d)
---|---|---|---|---|---|---
TIC 30631031 | Single Star | $2.11\pm 0.21$ | $1.93\pm 0.19$ | $1.97\pm 0.19$ | $1.99\pm 0.20$ | $1.93^{+0.04}_{-0.04}$
TIC 41591638 | Spectroscopic Binary | $7.31\pm 0.73$ | $6.11\pm 0.61$ | $10.23\pm 1.02$ | $7.42\pm 1.51$ |  $\cdots$
TIC 231072603 | RUWE Binary | $3.70\pm 0.37$ | $3.53\pm 0.35$ | $3.30\pm 0.33$ | $3.52\pm 0.35$ | $4.38^{+0.13}_{-0.12}$

Our final detection sample in TESS includes 169 rotation periods, which
represents $7.9\%$ of our original TESS sample of 2115 stars. This sample
includes stars that have reliable $P_{\mathrm{rot}}/\sin i$ measurements,
where $v\sin i$ is greater than 5 $kms^{-1}$. $P_{\mathrm{rot}}$ can therefore
be vetted to be consistent with $P_{\mathrm{rot}}/\sin i$. Of these stars with
confidently recovered rotation periods, 20 are flagged as RUWE binaries and 1
is flagged as a spectroscopic binary. We list our measured rotation periods in
Table 2.

Figure 9: Distribution of spectroscopic rotation periods in our TESS (top) and
Kepler (bottom) samples for both detections (orange) and nondetections
(turquoise). Here, nondetections refer to stars which did not exhibit spot
modulation in their lightcurves. Additionally, these distributions only
include stars with reliable $v\sin i$ values (i.e. $v\sin i>5$ $kms^{-1}$).
The dashed lines correspond to the median of each distribution. TESS rotation
periods are clearly biased to those less than 13.7 days. The Kepler sample
also shows a bias to shorter rotation periods, but reaches rotation periods
out to 30 days.

We posit that our ability to detect rotation periods greater than $13.7$ days
is suppressed by TESS systematics, thus biasing our measured rotation periods
to more rapidly rotating stars. This detection limit is shown in Figure 9,
where we compare the distribution of rotation periods detected in TESS to
those detected in Kepler, where rotation periods up to tens of days were
measured. Although other attempts have been made to measure rotation periods
for stars that span multiple TESS observing sectors (e.g. Lu et al., 2020;
Martins et al., 2020; Claytor et al., 2021), up to this point, only studies
using machine learning-based rotation period extraction methods, i.e. neural
networks, have successfully recovered rotation periods longer than $13.7$ days
(Lu et al., 2020; Claytor et al., 2021).

###  4.2 Comparing TESS and Kepler

Figure 10: Kiel diagram of stars with detected rotation periods in TESS (left)
and the McQuillan et al. (2014) Kepler sample (right). The top panels show the
number of stars in the full (combined detection and nondetection) samples in
each $T_{\mathrm{eff}}$ \- $\log g$ bin, while the bottom panels show the
detection fraction of photometric rotation periods in each bin. The detection
fraction distribution for the Kepler sample (bottom right) is calculated only
for stars whose detected rotation periods are less than $13.7$ days. The lines
correspond to a constant age of $500$ Myr (blue) and a constant rotation
period of $13.7$ days (red) as determined by kiauhoku from YREC models at
solar metallicity.

Prior to analyzing distributions of stellar parameters as a function of
rotation in our TESS and Kepler samples, we first examine the differences
present in these two populations. We compare the detection fraction of stars
across the Kiel diagram in the McQuillan et al. (2014) sample and our TESS
sample. We restrict our comparison in the McQuillan et al. (2014) sample to
only include stars with detected rotation periods less than $13.7$ days to
enable a more direct comparison to our biased TESS sample. Our comparison is
shown in Figure 10. The top two panels show the number density of stars in our
entire TESS sample and the entire McQuillan et al. (2014) sample. The bottom
two panels show the distribution of detection fractions for stars in our TESS
sample with detected photometric rotation periods (left) and stars in the
McQuillan et al. (2014) sample with detected rotation periods less than $13.7$
days. We find that we are more likely to detect rotation periods for cooler,
less evolved stars in TESS. This is consistent with McQuillan et al. (2014),
who observed that the amplitude of periodic variability ($R_{per}$) increased
for cooler stars.

We maximize our detection fraction for the coolest main sequence stars in our
TESS sample ($T_{\mathrm{eff}}$ = $5600K-5720K$, $\log g$ = $4.55-4.62$) at
$32\%$ of stars with detected rotation periods. The McQuillan et al. (2014)
sample contains 4 stars with detected rotation periods less than 13.7 days in
this bin. To more accurately compare the detection fractions of our TESS
sample and the McQuillan et al. (2014) sample, we take a wider range in
$T_{\mathrm{eff}}$ and $\log g$ of $5600K-5800K$ and $4.5-4.7$ dex. The
detection fraction for TESS stars in these bins is 25%, compared to the 40%
detection fraction seen in the McQuillan et al. (2014) sample, which includes
measured rotation periods out to tens of days. This discrepancy in rotation
period detection fractions is likely due to difficulties in measuring rotation
periods longer than 13.7 days in TESS lightcurves, where any star in our TESS
sample with a rotation period longer than 13.7 days is likely to have its
rotation signal suppressed by TESS instrument signals.

The distribution of detection fractions also differ substantially between our
TESS sample and the McQuillan et al. (2014) sample. We argue that these
differences in rotation distributions between the McQuillan et al. (2014)
sample and the TESS sample are likely caused by differences in stellar
populations along the sightlines of Kepler and TESS. The blue line in Figure
10 is a 500 Myr solar metallicity isochrone calculated by kiauhoku. The Kepler
sample includes few stars below the 500 Myr isochrone (Berger et al., 2020; Lu
et al., 2021). This makes sense given the location of the Kepler observing
field above the galactic midplane, where few young stars reside. The existence
of short-lived massive O and B stars (Pedersen et al., 2019; Bowman et al.,
2020) as well as the existence of stars identified with young ($<200$ Myr)
clusters and moving groups (e.g. Gagné et al., 2018) in the TESS SCVZ,
suggests a young population is present in the direction of the TESS SCVZ that
was not present in the Kepler field. While we leave the creation of a full
population model in the TESS SCVZ and analysis of age proxies (e.g.
kinematics) to future studies, a rough comparison of our stars with existing
rotation-age relationships (Claytor et al., 2020) suggests that this
population likely covers a large age range, including a significant number of
young stars.

Figure 11: Rotation as a function of evolution in our combined TESS and Kepler
sample. When we show our entire sample (left), we recover the broad trend that
more massive stars rotate more rapidly and that stars rotate faster with
increasing $T_{\mathrm{eff}}$. However, when we include stars within a
narrower range of zero-age main sequence $T_{\mathrm{eff}}$, we can see the
declining rotation rate as stars evolve up the subgiant branch, expanding and
losing angular momentum.

###  4.3 Inconsistencies with Models

We find inconsistencies between detected rotation periods in Kepler and the
predictions of models that include rotation. The red line in Figure 10 is the
predicted location of stars with a rotation period of 13.7 days (van Saders &
Pinsonneault, 2013). Under the conditions of this model, no periods shorter
than this should be detected to the right of this line. The detection fraction
in our TESS sample does drop significantly at around this line (bottom left
panel of Figure 10), suggesting consistency with the van Saders & Pinsonneault
(2013) prediction. The Kepler sample (bottom right panel), however, is
inconsistent with the van Saders & Pinsonneault (2013) prediction. In fact, we
see a significant overdensity in Kepler short period rotators in this regime,
comprising $\sim$70% of stars with detected rotation periods less than $13.7$
days. Visual inspection of Kepler target pixel files and our analysis of
$v\sin i$ does not suggest an obvious abundance of background stars which
could be responsible for this signal. The overdensity also does not coincide
with the presence of low-amplitude rotation signals, where measurement errors
are more likely to occur. Theoretical explanations for this overdensity of
rapidly rotating slightly evolved stars are also unable to explain the data.
We find that these stars are spinning too fast even when compared to models
with weakened magnetic braking (van Saders et al., 2016). Additionally, the
metallicity distribution of these anomalously fast rotating stars is
consistent with the general sample, so explanations relying on metallicity-
dependent rotational evolution are unlikely (Amard et al., 2020).

We therefore posit that the overdensity at rotation periods longer than $13.7$
days in the McQuillan et al. (2014) sample is most likely due to unidentified
binaries in the Kepler field. As stated in Section 3.3, an estimated 40% of
astrometric binaries and binaries with separations below 20 milliarcseconds
will not be detected in Gaia DR2 (Gaia Collaboration et al., 2018). Although
upon removing RUWE and spectroscopic binaries from the McQuillan et al. (2014)
sample, we find no significant change in the detection fractions we compute,
given the high fraction of undetected binaries in Gaia, we still suspect that
this overdensity is most likely due to unidentified binaries. This conclusion
is consistent with previous work that attributed anomalously rapid rotation in
Kepler to undetected binary companions from comparisons with Gaia (Simonian et
al., 2019).

##  5 Probing Rotation with $P_{\mathrm{rot}}/\sin i$

Due to the substantial detection bias towards stars with rapid rotation in our
TESS sample, any distributions we derive from photometric rotation periods
measured with TESS will be incomplete. Rotation periods derived from
spectroscopic $v\sin i$, however, are distributed similarly in the TESS and
Kepler samples (see Figure 9). We therefore use $P_{\mathrm{rot}}/\sin i$ for
both our TESS and Kepler samples for the remainder of our analysis.

Although TESS and Kepler probe different stellar populations, we expect the
fundamentals of angular momentum acquisition and loss to be the same. We
therefore combine our entire TESS and Kepler samples into one larger sample
for this section of our analysis. This combined sample includes all stars in
our original Kepler and TESS samples defined in Section 2.1 with $v\sin i$
values above a detection threshold of $5\,kms^{-1}$.

###  5.1 Rotation on the Kiel Diagram

In the regime probed by our combined Kepler and TESS sample, we expect
rotation to be a stronger function of $T_{\mathrm{eff}}$ than age. As
described in Section 1, hot stars ($T_{\mathrm{eff}}\,>$ 6250 K) do not
experience substantial spin down on the main sequence, and therefore exhibit
rapid rotation at the end of the main sequence. Cool stars
($T_{\mathrm{eff}}\,<$ 6250 K), however, are able to develop deep convective
zones and thus sustain a magnetized wind and shed angular momentum on the main
sequence.

We find that when we observe rotation across the Kiel diagram (see Figure 11)
for our entire sample, rotation is indeed a strong function of
$T_{\mathrm{eff}}$. At the Kraft Break, we find that the transition between
slow ($P_{\mathrm{rot}}>10$ days) and fast ($P_{\mathrm{rot}}<10$ days)
rotators is apparent. We narrow the range of probed zero-age main sequence
$T_{\mathrm{eff}}$ to include only stars within $100$ K of the Kraft Break.
When we limit our sample to this range, we find that rotation becomes a weaker
function of $T_{\mathrm{eff}}$ and a stronger function of evolutionary state.
We additionally find that these rotation periods are inconsistent with models
which include standard angular momentum loss (van Saders & Pinsonneault, 2013)
and weakened braking (van Saders et al., 2016), with our stars, on average,
rotating faster than predictions by both models. We suspect that the
challenges with detecting slow rotational velocities may be the culprit (see
Figure 9). Future studies focused on testing model accuracy would benefit from
samples which probe stars with a wide range of zero-age main sequence
$T_{\mathrm{eff}}$ and from the detection of rotation periods beyond those
probed by our sample.

###  5.2 Metallicity Dependence

Figure 12: Distribution of rotation as a function of zero-age main sequence
$T_{\mathrm{eff}}$ and metallicity. We divide our sample into a bin above the
Kraft Break (top) and a bin below the Kraft Break (bottom), and divide our
sample into low (orange), solar (green), and high (purple) metallicity bins.
We find that neither zero-age main sequence $T_{\mathrm{eff}}$ bin exhibits a
correlation between rotation and metallicity.

Amard et al. (2020) observed a correlation between rotation and metallicity in
their sample of 4055 Kepler stars between $0.85M_{\odot}$ and $1.3M_{\odot}$,
with metal-rich stars rotating more slowly than metal-poor stars within this
mass range. This result is presented as evidence for metallicity-dependent
magnetic braking, and is consistent with trends between rotation and
metallicity that are predicted for cool, old dwarfs (van Saders &
Pinsonneault, 2013; Claytor et al., 2020; Simonian et al., 2020). The size of
convection zones in these stars are sensitive to metallicity, where higher
metallicity causes deeper convection zones at fixed mass (van Saders &
Pinsonneault, 2013). Recently, the relation between stellar activity,
convection zone depth, and metallicity has been observed in Kepler by See et
al. (2021), who found that the amplitude of stellar variability exhibits a
positive correlation with metallicity and the convective turnover timescale.

Our sample, however, shows no correlation between rotation and metallicity. A
Kolmogorov-Smirnov test supports that the metallicity-binned
$P_{\mathrm{rot}}/\sin i$ distributions in Figure 12 are drawn from the same
distribution, indicating that our sample does not provide evidence for
metallicity-dependent magnetic braking.

This lack of a correlation between rotation and metallicity suggests that the
rotational evolution of stars in our sample is not highly metallicity-
dependent and suggests that our stars are experiencing weak magnetic braking
on the main sequence. Although our sample includes a large number of stars
below the Kraft break, where stars are expected to experience substantial
spin-down, Amard et al. (2020) searches a cooler and lower-mass population of
stars than us. When we limit our sample to stars cooler than $6000$K, we
observe suggestions of a correlation between rotation and metallicity.
However, these trends are not statistically significant. Further analysis of
samples that specifically probe stars at a fixed mass and a wide range of
metallicities is needed to determine the degree of correlation between
metallicity and rotation.

###  5.3 Impact of Binarity

Figure 13: Distribution of spectroscopic rotation periods for single stars
(blue), RUWE (wide) binaries (orange), and spectroscopic (close) binaries
(green). The dashed lines correspond to the median of each distribution.

Binarity can impact a star’s rotational history, from formation to tidal
synchronization and mergers (see e.g. Douglas et al., 2017; Lurie et al.,
2017; Fleming et al., 2019, using Kepler). We therefore use APOGEE spectra to
identify double-lined close binaries (within $2"$, $\sim 500$ AU) and the RUWE
statistic from Gaia DR2 to identify wide binaries (within $4"$, $\sim 1000$
AU) to determine if their rotation rates differ from a population of single
stars. We show in Figure 13 that presumed single stars have a wide
distribution of spectroscopic rotation periods ($P_{\mathrm{rot}}/\sin i$),
out to 40 days, which is consistent with previous figures (see e.g., Section
4) and past studies with Kepler (see e.g. Nielsen et al., 2013; Reinhold et
al., 2013; McQuillan et al., 2014; Amard et al., 2020). Looking at
spectroscopic (close) binaries, we observe a much more rapid median rotation
rate of 2.3 days. Previous work suggests that rapid rotation in the Kepler
field is caused by a combination of rapidly rotating near-equal luminosity
spectroscopic (close) binaries that have been spun up via tides and unresolved
spectroscopic (close) binaries with velocity displacements on the order of
APOGEE’s resolution (Simonian et al., 2019, 2020).

We additionally find that RUWE (wide) binaries have the same distribution in
rotation as single stars at a $\approx 99\%$ significance, as supported by a
Kolmogorov-Smirnov test (see also Mamajek & Hillenbrand, 2008; Fleming et al.,
2019; Godoy-Rivera et al., 2021). By showing that single stars and the
components of wide binaries have the same rotational distribution, Figure 13
validates the use of wide separation binaries as empirical constraints for the
rotational evolution of single stars (e.g. Mamajek & Hillenbrand, 2008; Godoy-
Rivera & Chanamé, 2018; Janes, 2017; Fleming et al., 2019; Otani et al.,
2021).

##  6 Summary & Conclusions

Using a sample of stars observed with TESS, rotation periods measured with
Kepler, spectroscopic parameters from APOGEE, and radii estimated using Gaia
parallaxes, we analyze rotation around the Kraft Break. With this sample, we
show:

  * •

It is possible to detect rotation from starspot modulation in 2 minute cadence
TESS SAP lightcurves using Lomb-Scargle, wavelet, and ACF analysis. However,
TESS systematics suppress rotation periods longer than the TESS orbital period
of $13.7$ days.

  * •

The detection fraction for rotation periods in TESS is $7.6\%$, compared to
the $20\%$ detection fraction seen in Kepler by Simonian et al. (2020). We
suspect that this large discrepancy in detection fractions is primarily due to
instrument systematics in TESS lightcurves.

  * •

Spectroscopy can be sensitive to longer rotation periods that are suppressed
by systematics in TESS lightcurves.

  * •

We can identify other classes of stellar variability in TESS lightcurves,
specifically pulsations (see Table 1). These identified candidate pulsators
show that there is a significant overlap in $T_{\mathrm{eff}}$ between
rotating and pulsating stellar populations.

  * •

Fundamental differences exist between the stellar populations probed by the
Kepler and TESS missions, with the TESS SCVZ field containing a significant
number of young stars (Age < 500 Myr).

  * •

We recover broad trends between rotation and mass. From these trends, we find
that tracing rotational evolution is challenging unless individual stars have
been carefully characterized.

  * •

Our sample does not appear to exhibit a correlation between rotation and
metallicity; we argue that this is a result of the differences between the
populations probed by our sample and the sample in Amard et al. (2020). Our
sample probes fewer cool low-mass stars and thus cannot probe metallicity-
dependent spin down.

  * •

A significant population of RUWE (wide) and spectroscopic (close) binaries are
present in TESS SCVZ and Kepler, comprising 24.6% and 16.9% of stars in these
samples. Consistent with previous results, we find that spectroscopic (close)
binaries rotate slightly faster than presumed single stars, whereas the
rotation distribution of wide (RUWE) binaries covers the same range of
rotation periods as single stars.

It is clear that stellar rotation is a complex physical process that has
connections to our understanding of stellar interiors, activity, magnetism,
and our own host star (the Sun). Past studies of rotational evolution have
been successful in improving our understanding of rotation across stellar
populations (e.g. McQuillan et al., 2014; Amard et al., 2020; Simonian et al.,
2020). The advent of comprehensive spectroscopic follow-up with APOGEE and
photometric characterization with Gaia have improved our ability to understand
the multi-faceted nature of stellar rotation. Future spectroscopic surveys
(e.g. SDSS-V and the Rubin Observatory) and future Gaia releases (e.g. EDR3;
Gaia Collaboration et al., 2021) will enable precise characterization of stars
across the entire sky. Therefore, further studies will be able to utilize a
combination of photometric and spectroscopic parameters for a variety of
stellar populations.

Strides are additionally being made in detecting long rotation periods from
TESS lightcurves using more modern techniques like machine learning (Lu et
al., 2020; Claytor et al., 2021). Improved characterization of the TESS
instrument as well as improved understanding of scattered light across all
TESS observing sectors will improve our ability to detect rotation signals
beyond tens of days. Additionally, future surveys would benefit from long
baselines which allow studies to probe stars with a wide range of rotation
periods. Detailed surveys (e.g. the European Space Agency PLAnetary Transits
and Oscillations (PLATO) mission) and follow-up missions will improve our
coverage of rotation across the entire sky, revealing the precise dependence
of rotation on mass, metallicity, evolutionary state, and binarity.

E.A.A., J.N.T., and J.V.S. acknowledge support from NASA Award 80NSSC20K0056.
J.N.T. acknowledges that support for this work was provided by NASA through
the NASA Hubble Fellowship grant No.51424 awarded by the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Inc., for NASA, under contract NAS5-26555. T.A.B.
acknowledges support by a NASA FINESST award (80NSSC19K1424). A portion of
T.A.B.’s research was supported by an appointment to the NASA Postdoctoral
Program at the NASA Goddard Space Flight Center, administered by Universities
Space Research Association and Oak Ridge Associated Universities under
contract with NASA. Support for this work was provided by NASA through Hubble
Fellowship grant #51386.01 awarded to R.L.B. by the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., for NASA, under contract NAS 5-26555. This paper includes
data collected by the TESS mission, which are publicly available from the
Mikulski Archive for Space Telescopes (MAST). Funding for the TESS mission is
provided by NASA’s Science Mission directorate. Funding for the Sloan Digital
Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S.
Department of Energy Office of Science, and the Participating Institutions.
SDSS acknowledges support and resources from the Center for High-Performance
Computing at the University of Utah. The SDSS website is www.sdss.org. SDSS is
managed by the Astrophysical Research Consortium for the Participating
Institutions of the SDSS Collaboration including the Brazilian Participation
Group, the Carnegie Institution for Science, Carnegie Mellon University, the
Chilean Participation Group, the French Participation Group, Harvard-
Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The
Johns Hopkins University, Kavli Institute for the Physics and Mathematics of
the Universe (IPMU) / University of Tokyo, the Korean Participation Group,
Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik
Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-
Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für
Extraterrestrische Physik (MPE), National Astronomical Observatories of China,
New Mexico State University, New York University, University of Notre Dame,
Observatório Nacional / MCTI, The Ohio State University, Pennsylvania State
University, Shanghai Astronomical Observatory, United Kingdom Participation
Group, Universidad Nacional Autónoma de México, University of Arizona,
University of Colorado Boulder, University of Oxford, University of
Portsmouth, University of Utah, University of Virginia, University of
Washington, University of Wisconsin, Vanderbilt University, and Yale
University. This work has made use of data from the European Space Agency
(ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the _Gaia_
Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the _Gaia_ Multilateral Agreement. This publication makes use
of data products from the Two Micron All Sky Survey, which is a joint project
of the University of Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the National Aeronautics
and Space Administration and the National Science Foundation.

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