# TESS Stellar Rotation up to 80 days in the Southern Continuous Viewing Zone

Zachary R. Claytor Department of Astronomy, University of Florida, 211 Bryant
Space Science Center, Gainesville, FL 32611, USA  Institute for Astronomy,
University of Hawai‘i at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
Jennifer L. van Saders Institute for Astronomy, University of Hawai‘i at
Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Lyra Cao Department of
Astronomy, The Ohio State University, Columbus, OH 43210, USA  Marc H.
Pinsonneault Department of Astronomy, The Ohio State University, Columbus, OH
43210, USA  Johanna Teske Earth & Planets Laboratory, Carnegie Institution for
Science, 5241 Broad Branch Road, NW, Washington, DC 20015, USA  Rachael L.
Beaton Space Telescope Science Institute, Baltimore, MD, 21218, USA
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD
21218, USA

###### Abstract

The TESS mission delivers time-series photometry for millions of stars across
the sky, offering a probe into stellar astrophysics, including rotation, on a
population scale. However, light curve systematics related to the satellite’s
13.7-day orbit have prevented stellar rotation searches for periods longer
than 13 days, putting the majority of stars beyond reach. Machine learning
methods have the ability to identify systematics and recover robust signals,
enabling us to recover rotation periods up to 30 days for FGK dwarfs and 80
days for M dwarfs. We present a catalog of 7,971 rotation periods for cool
dwarfs in the Southern Continuous Viewing Zone, estimated using convolutional
neural networks. We find evidence for structure in the period distribution
consistent with prior Kepler and K2 results, including a gap in 10–20-day cool
star periods thought to arise from a change in stellar spin-down or activity.
Using a combination of spectroscopic and gyrochronologic constraints, we fit
stellar evolution models to estimate masses and ages for stars with rotation
periods. We find strong correlations between the detectability of rotation in
TESS and the effective temperature, age, and metallicity of the stars.
Finally, we investigate the relationships between rotation and newly obtained
spot filling fractions estimated from APOGEE spectra. Field star spot filling
fractions are elevated in the same temperature and period regime where open
clusters’ magnetic braking stalls, lending support to an internal shear
mechanism that can produce both phenomena.

††journal: AAS Journals††software: butterpy (Claytor et al., 2021, 2022),
kiauhoku (Claytor et al., 2020a, b), NumPy (Harris et al., 2020), Pandas (Wes
McKinney, 2010), Matplotlib (Hunter, 2007), AstroPy (Astropy Collaboration et
al., 2013, 2018), SciPy (Virtanen et al., 2020), PyTorch (Paszke et al.,
2019), Lightkurve (Lightkurve Collaboration et al., 2018), TESScut (Brasseur
et al., 2019), iPython (Perez & Granger, 2007), starspot (Angus, 2021),
Astroquery, (Ginsburg et al., 2019)

##  1 Introduction

Rotation, activity, and magnetism are all deeply connected to the structure
and evolution of stars. In stars similar to and less massive than the Sun,
rotation and convection power magnetism, which influences stellar winds and
causes flares. Magnetized winds create torque on stars, causing them to spin
down over time (Weber & Davis, 1967); this allows us to infer stellar ages
from rotation periods using gyrochronology (Skumanich, 1972; Barnes, 2003).
Stellar magnetism is the source of space weather, which directly affects life
on Earth as well as the habitability of planets around other stars. Because of
the inextricable links to rotation, a complete picture of stellar activity and
magnetism demands a grasp of rotation across all types of stars.

The Kepler mission (Borucki et al., 2010) enabled rotation period estimates
for more than 50,000 stars in a single 100-square-degree patch of sky
(McQuillan et al., 2014; Santos et al., 2019, 2021), revolutionizing our
understanding of stellar rotation. Kepler’s rotation periods enabled precise
age estimates for field stars (e.g., Claytor et al., 2020a; Lu et al., 2021)
and investigations of changing stellar activity with time (Mathur et al.,
2023). The mission also revealed departures from expected rotational behavior,
such as a gap in the period distribution of cool stars (McQuillan et al.,
2014) and a halt of magnetic braking in middle-aged Solar-like stars (Angus et
al., 2015; van Saders et al., 2016; David et al., 2022). Stellar evolution and
population synthesis models failed to predict these behaviors (van Saders et
al., 2019), highlighting the need for updated theory as well as more period
measurements.

As successful as Kepler was in measuring rotation periods, the survey design
imposed tight limitations on the kinds of stars that could be studied in
rotation. The spacecraft’s goal of finding Earth-like planets around Sun-like
stars resulted in a complex selection function that biased observed stellar
samples in comparison to the underlying population (e.g., Borucki et al.,
2010; Wolniewicz et al., 2021). For example, the choice to avoid young stellar
populations biases the Kepler sample toward less active, slowly rotating stars
intrinsically more difficult to detect in rotation. Any Kepler study preserves
these biases, and the selection function is difficult to correct for.
Furthermore, the small observing footprint means that any new rotational
physics inferred from Kepler must be tested against other samples across the
sky. The solution is an untargeted, all-sky survey.

The Transiting Exoplanet Survey Satellite (TESS Ricker et al., 2015) stares at
millions of stars in its search for transiting planets, surveying the entire
sky in 27-day sectors. In addition to short-cadence light curves for pre-
selected targets, TESS delivers full-frame images (FFIs), enabling high-
precision photometry for any source brighter than $\sim$15th magnitude.
Importantly, TESS does not rely only on postage stamps for selected targets as
Kepler did; the FFIs permit investigators to design their own surveys. While
primarily a planet-finding mission, the mission’s short cadence and long
temporal baseline also make it suitable for studying stellar variability due
to oscillations, pulsations, and rotational spot modulation. While studies of
stellar oscillations and pulsations have achieved some success (e.g., Silva
Aguirre et al., 2020; Mackereth et al., 2021; Chontos et al., 2021; Hon et
al., 2021, 2022; Stello et al., 2022), systematics related to TESS’s observing
strategy and data processing have slowed the quest for rotation periods
(Oelkers & Stassun, 2018; Canto Martins et al., 2020; Avallone et al., 2022;
Kounkel et al., 2022). It is worth noting that the Kepler mission faced
similar challenges; the seminal stellar rotation paper (McQuillan et al.,
2014) was published 5 years after the satellite was launched.

TESS’s unique 2:1 resonance orbit of the Earth-Moon system subjects the
detectors to earthshine and moonlight on the timescale of the orbit, 13.7 days
(Vanderspek et al., 2018). The earthshine itself has time-varying signals
within it, such as a 1-day modulation from the rotation of the Earth (Luger et
al., 2019). Besides earthshine, TESS encounters systematics related to angular
momentum dumps, detector heating, data downlinks, and more, all on timescales
that interfere with astrophysical signals. The telescope’s large field of view
(24∘ by 96∘ total) makes the background spatially non-uniform as well. Because
of these effects, throughout a sector the TESS detectors encounter systematics
on different pixels at different times and with varying intensity, making them
difficult to correct.

Attempts to remove or correct spurious instrumental signals may also attenuate
astrophysical signals, particularly those on the timescales of the telescope’s
orbital period (13.7 days) and longer. Rapid rotators, which also have larger
spot modulation amplitudes, are affected less, and conventional rotation
searches with TESS have been largely successful at measuring periods shorter
than 13 days (see, for example, Canto Martins et al. 2020 with 131 periods,
Avallone et al. 2022 with 169, Holcomb et al. 2022 with 13,504, and Kounkel et
al. 2022 with $\sim$100,000). However, the same searches have struggled to
recover longer periods, instead catching periods associated with the TESS
systematics. So far, only Lu et al. (2020) have claimed to recover long
periods in TESS, but they relied heavily on priors from the observed Kepler
period distribution.

The efforts to correct TESS systematics have yielded broadly useful public
pipelines and tools like eleanor (Feinstein et al., 2019), TESS-SIP (Hedges et
al., 2020), Unpopular (Hattori et al., 2022), and T’DA (Handberg et al., 2021;
Lund et al., 2021). While each pipeline makes different but well-motivated
choices to handle the systematics, each decision runs the risk of accidentally
removing stellar signals (Hattori et al., 2022; Kounkel et al., 2022). Rather
than trying to remove the systematics at the risk of removing astrophysical
signals, we adopt deep machine learning methods that see the periodicity with
the noise and disentangle them.

Deep learning methods are now widely used in stellar astrophysics. Breton et
al. (2021) used random forests to classify and detect rotation signals in
Kepler light curves, while Lu et al. (2020) used random forests to draw
connections between stellar parameters to estimate TESS rotation periods.
Feinstein et al. (2020) employed convolutional neural networks (CNNs) to
identify stellar flares in light curves, while Hon et al. (2021) applied
similar techniques to detect oscillations in red giant stars. CNNs are
particularly powerful when working with images or image-like data, which are
ubiquitous in astronomy. CNNs can be trained to identify images of many
different classes despite contaminating features, making them particularly
attractive for our problem with TESS systematics.

In a pilot study of 21 objects, Claytor et al. (2022) demonstrated that long
periods can be recovered from TESS data using CNNs trained on simulated data.
In this work we apply the Claytor et al. (2022) approach to a greatly expanded
sample to infer rotation periods with uncertainties for cool, main-sequence
stars in the TESS Southern Continuous Viewing Zone (SCVZ). We employ new
training sets tailored to specific period ranges and the specific light curves
in which we search for periods. We present the periods, their associated
uncertainties, and model-inferred stellar parameters in the first catalog
probing long stellar rotation periods with TESS.

The paper is outlined as follows. In Section 2 we describe our data and sample
selection. In Section 3 we outline our deep learning framework, including the
training sets and model architectures. Section 4 details our method to fit
stellar evolutionary models to stars with reliable rotation periods. In
Section 5 we present the rotation periods and analyze the TESS SCVZ period
distribution, comparing and contrasting with Kepler and K2. In Section 6 we
explore the detectability of periods as a function of temperature,
metallicity, age, and convection zone properties to understand the effects of
detection limits on the period distribution. In Section 7 we use new spot
filling fraction measurements from infrared spectroscopy to examine the
effects of spottedness on the detectability of rotation, and we finally
conclude in Section 8.

##  2 Data and Sample Selection

Table 1: Sample Selection

Designation | Description | Criteria | # targets
---|---|---|---
A1 | TESS-SPOC Dwarfs |  eclat $\leq-78^{\circ}$ & Tmag $\leq 15$ & Teff $\leq 10,000$ K | 38,215
A2 | TASOC Dwarfs |  eclat $\leq-78^{\circ}$ & Tmag $\leq 15$ & Teff $\leq 5,000$ K | 29,609
B1 | APOGEE–TESS-SPOC | A1 & APOGEE DR17 | 16,545
B2 | APOGEE–TASOC | A2 & APOGEE DR17 | 3,156
C1 | Rotators | Either A1 or A2 & reliable period | 7,971
C2 | Non-Rotators | Either A1 or A2 & no reliable period | 53,141
D | APOGEE Rotators | Either B1 or B2 & C1 | 2,654
Gold | Binary-Cleaned Rotators | D & STAR_BAD $=$ SNR_BAD $=0$ & RUWE $<1.2$ & contratio $<0.1$ | 1,227
Platinum | Single Cool Dwarfs | Gold & $M_{G}>4.4$ & $G_{BP}-G_{RP}>1$ & $|\Delta M_{G}|<0.4$ | 566

Note. — Our sample selections using TIC version 8.2 (Stassun et al., 2019;
Paegert et al., 2021), APOGEE DR17 (Abdurro’uf et al., 2022), and Gaia DR3
(Gaia Collaboration et al., 2022). Selection criteria are given as table
column names where convenient and include ecliptic latitude (TIC eclat), TESS
magnitude (TIC Tmag), effective temperature (TIC Teff), ASPCAP (García Pérez
et al., 2016) spectral fit flags STAR_BAD and SNR_BAD, absolute $G$-magnitude
(Gaia $M_{G}$), color index (Gaia $G_{BP}-G_{RP}$), and Gaia photometric
excess above the main sequence ($|\Delta M_{G}|$). The lettered samples are
described in Section 2, while the Gold and Platinum science samples are
detailed in Sections 5.5 and 7 respectively.

For the period search, we targeted stars cool enough to maintain surface
convection zones and dynamos capable of producing surface spots. The steps of
our full sample selection our outlined in Table 1. We excluded evolved red
giants, of which the large majority are slow rotators (Ceillier et al., 2017).
The 1–2% that rotate rapidly are typically the products of binary star
interactions (Carlberg et al., 2011), and not reliable age tracers. We
selected relatively bright, cool dwarf and subgiant stars in the TESS SCVZ, a
450 square degree field centered around the southern ecliptic pole. TESS
observed the SCVZ continuously for 350 days in its first year, taking FFIs
every 30 minutes. The long baseline ensures sufficient coverage for the most
slowly-rotating stars we might hope to detect. For example, an M-dwarf
rotating once every 100 days will complete 3.5 rotations under observation in
the CVZs. In the same interval, an old K-dwarf rotating at 45 days will rotate
nearly 8 times, and a G-dwarf at 30 days will rotate more than 10.

We selected stars from the TESS Input Catalog (TIC, Stassun et al., 2019;
Paegert et al., 2021) with effective temperature $\leq 10,000$ K, TESS
magnitude $\leq 15$, and ecliptic latitude $\leq-78^{\circ}$ to target the
SCVZ. There are 398,977 such stars in the TIC, but requiring public photometry
narrowed the sample considerably. 38,215 targets had public FFI photometry
from the TESS Science Processing Operations Center (TESS-SPOC, Jenkins et al.,
2016; Caldwell et al., 2020). We also used FFI data products from the TESS
Asteroseismic Science Operations Center (TASOC, Handberg et al., 2021; Lund et
al., 2021), but we selected only the 29,609 targets with TIC
$T_{\mathrm{eff}}<5,000$ K to prioritize the most likely rotation detections.
We motivate the choice to use both TESS-SPOC and TASOC products in Section
2.3, and we detail each pipeline’s target selections in Sections 2.3.1 and
2.3.2.

###  2.1 APOGEE Spectroscopy

While the TESS Input Catalog has metallicities and surface gravities for all
the stars in our sample, the sources are a heterogeneous combination of
photometry and spectroscopy, observations and models. Furthermore, the TIC has
no information on detailed abundances, which are useful when investigating
changing Galactic chemistry with time, and which are important to the
connection between rotation and magnetism (e.g., Claytor et al., 2020a). We
therefore supplement TESS photometric rotation periods with spectroscopic
parameters from the Apache Point Observatory Galactic Evolution Experiment
(Majewski et al., 2017, APOGEE).

APOGEE collects high-resolution ($R\sim 22,500$), near-infrared (1.51–1.70
$\mathrm{\mu}$m) spectra and provides calibrated, model-dependent estimates of
effective temperature, surface gravity, metallicity, and detailed abundances
for hundreds of thousands of stars across the entire sky. The TESS/APOGEE
survey within APOGEE-2S (Section 5.8 of Santana et al., 2021) targeted 38,000
stars in the TESS SCVZ with 2MASS color and magnitude ranges $7<H<11$ and
$J-K>0.3$, and about 9,000 other SCVZ stars were observed for other programs.
We cross-matched the TIC SCVZ cool dwarfs with APOGEE Data Release 17
(Abdurro’uf et al., 2022) to obtain spectroscopic parameters for 47,142 stars.
Of those, 16,545 have TESS-SPOC data products, and 3,156 have data products in
our TASOC subsample. These combine to yield 17,796 unique targets with APOGEE
spectroscopy and either TESS-SPOC or TASOC photometry.

We adopted calibrated effective temperatures, metallicities, and
$\alpha$-element abundances estimated by the APOGEE Stellar Parameters and
Abundances Pipeline (ASPCAP, García Pérez et al., 2016). Comparisons between
APOGEE stellar parameters and high-fidelity measurements have demonstrated the
ASPCAP-derived uncertainties to be underestimated for giants (Serenelli et
al., 2017), and dwarfs alike (Birky et al., 2020; Sarmento et al., 2021).
Pinsonneault et al. (in prep.) find temperature errors of 30 K in giants,
larger for dwarfs, and scatter in clusters of 0.05 dex in metallicity and 0.03
dex in $\alpha$ enhancement. We therefore set minimum uncertainty floors of 50
K for $T_{\mathrm{eff}}$, 0.05 dex for [M/H], and 0.03 dex for [$\alpha$/M].
While these likely still underestimate the error in the ASPCAP measurements,
they were large enough for our fitting routines to find self-consistent models
that successfully predicted other stellar parameters, e.g., luminosity or
surface gravity.

###  2.2 Gaia

We supplemented our sample with data from Gaia DR3 (Gaia Collaboration et al.,
2022), including $G$, $G_{BP}$, and $G_{RP}$ magnitudes, parallaxes, and
Renormalized Unit Weight Error (RUWE). Gaia data were available for all
targets. Computing the absolute magnitude $M_{G}$ from $G$ and parallax, we
use a photometric excess and RUWE to identify and remove likely binaries
before population analysis.

###  2.3 Photometry

There are several publicly available light curve sets, pipelines, and tools
designed and optimized for TESS data. We review some of the most widely used
in Appendix A. After trying several systematics removal pipelines and data
products, we found that all pipelines were too aggressive and removed stellar
signal. Instead, we used the apertures from two public pipelines and performed
our own minimal corrections. Due to data availability and lightweight data
products, we determined the apertures from the TESS-SPOC (Jenkins et al.,
2016; Caldwell et al., 2020) and TASOC (Handberg et al., 2021; Lund et al.,
2021) to be the best available for a rotation search at the time of writing.

TESS-SPOC provides data products for fewer stars over a longer baseline, while
TASOC provides products for a larger sample, but over a shorter baseline in
TESS year 1. The two pipelines feature different target and aperture
selections, providing two slightly overlapping stellar samples so that we can
maximize the number of rotation periods while testing for robustness of
periods against the pipelines’ different apertures. We summarize the
pipelines’ key differences in the next two sections; we then describe our
custom photometry using the pipeline apertures in Section 2.3.3. Both
pipelines’ target pixel file (TPF) and aperture data are publicly available on
MAST111TESS-SPOC data are available at 10.17909/t9-wpz1-8s54 (catalog
10.17909/t9-wpz1-8s54), while TASOC data are available at 10.17909/t9-4smn-
dx89 (catalog 10.17909/t9-4smn-dx89).

####  2.3.1 TESS-SPOC

The SPOC pipeline (Jenkins et al., 2016) was initially used to calibrate the
TESS FFIs and generate TPFs and light curves for all two-minute cadence
targets. Caldwell et al. (2020) more recently used the SPOC pipeline to create
TPFs and light curves for FFI targets, providing the TESS-SPOC light curves on
MAST.

Caldwell et al. (2020) selected a maximum of ten thousand targets per CCD from
the TIC for a maximum of 40,000 stars in the SCVZ. For each CCD, the selection
order was (1) all two-minute cadence targets; (2) potentially high-value
planet host candidates with $H$ magnitude $\leq 10$ or distance $\leq 100$ pc,
flux contamination $\leq 50\%$, and TESS magnitude $Tmag\leq 16$; (3) field
star targets brighter than $Tmag\leq 13.5$, log surface gravity $\geq 3.5$
(CGS units), and flux contamination $\leq 20\%$. The depth $Tmag\leq 13.5$ was
chosen to ensure sufficient signal-to-noise. We estimated the 6-hour CDPP of
our custom TESS-SPOC light curves to be about 4,000 ppm at $Tmag=13.5$. At
this faint limit, a 5$\sigma$ detection should vary at the 2% level. About
0.3% of Kepler rotators varied at this level (Santos et al., 2019, 2021).

TESS-SPOC computed photometric apertures using the same module as was used for
Kepler. Briefly, the module uses a synthetic FFI produced from the input
catalog and the real pixel response function to compute the optimal aperture
for each target. Caldwell et al. (2020) detail the full FFI target selection,
Jenkins et al. (2016) describe the SPOC pipeline, and Smith et al. (2016)
outline the aperture selection. The TESS-SPOC pipeline has produced TPFs,
which include target apertures, for all sectors in year 1. We queried all TPFs
available for our sample, yielding time-series images and photometric
apertures for 38,215 targets.

####  2.3.2 TASOC

TASOC has performed photometry for all stars brighter than TESS magnitude
$\leq 15$ for use in asteroseismology (Handberg et al., 2021; Lund et al.,
2021). To date, only sectors 1–6 from the first year have been processed,
yielding time-series FFI photometry with a 160-day baseline and 30-minute
cadence. While fewer sectors of data are available from TASOC, limiting us to
shorter rotation periods than TESS-SPOC, TASOC’s fainter magnitude limit and
lack of number cap (i.e., TESS-SPOC processed not more than 10,000 stars per
CCD, but TASOC has no such limit) complements the TESS-SPOC data. To compute
light curves, we downloaded the TASOC apertures and applied them to cutouts
from the calibrated FFIs.

The TASOC pipeline computed apertures for all TIC targets brighter than
$Tmag\leq 15$. Aperture selection is fully described by Handberg et al.
(2021), but uses the clustering algorithm DBSCAN (Ester et al., 1996) to find
clusters of pixels associated with TIC targets. The watershed image
segmentation routine from scikit-image (van der Walt et al., 2014) is then
used to segment apertures containing more than one target. In general, the
apertures created by the TASOC pipeline are larger than those created by TESS-
SPOC, resulting in light curves with higher photometric precision. We
estimated our custom TASOC light curves to have 6-hour CDPP of 3,000 ppm at
$Tmag=13.5$. A 5$\sigma$ detection at this magnitude will vary at the 1.5%
level. In Kepler, about 0.8% of rotating stars varied at this level (Santos et
al., 2019, 2021).

TASOC data products are also available on MAST. To obtain the likeliest
targets for detecting rotation, we queried data for TIC dwarf stars cooler
than 5,000 K, yielding FFI cutouts for 29,609 targets spanning the first 6
sectors of the TESS mission.

####  2.3.3 Custom Light Curves and Wavelet Transform

For both datasets, we began with the publicly available TPF cutouts from
calibrated FFIs. The FFI calibrations include traditional bias, dark, and flat
field corrections, cosmic ray removal, corrections for variations in pixel
sensitivity, and removal of smear signals resulting from the cameras’ lack of
shutters (Jenkins et al., 2016). After FFI calibration, both the TESS-SPOC and
TASOC pipelines perform background subtraction and systematics correction to
produce light curves; we opt not to use this next level of data correction, as
they can have the unintended consequence of removing or attenuating the
stellar signals. To mitigate the removal of stellar rotation signals, we
performed custom photometry using the apertures supplied by the pipelines. For
each available TPF, we computed light curves as follows:

  1. 1.

reject cadences with bad quality flags, which are usually associated with
cosmic rays, data downlinks, or angular momentum dumps

  2. 2.

compute a raw light curve using simple aperture photometry, adding all pixels
within the aperture

  3. 3.

remove the first three principal components of the time series using
lightkurve.RegressionCorrector

  4. 4.

reject 5$\sigma$ outliers from light curve.

Although neural networks can perform regression in spite of systematics to
some extent, _some_ systematics removal is necessary. We sought to perform as
little systematics correction as possible in order to preserve the underlying
stellar signals. Removing the first three principal components corrected the
largest TESS systematics—Earthshine and angular momentum dumps—while leaving
smaller systematics and stellar signals mostly intact. To determine the
optimal number $n_{\mathrm{pca}}$ of principal components to remove, we
removed 1, 2, 3, 4, and 5 components from a set of 10 randomly selected light
curves. We then visually inspected the resulting light curves to determine for
what value of $n_{\mathrm{pca}}$ the largest systematics were removed.
Meanwhile, removing 5$\sigma$ outliers cleaned the light curves from
systematic jumps and stellar flares. Next, we median-divided the light curves
for each target and stitched them together, linearly interpolating to fill any
gaps. Finally, we computed Morlet wavelet transforms following Claytor et al.
(2022) and binned them to $64\times 64$ pixels to be used as input to the
convolutional neural network.

####  2.3.4 Variability Amplitudes

We computed the photometric variability amplitudes $R_{\mathrm{per}}$ and
$S_{\mathrm{ph}}$ for all our stars with estimated periods. Like McQuillan et
al. (2014), to compute $R_{\mathrm{per}}$ we measured the interval between the
5th and 95th percentile of normalized flux in each period bin, then took the
median of those values. We computed $S_{\mathrm{ph}}$ as in Mathur et al.
(2023), by partitioning the light curve into segments of duration
5$P_{\mathrm{rot}}$, then taking the standard deviation of the light curve
flux over each segment. This creates a time series of standard deviations;
$S_{\mathrm{ph}}$ is taken to be the median value. For different analyses we
use either $R_{\mathrm{per}}$ or $S_{\mathrm{ph}}$, but in theory the two
metrics are related by $S_{\mathrm{ph}}\approx 0.35R_{\mathrm{per}}$222This
approximation holds true for perfect sinusoids observed for an integer number
$N$ of cycles or in the limit of large $N$. However, our measured
$S_{\mathrm{ph}}$ and $R_{\mathrm{per}}$ follow this relation remarkably
well.. We verified this relation in our measurements, so for this work we
consider the two metrics to be interchangeable.

We emphasize that due to sector-to-sector stitching and detector sensitivity
changes during momentum dumps, variability amplitudes for periods longer than
about 13 days and especially 27 days will inevitably be suppressed. To attempt
to account for this, we ran a series of noiseless, sinusoidal light curve
simulations through a renormalization and stitching algorithm and compared the
measured amplitudes to the true input amplitudes. For perfect sinusoids, we
found that the amplitude suppression factor decays exponentially with period
past 27 days. However, applying a correction to our measured amplitudes did
not affect our results; to avoid artificially injecting period–amplitude
biases we leave our reported amplitudes uncorrected.

Finally, we also measured the 6-hour combined differential photometric
precision (CDPP, Christiansen et al., 2012), which quantifies the photometric
noise, for each of our light curves. Since the CDPP is measured on timescales
shorter than the typical TESS systematics, it should be unaffected by momentum
dumps and sector-to-sector stitching.

##  3 Deep Learning Framework

To infer rotation periods from TESS light curves, we applied the method of
Claytor et al. (2022) with a few modifications. Namely, we generated new
training sets tailored to both the TESS-SPOC and TASOC samples (mostly to
represent the different light curve lengths), and we optimized different
neural networks for each data set.

###  3.1 Training Set

In Claytor et al. (2022) we trained a convolutional neural network on a set of
synthetic light curves made from physically realistic spot evolution
simulations, combined with real TESS noise from SCVZ galaxy light curves.
Inactive galaxies do not vary on yearlong timescales or shorter, and thus they
are a robust standard sample that can be useful to infer systematics. Other
quiescent objects can serve the same role, such as hot stars, which we employ
here.

Two weaknesses of our previous approach were that (1) we were not successful
in recovering periods of less than 10 days from our held-out test set, and (2)
the neural network overfit within a few (of order 10) iterations over the
training set. The first weakness was due to the choice of a loss function that
enabled the network to estimate period uncertainty. In the presence of
uncertainty, inferred periods are biased toward the mean of the distribution
and away from the upper and lower limits. The effect is most pronounced for
the top and bottom 10% of the training set period range, affecting the ranges
from 0–18 days and 162–180 days. Since the ability to estimate the period
uncertainty is a key strength of our approach, we worked around this problem
by using multiple training sets with different period ranges.

We created four separate simulated training sets using butterpy (Claytor et
al., 2021, 2022) with periods ranging from 0.1 day to 30, 60, 90, and 180
days. Having a shorter upper limit such as 30 days allows us to more
successfully probe the short-period range—here only 0–3 days and 27–30 days
are severely affected—while having multiple training sets with increasing
upper limits gives us multiple period estimates that we can mutually compare
for extra tests of reliability. The distributions of all simulation input
parameters besides period were the same as in Claytor et al. (2022) (the
simulations for the 180 day upper limit _are_ the same as in the previous
work), and the same simulations were used for both the TESS-SPOC and TASOC
training sets. The only other difference was the source of the light curves
combined with the simulations to emulate instrumental noise and systematics.
We note that using multiple training realizations yields multiple period
estimates for the same star; we discuss the breaking of ties in Section 5.

The second shortcoming was simply due to the small number ($\sim 2,000$) of
galaxy light curve examples. If there are too few examples of noise in the
training set, the neural network learns the noise quickly and overfits the
data. Since there are many more bright stars than galaxies in TESS, we
addressed this by combining our simulations with light curves of stars in
temperature ranges that should be comparatively quiescent to emulate TESS
noise. McQuillan et al. (2014) detected periods in Kepler stars hotter than
the Sun half as often as in cooler stars. Given TESS’s slightly worse
photometric precision and redder pass band than Kepler, we expect TESS stars
hotter than the Sun to be even harder to detect in rotation. This makes stars
in the temperature range above $\sim 5,800$ K ideal for use as quiescent light
curve sources. At first we queried TPFs and computed light curves for TASOC
stars in the range 5,800 K $\leq T_{\mathrm{eff}}\leq$ 6,000 K. We kept light
curves with at least 4 sectors to allow for gaps in the data while ensuring
that there were data for more than half the time baseline. This yielded a set
of 23,332 TASOC noise templates, an order of magnitude more than the number of
galaxy sources used in the previous exercise. The same range of temperatures
in TESS-SPOC, requiring that light curves have at least 7 sectors to cover
more than half of the time baseline, has only 6,000 targets, so a larger
temperature range was required. We used the range 6,000 K $\leq
T_{\mathrm{eff}}\leq$ 8,000 K, which contained 17,637 sources. Table 2 details
the noise light curves samples that make up the TESS-SPOC and TASOC training
sets.

Table 2: TESS-SPOC and TASOC Noise Light Curves for Training Set | TESS-SPOC | TASOC
---|---|---
Sectors included | 1–13 | 1–6
Minimum number of sectors | 7 | 4
Time baseline (days) | 350 | 160
$T_{\mathrm{eff}}$ range of sources (K) | 6,000–8,000 | 5,800–6,000
Number of sources | 17,637 | 23,332

We note that the temperature range for the TESS-SPOC noise light curves
overlaps with the $\delta$ Scuti instability strip (e.g., Murphy et al., 2019)
and with the $\gamma$ Doradus strip (e.g., Balona et al., 2011; Bradley et
al., 2015). Of our TESS-SPOC noise targets, 1,724 ($\sim$10%) fall within the
$\delta$ Scuti strip, and depending on the criterion used, as few as 30%
(Balona et al., 2011) and as many as two-thirds (Bradley et al., 2015) are
within the $\gamma$ Dor strip. Because $\delta$ Scuti stars pulsate with
periods on the order of hours and $\gamma$ Dor less than about 3 days, we do
not expect significant contamination from pulsation in our training light
curves. The TASOC noise sample does not overlap with either instability strip.
The presence of contaminants in the training set should make the CNN more
robust against contamination (i.e., misidentifying a pulsator as a rotator),
but thoroughly testing this is beyond the scope of this work.

###  3.2 Convolutional Neural Network

We began with the same CNN as in Claytor et al. (2022), which uses the Adam
optimizer (Kingma & Ba, 2014) and negative log-Laplacian loss, enabling the
estimation of uncertainty along with the rotation period. The loss function
has the form

$\mathcal{L}=\ln\left(2b\right)+\frac{|P_{\mathrm{true}}-P_{\mathrm{pred}}|}{b},$
(1)

where $b$, the median absolute deviation, is taken to represent the
uncertainty.

We experimented with different architectures to optimize different networks to
the TESS-SPOC and TASOC training sets. The original architecture had three
convolution layers with (A) 8, 16, and 32 kernels, respectively, but we also
tried (B) 16, 32, and 64 kernels; (C) 32, 64, and 128; and (D) 64, 128, and
256. More kernels or filters per layer allow the network to learn more
features if they are present in the data, but they may also cause the network
to overfit the data faster. We trained each architecture individually on each
training set and chose the architecture with the best overall recovery on a
held-out test sample. For the TESS-SPOC set, architecture C performed best
overall, but architecture A was optimal for the TASOC set. We discuss the
details of architecture optimization in Appendix B.

##  4 Rotational Modeling

With newly obtained TESS rotation periods, we will be able to look for trends
of rotation detectability and variability across fundamental stellar
parameters. Stars spin down and become less active as they age (Skumanich,
1972), so we expect both detectability and variability to decrease with age.
We also know activity to vary with Rossby number, the ratio of rotation period
to the convective overturn timescale (e.g., Noyes et al., 1984; Wright et al.,
2011). To validate these relationships and look for potential departures from
expected behavior in TESS, we will need to derive ages and Rossby numbers for
our sample. We employ the stellar evolution and rotational modeling using
kiauhoku (Claytor et al., 2020a, b) to infer ages, masses, convective
timescales, and Rossby numbers for our stars with rotation periods and APOGEE
spectroscopy.

Table 3: Input Physics to Stellar Evolution Models Parameter | Value/Source
---|---
Atmosphere | Castelli & Kurucz (2004)
Convective overshoot | False
Diffusion | True
Equation of state | OPAL (Rogers & Nayfonov, 2002)
High-temperature opacities | OP (Mendoza et al., 2007)
Low-temperature opacities | Ferguson et al. (2005)
Mixing length $\alpha$ | 1.86
Mixture and solar $Z/X$ | Grevesse & Sauval (1998)
Nuclear reaction rates | Adelberger et al. (2011)
Solar $X$ | 0.7089
Solar $Y$ | 0.2728
Solar $Z$ | 0.0183
$\Delta Y/\Delta Z$ | 1.4
Surface $(Z/X)_{\odot}$ | 0.02289
Angular momentum evolution | van Saders et al. (2016)
Initial rotation period | 8.134 d
Critical Rossby number | 2.16
Critical $\omega$ for saturation | 3.394$\times 10^{-5}$ s-1
$f_{k}$ | 6.575
Disk coupling timescale | 0.281 Myr

The stellar evolution tracks were generated using the non-rotating version of
the Yale Rotating Stellar Evolution Code (YREC, Demarque et al., 2008), then
global stellar properties were used to calculate angular momentum evolution
following the magnetic braking law of van Saders et al. (2016). The models are
fully described by Claytor et al. (2020a), but we list the input physics and
solar calibration here in Table 3. The angular momentum evolution includes
weakened magnetic braking beginning about halfway through the main sequence
(van Saders et al., 2016), but does not include core-envelope decoupling
(e.g., Spada & Lanzafame, 2020) or the apparent stalling of spin-down that
appears to occur in young, cool stars (Curtis et al., 2019).

Using the Markov-chain Monte Carlo (MCMC) tools in kiauhoku, we interpolated
and fit stellar evolution models to observational data. For the MCMC we used a
$\chi^{2}$ log-likelihood of the form

$\mathcal{L}_{\chi^{2}}=-\frac{1}{2}\sum_{i}\frac{\left(x_{i}-x_{i}^{\prime}\right)^{2}}{\sigma_{x_{i}}^{2}},$

where $x_{i}$ and $\sigma_{x_{i}}$ are the observational input parameters and
uncertainties, respectively, $x_{i}^{\prime}$ is the computed value from the
model, and $i$ iterates over the input parameters. The observables used in
this computation were the CNN-inferred rotation periods, APOGEE calibrated
temperatures, metallicities ([M/H]) and $\alpha$-element abundances
([$\alpha$/M]). All MCMC input data are provided with uncertainties in Table
4.

##  5 Rotation Periods of TESS Stars

We estimated periods for TESS-SPOC targets with at least 7 sectors and TASOC
targets with at least 4 sectors, the same minimum numbers of sectors as for
the training light curves. To determine reliability we followed the method of
Claytor et al. (2022) and used a cut in fractional uncertainty to denote
reliability. We do not treat the estimated $\sigma_{P}$ ($=b$ in Eq. 1) as a
formal uncertainty. Rather, the quantity $\sigma_{P}/P$ serves as a metric of
relative credibility of period estimates. $\sigma_{P}/P\leq 35\%$ translated
to $\sim$10% median percent error in the training set recovery, so we adopt
this uncertainty cut as our baseline for reliability. Since there are four
neural networks, each with its own period range, we obtained four sets of
period candidates for both the TESS-SPOC and TASOC data sets. If two or more
neural networks yielded estimates passing our reliability cut for the same
star, we averaged the estimates and added their standard deviation in
quadrature to the uncertainty. If the newly combined fractional uncertainty
was larger than 35%, we discarded the star.

Table 4: Properties of 7,971 Rotationally Detected TESS SCVZ Stars, MCMC Input
& Fit Parameters

Label | Description
---|---
TIC | TESS Input Catalog ID
prot | CNN-inferred rotation period
e_prot | rotation period uncertainty
prov | period provenance: TESS-SPOC or TASOC
rper | photometric activity range $R_{\mathrm{per}}$
sph | photometric activity index $S_{\mathrm{ph}}$
cdpp | combined differential photometric precision
Tmag | TESS magnitude
contratio | TIC flux contamination ratio
instability | instability strip flag
parallax |  Gaia DR3 parallax
ruwe |  Gaia DR3 renormalized unit weight error
phot_g_mean_mag |  Gaia DR3 apparent $G$ magnitude
bp_rp |  Gaia DR3 $G_{BP}-G_{RP}$ color index
teff | APOGEE DR17 effective temperature
teff_err | temperature uncertainty
m_h | APOGEE DR17 metallicity [M/H]
m_h_err | metallicity uncertainty
alpha_m | APOGEE DR17 $\alpha$ enhancement [$\alpha$/M]
alpha_m_err |  $\alpha$ enhancement uncertainty
snr_bad | APOGEE DR17 spectral signal-to-noise flag
fspot | spot filling fraction
age | MCMC gyrochronological age
e_age+ | 1$\sigma$ age upper credible limit
e_age- | 1$\sigma$ age lower credible limit
mass | MCMC-inferred stellar mass
e_mass+ | 1$\sigma$ mass upper credible limit
e_mass- | 1$\sigma$ mass lower credible limit
rad | MCMC-inferred stellar radius
e_rad+ | 1$\sigma$ radius upper credible limit
e_rad- | 1$\sigma$ radius lower credible limit
Ro | MCMC-inferred Rossby number
fconv | MCMC convergence flag

Note. — The “snr_bad” flag represents the APOGEE spectral signal-to-noise flag
and is set for only 21 stars. The “instability” flag marks stars whose TIC
temperatures and luminosities place them in the instability strip. It is set
to 1 for 98 stars within the $\delta$ Scuti instability strip characterized by
Murphy et al. (2019), 2 for 105 stars in the $\gamma$ Doradus strip of Balona
et al. (2011), and 3 for 243 stars in the $\gamma$ Dor strip of Bradley et al.
(2015). This table is available in its entirety in machine-readable format.

We obtained 4,853 TESS-SPOC stars with reliable periods and 3,545 reliable
TASOC periods. These combine for a total of 7,971 unique targets, 427 of which
overlap between the two samples. We discuss the overlap sample in Section 5.3.
The rotation periods up to 80 days, their photometric amplitudes, selected
spectroscopic parameters, and associated flags are presented in Table 4. We
also list the stellar parameters for the 53,141 rotationally non-detected
stars in Table 5. For stars with periods from both TESS-SPOC and TASOC data,
we favored the TESS-SPOC period in the final table due to the light curves
having twice the duration as the TASOC light curves. We note that while the
CNN estimated periods longer than 80 days that passed the uncertainty cut,
this regime is highly contaminated by obviously spurious detections. We leave
the vetting of periods longer than 80 days to future work, and for now we
consider only shorter periods to be reliable. Figure 1 shows a small selection
of light curves for which we obtained periods. The periods are plotted against
TIC effective temperature in Figure 2 to illustrate, for the first time, the
distribution of main sequence stellar rotation periods longer than 13 days in
TESS.

Table 5: Properties of 53,141 Rotationally Nondetected TESS SCVZ Stars

Label | Description
---|---
TIC | TESS Input Catalog ID
prov | period provenance: TESS-SPOC or TASOC
rvar | photometric activity range $R_{\mathrm{var}}$
cdpp | combined differential photometric precision
Tmag | TESS magnitude
contratio | TIC flux contamination ratio
instability | instability strip flag
parallax |  Gaia DR3 parallax
ruwe |  Gaia DR3 renormalized unit weight error
phot_g_mean_mag |  Gaia DR3 apparent $G$ magnitude
bp_rp |  Gaia DR3 $G_{BP}-G_{RP}$ color index
teff | APOGEE DR17 effective temperature
teff_err | temperature uncertainty
m_h | APOGEE DR17 metallicity [M/H]
m_h_err | metallicity uncertainty
alpha_m | APOGEE DR17 $\alpha$ enhancement [$\alpha$/M]
alpha_m_err |  $\alpha$ enhancement uncertainty
star_bad | APOGEE DR17 stellar parameter fit flag
snr_bad | APOGEE DR17 spectral signal-to-noise flag

Note. — The “star_bad” flag represents the APOGEE stellar parameter fit flag,
set when a best-fit model is close to a grid edge. It is set for 772 stars.
The “snr_bad” flag represents the APOGEE spectral signal-to-noise flag and is
set for only 123 stars. The “instability” flag marks stars whose TIC
temperatures and luminosities place them in the instability strip. It is set
to 1 for 1,997 stars within the $\delta$ Scuti instability strip characterized
by Murphy et al. (2019), 2 for 3,412 stars in the $\gamma$ Doradus strip of
Balona et al. (2011), and 3 for 6,161 stars in the $\gamma$ Dor strip of
Bradley et al. (2015). This table is available in its entirety in machine-
readable format.

Figure 1: A selection of light curves for which we successfully estimated
rotation periods using the neural network. Top: an M dwarf with strong,
coherent spot modulation. Middle: a slowly-rotating M dwarf with several
missing sectors. Bottom: a rapidly rotating G dwarf. Figure 2:
Period–temperature distribution for the TESS-SPOC (left) and TASOC (right)
samples. Estimating periods using CNNs we recover the short-period slope and
intermediate-period gap seen in other rotating populations (e.g., McQuillan et
al., 2014). Detection biases create a sharp edge at 27 days, above which the
period uncertainties are larger due to the sector-to-sector stitching
necessary for long-baseline TESS light curves. The vertical edge at 6,000 K
occurs at the lower temperature edge of the training set. The drop in period
detections for hotter stars may be due to conflicts with the training set or a
true drop in amplitudes. Toward $P_{\mathrm{rot}}=90$ days, contamination
increases because less certain period estimates are biased toward the median
period of the training set. The detections above $P_{\mathrm{rot}}>30$ d and
$T_{\mathrm{eff}}>5,000$ K are likely to be spurious, but most of the M dwarf
periods up to 80 days appear to be real.

###  5.1 Features of the Period Distribution: Biases

Since the TESS-SPOC sample spans a wider range in temperature than our TASOC
sample, we will focus our main discussion of the period distribution on the
TESS-SPOC sample. In this period distribution there are two apparent edges
worth noting. First, there is temperature edge at 6,000 K. The underlying
sample distribution has no such edge, so it must be produced by the period
search. 6,000 K is the lower bound of the noise source sample used for the
TESS-SPOC training set, so above this temperature there is some overlap
between the training light curves and the “real” data. It is possible that
inclusion in the training set as a noise template (multiple instances with
varying injected simulated rotation signals) confuses the neural network and
causes it to assign a large uncertainty to these targets. Another possibility
is that spot modulation amplitudes drop above 6,000 K, where the convective
envelope disappears and stars become less active. This drop in amplitude is
seen in the Kepler stars of Fig. 3. The drop in detections above 6,000 K is
likely a combination of these effects.

The other edge is in rotation period and occurs at roughly 27 days. While slow
rotators tend to be less active than fast rotators at fixed temperature, the
spot modulation amplitudes at which we expect to lose detections vary in
period across temperature. In other words, a period detection edge produced by
astrophysical variability should not be flat. Rather, the 27-day detection
edge is likely related to the 27-day sector length in TESS. Without a reliable
absolute flux calibration in each sector, stitching sector light curves
together can destroy coherent signals longer than 27 days in period. While we
include sector-to-sector stitching in all our training sets, the 27-day edge
suggests that the training sets do not fully capture the sector effects in
TESS, or at the very least the sector effects make period estimates much less
certain beyond 27 days.

###  5.2 Features of the Period Distribution: Populations

The period–temperature distribution displays a sloped, short-period edge,
similar to what was seen in Kepler (McQuillan et al., 2014; Santos et al.,
2019, 2021). This edge represents the point at which field stars converge onto
the slowly-rotating sequence (Curtis et al., 2020).

The distribution also displays a gap in rotation period, occurring at roughly
12 days at 5,000 K and increasing to 20 days at 4,000 K. McQuillan et al.
(2013) first identified this gap in the Kepler field, and it has also been
recovered in other field star samples using K2 (Reinhold & Hekker, 2020;
Gordon et al., 2021). Lu et al. (2022) showed that the gap may close in fully
convective star samples. We present here the first detection of the rotation
period gap using TESS.

Figure 3: The rotation period distribution versus temperature for both the
Kepler field (left, Santos et al., 2019, 2021) and TESS SCVZ (right, this
work). The bins are colored by photometric variability index
$S_{\mathrm{ph}}$. As expected, amplitudes generally decrease with increasing
period at fixed temperature. Interestingly, the amplitudes near the rotation
gap are smaller than those away from the gap in the same temperature and
period ranges, in agreement with Kepler as shown left, and with K2 Reinhold et
al. (2019); Reinhold & Hekker (2020). The TESS bins above a temperature of
5,000 K and above a period of 30 days nearly all have only a single star; this
region is sparsely populated, and most periods here are likely to be spurious
detections.

Figure 3 shows another look at the rotation period distribution, now colored
by the photometric variability amplitude $S_{\mathrm{ph}}$, in comparison with
the distribution from the Kepler field (Santos et al., 2019, 2021). As we
expect, stellar variability generally decreases with increasing periods at
fixed temperature, since slowly rotating stars are less magnetically active
than faster stars. There is a significant dip in the variability between 3,500
K and 4,500 K, most notably near the location of the rotation period gap,
which goes from about (5,000 K, 12 d) and curves upward to (4,000 K, $\sim$20
d) (refer to Figure 2). This is consistent with Reinhold et al. (2019);
Reinhold & Hekker (2020), who found a similar dip in variability near the
period gap in Kepler and K2 stars. They argued that the dip in variability
causes the apparent gap in rotation periods, where stars in the gap exhibit
modulation too small to be detected in rotation.

Figure 4 shows the TESS-SPOC period–temperature distribution using different
variability range $R_{\mathrm{per}}$ floor values. Requiring
$\log(R_{\mathrm{per}}/\mathrm{ppm})>3.5$ removes many stars from the top-left
corner of the diagram, which are hot but have apparently long rotation
periods. While we do not expect to find many stars in this regime based on
Galactic population synthesis (e.g., van Saders et al., 2019), the stars that
are here should have low variability because they are hot and therefore have
thin-to-nonexistent outer convective envelopes, and because they spin
relatively slowly. The stars that are lost from the top panel to the middle
panel of Figure 4 are likely mostly spurious detections whose measured
$R_{\mathrm{per}}$ is actually the photometric noise, as well as a handful of
correctly measured, low-variability stars. As we continue to increase the
$R_{\mathrm{per}}$ floor, we see two effects. First, we lose more low-
variability stars on the hot, long-period edge. This is precisely what we
expect to see in a period sample: raising the variability floor, we should
lose the highest-Rossby number stars first. These are the slowly rotating, hot
stars in the top left “corner” of the distribution. Second, the gap becomes
more apparent, consistent with Reinhold & Hekker (2020), although stars are
not lost from the gap at a significantly higher rate than stars outside the
gap.

Figure 4: Distribution of periods and TIC temperatures for the TESS-SPOC
sample, with varying lower bound on photometric variability range
$R_{\mathrm{per}}$. All stars shown pass our reliability criterion
($\sigma_{P}/P\leq 35\%$).

###  5.3 Comparison between TESS-SPOC and TASOC

In the TASOC sample (e.g., in the right panel of Figure 2), we again see a
weak presence of the period gap as well as the sloped short-period edge. the
TASOC sample also show the 27-day horizontal detection edge exhibited by the
TESS-SPOC sample, resulting from the increase in uncertainty past 27 days from
sector-to-sector stitching.

There are 427 stars in common between the TASOC and TESS-SPOC samples. We
estimated two periods for each of these stars using different neural networks
fit to different training sets tailored to the different light curve lengths.
While the underlying pixel data between the two samples were the same, the
apertures used to perform photometry were different, and the TESS-SPOC light
curves were more than twice as long (13 sectors) as the TASOC light curves (6
sectors). In addition, the two training sets used different underlying samples
of stars for noise and systematics. This gives us a sample to compare period
estimates for robustness against photometric aperture, training set, and
duration of observation.

Figure 5: Period comparisons for the 427 stars in both the TESS-SPOC and TASOC
samples. The solid black line represents perfect agreement, while the dashed
red lines are $\pm 50\%$ error. The black dotted lines are at 27 days on
either axis, showing the TESS sector length. There is generally good agreement
for most stars, with median percent error of 7%, and most of the disagreeing
estimates have relatively large uncertainty. We note that while we do not
include periods greater than 80 days in our analysis or table, we show them
here to illustrate at what periods the agreement worsens.

In Figure 5 we compare the period estimates for the overlap sample. They
mostly agree, with a median relative error of 7%. The estimates that disagree
have relatively large uncertainties, though the fact that they make our 35%
reliability cut means that there will be some contamination in our period
sample. 76% of stars in the overlap sample have period estimates agreeing to
within 20%. The discrepancies likely arise from the different aperture
selection, different light curve durations, or differences in the underlying
training sets, although here we do not attempt to isolate the main
contributor.

###  5.4 Comparison with other large field rotation samples

The TESS rotation period distribution is the product of the underlying
distribution of periods, the presence of modulation in the light curve, the
availability of data products, and the ability to detect periods across
various stellar parameters. To try and understand the relative influence of
these effects, we compare the TESS period distribution with other large period
data sets, particularly Kepler and K2. Figure 6 shows the period distributions
from Kepler and K2, while Figure 7 shows our newly obtained TESS distribution.
We represent temperature bins as vertical histograms in the style of McQuillan
et al. (2014) to increase the clarity of the period gap in the cool-
temperature regime. The number of temperature and period bins is adjusted in
each panel to account for the total number of stars in each sample.

Figure 6: Histogram representations for the period–temperature distribution of
Kepler (Santos et al., 2019, 2021, top) and K2 (Reinhold & Hekker, 2020,
bottom).

Figure 7: Histogram representations for the period–temperature distribution of
TESS, with no photometric variability cut (top), and restricting to
$\log(R_{\mathrm{per}}/\mathrm{ppm})>3.5$ (bottom). Temperatures are from the
TIC.

The top panel of Figure 6 displays 52,338 carefully vetted Kepler rotation
periods from Santos et al. (2021). The Kepler period distribution exhibits a
pileup on its upper edge for stars hotter than $\sim 5,500$ K, which is a
prediction of the weakened magnetic braking hypothesis (van Saders et al.,
2016, 2019) and has been well-studied in the Kepler field (Hall et al., 2021;
Masuda et al., 2022; David et al., 2022). Also present is the rotation period
gap, clearly visible at $\sim 15$ days at 5,000 K, $\sim 17$ days at 4,500 K,
and continuing to increase and widen at cooler temperatures.

The bottom panel of Figure 6 shows 13,847 rotation periods from stars in K2
measured by Reinhold & Hekker (2020). These represent a high-fidelity
subsample with normalized Lomb-Scargle peak height $>0.5$ and variability
range $R_{\mathrm{var}}>0.1\%$. Peak heights range from 0 to 1 and quantify
how sinusoidal a light curve is, with a perfect sinusoid returning unit peak
height, and noisy, non-sinusoidal data returning values close to zero.
$R_{\mathrm{var}}$ is defined similarly to $R_{\mathrm{per}}$, except that
$R_{\mathrm{var}}$ is the variability range over the entire light curve,
rather than a median of ranges per period bin. The K2 distribution shows the
period gap most strongly between 5,000 K and about 4,250 K, but it is weakly
visible in cooler stars, where it appears to increase in period and widen as
in Kepler. The hot star pileup is not apparent here. This is likely due to the
relatively large temperature uncertainty in the K2 Ecliptic Plane Input
Catalog (median 140 K, Huber et al., 2016), which blurs out features in the
temperature distribution. (van Saders et al., 2019; David et al., 2022).
Finally, periods longer than about 35 days are largely absent from the K2
distribution because of K2’s 80-day observing campaigns in each field.

The TESS distribution in the top panel of Figure 7 shows periods for 5,056
TESS-SPOC stars with $\sigma_{P}/P\leq 35\%$, representing the most credible
detections. The period gap is present and is most apparent at temperatures
between 4,000 and 5,000 K. It is still visible at temperatures cooler than
4,000 K, but the dearth of reliable detections at periods nearing 30 days
makes the gap more difficult to detect. In addition to the gap, we detect a
handful of M-dwarfs rotating with periods between 40 and 60 days; similar
stars were also observed in the Kepler period distribution. We visually
inspected the light curves for these stars and confirmed them to be true
rotation detections with photometric variability $R_{\mathrm{per}}$
approaching 1%. On the hot end, the distribution lacks the long-period edge
seen in Kepler because of the abundance of hot stars apparently rotating with
$\sim$20-day periods. These are likely spurious detections, as their measured
amplitudes are close to the noise floor (close to 100 ppm for $Tmag=8$ and 1%
for $Tmag=15$). When we raise the variability floor in the bottom panel of
Figure 7, the hot, slow rotators mostly disappear, but the gap and the slowly
rotating M-dwarfs remain.

Figure 8: The Kepler period–temperature distribution from Santos et al. (2019,
2021) (black points) with our new TESS rotation periods overplotted (red
points). TESS temperatures are from the TIC.

We offer one final view of the TESS period distribution, now plotted over the
Kepler distribution of Santos et al. (2019, 2021), in Figure 8. The short-
period edge of the TESS distribution has the same location and slope of
Kepler’s, suggesting that the edge is a result of rotational evolution, rather
than arising from details of the star formation history (Davenport, 2017;
Curtis et al., 2020). The rotation period gap agrees as well, following
Kepler’s for as long as the TESS gap remains visible into the hot stars. TESS
appears to see stars in regions Kepler does not: the slowly rotating, hot
stars ($T_{\mathrm{eff}}>5000$ K and $P_{\mathrm{rot}}>30$ d) have amplitudes
close to the noise floor and are likely spurious detections. On the other
hand, the slowly rotating M dwarfs, with TESS periods up to 80 days, have been
vetted by eye and are mostly real rotation detections. Interestingly, the
branch of stars beneath the period gap turns over at temperatures below 3,500
K, which is not seen in Kepler but is seen in some younger samples observed by
K2 and MEarth (Reinhold & Hekker, 2020; Newton et al., 2016, 2018).

###  5.5 Modeling Results

Taking the stars with reliable periods from either TESS-SPOC or TASOC, we
cross matched with APOGEE DR17 spectroscopic parameters estimated with ASPCAP
(García Pérez et al., 2016). To ensure a high quality sample, we removed
objects with the ASPCAP STAR_BAD and SNR_BAD flags set. We also checked the
MULTIPLE_SUSPECT flag for possible double-lined spectroscopic binaries, but
none of our APOGEE rotators were flagged. Some stars in our sample had
multiple visits and therefore multiple ASPCAP measurements. For targets with
multiple measurements, we averaged the temperatures, metallicities, and
$\alpha$ abundances, then added the standard deviation of those measurements
in quadrature with the formal ASPCAP uncertainties to obtain an uncertainty
for each measurement. This affected 201 targets out of 2,654 with APOGEE
spectroscopy. We then filtered out targets with large Renormalized Unit Weight
Error (RUWE $>$ 1.2, Gaia Collaboration et al., 2022) and high flux
contamination ratio (TIC contratio $>$ 10%) to clean the sample of potential
binary or nearby flux contaminants. This yielded a sample of 1,227 stars,
which we designate as our “Gold” sample. We fit models to these stars
according to the procedure in Section 4, taking the posterior medians as the
nominal fit parameters. The fit parameters and their uncertainties are
presented in Table 4.

Figure 9: The distribution of rotation-based stellar ages in the TESS SCVZ
(the Gold sample in Table 1). The stars shown span a temperature range of
3400–6400 K. We separate stars with periods less than 10 days, which are more
likely to be tidally synchronized binaries than true rapid rotators (Simonian
et al., 2019).

####  5.5.1 The TESS SCVZ age distribution

The ages for our stars, which are estimated using our TESS rotation periods,
are shown in Figure 9. We separate stars with rotation periods less than 10
days, which in Kepler were more likely to be spun-up by close binary
companions than be true rapid rotators Simonian et al. (2019). The age
distribution peaks between 2 and 4 Gyr, which is consistent with other age
distributions of Solar neighborhood stars: Buder et al. (2019) used isochrones
for GALAH stars, Claytor et al. (2020a) used rotation-based ages for Kepler
dwarfs, Berger et al. (2020) used isochrones for Kepler dwarfs, Silva Aguirre
et al. (2018) used asteroseismology for Kepler giants, and Mackereth et al.
(2021) used seismology in the TESS CVZs; all obtained a distribution peaking
between 2 and 4 Gyr. We note that our age distribution lacks many of the old
($>6$ Gyr) stars seen in other samples. This is a consequence of two detection
biases: (1) our 27-day detection edge prevents the reliable detection of stars
hotter than 4,000 K, and (2) old stars rotate more slowly and are less active,
further complicating their detection in rotation.

Figure 10: Stellar $\alpha$-element abundances vs. age in the TESS SCVZ (the
Gold sample in Table 1), tracing Galactic chemical evolution of $\alpha$
enhancement from supernovae. Consistent with trends seen in the Kepler field
(e.g., Silva Aguirre et al., 2018; Claytor et al., 2020a), there is a weak
trend of increasing $\alpha$ abundance with age below 8 Gyr associated with
the classical “thin” Galactic disk. We expect but do not see a strong trend of
increasing $\alpha$ abundance with age above 8 Gyr (the classical “thick”
disk), but this because we detect very few older stars due to (1) our 27-day
detection edge, and (2) slow rotation and weak activity making stars more
difficult to detect.

####  5.5.2 Galactic chemical evolution

With rotation-based ages and high-resolution APOGEE spectroscopic abundances,
we can also look for Galactic chemical evolution trends in TESS (e.g., Silva
Aguirre et al., 2018; Claytor et al., 2020a). Stars’ initial composition
patterns are set by the compositions of the clouds in which they form, which
are in turn enriched by stars that have lived and died before them. Galactic
chemical evolution is often traced using the relative abundances of $\alpha$
elements (e.g., O, M, Ca) to metals and metals to hydrogen (Bensby et al.,
2014). APOGEE provides values for [$\alpha$/M] and [M/H], which we adopt. The
background Galactic composition was governed by the dominance of core-collapse
supernovae in the early Milky Way, followed by dominance of type Ia supernovae
beginning about 8 Gyr ago (Feltzing et al., 2017). Both types of supernovae
enriched the interstellar medium with metals, but in different ratios.
Consequently, stars display decreasing [$\alpha$/M] and increasing [M/H] with
time (reversed as a function of age). We therefore expect old stars to have
low metallicity but high $\alpha$ enhancement, while young stars should have
higher metallicity and lower $\alpha$-element abundances (Haywood et al.,
2013; Bensby et al., 2014; Martig et al., 2016; Feltzing et al., 2017; Buder
et al., 2019). These young and old populations are representative of the
classical Galactic “thin” and “thick” disks, respectively.

Figure 10 shows stellar $\alpha$-element abundance as a function of rotation-
based age in the TESS SCVZ. As expected, young stars are generally
$\alpha$-poor and metal-rich. There is a slight increasing trend of $\alpha$
enhancement with age. We detect very few stars in rotation older than 6 Gyr
due to the detection biases discussed in Section 5.5.1. Finally, we also
detect a few young $\alpha$-rich stars. These are known from other samples
(e.g., Martig et al., 2015; Silva Aguirre et al., 2018; Claytor et al., 2020a)
and are likely to be the products of stellar mergers. In this scenario, two
old, $\alpha$-enhanced stars merge, destroying the stars’ rotation histories
and yielding a fast-rotating, apparently young, $\alpha$-enhanced product
(Zhang et al., 2021).

####  5.5.3 Stellar activity

Finally, with rotationally characterized stars we can begin to investigate
trends of photometric activity with model-derived parameters like age and
Rossby number. We define the Rossby number as the ratio of the rotation period
over the convective overturn timescale $\tau_{\mathrm{cz}}$, where the
convective timescale is computed from our models as the pressure scale height
$H_{P}$ divided by the convective velocity evaluated at a distance $H_{P}$
above the base of the convection zone. To quantify the photometric activity,
we use the photometric activity index $S_{\mathrm{ph}}$ rather than
$R_{\mathrm{per}}$ so that we can compare the trends in the TESS SCVZ with
those in the Kepler field observed by Santos et al. (2019, 2021), and Mathur
et al. (2023). The ages and and Rossby numbers for the Mathur et al. sample
were computed using the same procedure and models underlying this work, so we
can directly compare the Kepler and TESS distributions. We start with the Gold
sample and discard stars with periods less than 10 days as before, leaving
1,065 stars with TESS periods, APOGEE spectroscopic parameters, and well-
determined Rossby numbers and ages.

Figure 11: The stellar activity index $S_{\mathrm{ph}}$ versus Rossby number
for 1,065 stars in our Gold sample with periods greater than 10 days, plotted
over the Kepler sample of Santos et al. (2019, 2021) and Mathur et al. (2023).
As expected from theory and as seen in the Kepler field, photometric activity
decreases with increasing Rossby number. The gray region denotes our measured
TESS noise floor, ranging from 250 ppm at 8th magnitude to 1,100 ppm at 13th
magnitude. We do not detect TESS stars with amplitudes as low as Kepler due to
TESS’s worse photometric precision and therefore higher noise floor. Figure
12: The stellar activity index $S_{\mathrm{ph}}$ versus rotation-based age for
1,065 stars in our Gold sample with periods greater than 10 days, plotted over
the Kepler sample of Santos et al. (2019, 2021) and Mathur et al. (2023).
Photometric activity decreases as stars age, an effect of spin-down seen in
the Kepler field and predicted by theory.

Figure 11 shows the photometric activity index $S_{\mathrm{ph}}$ versus the
Rossby number for our binary-cleaned stars, plotted over the distribution of
stars from Kepler. Activity decreases with increasing Rossby number, as
expected. The TESS distribution generally agrees with the Kepler distribution.
We have a few stars close to the high-activity saturated regime (e.g., Wright
et al., 2011), but most of our stars are magnetically unsaturated. The TESS
detection limits are clear here, as our lowest-activity star with a period
detection has $S_{\mathrm{ph}}=345$ ppm, compared to Kepler’s lower limit in
the tens of ppm. We have a few hot stars at Ro $\gtrsim 2$ where Kepler has
almost none. These are the likely spurious period detections from before
(e.g., Figure 2).

We show $S_{\mathrm{ph}}$ as a function of rotation-based age in Figure 12.
Photometric activity decreases with age, an effect of stellar spin-down. The
TESS distribution follows the range and morphology of the Kepler distribution
all the way down to the TESS rotation detection limit of
$S_{\mathrm{ph}}\approx 350$ ppm.

##  6 Detectability of Rotation

Here we consider the detectability of rotation as a function of fundamental
stellar parameters. At a fixed rotation period, at lower temperature and
higher metallicity, we expect deeper convective envelopes, stronger magnetism,
more surface spots, and more easily detectable rotational modulation. Besides
changing with static stellar parameters, the strength of a star’s magnetism
changes as the star ages. Main-sequence stars with outer convective envelopes
are born spinning fast and spin down as they age as they lose angular momentum
to magnetized stellar winds (Kraft, 1967; Skumanich, 1972). The decrease in
rotation speed results in a weakening magnetic field, fewer or smaller spots,
and less flux modulation, making rotation more difficult to detect in older
stars than in younger stars at fixed mass and composition.

While we might expect rotation to be harder to detect in lower metallicity
stars, an age dependence enters the picture because of the variation of
Galactic composition with age. Because the background abundance ratios change
with time, any apparent changes in rotation detectability with stellar
abundances may actually be caused by the decreasing detectability with age.

To investigate the detectability of rotation with fundamental stellar
properties, we consider the fraction of targets for which we detected periods
in stellar parameter bins. While the CNN infers a period for each target, we
can use the estimated uncertainty to determine whether those periods are
reliable. As in Claytor et al. (2022), we label targets with
$\sigma_{P}/P<0.35$ (corresponding to $\sim$10% median percent error) as
successful detections, and anything else as a nondetection.

Figure 13: The detectability of rotation across APOGEE temperature and
metallicity. We preferentially detect rotation in cool stars, which have
deeper convective envelopes and are more active. When the detectability drops
off as a function of temperature, we see a weak trend of increasing
detectability with increasing metallicity. This may be due to metallicity
increasing the convective depth at fixed temperature. It may also be an age
effect, as young, active stars tend to be more metal-rich.

Figure 13 shows the rotation detection fraction versus temperature and
metallicity for all our rotationally-searched stars with APOGEE spectroscopy.
Only bins with at least five targets are shown so that the diagram is not
muddled by small number fluctuations. As expected, cooler stars, especially
cooler than 5,000 K, are detected in period significantly more often than
hotter stars. In the range $5,000~{}\mathrm{K}<T_{\mathrm{eff}}\lesssim
6,000~{}\mathrm{K}$, where the detections begin to decrease as a function of
temperature, there appears to be a weak trend in metallicity, with higher-
metallicity ([M/H] $\gtrsim-0.1$) stars being detected in period more
frequently than lower-metallicity stars. This is consistent with Claytor et
al. (2020a); Amard et al. (2020); See et al. (2021), who found that rotation
is more easily detected in Kepler stars with higher metallicity at fixed mass.
We see the same bias toward higher metallicity among our rotators, which may
be due either to the deeper convective envelope resulting from enhanced
opacity, or to more rapid rotation (and therefore higher activity) from
increased moment of inertia and slower spin-down.

Figure 14: The detectability of rotation across APOGEE metallicity and
$\alpha$-element abundace. We detect fewer stars in rotation at high
[$\alpha$/M] because these stars tend to be old, slowly rotating, and less
active. At fixed [$\alpha$/M], there is scatter in the trend of detectability
with metallicity from bin to bin.

Another view of the detection fraction is shown in Figure 14, this time as a
function of metallicity and $\alpha$-element enhancement. At fixed
metallicity, we detect fewer stars in rotation at high [$\alpha$/M] due to the
underlying relationship between age and $\alpha$ enhancement. High-$\alpha$
stars tend to be older, spin more slowly, and are less active, so we expect
them to be more difficult to detect in rotation. This view also allows us to
inspect the period detection fraction across metallicity at fixed $\alpha$
enhancement. At fixed [$\alpha$/M], there is significant scatter in the
detection fraction across metallicity. Some bins (e.g., 0 $<$ [$\alpha$/M] $<$
0.05) show gradually increasing detectability with increasing metallicity,
while others (e.g., -0.05 $<$ [$\alpha$/M] $<$ 0) worsen in detection at
higher metallicity. Due to the amount of noise in the bins, it is difficult to
conclude whether the apparently enhanced detection fraction at higher
metallicity is caused by higher activity from deeper convection zones or by
the underlying age distribution.

##  7 Spot Filling Fraction

The links between temperature, metallicity, age, convection, rotation, and
photometric variability shed light on the generation of magnetism in cool,
main-sequence stars. The strength of rotational modulation in the light curve,
and therefore the detectability of rotation, hint at the presence of cool
spots created by magnetic fields concentrated near the stellar surface.
Because spots are created by the same dynamo that rotation and convection
drive, we can use the prevalence of spots in different temperature and
rotation ranges to infer dynamo properties in those regimes.

Cao & Pinsonneault (2022) found that temperature-sensitive spectral features
include contributions from the quiet photosphere and cooler spots. Thus,
fitting APOGEE spectra with two temperature components, they inferred the
surface spot filling fractions and the temperature contrasts of a sample of
stars. They used a modified version of the FERRE code (Allende Prieto et al.,
2006), the spectral fitting code used by the ASPCAP pipeline, to infer spot
filling fractions for all stars in APOGEE DR17. Following this method, we
obtained spot filling fractions and updated effective temperatures for the
stars in our sample with APOGEE spectra.

We began with the 1,227 stars in our Gold sample (described in Section 5.5).
We made cuts in Gaia DR3 magnitudes and colors using $M_{G}>4.4$ and
$G_{BP}-G_{RP}>1$ to target below the field main-sequence turnoff and ensure
all our stars are securely on the main sequence. This yielded 585 cool, main-
sequence stars, but a few (less than 20) showed an excess in $M_{G}$,
indicating that they were likely leftover binary systems (e.g., Berger et al.,
2018). To remove these, we fit a line to the main sequence and computed the
magnitude excess as $\Delta M_{G}=M_{G}-\langle M_{G}\rangle$, where $\langle
M_{G}\rangle$ was the fit main-sequence magnitude. The distribution of
magnitude excesses had two clear peaks, with a trough we visually identified
at $\Delta M_{G}=-0.4$. We removed stars with $|\Delta M_{G}|<0.4$, leaving
566 stars. We designate these as our “Platinum” sample, a pure, cool, main-
sequence sample robustly free from binary contamination.

With spot filling fractions, we can now investigate the detectability of
rotation as a function of surface spot coverage. We might expect more spotted
stars to be easier to detect in rotation, as they should have higher
photometric variability. Figure 16 shows the 268 Platinum sample K-dwarfs with
$1.5<G_{BP}-G_{RP}<2$, along with the 209 stars in the same regime but with no
rotation detection. The left panel shows the subsamples’ distributions of spot
filling fractions, while the right panel shows the cumulative frequency
distributions. A Kolmogorov–Smirnov test returns a $p$-value of 0.3, rejecting
the null hypothesis that the two samples are drawn from the same underlying
distribution with only 70% (i.e., just over 1$\sigma$) significance. There are
too few stars in this regime to confirm any difference in spot filling
fraction between the period detection and nondetection samples.

Figure 15: Gaia DR3 color-magnitude diagram of our Platinum cool main-sequence
sample, colored by surface spot filling fraction. The sample is carefully
cleaned of potential binary systems, which can interlope as falsely high spot
filling fractions.

We show the Platinum sample on a Gaia color-magnitude diagram in Figure 15,
with points colored by spot filling fraction. While most stars in our sample
have low spot filling fractions less than 10%, the mid-K range
($1.5<G_{BP}-G_{RP}<2$) exhibits elevated fractions. Here, filling fractions
reach $\approx$ 0.3–0.4, behavior that was first observed by Cao et al.
(2023), which they attributed to internal differential rotation. There is a
clear gradient of increasing filling fraction with increasing $M_{G}$. This
may represent an increase of spot coverage with increasing metallicity in this
temperature regime; the correlation between spot filling fraction and
metallicity is still present after strong binary rejection, and the trend
disappears outside this temperature range.

Figure 16: The distribution of spot filling fractions separated by whether
rotation was detected in the Gaia color range $1.5<G_{BP}-G_{RP}<2$. While we
might expect more spotted stars to be easier to detect in rotation, there are
too few stars to draw any statistically significant conclusions.

Cao et al. (2023) suggested that core-envelope decoupling gives rise to
anomalous rotation behavior in cool stars, evidenced by elevated spot filling
fractions in cluster K-dwarfs between 4,000 and 4,500 K. The process of
decoupling and recoupling drives a radial shear layer and enhanced surface
magnetism. With field star rotation periods up to 27 days, we can investigate
the behavior of rotation and spottedness in the TESS SCVZ. Figure 17 shows the
period–temperature distribution of our Platinum sample, again colored by spot
filling fraction, with the rotation sequences from benchmark open clusters
Pleiades (Rebull et al., 2016), Praesepe (Douglas et al., 2017, 2019), NGC
6811 (Curtis et al., 2019), Ruprecht 147 (Curtis et al., 2020), and M67
(Barnes et al., 2016; Dungee, 2022). Here we use the two-component fit
effective temperature, rather than the TIC or ASPCAP values, for consistency
with the spot filling fractions. As a function of temperature, spot filling
fractions increase in the mid-K range—the same behavior Cao et al. (2023)
identified in Praesepe. At fixed temperature, we might expect filling
fractions to be higher at shorter periods, where stars rotate faster and are
more magnetically active. Instead, spot filling fractions in the mid-K range
appear to be elevated across the entire span of recovered periods ($\sim$10–30
d). Cao et al. (2023) predict shear-enhanced magnetism to persist in this
temperature range until ages of a few Gyr (temperature-dependent), so it is
likely we do not reach long enough periods for the spot filling fraction to
decrease.

Figure 17: The period distribution of our Platinum sample, colored by spot
filling fraction, and plotted with rotation sequences of benchmark open
clusters. In the K temperature range (between 4,000 and 4,500 K), NGC 6811 and
Rup 147 exhibit stalled rotational braking, a departure from current
gyrochronological models. The spot filling fractions are elevated here as
well.

The increase in spot filling fraction occurs in the color and temperature
range where open clusters NGC 6811 and Rup 147 were shown to exhibit an
unexpected epoch of stalled rotational braking (Curtis et al., 2019, 2020).
NGC 6811 is 1 Gyr old (Curtis et al., 2019), but for temperatures cooler than
5,000 K its rotation sequence rests upon that of the 670-Myr-old Praesepe
(Douglas et al., 2017, 2019). Somewhere between the ages of these clusters,
stellar spin-down departs from the classical picture from gyrochronology. By
2.7 Gyr (Rup 147, Curtis et al., 2020), stars at the hot end have resumed
braking, but the cooler stars lag behind, suggesting that the epoch of stalled
braking lasts longer for lower-mass stars (e.g., Gallet & Bouvier, 2015;
Lanzafame & Spada, 2015; Somers & Pinsonneault, 2016; Spada & Lanzafame, 2020;
Cao et al., 2023).

Spada & Lanzafame (2020) showed that a two-zone interior model, which allows
the core and envelope to rotate at different rates, can nearly reproduce the
stalled spin-down behavior exhibited by these clusters. In these models the
core and envelope decouple, and the envelope continues to spin down from
magnetic braking while the core maintains its rotation speed. During
recoupling, angular momentum is transferred from the core to the envelope, and
the apparent spin-down is temporarily slowed or halted. After recoupling, the
star again behaves as a solid body and undergoes classical (i.e., power law,
Skumanich, 1972) braking. While Curtis et al. (2020) argued in favor of the
two-zone model, they could not rule out a temporary reduction in the braking
torque, either from reduced wind or weakening of the magnetic field, as a
possible cause. We suggest that the coincidence of elevated spot filling
fractions in field stars with the stalled braking seen in open clusters
supports the shear-driven dynamo hypothesis argued by Cao et al. (2023).

##  8 Summary & Conclusion

We used deep learning to infer reliable periods for 7,971 main sequence stars
near the southern ecliptic pole from year-long TESS full-frame image light
curves. Our periods represent the first large-scale recovery and measurement
of rotation in TESS stars rotating more slowly than 13.7 days, the limit
previously imposed by TESS’s complicated systematics. We fit stellar
evolutionary models to the stars using rotation and high-resolution
spectroscopic parameters to determine stellar ages, masses, convection
timescales, Rossby numbers, and more. We investigated the detectability of
rotation as a function of fundamental stellar parameters as well as new spot
filling fractions inferred from spectroscopy. Our key results and conclusions
are as follows:

  * •

We find evidence for the intermediate rotation period gap, first discovered in
the Kepler field and seen in K2 field star samples across the ecliptic plane,
the first such detection from TESS stars. The period gap in TESS closely
aligns with the gaps from previous missions, cementing the conclusion that the
gap is a product of stellar structure and evolution and not star formation
history.

  * •

The rotation period gap coincides with a dip in photometric variability,
consistent with the findings of Reinhold et al. (2019); Reinhold & Hekker
(2020) in other field star populations.

  * •

The distribution of rotation periods in TESS closely resembles the
distributions seen by Kepler and K2. Its lower edge features a slope of
increasing period with decreasing temperature, similar to the distributions
from previous missions, and we detect slowly rotating M-dwarfs with a similar
location and distribution as in Kepler.

  * •

We detect a higher fraction of stars in rotation at cooler effective
temperatures, where stars rotate faster at fixed age and have deeper
convective envelopes resulting in higher activity amplitudes. We also
preferentially detect rotation in stars at higher metallicities at fixed
temperature. This may owe to deepening convective envelopes with increasing
metallicity, or to increased moment of inertia with increasing metallicity
resulting in slower spin down, and faster period (and therefore higher
activity) at fixed age.

  * •

In Gaia color regimes with a range of spot filling fractions, stars detected
in rotation showed no significant difference in spot filling fraction compared
to stars with no period detection.

  * •

Field stars with elevated spot filling fractions coincide with open cluster
stars that exhibit a temporary stall in magnetic braking. These coincide at
least partly with the period gap and its variability depression, suggesting a
common cause.

While TESS systematics have presented unique challenges that remain difficult
to solve with conventional period-finding techniques, deep learning presents a
way to circumvent instrument systematics without having to solve systematics
removal for every individual case. Since first observing the southern
hemisphere in 2018, TESS has also observed the North, revisited both
hemispheres, and continues to observe the entire sky in its search for
transiting exoplanets. As it does, it continues to build a vast trove of
stellar light curves to search for rotation in stars across the entire sky.

Our simulation-driven CNN approach enables the inference of more than just
rotation. The existing training sets include activity level, latitudinal
differential rotation, spot lifetimes, and activity cycles. These quantities
can be probed with minimal modification to our CNN framework and would provide
new avenues of investigation of stellar rotational, activity, and magnetic
evolution.

Understanding the complicated rotational evolution of low-mass stars and the
related anomalies in activity and spot coverage will require more rotation
periods for more diverse populations of stars. As we grow the number of
rotation periods obtained with TESS, precise and homogeneously derived
temperatures and metallicities will be imperative to pinpoint the regimes
where stellar rotation and activity processes change. The Milky Way Mapper
(MWM) of the Sloan Digital Sky Survey V (Kollmeier et al., 2017) is obtaining
APOGEE spectroscopy for 6 million stars across the whole, including 300,000
observed with TESS two-minute cadence in the SCVZ. MWM will provide
homogeneous temperatures, metallicities, and detailed chemical abundances for
all these stars, offering unprecedented precision on the fundamental
parameters of a large rotation sample.

Upcoming space missions will provide crucial avenues to rotation periods as
well. The methods in this work will be applicable to photometry obtained by
the Nancy Grace Roman Space Telescope (Spergel et al., 2015). Roman will
perform a Galactic Bulge Time Domain Survey (Penny et al., 2019; Johnson et
al., 2020) with cadence similar to TESS with the addition of lower cadence
photometry in at least one secondary band. Not only will a rotation be made
accessible in a relatively unprobed population of stars toward the Galactic
bulge, but the multi-band coverage will provide access to time-domain
temperature resolution, enabling the study of stellar spot and facula
distributions for hundreds of thousands of stars. Furthermore, the potential
to observe two globular clusters near the Galactic center with Roman
(Grunblatt et al., 2023) would provide the first large gyrochronology anchors
at both old ages and sub-Solar metallicities.

We gratefully acknowledge Gagandeep Anand, Ashley Chontos, Monique Chyba, Ryan
Dungee, Rafael Garcia, Daniel Huber, Corin Marasco, Savita Mathur, Peter
Sadowski, Ângela Santos, Benjamin Shappee, Xudong Sun, and Jamie Tayar for
helpful conversations that improved the quality of this manuscript. The
technical support and advanced computing resources from the University of
Hawai‘i Information Technology Services – Cyberinfrastructure are gratefully
acknowledged. J.v.S. and Z.R.C. acknowledge support from the National
Aeronautics and Space Administration (80NSSC21K0246, 80NSSC18K18584) This
paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the 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-IV acknowledges support and resources from
the Center for High Performance Computing at the University of Utah. The SDSS
website is www.sdss4.org. SDSS-IV is managed by the Astrophysical Research
Consortium for the Participating Institutions of the SDSS Collaboration
including the Brazilian Participation Group, the Carnegie Institution for
Science, Carnegie Mellon University, Center for Astrophysics — Harvard &
Smithsonian, the Chilean Participation Group, the French Participation Group,
Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli
Institute for the Physics and Mathematics of the Universe (IPMU) / University
of Tokyo, the Korean Participation Group, Lawrence Berkeley National
Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-
Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik
(MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE),
National Astronomical Observatories of China, New Mexico State University, New
York University, University of Notre Dame, Observatário Nacional / MCTI, The
Ohio State University, Pennsylvania State University, Shanghai Astronomical
Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma
de México, University of Arizona, University of Colorado Boulder, University
of Oxford, University of Portsmouth, University of Utah, University of
Virginia, University of Washington, University of Wisconsin, Vanderbilt
University, and Yale University. 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.

##  Appendix A Public TESS Photometry and Tools

There are several publicly available light curve sets, pipelines, and tools
designed and optimized for TESS data. We list some of the most widely used in
Table 6. Tools like Lightkurve (Lightkurve Collaboration et al., 2018) and
eleanor (Feinstein et al., 2019) are general tools to download, process, and
analyze TESS data. eleanor is a flexible tool that allows for several
different systematics correction routines to be used on the same light curves.
However, it requires large downloads, making it somewhat inconvenient for
working with large data. Unpopular (Hattori et al., 2022) is a light curve
processing pipeline optimized for systematics removal while preserving multi-
sector astrophysical signals. It may be ideal for the problem of rotation, but
it requires downloading large FFI cutouts, or the entire set of FFIs, for it
to work optimally. Lightkurve does no automatic processing and provides simple
tools for downloading and interacting with image and light curve data. We use
Lightkurve for all our photometry and light curve processing.

Among the many public light curve datasets, the TESS Quick-Look Pipeline (QLP,
e.g., Huang et al., 2020a) and DIAmante (Montalto et al., 2020) are designed
for planet searches, so their light curve processing is aggressive and can
remove the stellar signals we are interested in. The difference imaging
analysis (DIA) light curves of Oelkers & Stassun (2018) are for general use,
but only sectors 1–5 of the first year are available. The GSFC-ELEANOR-LITE
light curves (Powell et al., 2022) are a brand new data set using eleanor to
create general-use light curves for all TESS stars brighter than 16th
magnitude in the TESS band pass. They will be worth considering for large
scale investigations in TESS, but currently only four sectors are publicly
available. The TESS Science Processing Operations Center (TESS-SPOC, Caldwell
et al., 2020) has FFI light curves for nearly 40,000 bright SCVZ targets, with
background subtraction and systematics correction, as well as underlying pixel
data and apertures, available. They are suitable for general use and are
easily downloaded from MAST. Finally, the TESS Asteroseismic Science
Operations Center (TASOC, e.g., Handberg et al., 2021) is producing data
products for all targets brighter than 15th TESS magnitude. They provide two
different light curve products optimized for signals at different timescales
with varying levels of systematics correction.

Table 6: TESS Full Frame Image Light Curves, Pipelines, and Tools

Name | Reference(s) | Science Use
---|---|---
Lightkurve | Lightkurve Collaboration et al. (2018) | general
eleanor | Feinstein et al. (2019) | general
Unpopular | Hattori et al. (2022) | general
DIA | Oelkers & Stassun (2018) | general
QLP | Huang et al. (2020a, b); Kunimoto et al. (2021) | exoplanet detection
TESS-SPOC | Caldwell et al. (2020) | general
DIAmante | Montalto et al. (2020) | exoplanet detection
T’DA/TASOC | Handberg et al. (2021); Lund et al. (2021) | asteroseismology
GSFC-ELEANOR-LITE | Powell et al. (2022) | general

Note. — Software tools are listed first, followed by public light curve data
sets. Tools and data sets used in this work are highlighted in blue. All light
curve data sets are documented and publicly available as MAST High Level
Science Products at https://archive.stsci.edu/hlsp, except for DIA (Oelkers &
Stassun, 2018), which is available at https://filtergraph.com/tess_ffi.

##  Appendix B Optimizing the Neural Network Architecture

In Section 3.2 we lay out the various convolutional neural network (CNN)
architectures that we trained and assessed to optimize our network’s
performance. Here we discuss the details of that optimization and the
justification for our choices of architecture.

For both the TESS-SPOC and TASOC data products, we trained four different
CNNs, each with 3 convolution layers, but each CNN had different numbers of
convolution kernels to give the networks different flexibility in learning
features. The architectures were (A) 8, 16, and 32 kernels; (B) 6, 32, and 64
kernels; (C) 32, 64, and 128; and (D) 64, 128, and 256. We also used four
different training sets for both TESS-SPOC and TASOC, each with a different
upper limit on rotation period. The period upper limits were 30, 60, 90, and
180 days, intended to optimize different networks for different period ranges.
We trained all four architectures on each period range, compared performance
metrics, and chose the architecture that had the best performance on average
across all four training sets. For performance metrics, we considered, (1)
average test loss, (2) median relative uncertainty, (3) percentage of test
targets recovered to within 10% and 20% accuracy, and (4) the 1st and 99th
percentiles of the filtered period estimates. To illustrate the meaning of
these values, we will use the 180-day TESS-SPOC training set as an example.

During training, each training set is partitioned into a training, validation,
and test set. The training set is used to fit the network parameters, the
validation set is used to determine when to stop training to avoid
overfitting, and the test set is used to assess performance. We monitored the
average loss for all three partitions during training so that we can construct
learning curves, which show the loss values versus training epoch. Figure 18
shows the learning curves for all four architectures on the 180-day training
set. The solid lines represent the training loss, while the dashed lines
represent the test loss. Left unchecked, training loss will continue to
decrease, but the loss on a held-out validation set will plateau or begin to
increase once the network begins overfitting, which we use as our stopping
criterion. The test loss is highest for run A, the simplest architecture we
used. This indicates that run A is not complex enough to fully learn the
features in the data, or at least that it begins overfitting before it can
fully learn the features. Run B performs better, but is comparable to runs C
and D, which fully train in fewer epochs. We can rule out run A for this case,
but more metrics are needed to properly assess which run performs best.

Figure 18: Learning curves of all four CNN architectures for the 180-day
training set. The solid lines track the training loss, while the dashed lines
show the test loss, which was used to assess performance of the networks once
trained.

One of the strengths of our method is the ability to estimate an uncertainty,
which we can use as a metric of predicted reliability (Claytor et al., 2022).
Specifically, we use the fractional uncertainty $\sigma_{P}/P$ to normalize
for period dependence. A better-trained network should have lower values of
$\sigma_{P}/P$, indicating more reliable estimates. We use the median
$\sigma_{P}/P$ as an additional metric of performance in addition to using it
to filter out bad estimates. Figure 19 shows the _filtered_ period estimates
for each run, but note that the median fractional uncertainty listed in each
panel is computed over the _unfiltered_ periods. Run B has the lowest
estimated uncertainty, so by this metric it performs the best and has the most
reliable estimates.

Figure 19: Period detections for different CNN architectures, filtered by
relative uncertainty. Architectures are increasingly complex from A to D, and
recovery statistics are shown in the legend of each panel.

We also use accuracy metrics to assess performance. The “acc10” and “acc20”
metrics quantify what fraction of test targets are recovered to within 10% and
20% accuracy after filtering by uncertainty. The “acc10” metrics for each run
are near 50%, which also means that the median relative error on the period
estimates is near 10% for all runs. Run B has the highest accuracy metrics, so
it once again performs best.

Estimating uncertainty biases estimates toward the median of the distribution,
making period inference near the edges of the training set period range more
difficult (Claytor et al., 2022). We attempt to mitigate this by tabulating
the 1st and 99th percentiles of each (unfiltered and filtered) inferred period
range. Figure 20 shows the distribution of periods for both the unfiltered
(left) and filtered (right) estimates. Though it is difficult to assess by
eye, run A has the lowest 1st percentile (12.1 d) in the filtered sample,
although all runs have first percentiles in the 12–13 day range. This also
gives us a lower limit for where we can expect successful period estimates
from this training set: networks trained on the 180-day set struggle to infer
periods less than 12 days, motivating the need for training sets with smaller
period ranges.

Figure 20: Distributions of rotation period estimtaes for different CNN
architectures, for both the unfiltered (left) and filtered (right) samples.

We prioritized metrics as follows: we considered the average test loss to rule
out runs that failed to compete in loss value (e.g., runs B, C, and D achieved
comparable loss values, but run A fell short). We then prioritized the
accuracy metrics and uncertainty together, then if those were comparable we
used the 1st and 99th percentile values to break ties.

When considering all our metrics for the 180-day TESS-SPOC training set, run B
performs the best overall. We then repeated this process for each training set
and chose the architecture that performed best over all training sets.
Following this procedure, we chose architecture C for the TESS-SPOC data and
architecture A for TASOC. We note that it may be optimal to use the optimal
architecture for each training set, rather than adopt one architecture for all
sets. We will consider before publication and release of the final period
catalog.

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