# Seven Classes of Rotational Variables From a Study of 50,000 Spotted Stars
with ASAS-SN, Gaia, and APOGEE

Anya Phillips1, C. S. Kochanek1,2, Tharindu Jayasinghe3,4, Lyra Cao1, Collin
T. Christy5, D. M. Rowan1, and Marc Pinsonneault1
1Department of Astronomy, The Ohio State University, 140 West 18th Avenue,
Columbus, OH, 43210, USA
2Center for Cosmology and Astroparticle Physics, The Ohio State University,
191 W. Woodruff Avenue, Columbus, OH, 43210, USA
3Department of Astronomy, University of California, Berkeley, CA 94720
4NASA Hubble Fellow
5Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson,
AZ 85721 E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We examine the properties of $\sim 50,000$ rotational variables from the ASAS-
SN survey using distances, stellar properties, and probes of binarity from
Gaia DR3 and the SDSS APOGEE survey. They have high amplitudes and span a
broader period range than previously studied Kepler rotators. We find they
divide into three groups of main sequence stars (MS1, MS2s, MS2b) and four of
giants (G1/3, G2, G4s, and G4b). MS1 stars are slowly rotating (10-30 days),
likely single stars with a limited range of temperatures. MS2s stars are more
rapidly rotating (days) single stars spanning the lower main sequence up to
the Kraft break. There is a clear period gap (or minimum) between MS1 and
MS2s, similar to that seen for lower temperatures in the Kepler samples. MS2b
stars are tidally locked binaries with periods of days. G1/3 stars are heavily
spotted, tidally locked RS CVn with periods of tens of days. G2 stars are less
luminous, heavily spotted, tidally locked sub-subgiants with periods of $\sim
10$ days. G4s stars have intermediate luminosities to G1/3 and G2, slow
rotation periods (approaching 100 days) and are almost certainly all merger
remnants. G4b stars have similar rotation periods and luminosities to G4s, but
consist of sub-synchronously rotating binaries. We see no difference in
indicators for the presence of very wide binary companions between any of
these groups and control samples of photometric twin stars built for each
group.

###### keywords:

stars: variables: general – stars: rotation – stars: binaries: general –
stars: starspots

††pubyear: 2023††pagerange: Seven Classes of Rotational Variables From a Study
of 50,000 Spotted Stars with ASAS-SN, Gaia, and APOGEE–Seven Classes of
Rotational Variables From a Study of 50,000 Spotted Stars with ASAS-SN, Gaia,
and APOGEE

##  1 Introduction

Rotation provides a powerful stellar population diagnostic and is essential to
understanding stellar structure and evolution. In stars with convective
envelopes, rotationally-driven dynamos produce magnetic fields which in turn
lead to starspots on the stellar surface (e.g., Yadav et al. 2015). If the
star’s rotation is fast enough and the spot fraction is large enough, the
brightness of the star varies quasi-periodically, allowing a measurement of
the rotation rate of these "rotational variables." Surface spot coverage is
linked to mechanisms of interior angular momentum transport (Cao et al.,
2023), so studies of photometric modulation are well-suited to studying
stellar structure and evolution.

Rapid rotation in low mass stars is traditionally regarded as an indicator of
youth because the rotation rate in solar-mass stars, and their activity,
decrease with age (Skumanich, 1972) due to angular momentum loss from
magnetized solar-like winds (Weber & Davis, 1967). Lower mass stars take
longer to spin down than Solar analogs, and are consistently more active at a
given rotation period. Stellar activity is usually parameterized by a Rossby
number, the ratio of the convective overturn timescale to the rotation period.
High mass stars have much shorter overturn timescales than low mass stars, and
are therefore inactive; this explains why low mass stars have magnetized winds
and spin down, while higher mass stars do not (Durney & Latour, 1978). Stellar
spin down is consequently a potentially important age indicator (Barnes,
2007), especially in lower mass stars that experience little nuclear evolution
in a Hubble time.

We can model the correlation between rotation rate and age with
gyrochronology, where the rotation rate of a main sequence star is used as an
age estimator. This method has blossomed with the large samples of low-
amplitude rotational variables discovered by Kepler (Borucki et al., 2010;
Koch et al., 2010). For example, McQuillan et al. (2014) derived rotation
periods of $\sim 34,000$ Kepler main-sequence stars with amplitudes a low as
0.1% and applied gyrochronological models to estimate their ages. In practice,
it has proven challenging to quantify such gyrochronology relationships. For
example, magnetic braking ceases in the oldest, least active stars (van Saders
et al., 2016). There is also a transient phase where spin down pauses; this
was first discovered in Solar analogs (Krishnamurthi et al., 1997), but lasts
for a longer time in K dwarfs (Curtis et al., 2019), which complicates
gyrochronology (Bouma et al., 2023).

Binary stars provide a completely different channel for inducing rapid
rotation. Close binary systems are synchronized by tides, allowing low mass
stars to remain active for their entire main sequence lifetime (Wilson, 1966).
Angular momentum lost in winds is extracted from the orbits of sufficiently
short-period binaries, and this can produce mergers, sometimes referred to as
blue stragglers, on the main sequence (Andronov et al., 2006).

Once off the main sequence, single stars expand and slow down, even without
magnetized winds. As a result, most evolved giant stars are slow rotators.
However, when mergers occur on the giant branch, the merger products can
rotate rapidly. Daher et al. (2022) found that $0.8$–$3.5\%$ of $79,308$
APOGEE field giants in their sample rapidly rotate, depending on the chosen
threshold for what constitutes "rapid rotation." Other studies (including
Tayar et al. 2015, Carlberg et al. 2011, and Massarotti et al. 2008) find
rapid rotator fractions in this range with the exact values depending on
varying amplitude thresholds and physical differences in the selected stellar
populations (Patton et al., 2023). Many of these rapidly rotating giants are
apparently single (Patton et al., 2023) and are almost certainly merger
products.

Rapidly rotating giants can also result from tidal interaction in a binary,
and giants in binary systems can become tidally synchronized at a wide range
of periods (see Leiner et al. 2022 for a recent discussion). The combination
of long overturn timescales and relatively short rotation periods (either due
to tidal interaction or being a merger product) can produce extremely high
activity in a minority of stars. Almost all magnetically active giants are
therefore expected to either be merger products or currently interacting
binary stars. Ceillier et al. (2017) found a high rate of interacting binaries
and mergers on the red giant branch, showing 15% of 575 low mass
($M<1.1M_{\odot}$) red clump stars from Kepler to have detectable rotation
through brightness modulations, inconsistent with single stars which are not
merger products. Further, Gaulme et al. (2020) directly established the
connection between rotational modulation due to starspots and tidal
interaction for Kepler red giants, finding $\sim 85\%$ of non-oscillating red
giants with rotational modulation to be in spectroscopic binaries.

Two known populations of rapidly rotating, synchronized binary giants are the
RS Canum Venaticorum-type stars (RS CVn) (Hall, 1976), and a less-luminous and
shorter-period group of sub-subgiants (Leiner et al., 2022). Both populations
lie at the base of the giant branch. As giants become larger, the timescale
for their evolution becomes shorter, while the timescale needed to synchronize
increases (Verbunt & Phinney, 1995). Fully synchronized systems are therefore
not expected for luminous giants. However, merger products can appear at any
luminosity.

In this paper, we carry out a population survey of rotational variables based
on roughly 50,000 systems identified by the All-Sky Automated Survey for
Supernovae (ASAS-SN) (Jayasinghe et al., 2018, 2019b, 2019a, 2020, 2021;
Christy et al., 2023). These tend to be fairly high amplitude (10-30%) and
span a range of periods from $\sim$10 to 160 days (see Figure 3). The key to
our survey is the availability of distances through Gaia (Gaia Collaboration
et al., 2016, 2021, 2022) and a broad range of stellar properties from both
Gaia and SDSS APOGEE DR17 (Abdurro’uf et al., 2022). In particular, these
supply considerable information on the binarity of systems. The starting point
is the observation in Christy et al. (2023), for a version of the left panel
of Figure 1 showing the distribution of ASAS-SN rotational variables in
absolute magnitude and rotation period, that the rotational variables seemed
to lie in discrete groups. After dividing our sample, we examine each groups’
detailed properties, in particular radial velocity variability, binarity,
rotation rates, and spot coverage. We describe the data used in Section 2, and
then explore the properties of the empirically-divided groups in Section 3. We
conclude that there are seven distinct groups of rotational variables in
Section 4 and discuss future directions.

##  2 Observations and Methods

Figure 1: (Left panel) Distribution of the stars in absolute Ks magnitude and
period with the clusters assigned by HDBSCAN, and the boundaries used to
manually assign all the stars to groups. The dashed boundaries are numbered in
red, corresponding to those in Table 1. (Right panel) Distribution of the
rotational variables in absolute G magnitude versus B${}_{P}-$RP color, color-
coded by the manually assigned cluster. We also include 1, 3, and 10 Gyr Solar
metalicity PARSEC isochrones (Bressan et al., 2012; Marigo et al., 2013). The
main sequence is also shown at twice the luminosity as the sequence of "twin"
binaries. Stars above the sequence of twin binaries are probably young stellar
objects (YSOs). Figure 2: (Left panel) Absolute G magnitude versus
$B_{P}-R_{P}$ color for a random sample of $\sim$500,000 ASAS-SN sources
searched for variability where the bin color corresponds to the number of
sources. (Right panel) Absolute G magnitude versus $B_{P}-R_{P}$ color where
bin color corresponds to the fraction of stars which were identified as
rotational variables. Both panels are overlayed with the same isochrones as in
the right panel of Figure 1.

Here we consider the subsample of $48,298$ rotational variables shown in the
left panel of Figure 1. We restricted the sample to systems with Gaia EDR3
(Gaia Collaboration et al., 2016, 2021) parallax signal-to-noise ratios of
$>10$. We use distances from Bailer-Jones et al. (2021) and extinction
estimates from the mwdust (Bovy et al., 2016) ‘Combined19’ dust map (Drimmel
et al., 2003; Marshall et al., 2005; Green et al., 2019). We keep systems with
estimated extinctions $A_{V}<2$ and dispose of the small number of outliers
with either extinction corrected $B_{P}-R_{P}>3$ or $M_{G}<-1$. The ASAS-SN
variable catalog is dominated by the V band sample (Jayasinghe et al., 2018,
2019b, 2019a, 2020, 2021) which had significant systematic problems for
periods near one day, so we reject systems with $\rm{log_{10}(Period/days)}$
between $-.0018$ and $.0005$.

The left panel of Figure 1, the distribution of our sample in period and
absolute 2MASS (Skrutskie et al., 2006) Ks magnitude, appears to have $6$
clusters; four of giants and two of main sequence stars. To test this more
formally, we used the density-based clustering algorithm HDBSCAN (McInnes et
al., 2017) to identify clusters in this parameter space. HDBSCAN assigns each
source to one of the clusters or to noise. We first separated the main
sequence and giants using the input parameter min_cluster_size = 1000. We
further divided the main sequence using min_cluster_size = 1000 with the
additional parameters min_samples = 200, and cluster_selection_epsilon = .07,
and the giants using min_cluster_size = 1000, with the additional parameters
min_samples = 150, and cluster_selection_epsilon = .07.

This combination of parameters lead to the identification of $7$ clusters, the
$6$ identified by eye and a seventh associated with the 1 day period notch. We
ignore this grouping (it is not shown in Figure 1) and assign it to the
adjacent cluster. The combination of parameters we used in HDBSCAN were meant
to maximize the number of points assigned to clusters, but nonetheless many of
the stars were not assigned to any group, as can be seen in the left panel of
Figure 1. For our analysis, we divided the stars into 6 clusters using the
lines shown in the left panel of Figure 1 and presented in Table 1 to expand
the HDBSCAN clusters to include all of the stars. We label these initial
groups as MS1 and MS2 for the main sequence, G1 and G2 for the shorter period
giants, and G3 and G4 for the longer period giants. The right panel of Figure
1 shows these clusters in extinction-corrected absolute G magnitude and
B${}_{P}-$RP color and the groups also partially separate in this space. While
we begin with these six groups identified in period and absolute magnitude, we
find them to further subdivide using other parameters; we will discuss this in
Section 3.

We visually inspected 100 randomly selected light curves from each group. The
light curves overwhelmingly are those of rotational variables with very little
contamination. We had hoped that there would be some qualitative differences
between the light curves of the different groups, but no such differences were
apparent. The residual low-level contamination observed in the light curves
and the fact that we do not expect our manual divisions to be perfect will
lead to scatter in other parameter spaces. Nonetheless, these divisions
suffice for our purpose of highlighting the bulk properties of each group. We
also checked distributions in ASAS-SN amplitudes for each group, but the only
obvious trend is the selection effect that fainter stars need higher
amplitudes to be identified as variables.

Table 1: Boundaries used to separate groups (labeled in red in the left panel of Figure 1). Boundary # | Equation | Range
---|---|---
1 | $\rm{M_{K_{S}}=2.3}$ | $\rm{-1<\log(P/days)<3}$
2 | $\rm{M_{K_{S}}=5.5\log(P/days)-2}$ | $\rm{0.782<\log(P/days)<3}$
3 | $\rm{\log(P/days)=1.5}$ | $\rm{-0.25<M_{K_{S}}<2.3}$
4 | $\rm{M_{K_{S}}=-1.5\log(P/days)+2}$ | $\rm{1.33<\log(P/days)<3}$
5 | $\rm{M_{K_{S}}=0}$ | $\rm{-0.577<\log(P/days)<1.33}$
6 | $\rm{M_{K_{S}}=-2.6\log(P/days)+1.5}$ | $\rm{-0.308<\log(P/days)<3}$

We also extracted 500,000 random stars from the full sample of ASAS-SN stars
searched for variability in Christy et al. (2023), and the left panel of
Figure 2 shows their distribution in color and absolute magnitude. The right
panel shows the fraction of sources identified as rotational variables. This
uses the ratio of the number of rotational variables to the number of random
sources statistically corrected to be the fraction of the full input sample.
This makes no attempt to determine selection effects, but there is a clear
absence of rotational variables on the main sequence above the Kraft (1967)
break and on the upper giant branch. Rotational variables are more common
lower on the main sequence, along the binary main sequence and for the sub-
subgiants (Leiner et al., 2022).

Figure 3 compares our sample in amplitude and period to the McQuillan et al.
(2014) sample of rotational variables in Kepler. As a ground-based survey,
ASAS-SN probes a higher-amplitude sample and over a broader period range than
Kepler, which focused on a field dominated by old stars with low variability
amplitudes (Brown et al., 2011). The two samples have essentially no overlap.

Figure 3: Distribution in amplitude and rotation period for the ASAS-SN
(orange) and Kepler (blue) rotational variables from McQuillan et al. (2014).
We binned the data in this parameter space and the lines show contours of
constant bin count. The amplitudes are in the g-band for ASAS-SN stars and in
Kepler’s bandpass for Kepler stars. Kepler (ASAS-SN) amplitudes are the 5–95%
(2.5–97.5%) ranges of the light curves. Figure 4: Comparison between APOGEE’s
VSCATTER parameter and Gaia’s rv_amplitude_robust, including all rotators and
twins meeting the criteria for the Gaia RV-variability analysis and with
APOGEE NVISITS $>1$, and coded by whether they meet the Katz et al. (2022)
criteria to be considered RV-variable. The blue diagonal indicates where
VSCATTER $=\rm{\texttt{rv\\_amplitude\\_robust}}/2\sqrt{2}$. The vertical line
indicates rv_amplitude_robust=20 km s-1 and the horizontal line indicates
VSCATTER=3 km s-1.

We matched the rotational variables sample to APOGEE DR17 (Abdurro’uf et al.,
2022) and Gaia DR3 (Gaia Collaboration et al., 2022). Table 2 displays the
number of stars in each survey as well as in certain subsets, and Table 3
shows the fractions of stars in each group with characteristics derived from
this ancillary data. From Gaia, we include in Table 3 the fraction of sources
with RUWE $\geq$ 1.2, a conservative indicator for a wide binary or triple
companion (Pearce et al., 2020; Belokurov et al., 2020), flagged astrometric
binaries (7- and 9-parameter acceleration solutions and astrometric orbits)
(Holl et al., 2022), spectroscopic (SB1 and SB2) binaries (Babusiaux et al.,
2022), and systems with high dispersions in their Gaia radial velocities (see
below). The astrometric binaries, systems with astrometric accelerations, and
systems with high RUWE are all associated with long period orbits that should
not be directly associated with the rotational variability. We can see this
explicitly in the astrometric binaries, where the typical period is 100-1000
days. Wide binaries can, however, indirectly be associated with the rotational
variability if the system is really a triple and the long period companion
drives the evolution of a short period inner binary through Kozai-Lidov-type
interactions (e.g., Fabrycky & Tremaine 2007). We also use the Gaia
$v_{\rm{broad}}$ parameter as an estimate of the stellar rotation $v\sin i$.
Based on Frémat et al. (2022), $v_{\rm{broad}}>10$ km s-1 is indicative of
fast rotation while lower values are consistent with noise. However, Patton et
al. (2023) adopt the more conservative rapid rotation threshold
$v_{\rm{broad}}>20$ km s-1, which yields better agreement with their rapid
rotator fractions from APOGEE $v\sin i>10$ km s-1.

Gaia DR3 includes a number of variables which can be used to identify probable
binaries through the scatter of the individual radial velocity (RV)
measurements compared to the estimated noise, as described in Katz et al.
(2022). We considered stars with

  1. 1.

rv_nb_transits $\geq$ 5,

  2. 2.

rv_expected_sig_to_noise $\geq$ 5, and

  3. 3.

3900 $\leq$ rv_template_teff $\leq$ 8000.

We found that either the rv_renormalised_gof or rv_amplitude_robust variables
provided the clearest distinctions between probable binary and single (or wide
binary) stars, where rv_renormalised_gof is a measure of the goodness of fit
of a constant RV to the data, and rv_amplitude_robust is the peak-to-peak
velocity amplitude after clipping outliers. Katz et al. (2022) use a
conservative criterion for a binary of $\texttt{rv\\_renormalised\\_gof}>4$
and $\texttt{rv\\_chisq\\_pvalue}\leq 0.01$. For ease of comparison to the
APOGEE VSCATTER (see below), we will focus on rv_amplitude_robust. Based
either on the value of rv_renormalised_gof or the comparison to APOGEE, we
find that $\texttt{rv\\_amplitude\\_robust}\geq 20$ km s-1 is a good proxy for
binarity. Requiring more transits (10) or higher signal-to-noise (10) had
little effect on the results.

We use the stellar parameters $T_{\rm{eff}}$, $\log g$ and $v\sin i$ from the
APOGEE survey. Where APOGEE has multiple RV measurements (NVISITS>1), we can
use the root-mean-square scatter of the velocities VSCATTER as an indicator of
binarity. Stars with VSCATTER $\geq 3$ km s-1 (approximately equivalent to
having a maximum difference in individual radial velocity measurements $\Delta
RV_{\rm{max}}>$3–10 km s-1) are almost certainly binaries (Badenes et al.,
2018; Mazzola et al., 2020). We also include the number of stars with
estimates of starspot coverage from the LEOPARD spectroscopic analysis of Cao
& Pinsonneault (2022). This algorithm fits the APOGEE spectrum using models
with two different $T_{\rm{eff}}$ to estimate a temperature difference and a
spot fraction $f_{\rm{spot}}$ for the fraction of the stellar surface
associated with the cooler temperatures. This analysis can also interpret
SB2’s as spot contributions if the spectral types of each star are similar,
which is only likely for similar mass main sequence binaries.

Table 2: Statistics of available data. Rotators Sample | Total | MS1 | MS2 | G1 | G2 | G3 | G4
---|---|---|---|---|---|---|---
All Gaia | 48298 | 3258 | 20140 | 2342 | 10619 | 7201 | 4738
Viable for Gaia RV analysis | 38446 | 2860 | 14951 | 2177 | 7356 | 6867 | 4235
All APOGEE | 2133 | 221 | 1073 | 64 | 395 | 219 | 161
APOGEE NVISITS > 1 | 1438 | 139 | 711 | 44 | 273 | 156 | 115
$f_{\rm{spot}}$ Estimates | 2121 | 219 | 1069 | 64 | 392 | 218 | 159
Twins Sample | Total | MS1 | MS2 | G1 | G2 | G3 | G4
All Gaia | 44836 | 3191 | 19665 | 2254 | 8224 | 6945 | 4557
Viable for Gaia RV analysis | 36879 | 2857 | 14965 | 2100 | 6368 | 6559 | 4030
All APOGEE | 1892 | 206 | 1039 | 66 | 188 | 238 | 155
APOGEE NVISITS > 1 | 1279 | 132 | 694 | 44 | 114 | 178 | 117
$f_{\rm{spot}}$ Estimates | 1873 | 203 | 1025 | 66 | 186 | 238 | 155
Table 3: Fractions of stars in each group with the characteristics listed in the left column. Fractions are for the subsample in Table 2 for which the measurement exists. Rotators Sample | MS1 | MS2 | G1 | G2 | G3 | G4
---|---|---|---|---|---|---
Gaia RUWE $\geq$ 1.2 | 43.0% | 35.0% | 9.8% | 12.2% | 12.4% | 12.6%
Gaia Astrometric Binaries | 4.3% | 3.5% | 0.5% | 0.5% | 0.3% | 0.7%
Gaia Spectroscopic Binaries | 1.0% | 0.8% | 3.5% | 1.4% | 14.0% | 9.9%
Gaia Variable Radial Velocity | 13.9% | 58.6% | 98.4% | 94.8% | 87.1% | 58.8%
rv_amplitude_robust$\geq 20$ km s-1 | 13.3% | 58.4% | 97.1% | 94.3% | 79.7% | 51.9%
APOGEE VSCATTER $\geq$ 3 km s-1 | 4.3% | 34.3% | 86.4% | 85.0% | 73.7% | 63.5%
Twin Sample | MS1 | MS2 | G1 | G2 | G3 | G4
Gaia RUWE $\geq$ 1.2 | 38.0% | 43.1% | 11.6% | 15.4% | 12.8% | 14.6%
Gaia Astrometric Binaries | 5.3% | 6.3% | 0.3% | 1.2% | 0.6% | 1.2%
Gaia Spectroscopic Binaries | 1.4% | 0.7% | 1.3% | 0.8% | 1.7% | 1.5%
Gaia Variable Radial Velocity | 14.7% | 26.3% | 11.5% | 16.3% | 10.6% | 13.7%
rv_amplitude_robust$\geq 20$ km s-1 | 13.7% | 27.0% | 6.4% | 16.5% | 5.3% | 9.9%
APOGEE VSCATTER $\geq$ 3 km s-1 | 10.6% | 11.2% | 4.6% | 10.5% | 1.7% | 7.7%

To explore how the rotational variables compared to similar stars which are
not known rotators, we constructed a sample of twins. For each star we
selected all Gaia stars with

  1. 1.

a parallax within 0.9 and 1.1 times the parallax of the rotator,

  2. 2.

a difference in G magnitude less than $0.025n$,

  3. 3.

a difference in BP magnitude less than $0.025n$, and

  4. 4.

a difference in RP magnitude less than $0.025n$,

where $n$ is an integer starting at $n=1$. We assign each star a metric which
is simply the unweighted quadrature sum of the differences in parallax, $G$,
$B_{P}$, and $R_{P}$ magnitudes and keep the lowest 16. If we find fewer than
16 stars, we iteratively increase $n$ until we have 16 stars. In most cases we
succeed with $n=1$ and the overwhelming majority succeed for $n\leq 2$. We
then get the mwdust extinction estimates for all 16 candidates and keep the
one whose extinction is closest to the extinction of the rotator. We finally
use the $\sim$93% of twins whose extinctions agree to $|\Delta A_{V}|\leq 0.2$
mag, which means that the extinction-corrected magnitudes and colors will have
maximum differences due to the extinction mismatch of 0.2 and 0.1 mag,
respectively. By keeping more than 16 candidates we could still better match
the extinctions, but this seemed good enough for our purposes given the small
discrepancies in extinction-corrected photometry. We then extracted all of the
ancillary data for the twins that we obtained for the rotators. For all of the
rotator classes except G2 this provided twins for $\simeq 97\%$ of the stars,
while for G2 we are left with twins for only 77% of the stars. Much of the G2
group lies brighter than the main sequence but redwards of the red giant
branch. Such sub-subgiants are relatively rare, so it is not surprising that
it is more difficult to find twins.

The extinction-corrected absolute magnitude and color distributions of the
twins and their corresponding rotational variables are very similar, as are
their Gaia $\log g$ and $T_{\rm{eff}}$ distributions. The APOGEE $\log g$ and
$T_{\rm{eff}}$ distributions show several notable differences as can be seen
from the summary statistics in Table 4. The two MS samples are fairly similar,
although the MS2 $T_{\rm{eff}}$ distribution of the twins extends to modestly
(a few $100$ K) hotter temperatures. There are clear shifts for all of the
giant groups, where the twins have systematically higher $T_{\rm{eff}}$ and
lower $\log g$ than their corresponding rotators. This is a known bias in the
APOGEE parameters for active stars: APOGEE’s analysis pipeline does not
include rotation as a free parameter when fitting giant spectral templates, so
the broadening of the lines created by spots and rotation in rapidly rotating
giants strongly influence the derived stellar parameters, leading to
underestimates of effective temperature and overestimates of surface gravity
(Patton et al., 2023).

Table 4: Median APOGEE $\log g$ and $T_{\rm{eff}}$ for each group of rotators and its twin sample, with 16th and 84th percentile uncertainties. | $\log g$ |  $T_{\rm{eff}}$ (K)
---|---|---
MS1 rotators | $4.57^{+0.03}_{-0.05}$ | $4735^{+233}_{-437}$
MS1 twins | $4.57^{+0.03}_{-0.04}$ | $4748^{+224}_{-364}$
MS2 rotators | $4.57^{+0.07}_{-0.16}$ | $4234^{+427}_{-390}$
MS2 twins | $4.59^{+0.05}_{-0.05}$ | $4431^{+375}_{-538}$
G1 rotators | $2.74^{+0.19}_{-0.13}$ | $4616^{+47}_{-77}$
G1 twins | $2.61^{+0.34}_{-0.19}$ | $4692^{+149}_{-106}$
G2 rotators | $3.57^{+0.36}_{-0.28}$ | $4435^{+197}_{-233}$
G2 twins | $3.42^{+0.27}_{-0.21}$ | $4901^{+156}_{-165}$
G3 rotators | $2.99^{+0.21}_{-0.30}$ | $4560^{+106}_{-121}$
G3 twins | $2.61^{+0.25}_{-0.14}$ | $4692^{+162}_{-120}$
G4 rotators | $3.30^{+0.23}_{-0.16}$ | $4682^{+321}_{-160}$
G4 twins | $3.12^{+0.26}_{-0.26}$ | $4821^{+162}_{-131}$

Figure 4 compares the APOGEE VSCATTER to the Gaia rv_amplitude_robust for both
the twins and the rotators. The points are coded by whether they meet the Katz
et al. (2022) criteria for RV variability. For a sine wave, the radial
velocity amplitude rv_amplitude_robust would be $2\sqrt{2}$ larger than
VSCATTER. The two estimates of velocity scatter are reasonably well
correlated, but the overall scatter is large because both are based on a small
(APOGEE) or modest (Gaia) number of measurements. Nonetheless,
rv_amplitude_robust>20 km s-1 is a reasonable proxy for binarity.

##  3 Discussion

Figure 5: Distributions of each group of rotators and their twins in Gaia
rv_amplitude_robust (top) and in APOGEE VSCATTER (bottom). Stars to the right
of the vertical lines at $20$ km/s (Gaia) and $3$ km/s (APOGEE) are almost
certainly binaries.

We expect binarity to play a key role in producing rotational variables, so we
start with the distributions in Gaia rv_amplitude_robust and APOGEE VSCATTER,
shown for each group and its twin in Figure 5. We see that the MS1
distribution is single-peaked at low rv_amplitude_robust and VSCATTER, with a
nearly identical distribution to its twins, and so MS1 consists largely of
single stars (or sufficiently wide binaries). The rotators in MS2 are strongly
bimodal with one group of single stars and one group of binaries which we
label MS2s and MS2b, respectively. In the right panel of Figure 1, we see that
the MS2 group lies both on the main sequence (MS2s) and on the "binary main
sequence," above the main sequence in magnitude where the luminosities of the
PARSEC isochrones have been doubled (MS2b). The MS2 twins also show a
significant tail of RV variables, almost certainly because twins of MS2b stars
on the binary main sequence are also binaries.

Separating MS2s/b based only on radial velocity scatter yields a small sample,
limited to stars with multiple radial velocity measurements, but we can also
divide MS2 photometrically. We use the criteria from Cao & Pinsonneault
(2022), who fit a polynomial to the observed main sequence and defined
photometric binaries as those at least 0.25 mag brighter than this fit. This
method implies a binary fraction for MS2 of $\sim 43\%$. For comparison, if we
just split the APOGEE VSCATTER sample at 3 km s-1, we would have a binary
fraction of $\sim 34$%. Since this is incomplete because it does not account
for binaries missed due to inclination, the two estimates are reasonably
consistent. We use the photometric division of MS2s/b in §4 when comparing our
main sequence sample to that of McQuillan et al. (2014).

To compare the results of separating MS2s/b photometrically and based on
radial velocity scatter, Figure 6 shows a color-magnitude diagram of MS2 stars
colored by whether VSCATTER$\geq 3$ km s-1. We see a band of low-VSCATTER
stars on the main sequence, a band of high-VSCATTER stars on the binary main
sequence, and a third band of low-VSCATTER stars above the binary main
sequence. This upper band of stars with low RV scatter is not present in
color-magnitude diagrams where we split instead by whether
rv_amplitude_robust$\geq 20$ km s-1, or based on RV variability according to
criteria from Katz et al. (2022). Unresolved triple systems could lie above
the binary main sequence without having significant radial velocity scatter,
but we saw no indication in Gaia RUWE that this upper band of low-VSCATTER
stars are triple systems. Instead, these are probably young stellar objects
(YSOs), which also have quasi-periodic rotational modulation, but lie well
above the main sequence (Rebull et al., 2016). To confirm this, we verified
that that the YSO candidates have higher than average $A_{V}$.

Figure 6: Absolute G magnitude versus $\rm{B_{P}-R_{P}}$ color for MS2 stars
with valid VSCATTER measurements with the single and binary 1 Gyr PARSEC
isochrones from Figure 1 overlayed.

The G1-G3 rotators are all clearly binaries based on their rv_amplitude_robust
and VSCATTER distributions, whereas their twins are predominantly single
stars. While the G4 twins are overwhelmingly single, the G4 distribution is
bimodal, indicating subpopulations of both single (G4s) and binary (G4b)
stars. Note that the bimodality seen in MS2 and G4 is real and not due to
inclination. Inclination effects produce distributions with the rapidly
dropping tails to lower velocity seen for G3. Formally, for a true binary of
orbital velocity $v_{T}$, the observed velocity $v_{0}$ is distributed as
$x(1-x^{2})^{-1/2}$, with $x=v_{0}/v_{T}$ for a uniform distribution in $\cos
i$. The single star subset of G4 likely consists of merger products. The
distributions in Gaia rv_renormalised_gof confirm the results of the
rv_amplitude_robust and VSCATTER distributions, in particular the existence of
the G4 merger subpopulation.

Figure 7: Distributions in $v_{\rm{broad}}$ (top) and $v\sin i$ (bottom) for
each group and their twins. The $v\sin i$ distributions are split by VSCATTER,
where the solid histograms are for VSCATTER<3 and the dashed histograms are
for VSCATTER$\geq$3 km s-1. The G1 and G3 twins, as well as the high-VSCATTER
population of MS1 and low-VSCATTER population of G1 have $<10$ APOGEE $v\sin
i$ measurements and are not shown because there are too few systems. Figure 8:
Distributions in APOGEE $v\sin i$ and VSCATTER for each group.

We also expect rotation rates to be an important physical probe of rotational
variables, and Figure 7 shows the twin and rotator distributions in APOGEE
$v\sin i$ and Gaia $v_{\rm{broad}}$, as an estimator of $v\sin i$. The $v\sin
i$ distributions include only stars with NVISITS>1 and are split by whether
VSCATTER$\geq 3$ km s-1. The median $v\sin i$ values for each subgroup are
given in Table 5. While the APOGEE $v\sin i$ should be accurate even for small
velocities, comparisons with APOGEE show that values of Gaia
$v_{\rm{broad}}<10$ km s-1 should be regarded as upper limits (Frémat et al.,
2022). MS1, and its twin are comprised of slow rotators. In Table 5, we see
that the high VSCATTER stars of MS1 have a much higher median $v\sin i$, but
the small number of them means that they are not included in Figure 7. MS2,
particularly in $v_{\rm{broad}}$, again seems bimodal, while its twin is
comprised predominantly of slower rotators. In Figure 8, where we show the
distribution of MS1 and MS2 in both VSCATTER and $v\sin i$, there is a clear
separation of MS2 into two populations, where the high-VSCATTER systems all
have high $v\sin i$. MS2’s bimodality is less obvious in the $v\sin i$
distributions because there are low-VSCATTER systems with high $v\sin i$. The
slower MS2 rotators still seem to have larger $v\sin i$ than the MS2 twins and
MS1.

The giants are in three groups. G1 has high $v\sin i$ and $v_{\rm{broad}}$,
and while its twin group has very few $v\sin i$ measurements, it tends to have
lower $v_{\rm{broad}}$. G4 has the slowest rotation rates of the giants, and
while it has a similar $v_{\rm{broad}}$ distribution to its twin, the G4 twins
have still slower $v\sin i$. Of the giant groups, G4 is the only one where the
high-VSCATTER stars do not have a high $v\sin i$. The G4 stars have long
rotational periods, so the smaller $v\sin i$ are expected, but the cause is
physically interesting. The G4 binary stars are not tidally locked, but are in
sub-synchronous orbits (see the discussion associated with Figure 10). G2 and
G3 have intermediate rotation rates to G1 and G4, and their twins tend to have
slower rotation rates (though the G3 twins have very few $v\sin i$
measurements). Note that because of the crudeness of the $v_{\rm{broad}}$
parameter, the Gaia equivalent of Figure 8, $v_{\rm{broad}}$ versus
rv_amplitude_robust, is uninformative.

Figure 9: Distributions of the spot fraction $f_{\rm{spot}}$ for each group.
The solid histograms are for rotational variables with APOGEE VSCATTER < 3 km
s-1 and the colorful dashed histograms are distributions for stars with
VSCATTER$\geq$3 km s-1. The black dashed histograms are for the twin groups.
Only distributions with $\geq 10$ members are shown.

Figure 9 shows the distributions of the stars in the starspot filling fraction
($f_{\rm{spot}}$) where the rotational variables are also divided into likely
binaries with $\texttt{VSCATTER}\geq 3$ and likely non-binaries with
$\texttt{VSCATTER}<3$. Cases when there are fewer than $10$ stars are not
shown. Table 5 gives the median spot fractions of each group. Except for the
MS2 twins, the twin populations have $f_{\rm{spot}}$ distributions strongly
peaked near zero. It seems likely that many of the tails of the twin
distributions towards higher spot fractions are due to the presence of spotted
stars which have not been recognized as rotational variables.

The spot fractions are estimated from the individual APOGEE epochs, and so are
unaffected by the presence of orbital Doppler shifts, but they can be affected
by the presence of spectral contamination from the companion. This means that
only the main sequence binary sub-population MS2b, which tends to lie close to
the binary main sequence, can have significant biases in $f_{\rm{spot}}$ due
to the presence of a binary companion. The giant binaries will generally have
a much lower luminosity main sequence companion which cannot significantly
contaminate the giant’s spectrum.

MS1 contains few binaries and has a spot fraction distribution nearly
identical to its twins. There is, however, a modest binary sub-population that
is more heavily spotted (see Table 5). MS2 has significantly higher spot
fractions than its twins, and the non-binary MS2s sub-sample is modestly less
spotted than the binary MS2b sub-sample. The MS2s stars are, however, much
more spotted than the MS1 stars supporting the argument than they are two
different populations rather than a continuum. Here the long tail on the twin
distribution is due to the tendency of MS2b twins to also be binaries. We
confirmed that the high-$f_{\rm{spot}}$ MS2 twins also tend to have high
VSCATTER.

G1 has a very broad range of spot fractions, although the median of
$f_{\rm{spot}}\sim 28\%$ is less than that of the MS2 and G2 groups and
similar to that of the MS1 group. Except for the likely merger sub-population
of G4, the spotted giants are all binaries based on their VSCATTER
distributions. This means that the low-VSCATTER systems should be dominated by
binaries viewed at high inclinations rather than being a physically distinct
population, so we expect similar spot fractions at high- and low-VSCATTER for
G1, G2 and G3 but not G4, as we see in Figure 9 and Table 5. The G2 group has
the highest median spot fractions with the low-VSCATTER group having a
modestly higher median ($f_{\rm{spot}}\simeq 40\%$ versus $36\%$). The G3
group has some of the lowest spot fractions with little difference between the
high- and low-VSCATTER sub-samples. Like G2, the VSCATTER distributions again
argue for a purely binary population. Finally, the G4 group has some of the
lowest spot fractions, but the high-VSCATTER systems are significantly more
spotted than the low-VSCATTER systems. Unlike G1, G2, and G3, G4 does have a
large number of non-binary, low-VSCATTER systems, so it is not surprising that
they also have different spot fractions.

Table 5: Median $f_{\rm{spot}}$, $v\sin i$, $v_{\rm{broad}}$, $P_{\rm{rot}}$, and $P_{\rm{orb}}$ of each group and their twin as a whole, and each group of rotators separated by whether VSCATTER$\geq 3$ km s-1 (for stars with >1 APOGEE visit) or whether rv_amplitude_robust$\geq 20$ km s-1 (for stars meeting the criteria for Gaia radial velocity analysis outlined in Section 2). For some subsets, there were no stars with orbital period estimates from Gaia, so their $P_{\rm{orb}}$ entries are left blank. Stars from the "twin" sample are not ASAS-SN rotational variables and do not have ASAS-SN rotational periods, so their $P_{\rm{rot}}$ entries are marked "n/a." median: | $f_{\rm{spot}}$ |  $v\sin i$ (km/s) |  $v_{\rm{broad}}$ (km/s) |  $P_{\rm{rot}}$ (days) |  $P_{\rm{orb}}$ (days)
---|---|---|---|---|---
All MS1 Rotators | 0.09 | 3.4 | 9.9 | 18.1 | 13.9
MS1 VSCATTER$<3$ km s-1 | 0.09 | 3.3 | 9.4 | 18.1 |
MS1 VSCATTER$\geq 3$ km s-1 | 0.28 | 10.2 | 24.5 | 12.7 |
MS1 rv_amplitude_robust$<20$ km s-1 | 0.06 | 2.7 | 9.8 | 18.5 | 180.6
MS1 rv_amplitude_robust$\geq 20$ km s-1 | 0.27 | 7.7 | 12.2 | 16.7 | 14.1
MS1 twins | 0.06 | 2.2 | 9.3 | n/a | 25.3
All MS2 Rotators | 0.35 | 12.8 | 14.1 | 3.7 | 6.1
MS2 VSCATTER$<3$ km s-1 | 0.30 | 10.6 | 15.0 | 5.0 |
MS2 VSCATTER$\geq 3$ km s-1 | 0.41 | 20.2 | 15.0 | 2.3 | 6.1
MS2 rv_amplitude_robust$<20$ km s-1 | 0.17 | 6.6 | 11.9 | 8.5 | 4.0
MS2 rv_amplitude_robust$\geq 20$ km s-1 | 0.37 | 16.5 | 20.2 | 3.3 | 6.1
MS2 twins | 0.14 | 3.6 | 10.7 | n/a | 23.8
All G1 Rotators | 0.24 | 37.8 | 30.2 | 5.4 | 13.4
G1 VSCATTER$<3$ km s-1 | 0.27 | 48.1 | 41.5 | 7.9 |
G1 VSCATTER$\geq 3$ km s-1 | 0.28 | 36.3 | 32.3 | 5.2 | 9.6
G1 rv_amplitude_robust$<20$ km s-1 | 0.31 | 48.7 | 34.3 | 10.2 |
G1 rv_amplitude_robust$\geq 20$ km s-1 | 0.16 | 37.8 | 30.0 | 5.8 | 13.4
G1 twins | 0.01 | 2.2 | 9.2 | n/a | 240.6
All G2 Rotators | 0.37 | 20.0 | 20.3 | 9.0 | 12.4
G2 VSCATTER$<3$ km s-1 | 0.40 | 18.1 | 13.2 | 8.1 |
G2 VSCATTER$\geq 3$ km s-1 | 0.36 | 19.2 | 27.1 | 9.0 | 13.1
G2 rv_amplitude_robust$<20$ km s-1 | 0.15 | 8.5 | 11.6 | 23.5 | 695.2
G2 rv_amplitude_robust$\geq 20$ km s-1 | 0.35 | 19.7 | 21.1 | 9.0 | 12.3
G2 twins | 0.02 | 2.1 | 9.1 | n/a | 35.7
All G3 Rotators | 0.15 | 21.3 | 14.0 | 30.9 | 41.1
G3 VSCATTER$<3$ km s-1 | 0.14 | 17.9 | 13.3 | 41.9 | 48.9
G3 VSCATTER$\geq 3$ km s-1 | 0.16 | 21.2 | 17.0 | 29.4 | 36.0
G3 rv_amplitude_robust$<20$ km s-1 | 0.13 | 15.2 | 11.7 | 49.3 | 45.4
G3 rv_amplitude_robust$\geq 20$ km s-1 | 0.14 | 23.2 | 14.8 | 32.0 | 41.1
G3 twins | 0.01 | 20.0 | 9.3 | n/a | 352.7
All G4 Rotators | 0.13 | 5.4 | 9.0 | 64.4 | 28.9
G4 VSCATTER$<3$ km s-1 | 0.10 | 5.3 | 9.5 | 64.9 | 26.2
G4 VSCATTER$\geq 3$ km s-1 | 0.17 | 6.0 | 7.2 | 59.8 | 36.9
G4 rv_amplitude_robust$<20$ km s-1 | 0.01 | 3.5 | 8.9 | 74.8 | 17.4
G4 rv_amplitude_robust$\geq 20$ km s-1 | 0.14 | 5.8 | 9.0 | 66.8 | 29.7
G4 twins | 0.00 | 1.9 | 9.1 | n/a | 175.5
Figure 10: Ratio of the Gaia SB1 $P_{\rm{orb}}$ to the ASAS-SN $P_{\rm{rot}}$
versus $P_{\rm{orb}}$. The lower dashed line is for
$P_{\rm{orb}}=P_{\rm{rot}}$ and the upper dotted line is for
$P_{\rm{orb}}=2P_{\rm{rot}}$. The colorful points have orbital scores $>0.587$
and the grey background points have orbital scores $\leq 0.587$.

Figure 10 shows the ratio of the Gaia SB1 orbital period, $P_{\rm{orb}}$, to
the ASAS-SN rotational period, $P_{\rm{rot}}$, as a function of $P_{\rm{orb}}$
for each group. We expect many rotational variables to be tidally synchronized
binaries so we include lines at $P_{\rm{orb}}=P_{\rm{rot}}$ and
$P_{\rm{orb}}=2P_{\rm{rot}}$. There is no physics that would yield the
$P_{\rm{orb}}=2P_{\rm{rot}}$ ratio, but it is difficult to measure an orbital
period to be half it’s actual value. Therefore, rotators lying on this line
likely have a reported rotational period aliased to half the true value (see
below).

Many of the widely-scattered points are likely due to incorrect Gaia periods
(Jayasinghe et al. 2022 found for $\sim 11$% of their detached eclipsing
binary sample that the orbital periods from Gaia SB1 disagreed with values
from ASAS-SN). To verify this, we used orbital score values from Bashi et al.
(2022) assigned to each Gaia SB1, where the score (ranging from 0 to 1)
corresponds to the validity of the orbital solution. They recommend a "clean
score limit" of $0.587$, which yields a sensitivity of 80% (that is, a sample
of SB1’s with scores $>0.587$ should have fewer than 20% false orbits). Figure
10 separates each group by whether their score is $>0.587$. For most groups,
the systems with higher scores lie overwhelmingly on one of the horizontal
lines, while the systems with lower scores account for a majority of the
scatter. Additionally, we assume that systems with reliable Gaia periods also
have reliable Gaia eccentricities, and expect the spectroscopic binaries in
our sample to be in close circular orbits after tidal synchronization. Most of
the non-synchronous systems are also reported to be eccentric ($e>0.1$).

MS1 tends to have somewhat shorter $P_{\rm{orb}}$ than $P_{\rm{rot}}$ and MS2
tends to be synchronized. G1 is strongly clustered on
$P_{\rm{orb}}=2P_{\rm{rot}}$. While this is typical of ellipsoidal variables
and contact binaries, our visual inspection of the light curves rules out
misclassification of such stars. Instead, G1 likely consists of rotators in
synchronized binaries with two dominant and roughly symmetrically-placed
starspots such that the frequency of their observed change in brightness is
doubled compared to their actual rotation period. G2 is strongly clustered on
$P_{\rm{orb}}=P_{\rm{rot}}$ and so consists of synchronized systems. G3 also
consists largely of stars with $P_{\rm{orb}}=P_{\rm{rot}}$. G3 systems with
low orbital scores and non-zero eccentricity contribute significant scatter,
especially for $P_{\rm{orb}}>P_{\rm{rot}}$. The G4b systems with both reliable
and unreliable periods are strongly clustered at $P_{\rm{orb}}<P_{\rm{rot}}$,
which means that G4b consists of younger giants in the process of tidal
synchronization with their companions ("subsynchronous binaries"). This is
consistent with the high-VSCATTER population of G4 having the lowest median
$v\sin i$ of all the high-VSCATTER rotators in Table 5.

We also checked the number of stars with RUWE $\geq$ 1.2 and the number of
Gaia astrometric binaries within each group. High RUWE and astrometric
binaries are fairly common in the main sequence groups, but, as shown in Table
3, there is little difference in the fractions with RUWE $\geq$ 1.2 or in the
fractions of flagged astrometric binaries between the rotators and their
twins. High RUWE and astrometric binaries are uncommon for all giant groups
and their twins, as is expected given that they are generally more distant
than the main sequence sample. Overall, the presence of a widely orbiting
companion or tertiary seems to be unimportant in creating rotational
variables.

##  4 Conclusions

Based on these results, we hypothesize that we have seven distinct groups of
rotational variables: MS1, MS2s, MS2b, G1/G3, G2, G4s and G4b. We summarize
our major findings about each group below:

  1. 1.

MS1 consists of main sequence K-M dwarfs with typical masses of $0.6$ to
$0.8M_{\odot}$ based on the PARSEC isochrones. They generally are not
(detectable) binaries, which for APOGEE means that the can only be in binaries
with periods $>10^{3}$ days (see Mazzola et al. 2020). That they generally lie
close to the main sequence means that few can have similar mass companions of
any period unless they are sufficiently separated to be a spatially resolved
binary. They rotate relatively slowly (median period of 18.12 days), although
not quite as slowly as their APOGEE twins. The majority are not very heavily
spotted.

  2. 2.

MS2s also consists of main sequence stars but with masses extending from the
bottom of the main sequence to $0.8M_{\odot}$. They are also not binaries up
to the same caveats as for MS1. They have faster rotation periods (median
period of 8.08 days) and this is reflected in the different $v\sin i$
distributions. They also have higher spot fractions.

  3. 3.

The properties of MS1 and MS2s are sufficiently disjoint that we are
reasonably confident they are distinct. Figure 11 shows the distribution of
MS1, MS2s, MS2b, and the Kepler sample of rotational variables from McQuillan
et al. (2014) in rotation period and $T_{\rm{eff}}$. In this figure, we have
separated MS2s from MS2b photometrically, as described in Section 3, to
maximize the sample size. The Kepler sample notably bifurcates in rotation
period at $T_{\rm{eff}}<4500$K, possibly due to a transient phase of rapid
mass-dependent spin-down across the gap (McQuillan et al., 2014). MS1 and MS2s
seem to follow the direction of this "McQuillan gap," and while the MS2s stars
span a wide range of temperatures, MS1 stars lie only at higher temperatures
where McQuillan et al. (2014) did not find a bimodality in the Kepler period
distribution.

The origin of the restricted temperature or mass range of MS1 as compared to
MS2 is not presently clear. One possibility is that it is simply an amplitude-
dependent selection effect against cool, long-period rotators, but a
completeness study of the ASAS-SN sample is a major undertaking beyond the
scope of this work. That they appear to lie on an extension of the McQuillan
et al. (2014) period gap suggests that one difference between MS1 and MS2s is
age, but we have no direct means of testing this other than the standard
gyrochronological assessment.

Figure 11: Distribution of the Kepler rotational variables from McQuillan et
al. (2014), MS1, MS2s (top) and MS2b (bottom) in period and $T_{\rm{eff}}$.

  4. 4.

MS2b clearly is a different population than MS1 or MS2s. They are
overwhelmingly synchronous binaries and many lie near the binary main
sequence, indicating that the companion is of similar mass. Like the MS2s
population, they seem to span the full dwarf mass range. They rotate a little
faster than the MS2s stars, and have higher estimated spot filling fractions.
In this case, however, the spectrum of the companion star may be contributing
to the inferred spot fraction. That most of the MS2b stars have short periods
suggests that the scatter of Kepler stars to similar periods is also dominated
by binaries, consistent with the findings in Simonian et al. (2019).

  5. 5.

We suspect that G1 and G3 are really a single group of heavily spotted red
giants in synchronized binaries (which we label G1/3 going forward). The G1
systems all have $P_{\rm{orb}}/P_{\rm{rot}}=2$ (see Figure 10), which means
that the true rotational period is twice that reported. If we shift G1 in
rotational period, they largely overlap with G3, and they have very similar
luminosities, temperatures and gravities. The are not, however, truly
identical after correcting the rotation period. The G1 stars are more heavily
spotted (median $f_{\rm{spot}}$ of 28% versus 16%) and have larger rotation
velocities (median $v\sin i$ of 38 versus 21 km/s). Since the rotational
periods are similar, the higher $v\sin i$ suggests that the G1 stars are
probably viewed more edge on. G1/3 stars are likely RS CVn stars (Hall, 1976).
Leiner et al. (2022) noted that the rotation periods of RS CVn lie between
1-100 days and with magnitudes $-1<M_{G}<5$, both consistent with our sample
of G1/3 stars.

  6. 6.

The G2 group likely consists of sub-subgiants (SSGs). Leiner et al. (2022)
found SSGs to have rotation periods overwhelmingly $<20$ days, consistent with
our G2 sample, and with $3<M_{G}<5$. The G2 sample spans a somewhat broader
range of $2<M_{G}<5$. Leiner et al. (2022) also noted that their total RS CVn
and SSG sample had increasing luminosity with rotation period, due to the
necessity of a more massive companion to maintain tidal interaction in wider
systems. In the left panel of Figure 1 we see a similar upward trend in the
G1/3 and G2 groups. Finally, Leiner et al. (2022) suggests a limit of
$P\lesssim 30$ days as a tidal circularization period for SSGs and the least
luminous RS CVn, consistent with our boundary between the G2 and G4 groups
(boundary #3 in Table 1) at a period of $10^{1.5}\simeq 30$ days.

  7. 7.

The G4s stars with low radial velocity scatter are likely recent merger
products.

  8. 8.

The G4b stars with high radial velocity scatter are sub-synchronous binaries
(Figure 10), beginning to tidally interact as they expand toward their
companion. They are of intermediate luminosity (and so have intermediate
evolution timescales) compared to G1/3 and G2, but have sufficiently wide
orbits that their spin-up timescales are shorter than their evolutionary
timescales. Neither the G4s recent mergers nor the G4b sub-synchronous
binaries seem to have been previously recognized.

There is enormous scope for expanding on this population study of rotational
variables. First, the ASAS-SN sample itself has considerable room for growth
since the current sample is largely based on the older V-band ASAS-SN data and
a small portion of the newer g-band data. With the additional data it should
not only be possible to expand the sample considerably but it should also be
possible to push to lower amplitudes. Rotational variables from brighter
surveys like ASAS (Pojmanski, 2002) or KELT (Oelkers et al., 2018) could also
be added, as well as the lower amplitude systems found by Kepler (e.g.
McQuillan et al. 2014) or TESS (Ricker et al., 2014). Except for the well-
studied Kepler field, there is less reason to expand to fainter stars because
these will generally lack the ancillary spectroscopic data needed to study
rotation rates, binarity, or composition. The biggest immediate return is
likely from including brighter systems, since this should significantly
increase the numbers of systems with Gaia spectroscopic orbits, the area of
comparison where our samples are smallest.

Many of our conclusions rely primarily on the availability of APOGEE
parameters, namely, VSCATTER, $v\sin i$, and $f_{\rm{spot}}$. APOGEE had
specific targeting criterion which could bias our results, but we believe we
cover any such biases both through confirming our observations in APOGEE data,
if with lower precision, with information from Gaia, and through comparison
with the twin sample. APOGEE targeting was based on photometry (Majewski et
al., 2017), so comparisons with the twins should be independent of APOGEE
targeting choices.

Additionally, APOGEE’s spectroscopic data is significantly limited in
availability compared to the size of our initial sample (APOGEE includes only
$\sim 4.4$% of the ASAS-SN rotational variables from this work). However, the
numbers of stars with ancillary spectroscopic data will rapidly increase with
the SDSS Milky Way Mapper (Kollmeier et al., 2017) extension of the APOGEE
survey (Majewski et al., 2017) and Gaia DR4. The sources of the spectroscopic
data could also be expanded to include surveys such as LAMOST (Cui et al.,
2012) and DESI (Cooper et al., 2022; DESI Collaboration et al., 2022). The
biggest impact will likely be Gaia DR4 because it will provide a huge
expansion of the numbers of systems with spectroscopic orbits and provide the
actual RV measurements, which are needed to understand many of the rejected
orbital solutions from DR3. In particular, there are suggestions from Figure
10 that the G2 group might be slightly sub-synchronous, that the G3 group is a
mixture of synchronized and unsynchronized systems, and that the sub-
synchronous rotation of the G4b group is correlated with period. For now, we
conservatively call the G2 and G3 groups synchronized, but with better orbital
solutions we may find that the situation is more complex.

We also did not explore the compositions of the variables both between the
groups and with their twins. The underlying reason is that APOGEE does not
include stellar rotation in its models of the giants for computational reasons
(Holtzman et al., 2018). This leads to biases on the inferred parameters
because the broader lines created by the rapid rotation are interpreted as
some other physics. We see this here in the temperature and $\log g$ offsets
between the giant groups and their twins. However, this problem also biases
the abundances, and Patton et al. (2023) find that apparent abundance
anomalies are a means of flagging rapidly rotating APOGEE giants. Trying to
understand how this problem would affect any elemental comparison seemed
beyond the scope of this paper. This issue might also be one reason to include
information from the large, lower resolution spectroscopic surveys like LAMOST
and DESI, where the line broadening in giants due to rotation would be
irrelevant.

## Acknowledgements

The authors thank Las Cumbres Observatory and its staff for their continued
support of ASAS-SN. CSK is supported by NSF grants AST-1814440 and
AST-1908570. Support for TJ was provided by NASA through the NASA Hubble
Fellowship grant HF2-51509 awarded by the Space Telescope Science Institute,
which is operated by the Association of Universities for Research in
Astronomy, Inc., for NASA, under contract NAS5-26555. LC acknowledges support
from TESS Cycle 5 GI program G05113 and NASA grant 80NSSC19K0597.

ASAS-SN is funded in part by the Gordon and Betty Moore Foundation through
grants GBMF5490 and GBMF10501 and by the Alfred P. Sloan Foundation through
grant G-2021-14192 to the Ohio State University, the Mt. Cuba Astronomical
Foundation, the Center for Cosmology and AstroParticle Physics (CCAPP) at OSU,
the Chinese Academy of Sciences South America Center for Astronomy (CAS-SACA),
and the Villum Fonden (Denmark). Development of ASAS-SN has been supported by
NSF grant AST- 0908816, the Center for Cosmology and Astroparticle Physics,
Ohio State University, the Mt. Cuba Astronomical Foundation, and by George
Skestos.

## Data Availability

All data used in this study are public.

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