# Stellar Cycles in Fully Convective Stars and a New Interpretation of Dynamo
Evolution

Zackery A. Irving  School of Physics and Astronomy,
University of Southampton,
University Road, Southampton SO17 1BJ, UK  Center for Astrophysics — Harvard &
Smithsonian,
60 Garden Street, Cambridge, MA 02138, USA  Steven H. Saar  Center for
Astrophysics — Harvard & Smithsonian,
60 Garden Street, Cambridge, MA 02138, USA  Bradford J. Wargelin  Center for
Astrophysics — Harvard & Smithsonian,
60 Garden Street, Cambridge, MA 02138, USA  José-Dias do Nascimento, Jr
Center for Astrophysics — Harvard & Smithsonian,
60 Garden Street, Cambridge, MA 02138, USA  Univ. Federal do Rio G. do Norte,
UFRN, Dep. de Física,
CP 1641, 59072-970, Natal, RN, Brazil

###### Abstract

An $\alpha\Omega$ dynamo, combining shear and cyclonic convection in the
tachocline, is believed to generate the solar cycle. However, this model
cannot explain cycles in fast rotators (with minimal shear) or in fully
convective stars (no tachocline); analysis of such stars could therefore
provide key insights into how these cycles work. We reexamine ASAS data for 15
M dwarfs, 11 of which are presumed fully convective; the addition of newer
ASAS-SN data confirms cycles in roughly a dozen of them, while presenting new
or revised rotation periods for five. The amplitudes and periods of these
cycles follow $A_{\rm cyc}\propto P_{\rm cyc}^{0.94\pm 0.11}$, with $P_{\rm
cyc}/P_{\rm rot}\propto{\rm Ro}^{-1.02\pm 0.06}$ (where Ro is the Rossby
number), very similar to $P_{\rm cyc}/P_{\rm rot}\propto{\rm Ro}^{-0.81\pm
0.17}$ that we find for 40 previously studied FGK stars, although $P_{\rm
cyc}/P_{\rm rot}$ and $\alpha$ are a factor of $\sim$20 smaller in the M
stars. The very different $P_{\rm cyc}/P_{\rm rot}$-Ro relationship seen here
compared to previous work suggests that two types of dynamo, with opposite Ro
dependences, operate in cool stars. Initially, a (likely $\alpha^{2}$ or
$\alpha^{2}\Omega$) dynamo operates throughout the convective zone in mid-late
M and fast rotating FGK stars, but once magnetic breaking decouples the core
and convective envelope, a tachocline $\alpha\Omega$ dynamo begins and
eventually dominates in older FGK stars. A change in $\alpha$ in the
tachocline dynamo generates the fundamentally different $P_{\rm cyc}/P_{\rm
rot}$-Ro relationship.

M dwarf stars(982) — Stellar activity(1580) — Stellar magnetic fields(1610) —
Stellar rotation(1629)

##  1 Introduction

Stellar cycles are (quasi-)periodic fluctuations in the intrinsic brightness
of stars due to changing internal magnetic fields. Such cycles are most
clearly manifest visually by the appearance of starspots, however are more
easily observed by studying chromospheric (Ca ii H and K) or coronal (X-ray)
variability - both of which are more monotonically correlated with magnetic
activity. Indeed, in the optical waveband the solar cycle has an amplitude of
the order of just $0.1\%$, while the coronal (X-ray) variability varies by a
factor of $\sim$6 (Judge et al., 2003). Sunspot observations first led to the
discovery of the 11-yr solar cycle almost 200 years ago (Schwabe, 1844). Over
half a century later, it was realised that sunspots are a manifestation of
solar magnetic activity (Hale, 1908, 1913), and it is now understood that this
11-yr cycle is just one-half of the 22-yr polarity reversal cycle (Hale et
al., 1919; Babcock, 1959, 1961). However, while the solar cycle has an average
period of 11 years, due to the quasi-periodic nature of stellar cycles it is
not uncommon for a given cycle to have a period between 9-13 years (see, for
example, Donahue & Baliunas, 1992).

Presently, stellar cycles are believed to be a result of dynamo processes
(see, for example, Parker, 1955; Baliunas et al., 1995; Dikpati & Charbonneau,
1999; Kitchatinov & Rüdiger, 1999). A dynamo process is a physical mechanism
in which kinetic energy is converted into magnetic energy via inductive
effects of motions in an electrically conducting fluid, in a self-regenerating
process. The solar cycle can be well described by an $\alpha\Omega$ dynamo,
where shearing (due to differential rotation) transforms a poloidal field into
a toroidal one ($\Omega$ effect), and cyclonic convection (due to Coriolis and
magnetic forces) then restores the poloidal field with reversed polarity
($\alpha$ effect) (Parker, 1955; Babcock, 1961; Leighton, 1969). Comparing our
Sun’s cycle to the cycles of 4454 cool stars, Boro Saikia et al. (2018) found
that the solar dynamo cycle is not uncommon. Many of the finer details of
these dynamo processes, however, remain poorly understood - even in the case
of our Sun (see, for example, Ossendrijver, 2003; Charbonneau, 2010).

Differential rotation (DR) is a key component of the solar $\alpha\Omega$
dynamo model, with current models suggesting that the $\Omega$ effect takes
place in the tachocline layer at the base of the convective envelope (e.g.,
Gilman & Fox, 1997; Dikpati & Charbonneau, 1999). In the case of our Sun, DR
causes the equator to rotate more quickly than the poles; this is referred to
as solar-like DR. Some theoretical work (e.g., Kitchatinov & Rüdiger, 1999)
and observations (e.g., Barnes et al., 2005) suggest that DR has little or no
dependence on rotation, while other observations indicate that the two are
closely tied (e.g., Donahue et al., 1996). Saar (2011) updated the work of
Donahue et al. (1996) and confirmed that DR appears to scale almost linearly
with rotation period, at least for relatively slow rotators. Saar (2011)
further argued that the results of Barnes et al. (2005) were misleading as
they included close binaries, where large tidal forces can freeze out DR and
cause solid-body rotation.

For stars with very slow rotation, numerical simulations (e.g., Karak et al.,
2020; Käpylä, 2021) suggest that DR should be antisolar, where the poles
rotate more quickly than the equator, although the point of transition from
solar to antisolar DR is still debated. Antisolar DR in the classical
$\alpha\Omega$ dynamo model does not produce polarity reversal, and so such
stars would not be expected to have solar-like cycles. Nonetheless, there is
strong evidence that some slowly rotating stars do have cycles (see, for
example, Suárez Mascareño et al., 2016).

Whether or not DR depends on rotation rate, dynamo processes in more rapidly
rotating stars are expected to differ fundamentally from those in the Sun,
possessing $\alpha^{2}\Omega$ or even $\alpha^{2}$ dynamos (e.g., Kitchatinov
& Rüdiger, 1999), where the dynamo actions originate from surface shear or in
the convection zone, respectively. Kitchatinov & Rüdiger (1999) suggested that
these $\alpha^{2}$ dynamos should not result in cyclic activity, and could
explain why Baliunas et al. (1995) found an absence of activity cycles in
rapidly rotating young stars. However, more recent studies (e.g., Suárez
Mascareño et al., 2016; Lehtinen et al., 2016) have shown that activity cycles
are observed in fast and slow rotating stars alike. In addition, it is
difficult to explain the observed increase in magnetic activity with rotation
under the assumption that DR is constant (e.g., Durney et al., 1993; Saar,
2011; Wright et al., 2011). It should be noted, however, that DR studies can
only measure surface DR, not DR in the tachocline, and there is no guarantee
that the two are the same.

A convective envelope is another key ingredient of the $\alpha\Omega$ dynamo.
Convective envelopes surrounding a radiative core are a staple of cool stars
earlier than type $\sim$M3.5, while late M-type stars are believed to be fully
convective (Chabrier & Baraffe, 1997); as such, these stars cannot support
solar-like $\alpha\Omega$ dynamos (since they have no tachocline). Instead,
late M-type stars are expected to support $\alpha^{2}$ dynamos, which, as
already discussed, are not expected to exhibit activity cycles. However, there
is some evidence suggesting that $\alpha^{2}$ dynamos may exhibit activity
cycles under certain conditions (e.g., Rüdiger et al., 2003; Gastine et al.,
2012; Käpylä et al., 2013). In addition to this, activity cycles have also
been observed in several stars which are believed to be fully convective (see,
for example, Suárez Mascareño et al., 2016; Wargelin et al., 2017).

Interestingly, there is evidence suggesting that stellar activity cycle
periods are (weakly) correlated with rotation period (see, for example, Noyes
et al., 1984; Brandenburg et al., 1998; Saar & Brandenburg, 1999; Böhm-
Vitense, 2007; Suárez Mascareño et al., 2016; Boro Saikia et al., 2018; Wright
et al., 2018) \- M-type stars included. In their two paper series, Brandenburg
et al. (1998); Saar & Brandenburg (1999) found that FGK stars typically occupy
one of three branches: termed the “superactive” (S), “active” (A), and
“inactive” (I) branches, with our Sun residing approximately between the A and
I branches. Saar & Brandenburg (1999) also found hints of a tentative fourth,
“transitional” (T), branch connecting the A and S branches. Recent studies
(e.g., Lehtinen et al., 2016; Distefano et al., 2017; Boro Saikia et al.,
2018) have found further evidence for the existence of this T branch, while
raising doubts about the A branch, which has become increasingly less clear as
more cycles have been found.

To develop our understanding of the dynamo processes responsible for stellar
activity cycles, active stars with a wide range of physical parameters should
be observed. Active late M-type stars are of particular interest since their
cycles cannot be adequately explained by current dynamo models. Due to how
intrinsically faint M stars are, however, they were underrepresented in
earlier surveys (e.g., Wilson, 1978; Baliunas et al., 1995). Furthermore, even
in recent studies (e.g., Suárez Mascareño et al., 2016; Wright et al., 2018),
rapidly rotating M stars are still poorly represented. Rapidly rotating late M
stars could be especially revealing, as their deep convective zones combined
with short rotation periods may allow for unique dynamos.

Herein, we reexamine the 15 latest type stars listed in Suárez Mascareño et
al. (2016) (excluding GJ 526, discussed below), who analyzed $\sim$9 years of
ASAS-3 (Pojmanski, 1997, 2002) photometric monitoring data and suggested that
13 of them, several of which are fully convective, showed evidence for cyclic
activity. We use those same data and include new data from ASAS-SN (Shappee et
al., 2014; Kochanek et al., 2017), approximately doubling the period of
monitoring; in some cases, we also include TESS data to constrain shorter
rotation periods. Note, however, that there is some uncertainty regarding the
spectral types, and some stars may not be fully convective. GJ 526, for
example, is listed in Suárez Mascareño et al. (2016) as M4, but M2 in SIMBAD
and Suárez Mascareño et al. (2015). Moreover, its V–KS color (see Section 2.6)
also indicates it is earlier than M4. In any case, GJ 526 is often too bright
for ASAS-SN monitoring (g=7.6). A few stars may also be unresolved binary
systems (see Section 2.3), further complicating spectral type determinations,
but roughly a dozen of our stars are almost certainly fully convective (see
Section 2.4).

While photometric measurements are not as tightly correlated with magnetic
activity as other metrics (e.g. chromospheric line intensities and coronal
X-ray emission), and can suffer from counterbalancing effects between
starspots and the surrounding faculae, they are technologically easier to
make. Furthermore, there is evidence suggesting that in active, rapidly
rotating stars, chromospheric emission can become saturated (e.g., Vilhu,
1984), while starspot (and hence photometric) variability can continue to rise
with increasing rotation (e.g., O’dell et al., 1995; Krishnamurthi et al.,
1998).

In Section 2, we discuss the data used and the analysis performed to identify
rotation periods and activity cycles in 15 M-type stars from their photometric
time series. Results are presented in Section 3, and our interpretation of
these results is provided in Section 4. Our final thoughts are then given in
Section 5.

##  2 Method

###  2.1 Data Collection

ASAS-3 optical photometry data were downloaded from the ASAS All Star
Catalogue.111http://www.astrouw.edu.pl/asas/?page=aasc All ASAS-3 data use
V-band filters, and individual measurements are ascribed one of four grades: A
- best data; B - mean data; C - A and B with not measured indication; D -
worst data. We opted to use only measurements with an A or B grade. For most
stars in our sample, the MAG_2 (four pixel) aperture data had the least
scatter, and therefore the lowest empirical uncertainty; to be consistent, we
therefore used this aperture for all our stars. We note, however, that the
choice of aperture can sometimes have a significant effect on the inferred
cycle period, and could explain (at least partly) why our results sometimes
differ from those of Suárez Mascareño et al. (2016).222We further note that
the aperture columns are sometimes arranged inconsistently between different
stars, and so care needs to be taken to ensure the same aperture is being used
for all stars. ASAS-3 has a FWHM PSF of typically 23”, corresponding to
approximately 1.5 pixels.

ASAS-SN optical photometry data were collected using the light curve
server.333https://asas-sn.osu.edu Early ASAS-SN data were gathered using
V-band filters, while the most recent data use g-band. The ASAS-SN light curve
server does not perform proper motion corrections, unlike the ASAS All Star
Catalogue, so proper motion corrected coordinates had to be specified for each
query. ASAS-SN has a 16” FWHM PSF, so we allowed for a proper motion of up to
1” either side of the input coordinates; extractions use a two pixel (16”)
aperture. We then compiled these proper motion corrected light curves into
single light curves for each star once we had downloaded all available data.

TESS optical photometry data were downloaded using the Lightkurve Python
package (Lightkurve Collaboration et al., 2018; Astropy Collaboration et al.,
2018; Brasseur et al., 2019; Ginsburg et al., 2019), which accesses publicly
available data directly from the Mikulski Archive for Space Telescopes (MAST).
Only data products produced via their Science Processing Operations Center
Science Analysis Pipeline (Jenkins et al., 2016) were used. For each star, we
also used the pipeline aperture.

###  2.2 Data Preparation

####  2.2.1 ASAS-SN

Initially, the early V-band and recent g-band data were disjointed in the
compiled ASAS-SN light curves. We therefore needed to color correct these
data. Using the empirical UBVR-uvgr relation found by Kent (1985), and
improved by Windhorst et al. (1991), we can convert from V- to g-band
magnitudes using:

${\rm V}={\rm g}-0.03-0.42({\rm g-r}),$ (1)

where V is the V-band magnitude, g is the g-band magnitude, and ${\rm g-r}$ is
the g–r color index, given by:

${\rm g-r}=1.02({\rm B-V})-0.22,$ (2)

where ${\rm B-V}$ is the B–V color index. However, M stars are relatively cool
objects, and emit mostly towards the red end of the optical spectrum; as such,
the B–V color index is not the best diagnostic for these stars. In addition,
the relation given in Equation 1 was derived using predominantly O–K stars.
After color correcting V-band data using these formulae, residual offsets were
therefore required. These offsets were computed by comparing the means of
overlapping g- and V-band data.

Further internal cross-calibration is then required because ASAS-SN data are
collected using multiple telescopes positioned around the world, with small
but sometimes significant differences between nominally identical instruments.
The ASAS-SN pipeline automatically adjusts for these differences, but color-
dependent residuals may remain, as we found for our reddish M stars.

To correct for differences between different ASAS-SN telescopes, we first
selected the telescope with the largest number of measurements, thus defining
our reference data set. We then calibrated the remaining telescopes
sequentially in descending order of relative overlap. To calibrate a
telescope, we applied an offset such that the averages of overlapping data
points between the telescope and the reference data set were equal. Once a
telescope had been calibrated, it then became part of the reference data set
for subsequent calibrations. Fortunately, we did not find any instances of
telescopes with no overlapping data. However, if a particular telescope was
found to show unusual behavior within a given light curve (relative to the
other telescopes) then this telescope’s data were removed. For example,
telescope “bA” showed an unusually large scatter for GJ 447, obscuring its
cyclic modulations, but appeared perfectly normal for GJ 581, GJ 729, and GJ
849; this telescope’s measurements were therefore removed in the former case,
but kept in the latter three.

####  2.2.2 TESS

Uncorrected TESS light curves are dominated by scattered light background
caused by spacecraft motions. To reduce this background, we used Lightkurve to
perform pixel level decorrelation (PLD), which has been shown to be effective
at removing systematic noise from both Spitzer (Deming et al., 2015) and K2
(Luger et al., 2016, 2018) data. PLD works by creating masks of the object of
interest and the surrounding background, and then using linear regression to
model trends in the background. After subtracting this noise model from the
uncorrected light curve, systematic noise is greatly reduced.

####  2.2.3 Outlier Removal

After performing the analysis described above on ASAS-SN and TESS data, we
then removed outliers from all our data sets. For ASAS-3 data, we clipped each
observing season to remove measurements more than two standard deviations from
the mean.

For ASAS-SN data, we initially separated the data by telescope (and
consequently, by filter). We then binned each filter’s telescope-separated
data according to the star’s rotation period (since observing seasons were not
always well-defined when data were separated by telescope). ASAS-SN typically
makes between 1–3 observations of each region of the visible sky each night (3
being the nominal number); binning ASAS-SN data into bins of less than 2d
therefore becomes volatile and unreliable due to the small number of data
points per bin. Consequently, we chose to bin stars with rotation periods of
less than 24d differently, instead using bin sizes of
10$P_{\textrm{rot}}$.444For a star with $P_{\rm rot}=24$d, this is
approximately equal to an observing season. In all other cases, we used the
rotation period. We then clipped these data using a tolerance of two standard
deviations from the mean. Next, we repeated this process using bin sizes of
$P_{\rm rot}$ if $P_{\rm rot}\leq 24$d, or $P_{\rm rot}/12$ otherwise, with a
tolerance of two and a half standard deviations from the mean (note we used a
looser tolerance for these smaller bins).

After removing outliers from the telescope-separated ASAS-SN data, we
recalibrated these telescopes and recorrected any residual offset between the
V- and g-band data resulting from outlier removal. We then binned the entire
calibrated light curve using bin sizes of $P_{\rm rot}$ if $P_{\rm rot}\leq
24$d, or $P_{\rm rot}/12$ otherwise, and removed measurements more than two
and a half standard deviations from the mean. In total, this usually resulted
in $\sim$10% of the data being removed. Finally, we recalibrated these
telescopes again and recorrected any residual offset between the V- and g-band
data.

Since TESS data have such high cadence, we simply binned these data into 0.1d
bins and removed outliers using a tolerance of two and a half standard
deviations from the mean.

###  2.3 Binary Systems and Crowded Fields

When assessing our results it is important to keep in mind that some target-
star intensity extractions may suffer from contamination by binary companions
or other stars that appear nearby in the sky. Winters et al. (2019) list data
from an all-sky volume-limited (25pc) survey for stellar companions to 1120
M-dwarf primaries, and serves as our principal reference regarding binarity
since it includes all our stars of interest except for Proxima (GJ 551). Two
systems are listed as binaries: GJ 234 and GJ 896.

The GJ 234 pair has a semi-major axis of 1.1” (e.g., Gatewood et al., 2003;
Kervella et al., 2019) but contamination by the secondary is small, with
estimated intensity ratios of 100:5.3 in the B band (Gatewood et al., 2003),
100:5.9 in V (Henry et al., 1999), and 100:25 in K (Coppenbarger et al.,
1994). GJ 896 A and B had a separation of 5.35” in 2004 (Winters et al., 2019)
with an estimated V magnitude of 10.29 for component A and 2.12 magnitudes
dimmer for B (100:14 ratio). Davison et al. (2015) measured a separation of 7”
and list the spectral types as M3.5+M4.0. These stars are mostly blended in
ASAS and ASAS-SN data, and it is difficult to determine if the lack of an
obvious cycle in GJ 896A is intrinsic, from contamination by the secondary
star, or variable extraction efficiencies as the pair’s center of emission
moves around its center of mass and proper motion.

The multiplicity of GJ 273 is somewhat unclear, with Vrijmoet et al. (2022)
listing it as a double based on Gaia data, while Winters et al. (2019)
conclude that the putative companion is probably an unassociated background
object. Astudillo-Defru et al. (2017) analyzed HARPS RV data and report that
GJ 273 has four planets, without any mention of spectral evidence for a
companion; in our analysis, we treat this star as a single.

Although GJ 273, along with GJ 317 and GJ 406, were not included in their
study of binarity in nearby stars using data from HIPPARCOS and Gaia DR2,
Kervella et al. (2019) calculated minimum (inclination dependent) masses for
companions to the rest of the stars in our sample. Apart from the two known
binary systems, minimum masses were no larger than $0.25M_{J}$ assuming 0
inclination and orbital radii of $r=1$ AU, with mass scaling as $\sqrt{r}$.
All our stars have clear rotational modulation, indicating that inclination
cannot be too small (assuming rotational and orbital co-planarity), so it
seems unlikely that any of the stars under study here (apart from GJ 234 and
GJ 896) have stellar mass companions. Many of these stars are also the
subjects of RV studies, which would readily reveal binarity unless the orbital
inclination was very small. We particularly mention GJ 358 because of its
remarkable changing cycle periods: Bonfils et al. (2013) explicitly treat this
star as a single after excluding known and discovered spectroscopic binaries
and visual pairs from their analysis of HARPS data.

Intensity extractions can also be affected by stars that appear nearby on the
sky even if they are not physically associated. GJ 406 has a very large proper
motion (4.72”/yr) and has recently passed quite close to two objects that may
provide contaminating flux. Likewise, Proxima lies in a very crowded part of
the sky and is currently moving at 3.86”/yr within a group of stars that,
although dimmer, may be bright enough to cause significant contamination
within the ASAS-3 and ASAS-SN extraction regions. Those effects should be
slowly varying but complicate interpretation of observations and may be at
least partly responsible for the general brightening seen in Proxima’s ASAS
and ASAS-SN data.

###  2.4 Separating Partially from Fully Convective Stars

A further consideration to make when assessing our results is which stars are
fully convective. As mentioned in Section 1, spectral type is often used to
distinguish partially from fully convective M stars. However, spectral type
determinations are not always a robust metric, and the boundary between sub-
types is often blurred. Indeed, some of these difficulties were mentioned in
Section 1, and discussed in detail in Section 2.3. Fortunately, recent studies
have alleviated some of the ambiguity in identifying fully convective M stars.

Using data from Gaia DR2, Jao et al. (2018) identified a gap in the color-
magnitude diagram for the lower main sequence, which they proposed could be
used to distinguish partially from fully convective M stars. van Saders &
Pinsonneault (2012) first predicted this gap as the result of nonequilibrium
3He fusion prior to stars becoming fully convective; using numerical
simulations, Feiden et al. (2021) were able to reproduce this gap. Using this
distinction, specifically in the V–KS vs. MV plane, we identify 11 of our 15 M
stars as being fully convective (see Table 1). We note, however, that for a
few high proper motion stars not included in Gaia DR2, we had to use DR3
parallaxes.

###  2.5 Time Series Analysis Techniques

Herein, we predominately use traditional time series analysis techniques,
specifically Lomb–Scargle (L–S) periodograms (Lomb, 1976; Scargle, 1982) for
two key reasons. Firstly, more advanced techniques, such as Gaussian processes
(GPs), come with a significant computational expense. Depending on the number
of covariance function hyperparameters, the number of mean function
parameters, the priors, and the size of the data set itself, fitting a GP
model can easily take several orders of magnitude times the time it takes to
obtain an L–S periodogram. Secondly, as we discuss later, the literature shows
that stellar cycles often appear to be well approximated by sines. We note,
however, two exceptions: GJ 358 and GJ 551 (Proxima), whose cyclic modulations
cannot adequately be described by one or two single-period sinusoids; these
stars are therefore better suited to a GP analysis.

To compute the false alarm probabilities (FAPs) on our detections, we use the
approximation proposed by Baluev (2008), often referred to as the Baluev
estimate. We compared this method with a more robust bootstrap simulation
method for half a dozen ASAS-3 and ASAS-SN light curves, and found that
results were usually within 10%, and many times within 5%; the largest
difference we found was ${\rm FAP}_{\rm Baluev}:{\rm FAP}_{\rm
bootstrap}\approx 0.79$.555Note, however, this result was obtained for GJ
285’s ASAS-SN light curve, where the peak in the periodogram was an order of
magnitude above either method’s 0.1% FAP power threshold. Due to the
computational expense of bootstrap simulations, without much improvement to
performance, we prefer the Baluev estimate. In our work, we consider a FAP of
$\leq 0.1\%$ to constitute a significant detection, the same threshold used in
Suárez Mascareño et al. (2016).

In theory, typical time series analysis techniques, such as fast Fourier
transforms (FFTs) and L–S periodograms, may not necessarily be the best tools
for the analysis of stellar cycles. Both are limited in that they model
signals with sinusoids; while this may be a good model for idealized systems,
it often fails in reality. Indeed, stellar cycles need not be sinusoidal in
nature, nor strictly periodic.

To identify stellar cycles in unevenly sampled light curves, Olspert et al.
(2018b) introduced a Bayesian generalized Lomb–Scargle periodogram with trend
(BGLST); using synthetic quasi-periodic data, they showed that their BGLST
performed better than the standard L–S periodogram. Moreover, in their follow-
up paper, Olspert et al. (2018a) further showed that GPs can be used
effectively in the analysis of stellar cycles - in their case using Ca ii H
and K data.

The greatest benefit of GPs over traditional time series analysis techniques,
such as the L–S periodogram, is their flexibility. Much like L–S periodograms,
GPs can be used with unevenly sampled data. However, GPs need not assume the
data contain a periodic, sinusoidal signal. The assumptions of a GP are
encoded in its mean and covariance functions, which, in theory, allow for any
number of arbitrary assumptions to be made.

Olspert et al. (2018a), in their study of stellar cycles, compared the
performance of a Bayesian harmonic (i.e. sinusoidal) regression model, and GP
models with periodic and quasi-periodic covariance functions. They found that
where traditional methods suggest double cycles cycles, their quasi-periodic
GP model often found only a single cycle; as the assumption of a quasi-
periodic signal has more physical justifications, Olspert et al. (2018a)
concluded that double cycles are rarer than initial results (e.g., Wilson,
1978; Baliunas et al., 1995; Brandenburg et al., 1998; Saar & Brandenburg,
1999) suggest.

However, while a quasi-periodic model is more physically justified, its
results are often not much of an improvement over a simple harmonic model.
Olspert et al. (2018a) identified harmonic signals in the light curves in 36
of stars; of these 36 stars, quasi-periodic signals were also identified in
26. Comparing these 26 quasi-periodic periods with their harmonic equivalents,
22 were comfortably within $3\sigma$. This suggests that while stellar cycles
need not be strictly periodic, nor sinusoidal, they can often be well
approximated by sines.

###  2.6 Determining Periods

####  2.6.1 Rotation Periods from Photometric Time Series

To determine rotation periods we used L–S periodograms, as well as data from
both ASAS-SN and TESS. ASAS-3 data are noisier and lower cadence than ASAS-SN
data, and therefore provide no additional benefit for rotational studies; for
these reasons, ASAS-3 data are not included. The first step in our analysis
was to separate the ASAS-SN data by observing season. We then computed L–S
periodograms for each of these seasons, and, provided the period of the peak
was greater than 15d, used the tallest peak in each season’s L–S periodogram
to fit a sine model using the Levenberg–Marquardt (L–M) algorithm (a
sophisticated non-linear least-squares fitting algorithm, see Moré, 1978) \-
as shown in Figure 1 for GJ 54.1. Note that there were often no significant
peaks at $P_{\rm rot}/2$.

Figure 1: L–S periodograms (left) computed from ASAS-SN observing seasons
(right) on GJ 54.1. Dashed red lines (left) show the power thresholds for FAPs
of 0.1%. Dashed black lines (right) show optimized sine fits using the period
of the tallest peak in the corresponding periodogram.

We discarded periodograms computed from seasons with low data density, or
incomplete seasons,666In some cases, incomplete seasons were included if
multiple rotations were clearly shown. and identified significant peaks in
each periodogram. After disregarding anomalous peaks, like the broad peak at
approximately 180d in the third row of Figure 1, we inferred the rotation
period by computing the mean period of the remaining peaks, and estimated the
uncertainty as the standard deviation on this mean.

However, if the rotation period of a star was sufficiently small (below
$15$d), we used the higher cadence but shorter duration TESS data (TESS
observation windows typically span $\sim$27d) to better constrain this period.
If a star had multiple TESS observations, we found that in every case the
inferred periods were identical to within the (tightly constrained)
uncertainties.

Our analysis using TESS data proceeded in much the same way as our analysis
using ASAS-SN data; an example is shown in Figure 2 for GJ 234. Due to our
currently limited understanding of TESS systematics, however, the
identification of rotation periods is restricted to below approximately half a
TESS observing window (hence our 15d cutoff) - see, for example, Anthony et
al. (2022); Claytor et al. (2022).

Figure 2: L–S periodogram (top) computed from the 2020 TESS observation of GJ
234 (bottom). The dashed red line (top) shows the power threshold for a FAP of
0.1%; dashed black line (bottom) shows the optimized sine fit using the period
of the tallest peak in the periodogram.

To infer the rotation period of a star from its TESS light curve, we fit a
Gaussian to the tallest peak in its periodogram. Our estimate for the rotation
period was thus the mean of this Gaussian, with the uncertainty given by its
standard deviation. Since we were often limited to one or two TESS
observations, following the same procedure as described above for ASAS-SN
resulted in unrealistically small errors in some cases, and no errors in
others; we therefore adopted this alternative method. We note that when we
tested this approach on ASAS-SN data, it resulted in larger uncertainties due
to the broad peaks in these periodograms. However, TESS observations only span
$\sim$27d, and so these broadening effects (e.g., resulting from differential
rotation and multiple star spot generations) are minimal. TESS is also a
space-based observatory, with a much higher signal-to-noise ratio than ASAS-
SN. Given these differences, we deem this alternative method to be
appropriate. Results are presented in Table 1 and discussed in Section 3.

####  2.6.2 Cycle Periods from Photometric Time Series

To determine cycle periods we typically use L–S periodograms incorporating
data from both ASAS-3 and ASAS-SN; exceptions to this are Proxima (for which
we also have ASAS-4 data and use GPs), and GJ 358 (for which we also use GPs)
- see Section 2.5 for more details.

For each star, we computed a L–S periodogram from its light curve. We then
identified significant peaks in those periodograms with periods of more than
1.5 years, and used the periods of those peaks to fit sine functions to the
light curve; see Figures 3 and 4 for examples. All ASAS-3/-4 light curves and
associated periodograms (16 images) are provided in Figure Set 1; likewise for
ASAS-SN (15 images) in Figure Set 2.

Fig. Set1. ASAS-3/-4

Figure 3: L–S periodogram (top) computed from ASAS-3 data on GJ 628 (bottom).
The FAP $=0.1\%$ power threshold is shown by the dashed red line, while the
dashed blue line shows the time-span of the data. The blue markers show the
mean values (dates and magnitudes) for each observing season. The period of
the labeled peak in the periodogram was used to create the sine fit in the
bottom plot. The complete figure set (16 images) is available in the online
journal (Figure Set 1).

If a periodogram contained multiple significant peaks, we fit sine functions
using each peak and then combined these functions to create a superposition.
If the fit of the superposition was worse than the fit of any single
component, then the worst fitting components were removed until the fit of the
superposition was better than the fit of any single component, or only the
best fitting component was remaining. If the period(s) of the remaining
component(s) was within the period of observation, then we deem this cycle to
be “well-defined” - and poorly constrained otherwise.

Fig. Set2. ASAS-SN

Figure 4: Periodogram and ASAS-SN data for LP 816-60. The complete figure set
(15 images) is available in the online journal (Figure Set 2).

##  3 Results

Rotation and cycle periods are presented in Table 1. All results from this
work have FAPs $\leq 0.1\%$.

.
ASAS-3 Previous ASAS-3 Results ASAS-SN Star Sp. Type V–KS $P_{\rm rot}$
$P_{\rm rot}$ Lit. Ref. $P_{\rm cyc}$ $A_{\rm cyc}$ $P_{\rm cyc}$ $A_{\rm
cyc}$ $P_{\rm cyc}$ $A_{\rm cyc}$ Phase (d) (d) (yr) (mmag) (yr) (mmag) (yr)
(mmag) Match GJ 358∗ M3 4.638 $25.2\pm 1.1$ $25.2\pm 0.1$ 2 4.7 $\pm$ 0.9 21.9
$\pm$ 1.9 $4.9\pm 0.1$ $20.5\pm 0.9$ 1.7 $\pm$ 0.2 10.7 $\pm$ 0.6 poor
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
$\cdots$ $\cdots$ 3.2 $\pm$ 0.5 4.0 $\pm$ 0.6 $\cdots$ GJ 581 M3 4.723
$141.6\pm 3.5$ $132.5\pm 6.3^{\rm c}$ 1 $\cdots$ $\cdots$ $6.2\pm 0.9$ 3.8
$\pm$ 0.9 3.8 $\pm$ 0.6 6.7 $\pm$ 0.6 $\cdots$ GJ 628 M3 4.997 $79.6\pm 2.6$
$119.3\pm 0.5$ 2 4.5 $\pm$ 1.3 8.1 $\pm$ 1.2 $4.4\pm 0.2$ 8.3 $\pm$ 1.1 3.0
$\pm$ 0.5 5.0 $\pm$ 0.5 good $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ 5.0 $\pm$ 1.2 3.8 $\pm$ 0.5
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
$\cdots$ $\cdots$ $\cdots$ long $\cdots$ $\cdots$ GJ 849∗ M3.5 4.772 $41.4\pm
1.3$ $39.2\pm 6.3^{\rm c}$ 1 10.3 $\pm$ 3.0 25.6 $\pm$ 1.5 $10.2\pm 0.9$ 12.4
$\pm$ 1.0 3.7 $\pm$ 0.6 6.7 $\pm$ 0.5 poor $\cdots$ $\cdots$ $\cdots$ $\cdots$
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ 10.8 $\pm$ 3.0 9.5 $\pm$
0.5 $\cdots$ GJ 896A∗ M3.5 4.847 $15.58\pm 0.06$ $15.3\pm 0.1$ 2 long $\cdots$
$\cdots$ $\cdots$ 10.1 $\pm$ 2.7 8.7 $\pm$ 0.7 $\cdots$ GJ 317∗ M3.5 4.947
$57.5\pm 1.7$ $\cdots$ $\cdots$ long $\cdots$ $5.2\pm 0.3$ 12.4 $\pm$ 1.1 2.3
$\pm$ 0.3 5.7 $\pm$ 0.6 $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ 10.1 $\pm$ 4.4 9.6 $\pm$ 0.6
$\cdots$ GJ 273 M3.5 5.015 $\cdots$ $115.9\pm 19.4^{\rm c}$ 1 5.8 $\pm$ 1.3
9.4 $\pm$ 1.2 $6.6\pm 1.3$ 7.7 $\pm$ 1.5 5.9 $\pm$ 1.4 10.1 $\pm$ 0.8 good GJ
729 M3.5 5.06 $2.9\pm 0.1$ $2.9\pm 0.1$ 2 5.7 $\pm$ 1.1 10.4 $\pm$ 1.3 $7.1\pm
0.1$ 5.3 $\pm$ 0.7 3.5 $\pm$ 1.1 6.8 $\pm$ 0.7 fair $\cdots$ $\cdots$ $\cdots$
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ 12.2 $\pm$ 6.7
8.1 $\pm$ 0.7 $\cdots$ LP 816-60 M4 5.259 $86.3\pm 02$ $67.6\pm 0.1$ 2
$\cdots$ $\cdots$ 10.6 $\pm$ 1.7f 8.2 $\pm$ 1.2f 2.0 $\pm$ 0.2 6.0 $\pm$ 0.8
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
$\cdots$ $\cdots$ $\cdots$ 4.2 $\pm$ 0.7 5.9 $\pm$ 0.6 $\cdots$ GJ 447 M4
5.499 $175.9\pm 3.4$ $165.1\pm 0.8^{\rm e}$ 2 5.3 $\pm$ 1.3 9.8 $\pm$ 1.4
$4.1\pm 0.3^{\rm g}$ 7.1 $\pm$ 1.1g 5.0 $\pm$ 1.1 19.6 $\pm$ 1.0 poor $\cdots$
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
$\cdots$ 10.6 $\pm$ 6.9 21.5 $\pm$ 1.0 $\cdots$ GJ 285 M4 5.527 $2.775\pm
0.001$ $2.8\pm 0.1$ 2 10.0 $\pm$ 4.1 39.1 $\pm$ 2.2 $10.6\pm 0.4$ 37.2 $\pm$
1.1 10.9 $\pm$ 4.0 37.2 $\pm$ 1.1 good GJ 54.1 M4 5.654 $71.4\pm 2.4$ $69.2\pm
0.1$ 2 $\cdots$ $\cdots$ $\cdots$ $\cdots$ 2.2 $\pm$ 0.3 6.8 $\pm$ 0.6
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
$\cdots$ $\cdots$ $\cdots$ 3.7 $\pm$ 0.8 5.8 $\pm$ 0.7 $\cdots$ $\cdots$
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
$\cdots$ long $\cdots$ $\cdots$ GJ 234a M4.5 5.585 $1.582\pm 0.002$ $8.1\pm
0.1^{\rm d}$ 2 5.6 $\pm$ 1.5 10.9 $\pm$ 1.4 $5.9\pm 0.5$ 10.1 $\pm$ 1.4 5.2
$\pm$ 1.0 13.1 $\pm$ 0.5 fair GJ 551b M5.5 6.75 $85.1\pm 1.2$ $83.2\pm
0.1^{\rm c}$ 2 6.7 $\pm$ 1.4 16.3 $\pm$ 1.5 $6.8\pm 0.3$ 15.5 $\pm$ 0.9 18.3
$\pm$ 14.2 39.5 $\pm$ 1.1 poor $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
$\cdots$ 5.1 $\pm$ 1.0h 20.0 $\pm$ 3.0h $\cdots$ $\cdots$ $\cdots$ $\cdots$
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ longh $\cdots$
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ GJ 406b M6 7.423 $2.6\pm 0.2$
$\cdots$ $\cdots$ 13.7 $\pm$ 7.3 128.4 $\pm$ 22.4 $8.9\pm 0.2$ 64.8 $\pm$ 1.2
12.0 $\pm$ 4.4 93.3 $\pm$ 1.1 fair

Table 1: Cycle Periods Inferred From Photometric Time Series.

###  3.1 Rotation Periods

Using the analysis procedures described in Section 2.6.1, high-confidence
rotation periods were determined for all but two of our stars. For one
exception, GJ 729, $P_{\rm rot}$ is less than 15d but TESS data were not
available for our work. We therefore used the same method as for stars with
longer rotation periods, computing the mean and standard deviation of similar
significant peaks in each ASAS-SN observing season’s periodogram; these
periodograms are shown in Figure 5. This resulted in a rotation period of
$2.9\pm 0.1$d, in agreement with Suárez Mascareño et al. (2016); Ibañez Bustos
et al. (2020), who found a rotation periods of $2.9\pm 0.1$d (from photometric
monitoring) and $2.848\pm 0.001$d (from chromospheric indicators),
respectively.

Figure 5: L–S periodograms (left) computed from ASAS-SN observing seasons
(right) on GJ 729. Due to the short inferred rotation periods, no sine fits
are shown as the modulations would be unresolvable.

For the second exception, GJ 273, we could not determine a rotation period
from our data sources. Breaking ASAS-SN data down by observing season, as
described in Section 2.6.1, and shown in Figure 6, produced only one
significant peak at $\sim$42d. However, no other observing seasons showed any
significant peaks at all. Computing periodograms from the entire ASAS-SN light
curve, as shown in Figure 7, also produced no significant peaks with a period
of less than one year. Figures 6 and 7 show that this star has fewer ASAS-SN
observations than many of the other stars in our sample, especially in the
early seasons (which use V-band filters). The sparsity of these observations
undoubtedly contributed to our inability to identify rotational modulations
within this light curve. For these reasons, we adopted the $P_{\rm rot}$ of
$115.9\pm 19.4$d measured by Suárez Mascareño et al. (2015) using HARPS Ca ii
H and K and H$\alpha$ data.

Figure 6: L–S periodograms (left) computed from ASAS-SN observing seasons
(right) on GJ 273. Figure 7: Periodogram and ASAS-SN data for GJ 273.

Given analysis uncertainties and the inherent variation in rotational
measurements caused by starspot evolution, migration in latitude and
longitude, and differential rotation, our results are generally in good
agreement with previous measurements. Where there are significant
disagreements (GJ 234, GJ 628, and LP 816-60) we believe our measurements are
more trustworthy given the higher quality and cadence of TESS and ASAS-SN data
compared to the ASAS-3 data used by Suárez Mascareño et al. (2016). Note that
two of our stars, GJ 317 and GJ 406, have $P_{\rm rot}$ values reported for
the first time.

###  3.2 Cycle Periods

L–S analysis yields FAPs of less than 0.1% using ASAS-3/-4 and/or ASAS-SN data
for 12 stars, not including a few cases where the significant peak corresponds
to a cycle period exceeding the period of observation. Below, we comment on
three stars of particular note. For stars not mentioned below, light curves
and associated L–S periodograms are provided in Figure Sets 1 (ASAS-3/-4) and
2 (ASAS-SN).

####  3.2.1 LP 816-60

Figure 4 provides strong evidence for at least one activity cycle in LP
816-60. From this figure, it is clear that there are two strong periodic
signals in LP 816-60’s ASAS-SN light curve, with periods of $2.0$\- and
$4.2$-yr. Upon first inspection, one of these peaks could be interpreted as a
harmonic of the other. Indeed, 4.2 is very close to double 2.0, and taking
into account the uncertainties on these periods, one could well be a harmonic
of the other.

Evidence presented by do Nascimento et al. (in preparation) suggests that it
is not uncommon to measure cycles with $P_{\rm cyc}$ and $P_{\rm cyc}/2$. In
that paper, they interpret such results as the separate Hale (magnetic
polarity) and Schwabe (activity, starspots) cycles having different amplitudes
due to (for reasons which are currently unclear) one polarity being stronger.
Given the two clear cycles visible in LP 816-60’s ASAS-SN data, this star may
therefore provide evidence for M dwarfs also possessing (or, at least,
appearing to have) asymmetric polarity strengths. Another possibility is that
both peaks are the result of a single quasi-periodic cycle, as discussed in
Olspert et al. (2018a).

Alternatively, since these cycles have been inferred using optical photometry
data, they could be the result of an observing effect. Assuming a 4.2-yr cycle
period, it could be the case that for one half of this cycle, starspots are
preferentially generated out of our line of sight (i.e., sin(i) $\ll$ 1) -
although the clear rotational modulations make this less likely. Without
Zeeman-Doppler imaging (ZDI) (e.g., Hébrard et al., 2016), or X-ray
observations (e.g., Wargelin et al., 2017), however, it is difficult to
determine whether an observed cycle is a result of magnetic activity
(specifically a Schwabe cycle) or some other phenomenon.

####  3.2.2 GJ 358

GJ 358’s ASAS-SN light curve (Figure 8) shows that the cyclic modulations
appear to be reasonably well described by two sinusoidal functions, following
the $P_{\rm cyc}$ and $P_{\rm cyc}/2$ rule suggested by do Nascimento et al.
(in preparation). Interestingly, both the ASAS-3 and ASAS-SN light curves
(Figures 9 and 8) show long-term “linear” trends whose gradients are of
opposite sign; this may suggest that GJ 358 exhibits a longer-term,
Gleissberg-esque cycle, although many more years of data will be needed to
confirm this.

Figure 8: Periodogram and ASAS-SN data for GJ 358. Figure 9: Periodogram and
ASAS-3 data for GJ 358.

Further study of the lightcurves from ASAS-3 and ASAS-SN, both of which are
shown in a single plot in Figure 10, reveals that GJ 358’s cycle period has
been steadily decreasing for at least the past 20 years. ASAS-3 data show
cyclic modulations with periods greater than four years, while the most recent
modulation seen in ASAS-SN data has a period of just 1.5 years. This star
therefore appears to be quite unusual. Unfortunately, there is a $\sim$6-yr
gap between ASAS-3 and ASAS-SN data on this object. This gap could be filled
with ASAS-4 data, however these data are not publicly available. Analysis of
these data would prove very interesting and reveal, for example, whether the
cycle period has changed monotonically, and when and how the longer-term
trends seen in ASAS-3 and ASAS-SN data intersected.

Figure 10: GP fits (dashed black line) to ASAS-3 and ASAS-SN data (blue and
red markers, respectively) on GJ 358. ASAS-3 data were color corrected using
Equation 1, and then further offset, to align vertically with ASAS-SN data.
Local minima in these GP fits are marked by vertical dashed lines and the
periods between successive minima are labeled.

At present, it is also unclear whether GJ 358’s cycle is approaching some kind
of (quasi-)steady-state, or if this cycle period will reach a local minimum
before increasing again. It is also unclear if this changing period is related
to the long-term cycle suggested by the “linear” trends seen in GJ 358’s
ASAS-3 and ASAS-SN data. Future observations on this object are therefore
needed to better understand this seemingly exceptional cyclic activity.

####  3.2.3 GJ 551

GJ 551 (Proxima Centauri) is an interesting case for several reasons. It is
the most well-studied example of a fully convective M star exhibiting cyclic
activity, and is the only fully convective star (in fact, the only M star) to
have undergone long-term X-ray monitoring. Suárez Mascareño et al. (2016)
found a 7-yr cycle in ASAS-3 data, and Wargelin et al. (2017) included an
additional $\sim$5 years of ASAS-4 data, plus UV and X-ray observations to
further support that conclusion.

More recent optical monitoring data, however, are less clear, as reflected in
the ASAS-SN results listed in Table 1, and shown in Figure 11, which includes
ASAS-4 data extending into 2019 that were presented in Damasso et al. (2020).
ASAS-3 and ASAS-4 data were cross calibrated (private communication, G.
Pojmański), and then calibrated versus ASAS-SN data following the “overlapping
data” procedures described in Section 2.2.1. As seen in Figure 11, periodic
behavior is quite apparent at the beginning, but steadily weakens while
average brightness increases. We suspect that the increasing stellar
contamination noted in Section 2.3 is significantly affecting the data, but
attempting to model and correct the contamination is beyond the scope of this
paper. In contrast, X-ray data from Swift, now spanning more than 12 years
(with some gaps), indicate well behaved cyclic behavior, currently with a
period of $\sim$9 years (Wargelin, et al., in preparation).

Figure 11: GP fit (dashed black line) to Proxima’s ASAS-3, ASAS-4, and ASAS-SN
data.

###  3.3 Periodicities and Magnetic Cycles

It should be noted that some of the cycles presented in Table 1 may not be
related to a star’s global magnetic field, but could be the result of other
physical phenomena (or, if they are, may not be Schwabe cycles; do Nascimento
et al. in preparation). For example, Rossby waves (horizontally flowing eddies
in the convective zone/photosphere), as seen in the Sun, can act as dynamos
(e.g., Gilman, 1969), leading to perceived cyclic activity which is not
necessarily related to the star’s global magnetic field.

In the case of our Sun, there is also the Rieger cycle (Rieger et al., 1984),
where hard solar flares are observed to occur in groups with a mean spacing of
$\sim$154d. Similar cycles surely exist in other stars; while flares are a
result of magnetic activity, they are local effects and it is not well
understood how they relate to a star’s global magnetic field - even in the
case of our Sun (e.g., Toriumi & Park, 2022).

###  3.4 Phase Matching

One way to assess the validity of a star’s apparent cycles is to see if they
are consistent through ASAS-3 and ASAS-SN data. If the best fitting cyclic
models from both data sets show a reasonable level of agreement (in terms of
both phase and amplitude) where they overlap, then these cycles are more
likely to be real. Note, however, that this method assumes a roughly constant
cycle period, which, as discussed in Section 1, may not always be the case.

To investigate the amplitude and phase match-ups, we extend the cycles
identified in Table 1 such that the ASAS-3 and ASAS-SN cyclic models for each
star overlap by one year. In addition to this, we also offset the ASAS-3 data
such that the mean magnitude of this light curve is the same as the mean
magnitude of the ASAS-SN light curve, thus allowing easy comparison of the
models’ phases and amplitudes. Example plots are shown in Figure 12, showing
the results for GJ 273, GJ 234, and GJ 447. All phase match plots (10 images)
are provided in Figure Set 3. Note, however, that we place little emphasis on
these plots; indeed, they are primarily included so that the inferred cycles
from both data sets can be more easily compared.

Fig. Set3. Phase Check

Figure 12: Plots of the cyclic models found from ASAS-3 (blue line) and ASAS-
SN (red line) data for GJ 273, GJ 234, and GJ 447 from Table 1, showing good,
fair, and poor agreement, respectively. The complete figure set (10 images) is
available in the online journal (Figure Set 3).

As can be seen in the top panel of Figure 12, both the ASAS-3 and ASAS-SN
cyclic models on GJ 273 agree very well, in terms of both phase and amplitude,
where they overlap. We interpret this as enhanced evidence for these cycles
being real. On the other hand, GJ 234 (middle panel of Figure 12) shows good
amplitude agreement, but poorer phase agreement, while GJ 447 (bottom panel of
Figure 12) shows poor agreement in both cases. In the final column of Table 1,
we assign each star’s ASAS-3 and ASAS-SN cycles a subjective “phase match”
grade of either good, fair, or poor. For reference, we judge that GJ 273, GJ
234, GJ 447 show good, fair, and poor phase agreement, respectively. The
remaining plots are provided in Figure Set 3.

##  4 Discussion

With rotation and cycle parameters in hand, we can investigate correlations
among various physical quantities. First, however, we must determine a largely
theoretical quantity for each star: its convective turnover time, $\tau$.

###  4.1 Determining Convective Turnover Times and Rossby Numbers

The convective turnover time is a measure of how long it takes for convection
to move material from the bottom of the convective zone to the top, or vice
versa, and is related to the ratio of the thickness of the convective envelope
(thin in F stars, fully convective by $\sim$M3.5) and the mean convective
velocity (which increases with temperature). To compare stars with different
sized convective zones, temperatures, and rotation periods, a dimensionless
quantity known as the Rossby number, Ro, is commonly used. Another
dimensionless quantity is the linear $\alpha\Omega$ dynamo number, D, a
measure of the dynamo’s strength, which is proportional to $\textrm{Ro}^{-2}$.
The Rossby number of a star is defined as:

$\textrm{Ro}=\frac{P_{\rm rot}}{\tau},$ (3)

where $P_{\rm rot}$ is the rotation period and $\tau$ is the convective
turnover time. Note, however, that there are two definitions for the
convective turnover time: a local and a global definition. The local
convective turnover time, $\tau_{L}$, applies to the bottom of the convective
zone (in the tachocline, where the main dynamo amplification takes place in
some models), while the global convective turnover time, $\tau_{G}$, is an
average over the entire convective zone. The global convective turnover time
is therefore a better quantity for M stars, where the tachocline is negligible
or absent.

The choice of a convective turnover time is important. We need both a local
(for tachocline-based dynamos) and a global (for full convection zone dynamos)
$\tau$, which we denote $\tau_{\rm L}$ and $\tau_{\rm G}$, respectively. Since
we are interested in the operation of dynamos, a “$\tau$” derived from any
indirect activity diagnostic (e.g., Ca ii HK or X-ray emission) is of dubious
utility. This is because such empirical $\tau$s implicitly assume that the
function $f(M,\log g,[Fe/H])$ which best connects the given emission to
rotation can be defined to be “$\tau$,” ignoring the complex physics
connecting magnetic flux to the heating which produces the emission in
question (e.g., Cuntz et al., 1999). The hidden inclusion of this additional
heating physics makes these empirical “$\tau$s” inappropriate for
understanding how dynamos themselves operate. Note that a $\tau$ derived by
best fitting rotation with unsigned magnetic flux measurements would be more
appropriate. This has not yet been attempted to our knowledge, likely because
data on unsigned fluxes are sparse and generally come with large errors (e.g.,
Reiners, 2012; Saar et al., 1994). Calculating such a $\tau$ is outside the
scope of the current work; however new flux measurement methods may make it
feasible in the near future (e.g., Lehmann et al., 2015; Mortier, 2016).

Unfortunately, theoretically based $\tau$s are also problematic. Due to
uncertainties concerning M dwarf internal structure near the fully convective
limit (Baraffe & Chabrier, 2018; Jao et al., 2022), any purely theoretical
$\tau$ is at best approximate at low masses (Jao et al., 2022). There are also
problems defining a local $\tau$ in the center of fully convective stars.
Indeed, with no convection zone bottom, the tachocline concept driving the
need for a local $\tau$ requires modification itself. Fortunately, a recent
paper has worked to improve $\tau$ models in low mass stars (Landin et al.,
2023). We adopt their values here at age $\approx$1Gyr. This is post-ZAMS for
all masses here except 0.1 $M_{\odot}$, but we note from their Figure 1 that
$\tau_{\rm G}(0.1M_{\odot})$ evolves negligibly for ages $>$0.3 Gyr. Indeed,
over the mass range considered here, $\tau$ varies little along the main
sequence except at the highest masses, where differences of $\sim$30% may
accrue at age extremes. We use the V–KS vs. $T_{\rm eff}$ relation of Pecaut &
Mamajek (2013) (and later improvements; Mamajek, E.E. 2021, private
communication) to match to our stars. As a cross-check, we compare the
calculations at low mass with a different kind of empirical $\tau$ calculation
based purely on stellar properties - derived equating the star’s output
bolometric flux with convective flux (Corsaro et al., 2021). They find
$\tau_{\rm G}\propto R[M/(LR)]^{1/3}\propto M^{1/3}T_{\rm eff}^{-4/3}$ for
fully convective stars, where $M,L,$ and $R$ are the stellar mass, luminosity
and radius, respectively (taken from Pecaut & Mamajek, 2013). We find a scaled
version of these physically-based empirical $\tau_{\rm G}$ for the most part
compare well with the new $\tau_{\rm G}$ of Landin et al. (2023) (Figure 13;
where we also display their $\tau_{\rm L}$). The exception is in the narrow
temperature range 3250K $<T_{\rm eff}<3400$K, very near the fully convective
boundary, where $\tau_{\rm G}$(Landin) shows a spike that $\tau_{\rm
G}$(Corsaro) does not. (We also note that $T_{\rm eff}(0.1M_{\odot})$ is
cooler in Landin et al. (2023).) We suggest that $\tau_{\rm G}$ calculations
near the stellar center may still have problems (Jao et al., 2022), and in
keeping with the idea that $\tau_{\rm L}$ and $\tau_{\rm G}$ should roughly
scale with each other (e.g., Montesinos et al., 2001), we adopt the Corsaro
values for $T_{\rm eff}\leq 3400$ K. We use $\tau_{\rm G}$ for fully
convective stars, and either $\tau_{\rm G}$ or $\tau_{\rm L}$ otherwise,
depending on the circumstances.

Figure 13: Convective turnover time (in days) vs. V–KS color index from Landin
et al. (2023). The dashed red line shows the global turnover time
($\tau_{G}$); the solid green line shows the local turnover time ($\tau_{L}$);
the dotted blue line shows the scaled version of an independent $\tau_{G}$
calculation for fully convective stars Corsaro et al. (2021) which is in good
agreement except in the narrow zone 3250 K $\leq T_{\rm eff}\leq$ 3400 K. We
adopt $\tau$(Landin) everywhere except for $\tau_{\rm G}$ for $T_{\rm eff}<$
3400K (V–K${}_{\rm S}\approx$ 4.6), where we transition to the scaled
$\tau_{\rm G}$(Corsaro); see text for details. $\tau$(Landin) have been
truncated (using linear interpolation) at V–K${}_{\rm S}=8$; see Table 1.

With estimated convective turnover times and Rossby numbers for our 15 M-type
stars, we then used results from Saar & Brandenburg (1999), Lehtinen et al.
(2016), and Olspert et al. (2018a) to gather a comparison sample of 40 F, G,
and K type stars with robust, well-defined cycles. Of these 40 stars, listed
in Table 2, 10 have double cycles. Note, however, Olspert et al. (2018a) use
multiple different methods to identify stellar cycles; for consistency with
our approach, we use results obtained via their harmonic model (see Section
2.5 for more details). To estimate the convective turnover times of these
stars, we use the method described above.777Note: while we do not include a
plot of $\tau_{L}$ vs. B–V, the stars from Table 2 correspond to roughly the
range $1\leq$ V–KS $\leq 3.5$ in Figure 13. We note, however, that we exclude
HD 18256 (F6) as it is too hot for our $\tau$ estimations; interestingly, in
their study of Ca ii H and K cycles in 15 stars, Baliunas et al. (2004) found
that this star had by the far the largest “anharmonicity,” a measure of
deviation from single-period sinusoidal behavior, in their sample.

Table 2: Activity Cycles For FGK Type Stars Taken From Previous Works. Star | Spectral | B-V | $P_{\rm rot}$ | $P_{\rm cyc}$ | Ref. | Star | Spectral | B-V | $P_{\rm rot}$ | $P_{\rm cyc}$ | Ref.
---|---|---|---|---|---|---|---|---|---|---|---
| Type |  | (d) | (yr) |  |  | Type |  | (d) | (yr) |
HD 1835 | G2.5 | 0.66 | 7.78 | 9.1 | 1 | HD 18256 | F6 | 0.45 | 3 | 6.8 | 1
HD 20630 | G5 | 0.68 | 9.24 | 5.6 | 1 | HD 76151 | G2 | 0.67 | 15 | 2.52 | 1
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 14.4 | 15.9 | 3
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 5.09a | 3
HD 82443 | G9 | 0.77 | 5.38 | 3.89 | 1 | HD 115404 | K2 | 0.94 | 18.47 | 12.4 | 1
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 20 | 2 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
HD 149661 | K1 | 0.84 | 21.07 | 16.2 | 1 | HD 165341 | K0 | 0.86 | 19.9 | 15.5 | 1
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 4 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 5.1 |  $\cdots$
HD 100180 | F9.5 | 0.57 | 14 | 3.6 | 1 | HD 190406 | G0 | 0.61 | 13.94 | 2.6 | 1
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 16.9 |  $\cdots$
HD 1405 | K2 | 0.95 | 1.756 | 8 | 2 | HD 70573 | G1/2 | 0.62 | 3.314 | 6.9 | 2
HD 82558 | K1 | 0.93 | 1.604 | 14.5 | 2 | HD 116956 | G9 | 0.80 | 7.86 | 14.7 | 2
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 18 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
HD 135599 | K0 | 0.83 | 5.529 | 14.6 | 2 | HD 171488 | G2 | 0.62 | 1.345 | 9.5 | 2
Sun | G2 | 0.631 | 26.09 | 10.89 | 3 | HD 103095 | K1 | 0.75 | 34.3 | 7.13 | 3
HD 10476 | K1 | 0.84 | 35.6 | 10.6 | 3 | HD 10780 | K0 | 0.81 | 22.14 | 7.53 | 3
HD 114710 | F9.5 | 0.59 | 11.99 | 16.56 | 3 | HD 146233 | G2 | 0.65 | 22.62 | 11.2 | 3
HD 152391 | G8.5 | 0.76 | 10.62 | 9.03 | 3 | HD 155886 | K2 | 0.85 | 20.58 | 10.44 | 3
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 13.73 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 5.0 |  $\cdots$
HD 156026 | K5 | 1.16 | 16.69 | 17.89 | 3 | HD 160346 | K3 | 0.971 | 32.0 | 7.21 | 3
HD 16160 | K3 | 0.98 | 48.58 | 12.45 | 3 | HD 165341A | K0 | 0.86 | 19.33 | 5.19 | 3
HD 166620 | K2 | 0.87 | 42.1 | 16.16 | 3 | HD 185144 | K0 | 0.87 | 27.7 | 6.66 | 3
HD 201091 | K5 | 1.18 | 35.54 | 7.16 | 3 | HD 201092 | K7 | 1.37 | 34.55 | 11.65 | 3
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 21.09 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
HD 219834B | K2 | 0.91 | 34.78 | 9.32 | 3 | HD 26965 | K0 | 0.82 | 38.65 | 10.66 | 3
HD 32147 | K3 | 1.06 | 33.7 | 11.13 | 3 | HD 3651 | K0.5 | 0.83 | 37.0 | 16.98 | 3
HD 37394 | K1 | 0.84 | 11.49 | 5.83 | 3 | HD 4628 | K2.5 | 0.90 | 37.14 | 8.56 | 3
$\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 5.79 |  $\cdots$
HD 78366 | G0 | 0.60 | 9.519 | 14.63 | 3 | HD 81809 | G1.5 | 0.80 | 41.66 | 8.11 | 3

####  4.1.1 Data Selection

When gathering our sample of FGK stars from other works, we only included
cycles in known (or likely) dwarfs. Comparing the results of Saar &
Brandenburg (1999) with Olspert et al. (2018a), there are 17 stars in common.
Since Olspert et al. (2018a) had longer time series for their analysis, we
favor their results in most cases. We note, however, an exception below.

Saar & Brandenburg (1999) found a single 2.52-yr cycle in HD 76151, while
Olspert et al. (2018a) found 5.0-yr and 15.9-yr cycles using their harmonic
model. We interpret this 5.0-yr cycle as a Hale cycle (born of a polarity
asymmetry in HD 76151’s magnetic field; do Nascimento et al. in preparation).
For this reason, we reject this 5.0-yr cycle. We also note that this 5.0-yr
cycle resides between the I and A branches when plotted in $P_{\rm cyc}/P_{\rm
rot}$–Ro-1 space, which we interpret as stronger evidence for this not being a
Schwabe cycle (see Section 4.7).

###  4.2 Acyc versus Pcyc

Figure 14 shows a plot of cycle amplitude against period for the well-defined
cycles in Table 1, along with a least-squares fit following $A_{\rm
cyc}\propto P_{\rm cyc}^{0.94\pm 0.11}$. When fitting this power law, only
stars with a single, well-defined cycle per data set were used (i.e., if a
star had a single cycle in both ASAS-3 and ASAS-SN data, then both cycles were
used). Proxima’s suspect 5.1-yr ASAS-4 cycle was ignored during fitting (see
Section 2.3).

Figure 14: Cycle amplitude (in millimagnitudes) against cycle period (in
years) using the well-defined cycles from Table 1. Color-only markers show
cycles inferred from ASAS data; markers with dark outlines are from ASAS-SN
data; fully convective stars use circle markers; partially convective stars
use triangle markers; faded markers denote that these points were ignored
during fitting; concurrent cycles within the same star are connected with
dotted lines. The outlier points for GJ 358 and GJ 447 (with colored labels)
are discussed in the text.

As can be seen in Figure 14, cycles with longer periods generally also have
larger amplitudes (ignoring stars with more than one concurrent cycle, whose
cycles are connected with dotted lines and shown with faded markers). We
speculate that this may be caused by longer periods allowing more build up of
magnetic energy, resulting in a more pronounced cycle; this process is
interrupted in stars with more than one concurrent cycle, which is why they
appear to contradict this trend. This relation may also explain why short
period cycles are detected less frequently in ASAS-3 data, which are noisier
than their ASAS-SN successor. It may be the case that short period cycles are
present in ASAS-3 data, but cannot easily be resolved as their amplitudes are
comparable to the ASAS-3 noise level.

Since stars with more than one well-defined cycle per data set were excluded
from the power law fit shown in Figure 14, it is interesting to note where
these excluded cycles reside in relation to this fit. Indeed, it can be seen
that for stars with two concurrent cycles, one is usually closer to this power
law than the other; these other cycles may therefore be “false cycles”
(Section 3.3) or hints of a multi-branched relationship as seen by some in the
$P_{\textrm{cyc}}$–$P_{\textrm{rot}}$ relation (e.g., Saar & Brandenburg,
1999; Böhm-Vitense, 2007; Lehtinen et al., 2016). It should also be noted that
$A_{\rm cyc}$ can be quite variable (more so in the Sun than $P_{\rm cyc}$),
and so some outliers may be expected. That said, the (somewhat tentative)
agreement among these excluded cycles is interesting, and we interpret this as
further evidence for this relation between cycle amplitude and period.

Multiple branches in the $A_{\rm cyc}$–$P_{\rm cyc}$ relation may explain why,
for example, the ASAS-3 cycle of GJ 358 and ASAS-SN cycle of GJ 447 are
outliers from the power law shown in Figure 14. As discussed in Section 3.2.2,
however, GJ 358 already appears to be a highly unusual star. GJ 447, on the
other hand, does have a second cycle period in Table 1, although it is not
well constrained. This second cycle has an estimated period of $10.6\pm
6.9$yr, and estimated amplitude of $21.5\pm 1.0$mmag, which would put it very
close to the power law shown in Figure 14, although with a large period
uncertainty.

Using chromospheric Ca ii H and K data on FGK stars, Saar & Brandenburg (2002)
found stronger evidence for multiple branches in the $A_{\rm cyc}$–$P_{\rm
cyc}$ relation owing to their larger sample of stars. However, as the
amplitudes from Saar & Brandenburg (2002) are from chromospheric indicators,
they are not directly comparable to our photometric amplitudes. It would be
very interesting to see how the photometric cycle amplitudes of F–M type stars
compare, however this is left to future work.

In the case of our Sun, the solar cycle does not have a strict 11-yr
periodicity; indeed, the $\sim$4 year range of the solar $P_{\textrm{cyc}}$
(Donahue & Baliunas, 1992) is fully $\pm$18% of the average value. Moreover,
if a given solar cycle has a shorter than average period, then it will usually
have a larger than average amplitude, and vice versa (related to the Waldmeier
effect; Waldmeier, 1935). For the M-type stars analyzed in this work, the
observation intervals cover only a handful of cycles in the best case, so it
is difficult to determine if these cycles reflect a natural range of $P_{\rm
cyc}$ as seen in the Sun. However, from Figure 14, it is clear that for M-type
stars which exhibit two apparent cycles, the shorter cycle usually has a
greater amplitude; in cases where the shorter $P_{\rm cyc}$ is approximately
half the longer $P_{\rm cyc}$, this is consistent with one polarity being
stronger than the other (i.e., a Hale, rather than Schawbe, cycle as do
Nascimento et al. in preparation, suggest). Alternatively, if as Olspert et
al. (2018a) suggest, these double cycles are the result of a single, quasi-
periodic cycle, then perhaps this suggests that M dwarfs behave similarly to
the Sun - with shorter than average cycles having greater than average
amplitudes, and vice versa.

###  4.3 Pcyc versus Prot and Ro

Plotting cycle period against rotation period for the well-defined cycles in
Table 1, along with previously detected cycles in FGK stars presented in Table
2, shows high scatter, with no strong correlations. However, convective zones
are believed to be a key ingredient of stellar cycles, and these are not
accounted for in this plot. We therefore place little emphasis on this plot
(and choose to not include it here) as it ignores a major physical parameter
of current stellar cycle models: Rossby number.

Figure 15 plots cycle period against Rossby number for the well-defined cycles
presented in Table 1, along with the previously detected cycles in FGK stars
presented in Table 2. This figure shows that when the convective turnover time
is accounted for, in the form of the Rossby number, M-type stars appear to
have similar cycles to those of FGK stars at equivalent Ro. However, it is
clear that our M dwarfs favor shorter cycle periods, extending the lower
cluster of FGK stars to lower $P_{\rm cyc}$. M dwarfs also isolate themselves
at low Rossby numbers, though this may be a sampling effect; perhaps M dwarfs
extend the upper cluster of FGK stars to lower Ro too.

Figure 15: Plot of cycle period (in years) against Rossby number, including
FGK type stars from Table 2. For the M dwarfs analyzed in this work, filled
markers represent cycles inferred from ASAS-SN data; empty markers are from
ASAS-3/-4 data; fully convective M stars use circle markers; partially
convective M stars use triangle markers. Multiple cycles within the same star
(and, for our M dwarfs, found within the same data set) are connected by
dotted lines.

In Figure 15, the longer cycle periods seen in FGK stars likely reflect the
fact that these stars have generally been monitored over longer periods than M
dwarfs. The evidence for M dwarfs exhibiting shorter $P_{\rm cyc}$ (relative
to FGK stars) is strong, but we also found indications of longer, presently
unresolvable or poorly constrained cycles in 11 of our M dwarfs. Reproducing
these figures with another 10 years of ASAS-SN data may therefore prove
interesting.

###  4.4 Pcyc/Prot versus Ro-1

Figure 16 shows a plot of cycle period divided by rotation period against
inverse Rossby number for the well-defined cycles presented in Table 1, along
with previously detected cycles in FGK stars presented in Table 2.

Figure 16: Plot of $P_{\rm cyc}/P_{\rm rot}$ against inverse Rossby number,
including FGK type stars from Table 2. The dashed grey fits show the A and I
branches from Saar & Brandenburg (1999) (which have been re-fit using our
updated sample), while the black power law fits were found in this work. The
global convective turnover time was used to compute Ro for all cycles in this
figure. For the FGK stars, since some of our sample did not include
uncertainties on their cycle/rotation periods, we weight all FGK cycles
equally; for our M dwarfs, we assume a $\pm 20\%$ uncertainty on $\tau$.

The power law fit to the M-dwarf data in Figure 16 shows that these data
follow the relation:

$\frac{P_{\rm cyc}}{P_{\rm rot}}=(3.6\pm 1.2)\times\textrm{Ro}^{-1.02\pm
0.06}.$ (4)

Similarly, Figure 16 shows that FGK stars follow the relation:

$\frac{P_{\rm cyc}}{P_{\rm rot}}=(63.6\pm 1.1)\times\textrm{Ro}^{-1.06\pm
0.09},$ (5)

suggesting that $P_{\rm cyc}\propto\tau$ for all stars, in contrast to the
results of Saar & Brandenburg (1999) \- shown by the A and I branches (which
we have re-fit using our updated sample).

Brandenburg et al. (1998); Saar & Brandenburg (1999) interpreted the ratio of
$P_{\rm cyc}/P_{\rm rot}$ as a measure of the $\alpha$ effect, based on a
simple $\alpha\Omega$ dynamo model proposed by Robinson & Durney (1982); Noyes
et al. (1984). In this framework, Equations 4 and 5 suggest that M dwarfs have
weaker $\alpha$ effects than FGK type stars at equivalent Rossby numbers (note
the scaling constant in Equation 4 is 17.7 times smaller than in Equation 5).
This result could be explained by considering the physical properties of these
stars. Since M-type stars have deeper convective zones, and consequently
larger turnover times, longer rotation periods are needed to get similar
Rossby numbers. Longer rotation periods therefore mean the Coriolis forces
acting within these convective zones will be smaller, resulting in less
helical turbulent convection. Furthermore, the average convective velocities
themselves are lower in these cooler stars; both these properties act to
reduce the $\alpha$ effect.

Note, however, that the above explanation would suggest a smooth transition
from F to M in Figure 16. In contrast, this figure shows a gap between FGK and
M stars. Perhaps, since M stars are (almost) fully convective, and therefore
have (almost) no tachocline, this causes a significant change in the $\alpha$
effect. Alternatively, this gap could simply be the result of our limited
sample of stars, which includes two K5s, a K7 but then no stars until M3.
Clearly, Figure 16 needs to be extended to cover a wider range of stars, with
a wide range of physical properties and rotation rates, to better understand
if this gap is real - and if so, and what may be causing it.

###  4.5 $\alpha$ and $\Omega$ Effect Scaling

The axes of Figure 16 have the benefit of being dimensionless, and Ro in
particular is directly related to the mean-field dynamo number, D, as
mentioned in Section 4.1. Clusters and trends within the data can thus provide
insights into the underlying dynamo properties. Indeed, we have already
discussed the implications of the scaling constants of Equations 4 and 5;
below, we discuss the implications of the power law indices of these
equations.

According to most theoretical models (e.g., Ruediger & Kichatinov, 1993), the
$\alpha$ effect should be quenched with growing magnetic fields, with the
reasoning that the stronger a star’s magnetic field, the more that cyclonic
convection (i.e., the $\alpha$ effect) is suppressed. This paradigm persisted
until Saar & Brandenburg (1999) found evidence suggesting that the $\alpha$
effect may be anti-quenched on their A and I branches. Here, with the benefit
of a more focused sample of stars (no binary or evolved stars), we
reinvestigate these claims.

Using a 2-D non-linear dynamo model, Tobias (1998) found:

$P_{\rm cyc}\propto D^{\gamma},$ (6)

where D is the dynamo number, and $-0.67\leq\gamma\leq-0.38$. To explore the
role of quenching in a simpler, linear 1-D dynamo model, Saar & Brandenburg
(1999) defined a quenching index, $q$, such that:

$D\propto\alpha\Omega^{\prime}\propto{\rm Ro}^{-q-2},$ (7)

where $\Omega^{\prime}$ is the radial DR and:

$q=q_{\alpha}+q_{\Omega}-2,$ (8)

where $q_{\alpha}$ is the quenching index of the $\alpha$ effect (i.e.,
$\alpha\propto\Omega^{q_{\alpha}}$, with $\Omega\equiv\frac{2\pi}{P_{\rm
rot}}$), and $q_{\Omega}$ is the quenching index of the $\Omega$ effect (i.e.,
$\Omega^{\prime}\propto\Omega^{q_{\Omega}}$). From solution of the standard
dynamo equations (assuming a fixed wavenumber, $k$), Brandenburg et al. (1998)
noted that the cycle frequency was given by:

$\omega_{\rm cyc}\equiv\frac{2\pi}{P_{\rm
cyc}}\propto\Big{(}\frac{\alpha\Omega^{\prime}kL}{2}\Big{)}^{\frac{1}{2}},$
(9)

where $L$ is the dynamo length scale. Combining Equations 7 and 9, we have:

$P_{\rm cyc}\propto(\alpha\Omega^{\prime})^{-\frac{1}{2}}\propto{\rm
Ro}^{\frac{q}{2}+1},$ (10)

or:

$\frac{P_{\rm cyc}}{P_{\rm rot}}\propto{\rm Ro}^{\frac{q}{2}}.$ (11)

Since Figure 16 suggests that the relationship between $P_{\rm cyc}/P_{\rm
rot}$-Ro can be described by a power law, where the power law index can take
on various values, we can thus write:

${\rm Ro}^{\frac{q}{2}}\propto{\rm Ro}^{\delta},$ (12)

where $\delta$ is the power law index (i.e., $\delta=-1.04$ from Equation 4).
Thus we find (combining Equations 8 and 12):

$q_{\alpha}=2\delta-q_{\Omega}+2.$ (13)

With this equation, we can infer the scaling of the $\alpha$ effect using the
power laws plotted in Figure 16. First, however, we must estimate $q_{\Omega}$
(the quenching index for radial DR).

DR measurements are difficult, and fraught with misidentifications (Aigrain et
al., 2015; Basri, 2018), but since the good evidence suggests that surface DR
scales linearly with rotation period for slower rotators (e.g., Saar, 2011),
it is reasonable to assume $q_{\Omega}\approx 1$; for saturated activity stars
(discussed in Section 4.7), the dependence reverses, and
$q_{\Omega}\approx-1.7$ (Saar, 2011). We note that we are implicitly assuming
that radial DR scales linearly with the measured surface DR.888Note, also,
some observers (e.g., Barnes et al., 2005) argue DR has little/no dependence
on rotation period, so our assumed $q_{\Omega}$ may be incorrect. With this
assumption, and using these $q_{\Omega}$, we thus have:

$q_{\alpha}=2\delta+1,$ (14)

in the unsaturated regime, and:

$q_{\alpha}=2\delta+3.7,$ (15)

in the saturated regime. Using the power law indices from Equations 4 and 5,
we find: $q_{\alpha,M}=-1.03$ and $q_{\alpha,FGK}=-1.13$ in the unsaturated
regime, and $q_{\alpha,M}=1.67$ and $q_{\alpha,FGK}=1.57$ in the saturated
regime. This suggests that the $\alpha$ effect is quenched with faster
rotation in the unsaturated regime (but anti-quenched in the saturated regime)
similarly for FKG and M stars.

As mentioned previously, quenching of the $\alpha$ effect (in the unsaturated
regime) is an expected result. In contrast, the $\alpha$ effect anti-quenching
that we find in the saturated regime is surprising - though anti-quenching is
supported by some MHD simulations (e.g., Chatterjee et al., 2011). We note
that our findings in the unsaturated regime disagree with the results of Saar
& Brandenburg (1999), whose limited sample of stars (and different assumptions
about $\Omega$ effect scaling) led to the opposite conclusion for their A and
I branches (note that the fit in Figure 16 is more consistent with their
tentative transitional branch for faster rotators; cf. Lehtinen et al. 2016).
That said, it should be noted that these results are based on a very simple
dynamo model (Brandenburg et al., 1998), along with some debated assumptions
about DR and $\alpha$ and $\Omega$ effect quenching (e.g., Barnes et al.,
2005; Saar, 2011).

###  4.6 Refining Our Selection of Convective Turnover Times

The clustering of stars at the end of the FGK sequence in Figure 16
(particularly around the putative I branch), together with the spread of the
sequence in Ro, suggest that perhaps the concept of A and I branches is not
completely without merit. If we consider possible A and I branches, though, we
must note that we have not been self-consistent in our use of $\tau$. If the I
branch is dominated by a tachocline dynamo (as assumed for the Sun), a local
$\tau$, $\tau_{L}$, specifically computed for that location, is more suitable.
Further, a fit to all the FGK stars together becomes inappropriate; the faster
rotators (using $\tau_{G}$, as before) and the slower, I branch stars (using
$\tau_{L}$) should be treated separately. Using this distinction, we present
the results in Figure 17.

Figure 17: Similar to Figure 16 but with convective turnover times for cycles
on the I branch calculated at the base of the convective envelope (i.e., using
$\tau_{L}$ instead of $\tau_{G}$). The A and I branches were also ignored when
fitting a power law to the FGK stars.

Due to reasons which will become clear in Section 4.7, for stars with cycles
on both the A and I branches, we deem the I branch cycle to be the “primary”
and the A branch cycle the “secondary.” In a preliminary version of Figure 17,
secondary cycles were plotted also using $\tau_{L}$, but we noted this caused
larger scatter in the “A branch” fit ($\sigma=1.07$ in log space). If we use
$\tau_{G}$ for these secondary cycles, effectively assuming they arise from a
convection zone dynamo still in operation, the “A branch” scatter is reduced
by $\sim$20% ($\sigma$= 0.81; Figure 17). These new values modify Equation 5
to be:

$\frac{P_{\rm cyc}}{P_{\rm rot}}=(162.6\pm 0.2)\times\textrm{Ro}^{-0.81\pm
0.17},$ (16)

but do little to change the spirit of Section 4.5’s discussion. A list of
different model parameters, their goodness-of-fit metrics, and their implied
$q_{\alpha}$ values is provided in Table 3.

Table 3: Model Parameters for $P_{\rm cyc}/P_{\rm rot}={\rm a}\times{\rm Ro}^{-\rm b}$ Models. Stars | $\tau$ | a | $\sigma_{\rm a}$ | b | $\sigma_{\rm b}$ | RMSE | $q_{\alpha}$ | Used in
---|---|---|---|---|---|---|---|---
Included | Excluded |  |  |  |  |  |  | Unsat. | Sat. | Figures
M |  $\cdots$ | G | 3.6 | 1.2 | 1.02 | 0.06 | 0.17 | -1.03 | 1.67 |  16,17,18
FGK |  $\cdots$ | G | 63.6 | 1.1 | 1.06 | 0.09 | 0.24 | -1.13 | 1.57 | 16
I branch |  $\cdots$ | G | 112.1 | 1.2 | -0.25 | 0.21 | 0.13 | 1.49 |  $\cdots$ | 16
I branch |  $\cdots$ | L | 101.1 | 1.1 | -0.05 | 0.24 | 0.15 | 1.10 |  $\cdots$ |  17,18
A branch |  $\cdots$ | G | 380.3 | 1.4 | -0.15 | 0.24 | 0.15 | 1.30 |  $\cdots$ |  16,17,18
A branch |  $\cdots$ | G,La | 386.3 | 1.2 | -0.12 | 0.14 | 0.14 | 1.25 |  $\cdots$ |  $\cdots$
FKG | I branch | G | 124.0 | 1.2 | 0.83 | 0.11 | 0.20 | -0.67 | 2.03 |  $\cdots$
FKG | A and I branches | G | 162.6 | 1.6 | 0.81 | 0.17 | 0.12 | -0.61 | 2.09 |  17,18

####  4.6.1 GJ 358

If, as we propose in Section 4.6, F–K stars can have convective zone and
tachocline dynamos operating simultaneously, then, for consistency, we should
also consider this possibility for our partially convective M stars; we note,
once again, the exceptional cyclic behavior of GJ 358. GJ 358 is one of
(likely) four partially convective M stars in our sample, and is the only such
star to show strong evidence for multiple concurrent cycles. Giving this star
the same treatment as the F–K stars from the previous section, we might
conclude that these cycles are from different dynamos. However, from a sample
size of one, the division between the “A” and “I branches” for M stars is
somewhat unclear.

Assuming GJ 358’s cyclic behavior is the result of separate tachocline and
convective zone dynamos, then its changing (notably, decreasing) cycle period
makes some sense. Since these two dynamos are (in our simple scenario)
independent, then the cyclic behavior that we see is actually a superposition
of two independent cycles; if these cycles have different
(quasi)periods,999Note: we know of no theoretical or observational motivation
for these cycles to have the same period. then observing a “cycle with a
changing period” is not totally surprising.

Using the I branch and M dwarf fits from Figure 17, we can infer theoretical
cycle periods for GJ 358’s tachocline and convective zone dynamos. Starting
with the tachocline dynamo (i.e., the I branch), we find $P_{\rm cyc}=P_{\rm
rot}\times 101.1\times{\rm Ro}^{0.05}$, which, when plugging in the values
(using $\tau_{L}$ to compute Ro), gives $P_{\rm cyc}=6.5{\rm yr}$. This is
$2\sigma$ above the $4.7\pm 0.9$yr period we infer from GJ 358’s ASAS-3 data,
which is the longest period we were able to constrain for this star, and
shifts it left in Figure 17 to Ro${}^{-1}\approx 4$ (extending the I branch to
larger Ro-1). Now considering the convective zone dynamo, using the fit to all
M stars, we find $P_{\rm cyc}=P_{\rm rot}\times 3.6\times{\rm Ro}^{-1.02}$,
which gives $P_{\rm cyc}=4.4{\rm yr}$. This is consistent with the
aforementioned ASAS-3 cycle period, and with the same Ro as shown in Figure
17. Perhaps these two dynamos are not totally independent, and interactions
between the two cause more significant deviations from true periodic behavior
than is typically observed in other stars, resulting in the decreasing cycle
period we observe in Figure 10.

###  4.7 A New Interpretation of Dynamo Evolution

Indeed, our results here and those of Saar & Brandenburg (1999) may not
necessarily be at odds; perhaps we are seeing a switch in dominance between
two different types of dynamos with different properties. We largely set aside
discussion of the A branch here, since more recent work casts doubt on its
robustness/reality (e.g., Boro Saikia et al., 2018). We also skip discussion
of the “superactive” (S) branch, which is occupied almost solely by binaries
where tidal rotational locking may alter the dynamo fundamentally and place it
beyond the scope of the presumed single star dynamos considered here.

We propose the following scenario. The most rapid rotators seem to have nearly
solid-body rotation, for example GJ 1243 (M4, $P_{\rm rot}=0.59$d,
$k/k_{\odot}=(\Delta\Omega/\Omega)/(\Delta\Omega_{\odot}/\Omega_{\odot})\approx
0.01$; Davenport et al. 2020), LQ Hya (K2, $P_{\rm rot}=1.597$d,
$k/k_{\odot}=0.028$; Kovári et al. 2004), or V889 Her (G2, $P_{\rm
rot}=1.337$d, $k/k_{\odot}=0.045$; Kővári et al. 2011). With so little DR,
their dynamos are likely a turbulent $\alpha^{2}$ type or at best
$\alpha^{2}\Omega$, with the dynamo extant throughout the convection zone.
This concept is consistent with models (e.g., Guerrero et al., 2019), and some
detailed MHD models get roughly the same power law index for $P_{\rm
cyc}/P_{\rm rot}\propto$ Roδ: for example, Warnecke (2018) obtain $-0.89\pm
0.04$, and if one does a least-squares fit of the five long cycle models in
Brun et al. (2022) (which are clearly not on the I branch) using their
“stellar” Ro, the slope is $-0.90\pm 0.47$. These both compare well with our
values of $-1.06\pm 0.09$ and $-0.81\pm 0.17$.

We therefore identify the fits found here as representing the initial, more
turbulent, $\alpha$-anti-quenched, full convection zone dynamos. Stars on the
ZAMS start with rapid rotation, low DR and saturated magnetic activity, likely
with predominantly poloidal large scale field structure (Donati & Landstreet,
2009). As stars lose angular momentum, they evolve from upper right to lower
left in Figure 16, with $P_{\rm cyc}/P_{\rm rot}$ decreasing, and (if Saar
2011 is correct) increasing DR (sketched in Figure 18; red arrows). DR
increases to a maximum point, at which point saturated magnetic activity
stops, the $\alpha$ effect starts becoming quenched, and further spindown
leads to activity reduction along the well-known rotation-activity relations.

(As a brief aside, we note that an interesting alternative to the full-
convection zone dynamo, particularly in M stars, is the idea of dynamo cycles
driven by thermal fluctuations (e.g., Nigro, 2022). This could potentially
explain the lack of rotational dependence in M stars, as the distribution of
thermal fluctuations would primarily depend on mass/$T_{\rm eff}$, echoed in
the trend seen here of $P_{\rm cyc}\propto\tau$. The similar trend in faster
rotating FGK stars could also be influenced, at least in part, by such a
thermally driven dynamo. More work on these models would be useful.)

Figure 18: Symbols and gray lines as in Figure 17, with sketch of proposed
dynamo evolution and MHD model trends overlayed. Evolution begins on the ZAMS
at high Ro-1 in the upper right. Both FGK and M stars begin with $\alpha$
anti-quenching $\alpha^{2}$ or $\alpha^{2}\Omega$ dynamos and initially low
but increasing DR (red arrows). Core/envelope decoupling in FGK stars
(possibly at at a peak in DR and the end of activity saturation; right magenta
dashed line) starts a phase of $\alpha$ quenching, and the beginning of the
formation of a tachocline, and likely an $\alpha$ anti-quenched, tachocline
based $\alpha\Omega$ dynamo as well. This dynamo does not dominate, however,
until the star reaches the I branch (blue), though there may be a regime of
mixed dynamo dominance first (purple), around the time that large-scale
poloidal fields become dominant (left magenta dashed line). After further
evolution on the I branch, eventually the star slows to the point that the
dynamo sputters, Maunder-like magnetic minima appear, $P_{\rm cyc}$ lengthens,
and in the end the cycle fails entirely, rotational evolution ceases and the
star evolves vertically off the diagram (gold). In M stars, once activity
saturation ends (right magenta dashed line), an $\alpha$-quenched $\alpha^{2}$
or $\alpha^{2}\Omega$ dynamo takes over. 3D MHD models generally agree with
the trends found here for the Ms and fast rotating FGKs (e.g., Brun et al.,
2022, dashed black) and the I branch (Warnecke, 2018, dotted black). See text
for more details.

At some point in partially convective stars, the radiative core and convective
envelope decouple (e.g., MacGregor & Brenner, 1991), allowing a region of
strong shear to form at the interface (the tachocline). The Sun and slower FGK
rotators are thought to have $\alpha\Omega$ dynamos with the primary
generation region in this strong shear layer. M stars, lacking significant (or
any) tachoclines, cannot make this transition.

However, once the tachocline dynamo dominates in older FGK stars, which
happens at least by the time the stars reach the I branch around Ro-1 = 1 to
3, further cycle evolution proceeds along the I branch, with $P_{\rm
cyc}/P_{\rm rot}$ slightly increasing, or maybe remaining roughly constant
(shown by a blue arrow in Figure 18). Perhaps there is a transitional phase of
mixed dynamo dominance before this, explaining the diffuse A “branch” and
abundance of stars with multiple cycles in this region. The recently noted
“bend” in the chromospheric rotation-activity relation (Lehtinen et al., 2021)
may be another reflection of this change. ZDI shows that large scale toroidal
fields already dominate well before the I branch is reached, at around
Ro${}^{-1}\approx 15$ (on the $\tau_{C}$ scale used here, converted from
Donati & Landstreet, 2009), about where DR peaks and activity saturation ends
(shown by the right magenta dashed line in Figure 18). Perhaps this is where
core/envelope decoupling also takes place. Large scale toroidal field
dominance only ends at Ro${}^{-1}\approx 2$Ro${}^{-1}_{\odot}\approx 1$
(Donati & Landstreet, 2009) \- i.e., at about the time the star first reaches
the I branch. Note that no further stars are seen along the $-1.06$ slope in
Figure 16 among the FGK stars once the I branch is reached. We view this as
supporting the final dominance of the tachocline dynamo.

Following the arguments using the same simple dynamo model, Saar & Brandenburg
(1999) suggest the change in slope at the I branch is due to $\alpha$ anti-
quenching, which implies (surprisingly) that here $\alpha$ increases with the
magnetic field ($q_{\alpha}$ = 1.10 on the I branch in Figure 17; Table 3).
This might be due to $\alpha$ in this regime being driven by magnetic buoyancy
and pumping effects (Chatterjee et al., 2011). Here again, some detailed MHD
models are in reasonably good agreement: a power law fit to the results of
models M1–4 of Warnecke (2018) (their slow-moderate rotators, ignoring the
very slowest) yields $P_{\rm cyc}/P_{\rm rot}=213$Ro0.514, which lies between
the A and I branches and is of similar slope (shown by a dotted black line in
Figure 18). As noted above, we further suggest that some secondary cycles are
remnant convection zone dynamos still functioning as the tachocline dynamo
ramps up to dominance. This dual $P_{\rm cyc}$ \- dual dynamo transitional
regime corresponds to the purple transitional epoch arrow in the dynamo
evolution in Figure 18.

In the end, though, after further evolution along the I branch, below a
critical Ro-1, the dynamo may begin to sputter (van Saders et al., 2016), the
star begins to experience magnetic grand minima (Metcalfe et al., 2016),
differential rotation weakens (Metcalfe et al., 2022) or may even reverse sign
(Gastine et al., 2014; Karak et al., 2020), and the dynamo dies, with $P_{\rm
cyc}$ going to infinity (see, e.g., the lengthening $P_{\rm cyc}$ in the
evolutionary sequence of G2 stars 18 Sco, the Sun, and $\alpha$ Cen A Judge et
al., 2017). With the cycling dynamo shut down, large-scale fields driving
spin-down vanish, stopping further rotational evolution, and the star evolves
vertically off the diagram at constant Ro (e.g., Metcalfe & van Saders, 2017,
shown by a vertical gold arrow in Figure 18).

##  5 Conclusions

We have found evidence for magnetic activity cycles in at least 12 M-type
stars, and traces of cyclic behavior in three more, using photometric time
series. Of these 15 M dwarfs, approximately 12 are type M3.5 or later and
likely fully convective. For these cycles, we investigated the correlation
between cycle amplitude and period, and found that this is reasonably well
described by a power law with an exponent of $0.94\pm 0.11$. We also found
indications of multiple branches in this relation, potentially in agreement
with the findings of Saar & Brandenburg (2002). However, the unclear nature of
these additional “cycles” (see Section 3.3) makes the latter less certain.

We further investigated correlations between cycle and rotation period, cycle
period and Rossby number, and the ratio of cycle to rotation period and
inverse Rossby number. Our results suggest that the dynamo processes acting
within M dwarfs may not be as different from those acting in FGK type stars
(at least the more rapidly rotating ones with Ro${}^{-1}<3$) as some previous
models (e.g., Kitchatinov & Rüdiger, 1999) suggest. Some more recent models
(e.g., Warnecke, 2018; Brun et al., 2022, shown in Figure 18) appear to
capture the overall trend for faster rotating FGK stars in Figure 17; if
similar models in fully convective M stars show similar trends, theory and
observations may be in reasonable agreement.

Using Rossby number to parameterize magnetic activity, and a simple dynamo
model (Saar & Brandenburg, 1999), we find, at equivalent Rossby numbers, the
$\alpha$ effect is similarly quenched with rotation ($q_{\alpha}\approx-1$ in
the unsaturated regime), but is of reduced amplitude in M-dwarfs compared to
that of FGK stars. These findings are in agreement with some models (e.g.,
Ruediger & Kichatinov, 1993), however are in apparent disagreement with the
results of Saar & Brandenburg (1999), whose limited sample suggested a
contrary relation (at least for their A and I branches). However, we propose
that what we are actually seeing is the result of one type of dynamo (an
$\alpha$-quenched $\alpha^{2}$ or $\alpha^{2}\Omega$ full convection zone
dynamo), present in all fast rotators (a thermally driven dynamo is an
alternative possibility). In FGK stars, once they have spun down sufficiently,
a tachocline-based $\alpha$ anti-quenched $\alpha\Omega$ dynamo takes over
(forming the I branch of Brandenburg et al., 1998). Fully convective M stars,
lacking tachoclines, spin down retaining their full convective zone dynamos.
We also find that the $\alpha$ effect is again anti-quenched in the saturated
regime, which some theoretical models support under certain conditions (e.g.,
Chatterjee et al., 2011).

Clearly, stellar cycles and the underlying physical mechanisms are not yet
fully understood. To follow up this work, we suggest these stars be monitored
using ZDI, which can be used to determine large-scale magnetic field topology
and polarity, and in the UV or X-rays, which are much more sensitive and
directly related to magnetic activity. These follow-up observations would
reveal whether the cycles presented in Table 1 are a result of global magnetic
activity, or some other phenomena (i.e. Rossby waves, etc.). It will be
particularly useful to search for more cycles in K7-M2 stars, which are
lacking in our sample. These cycles may reveal whether the $\alpha$ effect
transition from FGK to fully convective M stars (Figure 16) is sharp or
gradual. It is also intriguing that $\delta$ (i.e, the power law slopes)
appears to remain constant through the joint $\alpha$ and DR transition when
activity saturation ends (Section 4.7); we propose that the implications of
this for the underlying dynamo should be explored.

## Acknowledgments

This paper includes data collected by the TESS mission, which are publicly
available from the Mikulski Archive for Space Telescopes (MAST). Funding for
the TESS mission is provided by NASA’s Science Mission directorate. We
gratefully acknowledge use of data provided by the ASAS and ASAS-SN
collaborations.

Z.A.I thanks the University of Southampton’s Center for Astrophysics $|$
Harvard & Smithsonian exchange program, with special thanks to program
coordinators Diego Altamirano and Jeremy Drake. S.H.S. gratefully acknowledges
support from NASA Heliophysics LWS grant NNX16AB79G, NASA XRP grant
80NSSC21K0607, and NASA EPRV grant 80NSSC21K1037. We thank Axel Brandenburg
for helpful discussions. This work was also supported by NASA’s Swift Guest
Investigator program under Grant 80NSSC20K1111, and B.J.W was supported by
NASA contract NAS8-03060 to the Chandra X-Ray Center. We thank the anonymous
referee for many helpful suggestions which have improved the paper.

This research also made use of Lightkurve, a Python package for Kepler and
TESS data analysis (Lightkurve Collaboration et al., 2018); NumPy (Harris et
al., 2020); Astropy,101010http://www.astropy.org a community-developed core
Python package for Astronomy (Astropy Collaboration et al., 2013, 2018); SciPy
(Virtanen et al., 2020); Matplotlib, a 2D graphics environment (Hunter, 2007).

## Data Availability

Data analyzed herein are publicly available via the ASAS All-Star
Catalog,111111http://www.astrouw.edu.pl/asas the ASAS-SN light curve
server,121212https://asas-sn.osu.edu and the Mikulski Space Archive for Space
Telescopes (MAST): http://dx.doi.org/10.17909/xdwe-q663 (catalog
10.17909/xdwe-q663). ASAS-4 data on Proxima were provided by G. Pojmański.

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