11institutetext: Armagh Observatory and Planetarium, College Hill, Armagh,
BT61 9DG, N. Ireland, UK22institutetext: Finnish Centre for Astronomy with ESO
(FINCA), Quantum, Vesilinnantie 5, FI-20014, University of Turku, Finland
22email<EMAIL_ADDRESS>
# Searching for stellar cycles on low mass stars using TESS data

Gavin Ramsay 11 Pasi Hakala 22 J. Gerry Doyle 11

(Accepted 9 July 2024)

We have searched for stellar activity cycles in late low mass M dwarfs (M0–M6)
located in the TESS north and south continuous viewing zones using data from
sectors 1–61 (Cycle 1 to part way through Cycle 5). We utilise TESS-SPOC data
which initially had a cadence of 30 min but reducing to 10 min in Cycles 3. In
addition, we require each star to be observed in at least 6 sectors in each
North/South Cycle: 1,950 low mass stars meet these criteria. Strong evidence
was seen in 245 stars for a very stable photometric variation which we assume
to be a signature of the stars rotation period. We did a similar study for
Solar-like stars and found that 194 out of 1432 stars had a very stable
modulation. We then searched for evidence of a variation in the rotational
amplitude. We found 26 low mass stars showed evidence of variability in their
photometric amplitude and only one Solar-like star. Some show a monotonic
trend over 3–4 yrs whilst other show shorter term variations. We determine the
predicted cycle durations of these stars using the relationship found by
Irving et al. (2023) using an estimate of the stars Rossby number. Finally we
find a marginally statistically significant correlation between the range in
the rotational amplitude modulation and the rotation period.

###### Key Words.:

Physical data and processes: magnetic fields – Stars: activity, flares, low-
mass, late-type, rotation

##  1 Introduction

Low mass stars account for nearly 3/4 of the stars in our local neighbourhood
(Henry et al., 2006). In spite of this, it is only in the last few decades
that a more intense focus has returned to these low luminosity stars. Within
the MV spectral type, stars have masses in the range 0.08–0.6 $M_{\odot}$
(M9V–M0V), becoming fully convective around early-to mid M. The reason for the
renewed interest in these stars is due to exoplanets producing a greater
transit depth in these small stars making them easier to detect. Furthermore,
many of these stars are active, especially fully convective stars (e.g.
Pettersen (1989)), which provides insight into how fully convective stars
generate and sustain a global magnetic field, which is of interest to the
field of magnetism in general.

Moreover, the instruments to study stars of all kinds has radically changed
since the first exoplanets were detected. Prior to the launch of Corot and
Kepler, several groups attempted ground-based multi-telescope observations of
single stars (e.g. Rodono et al. (1986); Doyle et al. (1988a, b, 1993)). What
these series of space missions facilitated (amongst many other things) was the
measurement of the rotation period for a large number of stars (McQuillan et
al., 2014) and a comprehensive search for flares from stars with a wide range
of spectral type (Davenport, 2016). In turn, this allows for the determination
of how the stellar rotation period varies as a function of age and mass
(Rebull et al., 2016) and how the rate of flares from an active host star can
affect the atmosphere of any orbiting exoplanet (Konings et al., 2022).

Determining the rotation period of low mass stars is usually based on the
assumption that any modulation in the stars light curve is due to the rotation
of a starspot(s) which come into and out of view as the star rotates. For
stars whose rotation axis is viewed near face-on, it is expected that
starspot(s) would remain in view and show no modulation unless the spot
coverage varied over time. Initially it was assumed that a single large spot
group caused the modulation in a star’s flux, with any variations in the
rotation flux profile due to changes in the spot distribution or that
additional starspots emerged. However, it has been shown that single filter
photometry data can be reproduced by a wide range of spot distribution, sizes,
inclinations and spot temperatures (Jackson & Jeffries, 2013; Luger et al.,
2021a). On the other hand, Luger et al. (2021b) argue that some degeneracies
in the spot distributions can be broken by analysing the light curve of many
stars which have broadly similar properties.

Although there is still some uncertainty in mapping the distribution of spots
over the surface of these stars, progress has been made in searching for
activity cycles on stars other than the Sun. The Solar cycle lasts $\sim$11 yr
(strictly speaking $\sim$22 yr) and manifests itself through the changing
number of sunspots; emission in the core of the Ca ii H&K lines; X-ray flux
and the number of flares and Coronal Mass Ejections. Given the timescales
expected (years), finding evidence for stellar activity cycles took time to
emerge. Systematic and regular spectroscopic observations of Solar-like stars
began during the 1960’s with a focus on using the Ca ii H&K lines as an
activity indicator (Wilson, 1968). Subsequent studies using data such as these
showed that other stars exhibit activity cycles on a similar timescale to the
Sun (e.g. Baliunas et al. (1995) and references therein). For lower mass
stars, an activity cycle of $\sim$7 yr was detected in the star nearest to the
Sun, Proxima Centauri (M5.5Ve), using optical, UV and X-ray observations
(Wargelin et al., 2017). Furthermore, Ibañez Bustos et al. (2020) used Ca ii
H&K lines to determine a stellar cycle of $\sim$4 yr in the M4V Gl 729. Kuker
et al (2019) reviewed observations which appear to show that fast rotating M
dwarfs have cycles of a year but increase to around four years for slower
rotators. However, fast rotators cannot be understood by an advection-
dominated dynamo model.

In principle, ground based optical photometric observations which are stable
and free from systematic trends can be used to detect activity cycles as long
as the timeline is sufficiently long. Since 2018 the TESS satellite has
gradually been building up a set of observations of stars covering most of the
sky. This is particulary true for stars within $\sim 11^{\circ}$ of the north
and south ecliptic poles (the ‘continuous viewing zones’) since they can be
observed for nearly one year with a return visit one year later. However,
since these light curves are not flux calibrated, low amplitude, long term
modulations or trends can be difficult to detect in a robust manner.

An alternative means is to search for variations in the amplitude of the
rotational modulation over time (Berdyugina & Järvinen, 2005). Using this
approach, Suárez Mascareño et al. (2016) determined that two K type stars
showed periods of 10.2 and 8.7 yr. Reinhold et al. (2017) used Kepler data to
study the light curves of 23601 stars and used the Generalised Lomb Scargle
Periodgram (LSP) to search for variations in the rotational amplitude with
time and found amplitude periodicities in 3203 stars, with weak evidence that
the cycle period increased for longer rotation periods. Most of the stars in
that sample had spectral types earlier than MV stars. For a recent review of
activity cycles on stars other than Sun see Jeffers et al. (2023).

In this exploratory study we use TESS data of low mass stars located in the
north and south continuous viewing zones on two occasions to assess how stable
the dominant period in their light curves is. For stars which showed very
stable periods we searched for amplitude variations which may reveal evidence
of stellar activity cycles.

##  2 The TESS sample

The TESS satellite was launched into an Earth orbit with a period of 13.7 d.
It has four 10.5 cm telescopes that observe a 24${}^{\circ}\times 90^{\circ}$
instantaneous strip of sky (a sector) for $\sim$27 d (see Ricker et al., 2015,
for details). In the first year (Cycle 1) TESS covered most of the southern
ecliptic hemisphere and in the second year (Cycle 2) it covered the northern
ecliptic hemisphere, although with less sky coverage than Cycle 1 to avoid
contamination from stray Earth and Moon-light. In both Cycles, there is an
area around the ecliptic poles which are visible in each sector. In Cycle 3
the field pointings mirrored Cycle 1 and in Cycle 4, some sectors observed in
Cycle 2 were observed again in addition to some fields not previous observed.
In Cycles 1 and 2, full frame images were obtained every 30 min, which
increased to every 10 min in Cycles 3 and 4. From the start, the Guest
Observer programme was able to populate targets which were observed in a 2-min
cadence, with a 20 s cadence option becoming available in Cycles 3 and 4.

We used the Gaia EDR3 catalogue of stars within 100 pc (Smart et al., 2021)
and using their $(BP-RP)$ colour and $MG$ absolute magnitude we selected those
which had an equivalent spectral type later than $\sim$M0V and were on, or
close to, the main sequence. We then cross matched this sample with those
stars which had TESS light curves derived using the TESS Science Processing
Operations Center FFI Target List Products (SPOC, Caldwell et al. (2020))
which uses the full-frame images to extract light curves (we searched light
curves from TESS sectors 0–61). Since our aim was to search for variations in
the amplitude of the main period, we then restricted our sample to require a
star to have been observed in at least 6 sectors. Finally, since the pixel
size of TESS (21`` per pixel) is sufficiently large to include spatially
nearby bright stars to contaminate the resulting light curve, we filtered
stars so the Contamination Ratio (Stassun et al., 2018) (i.e. the fraction of
the flux from other spatially nearby stars) is $<$0.1 . We show in Figure 1
the 1950 stars which are our M dwarf sample. For a comparison, we also
obtained a Solar-like sample using the same criteria as for M dwarfs, where we
took stars close to the main sequence for $0.6<(BP-RP)<$1.1 (F5V–K2V, Pecaut &
Mamajek (2013)): we identified 1432 stars and these are shown as green dots in
Figure 1.

Figure 1: The Gaia HRD ($BP-RP$, $M_{G}$) of low mass stars (red) and Solar
type stars (green) which were observed for at least six sectors in each TESS
Cycle.

##  3 Data analysis

Each sector of data was normalised so its mean was unity and then clipped to
remove data which was 5$\sigma$ deviation from the mean. For each sector of
data we used the LSP, as implemented in the VARTOOLS suite of tools (Hartmann
& Bakos, 2016; Zechmeister & Kürster, 2009), to search for evidence for
periodic variability. For each star we determined the period of the highest
peak in the LS power spectrum in each sector of data in the range 0.08–10 d;
its False Alarm Probability (FAP) and the full amplitude (which we shorten to
amplitude for the rest of the paper) of the modulation for each sector was
determined. We then obtained the median value for these parameters for each
star. Since the lightcurves can still have systematic trends present even
after a global trend has been removed, this can cause peaks in the LSP which
are not astrophysical. In addition, the observing window can cause peaks in
the LSP, even at periods which are a higher harmonic of the window length.
Setting a threshold for the FAP to determine if a star shows significant
variability is therefore not always clear cut (e.g. Anthony et al (2022)) and
red-noise can result in peaks in the LSP which technically have a FAP$<<$0.01
(e.g. Dorn-Wallenstein et al. (2019)).

Figure 2: Left hand panel: For all the 1950 M dwarf stars in the sample we
determined the median period from their individual light curves and calculated
$\Delta_{LS}$. Higher values of $\Delta_{LS}$ indicate a greater spread in the
values of the period over the sectors in which they were determined. The
colours of the points indicate how many sectors of data were analysed. Right
hand panel: The same distribution but for the 1432 stars in our Solar-like
sample.

We therefore use the approach of Lu et al. (2022), who determined the spread
in the period in different quarters of Kepler data using:

$\Delta_{\rm LS}=\frac{1}{N_{S}\langle P_{{\rm
LS},s}\rangle}\sum_{s}\left|P_{{\rm LS},s}-\langle P_{{\rm
LS},s}\rangle\right|,$ (1)

where $N_{S}$ is the number of sectors with measured $P_{{\rm LS},s}$, and
$\langle\cdots\rangle$ denotes the median value operator. We do not
necessarily expect the rotation period of active stars to be strictly
periodic, and therefore have very low values of $\Delta_{\rm LS}$, since
active regions can emerge and vanish at different latitudes over the sequence
of the observations, which together with differential rotation of the star can
lead to slightly different observed periods.

Since the LSP can identify half the real period when pulse profiles are double
peaked, we determined the median period and $\Delta_{\rm LS}$ and then
recomputed $\Delta_{\rm LS}$ so that if an individual sector was half the
median period (within a few percent) we used twice this value and redetermined
$\Delta_{\rm LS}$. We show the median period and its spread ($\Delta_{\rm
LS}$) for all 1950 stars in the left hand panel of Figure 2. There are two
clear grouping: a band of points starting at the lower left which have very
low $\Delta_{\rm LS}$ values which progress to higher values as the period
increases. A second group of points starts in the top left hand corner which
have a large spread in $\Delta_{\rm LS}$ which decreases as the period
increases.

We examined the light curves and periodograms for stars which had median
periods shorter than 2 d but had values of $\Delta_{\rm LS}$ higher than the
strip of sources with low $\Delta_{\rm LS}$: we found evidence for some stars
appearing to be pulsators and others which had a longer period modulation
super-imposed on the shorter period seen in an earlier Cycle of observations.
This suggests that at least some stars above the band of stars seen in the low
$\Delta_{\rm LS}$ strip are probably real variable stars but not isolated
single M dwarfs which have starspots. Of course many others are likely not
real variable stars and the LSP has just picked random peaks in the power
spectrum which are likely due to red noise.

We now make a comparison with the Solar-like sample discussed in Sect. 2\. We
applied exactly the same procedures as we did for the M dwarf sample. Given
Solar-type stars are expected to show (in general) rotation periods much
longer than a few days (e.g. McQuillan et al. (2014)) we would not expect to
see the same band of points at the lower left – unless their periods were
spurious. We show the distribution of the 1432 Solar-type stars in the right
hand panel of Figure 2: we find the same spread of points with high
$\Delta_{\rm LS}$, but only a few stars in the lower left. There are 17 Solar-
like stars which appear to show very stable periods (Figure 2). We searched
the SIMBAD111https://simbad.cds.unistra.fr/simbad/ database for information on
these stars and found they were a mixture of eclipsing binaries; eruptive
variables and Red Clump low mass core He burnging stars (e.g. Ruiz-Dern et al.
(2018)). We conclude they are not single star solar-like pulsators but other
types of variable star, which gives support to the view that the majority of
stable periodic low mass stars (left hand panel of Figure 2) are single low
mass stars which have long lived starspots.

In selecting those sources with which to search for changes in the amplitude
of variability we observe that in both samples, there is a ‘natural’ split
between most sources which have $\Delta_{\rm LS}$$>$0.05 and $\Delta_{\rm
LS}$$<$0.05. We therefore identified 245/1950 M dwarfs and 194/1432 solar-type
stars which have $\Delta_{\rm LS}$$<$0.05 for the next stage of our analysis.

Figure 3: Three examples of low mass stars, TIC 142015852, TIC 160118032, TIC
441811894, which have a variable rotational amplitude.

Figure 4: Left hand panel: The Gaia HRD ($BP-RP$, $M_{G}$) of low mass stars
(grey) with those identified as showing significant variability of the
rotational amplitude over time shown as circles whose size indicates their
Gaia RUWE value (see text for details). Right hand panel: The low mass star
sample (orange dots) with the 29 stars which were identified as having a
rotational amplitude which varies significantly over time shown as a red
$\times$ symbol.

##  4 Searching for variations in the rotational amplitude

Since the number of amplitude measurements per star is small (a maximum of 30
spread over five years) we need to identify a suitable statistical test which
can robustly identify variations or trends in the data. In order to achieve
this, we have used two different non-parametric tests for variability in the
time series i.e. the Mann-Kendall ($MK$) test for monotonic trends in time
series (Mann, 1945; Kendall, 1975) and a modified (non-parametric) version of
the $R$-statistic introduced in Baptista & Steiner (1993). The $MK$ test is
sensitive to monotonic (both linear and non-linear) trends in the data, whilst
the $R_{np}$-test measures the temporal distribution of the signs of the
residuals $r_{i}$ after subtracting the mean level from the time series. We
define $R_{np}$ as:

$R_{np}={1\over\sqrt{N-1}}\sum_{i=1}^{N-1}{r_{i}\over|r_{i}|}{r_{i+1}\over|r_{i+1}|}$
(2)

where $r_{i}$ are the time series residuals after subtracting the mean level.
$R_{np}$ follows a discrete normal distribution with a mean 0.0 and
$\sigma=1.0$. Large positive values of $R_{np}$ correspond to cases where a
number of consecutive datapoints show the same sign (either + or -) in their
residuals after subtraction of the mean, i.e. the constant value model does
not fit the data. The large negative values of $R_{np}$ would imply
systematically alternating consecutive residuals with opposite signs and could
only arise if the data contained a short period that roughly matches the
sampling period. This case is not relevant here. The probability distribution
for $R_{np}$ is discrete Gaussian, even if the data distribution would not be
known. This enables us to obtain reliable probabilities for $R_{np}$ values,
even in the case of a small number of points.

These two tests are sensitive to different types of variability (monotonic
trend vs. correlated/systematic residuals for a constant model), that can be
present in the data. We therefore selected those stars whose amplitude
variation over time gave a FAP $<$0.0001 with either the $MK$ stat or the
$R_{np}$-stat and had a minimum of 6 data points: this resulted in 29 low mass
stars (if we choose FAP $<$0.001 we obtain 53 stars). The details of these 29
stars are shown in Table 2 with three example light curves showing the
amplitude variation over time is shown in Figure 3 (the amplitude variation of
all 29 stars are shown in Figure 7–9). There is a range of light curve
characteristics such as TIC 142015852 which shows a rise to maximum followed
by a dip 1.5 yrs later; TIC 160118032 which shows a continuous increase
amplitude over 3 yrs and TIC 441811894 which shows a rise followed by a
decline several years later. In contrast, we found evidence for only one star
in our solar-like sample (TIC 38707949) which showed a general decrease in
rotational amplitude over a timescale of 2.5 yrs.

We also show the 29 stars in the Gaia HRD in the left hand panel of Figure 4
which shows that they are biased towards the red or more luminuous parts of
the lower main sequence. This may imply the stars are younger compared to the
overall sample or they are members of binaries. To explore this further, we
indicate using the colour of the points the Gaia Renormalised Unit Weight
Error (RUWE) parameter for each of the 29 stars. This is a measure of how much
the photo-center of a star moves over the course of the Gaia observations
(Lindegren et al., 2021a). Initial indications suggest that for stars with
RUWE$>$1.4 the star is an unresolved binary system (Lindegren et al., 2021b),
although this does not preclude that binaries can have RUWE$<$1.4 (Stassun &
Torres, 2021) and that the distribution of RUWE can vary over the Gaia HRD
(Penoyre et al., 2022). We cross matched our 29 low mass stars with the
resulting catalogue of binaries within 100 pc from (Penoyre et al., 2022)
finding that only two sources with the highest RUWE ($>$5.5) were in their
catalogue. However, 7 out of these 29 stars have RUWE$>$1.4 suggesting they
are possibly binary systems.

We folded and phase binned the data on the period identified for that sector
and show six examples in Figure 5. Two of the stars were in the binary sample
derived by Penoyre et al. (2022). Another one star, TIC 38586438, is an
eclipsing binary. We show the folded light curves for three stars which are
representative of the other stars in the sample.

Table 1: Details of the 29 low mass stars which show a significant variability
in their rotational amplitude over time.

TIC | RAJ2000 | DEJ2000 | Gmag | imag | MG | BP-RP | MedPer (d) | RUWE | $(V-K_{s})$ |  $\tau$ (d) |  $P_{predict,cyc}$ (yr)
---|---|---|---|---|---|---|---|---|---|---|---
33881250 | 66.342433 | -76.501968 | 14.23 | 13.35 | 10.4 | 3.06 | 1.141 | 1.33 | 5.72 | 360 | 3.1
38515801 | 62.551500 | -63.856128 | 14.57 | 13.68 | 10.0 | 2.86 | 1.072 | 1.19 | 5.50 | 370 | 3.7
38586438 | 64.090211 | -62.012890 | 12.83 | 12.03 | 8.9 | 2.43 | 0.556 | 1.53 | 4.65 | 420 | 6.9
142015852 | 96.591637 | -75.277925 | 12.45 |  | 8.1 | 2.08 | 2.622 | 1.24 | 4.05 | 60 | 1.3
142082942 | 98.259771 | -72.762015 | 14.23 | 13.34 | 10.9 | 2.81 | 0.924 | 1.10 | 5.42 | 370 | 3.8
149347629 | 83.648302 | -60.182846 | 14.13 | 13.25 | 9.8 | 3.03 | 0.566 | 1.10 | 5.68 | 360 | 3.2
160118032 | 231.210928 | 83.991618 | 14.61 | 13.93 | 10.1 | 2.66 | 0.351 | 1.16 | 5.03 | 400 | 5.2
176980970 | 101.601200 | -70.634179 | 14.57 | 13.72 | 10.0 | 2.84 | 0.611 | 1.18 | 5.34 | 380 | 4.1
177313167 | 103.849793 | -74.048574 | 14.21 | 13.33 | 11.5 | 2.97 | 0.300 | 1.20 | 5.53 | 370 | 3.6
198557860 | 238.295269 | 62.595799 | 14.50 | 13.79 | 11.1 | 3.20 | 0.423 | 1.28 | 5.69 | 360 | 3.2
219773818 | 263.311581 | 63.207745 | 14.56 | 13.86 | 12.7 | 3.42 | 1.138 | 1.11 | 6.26 | 350 | 2.2
230384382 | 285.730684 | 60.586065 | 14.72 | 14.04 | 10.2 | 2.77 | 0.497 | 5.56∗ | 4.93 | 400 | 5.5
230395228 | 286.673233 | 63.754117 | 13.69 | 12.82 | 10.6 | 3.00 | 0.780 | 1.17 | 5.23 | 380 | 4.4
233547261 | 281.738333 | 60.895959 | 14.46 | 13.77 | 10.0 | 2.61 | 2.451 | 1.14 | 4.98 | 400 | 5.4
271693722 | 109.549273 | -75.347401 | 14.71 | 13.86 | 10.3 | 2.72 | 1.923 | 1.19 | 4.97 | 400 | 5.4
278825715 | 100.316544 | -56.275061 | 13.46 | 12.56 | 9.5 | 2.79 | 6.503 | 1.24 |  |  |
294329266 | 109.049841 | -58.814441 | 13.50 | 12.64 | 9.5 | 2.73 | 0.370 | 2.44 | 5.32 | 380 | 4.2
300866996 | 117.229923 | -67.359987 | 14.32 | 13.37 | 10.7 | 3.45 | 0.355 | 1.84 | 6.64 | 350 | 1.8
306779230 | 121.428627 | -71.190802 | 14.41 | 13.52 | 9.6 | 2.92 | 1.329 | 1.16 | 5.73 | 350 | 3.0
339770560 | 111.121245 | -59.372994 | 13.86 | 12.99 | 9.9 | 2.89 | 1.098 | 1.15 |  |  |
341870349 | 276.732905 | 71.413962 | 13.22 | 12.61 | 8.7 | 2.43 | 4.041 | 1.27 | 4.61 | 420 | 7.0
349192028 | 108.836918 | -60.377753 | 12.86 | 12.14 | 8.2 | 2.11 | 2.059 | 1.30 | 4.08 | 60 | 1.3
350559457 | 86.563846 | -55.797830 | 14.66 | 13.74 | 11.4 | 3.28 | 0.571 | 1.05 | 6.16 | 350 | 2.4
353898013 | 271.899854 | 56.324229 | 13.94 | 13.35 | 10.7 | 2.91 | 1.659 | 1.19 | 5.29 | 380 | 4.3
375035131 | 99.647221 | -62.560617 | 14.77 | 13.85 | 10.0 | 2.90 | 0.928 | 1.09 | 5.56 | 370 | 3.5
382147080 | 80.523154 | -55.826228 | 13.52 | 12.65 | 9.0 | 2.68 | 0.473 | 6.24∗ | 4.95 | 400 | 5.5
402897917 | 302.768145 | 68.500219 | 14.69 | 13.99 | 11.1 | 2.91 | 1.088 | 1.03 | 5.33 | 380 | 4.2
441734910 | 258.253191 | 73.934726 | 12.72 | 12.13 | 10.2 | 2.58 | 1.359 | 1.43 | 4.57 | 420 | 7.2
441811894 | 266.958528 | 73.176166 | 14.34 | 13.40 | 11.3 | 3.10 | 0.236 | 1.72 | 5.42 | 380 | 4.0

222An asterix in the RUWE column implies it has been classed as a binary in
Penoyre et al. (2022), while the folded light curve of TIC 38586438 shows it
is an eclipsing binary (Figure 5). We also show the $(V-K_{s}$) value from the
TIC, the convective turnover time, $\tau$, and the predicted length of the
activity cycle both derived from the relationships in Irving et al. (2023).
Stars TIC 33881250, 149347629 and 441811894 are the stars with periods $<$2 d
showing higher values of $\Delta_{\rm LS}$ than the other stars with crosses
in the right hand panel of Figure 4.

In the right hand panel of Figure 4 we show a similar plot as the left hand
panel of Figure 2 but highlighting the location of the 29 low mass stars which
we identified as having variable amplitudes. Out of the 29 stars, 26 are
located in the band of stars with low $\Delta_{\rm LS}$ values. All the stars
have periods shorter than 7 days. There are three stars which have periods
shorter than 2 days and have $\Delta_{\rm LS}$ values much higher than the
others. One of these (TIC 33881250) has one sector where the period of the
highest peak differs from the others and has a much higher FAP, although
several flares could have interfered with the period determination. Another
two stars (TIC 149347629 and 441811894) show two periods with the highest peak
changing: this may indicate they are pulsators. We therefore do not consider
these stars in the next stage of the analysis and are left with 26 stars.

Figure 5: The first sector of data for six stars in our M dwarf sample which
have been folded and binned on the period of the highest peak in the LSP – the
phase is arbitrary. The left hand and middle stars in the upper panel are in
the binary sample of Penoyre et al. (2022) while TIC 38586438 appears to be an
eclipsing binary. The shape of the folded light curves of TIC 160118032,
271693722 and 142082942 are representative of other stars in the sample.

##  5 Expected length of stellar cycles in low mass stars

Using measurements of the Ca ii H&K lines using data taken at Mt Wilson from
the 1960’s (Wilson, 1968), Brandenburg et al. (1998) found evidence that the
ratio of the period of the activity cycle, $P_{cyc}$, and the stellar rotation
period, $P_{rot}$, was related to the Rossby number ($R_{o}=P_{rot}/{\tau})$,
where $\tau$ is the convective turnover time. More recently, Irving et al.
(2023) searched for a relationship between $P_{cyc}/P_{rot}$ and $R_{o}$ for
Solar-like and M dwarf stars. For the latter they determined $P_{cyc}/P_{rot}$
= 3.6 R${}_{o}^{-1.02}$ (see Fig 16 and eqn 4 of Irving et al. (2023)). Based
on this relationship we calculated what the predicted $P_{cyc}$ for the stars
in our sample would be. We assume that the period we measure from the TESS
data is the stars rotation period.

Because $R_{o}$ is not an observable quantity, we have to use a proxy. One
established means is that derived by Wright et al. (2018) who use the
$(V-K_{s})$ colour to derive $\tau$. However, Irving et al. (2023) using work
of Corsaro et al. (2021), Landin et al. (2023) and Pecaut & Mamajek (2013),
derived new relationships for $(V-K_{s})$ and $\tau$. For $(V-K_{s})<4.6$
(stars earlier than M3V) the resulting values of $\tau$ are similar to that of
Wright et al. (2018). Following Irving et al. (2023) we use the $\tau_{L}$
relationship for $(V-K_{s})<4.6$ and $\tau_{G}$ for $(V-K_{s})>4.6$ (see
Figure 13 of Irving et al. (2023): for those stars near the boundary we chose
the higher value). The values of $\tau$ for each star are shown in Table 2.
Using $\tau$ and the period for each star we compute $R_{o}$ and use the
relationship of Irving et al. between $P_{cyc}/P_{rot}$ and Ro to determine
the length of the activity cycle of our stars. We show the predicted cycle
lengths in Table 2 which range from 0.6–4.6 yrs: they are longer by a factor
of $\sim$2–7 compared to the values determined by simply using the
relationship of Wright et al. (2018).

The amplitude variation over time is shown for all stars in Figures 7 and 9.
In order to estimate the length of the activity cycles from the amplitude time
series data, we experimented with fitting the amplitude time series with a
Fourier series models with different number of harmonic terms and also the
period as a free parameter. Such a problem is non-linear and thus we carried
out extensive MCMC fitting to derive limits/probability distribution for any
activity periods. However, we were not able to obtain repeatable and reliable
results. This is due to the gaps in the time series, and the length of the
trends. However, it is clear that some stars such as TIC 441734910 have a
downward trend lasting $\sim$3.5 yrs implying any activity cycles could be
$\sim$8–10 yrs. The predicted activity cycle based on the relationship of
Irving et al. (2023) is 7.2 yrs. In contrast, TIC 142015852 shows an amplitude
variation which is consistent with a timescale of $\sim$3 yrs: the predicted
cycle length is 1.3 yrs. This gives us modest confidence that the variations
which we observe are of the same order as the predicted timescales for
activity cycles.

Figure 6: The variation in the amplitude of the modulation over the course of
their observation as a function of modulation period. There is a marginally
significant trend in the size of the amplitude variation over period.

##  6 Amplitude variation as a function of rotation period

We determined the range in the amplitude over time by simply taking the
difference between the maximum and minimum values for each of the 26 stars.
The fractional mean amplitude ‘Max-Min’ is 0.022, with a range between 0.005
and 0.063. We then searched for a correlation between Max-Min and the stars
mass, temperature (from the TIC, Stassun & Torres (2021)) and rotation period.
We used the Kendall $\tau$ test to determine the significance of any
correlation. We found no evidence for a correlation between mass or
temperature. However, we found a FAP for the rank correlation between the
amplitude variation and modulation period of 0.00587 (Figure 6). (A similar
result was found when we compare the amplitude variation with Rossby number).
Although this implies a 99.41 percent probability that the correlation is
real, it does not formally reach the 3$\sigma$ level. The one outlying point
is TIC 341870349 ($P_{rot}\sim$4 d), although it is not obvious why this is
the case.

Although statistical confidence in the correlation does not formally reach the
3$\sigma$ level, we note that stars showing a larger range in the amplitude of
modulation are likely to have a greater asymmetry in the distribution of spots
than stars showing a smaller modulation. It may imply stars which are faster
rotating (and therefore likely young) have spots more uniformly distributed
and covering a larger fraction of their photosphere.

##  7 Discussion

There is now robust evidence from spectroscopic observations that Solar-like
stars can show stellar activity cycles similar in duration to that found in
the Sun. This is because of the length of observations, such as initiated at
Mt Wilson (Wilson, 1968), but also more recent high precision spectroscopy
from instruments such as HARPS (Lovis et al, 2011). Obtaining similarly robust
claims for other stars such as M dwarfs using photometric measurements is
difficult since all-sky precise photometry is still challenging (e.g. Stubbs &
Tonry (2006)). We also note that most claims of significant detections of
periods in photometric data are made using the LSP which assume that data have
a Gaussian noise profile whereas almost all such data streams contain red
noise which result in FAP’s which are far lower than they are in reality (e.g.
Littlefair et al. (2017)). Similarly, Basri & Shah (2020), using simulations
of starspot distributions, show that ‘short’ activity cycles (of the order of
several tens or more rotation cycles) are simply due to random processes.

In this exploratory study, we have gone to great lengths to identify stars
which have very stable photometric periods over a timescale of several years.
As a result we have found 26 low mass M dwarfs which show evidence for a
significant variation in the amplitude of their rotational modulation over a
time interval of nearly five years. There are some stars whose amplitude
variation points to a long term cycle of up to ten years and there are other
stars whose timescale is $\sim$3–4 yrs. Given the relatively short rotation
periods, we do not consider these to be equivalent of spurious activity cycles
noted by Basri & Shah (2020).

Compared to Solar-like stars, the number of low mass stars (MV and later)
which have robustly determined activity cycles from spectra is much fewer,
partly because they are fainter in the blue where the activity marker Ca ii
H&K lines are present. Suárez Mascareño et al. (2016) used photometric data
obtained using the All Sky Automated Survey (ASAS) to search for activity
cycles and found 25 early M and 9 mid M spectral types – these samples were
biased towards relatively long rotational periods, a median of 33.4 d (early
M) and 86.2 d (mid M). In the sample of Irving et al. (2023) which was
mentioned in Sect. 5, of their 15 stars, four had periods shorter than 6 d and
only one with a period shorter than 2 d. Our sample of 26 stars therefore
reveals a set of low mass stars which are rapidly rotating and have potential
activity cycles.

##  8 Conclusion

In this pilot study, we examined the TESS light curves of 1950 low mass and
1432 Solar-type stars. We found that only 245 (12.6%) low mass stars showed
stable periods over many sectors compared to 194 (13.5%) Solar-type stars – we
assume that these periods are a signature of the stars rotation period.
However, there are many more low mass stars which show very stable periods at
periods shorter than $\sim$4 d than Solar-type stars. This implies that the
low spread in measured periods of the short period M dwarfs is real and not
produced by sampling or any artefacts in the data. We note that the FAP of
periods detected using the LSP can give periods which may appear to be
significant at the 1% level (which is commonly used in the literature) which
are in fact due to red noise.

We searched for variations in the amplitude of the rotation period using two
statistical tests and find that 29 low mass stars and one Solar-type star
shows evidence for significant variability. The TESS light curves of stars in
the continuous viewing zones all have gaps of one year. This makes determining
any meaningful estimate of the period of variation difficult. However, they
are not widely inconsistent with the predicted cycle length which is based on
the stars Rossby number.

One mission which will observe the same field continuously is ESA’s PLATO
misssion which is due to be launched in Dec 2026 and will make an initial
pointing in the southern hemisphere lasting at least 2 yrs (Nascimbeni et al.,
2022) covering a continuous area of 1037 deg2. Since this field will have been
observed a number of times using TESS it will provide the means to study the
modulation amplitude of thousands of stars over a time interval comparable or
longer to the expected activity cycles.

Another means to fill the gaps is ground based wide-field optical surveys such
as GOTO (Steeghs et al., 2022; Dyer et al., 2022), or ATLAS (Tonry et al.,
2018), which have telescopes in both hemispheres are perfectly placed to fill
in the gaps when TESS is not able to observe parts of the sky, if they make
extended observations of the same fields every few months. It would also be
interesting to determine if there is any variation in the X-ray flux of the 26
stars found to show long term variability in their rotational amplitude. We
hope that eRosita (Predehl et al., 2021) can resume operations in the near
future.

###### Acknowledgements.

This paper includes data collected by the TESS mission, for which funding is
provided by the NASA Explorer Program. This work also presents results from
the European Space Agency (ESA) space mission Gaia. Gaia data is being
processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding
for the DPAC is provided by national institutions, in particular the
institutions participating in the Gaia MultiLateral Agreement (MLA). The Gaia
mission website is https://www.cosmos.esa.int/gaia. The Gaia archive website
is https://archives.esac.esa.int/gaia. Armagh Observatory & Planetarium is
core funded by the N. Ireland Executive through the Dept. for Communities. JGD
would like to thank the Leverhulme Trust for a Emeritus Fellowship.

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##  Appendix A Long term variability of amplitude

Figure 7: The amplitude of the rotational modulation of low mass stars which
we have identified as showing significant variability. Figure 8: The amplitude
of the rotational modulation of low mass stars which we have identified as
showing significant variability (Cont). Figure 9: The amplitude of the
rotational modulation of low mass stars which we have identified as showing
significant variability (Cont).