A&A, 689, A288 (2024)
https://doi.org/10.1051/0004-6361/202450523
c© The Authors 2024
Astronomy
&Astrophysics
Searching for the stellar cycles of low-mass stars using TESS data
Gavin Ramsay1,?, Pasi Hakala2, and J. Gerry Doyle1
1 Armagh Observatory and Planetarium, College Hill, Armagh, BT61 9DG, N. Ireland, UK
2 Finnish Centre for Astronomy with ESO (FINCA), Quantum, Vesilinnantie 5, FI-20014 University of Turku, Finland
Received 26 April 2024 / Accepted 9 July 2024
ABSTRACT
We carried out a search for stellar activity cycles in late low-mass M dwarfs (M0–M6) located in the TESS northern and southern
continuous viewing zones using data from sectors 1–61 (Cycle 1 to partway through Cycle 5). We utilised TESS-SPOC data, which
initially had a cadence of 30 min and was then reduced to 10 min in Cycle 3. In addition, we required for each star to be observed in at
least six sectors in each north and south Cycle: 1950 low-mass stars ultimately met these criteria. Strong evidence was seen in 245 stars
for a very stable photometric variation that we assumed to be a signature of the stars’ rotation period. We conducted 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 that showed evidence of variability in their photometric amplitude and only
one solar-like star. Some display a monotonic trend over 3–4 years, whilst others reveal shorter term variations. We determined the
predicted cycle durations of these stars using a relationship found in the literature and an estimate of the stars’ Rossby number. Finally,
we found a marginally statistically significant correlation between the range in the rotational amplitude modulation and the rotation
period.
Key words. stars: activity – stars: flare – stars: late-type – stars: low-mass – stars: rotation
1. Introduction
Low-mass stars account for nearly three-quarters of the stars
in our local neighbourhood (Henry et al. 2006). In spite of this,
over the last few decades, a more intense focus has been renewed
for these low-luminosity stars. Within the MV spectral type,
stars exhibit masses in the range of 0.08–0.6 M (M9V–M0V),
becoming fully convective around the early-to-mid M type. The
reason for the renewed interest in these stars is due to exoplan-
ets producing a greater transit depth in these small stars, thereby
making them easier to detect. Furthermore, many of these stars
are active, especially fully convective stars (e.g. Pettersen 1989),
which provides an insight into how fully convective stars gener-
ate and sustain a global magnetic field. This topic is of interest
to the field of magnetism in general.
Moreover, the instruments to study stars of all kinds has rad-
ically 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) as well as 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 star’s light
? Corresponding author; gavin.ramsay@armagh.ac.uk
curve is due to the rotation of a starspot(s) that come in and out
of view as the star rotates. For stars whose rotation axis is viewed
near face-on, it is expected that the 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 tempera-
tures (Jackson & Jeffries 2013; Luger et al. 2021a). On the other
hand, Luger et al. (2021b) argued that some degeneracies in the
spot distributions can be broken by analysing the light curve of
many stars with broadly similar properties.
Although there is still some uncertainty in terms of map-
ping 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 ∼11 yr (strictly speak-
ing ∼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
(CMEs). Given the timescales expected (years), finding evidence
for stellar activity cycles have taken time to emerge. System-
atic and regular spectroscopic observations of solar-like stars
began during the 1960s 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 ∼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 ∼4 yr in
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A288, page 1 of 10
Ramsay, G., et al.: A&A, 689, A288 (2024)
the M4V Gl 729. Küker et al. (2019) reviewed observations that
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
that 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 ∼11◦ 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 over time. They found
amplitude periodicities in 3203 stars, with some 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, we refer to 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 that showed very stable periods,
we searched for amplitude variations that may reveal evidence
of stellar activity cycles.
2. 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◦ × 90◦
instantaneous strip of sky (a sector) for ∼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 light from the Earth and Moon. 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 addi-
tion 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 becom-
ing 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) colours and MG
absolute magnitudes, we were able to select the ones with an
equivalent spectral type later than ∼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 2020) that uses the full-frame images to extract
light curves (we searched light curves from TESS sectors 0–61).
0.5 1.0 1.5 2.0 2.5 3.0 3.5
(BP RP)
2
4
6
8
10
12
14
M
G
Fig. 1. Gaia HRD (BP−RP, MG) of low-mass stars (red) and solar type
stars (green) that were observed for at least six sectors in each TESS
Cycle.
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 six sectors. Finally, since the pixel
size of TESS (21′′ per pixel) is sufficiently large to include spa-
tially 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, which are
shown as green dots in Figure 1.
3. Data analysis
Each sector of data was normalised so its mean was unity and
then clipped to remove data that were 5σ deviation from the
mean. For each sector of data we used the LSP, as imple-
mented in the VARTOOLS suite of tools (Hartman & Bakos 2016;
Zechmeister & Kürster 2009), to search for evidence for peri-
odic 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 deter-
mined. We then obtained the median value for these parame-
ters for each star. Since the light curves can still have systematic
trends present even after a global trend has been removed, this
can cause peaks in the LSP that are not astrophysical. In addi-
tion, 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).
We therefore use the approach of Lu et al. (2022), who deter-
mined the spread in the period in different quarters of Kepler
A288, page 2 of 10
Ramsay, G., et al.: A&A, 689, A288 (2024)
0 2 4 6 8 10
Period (d)
10 4
10 3
10 2
10 1
100
101
LS
5
10
15
20
25
30
Se
ct
or
s
0 2 4 6 8 10
Period (d)
10 4
10 3
10 2
10 1
100
101
LS
5
10
15
20
25
Se
ct
or
s
Fig. 2. Median period from their individual light curves and calculated ∆LS For all the 1950 M dwarf stars in the sample (left). Higher values of
∆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. The same distribution but for the 1432 stars in our solar-like sample is shown on the right.
data, using:
∆LS =
1
NS 〈PLS,s〉
∑
s
∣∣∣PLS,s − 〈PLS,s〉∣∣∣ , (1)
where NS is the number of sectors with measured PLS,s, and 〈· · · 〉
denotes the median value operator. We do not necessarily expect
the rotation period of active stars to be strictly periodic and there-
fore have very low values of ∆LS, since active regions can emerge
and vanish at different latitudes over the sequence of the obser-
vations, 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
∆LS and then recomputed ∆LS so that if an individual sector was
half the median period (within a few percent) we used twice this
value and redetermined ∆LS. We show the median period and its
spread (∆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 ∆LS values that 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 ∆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 ∆LS
higher than the strip of sources with low ∆LS: we found evidence
for some stars appearing to be pulsators and others that 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 ∆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 dis-
cussed in Sect. 3. 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 ∆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 SIMBAD1 database for infor-
mation on these stars and found they were a mixture of eclipsing
binaries; eruptive variables and red clump low-mass core He-
burning stars (e.g. Ruiz-Dern et al. 2018). We conclude they are
not single solar-like pulsator stars, but other types of variable star
instead; this lends support to the view that the majority of stable
periodic low-mass stars (left-hand panel of Figure 2) are single
low-mass stars that 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, with ∆LS > 0.05
and ∆LS < 0.05. We therefore identified 245/1950 M dwarfs and
194/1432 solar-type stars which have ∆LS < 0.05 for the next
stage of our analysis.
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 iden-
tify a suitable statistical test which can robustly identify vari-
ations or trends in the data. To achieve this, we have used
two different non-parametric tests for variability in the time
series, namely, the Mann-Kendall (MK) test for monotonic
trends in time series (Mann 1945; Kendall 1975) and a mod-
ified (non-parametric) version of the R-statistic introduced in
Baptista & Steiner (1993). The MK test is sensitive to mono-
tonic (both linear and non-linear) trends in the data, whilst the
Rnp-test measures the temporal distribution of the signs of the
residuals, ri, after subtracting the mean level from the time
series. We define Rnp as:
Rnp =
1√
N − 1
N−1∑
i=1
ri
|ri|
ri+1
|ri+1| , (2)
where ri are the time series residuals after subtracting the mean
level. Then Rnp follows a discrete normal distribution with a
1 https://simbad.cds.unistra.fr/simbad/
A288, page 3 of 10
Ramsay, G., et al.: A&A, 689, A288 (2024)
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.01
0.02
0.03
0.04
0.05
Am
p 
M
od
ul
at
io
n
TIC142015852
1800 2000 2200 2400 2600 2800
Time (Tess days)
0.025
0.030
0.035
0.040
0.045
0.050
Am
p 
M
od
ul
at
io
n
TIC160118032
1800 2000 2200 2400 2600 2800
Time (Tess days)
0.006
0.007
0.008
0.009
0.010
0.011
Am
p 
M
od
ul
at
io
n
TIC441811894
Fig. 3. Three examples of low-mass stars,
TIC 142015852, TIC 160118032, and TIC
441811894, which have a variable rota-
tional amplitude.
1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
(BP RP)
6
7
8
9
10
11
12
13
14
M
G
RUWE<1.4
1.4<RUWE<2.0
RUWE>2.0
0 2 4 6 8 10
Period (d)
10 4
10 3
10 2
10 1
100
101
LS
Fig. 4. Gaia HRD (BP − RP, MG) of low
mass stars (grey) with those identified as
showing significant variability of the rota-
tional amplitude over time shown as circles
whose size indicates their Gaia RUWE value
(see text for details) on the left. The low mass
star sample (orange dots) is on the right-hand
panel, with the 29 stars which were identi-
fied as having a rotational amplitude which
varies significantly over time shown as a red
× symbol.
mean 0.0 and σ = 1.0. Large positive values of Rnp corre-
spond to cases where a number of consecutive datapoints show
the same sign (either + or –) in their residuals after subtraction
of the mean, namely, the constant value model does not fit the
data. The large negative values of Rnp would imply systemati-
cally 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 Rnp is discrete Gaussian, even if the
data distribution would not be known. This enables us to obtain
reliable probabilities for Rnp values, even in the case of a small
number of points.
These two tests are sensitive to different types of variabil-
ity (monotonic trend vs correlated and 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 Rnp-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 1 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 Figures
A.1–A.3). 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 three yrs and TIC 441811894, which
shows a rise followed by a decline several years later. In con-
trast, we found evidence for only one star in our solar-like sam-
ple (TIC 38707949) that 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 fur-
ther, we indicate using the colour of the points the Gaia renor-
malised 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
2021a). Initial indications suggest that for stars with RUWE
>1.4 the star is an unresolved binary system (Lindegren 2021b),
although this does not preclude that binaries can have RUWE
values <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 exhibit RUWE >1.4,
suggesting they are possibly binary systems.
We folded and phase-binned the data on the period identified
for that sector and we 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 that are representative of
the other stars in the sample.
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 that we identified as having variable
amplitudes. Out of the 29 stars, 26 are located in the band of
stars with low ∆LS values. All the stars have periods shorter than
7 days. There are three stars with periods shorter than 2 days and
have ∆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 sev-
eral 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. Therefore, we did not consider these stars in the
next stage of the analysis and, thus, we were left with 26 stars.
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 1960s (Wilson 1968), Brandenburg et al.
A288, page 4 of 10
Ramsay, G., et al.: A&A, 689, A288 (2024)
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0.994
0.996
0.998
1.000
1.002
TIC382147080
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0.98
0.99
1.00
1.01
1.02
TIC230384382
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0.90
0.92
0.94
0.96
0.98
1.00
TIC38586438
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0.985
0.990
0.995
1.000
1.005
1.010
TIC160118032
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0.98
0.99
1.00
1.01
1.02
TIC271693722
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0.996
0.998
1.000
1.002
1.004
1.006
TIC142082942
Fig. 5. 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.
(1998) found evidence that the ratio of the period of the activ-
ity cycle, Pcyc, and the stellar rotation period, Prot, was related
to the Rossby number (Ro = Prot/τ), where τ is the convec-
tive turnover time. More recently, Irving et al. (2023) searched
for a relationship between Pcyc/Prot and Ro for solar-like and
M dwarf stars. For the latter, they determined Pcyc/Prot = 3.6
R−1.02o (see Fig. 16 and Eq. 4 of Irving et al. 2023). Based
on this relationship, we calculated what the predicted Pcyc
for the stars in our sample would be. We assume that the
period we measured from the TESS data is the star’s rotation
period.
Because Ro is not an observable quantity, we have to use a
proxy. One established means is the one derived by Wright et al.
(2018), who use the (V − Ks) colour to derive τ. How-
ever, Irving et al. (2023) using work of Corsaro et al. (2021),
Landin et al. (2023) and Pecaut & Mamajek (2013) derived new
relationships for (V − Ks) and τ. For (V − Ks) < 4.6 (stars ear-
lier than M3V), the resulting values of τ are similar to that of
Wright et al. (2018). Following Irving et al. (2023), we used the
τL relationship for (V −Ks) < 4.6 and τG for (V −Ks) > 4.6 (see
Figure 13 of Irving et al. 2023: for those stars near the bound-
ary we chose the higher value). The values of τ for each star
are shown in Table 1. Using τ and the period for each star, we
computed Ro and used the relationship of Irving et al. between
Pcyc/Prot and Ro to determine the length of the activity cycle of
our stars. We show the predicted cycle lengths in Table 1 which
range from 0.6–4.6 yrs: they are longer by a factor of ∼2–7 com-
pared 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 A.1 and A.3. To estimate the length of the activity cycles
from the amplitude time series data, we experimented with fit-
ting 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 the limits and the prob-
ability 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. How-
ever, it is clear that some stars such as TIC 441734910 have a
downward trend lasting ∼3.5 yrs, implying any activity cycles
could be ∼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 ∼3 yrs: the predicted cycle length is 1.3 yrs.
This gives us modest confidence that the variations we observe
are of the same order as the predicted timescales for activity
cycles.
6. Amplitude variation as a function of rotation
period
We determined the range in the amplitude over time by sim-
ply 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 the ‘max-min’ and the
stars mass, temperature (from the TIC, Stassun & Torres 2021)
and rotation period. We used the Kendall τ 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 varia-
tion and modulation period of 0.00587 (Figure 6). We note
that a similar result was found when we compared the ampli-
tude variation with Rossby number. Although this implies a
99.41 percent probability that the correlation is real, it does
not formally reach the 3σ level. The one outlying point is TIC
341870349 (Prot ∼4 d), although the reason behind this is not yet
clear.
Although the statistical confidence in the correlation does
not formally reach the 3σ 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 show-
ing a smaller modulation. It may imply that stars that are faster
rotating (and therefore likely to be young) have spots that are
more uniformly distributed and cover a larger fraction of their
photosphere.
A288, page 5 of 10
Ramsay, G., et al.: A&A, 689, A288 (2024)
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 − Ks) τ (d) Ppredict,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
Notes. An 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 − Ks) value from the TIC, the convective turnover time, τ, 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 ∆LS than the other stars with crosses in the right hand panel of Figure 4.
7. Discussion
There is now robust evidence from spectroscopic observations
that solar-like stars may exhibit stellar activity cycles that are
similar in duration to those of 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 sim-
ilarly robust claims for other stars such as M dwarfs using
photometric measurements is difficult since all-sky precise pho-
tometry 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 FAPs which are
far lower than they are in reality (e.g. Littlefair et al. 2017).
Similarly, Basri & Shah (2020), using simulations of star spot
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 that have very stable photometric periods over a
timescale of several years. As a result, we have found 26 low-
mass M dwarfs that show evidence for a significant variation in
the amplitude of their rotational modulation over a time inter-
val 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 ∼3–4 yrs. Given the relatively
0 1 2 3 4 5 6
Period (d)
0.01
0.02
0.03
0.04
0.05
0.06
M
ax
-M
in
 A
m
p
Fig. 6. 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 rise of the amplitude variation over
period.
short rotation periods, we do not consider these to be equivalent
to the spurious activity cycles noted by Basri & Shah (2020).
Compared to solar-like stars, the number of low-mass stars
(MV and later) that have robustly determined activity cycles
A288, page 6 of 10
Ramsay, G., et al.: A&A, 689, A288 (2024)
from spectra is much fewer, partly because they are fainter in
the blue, which is 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 rota-
tional periods, a median of 33.4 d (early M) and 86.2 d (mid
M). In the sample of Irving et al. (2023) mentioned in Sect. 5, of
their 15 stars, four had periods shorter than 6 d, and there was
only one with a period shorter than 2 d. Our sample of 26 stars
therefore reveals a set of low-mass stars that are rapidly rotating
and have potential activity cycles.
8. Conclusion
In this pilot study, we have 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. Thus, 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 ∼4 d than solar-
type stars. This implies that the low spread in the measured peri-
ods 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 that may appear
to be significant at the 1% level (commonly used in the litera-
ture), but that are, in fact, due to red noise.
We searched for variations in the amplitude of the rotation
period using two statistical tests and found 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 meaning-
ful 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 mission, which is due to be launched in Dec.
2026 and will make an initial pointing in the southern hemi-
sphere 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 that is 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 that are perfectly placed to fill in the gaps
when TESS is not able to observe parts of the sky. In that case,
they can 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) will be able to 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 Consor-
tium (DPAC). Funding for the DPAC is provided by national institutions, in par-
ticular 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 Obser-
vatory & 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 in the amplitude
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.002
0.004
0.006
0.008
0.010
0.012
Am
p 
M
od
ul
at
io
n
TIC33881250
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.01
0.02
0.03
0.04
0.05
Am
p 
M
od
ul
at
io
n
TIC38515801
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.026
0.028
0.030
0.032
0.034
0.036
0.038
0.040
Am
p 
M
od
ul
at
io
n
TIC38586438
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.01
0.02
0.03
0.04
0.05
Am
p 
M
od
ul
at
io
n
TIC142015852
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.008
0.010
0.012
0.014
0.016
0.018
0.020
Am
p 
M
od
ul
at
io
n
TIC142082942
1400 1600 1800 2000 2200 2400 2600 2800 3000
Time (Tess days)
0.008
0.010
0.012
0.014
0.016
Am
p 
M
od
ul
at
io
n
TIC149347629
1800 2000 2200 2400 2600 2800
Time (Tess days)
0.025
0.030
0.035
0.040
0.045
0.050
Am
p 
M
od
ul
at
io
n
TIC160118032
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.0125
0.0150
0.0175
0.0200
0.0225
0.0250
0.0275
Am
p 
M
od
ul
at
io
n
TIC176980970
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.006
0.008
0.010
0.012
0.014
0.016
Am
p 
M
od
ul
at
io
n
TIC177313167
1800 2000 2200 2400 2600 2800
Time (Tess days)
0.011
0.012
0.013
0.014
0.015
0.016
Am
p 
M
od
ul
at
io
n
TIC198557860
1800 2000 2200 2400 2600 2800 3000
Time (Tess days)
0.002
0.004
0.006
0.008
0.010
0.012
Am
p 
M
od
ul
at
io
n
TIC219773818
1800 2000 2200 2400 2600 2800 3000
Time (Tess days)
0.015
0.020
0.025
0.030
0.035
Am
p 
M
od
ul
at
io
n
TIC230384382
Fig. A.1. Amplitude of the rotational modulation of low-mass stars that we have identified as showing significant variability.
A288, page 8 of 10
Ramsay, G., et al.: A&A, 689, A288 (2024)
1800 2000 2200 2400 2600 2800 3000
Time (Tess days)
0.002
0.004
0.006
0.008
0.010
0.012
0.014
Am
p 
M
od
ul
at
io
n
TIC230395228
1800 2000 2200 2400 2600 2800 3000
Time (Tess days)
0.01
0.02
0.03
0.04
Am
p 
M
od
ul
at
io
n
TIC233547261
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
Am
p 
M
od
ul
at
io
n
TIC271693722
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Am
p 
M
od
ul
at
io
n
TIC278825715
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.005
0.010
0.015
0.020
0.025
Am
p 
M
od
ul
at
io
n
TIC294329266
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.0150
0.0175
0.0200
0.0225
0.0250
0.0275
0.0300
0.0325
Am
p 
M
od
ul
at
io
n
TIC300866996
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.015
0.020
0.025
0.030
0.035
0.040
Am
p 
M
od
ul
at
io
n
TIC306779230
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.0100
0.0125
0.0150
0.0175
0.0200
0.0225
0.0250
0.0275
0.0300
Am
p 
M
od
ul
at
io
n
TIC339770560
1800 2000 2200 2400 2600 2800 3000
Time (Tess days)
0.004
0.006
0.008
0.010
0.012
0.014
0.016
Am
p 
M
od
ul
at
io
n
TIC341870349
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.0025
0.0050
0.0075
0.0100
0.0125
0.0150
0.0175
0.0200
Am
p 
M
od
ul
at
io
n
TIC349192028
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.004
0.006
0.008
0.010
0.012
0.014
Am
p 
M
od
ul
at
io
n
TIC350559457
1800 2000 2200 2400 2600 2800 3000
Time (Tess days)
0.030
0.035
0.040
0.045
0.050
0.055
Am
p 
M
od
ul
at
io
n
TIC353898013
1400 1600 1800 2000 2200 2400
Time (Tess days)
0.0100
0.0125
0.0150
0.0175
0.0200
0.0225
0.0250
0.0275
0.0300
Am
p 
M
od
ul
at
io
n
TIC375035131
1500 1750 2000 2250 2500 2750 3000
Time (Tess days)
0.006
0.008
0.010
0.012
0.014
Am
p 
M
od
ul
at
io
n
TIC382147080
1800 2000 2200 2400 2600 2800 3000
Time (Tess days)
0.0175
0.0200
0.0225
0.0250
0.0275
0.0300
0.0325
0.0350
0.0375
Am
p 
M
od
ul
at
io
n
TIC402897917
Fig. A.2. Amplitude of the rotational modulation of low-mass stars that we have identified as showing significant variability (cont).
A288, page 9 of 10
Ramsay, G., et al.: A&A, 689, A288 (2024)
1800 2000 2200 2400 2600 2800 3000
Time (Tess days)
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
0.055
Am
p 
M
od
ul
at
io
n
TIC441734910
1800 2000 2200 2400 2600 2800
Time (Tess days)
0.006
0.007
0.008
0.009
0.010
0.011
Am
p 
M
od
ul
at
io
n
TIC441811894
Fig. A.3. Amplitude of the rotational modulation of low-mass stars that we have identified as showing significant variability (cont).
A288, page 10 of 10