E. Magaudda et al

<EMAIL_ADDRESS>
# First eROSITA-TESS results for M dwarfs: Mass dependence of the X-ray
activity rotation relation and an assessment of sensitivity limits

E. Magaudda  B. Stelzer  St. Raetz  Institut für Astronomie und Astrophysik,
Eberhard-Karls Universität Tübingen, Sand 1, D-72076, Tübingen, Germany  INAF-
Osservatorio Astronomico di Palermo, Piazza Parlamento 1, I-90134, Palermo,
Italy

###### Abstract

We present a study of the activity-rotation relation for M dwarf stars, using
new X-ray data from the ROentgen Survey with an Imaging Telescope Array
(eROSITA) on board the Russian Spektrum-Roentgen-Gamma mission (SRG), combined
with photometric rotation periods from the Transiting Exoplanet Survey
Satellite (TESS). The stars used in this work are selected from the superblink
proper motion catalog of nearby M dwarfs. We study the $135$ stars with both a
detection in the first eROSITA survey (eRASS1) and a rotation period
measurement from TESS jointly with the sample of $197$ superblink M dwarfs re-
adapted from our previous work. We fit the activity-rotation relation for
stars with rotation periods shorter than $\sim 10$ d (saturated regime) using
three mass bins. The surprising positive slope for stars in our lowest mass
bin ($M_{\star}\leq 0.4{\rm M_{\odot}}$) is due to a paucity of stars with
intermediate rotation periods ($\sim 1-10$ d), probably caused by fast period
evolution. The much higher fraction of eRASS1 detections compared to stars
that have also rotation periods from TESS shows that eROSITA is also sensitive
for slower rotating M dwarfs that are in the unsaturated regime with periods
inaccessible to TESS.

###### keywords:

stars:activity - stars:X-rays - stars:late-type - stars:rotation

††articletype: Article Type

##  1 Introduction

Solar- and later-type main sequence and giant stars experience magnetic
activity in the inner and outer atmosphere. The mechanism that powers and
enhances the magnetic field in solar-like stars is known to be an
$\alpha\Omega-$dynamo, i.e. it operates with combined contributions of the
convective motions of the outer stellar envelope ($\alpha$) and the stellar
rotation ($\Omega$). As a consequence, cooler regions than the surroundings,
so-called dark spots, form in the photosphere, while brighter and hotter
structures arise in chromosphere and corona. The latter ones produce copious
UV, X-ray, and radio emission. In later-type stars, close and beyond the fully
convective transition (SpT $\sim$ M3), the mechanism driving the continuous
formation and variation of magnetic fields is not well understood. An indirect
measure of this phenomenon is investigating the coronal activity-rotation
relation, typically expressed in terms of the X-ray luminosity ($L_{\rm x}$)
as a function of the rotational period ($P_{\rm rot}$). Pallavicini . (1981)
were the first to investigate the coronal X-ray emission as a function of
rotational velocity for a wide range of stellar spectral types (O3 to M).
Later, Pizzolato . (2003) focused their study on late-type main-sequence stars
with X-ray data from the ROSAT satellite and $P_{\rm rot}$ calculated from
$v\sin i$ measurements. In this early work the activity-rotation relation
presents two different regimes: (1) saturated, where the activity of fast
rotators does not depend on rotation, and (2) unsaturated, where the X-ray
activity of slowly rotating stars decreases with increasing rotation period.

In the most recent work from Magaudda . (2020) we presented a comprehensive
study of the relation between rotation, X-ray activity and age for $\sim 300$
M dwarfs. We homogenized data from the literature (Wright ., 2011, 2018;
Wright Drake, 2016; Stelzer ., 2016; González-Álvarez ., 2019) and added in
new very sensitive observations from dedicated observations with the X-ray
satellites XMM-Newton and Chandra and the photometry mission K2 from which we
derived rotation periods. We found a significantly steeper slope in the
unsaturated regime than previous works for fully convective stars and we
confirmed a non-constant X-ray emission level in the saturated regime, as
first proposed by Reiners . (2014).

In this article, we combine our previous results on the X-ray activity-
rotation relation (Magaudda ., 2020) with those obtained from new observations
with the extended ROentgen survey with an Imaging Telescope Array (eROSITA;
Predehl ., 2021) on the Russian Spektrum Roentgen-Gamma
(SRG)111http://srg.iki.rssi.ru/?page_id=676&lang=en mission and the Transiting
Exoplanet Survey Satellite (TESS; Ricker ., 2014).

We use the eROSITA and TESS sample compiled by Magaudda . (2021), on the basis
of the superblink proper motion survey by Lépine Gaidos (2011) (LG11, $\sim
9000$ M dwarfs), that we characterized with Gaia-DR2 data. We refer to
Magaudda . (2021) for more information on the derivation of our final catalog
of LG11-Gaia M dwarfs detected with eROSITA and with TESS observations. More
details on the sample used for the analysis of the activity-rotation relation
treated in this work are also given in Sect. 2, and we present the results in
Sect. 3, with an investigation of the possible observational biases related to
them. In Sect. 4 we summarize our conclusions and give an outlook to future
studies in this field.

##  2 Sample

We selected M dwarfs from the superblink proper motion catalog of nearby M
dwarfs from Lépine Gaidos (2011) choosing only stars with complete photometry
from the second data release of the Gaia mission (Gaia DR2, Gaia Collaboration
., 2018) and distance from Bailer-Jones . (2018) (hereafter LG11-Gaia). In
particular, we study the X-ray emission of our LG11-Gaia sample during the
first eROSITA All-Sky survey (eRASS1). We use the preliminary catalog produced
by the eROSITA consortium that is based on data collected from Dec 2018 to
June 2019. We retrieve $P_{\rm rot}$ from TESS light curves with $2$-min
cadence. A complete discussion of the eROSITA/TESS source identification and
data analysis is found in Magaudda . (2021).

The sample used in this work includes the LG11-Gaia M dwarfs detected with
eROSITA and with reliable $P_{\rm rot}$ from TESS light curves. We refer to it
as the LG11-Gaia/eRASS1/TESS sample. The relations we used to calculate
stellar parameters (Mann ., 2015, 2016) are valid for stars with $4.6<M_{\rm
Ks}<9.8$ and $0.1<R_{\star}/R_{\odot}<0.7$, therefore we keep only the sub-
sample for which these conditions apply and we call it the ‘validated’
LG11-Gaia/eRASS1/TESS catalog.

Next to the ‘validated’ LG11-Gaia/eRASS1/TESS sample we use the MagauddaGaia20
catalog, that was selected from the one used in Magaudda . (2020) where for
consistency we consider only X-ray detected sources (i.e. we disregard all
upper limits) with complete Gaia-DR2 and in the validity range of the
relations from Mann . (2015, 2016), amounting to $197$ stars.

##  3 Results & Discussion

###  3.1 The X-ray activity-rotation relation

Table 1: Best fit parameter results. Mass Bin | Nstars | $\beta$ | Intercept
---|---|---|---
|  |  | [erg/s]
Full Sample | 242 | +0.12$\pm$0.06 | 28.78$\pm$0.03
$M_{\star}/M_{\odot}>0.6$ | 56 |  $-0.05\pm$0.09 | 29.36$\pm$0.03
$0.4\leq M_{\star}/M_{\odot}\leq 0.6$ | 120 |  $-0.19\pm$0.06 | 28.90$\pm$0.03
$M_{\star}/M_{\odot}<0.4$ | 72 | +0.24$\pm$0.07 | 28.18$\pm$0.04

New data from the eROSITA and TESS satellites provide unprecedented statistics
of fast rotating M dwarfs located in the saturated regime of the X-ray
activity-rotation relation. In Fig. 8 from Magaudda . (2021) we can appreciate
the capability of eROSITA in detecting stars. In fact the amount of X-ray
detected stars we present in Magaudda . (2021) is the largest to date for M
dwarfs in the saturated regime ($P_{\rm rot}\lesssim 10$ d), and extends to
the lowest X-ray luminosities measured for fast rotators so far. The absence
of stars with higher $P_{\rm rot}$ is explained by the duration of the TESS
campaigns ($\sim 27$ d) which impedes the detection of periods of slower
rotators, and consequently a study of the non-saturated region of the
relation. Thus, our new analysis is focused on the saturated regime, where we
added the new ‘validated’ LG11-Gaia/eRASS1/TESS sample to MagauddaGaia20.

In Magaudda . (2020) we provided a statistical analysis of both the saturated
and the unsaturated regimes. Therein we analyzed the activity-rotation
relation first for the whole sample and then derived it within three mass
bins, (1) $M_{\star}/M_{\odot}>0.6$, (2) $0.4\leq M_{\star}/M_{\odot}\leq
0.6$, and (3) $M_{\star}/M_{\odot}<0.4$.

Following that work, we analyze here the $L_{\rm x}-P_{\rm rot}$ relation in
the saturated regime of the full new sample, i.e. ‘validated’
LG11-Gaia/eRASS1/TESS plus MagauddaGaia20 stars, and then in individual mass
bins.

In the analysis of our new sample across the whole ‘validated’ mass range we
use the transition from the saturated to the non-saturated regime found in
Magaudda . (2020) for the full mass sample ($P_{\rm sat}=8.5$ d) to define the
saturated region for the sample. We performed a power law fit to the $L_{\rm
x}-P_{\rm rot}$ data for the full new sample. The best fit parameters are
listed in Table 1 and the value for the slope is also shown in Fig. 1.
Surprisingly, a positive slope is found, i.e. $L_{\rm x}$ increases with
increasing $P_{\rm rot}$, contrary to all previous studies of the saturated
regime that found $L_{\rm x}\approx\rm const$ or a slightly decreasing slope
(Magaudda ., 2020; Reiners ., 2014).

Figure 1: The X-ray activity-rotation relation for the sample of this work,
where new X-ray data from eROSITA and photometric rotation periods from TESS
were added to the sample presented by Magaudda . (2020). The power law fit in
the saturated regime, $P_{\rm rot}\leq 8.5$ d (red dashed line), and its slope
$\beta$ are shown.

To examine potential observational biases, we analyzed the relation in three
different mass bins, each with its own maximum period for the saturated regime
($P_{\rm rot,sat}$) from the analysis performed by Magaudda . (2020). In
particular, $P_{\rm rot,sat}~{}(M_{\star}/M_{\odot}>0.6)=5.2\pm 0.7$ d,
$P_{\rm rot,sat}~{}(0.4\leq M_{\star}/M_{\odot}\leq 0.6)=11.8\pm 2.0$ d and
$P_{\rm rot,sat}~{}(M_{\star}/M_{\odot}<0.4)=33.7\pm 4.5$ d. In Fig. 2 we show
the results of the power law fits for the three mass bins, and the best fit
parameters are, as before, listed in Table 1. It is evident that the slope is
positive only in the lowest mass bin. In this bin $L_{\rm x}$ covers a wide
range, going from the minimum $L_{\rm x}-$level detected for M dwarfs with
very short $P_{\rm rot}$ measurements ($P_{\rm rot}\leq 0.5$ d) to almost its
maximum value. In the bottom panel of Fig. 2 we present the $L_{\rm x}-P_{\rm
rot}$ relation for the lowest mass bin with a $M_{\star}-$color code. Here we
can see that the period distribution of very low mass stars
($M_{\star}/M_{\odot}\leq 0.2$) is composed of stars with very short or very
long $P_{\rm rot}$ (in the saturated and unsaturated regime, respectively).
Observational biases are strong as already mentioned in Magaudda . (2020) and
Magaudda . (2021): very long rotation periods are taken from Wright . (2018)
who selected data from the MEarth project to study the slow rotator regime.
TESS observations provide periods shorter than $\sim 15$ d because TESS
monitors a given field for only about a month, and we considered reliable all
$P_{\rm rot}$ shorter than about half the duration of a sector light curve.
However, these biases can not explain the absence of stars with $P_{\rm
rot}\approx 1-10$ d that is present in these very low mass stars
($M_{\star}/M_{\odot}<0.2$). The paucity of very low mass M dwarfs with
intermediate rotation periods is probably caused by fast angular momentum loss
that carries the stars in little time from being fast to slow rotators (Newton
., 2017). Hence, in this scenario it is intrinsically rare to observe late M
dwarfs with rotation periods of $\approx 1-10$ d.

In the highest and the intermediate mass bins the $L_{\rm x}-P_{\rm rot}$
relation shows a marginally negative slope ($\beta_{high}=-0.05\pm 0.09$ and
$\beta_{int}=-0.19\pm 0.06$) more in line with previous estimates, although
slightly smaller than what we found in Magaudda . (2020), where
$\beta_{high}=-0.17\pm 0.14$ and $\beta_{int}=-0.39\pm 0.13$.

Figure 2: $L_{\rm x}-P_{\rm rot}$ relation and power law fits in the stellar
mass bins defined in Sect. 3.1.

###  3.2 eROSITA detection sensitivity

Figure 3: Distance distributions in the three stellar mass bins defined in
Sect. 3.2 for the full LG11-Gaia sample (in gray) in comparison with the M
dwarf sub-sample detected only with eROSITA during eRASS1 (in red) and the one
detected with eROSITA and with reliable $P_{\rm rot}$ from TESS light curves
(in green). In the highest mass bin there are four stars with $Dist_{\rm
BJ18}>100$ pc that we do not show to have a better visualization of the
distance distribution. Figure 4: Ratio between the red and gray samples of the
histograms in Fig. 3, calculated for each mass bin. We indicate the number of
stars in each bin for the eRASS1 detections, i.e. the LG11-Gaia/eRASS1 sample.
Figure 5: X-ray luminosity distribution for the highest mass bin of LG11-Gaia
stars detected in eRASS1 compared to that of the subsample with reliable
$P_{\rm rot}$. Figure 6: The relation between $L_{\rm x}$ and TESS magnitude
for the LG11-Gaia/eRASS1 sample shown with a mass color code and the same
relation for those detections with reliable $P_{\rm rot}$ (open red diamonds).

Through the study of the activity-rotation relation in Sect. 3.1 we
encountered observational biases that call for a thorough investigation of the
eROSITA detection statistics for the different mass bins.

The LG11-Gaia sample, with stellar parameters that are within the validation
range of the relations from Mann . (2015, 2016), counts $7319$ stars with
complete Gaia-DR2 data, i.e. photometry and distance. Eight percent of this
sample is identified with eRASS1 sources, and $\sim~{}6$ % is detected in
eRASS1 and has a TESS 2-min light curve. We extracted reliable $P_{\rm rot}$
from TESS light curves for $135$ stars. This corresponds to $\sim~{}2\,\%$ of
the whole LG11-Gaia sample and $23$ % of the eRASS1 detections. Thus, as we
mentioned in Magaudda . (2021), monitoring coronal X-ray emission with eROSITA
is more efficient in identifying stellar magnetic activity than photometric
starspots with TESS.

To investigate the eROSITA detection statistics we explored the distance
($Dist_{\rm BJ18}$) and mass distributions of the different samples, and how
they relate to the detection efficiency of the instrument.

We separated the LG11-Gaia sample into the three $M_{\star}-$bins used in
Sect. 3.1. In Fig. 3 we distinguish the complete LG11-Gaia sample (gray
histogram) from those sources detected with eROSITA and those having both an
eROSITA detection and reliable TESS $P_{\rm rot}$ (red and green histograms,
respectively). The LG11-Gaia catalog consists of nearby ($Dist_{\rm BJ18}<100$
pc) and bright ($J<10$ mag) M stars according to the selection criteria
adopted by Lépine Gaidos (2011). In Fig. 3 the maximum distance is indeed
$\sim 100$ pc with $0.05$ % outliers with distances up to $\sim 250$ pc for
massive M dwarfs and it decreases for lower mass stars that are fainter.

From Fig. 3 we see that the eRASS1 distance limit is only slightly lower than
that of the LG11-Gaia catalog as a whole, but the fraction of eRASS1
detections decreases with distance. This is quantified in Fig. 4 where we
present the fraction of X-ray detected LG11-Gaia M dwarfs vs distance
separately for the three $M_{\star}-$bins. In other words, we show the ratio
between the red and the gray histograms of Fig. 3. As a measure for the
statistics we indicate the number of X-ray detected stars in each distance bin
attached to the plotting symbol. From this representation it is evident that
at a given distance the fraction of eRASS1 detected stars decreases with
decreasing mass. Apart from one outlier in the intermediate mass bin, the
detection fraction for lower distances reaches up to $\approx 30\,\%$ while it
is $<10$ % for distances beyond $\approx 20$ pc.

Comparing the eRASS1 sample to the subsample that also has reliable rotation
periods from TESS (red and green histograms in Fig. 3) we can see that there
is no clear difference in their distance limit, but the amount of detected
stars in the eRASS1 sample is larger than the one in the subsample with
reliable $P_{\rm rot}$, especially for the highest mass bin. In Fig. 5 we show
the $L_{\rm x}$ distribution for M dwarfs with $M_{\star}/M_{\odot}>0.6$ for
the eRASS1 sample with the one for eRASS1 stars with reliable $P_{\rm rot}$
(same color code as in Fig. 3). We see that periods have been detected only in
the stars with the highest $\log L_{\rm x}$ ($\gtrapprox 29.0$ erg/s).
According to the activity-rotation relation for the highest mass bin (top left
panel in Fig. 2) stars with $L_{\rm x}\lesssim 29$ erg/s are expected to have
$P_{\rm rot}\gtrsim 10$ d. We have verified that the situation is similar for
the intermediate-mass bin. Therefore, except for the lowest-mass M dwarfs,
eRASS1 can detect X-ray emission from unsaturated stars, for which we can not
detect rotation periods with TESS.

To further investigate how the capability of TESS for detecting rotation
periods depends on other parameters we show in Fig. 6 a diagram of X-ray
luminosity versus TESS magnitude222We have calculated $T_{\rm mag}$ with the
conversion from Gaia photometry provided by Stassun . (2019). ($T_{\rm mag}$).
Stars for which we were able to derive $P_{\rm rot}$ values are highlighted in
red. We show the rest of the eRASS1 sample with a mass color code. From this
representation we see (1) that rotation periods are mostly found for stars
with high $L_{\rm x}$-level as already shown in Fig. 5 and (2) that very few
rotation periods are detected on optically bright stars ($T_{\rm mag}\lesssim
9$ mag). This is surprising at first sight, as rotational modulation should be
more easy to identify in stars with brighter $T$ magnitude due to the higher
photometric precision.

We first focus on early M dwarfs (yellow and green in Fig. 6). It is clear
that there is a lack of optically bright ($T_{\rm mag}\lesssim 9$ mag) stars
with $\log{L_{\rm x}}\,{\rm[erg/s]}\gtrsim 28$. The optically brightest stars
sample the shortest distances. Thus, by the absence of stars in the upper left
of Fig. 6 we conclude that the sample is devoid of very nearby early M dwarfs
in the saturated regime. This can be explained if at least for the two higher-
mass bins, the unsaturated regime holds many more stars than the saturated
regime (represented by the yellow and green objects at bright $T_{\rm mag}$
but low $L_{\rm x}$). In fact, in the large sample of M dwarfs analyzed by
McQuillan . (2013, 2014) and based on three years of Kepler data the period
distribution shows an upper envelope that increases for decreasing masses. In
particular, stars in the mass range studied in this work are largely detected
in the saturated regime ($P_{\rm rot}{\color[rgb]{0,0,1}\lesssim}10$ d). For
the lowest mass stars ($M_{*}\leq 0.4\,{\rm M_{\odot}}$) the $T$ magnitude
distribution is similar than that of the higher-mass stars but shifted to
lower $L_{\rm x}$. However, here the absence of period detections in the
optically brighter (more nearby) stars is due to the fact that these have low
$L_{\rm x}$ values, that is they are unsaturated and their periods are
undetectably long for TESS. Optically fainter – that is more distant – stars
at the same (low) X-ray luminosity are entirely absent from the sample (see
the void of objects in the lower right of Fig. 6) because they are beyond the
eRASS1 flux limit.

##  4 Conclusions

We investigated the X-ray activity-rotation relation for M dwarfs combining
the sample from Magaudda . (2020) with new X-ray data from eROSITA and
photometric rotation periods extracted from TESS light curves. We performed a
power law fit of the saturated regime in three mass bins, finding that stars
with $M_{\star}<0.4\,M_{\odot}$ show a positive slope due to the paucity of
mid-to-late M dwarfs with intermediate rotation periods (Newton ., 2017). Past
studies (e.g. Prosser ., 1996; Jeffries ., 2011) proposed a so-called ‘super-
saturation’ regime where at very fast rotation rates (and low Rossby number)
the coronal activity is reduced. The rising slope that we find in the right
panels of Fig. 2 is driven by the lowest mass stars ($M_{*}\lesssim 0.2\,{\rm
M_{\odot}}$). However, we can not ascertain whether some of them represent
supersaturated downward outliers because of the absence of stars with the same
mass at intermediate periods ($\approx 1-10$ d) mentioned above.

In the course of our investigation of sensitivity limits we explored the mass
and distance dependencies finding that: (1) at a given distance the fraction
of eRASS1 detections decreases with decreasing stellar mass, a result of the
mass-dependence of $L_{\rm x}$ and the eRASS1 sensitivity limit. (2) eROSITA
is sensitive to detect stars with larger rotation periods that are not
detectable with TESS, (3) for early M dwarfs the saturated regime is very
poorly sampled with a very small sky volume indicating that in this mass range
the majority of stars are unsaturated, i.e. they are already in an evolved
phase of their spin-down history consistent with Kepler $P_{\rm
rot}-$distributions.

Clearly, several observational biases mix with intrinsic distributions of
X-ray activity and rotation periods. Here we have presented a first assessment
of such effects that require a more detailed study in the future.

## Acknowledgments

EM is supported by the Bundesministerium für Wirtschaft und Energie through
the Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR) under grant number
FKZ 50 OR 1808.

This work is based on data from eROSITA, the soft X-ray instrument aboard SRG,
a joint Russian-German science mission supported by the Russian Space Agency
(Roskosmos), in the interests of the Russian Academy of Sciences represented
by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und
Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL)
and its subcontractors, and is operated by NPOL with support from the Max
Planck Institute for Extraterrestrial Physics (MPE).

This paper includes data collected with the TESS mission, obtained from the
MAST data archive at the Space Telescope Science Institute (STScI). Funding
for the TESS mission is provided by the NASA Explorer Program. STScI is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS 5–26555.

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