11institutetext:  Institut de Recherche en Astrophysique et Planétologie,
Université de Toulouse, CNRS, 14 avenue Edouard Belin, F-31400, Toulouse,
France, 11email<EMAIL_ADDRESS>
Laboratório Nacional de Astrofísica, Rua Estados Unidos 154, 37504-364,
Itajubá - MG, Brazil
Institut d’Astrophysique de Paris, CNRS, Sorbonne Université, 98 bis bd Arago,
75014 Paris, France
Departamento de Física-Icex-UFMG Antônio Carlos, 6627, 31270-901 Belo
Horizonte, MG, Brazil
Science Division, Directorate of Science, European Space Research and
Technology Centre (ESA/ESTEC), Keplerlaan 1, 2201 AZ, Noordwijk, The
Netherlands
Department of Earth Sciences, University of Hawai’i at Mānoa, 1680 East-West
Road, Honolulu, HI, 96822, USA
Laboratoire Univers et Particules de Montpellier, Université de Montpellier,
CNRS, F-34095 Montpellier, France
Canada-France-Hawaii Telescope, CNRS, Kamuela, HI 96743, USA
Université de Montréal, Département de Physique, IREX, Montréal, QC H3C 3J7,
Canada
Beijing Normal University, No.19, Xinjiekouwai St, Haidian District, Beijing,
100875, People’s Republic of China
Université Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France
Laboratoire d’Astrophysique de Marseille, Aix-Marseille Université, CNRS, 38
rue Frédéric Joliot-Curie, F-13388, Marseille, France
Institut UTINAM, CNRS, Université Bourgogne Franche-Comté, OSU THETA Franche-
Comté-Bourgogne, Observatoire de Besançon, BP 1615, 25010 Besançon Cedex,
France

# The SPIRou legacy survey

Rotation period of quiet M dwarfs from circular polarization in near-infrared
spectral lines: I. The SPIRou APERO analysis

P. Fouqué , Based on observations obtained at the Canada-France-Hawaii
Telescope (CFHT) which is operated from the summit of Maunakea by the National
Research Council of Canada, the Institut National des Sciences de l’Univers of
the Centre National de la Recherche Scientifique of France, and the University
of Hawaii. Based on observations obtained with SPIRou, an international
project led by Institut de Recherche en Astrophysique et Planétologie,
Toulouse, France.11 E. Martioli 2233 J.-F. Donati 11 L.T. Lehmann 11 B. Zaire
1144 S. Bellotti 1155 E. Gaidos 66 J. Morin 77 C. Moutou 11 P. Petit 11 S.H.P.
Alencar 44 L. Arnold 88 É. Artigau 99 T.-Q. Cang 1010 A. Carmona 1111 N.J.
Cook 99 P. Cortés-Zuleta 1212 P.I. Cristofari 11 X. Delfosse 1111 R. Doyon 99
G. Hébrard 33 L. Malo 99 C. Reylé 1313 C. Usher
881122334455667788991010111112121313

(Received ; accepted )

###### Abstract

Context. The rotation period of stars is an important parameter together with
mass, radius, and effective temperature. It is an essential parameter for any
radial velocity monitoring, as stellar activity can mimic the presence of a
planet at the stellar rotation period. Several methods exist to measure it,
including long sequences of photometric measurements or temporal series of
stellar activity indicators.

Aims. Here, we use the circular polarization in near-infrared spectral lines
for a sample of 43 quiet M dwarfs and compare the measured rotation periods to
those obtained with other methods.

Methods. From Stokes $V$ spectropolarimetric sequences observed with SPIRou at
the Canada-France-Hawaii Telescope and the data processed with the APERO
pipeline, we computed the least-squares deconvolution profiles using different
masks of atomic stellar lines with known Landé factor appropriate to the
effective temperature of the star. We derived the longitudinal magnetic field
to examine its possible variation in 50 to 200 observations of each star. To
determine the stellar rotation period, we applied a Gaussian process
regression, enabling us to determine the rotation period of stars with
evolving longitudinal field.

Results. We were able to measure a rotation period for 27 of the 43 stars of
our sample. The rotation period was previously unknown for 8 of these stars.
Our rotation periods agree well with periods found in the literature based on
photometry and activity indicators, and we confirm that near-infrared
spectropolarimetry is an important tool for measuring rotation periods, even
for magnetically quiet stars. Furthermore, we computed the ages for 20 stars
of our sample using gyrochronology.

###### Key Words.:

stars: planetary systems – stars: individual – techniques: polarimetry, radial
velocity

##  1 Introduction

The rotation period (hereafter $P_{\mathrm{rot}}$) is an important
characteristic of a star, as is its effective temperature ($T_{\mathrm{eff}}$
), mass, radius and corresponding gravity ($\log g$), metallicity
($\left[M/H\right]$), and age. In the context of exoplanet search by
velocimetry (measure of the radial velocity, RV), the knowledge of the stellar
rotation period avoids attributing a periodogram peak to an exoplanet orbital
period by taking the systematic noise that is introduced by stellar activity
into account (e.g., Queloz et al., 2001). Several publications identified a
periodic signal as due to activity and/or proposed methods to distinguish RV
jitter from Keplerian variation (see, e.g., Bonfils et al. 2007; Figueira et
al. 2010; Boisse et al. 2011) or identified a signal that was previously
announced as due to a planet as actually being caused by activity (e.g.,
Huélamo et al. 2008; Robertson et al. 2014; Robertson & Mahadevan 2014; Kane
et al. 2016; Faria et al. 2020).

Stellar rotation periods for magnetic stars have been measured using the
small-scale surface magnetic field, which is detected through the broadening
and intensification of some stellar lines due to the Zeeman effect (e.g.,
Babcock, 1947) or the large-scale surface magnetic field derived from the
polarimetric signal of spectral lines, as pioneered by Preston (1971);
Landstreet (1980) for early-type stars. A periodic variation in the
longitudinal magnetic field is interpreted as due to the periodic appearance
of magnetic regions at the surface of the star, which allows measuring its
rotation period; see Landstreet (1992) for a review. An application to M
dwarfs can be found in Donati et al. (2008); Morin et al. (2008, 2010);
Hébrard et al. (2016).

Rotation periods can also be measured from long photometric sequences obtained
with small telescopes such as ASAS 111All Sky Automated Survey(Pojmanski,
1997), APT 222Automatic Photoelectric Telescope(Henry, 1999), HATNet
333Hungarian-made Automated Telescope Network(Bakos et al., 2004), NSVS
444Northern Sky Variability Survey(Woźniak et al., 2004), SuperWASP 555Wide
Angle Search for Planets(Pollacco et al., 2006), and MEarth (Nutzman &
Charbonneau, 2008; Irwin et al., 2009): examples can be found in Kiraga &
Stepien (2007); Irwin et al. (2011); Suárez Mascareño et al. (2016); Díez
Alonso et al. (2019), among others, or more recently, from space telescopes
such as Kepler, K2, or TESS 666Transiting Exoplanet Survey Satellite(Díez
Alonso et al., 2019).

An alternative technique uses sequences of stellar activity indicators such as
H$\alpha$, Ca II H&K, the Ca II infrared triplet, or values computed at the
same time as the radial velocities using the cross-correlation function (CCF)
such as the full width at half maximum (FWHM), contrast, or the bisector
(e.g., Noyes et al., 1984; Queloz et al., 2001; Bonfils et al., 2007; West et
al., 2008; Boisse et al., 2011; Bonfils et al., 2013; Suárez Mascareño et al.,
2015; Nelson et al., 2016; Suárez Mascareño et al., 2018; Toledo-Padrón et
al., 2019; Lafarga et al., 2021).

In the framework of the SPIRou legacy survey (hereafter SLS), a sample of
about 50 nearby low-mass stars is regularly monitored to detect exoplanets
using the SPIRou spectropolarimeter (Donati et al., 2020). Some of these
targets lack a measured rotation period, and some have several measurements
that sometimes disagree. To clarify the situation and increase the sample of
known rotation periods, we focus in this paper on determining the rotation
period by exploiting the large-scale magnetic field information extracted from
the Stokes $V$ profiles of spectral lines.

The paper is organized as follows: Section 2 briefly describes SPIRou777
http://spirou.irap.omp.eu and https://www.cfht.hawaii.edu/Instruments/SPIRou/
and the SPIRou Legacy Survey
(SLS888http://spirou.irap.omp.eu/Observations/The-SPIRou-Legacy-Survey; id P40
and P42, PI: Jean-François Donati). Section 3 describes the generic software
called A PipelinE to Reduce Observations APERO
(999https://github.com/njcuk9999/apero-drs v0.7.232, Cook et al. 2022), which
is used to reduce the SPIRou observations, in particular, the
spectropolarimetric analysis and the derivation of the longitudinal magnetic
field. Section 4 explains the results for the stars in our sample, separating
them into three categories according to their spectral type. Two examples are
treated in detail, and a comparison is made with results from the literature.
Finally, Section 5 examines our general results and further perspectives.

##  2 Observations

Observations are part of the CFHT large program SPIRou Legacy Survey. SPIRou
is a stabilized high-resolution near-infrared (NIR) spectropolarimeter (Donati
et al., 2020) mounted on the 3.6 m CFHT at the top of Maunakea, Hawaii. It is
designed for high-precision velocimetry and spectropolarimetry to detect and
characterize exoplanets and stellar magnetic fields. It provides full coverage
of the NIR spectrum from 950 nm to 2500 nm at a spectral resolving power of
$\lambda/\Delta\lambda\sim 70\,000$.

The SLS was allocated 310 nights over seven semesters (February 2019 to June
2022). It covers three different science topics, called work packages (see
Donati et al. 2020 for details). The analysis of this paper is restricted to
work package 1 (WP1), dedicated to a blind planet search. More than 50 M
dwarfs were regularly monitored during the SPIRou runs (typically ten
contiguous nights per month). We excluded monitored active stars with short
and well-known rotation periods from this study ($P_{\mathrm{rot}}$¡ 5 d): Gl
388 (AD Leo), Gl 406 (CN Leo), Gl 873 (EV Lac), GJ 1111 (DX Cnc), GJ 1154, GJ
1245B, GJ 3622, and PM J18482+0741. We therefore selected 43 stars in the
original WP1 sample for which about 150 (50 at least, 250 at most)
polarimetric sequences (of four individual subexposures each) have been
secured. This represents about 6800 visits, corresponding to more than 27 000
individual spectra secured for the considered sample. The stars belonging to
our sample and their stellar characteristics are given in Table 1: Gaia
absolute magnitudes and colors use Gaia DR3 Gaia Collaboration et al. (2021);
effective temperatures and metallicities come from Cristofari et al. (2022),
who used the same set of observations and the same sample (except for Gl 581,
which does not satisfy our criterion of the minimum number of visits); and
masses were computed from absolute 2MASS $K_{\rm s}$ magnitude and metallicity
using the Mann et al. 2019 relations. The relative precision of these masses
is 2-3% according to Mann et al. (2019), and the internal errors amount to 30
K and 0.1 dex for $T_{\mathrm{eff}}$ and [M/H], according to Cristofari et al.
(2022). We divided our sample into three subsamples according to a proxy of
stellar mass. These three mass bins correspond to the three types of magnetic
behavior identified among active M dwarfs by Morin et al. (2010) (see Figure
15 therein). We prefer to define regions according to the absolute $G$
magnitude rather than using the spectral types. Although they match on
average, a low metallicity makes the star fainter at a given spectral type: Gl
412A (M1.0V, $\left[M/H\right]$$=-0.42$) is 1 mag fainter in $G$ than Gl 410
(M1V, $\left[M/H\right]$$=+0.05$), therefore both should not belong to the
same group. The horizontal lines in Table 1 separate the three subsamples. The
number of visits corresponds to the initial number, before some polarimetric
sequences were rejected for the reasons explained in Sec. 3.2. Finally, we
list the FWHM of the median Stokes $I$ profile of each star because it was
used to define the velocity range on which we measured the longitudinal
magnetic field from the Stokes $V$ profile, as explained in Sec. 3.2.

Table 1: Stellar characteristics of the sample of 43 M dwarfs. For uncertainties on mass, $T_{\mathrm{eff}}$ , and [M/H], see text. The three sections correspond to early-, mid-, and late-type M dwarfs from top to bottom Star | spectral type | $M_{\rm G}$ | $G_{\rm BP}-G_{\rm RP}$ | mass | $T_{\rm eff}$ | [M/H] | visits | FWHM Stokes $I$
---|---|---|---|---|---|---|---|---
Units |  | mag | mag | M⊙ | K | dex |  | km s-1
Gl 338B | M0V | 8.046 | 1.846 | 0.58 | 3952 | $-0.08$ | 58 | $6.95\pm 0.25$
Gl 846 | M0.5V | 8.282 | 1.967 | 0.57 | 3833 | 0.07 | 194 | $6.65\pm 0.26$
Gl 205 | M1.5V | 8.327 | 2.122 | 0.58 | 3771 | 0.43 | 160 | $6.54\pm 0.16$
Gl 410 | M1V | 8.426 | 2.006 | 0.55 | 3842 | 0.05 | 131 | $7.73\pm 0.21$
Gl 880 | M1.5V | 8.611 | 2.151 | 0.55 | 3702 | 0.26 | 166 | $6.37\pm 0.17$
Gl 514 | M1.0V | 8.793 | 2.088 | 0.50 | 3699 | $-0.07$ | 177 | $5.84\pm 0.29$
Gl 382 | M2.0V | 8.898 | 2.234 | 0.51 | 3644 | 0.15 | 124 | $6.34\pm 0.31$
Gl 752A | M3.0V | 9.240 | 2.379 | 0.47 | 3558 | 0.11 | 130 | $6.06\pm 0.23$
Gl 48 | M2.5V | 9.364 | 2.459 | 0.46 | 3529 | 0.08 | 194 | $5.99\pm 0.26$
Gl 617B | M3V | 9.459 | 2.483 | 0.45 | 3525 | 0.20 | 149 | $6.09\pm 0.21$
Gl 412A | M1.0V | 9.460 | 2.104 | 0.39 | 3620 | $-0.42$ | 224 | $5.50\pm 0.24$
Gl 15A | M1.5V | 9.460 | 2.164 | 0.39 | 3611 | $-0.33$ | 256 | $5.76\pm 0.26$
Gl 849 | M3V | 9.511 | 2.542 | 0.46 | 3502 | 0.35 | 203 | $6.17\pm 0.22$
Gl 411 | M2.0V | 9.522 | 2.216 | 0.39 | 3589 | $-0.38$ | 182 | $4.93\pm 0.18$
Gl 480 | M3.5V | 9.565 | 2.591 | 0.45 | 3509 | 0.26 | 104 | $6.14\pm 0.22$
Gl 436 | M3.0V | 9.631 | 2.449 | 0.42 | 3508 | 0.03 | 90 | $5.89\pm 0.33$
Gl 687 | M3.0V | 9.739 | 2.518 | 0.39 | 3475 | 0.01 | 227 | $5.69\pm 0.20$
Gl 408 | M2.5V | 9.831 | 2.430 | 0.38 | 3487 | $-0.09$ | 179 | $6.17\pm 0.33$
Gl 317 | M3.5V | 9.859 | 2.664 | 0.42 | 3421 | 0.23 | 79 | $5.39\pm 0.21$
GJ 4063 | M3.5V | 9.982 | 2.770 |  | 3419 | 0.42 | 220 | $5.96\pm 0.22$
Gl 725A | M3.0V | 10.119 | 2.461 | 0.33 | 3470 | $-0.26$ | 219 | $5.43\pm 0.22$
Gl 251 | M3.0V | 10.129 | 2.561 | 0.35 | 3420 | $-0.01$ | 187 | $5.73\pm 0.26$
GJ 4333 | M3.5V | 10.233 | 2.818 | 0.37 | 3362 | 0.25 | 193 | $5.70\pm 0.16$
GJ 1012 | M4V | 10.268 | 2.710 | 0.35 | 3363 | 0.07 | 142 | $5.26\pm 0.17$
GJ 1148 | M4V | 10.370 | 2.778 | 0.34 | 3354 | 0.11 | 105 | $5.45\pm 0.16$
Gl 876 | M3.5V | 10.527 | 2.809 | 0.33 | 3366 | 0.15 | 88 | $5.65\pm 0.28$
PM J09553-2715 | M3V | 10.629 | 2.667 | 0.29 | 3366 | $-0.03$ | 75 | $5.77\pm 0.25$
PM J08402+3127 | M3.5V | 10.739 | 2.703 | 0.28 | 3347 | $-0.08$ | 142 | $5.33\pm 0.25$
Gl 725B | M3.5V | 10.790 | 2.625 | 0.25 | 3379 | $-0.28$ | 212 | $5.34\pm 0.22$
GJ 1105 | M4V | 10.931 | 2.792 | 0.27 | 3324 | $-0.04$ | 171 | $5.44\pm 0.17$
Gl 445 | M3.5V | 10.948 | 2.702 | 0.24 | 3356 | $-0.24$ | 94 | $5.21\pm 0.19$
GJ 3378 | M3.5V | 10.975 | 2.791 | 0.26 | 3326 | $-0.05$ | 177 | $5.65\pm 0.22$
Gl 169.1A | M4V | 10.994 | 2.896 | 0.28 | 3307 | 0.13 | 185 | $5.66\pm 0.28$
GJ 1289 | M3.5V | 11.556 | 3.011 | 0.21 | 3238 | 0.05 | 209 | $7.60\pm 0.37$
PM J21463+3813 | M4V | 11.591 | 2.814 | 0.18 | 3305 | $-0.38$ | 187 | $5.21\pm 0.26$
GJ 1103 | M4.5V | 11.817 | 3.116 | 0.19 | 3170 | $-0.03$ | 70 | $5.44\pm 0.28$
Gl 699 | M4.0V | 11.884 | 2.834 | 0.16 | 3311 | $-0.37$ | 249 | $6.04\pm 0.32$
Gl 15B | M3.5V | 11.928 | 2.836 | 0.16 | 3272 | $-0.42$ | 189 | $6.33\pm 0.37$
Gl 447 | M4.0V | 11.960 | 3.033 | 0.18 | 3198 | $-0.13$ | 60 | $6.24\pm 0.24$
GJ 1151 | M4.5V | 12.158 | 3.142 | 0.17 | 3178 | $-0.16$ | 157 | $6.42\pm 0.37$
Gl 905 | M5.5V | 12.881 | 3.529 | 0.15 | 3069 | 0.05 | 220 | $6.13\pm 0.29$
GJ 1286 | M5.5V | 13.344 | 3.706 | 0.12 | 2961 | $-0.23$ | 113 | $7.46\pm 0.46$
GJ 1002 | M5.5V | 13.347 | 3.675 | 0.12 | 2980 | $-0.33$ | 146 | $6.92\pm 0.34$

##  3 SPIRou data reduction and analysis

###  3.1  APERO reduction

Our SPIRou data were reduced with the software APERO. APERO first performs
some initial preprocessing of the 4096$\times$4096 pixel images of the HAWAII
4RGTM (H4RG), applying a series of procedures to correct detector effects,
remove background thermal noise, and identify bad pixels and cosmic-ray
impacts.

It then uses exposures of a quartz halogen lamp (flat) to calculate the
position of 49 of the 50 échelle spectral orders recorded on the detector. It
optimally extracts (Horne, 1986) spectra of the two science channels (fibers A
and B) and the simultaneous reference channel (fiber C). This APERO extraction
takes the non-Gaussian shape of the instrument profile generated by the pupil
slicer into account. APERO corrects the spectra for the blaze signature of the
échelle orders obtained from the flat-field exposures, as described in Cook et
al. (2022). Both a 2D order-by-order and 1 D order-merged spectrum are
produced for each channel of each scientific exposure.

The pixel-to-wavelength calibration is obtained from exposures of both a
uranium-neon hollow cathode lamp and a Fabry-Pérot etalon, generally following
the procedure given in Hobson et al. (2021), but with differences described in
Cook et al. (2022). We refer to Sections 5.4 and 6.6 of the latter paper for
details about the use of a reference night for the whole survey and updates
for each survey night, with subtle differences in the wavelength calibration
of fibers AB with respect to fibers A and B separately. This procedure
provides wavelengths in the rest frame of the observatory, but APERO also
calculates the barycentric Earth radial velocity (BERV) and the barycentric
Julian date (BJD) of each exposure using the code
barycorrpy101010https://github.com/shbhuk/barycorrpy(Wright & Eastman, 2014;
Kanodia & Wright, 2018). These can then be used to reference the wavelength
and time to the barycentric frame of the Solar System.

APERO calculates the spectrum of the telluric transmission using a novel
technique based on a model obtained from the collection of standard star
observations carried out since the beginning of SPIRou operations in 2019 and
a fit made for each individual observation using a revised technique based on
the principles explained in Artigau et al. (2014), as briefly described in
Cook et al. (2022) and in more details in Artigau et al. (2022). APERO also
calculates the Stokes parameters describing the polarization state, as
defined, for instance, in Landi Degl’Innocenti (1992). Each orthogonal
polarization state is recorded on one science fiber. Four sub-exposures are
taken with different positions of the quarter-wave rhombs: the method is
described in Donati et al. (1997). The two polarization states of each
subexposure are combined to define the Stokes parameters, as explained in Sec.
10.1 of Cook et al. (2022)) using equations defined in Bagnulo et al. (2009):
the Stokes $I$ parameter measures the intensity, the Stokes $V$ parameter
measures the circular polarization, and the Stokes $Q$ and $U$ measure the
linear polarization (not used in thiswork). Another combination defines a null
measurement to quantify the quality of the Stokes $V$ detection. The continuum
for Stokes $I$ and $V$ can be modeled by an iterative sigma-clip algorithm to
fit a polynomial to the data. The spectra are normalized to this continuum
before an LSD analysis is performed. More details are given in Martioli et al.
(2020); Cook et al. (2022).

###  3.2 Spectropolarimetry analysis

We further analyzed the SPIRou polarized spectra using the code spirou-
polarimetry111111https://github.com/edermartioli/spirou-polarimetry , which we
applied to the telluric-corrected 2D spectra. We assumed that the APERO
telluric correction is good enough so that we did not need to exclude zones of
telluric water bands in this code. The Stokes $I$, Stokes $V$, and null-
polarization spectra were compressed to one-line profiles using the least-
squares deconvolution (LSD) method of Donati et al. (1997). The line masks
used in our LSD analysis were computed using the VALD3 catalog (Piskunov et
al., 1995; Ryabchikova et al., 2015) and a MARCS model atmosphere (Gustafsson
et al., 2008) with a grid of effective temperatures between 3000 K and 4000 K
by steps of 500 K, a constant surface gravity of $\log g=5.0$ dex, and a
micro-turbulent velocity $\upsilon_{\mathrm{m}ic}$ of 1 km s-1 . We selected
all lines deeper than 3% and with a Landé factor of $g_{\rm eff}>0$ for a
total of 2460, 1335, and 956 atomic lines for 4000 K, 3500 K, and 3000 K,
respectively. We normalized the Stokes $V$ profile using a mean wavelength, a
mean depth, and a mean Landé factor of the lines in the mask, but we only used
the mean depth to normalize the Stokes $I$ profile. For the 4000 K mask (3500
K and 3000 K), the mean values are 1651.2 nm (1617.9 nm and 1604.8 nm) for the
wavelengths, 0.135 (0.133 and 0.155) for the depths, and 1.249 (1.242 and
1.235) for the Landé factors. To ensure that Stokes $I$ LSD profiles have a
continuum at 1, we used a linear regression of a significant number of bins
located well outside of the line for each profile to improve the continuum
normalization. We measured the standard deviation of the values in these bins
and rejected the profiles for which this standard deviation significantly
deviated from the average over all the profiles. This linear regression gives
us a better defined Voigt profile, allowing a more precise measure of the FWHM
of the median Stokes $I$ LSD profile of each star.

Not all polarimetric sequences led to a useful measurement of the longitudinal
magnetic field. There are several steps of data rejection: the telluric
correction of some spectra may fail, some LSD profiles are noisy enough that
they cannot be fit by a Voigt profile, and others can be fit, but present a
noisier continuum than the rest (see above) and are thus rejected. We only
rejected data that might contaminate our results, especially, to build a clean
median Stokes $I$ profile on which we measured the FWHM and defined the
velocity limits to integrate the Stokes $V$ profile (using 6 FWHM as the
measuring window, meaning $\pm 3$ FWHM, generally about $\pm 20$ km s-1 ). The
final number of polarimetric sequences used in our study is about 6500,
compared to the 6800 initially available sequences for our sample of stars.
This is an acceptable loss of about 4% that generally corresponds to spectra
with a low signal-to-noise ratio (S/N).

As an example, we took the LSD profiles of the early-M dwarf Gl 410 (DS Leo),
which displays mild signs of magnetic activity. Fig. 1 shows the resulting LSD
profiles at each observing epoch, stacked vertically. Dynamical maps from left
to right correspond to the intensity (Stokes $I$), the circular polarization
(Stokes $V$), and the null LSD profiles. A nonzero Stokes $V$ profile
indicates the presence of magnetic fields, while the null profile allows us to
assess the reliability of the Stokes $V$ detection. Finally, Fig. 2 shows the
median profile of our time series, where the Zeeman signature is clearly
visible in the detection of the Stokes $V$ profile. We also display a median
null profile to assess that the Stokes $V$ signature is reliably detected. For
comparison, we display the Stokes $I$, $V$ and null dynamical maps (Fig. 3)
and median profiles (Fig. 4) of the late-M dwarf Gl 905, whose longitudinal
magnetic field has a similar amplitude as Gl 410. Here again, the Stokes $V$
signal is clearly detected, with a shape corresponding at first order to the
first derivative of the Stokes $I$ profile.

In the appendix, we display the Stokes $V$ temporal evolution for each star of
our sample. The Stokes $I$ temporal evolution and the median Stokes profiles
carry less important information for the present work, and the null profiles
are very similar to each other. We decided not to publish them for brevity.

To confirm the consistency of our measurements, we also obtained an
independent polarimetric reduction and LSD analysis of our SPIRou data using
the pipeline Libre-Esprit (Donati et al., 1997, 2020). The results of this
alternative analysis will be given in a forthcoming paper (Donati et al., in
prep).

Figure 1: Dynamical maps of Gl 410 constructed from individual observations
(horizontal bands), stacked vertically. The first plot shows Stokes $I$ LSD
profiles, where the continuum is given in yellow and the different absorption
depths are shown in green and blue shades. The following plots show the Stokes
$V$ and null LSD profiles, where positive values are given as red shades, and
negative values are shown as blue shades. Note that Stokes $V$ and null LSD
amplitudes are expressed in percent. All plots are given in the rest frame of
Gl 410 and display a velocity window of 10 FWHM (see Table 1).  Figure 2:
Median of all LSD profiles in the Gl 410 SPIRou time series. The top panel
shows Stokes $I$ LSD (red points) with a Voigt profile model fit (green line),
the middle panel shows Stokes $V$ (blue points), and the bottom panel shows
the null polarization profile (orange points).  Figure 3: Similar to Fig. 1,
but for the dynamical maps of Stokes $I$, $V$ and null LSD profiles collected
for Gl 905 using SPIRou.  Figure 4: Median of all LSD profiles in the Gl 905
SPIRou time series. The top panel shows Stokes $I$ LSD (red points) with a
Voigt profile model fit (green line), the middle panel shows Stokes $V$ (blue
points), and the bottom panel shows the null polarization profile (orange
points).

###  3.3 Longitudinal magnetic fields

To diagnose the magnetic field in our sample of stars, we calculated the
longitudinal magnetic field $B_{\ell}$ in the LSD profiles of SPIRou following
the same prescription as in Rees & Semel (1979); Donati et al. (1997); Moutou
et al. (2020); Martioli et al. (2020). $B_{\ell}$ is defined as the
brightness-weighted line-of-sight-projected component of the vector magnetic
field integrated over the visible hemisphere of the star, and is given in G by

$B_{\ell}=-2.142\times 10^{11}\frac{\int v\,V(v)\,dv}{\lambda_{0}\cdot g_{\rm
eff}\cdot c\cdot\int\left[1-I(v)\right]dv},$ (1)

where $c$ is the speed of light in the same unit as $v$ (km s-1 ), $I(v)$ is
the Stokes $I$ LSD profile normalized by the continuum, and $V(v)$ is the
Stokes $V$ LSD profile, both as functions of the velocity $v$ in the stellar
frame, $\lambda_{0}$ is the mean wavelength in nm, and $g_{\rm eff}$ is the
mean Landé factor of the lines included in the LSD analysis.

As the spatial distribution of the magnetic regions may be nonaxisymmetric,
$B_{\ell}$ can be modulated by the stellar rotation. This allows deriving the
stellar rotation period if any periodicity is detected in its time series
(e.g., Preston, 1971; Landstreet, 1980). Examples focusing more on K and M
dwarfs can be found in (e.g., Morin et al., 2008; Moutou et al., 2017; Petit
et al., 2021). Fig. 5 and Fig. 6 show the generalized Lomb-Scargle (GLS)
periodogram (Zechmeister & Kürster, 2009) for the $B_{\ell}$ data calculated
using the tool
astropy.timeseries121212https://docs.astropy.org/en/stable/timeseries/lombscargle.html
for our two example stars Gl 410 and Gl 905. We find a maximum power at a
period of 13.9 d for Gl 410 and at 108.6 d for Gl 905, both with a false-alarm
probability (FAP) below 0.001%. In the case of Gl 905, three other peaks below
an FAP of 0.001% can be noted in the periodogram, and they are due to
harmonics of the main period.

Figure 5: Generalized Lomb-Scargle periodogram analysis of the longitudinal
magnetic field ($B_{\ell}$) time series of Gl 410. The dashed blue line shows
the highest power at a period of 13.9 d, and the dot-dashed lines indicate
possible harmonics at half and twice the period.  Figure 6: GLS periodogram
analysis of the longitudinal magnetic field ($B_{\ell}$) time series of Gl
905. The dashed blue line shows the highest power at a period of 108.6 d, and
the dot-dashed lines indicate possible harmonics at half and twice the period.

As the magnetic field of M dwarfs is likely to evolve with time, this study
requires a flexible model to account for the variability. We employed a
Gaussian process (GP) regression analysis (e.g., Haywood et al., 2014; Aigrain
et al., 2015) using the code george131313http://dfm.io/george/current/
(Ambikasaran et al., 2015), where we assumed that the rotationally modulated
stellar activity signal in $B_{\ell}$ is quasi-periodic (QP). Thus, we adopted
a parameterized covariance function (or kernel) as in Angus et al. (2018),
which is given by

$k(\tau_{ij})=\alpha^{2}\exp{\left[-\frac{\tau_{ij}^{2}}{2l^{2}}-\frac{1}{\beta^{2}}\sin^{2}{\left(\frac{\pi\tau_{ij}}{P_{\rm
rot}}\right)}\right]}+\sigma^{2}\delta_{ij},$ (2)

where $\tau_{ij}=t_{i}-t_{j}$ is the time difference between data points $i$
and $j$, $\alpha^{2}$ is the amplitude of the covariance, $l$ is the decay
time, $\beta$ is the smoothing factor, $P_{\rm rot}$ is the star rotation
period, and $\sigma$ is an additional uncorrelated white noise, which adds a
jitter term to the diagonal of the covariance matrix. This kernel combines a
squared exponential component describing the overall covariance decay and a
component that describes the periodic covariance structure, the amplitude of
which is controlled by the smoothing factor141414The definition of the decay
timescale and smoothing factor vary among researchers. The former is sometimes
defined as $\sqrt{2}\,l$ (e.g., Petit et al., 2021), and $1/\beta^{2}$ may
appear as $\Gamma$ and be called harmonic complexity (e.g., Nicholson &
Aigrain, 2022).. Typical values of $\beta$ vary between 0.25 and 1.25. The
lower values correspond to multiple harmonics, and the higher values emphasize
single sinusoidal variation. Similarly, the decay parameter varies between a
few tens to a few hundred days. As pointed out by Angus et al. (2018), the
flexibility of this model can easily lead to an overfitting of the data. To
avoid this, we adopted a uniform prior distribution of the parameters (see,
e.g., Table 3) that restricts the search range to realistic values. In
addition, when the posterior distribution of the smoothing factor and decay
time did not display a clear peak, we restricted the fit from six to five
parameters by fixing the value of the smoothing factor at 0.7, which
corresponds to the middle of the range of explored values and to the median
value of stars with a constrained smoothing factor. We also explored a four-
parameter fit in which both the smoothing factor and the decay time were fixed
at 0.7 and 200 days, respectively. We found no significant differences with
respect to the five-parameter fit. The value of the fixed decay time is a
guess at best and may depend on the spectral type (early-M stars seem to have
a shorter decay time, about 70 days): Giles et al. (2017) reported based on
Kepler light curves that cooler stars have spots that last much longer, in
particular for stars with longer rotational periods.

We used this GP framework to model the $B_{\ell}$ data, where we first fit the
GP model parameters by maximizing the likelihood function defined in (Eq.
5.8)151515http://gaussianprocess.org/gpml/chapters/RW5.pdf of Rasmussen &
Williams (2006) and as implemented in george, and then we sampled the
posterior distribution of the free parameters using a Bayesian Markov chain
Monte Carlo (MCMC) framework with the package emcee (Foreman-Mackey et al.,
2013). We set the MCMC with 50 walkers, 1000 burn-in samples, and 5000
samples. The results of our analysis are illustrated in Fig. 7 and Fig. 9,
where we present the observed $B_{\ell}$ data and the best-fit GP model for Gl
410 and Gl 905, respectively. Fig. 8 and Fig. 10 show the MCMC samples and
posterior distributions of the GP parameters for a six-parameter fit of the
$B_{\ell}$ time series for Gl 410 and Gl 905, respectively.

Figure 7: GP analysis of the SPIRou $B_{\ell}$ data of Gl 410. In the top
panel, the black points show the observed $B_{\ell}$ data, and the orange line
shows the best-fit quasi-periodic GP model. The bottom panel shows the
residuals with an RMS dispersion of 7.6 G. Figure 8: MCMC samples and the
posterior distributions of parameters in the quasi-periodic GP analysis of the
stellar activity in the SPIRou $B_{\ell}$ data of Gl 410. The blue crosses
mark the mode of the distribution, and the vertical dashed lines in the
histograms indicate the median and the 16 and 84 percentiles of the posterior
PDF. The shaded regions correspond to uncertainties of 1, 2, and 3 $\sigma$ in
order of increasing radius. Figure 9: GP analysis of the SPIRou $B_{\ell}$
data of Gl 905. In the top panel, the black points show the observed
$B_{\ell}$ data, and the orange line shows the six-parameter best-fit quasi-
periodic GP model. The bottom panel shows the residuals with an RMS dispersion
of 13 G. Figure 10: MCMC samples and the posterior distributions of the six-
parameter fit of a quasi-periodic GP on the SPIRou $B_{\ell}$ data of Gl 905.
The same format is used as in Fig. 8.

##  4 Results

###  4.1 Early-type M dwarfs

Our sample includes seven early-M dwarfs (M0V to M2V) with absolute magnitudes
in the Gaia EDR3 $G$ band between 8 and 9. With masses between 0.5-0.6 M⊙ ,
they are partially convective. To compute the LSD profiles of these stars, we
used a mask with atomic lines of the known Landé factor at 4000 K and $\log
g$=5.0. For six of them, the longitudinal magnetic field clearly varies, and
accordingly, the rotation period is measured quite accurately. We used the
six-parameter fit of the quasi-periodic GP model for them, but the decay time
could not be well constrained for two of them. They are reported in Table 3,
which lists the best-fit parameters measured as the median of the
distribution, with uncertainties given by the 0.16 and 0.84 quantiles. Here
and in Tables 5 and 8, the number of visits corresponds to the final number
after poor polarimetric sequences were rejected, and it therefore differs from
the numbers listed in Table 1. Results for Gl 205 were already published in
Cortes-Zuleta et al. (2023) based on a similar data set, but with the Libre-
Esprit data reduction and analysis package: the agreement of the measured
parameters is encouraging (see Table 2). The rotation periods lie between 11
and 38 d for this sample. As the radius of early-type M dwarfs is in the range
0.5-0.6 R⊙, these rotation periods translate into equatorial velocities below
3 km s-1 . This $v\,\mathrm{sin}\,i$ is below the detectability threshold of
our spectrograph (2-3 km s-1 ). A broadening in the FWHM of the Stokes $I$
profile may be due to several factors when the rotation is negligible: for
active stars, it may come from a Zeeman broadening effect, but for the quiet
stars studied in this sample, it is probably due to other factors. In Sec. 5
we return to this topic after the full sample is analyzed. For the earliest
spectral-type star Gl 338B, the Stokes V signal is clearly detected (see Fig.
40). However, we failed to detect any periodicity in the longitudinal magnetic
field variation, probably due to the low number of visits (58 observations).

Table 2: Best six-parameter fit of a quasi-periodic GP model obtained in our analysis of the stellar activity from the SPIRou $B_{\ell}$ data of Gl 205, compared to the results from Cortes-Zuleta et al. (2023) using a similar data set, but the Libre-Esprit data reduction and analysis package.  Package | rotation period | mean $B_{\ell}$ | white noise | amplitude | decay time | smoothing factor | rms | $\chi^{2}_{\rm red}$ | visits
---|---|---|---|---|---|---|---|---|---
|  $P_{\rm rot}$ [d] |  $\mu$ [G] |  $\sigma$ [G] |  $\alpha$ [G] |  $l$ [d] | $\beta$ | [G] |  |
APERO | $34.3^{+0.4}_{-0.4}$ | $2.5^{+2.0}_{-2.0}$ | $1.0^{+0.5}_{-0.6}$ | $7.1^{+1.3}_{-1.0}$ | $67^{+15}_{-11}$ | $0.54^{+0.09}_{-0.07}$ | 2.5 | 0.79 | 152
Libre-Esprit | $34.4^{+0.5}_{-0.4}$ | $1.3^{+0.9}_{-0.9}$ | $0.4^{+0.2}_{-0.2}$ | $3.1^{+0.6}_{-0.4}$ | $63^{+13}_{-8}$ | $0.57^{+0.10}_{-0.08}$ | 0.94 | 0.84 | 153

The GP parameters appear to show that the decay time is pretty well
constrained, with values clustering around 75 d. The smoothing parameter
varies more widely, between 0.5 (several harmonics) and 1.0 (smooth
variation). For two stars with poorly constrained smoothing parameter and
decay time, we used a five-parameter fit by fixing the smoothing parameter to
0.7. The white noise is always compatible with 0 within $2\sigma$, meaning
that the GP does not need an additional variable to explain the dispersion,
and the reduced $\chi^{2}$are close to 1. We reach an rms of the fit of a few
G for these stars. The results are summarized in Table 3.

Table 3: Best six-parameter fit of a quasi-periodic GP model obtained in our analysis of the stellar activity from the SPIRou $B_{\ell}$ data of the early-type M dwarfs in our sample.  Star | rotation period | mean $B_{\ell}$ | white noise | amplitude | decay time | smoothing factor | rms | $\chi^{2}_{\rm red}$ | visits
---|---|---|---|---|---|---|---|---|---
|  $P_{\rm rot}$ [d] |  $\mu$ [G] |  $\sigma$ [G] |  $\alpha$ [G] |  $l$ [d] | $\beta$ | [G] |  |
Priors | $\mathcal{U}(2,300)$ | $\mathcal{U}(-\infty,+\infty)$ | $\mathcal{U}(0,+\infty)$ | $\mathcal{U}(0,+\infty)$ | $\mathcal{U}(50,1000)$ | $\mathcal{U}(0.25,1.25)$ |  |  |
Gl 846 | $11.01^{+0.17}_{-0.22}$ | $-0.6^{+2.0}_{-2.1}$ | $2.6^{+1.0}_{-1.3}$ | $6.2^{+1.4}_{-1.2}$ | $57^{+18}_{-5}$ | $0.96^{+0.19}_{-0.22}$ | 6.1 | 1.05 | 188
Gl 205 | $34.3^{+0.4}_{-0.4}$ | $2.5^{+2.0}_{-2.0}$ | $1.0^{+0.5}_{-0.6}$ | $7.1^{+1.3}_{-1.0}$ | $67^{+15}_{-11}$ | $0.54^{+0.09}_{-0.07}$ | 2.5 | 0.79 | 152
Gl 410 | $13.87^{+0.08}_{-0.07}$ | $-1.1^{+10.8}_{-10.5}$ | $3.7^{+2.0}_{-2.2}$ | $28.9^{+6.7}_{-4.7}$ | $78^{+16}_{-14}$ | $0.73^{+0.13}_{-0.10}$ | 7.6 | 0.86 | 126
Gl 880 | $37.7^{+0.8}_{-0.6}$ | $4.0^{+3.4}_{-3.1}$ | $0.8^{+0.8}_{-0.6}$ | $11.4^{+2.4}_{-1.7}$ | $94^{+24}_{-20}$ | $0.56^{+0.10}_{-0.08}$ | 3.8 | 0.65 | 162
Gl 514 | $30.45^{+0.13}_{-0.14}$ | $-4.2^{+5.9}_{-5.4}$ | $2.1^{+1.3}_{-1.4}$ | $8.8^{+3.6}_{-2.4}$ |  | $0.99^{+0.18}_{-0.25}$ | 8.2 | 1.10 | 165
Gl 382 | $21.32^{+0.04}_{-0.03}$ | $-0.7^{+4.2}_{-4.2}$ | $2.8^{+1.2}_{-1.5}$ | $7.9^{+3.2}_{-1.9}$ |  | $0.62^{+0.27}_{-0.18}$ | 6.7 | 1.11 | 114

Rotation periods were already known for all these stars, based on a variety of
indicators: ZDI analysis, photometry (noted ”phm” in Tables 4, 6, and 9), and
activity indicators (noted ”act” in the same tables). In Table 4 we compare
our new values to those from the literature. The agreement is good in general,
but some literature values are discrepant for unclear reasons (see, e.g., Gl
846) or reveal a harmonic (twice the frequency of rotation: Gl 382 for Sabotta
et al. (2021)). Photometry and activity measurements may be affected by
sporadic variability (flares), and poor sampling may lead to differences or
detection of harmonics of the true rotation period. A case-by-case discussion
is difficult as the literature values are based on different techniques with
varying precision, time sampling, and total time coverage. For our method, the
time sampling and total coverage are generally adequate, and the precision is
quite uniform after rejecting low S/N measurements.

Table 4: Comparison of the rotation periods given in the literature with our
measured values for the early-M stars.

Reference | Category | Gl 205 | Gl 382 | Gl 410 | Gl 514 | Gl 846 | Gl 880
---|---|---|---|---|---|---|---
This work | $B_{\ell}$ | $34.3\pm 0.4$ | $21.32\pm 0.04$ | $13.87\pm 0.08$ | $30.45\pm 0.14$ | $11.01\pm 0.20$ | $37.7\pm 0.7$
Kiraga & Stepien (2007) | phm | 33.61 | 21.56 |  |  |  |
Donati et al. (2008) | ZDI |  |  | 14.0 |  |  |
Bonfils et al. (2013) | act | 32.8, 39.3 |  |  |  | 10.7 |
Suárez Mascareño et al. (2015) | act | $35.0\pm 0.1$ | $21.7\pm 0.1$ |  | $28.0\pm 2.9$ | $31.0\pm 0.1$ | $37.5\pm 0.1$
Suárez Mascareño et al. (2016) | phm | $33.4\pm 0.1$ | $21.2\pm 0.1$ |  |  |  |
Hébrard et al. (2016) | ZDI | $33.63\pm 0.37$ |  | $13.83\pm 0.10$ |  | $10.73\pm 0.10$ |
Suárez Mascareño et al. (2017) | act | $34.8\pm 1.3$ | $21.8\pm 0.1$ |  | $30.0\pm 0.9$ | $26.3\pm 5.6$ | $37.2\pm 6.7$
Díez Alonso et al. (2019) | phm | $33.8\pm 0.6$ | $21.6\pm 0.2$ | $14.6\pm 0.2$ |  | $29.5\pm 0.1$ | $39.5\pm 0.2$
Sabotta et al. (2021) | act | 37.08 | 10.65, 21.4 |  |  |  |

###  4.2 Mid-type M dwarfs

The most numerous subcategory in our sample comprises 26 stars with MG between
9 and 11, spectral types ranging from M1V to M4V, and masses between 0.2 and
0.5 M⊙. This group crosses the so-called Jao gap (Jao et al., 2018) at about
MG=10.2, GBP-GRP=2.3, and includes the transition between partially convective
and fully convective M dwarfs. As the transition mass between these two
regimes is poorly defined (0.2 to 0.35 M⊙) and depends on metallicity (see
Feiden et al. 2021 for a more detailed discussion), we preferred not to try to
define a finer grid of magnitudes or spectral types with the risk of a
possibly inhomogeneous subcategory. To compute the LSD profiles of these
stars, we used a mask with atomic lines of the known Landé factor at 3500 K
and $\log g$=5.0.

In contrast with the group of early-M dwarfs, here only half of the stars have
a detected rotation period. We first tried a six-parameter fit of the quasi-
periodic GP model for them, but only three stars have a well-constrained decay
time at about 100 days. For the other stars for which we were able to
determine a rotation period, we use a five or even a four parameters fit,
fixing the smoothing parameter at 0.7 (five-parameter fit), and when
necessary, the decay time to 200 days (four-parameter fit). We detect a very
long periodic variation of about 450 days for Gl 411 in this group, which
clearly disagrees with the shorter period reported from photometry by Díaz et
al. (2019) of $56.15\pm 0.27$d, and is unexpectedly long compared to all the M
dwarfs with known stellar rotation periods (see Fig. 11). It would imply an
unknown mechanism of angular momentum loss. We show the GP fit in Fig. 36 that
corresponds to this particular star, and its associated corner plot is
presented in Fig. 38. The measured period may correspond to a cyclic variation
in the magnetic field that is more related to the variation of the activity
than to the stellar rotation.

We were able to measure the rotation period for half of the mid-M dwarfs
compared to 100% for the early-M dwarfs. Furthermore, a six-parameter fit
could only be measured for three stars compared to four out of six for the
early-M dwarfs. This may come from a longer decay time that is not constrained
enough by our three-year survey. The shape of the variation in the
longitudinal magnetic field varies strongly, from almost sinusoidal variations
for Gl 169.1A (smoothing factor of 1) to a variation featuring only a few
harmonics for Gl 48, GJ 3378 or GJ 4333 (smoothing factor of 0.6). The
amplitude of the variations ranges from very weak (5 G for Gl 411) to about 20
G for GJ 4333 and Gl 876. For this group, the white-noise component of the GP
is always compatible with 0 at $2\sigma$, as was the case for the early-M
dwarfs.

Table 5: Best six to four parameters fit of a quasi-periodic GP model obtained
in our analysis of the stellar activity from the SPIRou $B_{\ell}$ data of the
mid-range M dwarfs in our sample. The asterisk after a star name means that
the measured period is more uncertain and was obtained by a five or four
parameters fit only.

Star | rotation period | mean $B_{\ell}$ | white noise | amplitude | decay time | smoothing factor | rms | $\chi^{2}_{\rm red}$ | visits
---|---|---|---|---|---|---|---|---|---
|  $P_{\rm rot}$ [d] |  $\mu$ [G] |  $\sigma$ [G] |  $\alpha$ [G] |  $l$ [d] | $\beta$ | [G] |  |
Priors | $\mathcal{U}(2,300)$ | $\mathcal{U}(-\infty,+\infty)$ | $\mathcal{U}(0,+\infty)$ | $\mathcal{U}(0,+\infty)$ | $\mathcal{U}(50,1000)$ | $\mathcal{U}(0.25,1.25)$ |  |  |
Gl 752A | $53.2^{+5.5}_{-3.0}$ | $0.2^{+3.4}_{-3.2}$ | $1.6^{+1.3}_{-1.1}$ | $8.6^{+2.9}_{-1.9}$ | $93^{+55}_{-29}$ | $0.89^{+0.24}_{-0.29}$ | 6.2 | 0.87 | 128
Gl 48 | $51.2^{+1.4}_{-1.4}$ | $-7.0^{+3.6}_{-3.7}$ | $1.8^{+1.6}_{-1.2}$ | $10.2^{+3.2}_{-2.1}$ | $112^{+55}_{-36}$ | $0.62^{+0.22}_{-0.16}$ | 9.4 | 0.86 | 188
Gl 15A | $44.3^{+2.0}_{-2.0}$ | $-1.0^{+1.8}_{-1.7}$ | $1.0^{+0.9}_{-0.7}$ | $5.4^{+1.4}_{-1.0}$ | $85^{+36}_{-22}$ | $0.71^{+0.28}_{-0.23}$ | 5.6 | 0.82 | 246
Gl 849 | $41.4^{+0.4}_{-0.4}$ | $4.7^{+6.3}_{-5.9}$ | $2.4^{+1.6}_{-1.6}$ | $12.1^{+5.1}_{-3.2}$ |  | $0.87^{+0.24}_{-0.23}$ | 10.0 | 1.01 | 185
Gl 411 | $471^{+41}_{-40}$ | $5.9^{+2.4}_{-2.1}$ | $0.8^{+0.8}_{-0.6}$ | $4.2^{+1.8}_{-1.1}$ |  | $0.68^{+0.32}_{-0.27}$ | 6.3 | 0.94 | 212
Gl 687 | $56.5^{+1.3}_{-0.5}$ | $4.9^{+8.9}_{-6.4}$ | $1.7^{+1.3}_{-1.1}$ | $13.3^{+7.3}_{-4.3}$ |  | $0.97^{+0.19}_{-0.26}$ | 8.2 | 1.10 | 214
Gl 725A | $103.5^{+4.6}_{-5.1}$ | $-14.9^{+3.4}_{-3.1}$ | $1.2^{+1.2}_{-0.8}$ | $8.7^{+2.4}_{-1.6}$ |  |  | 8.1 | 0.81 | 214
Gl 251∗ | $98.7^{+11.5}_{-4.8}$ | $17.4^{+3.6}_{-3.6}$ | $1.7^{+1.6}_{-1.2}$ | $8.3^{+2.4}_{-1.9}$ |  |  | 9.9 | 0.85 | 177
GJ 4333∗ | $72.0^{+0.9}_{-1.2}$ | $6.8^{+8.9}_{-8.0}$ | $2.6^{+2.1}_{-1.7}$ | $18.5^{+8.9}_{-4.1}$ |  | $0.55^{+0.25}_{-0.17}$ | 12.7 | 0.83 | 190
Gl 876 | $82.8^{+2.0}_{-0.7}$ | $3.5^{+13.3}_{-13.5}$ | $2.0^{+1.9}_{-1.4}$ | $23.0^{+12.3}_{-7.1}$ |  | $0.72^{+0.24}_{-0.19}$ | 8.5 | 0.92 | 88
PM J09553-2715∗ | $70.5^{+5.7}_{-1.9}$ | $15.4^{+9.8}_{-11.7}$ | $2.7^{+2.4}_{-1.9}$ | $15.1^{+8.7}_{-5.1}$ |  | $0.82^{+0.30}_{-0.33}$ | 11.8 | 0.91 | 74
GJ 3378∗ | $92.1^{+4.1}_{-5.3}$ | $13.6^{+5.6}_{-5.4}$ | $4.6^{+2.3}_{-2.7}$ | $12.0^{+5.1}_{-2.8}$ |  | $0.61^{+0.32}_{-0.23}$ | 13.9 | 1.00 | 174
Gl 169.1A | $91.9^{+4.1}_{-2.6}$ | $2.3^{+8.4}_{-7.2}$ | $1.9^{+1.9}_{-1.3}$ | $11.4^{+5.3}_{-3.3}$ |  | $1.04^{+0.15}_{-0.23}$ | 13.4 | 0.91 | 172

In Table 6 we compare our new values of the rotation period to those from the
literature. The agreement is good in general, but some literature values are
discrepant. Excluding possible sampling problems or a low number of
measurements in one case (Gl 876), there may be more fundamental reasons for
some techniques to succeed or fail on a given star: photometric variations may
be affected by flares, while the longitudinal magnetic field of the star may
stay constant due to an axisymmetric field topology that prevents us from
detecting a periodic variation even when the Stokes $V$ profiles show a clear
detection. At least in one case, Gl 411, the very long measured period may in
fact reflect a cyclic variation of activity rather than the stellar rotation
period. We also show at the end of Sec. 4.3 that some stars have a clear
detection of the Stokes $V$ profile, but no rotation period detected when the
magnetic field is axisymmetric.

Table 6: Comparison of the rotation periods given in the literature with our
measured values for the mid-M stars.

Reference | Category | Gl 752A | Gl 48 | Gl 15A | Gl 849 | Gl 411 | Gl 687 | Gl 251 | GJ 4333 | Gl 876 | GJ 3378
---|---|---|---|---|---|---|---|---|---|---|---
This work | $B_{\ell}$ | $53\pm 4$ | $51.2\pm 1.4$ | $44.3\pm 2.0$ | $41.4\pm 0.4$ | $470\pm 40$ | $56.5\pm 0.9$ | $99\pm 8$ | $72.0\pm 1.0$ | $82.8\pm 1.4$ | $92\pm 5$
Rivera et al. (2005) | phm |  |  |  |  |  |  |  |  | $96.7\pm 1.0$ |
Bonfils et al. (2013) | act |  |  |  | 2000 ? |  |  |  |  | 61.0, 30.1 |
Burt et al. (2014) | phm |  |  |  |  |  | $61.8\pm 1.0$ |  |  |  |
Howard et al. (2014) | phm |  |  | $43.82\pm 0.56$ |  |  |  |  |  |  |
Suárez Mascareño et al. (2015) | act | $46.5\pm 0.3$ |  |  | $39.2\pm 6.3$ |  |  |  |  | $87.3\pm 5.7$ |
Nelson et al. (2016) | act |  |  |  |  |  |  |  |  | $95\pm 1$ |
Suárez Mascareño et al. (2016) | phm | $46.0\pm 0.2$ |  |  |  |  |  |  |  |  |
Moutou et al. (2017) | ZDI |  |  |  |  |  |  | $90\pm 10$ |  |  |
Suárez Mascareño et al. (2017) | act |  |  |  |  |  |  |  |  | $90.9\pm 16.5$ |
Suárez Mascareño et al. (2018) | act |  |  | $45.0\pm 4.4$ |  |  |  |  |  |  |
Díaz et al. (2019) | phm |  |  |  |  | $56.15\pm 0.27$ |  |  |  |  |
Díez Alonso et al. (2019) | phm | $46.0\pm 0.2$ | $51.5\pm 2.6$ |  |  |  |  | $18.1\pm 0.3$ | $74.7\pm 0.7$ | $81.0\pm 0.8$ |
Reinhold & Hekker (2020) | phm |  |  |  |  |  |  |  |  | $31.31\pm 8.15$ |
Stock et al. (2020) | phm |  |  |  |  |  |  | $122.1\pm 2.2$ |  |  |
Sabotta et al. (2021) | act | 174.48 | 43.39 |  |  |  |  | 67.59, 119.48 |  |  | 83.39

We also tested the results of the Libre-Esprit pipeline as given in Carmona et
al. (in prep) for Gl 388 (AD Leo), even though this star was discarded from
our sample because it is too active. Table 7 compares the results, which agree
excellently well.

Table 7: Best six-parameter fit of a quasi-periodic GP model obtained in our analysis of the stellar activity from the SPIRou $B_{\ell}$ data of Gl 388 (AD Leo), compared to the results from Carmona et al. (in prep) using a similar data set, but the Libre-Esprit data reduction and analysis package.  Package | rotation period | mean $B_{\ell}$ | white noise | amplitude | decay time | smoothing factor | rms | $\chi^{2}_{\rm red}$ | visits
---|---|---|---|---|---|---|---|---|---
|  $P_{\rm rot}$ [d] |  $\mu$ [G] |  $\sigma$ [G] |  $\alpha$ [G] |  $l$ [d] | $\beta$ | [G] |  |
APERO | $2.2301^{+0.0019}_{-0.0017}$ | $-156^{+43}_{-42}$ | $4.6^{+3.3}_{-3.0}$ | $68^{+21}_{-14}$ | $220^{+41}_{-40}$ | $1.35^{+0.11}_{-0.19}$ | 12.7 | 0.91 | 70
Libre-Esprit | $2.2304^{+0.0015}_{-0.0013}$ | $-156^{+47}_{-44}$ | $5.4^{+3.7}_{-3.6}$ | $67^{+23}_{-15}$ | $255^{+96}_{-77}$ | $1.36^{+0.10}_{-0.17}$ | 14.7 | 0.96 | 69

###  4.3 Late-M dwarfs

The last group contains ten stars with $M_{G}$ above 11 (up to 13.5), spectral
types from M3.5V to M5.5V, and masses between 0.1 and 0.2 M⊙. These stars are
fully convective. Stars with later spectral types (M6.0V and M6.5V) in the WP1
sample are all active stars with short rotation periods and are therefore
excluded from this study. To compute the LSD profiles of these late-type
stars, we used a mask with atomic lines of the known Landé factor at 3000 K
and $\log g$=5.0. We generally had to use a four-parameter fit of the quasi-
periodic GP model for them, fixing the smoothing parameter to 0.7 and the
decay time to 200 days, except for two stars in this group, for which a six-
parameter fit gives well-constrained values of the smoothing factor and the
decay time. The others are marked with an asterisk in Table 8. For only one
star, Gl 447 (Ross 128, FI Vir), are we unable to confirm the long rotation
period measured from ASAS photometry by Suárez Mascareño et al. (2016)
($165.1\pm 0.8$ d) and Díez Alonso et al. (2019) ($163\pm 3$ d). This is
probably due to the low number and distribution of the visits (57, clustered
into two groups) compared to the other late-M dwarfs, which prevents us from
determining the rotation period of Gl 447. This highlights once again that our
set limit to 50 visits with adequate sampling is essential. Below this value,
the determination of the rotation period becomes difficult.

Table 8: Best six, five, or four parameters fit of a quasi-periodic GP model obtained in our analysis of the stellar activity from the SPIRou $B_{\ell}$ data of the late-M dwarfs in our sample. The asterisk after a star name means that the measured period is more uncertain and was obtained by a four-parameter fit only. Star | rotation period | mean $B_{\ell}$ | white noise | amplitude | decay time | smoothing factor | rms | $\chi^{2}_{\rm red}$ | visits
---|---|---|---|---|---|---|---|---|---
|  $P_{\rm rot}$ [d] |  $\mu$ [G] |  $\sigma$ [G] |  $\alpha$ [G] |  $l$ [d] | $\beta$ | [G] |  |
Priors | $\mathcal{U}(2,300)$ | $\mathcal{U}(-\infty,+\infty)$ | $\mathcal{U}(0,+\infty)$ | $\mathcal{U}(0,+\infty)$ | $\mathcal{U}(50,1000)$ | $\mathcal{U}(0.25,1.25)$ |  |  |
GJ 1289 | $74.0^{+1.5}_{-1.3}$ | $47^{+25}_{-25}$ | $2.4^{+2.5}_{-1.7}$ | $67^{+21}_{-12}$ | $142^{+33}_{-26}$ | $0.64^{+0.17}_{-0.11}$ | 14.7 | 0.67 | 180
GJ 1103∗ | $139^{+22}_{-23}$ | $8.2^{+10.1}_{-10.1}$ | $4.1^{+4.0}_{-2.9}$ | $18.2^{+8.9}_{-5.9}$ |  |  | 16.4 | 0.83 | 62
Gl 699∗ | $137.1^{+6.8}_{-4.0}$ | $3.0^{+5.7}_{-5.8}$ | $1.1^{+1.1}_{-0.8}$ | $14.7^{+3.0}_{-2.4}$ |  |  | 8.7 | 0.77 | 243
Gl 15B∗ | $116.5^{+6.4}_{-4.7}$ | $-0.3^{+6.3}_{-6.2}$ | $2.1^{+2.1}_{-1.4}$ | $13.8^{+3.8}_{-2.8}$ |  |  | 16.5 | 0.78 | 184
GJ 1151∗ | $158^{+14}_{-9}$ | $-9.6^{+13.4}_{-13.2}$ | $2.7^{+2.6}_{-1.9}$ | $29.6^{+8.2}_{-5.5}$ |  |  | 15.7 | 0.83 | 153
Gl 905 | $109.5^{+4.9}_{-5.4}$ | $-14^{+11}_{-11}$ | $1.7^{+1.8}_{-1.2}$ | $29.4^{+8.1}_{-5.3}$ | $149^{+26}_{-25}$ | $0.71^{+0.21}_{-0.17}$ | 12.6 | 0.73 | 216
GJ 1286∗ | $203^{+14}_{-21}$ | $36^{+19}_{-20}$ | $5.7^{+4.5}_{-3.8}$ | $44^{+14}_{-10}$ |  |  | 25.0 | 0.89 | 108
GJ 1002 | $93.0^{+1.4}_{-1.7}$ | $-6.4^{+12.5}_{-14.2}$ | $4.1^{+3.4}_{-2.8}$ | $20.4^{+9.7}_{-5.9}$ |  | $0.90^{+0.24}_{-0.31}$ | 19.9 | 1.05 | 154

In Table 9 we compare our new rotation periods to those from the literature.
The agreement is not as good as for earlier M dwarfs, probably because their
rotation periods are longer and the uncertainties are larger. It becomes
naturally harder to determine very long periods as we are limited by the time
range of the SPIRou observations.

Table 9: Comparison of the rotation periods given in the literature with our measured values for the late Ms. Reference | Category | GJ 1289 | Gl 699 | GJ 1151 | Gl 905 | GJ 1286 | GJ 1002
---|---|---|---|---|---|---|---
This work | $B_{\ell}$ | $74.0\pm 1.4$ | $137\pm 5$ | $158\pm 12$ | $110\pm 5$ | $203\pm 18$ | $93.0\pm 1.6$
Benedict et al. (1998) | phm |  | 130.4 |  |  |  |
Irwin et al. (2011) | phm |  |  | 132 |  |  |
Suárez Mascareño et al. (2015) | act |  | $148.6\pm 0.1$ |  |  |  |
Moutou et al. (2017) | ZDI | $54\pm 4$ |  |  |  |  |
Newton et al. (2018) | phm |  |  |  |  | 88.92 |
Toledo-Padrón et al. (2019) | phm |  | $145\pm 15$ |  |  |  |
Díez Alonso et al. (2019) | phm | $83.6\pm 7.0$ |  | $125\pm 23$ | $106\pm 6$ |  |
Sabotta et al. (2021) | act |  | 311.25 |  | 178.74 |  |
Suárez Mascareño et al. (2022) | act |  |  |  |  |  | $126\pm 15$

Finally, in Table 10, we present the data of the 16 stars for which we were
unable to measure a rotation period. For these stars, we arbitrarily fixed the
decay parameter to 200 d and the smoothing factor to 0.7 to obtain some
indications about the mean $B_{\ell}$, residual white noise, and amplitude of
the field. The GP also gives a value of the rotation period, but we do not
list it as it is not well constrained. We assumed that the values of the other
parameters of the GP still bear some information, although the fact that the
rotation period is not constrained and that two parameters of the GP are
arbitrarily fixed limits the value of this information. For completeness, we
list a period and its reference, when available in the literature, but this
information was not used in our GP fit.

The Stokes $V$ profiles in Appendix C may tentatively explain these
nondetections by their magnetic topology: if the magnetic field is
axisymmetric, we cannot detect the rotation modulation. To test this
interpretation, we measured the rate of detection of Stokes V profiles for a
given star by comparison to the noise measured in a velocity region well
outside of the line. We find that this metric quantifies the visual impression
of the time series of Stokes V profiles quite well, as shown in Appendix C.
Then, we subtracted a median Stokes V profile for each star and measured the
detection rate again. For some stars, it changes dramatically, and we
tentatively interpret this change by the subtraction of a constant component
due to an axisymmetric magnetic field. This is the case of Gl 408, Gl 338B, Gl
436, GJ 4063, and Gl 617B. For other stars, the detection rate does not
significantly change when the median-subtracted profiles are compared to the
original Stokes V profiles. This is the case of PM J21463+3813, Gl 412A, PM
J08402+3127, Gl 480, and GJ 1148 among the stars without a detected rotation
period. We plan a more detailed study of the magnetic topology of the stars in
our sample to further investigate the reasons for these nondetections.

Table 10: Best four-parameter fit of a quasi-periodic GP model obtained in our
analysis of the stellar activity from the SPIRou $B_{\ell}$ data of the M
dwarfs in our sample without a clear periodic variation detection.

Star | mean $B_{\ell}$ | white noise | amplitude | rms | $\chi^{2}_{\rm red}$ | number of visits | rotation period from literature | reference
---|---|---|---|---|---|---|---|---
|  $\mu$ [G] |  $\sigma$ [G] |  $\alpha$ [G] | [G] |  |  |  $P_{\rm rot}$ [d] |
Priors | $\mathcal{U}(-\infty,+\infty)$ | $\mathcal{U}(0,+\infty)$ | $\mathcal{U}(0,+\infty)$ |  |  |  |  |
Gl 338B | $-8.1^{+1.6}_{-1.6}$ | $1.3^{+1.0}_{-0.9}$ | $3.1^{+1.6}_{-1.4}$ | 4.4 | 1.03 | 58 | 16.66 | Sabotta et al. (2021)
Gl 617B | $19.3^{+4.0}_{-4.0}$ | $1.4^{+1.5}_{-1.0}$ | $8.9^{+2.9}_{-2.1}$ | 8.2 | 0.71 | 142 |  |
Gl 412A | $13.7^{+4.1}_{-4.3}$ | $4.0^{+1.2}_{-1.4}$ | $10.7^{+3.5}_{-2.5}$ | 9.7 | 1.23 | 174 | $100.9\pm 0.3$ | Suárez Mascareño et al. (2018)
Gl 480 | $6.6^{+3.5}_{-3.3}$ | $5.5^{+1.8}_{-2.1}$ | $6.9^{+2.8}_{-2.3}$ | 11.0 | 1.28 | 104 |  |
Gl 436 | $-10.4^{+2.2}_{-2.0}$ | $1.6^{+1.6}_{-1.1}$ | $4.4^{+2.1}_{-1.7}$ | 7.5 | 0.81 | 85 | $44.09\pm 0.08$ | Bourrier et al. (2018)
Gl 408 | $-43.8^{+3.6}_{-3.6}$ | $1.4^{+1.4}_{-1.0}$ | $8.7^{+2.6}_{-2.0}$ | 9.1 | 0.79 | 168 |  |
Gl 317 | $-10.8^{+8.4}_{-8.8}$ | $2.7^{+2.5}_{-1.9}$ | $14.5^{+8.2}_{-4.1}$ | 11.9 | 0.97 | 76 |  |
GJ 4063 | $17.8^{+3.3}_{-3.2}$ | $2.9^{+1.9}_{-1.9}$ | $7.1^{+2.3}_{-1.8}$ | 11.4 | 0.96 | 204 | $40.2\pm 0.8$ | Díez Alonso et al. (2019)
GJ 1012 | $6.4^{+2.6}_{-2.6}$ | $3.2^{+2.7}_{-2.2}$ | $4.0^{+3.8}_{-2.7}$ | 15.9 | 0.99 | 135 |  |
GJ 1148 | $-9.1^{+3.9}_{-3.5}$ | $2.3^{+2.4}_{-1.6}$ | $7.1^{+4.1}_{-2.9}$ | 13.9 | 0.81 | 101 | $71.5\pm 5.1$ | Díez Alonso et al. (2019)
PM J08402+3127 | $16^{+20}_{-21}$ | $4.6^{+3.8}_{-3.1}$ | $39^{+16}_{-9}$ | 18.8 | 0.98 | 139 | $118\pm 14$ | Díez Alonso et al. (2019)
Gl 725B | $-4.6^{+5.2}_{-5.1}$ | $1.2^{+1.3}_{-0.8}$ | $12.1^{+3.7}_{-2.8}$ | 10.7 | 0.75 | 209 |  |
GJ 1105 | $1.2^{+3.6}_{-3.1}$ | $2.4^{+2.2}_{-1.7}$ | $7.3^{+3.2}_{-2.7}$ | 13.5 | 0.93 | 161 |  |
Gl 445 | $-1.8^{+2.7}_{-2.6}$ | $3.4^{+3.2}_{-2.4}$ | $3.2^{+3.7}_{-2.3}$ | 17.4 | 0.97 | 90 |  |
PM J21463+3813 | $-0.4^{+7.5}_{-7.7}$ | $3.1^{+3.1}_{-2.2}$ | $16.3^{+6.6}_{-4.6}$ | 20.7 | 0.84 | 176 |  |
Gl 447 | $20.5^{+9.1}_{-9.5}$ | $4.8^{+3.0}_{-3.0}$ | $16.6^{+8.5}_{-5.3}$ | 14.9 | 1.13 | 57 | $165.1\pm 0.8$ | Suárez Mascareño et al. (2016)

##  5 Discussion and conclusions

We have shown that spectropolarimetry is a useful technique for measuring the
rotation period of a star, even for quiet M dwarfs. In our sample of 43 such
stars, we were able to reliably measure the rotation period for 27 stars, 8 of
which were previously unknown. The rotation periods cover a wide range in this
sample of quiet stars, from 10 to 450 days. The agreement with other
techniques, such as photometry or stellar activity indicators is good, except
for a few stars (e.g., Gl 251, Gl 411, and Gl 846) for which some techniques
find a harmonic of the rotation period or converge toward very different
values. The amplitude of variation in the longitudinal magnetic field for
these quiet M dwarfs ranges from 3 G (Gl 338B, Gl 445) to 20 G for the
majority of the stars in our sample, but it can reach up to 70 G in some cases
(GJ 1289), which is comparable to more active stars such as AD Leo (Gl 388:
$45\pm 6$ G, Carmona et al., in prep) or EV Lac (Gl 873: $149\pm 17$ G) over
the same period of observations. The FWHM of the median Stokes $I$ profile is
another important measurement, as it may reveal a broadening due to the Zeeman
effect for active stars. As the FWHM is sensitive to the total magnetic field
(small and large scale),while the longitudinal magnetic field is more
sensitive to the large-scale field, a large amplitude of variation in the
longitudinal field does not necessarily correlate with a high FWHM, and both
measures are therefore important. The average value over our sample is 6.07 km
s-1 , with a standard deviation of 0.66 km s-1 . This is clearly larger than
the mean uncertainty of each FWHM in Table 1. This shows a real dispersion
among the mean FWHM values, which will be further investigated by using
different line lists (low and high Landé factors) and by studying the time
variation in the measured FWHM for a given star, as in Bellotti et al. (in
prep) for AD Leo.

We divided our sample into three sub-categories according to the absolute
magnitude in Gaia $G$ band, roughly corresponding to early-, mid- and late-M
dwarfs. There is a clear tendency to lower values of the FWHM of the median
Stokes $I$ profile toward mid-type stars, with averages and standard
deviations for the three groups of 6.65 and 0.59 km s-1 (early type), 5.73 and
0.34 km s-1 (mid type), and 6.49 and 0.75 km s-1 (late type), respectively.
The larger dispersion in the early-type and late-type groups may be due to a
few active outliers: removing Gl 410 from the early-type group gives new
average and standard deviation of 6.45 and 0.40 km s-1 , respectively, while
for the late-type group, removing GJ 1286 and GJ 1289 gives 6.14 and 0.52 km
s-1 , respectively, now more compatible with the mid-type group, given the
small size of the early- and late-type groups. The most extreme values
correspond to Gl 411 ($4.93\pm 0.18$ km s-1 ) and Gl 410 ($7.73\pm 0.21$ km
s-1 ). A test on GJ 1289 shows that the measured FWHM clearly depends on the
minimum adopted depth (0.03 in this work) and the LSD mask (3000 and 3500 K
give different results for this 3238 K star), and does not clearly show an
increase when using masks limited to low Landé factors ($g_{\rm eff}$
$\leqslant 1.2$) versus high Landé factors ($g_{\rm eff}$ $>1.2$). The
possible effect of the magnetic field on the Stokes I FWHM of the stars in
this sample will be studied in detail in Donati et al. (in prep) and
Cristofari et al. (in prep).

We observe that it is easier to constrain the rotation period and the other
parameters of the GP for early-type stars, even if the amplitude of variation
of the longitudinal magnetic field is similar in the other two subcategories.
This may be due to the fact that they are brighter than later M dwarfs on
average. Additionally, the decay time seems well constrained between 50 and
100 days for the early-type stars. For the mid-M dwarfs, it lies around 100
days, and for the late-M dwarfs, it seems to be longer at about 150 days,
although these results are only based on three and two measurements,
respectively. There is a tendency for longer decay times for a longer stellar
rotation period. Clearly, the quasi-periodic GP fit converges more easily for
the early-type stars than for the others: it converges for only 60% of the
mid-type stars, and although the convergence rate is similar for the late- and
early-type stars, we have to use a four-parameter fit for the late-type stars
(fixing the smoothing factor and the decay time), while we can use a six-
parameter fit for the early-type stars with good constraints on the smoothing
factor and on the decay time.

Figure 11: Rotation period vs stellar mass diagram. Stars with rotation
periods determined in this work are shown as larger circles, where the symbol
sizes and colors are proportional to the amplitude of variation of $B_{\ell}$
(see $\alpha$ values given in Tables 3, 5, and 8). Rotation periods and mass
determinations for inactive (white circles) and active (gray circles) stars
from Newton et al. (2017) are shown for comparison.

Our sample is too small to infer general properties of M stars. Therefore, we
compare in Fig. 11 the sample of stars analyzed in this paper to more complete
nonmagnetic studies in which active and inactive stars are well represented.
We include a total of 212 M stars from Newton et al. (2017), whose mass and
rotation periods are similar to the range explored in this paper. Our sample
of quiet stars nicely fits the sequence of inactive stars, as expected. The
long rotation period of Gl 411 appears as an exception, and may in fact not
reflect the stellar rotation period, as discussed in Sec. 4.2.

Low-mass stars arrive rapidly rotating on the main sequence and spin down with
time as angular momentum is lost through magnetized winds (Barnes, 2007).
Empirically calibrated relations between age and period can be used to
estimate the ages of individual field stars. Gaidos et al. (2023) combined the
4 Gyr M-dwarf gyrochrone of the open cluster M 67 (Dungee et al., 2022) with
those of younger, previously published gyrochrones (Curtis et al., 2020) to
assign ages to M dwarf host stars with $T_{\rm eff}=$3200-4200 K. We used the
same gyrochronology to estimate the ages of 20 stars in our sample with well-
measured rotation periods and $T_{\rm eff}$ in that range. Mean values and
standard deviations were calculated from distributions constructed by 10000
Monte Carlo calculations, incorporating uncertainties from $P_{\rm rot}$ (from
Tables 5 and 8), the gyrochrones, $T_{\rm eff}$ (assumed $\pm$75K), [Fe/H]
(assumed to be $\pm$0.1 dex), and variation in the initial rotation periods of
the stars on the zero-age main sequence. The results are listed in Table 11:
most stars are 4–10 Gyr old, and a few younger stars have ages from 0.5 to 2
Gyr. The rotation-age relations are only valid for stars of approximately
solar metallicity, but we list in the table (col. 6) a correction computed
using a theory-based model and the stellar metallicity, to be added to the age
given in col. 4. We report the age for members of binaries, but we emphasize
that these stars may have distinct rotational histories due to tides and rapid
dissipation of primordial disks (Fleming et al., 2019; Messina, 2019).

Table 11: Gyrochronological ages of the stars in our sample. Errors on $T_{\rm eff}$ and [M/H] are assumed to be 75 K and 0.1 dex. The correction in the last column corresponds to the value to be added to the age due to the nonsolar metallicity of the star. Star | $T_{\rm eff}$ | $P_{\rm rot}$ | Age | [M/H] | Metallicity correction
---|---|---|---|---|---
| K | d | Gyr | dex | Gyr
Gl 846 | 3833 | $11.01\pm 0.20$ | $<0.53$ | 0.07 | 0.1
Gl 410 | 3842 | $13.87\pm 0.08$ | $0.89\pm 0.12$ | 0.05 | 0.1
Gl 382 | 3644 | $21.32\pm 0.04$ | $1.9\pm 0.6$ | 0.15 | 0.2
Gl 514 | 3699 | $30.45\pm 0.14$ | $3.8\pm 0.6$ | $-0.07$ | $-0.1$
GJ 1289 | 3238 | $74.0\pm 1.4$ | $3.8\pm 1.2$ | 0.05 | 0.0
Gl 849 | 3502 | $41.4\pm 0.4$ | $4.2\pm 1.0$ | 0.35 | 0.4
Gl 205 | 3771 | $34.3\pm 0.4$ | $5.2\pm 0.7$ | 0.43 | 0.6
PM J09553-2715 | 3366 | $70.5\pm 3.8$ | $5.2\pm 2.0$ | $-0.03$ | $-0.0$
Gl 880 | 3702 | $37.7\pm 0.7$ | $5.5\pm 0.8$ | 0.26 | 0.4
Gl 169.1A | 3307 | $91.9\pm 3.4$ | $5.5\pm 2.4$ | 0.13 | 0.1
Gl 687 | 3475 | $56.5\pm 0.9$ | $5.9\pm 1.8$ | 0.01 | 0.0
Gl 15A | 3611 | $44.3\pm 0.2$ | $6.0\pm 1.2$ | $-0.33$ | $-0.5$
GJ 3378 | 3326 | $92.1\pm 4.7$ | $6.0\pm 2.7$ | $-0.05$ | $-0.0$
Gl 48 | 3529 | $51.2\pm 1.4$ | $6.1\pm 1.6$ | 0.08 | 0.1
Gl 15B | 3272 | $116.5\pm 5.6$ | $6.1\pm 3.0$ | $-0.42$ | $-0.2$
Gl 876 | 3366 | $82.8\pm 1.4$ | $6.2\pm 2.7$ | 0.15 | 0.1
Gl 752A | 3558 | $53.2\pm 4.2$ | $7.1\pm 1.9$ | 0.11 | 0.2
Gl 699 | 3311 | $137.1\pm 5.4$ | $9.4\pm 5.0$ | $-0.37$ | $-0.2$
Gl 251 | 3420 | $98.7\pm 8.2$ | $10.8\pm 4.7$ | $-0.01$ | $-0.0$
Gl 725A | 3470 | $103.5\pm 4.8$ | $14.8\pm 5.4$ | $-0.26$ | $-0.3$

Our study only exploited the longitudinal magnetic field measured from the
spectropolarimetric data of the SPIRou SLS. The same data can be used to
achieve much more, for example, to investigate the magnetic topology of these
stars and its evolution using ZDI maps.

###### Acknowledgements.

Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT)
which is operated from the summit of Maunakea by the National Research Council
of Canada, the _Institut National des Sciences de l’Univers_ of the _Centre
National de la Recherche Scientifique_ of France, and the University of
Hawaii. The observations at the Canada-France-Hawaii Telescope were performed
with care and respect from the summit of Maunakea which is a significant
cultural and historic site.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular, the institutions
participating in the Gaia Multilateral Agreement.
This research has also made intensive use of the SIMBAD database and of the
VizieR catalog access tool, operated at CDS, Strasbourg, France and of NASA’s
Astrophysics Data System (ADS).
We made an extensive use of the DACE (Data Analysis Center for Exoplanets)
software from the University of Geneva (Díaz et al., 2014; Delisle et al.,
2016).
This work has made use of the VALD database, operated at Uppsala University,
the Institute of Astronomy RAS in Moscow, and the University of Vienna.
X.D., A.C., P.C.Z., E.M. and S.B. and more generally most of the French
authors of this paper acknowledge funding from the French National Research
Agency (ANR) under contract number ANR18CE310019 (SPlaSH). This work is
supported by the ANR in the framework of the _Investissements d’Avenir_
program (ANR-15-IDEX-02), through the funding of the “Origin of Life” project
of the Grenoble-Alpes University.
E.M. acknowledges funding from the _Fundação de Amparo à Pesquisa do Estado de
Minas Gerais_ (FAPEMIG) under the project number APQ-02493-22.
B.Z. acknowledges funding from the _Programa de Internacionalização da
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior_ (CAPES-PrInt
#88887.683070/2022-00) and FAPEMIG (APQ-01033-22).
J.-F.D. acknowledges funding from the European Research Council (ERC) under
the H2020 research & innovation programme (grant agreement #740651 NewWorlds).
N.J.C., E.A. and R.D. wish to thank the Natural Sciences and Engineering
Research Council of Canada and the _Fonds Québécois de Recherche - Nature et
Technologies_ , the _Observatoire du Mont-Mégantic_ and the Institute for
Research on Exoplanets and acknowledge funding from _Développement Economique
Canada, Quebec’s Ministère de l’Education et de l’Innovation_ , the Trottier
Family Foundation and the Canadian Space Agency.
This research made use of the hosting service github and, among others, of the
following software tools: `matplotlib` (Hunter, 2007); `NumPy` (Harris et al.,
2020); `SciPy` (Virtanen et al., 2020); `Astropy` (Astropy Collaboration et
al., 2013, 2018); `emcee` (Foreman-Mackey et al., 2013); `corner` (Foreman-
Mackey, 2016); `george` (Ambikasaran et al., 2015); `barrycorpy` (Wright &
Eastman, 2014).

## References

  * Aigrain et al. (2015) Aigrain, S., Hodgkin, S. T., Irwin, M. J., Lewis, J. R., & Roberts, S. J. 2015, MNRAS, 447, 2880
  * Ambikasaran et al. (2015) Ambikasaran, S., Foreman-Mackey, D., Greengard, L., Hogg, D. W., & O’Neil, M. 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence, 38, 252
  * Angus et al. (2018) Angus, R., Morton, T., Aigrain, S., Foreman-Mackey, D., & Rajpaul, V. 2018, MNRAS, 474, 2094
  * Artigau et al. (2014) Artigau, É., Astudillo-Defru, N., Delfosse, X., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9149, Proc. SPIE, 914905
  * Artigau et al. (2022) Artigau, É., Cadieux, C., Cook, N. J., et al. 2022, AJ, 164, 84
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33
  * Babcock (1947) Babcock, H. W. 1947, ApJ, 105, 105
  * Bagnulo et al. (2009) Bagnulo, S., Landolfi, M., Landstreet, J. D., et al. 2009, PASP, 121, 993
  * Bakos et al. (2004) Bakos, G., Noyes, R. W., Kovács, G., et al. 2004, PASP, 116, 266
  * Barnes (2007) Barnes, S. A. 2007, ApJ, 669, 1167
  * Benedict et al. (1998) Benedict, G. F., McArthur, B., Nelan, E., et al. 1998, AJ, 116, 429
  * Boisse et al. (2011) Boisse, I., Bouchy, F., Hébrard, G., et al. 2011, A&A, 528, A4
  * Bonfils et al. (2013) Bonfils, X., Delfosse, X., Udry, S., et al. 2013, A&A, 549, A109
  * Bonfils et al. (2007) Bonfils, X., Mayor, M., Delfosse, X., et al. 2007, A&A, 474, 293
  * Bourrier et al. (2018) Bourrier, V., Lovis, C., Beust, H., et al. 2018, Nature, 553, 477
  * Burt et al. (2014) Burt, J., Vogt, S. S., Butler, R. P., et al. 2014, ApJ, 789, 114
  * Cook et al. (2022) Cook, N. J., Artigau, É., Doyon, R., et al. 2022, PASP, 134, 114509
  * Cortes-Zuleta et al. (2023) Cortes-Zuleta, P., Boisse, I., Klein, B., et al. 2023, arXiv e-prints, arXiv:2301.10614
  * Cristofari et al. (2022) Cristofari, P. I., Donati, J. F., Masseron, T., et al. 2022, MNRAS, 516, 3802
  * Curtis et al. (2020) Curtis, J. L., Agüeros, M. A., Matt, S. P., et al. 2020, ApJ, 904, 140
  * Delisle et al. (2016) Delisle, J. B., Ségransan, D., Buchschacher, N., & Alesina, F. 2016, A&A, 590, A134
  * Díaz et al. (2014) Díaz, R. F., Almenara, J. M., Santerne, A., et al. 2014, MNRAS, 441, 983
  * Díaz et al. (2019) Díaz, R. F., Delfosse, X., Hobson, M. J., et al. 2019, A&A, 625, A17
  * Díez Alonso et al. (2019) Díez Alonso, E., Caballero, J. A., Montes, D., et al. 2019, A&A, 621, A126
  * Donati et al. (2020) Donati, J. F., Kouach, D., Moutou, C., et al. 2020, MNRAS, 498, 5684
  * Donati et al. (2008) Donati, J. F., Morin, J., Petit, P., et al. 2008, MNRAS, 390, 545
  * Donati et al. (1997) Donati, J. F., Semel, M., Carter, B. D., Rees, D. E., & Collier Cameron, A. 1997, MNRAS, 291, 658
  * Dungee et al. (2022) Dungee, R., van Saders, J., Gaidos, E., et al. 2022, ApJ, 938, 118
  * Faria et al. (2020) Faria, J. P., Adibekyan, V., Amazo-Gómez, E. M., et al. 2020, A&A, 635, A13
  * Feiden et al. (2021) Feiden, G. A., Skidmore, K., & Jao, W.-C. 2021, ApJ, 907, 53
  * Figueira et al. (2010) Figueira, P., Santos, N. C., Bonfils, X., et al. 2010, in EAS Publications Series, Vol. 42, EAS Publications Series, ed. K. Gożdziewski, A. Niedzielski, & J. Schneider, 131–135
  * Fleming et al. (2019) Fleming, D. P., Barnes, R., Davenport, J. R. A., & Luger, R. 2019, ApJ, 881, 88
  * Foreman-Mackey (2016) Foreman-Mackey, D. 2016, The Journal of Open Source Software, 1, 24
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1
  * Gaidos et al. (2023) Gaidos, E., Claytor, Z., Dungee, R., Ali, A., & Feiden, G. A. 2023, MNRAS[arXiv:2301.12109]
  * Giles et al. (2017) Giles, H. A. C., Collier Cameron, A., & Haywood, R. D. 2017, MNRAS, 472, 1618
  * Gustafsson et al. (2008) Gustafsson, B., Edvardsson, B., Eriksson, K., et al. 2008, A&A, 486, 951
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357
  * Haywood et al. (2014) Haywood, R. D., Collier Cameron, A., Queloz, D., et al. 2014, MNRAS, 443, 2517
  * Hébrard et al. (2016) Hébrard, É. M., Donati, J. F., Delfosse, X., et al. 2016, MNRAS, 461, 1465
  * Henry (1999) Henry, G. W. 1999, PASP, 111, 845
  * Hobson et al. (2021) Hobson, M. J., Bouchy, F., Cook, N. J., et al. 2021, A&A, 648, A48
  * Horne (1986) Horne, K. 1986, PASP, 98, 609
  * Howard et al. (2014) Howard, A. W., Marcy, G. W., Fischer, D. A., et al. 2014, ApJ, 794, 51
  * Huélamo et al. (2008) Huélamo, N., Figueira, P., Bonfils, X., et al. 2008, A&A, 489, L9
  * Hunter (2007) Hunter, J. D. 2007, Computing In Science & Engineering, 9, 90
  * Irwin et al. (2011) Irwin, J., Berta, Z. K., Burke, C. J., et al. 2011, ApJ, 727, 56
  * Irwin et al. (2009) Irwin, J., Charbonneau, D., Nutzman, P., & Falco, E. 2009, in Transiting Planets, ed. F. Pont, D. Sasselov, & M. J. Holman, Vol. 253, 37–43
  * Jao et al. (2018) Jao, W.-C., Henry, T. J., Gies, D. R., & Hambly, N. C. 2018, ApJ, 861, L11
  * Kane et al. (2016) Kane, S. R., Thirumalachari, B., Henry, G. W., et al. 2016, ApJ, 820, L5
  * Kanodia & Wright (2018) Kanodia, S. & Wright, J. 2018, Research Notes of the AAS, 2, 4
  * Kiraga & Stepien (2007) Kiraga, M. & Stepien, K. 2007, Acta Astron., 57, 149
  * Lafarga et al. (2021) Lafarga, M., Ribas, I., Reiners, A., et al. 2021, arXiv e-prints, arXiv:2105.13467
  * Landi Degl’Innocenti (1992) Landi Degl’Innocenti, E. 1992, in Solar Observations: Techniques and Interpretation, ed. F. Sanchez, M. Collados, & M. Vazquez, 71
  * Landstreet (1980) Landstreet, J. D. 1980, AJ, 85, 611
  * Landstreet (1992) Landstreet, J. D. 1992, A&A Rev., 4, 35
  * Mann et al. (2019) Mann, A. W., Dupuy, T., Kraus, A. L., et al. 2019, ApJ, 871, 63
  * Martioli et al. (2020) Martioli, E., Hébrard, G., Moutou, C., et al. 2020, A&A, 641, L1
  * Messina (2019) Messina, S. 2019, A&A, 627, A97
  * Morin et al. (2008) Morin, J., Donati, J. F., Petit, P., et al. 2008, MNRAS, 390, 567
  * Morin et al. (2010) Morin, J., Donati, J. F., Petit, P., et al. 2010, MNRAS, 407, 2269
  * Moutou et al. (2020) Moutou, C., Dalal, S., Donati, J. F., et al. 2020, A&A, 642, A72
  * Moutou et al. (2017) Moutou, C., Hébrard, E. M., Morin, J., et al. 2017, MNRAS, 472, 4563
  * Nelson et al. (2016) Nelson, B. E., Robertson, P. M., Payne, M. J., et al. 2016, MNRAS, 455, 2484
  * Newton et al. (2017) Newton, E. R., Irwin, J., Charbonneau, D., et al. 2017, ApJ, 834, 85
  * Newton et al. (2018) Newton, E. R., Mondrik, N., Irwin, J., Winters, J. G., & Charbonneau, D. 2018, AJ, 156, 217
  * Nicholson & Aigrain (2022) Nicholson, B. A. & Aigrain, S. 2022, MNRAS, 515, 5251
  * Noyes et al. (1984) Noyes, R. W., Weiss, N. O., & Vaughan, A. H. 1984, ApJ, 287, 769
  * Nutzman & Charbonneau (2008) Nutzman, P. & Charbonneau, D. 2008, PASP, 120, 317
  * Petit et al. (2021) Petit, P., Folsom, C. P., Donati, J. F., et al. 2021, A&A, 648, A55
  * Piskunov et al. (1995) Piskunov, N. E., Kupka, F., Ryabchikova, T. A., Weiss, W. W., & Jeffery, C. S. 1995, A&AS, 112, 525
  * Pojmanski (1997) Pojmanski, G. 1997, Acta Astron., 47, 467
  * Pollacco et al. (2006) Pollacco, D. L., Skillen, I., Collier Cameron, A., et al. 2006, PASP, 118, 1407
  * Preston (1971) Preston, G. W. 1971, PASP, 83, 571
  * Queloz et al. (2001) Queloz, D., Henry, G. W., Sivan, J. P., et al. 2001, A&A, 379, 279
  * Rasmussen & Williams (2006) Rasmussen, C. E. & Williams, C. K. I. 2006, Gaussian Processes for Machine Learning (MIT Press)
  * Rees & Semel (1979) Rees, D. E. & Semel, M. D. 1979, A&A, 74, 1
  * Reinhold & Hekker (2020) Reinhold, T. & Hekker, S. 2020, A&A, 635, A43
  * Rivera et al. (2005) Rivera, E. J., Lissauer, J. J., Butler, R. P., et al. 2005, ApJ, 634, 625
  * Robertson & Mahadevan (2014) Robertson, P. & Mahadevan, S. 2014, ApJ, 793, L24
  * Robertson et al. (2014) Robertson, P., Mahadevan, S., Endl, M., & Roy, A. 2014, Science, 345, 440
  * Ryabchikova et al. (2015) Ryabchikova, T., Piskunov, N., Kurucz, R. L., et al. 2015, Phys. Scr, 90, 054005
  * Sabotta et al. (2021) Sabotta, S., Schlecker, M., Chaturvedi, P., et al. 2021, A&A, 653, A114
  * Stock et al. (2020) Stock, S., Nagel, E., Kemmer, J., et al. 2020, A&A, 643, A112
  * Suárez Mascareño et al. (2022) Suárez Mascareño, A., González-Álvarez, E., Zapatero Osorio, M. R., et al. 2022, arXiv e-prints, arXiv:2212.07332
  * Suárez Mascareño et al. (2016) Suárez Mascareño, A., Rebolo, R., & González Hernández, J. I. 2016, A&A, 595, A12
  * Suárez Mascareño et al. (2015) Suárez Mascareño, A., Rebolo, R., González Hernández, J. I., & Esposito, M. 2015, MNRAS, 452, 2745
  * Suárez Mascareño et al. (2017) Suárez Mascareño, A., Rebolo, R., González Hernández, J. I., & Esposito, M. 2017, MNRAS, 468, 4772
  * Suárez Mascareño et al. (2018) Suárez Mascareño, A., Rebolo, R., González Hernández, J. I., et al. 2018, A&A, 612, A89
  * Toledo-Padrón et al. (2019) Toledo-Padrón, B., González Hernández, J. I., Rodríguez-López, C., et al. 2019, MNRAS, 488, 5145
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261
  * West et al. (2008) West, A. A., Hawley, S. L., Bochanski, J. J., et al. 2008, AJ, 135, 785
  * Woźniak et al. (2004) Woźniak, P. R., Vestrand, W. T., Akerlof, C. W., et al. 2004, AJ, 127, 2436
  * Wright & Eastman (2014) Wright, J. T. & Eastman, J. D. 2014, PASP, 126, 838
  * Zechmeister & Kürster (2009) Zechmeister, M. & Kürster, M. 2009, A&A, 496, 577

##  Appendix A Quasi-periodic Gaussian process regression for stars with a
measured rotation period

We display here the temporal evolution of the longitudinal magnetic field and
its GP fit leading to the measurement of the stellar rotation period for 25
stars of our sample (Figs. 12 to 35). GP fits are generally six-parameter
fits, except when stated in the caption.

Figure 12: GP analysis of the SPIRou $B_{\ell}$ data of Gl 846. The top panel
shows the observed $B_{\ell}$ data (black points) and the orange line shows
the best-fit quasi-periodic GP model. The bottom panel shows the residuals.
Figure 13: Same for Gl 205. Figure 14: Same for Gl 880. Figure 15: Same for Gl
514. Figure 16: Same for Gl 382. Figure 17: Same for Gl 752A. Figure 18: Same
for Gl 48. Figure 19: Same for Gl 15A. Figure 20: Same for Gl 849. Figure 21:
Same for Gl 687. Figure 22: Same for Gl 725A (five-parameter fit). Figure 23:
Same for Gl 251 (five-parameter fit). Figure 24: Same for GJ 4333. Figure 25:
Same for Gl 876. Figure 26: Same for PM J09553-2715. Figure 27: Same for GJ
3378. Figure 28: Same for Gl 169.1A. Figure 29: Same for GJ 1289. Figure 30:
Same for GJ 1103 (five-parameter fit). Figure 31: Same for Gl 699 (five-
parameter fit). Figure 32: Same for Gl 15B (five-parameter fit). Figure 33:
Same for GJ 1151 (five-parameter fit). Figure 34: Same for GJ 1286 (five-
parameter fit). Figure 35: Same for GJ 1002.

##  Appendix B Quasi-periodic Gaussian process regression for Gl 411

In this appendix, we study the particular case of the very long stellar
rotation period measured for Gl 411 (460 days). We display the temporal
evolution of the longitudinal magnetic field and its GP fit (Fig. 36). We then
display the associated corner plot showing the posterior distributions of each
parameter (Fig. 38).

At the suggestion of the referee, we compared the GP fit to a similar GP fit
to reshuffled data. We kept the same observation dates, but attributed the
$B_{\ell}$ measurements and their associated uncertainty randomly to a given
date. The GP fit is now flat and shows that the detected periodicity in the
actually measured data is real (Fig. 37). We also display the associated
corner plot for comparison. No periodic behavior can be detected at all (Fig.
39).

Figure 36: GP analysis of the SPIRou $B_{\ell}$ data of Gl 411. The top panel
shows the observed $B_{\ell}$ data (black points) and the orange line shows
the best-fit quasi-periodic GP model. The bottom panel shows the residuals of
the six-parameter GP fit.  Figure 37: GP analysis of reshuffled $B_{\ell}$
data at the same observation dates as Gl 411. Figure 38: Corner plot of the
posterior distribution of parameters of the GP fit for Gl 411. Figure 39:
Corner plot of the posterior distribution of parameters of the GP fit for the
reshuffled data at the observation dates of Gl 411.

##  Appendix C Stokes V profiles of stars without a detected rotation period

We display here the temporal evolution of the Stokes $V$ for the 16 stars of
our sample (Figs. 40 to 55) for which we could not detect a periodic variation
of the longitudinal magnetic field. This shows that for some stars, the Stokes
V is well detected, but is constant or does not vary periodically, while for
others, it is barely detected or not detected at all.

Figure 40: Temporal variation in Stokes $V$ profiles of the SPIRou
polarimetric series of Gl 338B. Figure 41: Same for Gl 617B. Figure 42: Same
for Gl 412A. Figure 43: Same for Gl 480. Figure 44: Same for Gl 436. Figure
45: Same for Gl 408. Figure 46: Same for Gl 317. Figure 47: Same for GJ 4063.
Figure 48: Same for GJ 1012. Figure 49: Same for GJ 1148. Figure 50: Same for
PM J08402+3127. Figure 51: Same for Gl 725B. Figure 52: Same for GJ 1105.
Figure 53: Same for Gl 445. Figure 54: Same for PM J21463+3813. Figure 55:
Same for Gl 447.