Optical and near-infrared stellar activity characterization of the early M dwarf Gl 205 with SOPHIE and SPIRou

Context. The stellar activity of M dwarfs is the main limiting factor in the discovery and characterization of the exoplanets orbiting them, because it induces quasi-periodic radial velocity (RV) variations. Aims. We aim to characterize the magnetic field and stellar activity of the early, moderately active M dwarf Gl 205 in the optical and near-infrared (NIR) domains. Methods. We obtained high-precision quasi-simultaneous spectra in the optical and NIR with the SOPHIE spectrograph and SPIRou spectropolarimeter between 2019 and 2022. We computed the RVs from both instruments and the SPIRou Stokes V profiles. We used Zeeman–Doppler imaging (ZDI) to map the large-scale magnetic field over the time span of the observations. We studied the temporal behavior of optical and NIR RVs and activity indicators with the Lomb-Scargle periodogram and a quasi-periodic Gaussian process regression (GPR). In the NIR, we studied the equivalent width of Al I, Ti I, K I, Fe I, and He I. We modeled the activity-induced RV jitter using a multi-dimensional GPR with activity indicators as ancillary time series. Results. The optical and NIR RVs show similar scatter but NIR shows a more complex temporal evolution. We observe an evolution of the magnetic field topology from a poloidal dipolar field in 2019 to a dominantly toroidal field in 2022. We measured a stellar rotation period of P rot = 34 . 4 ± 0 . 5days in the longitudinal magnetic field. Using ZDI, we measure the amount of latitudinal differential rotation (DR) shearing the stellar surface, yielding rotation periods of P eq = 32 . 0 ± 1 . 8 days at the stellar equator and P pol = 45 . 5 ± 0 . 3 days at the poles. We observed inconsistencies in the periodicities of the activity indicators that could be explained by these DR values. The multi-dimensional GP modeling yields an RMS of the RV residuals down to the noise level of 3ms − 1 for both instruments while using H α and the BIS in the optical and the full width at half maximum (FWHM) in the NIR as ancillary time series. Conclusions. The RV variations observed in Gl 205 are due to stellar activity, with a complex evolution and different expressions in the optical and NIR revealed thanks to an extensive follow-up. Spectropolarimetry remains the best technique to constrain the stellar rotation period over standard activity indicators, particularly for moderately active M dwarfs.


Introduction
M dwarfs are the most abundant stars in the Milky Way (Henry et al. 2006;Reylé et al. 2021) and they have become key-targets for exoplanetary surveys (e.g., Morley et al. 2017). Their low mass favors the detection of planets orbiting them as the Doppler variations are larger than for Solar-like stars for a given planetary mass and equilibrium temperature. Dedicated surveys to discover and characterize exoplanets around M dwarfs are currently being carried out using transit photometry (e.g., MEarth: Nutzman & Charbonneau 2008, TRAPPIST: Gillon et al. 2011, SPECULOOS: Delrez et al. 2018, Tierras: Garcia-Mejia et al. 2020 and Doppler spectroscopy (e.g., HPF: Mahadevan et al. 2012, HARPS: Bonfils et al. 2013, HARPS-N: Covino et al. 2013, IRD: Kotani et al. 2014, CARMENES: Quirrenbach et al. 2018, MAROON-X: Seifahrt et al. 2018, SOPHIE: Hobson et al. 2018, SPIRou: Donati et al. 2020. pia.cortes@lam.fr Transit and radial velocity (RV) surveys have shown that M dwarfs are Earth-like planets hosts (Bonfils et al. 2013;Kopparapu 2013;Dressing & Charbonneau 2013Sabotta et al. 2021;Pinamonti et al. 2022). However, despite the efforts to detect and characterize Earth-like planets around M dwarfs, the presence of stellar activity remains one of the main limitations in this endeavor. Cool stars such as M dwarfs are known to host magnetic fields causing the phenomena known as stellar activity (Reiners 2012).
Stellar magnetic activity produces quasi-periodic RV signals with amplitudes of a few meters per second that can easily lead to a false positive exoplanet detection (e.g., Queloz et al. 2001;Desidera et al. 2004;Huélamo et al. 2008;Carolo et al. 2014;Bortle et al. 2021). Features over the stellar surface, particularly dark spots and bright plages, can survive and evolve on time scales on a few stellar rotation cycles and thus, modulate the RVs at the period of the stellar rotation (Boisse et al. 2011;Scandariato et al. 2017 Several techniques have been developed during the last decade to mitigate the effects of stellar activity in order to improve the detection of exoplanets around active stars. When the stellar rotation period is known, one can model the RV variations induced by spots by fitting sinusoidal signals and their harmonics (Boisse et al. 2011) or by simulations of the active regions (Dumusque et al. 2014a). Another approach is to use activity indicators built from spectral information. These can be proxies of the shape of the cross-correlation function (CCF), for example the full width at half maximum (FWHM) and the bisector inverse slope (BIS) Boisse et al. 2011), or of the chromospheric emission in spectral lines sensitive to activity, such as the Ca II H&K and Hα (Boisse et al. 2009). A third approach, among others, is to model the stellar surface structures generating RV variations (e.g., Hébrard et al. 2016;Klein et al. 2021Klein et al. , 2022. The activity-induced behavior of the spectral lines in M dwarf spectra has been studied only for a handful of lines, such as Hα, Ca II H&K and IRT, Na I D, and He I, with the Hα and CaII H&K lines the most extensively explored. (e.g., Gomes da Silva et al. 2011;Newton et al. 2017;Fuhrmeister et al. 2019;Lafarga et al. 2021). The majority of the well-known activity tracers are located at optical wavelengths. Recently, some nearinfrared lines have been studied, for example the He I triplet (Schöfer et al. 2019;Fuhrmeister et al. 2019Fuhrmeister et al. , 2020 and the K I line (Fuhrmeister et al. 2022;Terrien et al. 2022). The He I triplet in absorption is not detected over the range of the whole M dwarf spectral type and disappears for stars later than M5. When in emission, it could be related to flaring events (Fuhrmeister et al. 2019). Moreover, Fuhrmeister et al. (2020) found that the variability in the He I triplet may be correlated with the Hα variability only for active M dwarfs. In the particular case of the active M dwarf AU Mic, the He I flux correlated well with the RVs (Klein et al. 2021). Regarding the K I line emission, it has been found that it is rarely correlated or anti-correlated with Hα (Fuhrmeister et al. 2022). Moreover, Terrien et al. (2022) found clear signals of Zeeman broadening in this line modulated by the stellar rotation in Gl 699 and Teegarden's star data.
More recently the use of data-driven techniques namely Gaussian Processes regression (GPR) (e.g., Haywood et al. 2014;Rajpaul et al. 2015;Jones et al. 2017;Gilbertson et al. 2020;Klein et al. 2021; Barragán et al. 2022a;Delisle et al. 2022) or Principal Component Analysis (PCA) (e.g., Davis et al. 2017;Cretignier et al. 2022) have shown good results on modelling the contribution of stellar activity in RVs. Nowadays the use of GPs has become the standard procedure for modeling and filtering stellar activity in RVs curves and transits exoplanet searches, since the GPR can easily fit the quasi-periodic activity signal. This technique is particularly successful when the information from activity indicators is used when fitting the GPs on the RV time series along with the Keplerian signal (Rajpaul et al. 2015; Suárez Mascareño et al. 2020;Faria et al. 2022;Zicher et al. 2022).
Spectropolarimetry data has been widely used to measure and constrain the properties of the large-scale magnetic field of M dwarfs Morin et al. 2008Morin et al. , 2010. In particular, it has been shown that the longitudinal magnetic field B is a reliable magnetic activity tracer for M dwarfs and therefore can be used to determine the stellar rotation period (e.g., Morin et al. 2008Morin et al. , 2010Folsom et al. 2016;Hébrard et al. 2016;Martioli et al. 2022).
Another way to mitigate stellar activity is to observe in nearinfrared wavelengths as for M dwarfs, the spot-induced RV jitter decreases as a function of wavelength (Martín et al. 2006;Desort et al. 2007;Reiners et al. 2010;Mahmud et al. 2011). The flux contrast between the dark spots and the stellar surface is smaller at near-infrared wavelengths than in the optical, generating less activity RV jitter. However, this effect strongly depends on the spectral type and the spot configuration on the stellar surface (Reiners et al. 2010;Andersen & Korhonen 2015). On the contrary, the RV jitter due to the Zeeman effect increases at longer wavelengths as the induced variation is proportional to the wavelength and the magnetic field strength (Hébrard et al. 2014). In cases of low-temperature contrast, the expected gain of the nearinfrared is hampered by the increasing Zeeman effect (Reiners et al. 2013;Klein & Donati 2020). Simultaneous observations at the optical and near-infrared can help to disentangle the activity contribution at both wavelength domains (Reiners et al. 2013;Robertson et al. 2020). Moreover, multiwavelength observations have been crucial to reject the planetary nature of RV signals (e.g., TW Hydrae: Huélamo et al. 2008;AD Leo: Carleo et al. 2020, Carmona et al. 2022 Gl 205 is an early (M1.5), nearby (5.7 pc), slow-rotating M1.5 dwarf with moderate levels of activity. It has a mass of 0.55 ± 0.03 M and a radius of 0.56 ± 0.03 R (Schweitzer et al. 2019). More stellar parameters are listed in Table 1. Previously, Gl 205 has been monitored with HARPS (Bonfils et al. 2013), and with HARPS-Pol and NARVAL (Hébrard et al. 2016). Using the Hα and Ca II H&K indices, Bonfils et al. (2013) found the stellar rotational period to be close to 33 d, and using spectropolarimetric data, Hébrard et al. 2016 determined a rotation period of P rot = 33.63 ± 0.37 d. Photometry data in the V band showed a periodic signal of 33 d and a possible long-trend magnetic cycle of ∼ 1500 d (Kiraga & Stepien 2007). The discrepancies between the periodicities of several activity indicators could be explained by differential rotation shearing the stellar surface and may induce a difference of ∼ 10 d in the rotation period between the equator and the pole (Hébrard et al. 2016).
In this work, we analyze the magnetic field and stellar activity of the early-M dwarf Gl 205, intensively monitored over a 2-yr period with the SOPHIE optical spectrograph and with the SPIRou near-infrared spectropolarimeter. This paper is structured as follows. In Section 2 we describe the observations and the reduction of the SOPHIE and SPIRou data. Section 3 is dedicated to the stellar characterization of Gl 205 and Section 4 to the analysis of TESS photometry. We describe the magnetic field properties using the SPIRou spectropolarimetric data in Section 5. In Section 6 we compare the optical and near-infrared RVs and in Section 7 we describe and analyse the activity indicators. We filter the RV variations due to activity using a multidimensional GP framework in Section 8. In Section 9 we discuss our RV limit detection of Keplerian signals. In Section 10 we discuss our results and present the principal conclusions of this work. Right: GLS periodogram of the full RVs time series, combining SOPHIE and SPIRou observations. The highest peak is marked with an arrow at 34.3 d. program include the detection of the exoplanets Gl 96b (Hobson et al. 2018), Gl 378b , and Gl 411b (Díaz et al. 2019), and the independent confirmation of Gl 617Ab (Hobson et al. 2018) The observations were gathered using the high-resolution (HR) mode, on which the spectrograph reaches a resolving power of λ/∆λ ∼ 75000. In order to measure the instrumental drift, simultaneous calibrations with a Fabry-Pérot (FP) étalon were performed. In total, we gathered 74 spectra of the star, with a median exposure time of 930 s and a median signal-to-noise ratio (SNR) per pixel at 550 nm of 95. After removing the observations with airmass>1.6, SNR<80, or affected by moonlight pollution, a total of 62 observations remained.
We used the SOPHIE Data Reduction Software (DRS, Bouchy et al. 2009) to reduce and extract the spectra. The SO-PHIE observations are corrected for the charge transfer inefficiency (CTI) effect of the CCD following Bouchy et al. (2009b) and Hobson et al. (2018). The DRS automatically computes the radial velocities using the CCF technique, which is obtained by cross-correlation of the spectra with an empirical M2V mask. The DRS also uses the CCF to deliver stellar activity indicators, such as the depth of the CCF defined as CCF contrast, CCF FWHM, and BIS.
However, in the case of M dwarfs spectra the CCF method is not an optimal approach due to the large number of absorption lines. To use most of the Doppler information available, we used a template-matching algorithm to obtain high-precision radial velocities (NAIRA, Astudillo-Defru et al. 2015. First, all the available spectra are normalized by the blaze functions and scaled to the unity by their median. Second, the spectra are shifted using the RVs computed by the DRS. Third, these spectra are co-added to build a high SNR stellar template. Finally, the maximum-likelihood RV is the minimum of the Chi-square profile obtained by shifting the stellar template over an array of RVs. A list of standard stars was used to correct the long-term variations of the RV zero-point in SOPHIE, an effect described by Courcol et al. (2015). To do so, we systematically monitored each possible night a group of G-type stars: HD185144, HD9407, HD89269A, and a group of M dwarfs: Gl 514, Gl 15A, and Gl 686 (Hobson et al. 2018).
The final radial velocities from NAIRA used in this work with their error bars are listed in the Table A.1, along with the activity indicators: CCF FWHM, CCF contrast, BIS, the S index, and Hα line (see Section 7). Figure 1 shows the SOPHIE RV time series whose average error bars are 1.9 m s −1 and the scatter is 4.8 m s −1 .

SPIRou
The Spectro-Polarimetre InfraRouge (SPIRou, Donati et al. 2020) is a high-resolution near-infrared spectropolarimeter and velocimeter mounted at the Canada-France-Hawaii Telescope (CFHT) in Hawaii. With a nominal spectral range from 9800 to 23500 Å, it covers the Y, J, H, and K bands of the infrared spectrum at a spectral resolving power of λ/∆λ ∼ 70 000. The observations of Gl 205 are part of the Planet Search program (WP1) of the SPIRou Legacy Survey (SLS, Donati et al. 2020), whose main goal is to perform a systematic RV monitoring of nearby M dwarfs.
SPIRou operates as a spectropolarimeter as well as spectrograph. A spectropolarimetric sequence consists of four subexposures, each one with a different rotation angle of the halfwave Fresnel rhombs in the polarimeter. We used the spectropolarimetric mode in order to obtain a set of Stokes I (unpolarized) and Stokes V (circularly polarized) profiles spectra per spectropolarimetric sequence of 4 sub-exposures (see Section 5).
The star was observed from September 2019 to January 2022, collecting a total of 156 sequences of 4 sub-exposures. The median exposure time of the observations is 61 s per rhomb position (244 s for a complete sequence) and the median SNR per pixel at 1670 nm is 290. We removed from the analysis 11 sequences with SNR<150 or airmass>1.7.
The data were reduced using the SPIRou data reduction software APERO 1 v0.7.194 (Cook et al. 2022). APERO performs an automatic reduction of the 4096×4096 pixels raw images, including correction for detector effects, and identification and removal of bad pixels and cosmic rays. The data are calibrated by performing flat and blaze corrections, then are optimally extracted (Horne 1986) from both science channels (fibers A and B that carry orthogonal polarimetric states of the incoming light) and from the calibration channel (fiber C).
The pixel-to-wavelength calibration is done using an UNe hollow cathode lamp and a Fabry-Pérot etalon, following Hobson et al. (2021), in order to obtain the wavelength at the observatory rest-frame. Then APERO utilizes the barycorrpy 2 (Kanodia & Wright 2018) Python code to compute the Barycentric Earth Radial Velocity (BERV) and Barycentric Julian Date (BJD) of each exposure. APERO performs a telluric and night emission correction in two steps. The first step is to obtain an atmospheric absorption model built with the TAPAS (Transmissions of the AtmosPhere for AStronomical data, Bertaux et al. 2014) which is applied to pre-clean the science frames. This model only leaves percent-level residuals in deep (> 50%) lines of H 2 O and dry absorption molecules (e.g., CH 4 , O 2 , CO 2 , N 2 O, and O 3 ). This procedure of building a TAPAS model is also done for a set of rapid rotating hot stars observed at different atmospheric conditions, in order to built a library of telluric residual models. This grid of telluric residual models have 3 degrees of freedom (optical depths of H 2 O and dry components, and a constant). The second step is to subtract this telluric residual model to the pre-cleaned data to obtain a telluric corrected spectra.
In a standard procedure, APERO computes automatically the radial velocities by cross-correlation of the telluric-corrected 1 https://github.com/njcuk9999/apero-drs 2 https://github.com/shbhuk/barycorrpy spectra with a given binary mask of stellar absorption lines. However, in this work, we made used of the RVs computed by the line-by-line (LBL) method based on the Bouchy et al. (2001) framework and optimized for SPIRou data. The LBL method is fully described in Artigau et al. (2022). This algorithm exploits the radial velocity content per line on the spectra to obtain one single RV measurement. Usually, an M dwarf observed by SPIRou will have ∼16 000 individual spectral lines. A finitemixture model approach deals with the high-sigma outliers in the RVs of the lines that come from cosmic rays or errors in the correction for tellurics. After outlier removal, the final RV is the mean of a Gaussian distribution containing the individual RVs of all the lines, and its uncertainty is derived following Bouchy et al. (2001). This method is also applied to the simultaneous calibration fiber to correct for the instrumental drift. The LBL RVs are corrected for the instrumental drift and for the long-term zero point using a Gaussian Process regression with data of the most observed stars in the SPIRou Legacy Survey. The details of this procedure will be described in Vandal et al. (in prep). The log of the observations and the RV measurements from the LBL method are listed in the Table A.2. Figure1 shows the SPIRou RV time series whose average error bar is 1.9 m s −1 and the scatter is 4.4 m s −1 .

TESS
Currently, the Transiting Exoplanet Survey Satellite (TESS; Ricker 2014) is in its extended mission, after successfully completing an all-sky survey of bright stars during its primary mission. TESS observed Gl 205 with 2-minute cadence during Sector 6, between December 12, 2018 and January 6, 2019, and again in Sector 32, from November 19 to December 16, 2020. We obtained the Presearch Data Conditiong (PDC) flux time series, processed by the TESS Science Processing Operations Center (SPOC), from the Mikulski Archive for Space Telescopes (MAST) 3 . The light curves are shown in Figure 2. We used the quality flags given by the pipeline to remove bad regions of the light curves thus, we only kept data points with quality flag equals zero.

Stellar characterization
We use the high-resolution spectra from SOPHIE and SPIRou, independently, to derive the atmospheric stellar parameters of Gl 205. For the SOPHIE optical part we made use of the ODUSSEAS (Antoniadis-Karnavas et al. 2020) code to compute the effective temperature, T eff , and the metallicity, [Fe/H]. For the SPIRou near-infrared spectra we follow Cristofari et al. (2022b) to derive T eff , surface gravity (log g), overall metallicity ([M/H]), and alpha-enhancement ([α/H]). The T eff measured from the optical and near-infrared spectra are in good agreement. The values of stellar parameters derived for Gl 205 are listed in Table 1. In this section we describe in detail both techniques.

SOPHIE spectra
A detailed description of the machine learning tool ODUSSEAS can be found in Antoniadis-Karnavas et al. (2020). The method is based on measuring the pseudo equivalent widths (pEWs) of absorption lines and blended lines in the range between 5300 Å and 6900 Å. The line list consists of 4104 absorption features, the same as used by Neves et al. (2014).
ODUSSEAS receives 1D spectra and their resolutions as input. The tool contains a supervised machine learning algorithm based on the scikit-learn Python package (Pedregosa et al. 2011), in order to determine the T eff and [Fe/H] of M dwarf stars.
Applied to new spectra, ODUSSEAS measures the pEWs of their lines and compares them to the models generated from the reference HARPS spectra sample, convolved to the respective resolution of the new spectra to be analyzed. The reference data set is built with spectra taken from the HARPS M dwarf sample. In the case of the SOPHIE spectrum of star Gl 205, the HARPS reference spectra are convolved from their original resolution of 115000 to the SOPHIE resolution of 75000.
For the analysis of Gl 205, the new reference data set of the upgraded ODUSSEAS version has been applied and includes spectra from 47 M dwarfs. The references for training and testing the models are the pEWs of the 47 HARPS spectra, used together with interferometry-based T eff (Rabus et al. 2019;Khata et al. 2021) and [Fe/H] derived by applying the method by Neves et al. (2012) using updated values of parallaxes from Gaia DR3.
The resulting stellar parameters of the star are calculated from the mean values of 100 determinations obtained by randomly shuffling and splitting each time the training (80% of the reference sample, i.e. 37 stars) and testing groups (remaining 20%, i.e. 10 stars). We report parameter uncertainties derived by quadratically adding the dispersion of the resulting stellar parameters and the mean absolute errors of the machine learning models at this resolution. We obtained an T eff = 3878 ± 81 K and [Fe/H] = 0.21 ± 0.06 dex.

SPIRou spectra
We estimate the T eff , log g, [M/H], and [α/H] from a high resolution template spectrum built from the over 500 spectra from sub-exposures acquired with SPIRou. The process relies on the direct comparison of the template spectrum to a grid of synthetic spectra computed from MARCS model atmospheres (Cristofari et al. 2022b,a). The comparison is performed on carefully selected spectral windows containing about 20 atomic lines and 40 molecular lines.
Prior to the comparison, synthetic spectra are broadened to account for instrumental effects, and the local continuum of the models is adjusted on windows built around the selected lines. Comparing the template spectrum to a grid of synthetic spectra with various T eff , log g, [M/H] and [α/H] results in the computation of 3 dimensional χ 2 grid on which we fit a 3D 2nd degree polynomial to retrieve a minimum.

Age and galactic population
We computed the age of Gl 205 with the stardate 4 (Angus et al. 2019) Python package which combines isochrones fitting with gyrochronology. The inputs of this code are the stellar parameters derived in Sections 3.2 and 3.1: T eff , log g, [Fe/H], and the parallax from Table 1. Since we obtained two estimations of T eff , we tested both values to see if we obtain results in agreement. Using the T eff from SOPHIE spectra we measure a stellar age of 3.7 +5.0 −1.6 Gyr. In the other hand, using the SPIRou T eff we estimate the age at 3.6 +5.0 −1.7 Gyr. Both estimations are in complete agreement. 4 https://github.com/RuthAngus/stardate The rotation period of Gl 205 is higher than that of stars of similar mass in the 4 Gyr-old open cluster M67 sequence (Dungee et al. 2022). Calibrating the gyrochronology relationship with this sequence results in an age determination of 5.2 ± 0.7 (Fouqué et al. 2023), within the error bars of our independent estimate.
In other to know to which galactic population (thin disk, thick disk, or halo) Gl 205 belongs, we followed Reddy et al. (2006) to obtain the probabilities of belonging to the three populations, based on the galactic velocities of the star. For Gl 205 the galactic velocities are (U, V, W) = (49, 13, 31) km s −1 , computed using the position, proper motion, and parallax from GAIA DR3 (Gaia Collaboration 2020). We found a 99% probability that Gl 205 belongs to the thin disk of the Milky Way.
The ages of the stars in the thin disk have a wide range. However, it has been stated that most of them have ages less than 5 Gyr, reaching up to 14 Gyr (Reddy et al. 2006;Haywood 2008;Holmberg et al. 2009). Our estimation of the age of Gl 205 is in agreement with these results. Regarding the metallicity, Allende Prieto et al. (2004) found that the mean metallicity of the stars in the thin disk is <[Fe/H]>=−0.09 ± 0.19. We estimated a highmetallicity for Gl 205 of 0.21 ± 0.06 which is located within 2σ of the mean value.

Comparison with the literature
We compared our results with previous studies of Gl 205. Schweitzer et al. (2019) derived the photospheric parameters T eff , log g, and [Fe/H] from CARMENES VIS spectra and the stellar radius using the Stefan-Boltzmann's law. Using the log g and the stellar radius, the authors derived the stellar mass. Their results of stellar mass and radius are listed in Table 1. They found T eff = 3891 ± 51 K, log g = 4.64 ± 0.07 dex, and [Fe/H] = 0.23 ± 0.16 dex. Our results of [Fe/H] and T eff computed from SOPHIE spectra and the log g from SPIRou spectra, are in particular good agreement within 1σ with the values of CARMENES. The T eff derived using SPIRou spectra is within 3σ to the CARMENES result. Neves et al. (2014) obtained [Fe/H] and T eff of Gl 205 from HARPS spectra with values of [Fe/H] = 0.19 ± 0.09 and T eff = 3670 ± 110 K. Our estimation of the metallicity from SOPHIE spectra is in good agreement with the one from HARPS and our T eff value lies within 3σ. However, it is in better agreement with our SPIRou T eff .
Using spectra of Gl 205 from the APOGEE survey, Souto et al. (2022) determined T eff = 3820 ± 110 K, log g = 4.67 ± 0.2 dex, and [Fe/H] = 0.29 ± 0.10 dex. Their result of T eff is in particular good agreement with our value derived from SPIRou spectra. Maldonado et al. (2015) used HARPS and HARPS-N spectra to derive stellar parameters of early-M dwarfs, finding a metallicity of Gl 205 of [Fe/H] = −0.03 ± 0.19 dex. Our estimation of [Fe/H] is within 2σ to their value.

Light curve analysis
In this section we describe the analysis of the TESS light curves of Gl 205 to identify stellar flares and search for exoplanet transits. Since each TESS sector has a duration of ∼ 27 d, the rotation period of Gl 205 (see Table 1) is not covered in one single sector and thus, the determination of the rotation period from the TESS data is not possible. Moreover, there is a time gap of 684 d between the sector 6 and sector 32. Notes. Time of the flare's peak t peak , equivalent duration of the flare event ED, Gaussian rise of the flare and its exponential decay.

Flares identification
For the identification of stellar flares we used the Python package stella 5 (Feinstein et al. 2020a). This open-source code uses convolutional neural networks (CNN) to identify flares events in the 2-minute cadence TESS light curves and delivers the probability of such event for a given time. The reported flare probability is the average prediction of the 10 models available in stella. After identification, stella uses an empirical flare model (Walkowicz et al. 2011;Davenport et al. 2014) to obtain the best-fit parameters through a χ 2 -fit. The flare model includes a sharp Gaussian rise and an exponential decay (Feinstein et al. 2020b). Candidates events with probability higher than 50% of being a flare were considered for modeling. In the modeling process, the part of the light curves that includes the flare is detrended to account for stellar variability.
We applied the algorithm in the available TESS sectors of Gl 205 using the available trained CNNs from Feinstein et al. (2020b). We identified two flares events during sector 6 and none during sector 32 (see Figure 2). We report in Table 2 the time of the flare's peak, its amplitude, the equivalent duration (ED) which is measured as the area of the flare event, the rise and fall parameters from the flares' model, and the probability.
The second flare identified seems more prominent than the first, reaching higher flux amplitude. However, both events have the same equivalent duration of about 10 hours meaning that the energy of these events are similar but with different time-scales. While the first flare lasted 2.5 hours, the second flare lasted 0.5 hours.
Günther et al. (2020) performed a study of stellar flares in the first data release of TESS. They found that mid to late M dwarfs show the highest fraction of flaring stars. However, the authors warned that this may be influenced by the TESS target selection. Only 10% of the early M dwarfs in their sample are flaring stars. Moreover, fast rotators (P<5 d) may have higher flares rates than slow rotators. Our findings in Gl 205 agree with the results of Günther et al. (2020) since we observed a low rate of flares for this star which is expected for early and slow-rotators M dwarfs.

Planet transit search
The TESS data validation report of Gl 205 identifies a planet candidate with an orbital period of 22.2 d. However, the transit event candidate proposed by the automatic pipeline is clearly affected by the flare of 2458471.056 BJD. We conduct our own analysis in order to identify possible transits in the light curves.
First, we removed outliers using a 3σ-clipping procedure and excluded the data affected by the flares. Since the light curves are affected by stellar variability we used the Python package wōtan 6  to remove the trends. This code in- cludes several methods to perform light curve detrending. We applied a method based on a time-windowed sliding filter with an iterative robust location estimator following the results of . For the detrending model, we excluded the edges of the light curves since these zones are usually affected by strong systematics. Figure 3 shows the detrending model of the light curves and the residuals which have a standard deviation of 210 ppm.
After detrending the light curves, we used the transit least squares 7 algorithm  to search for periodic transit events. The algorithm searches for transit features in an automatically built grid of orbital periods and transit durations. The grid of periods depends on the time-span of the data and the transit durations come from an empirical relation detailed in .
We first applied the algorithm in the whole data set, including sector 6 and sector 32. The grid of periods was automatically set between 0.6 to 365 d. The maximum signal detection efficiency (SDE) is found at an orbital period of 11.86 d but it has low significance (SDE = 8) and the periodogram is highly degenerate. The SNR of this stacked transit signal is 2.5 with only three events in the data set, one in sector 6 and two in sector 32. To confirm or rule-out this period we applied the algorithm in sector 32, independently, since two transits were identified in this sector. No transits were found in this sector thus, we discarded transit events in the two available TESS sectors of Gl 205.

Magnetic field analysis
The SPIRou spectropolarimetric products, in particular the Stokes I and V profiles were obtained using the Libre-ESpRIT pipeline described in . We applied leastsquares deconvolution (LSD;  to compute the average Stokes I (unpolarized) and Stokes V (circularlypolarized) line profiles for all our SPIRou observations. We used a mask of atomic lines, spanning SPIRou spectral domain, computed from a ATLAS9 local thermodynamical equilibrium model of atmosphere (Kurucz 1993), assuming an effective temperature of 3750 K and a surface gravity of log g = 5.0. Note that we only selected lines with a relative absorption larger than 3% (from the unpolarized continuum) to avoid an over-representation of weak lines in our final line list. Lines affected by strong tellurics, i.e. of relative absorption deeper than 20% within ±30 km s −1 from the line center, are masked out in the LSD process. The extracted Stokes I and V profiles feature a mean central wavelength of 1700 nm, an effective Landé factor of 1.25 and a relative depth of 12% with respect to the continuum.

Stellar rotation period
The disk-integrated longitudinal magnetic field B was computed using the Stokes V and Stokes I profiles, following the method of : where I and V are the unpolarized and circularly-polarized LSD profiles, I c is the continuum level, u is the velocity in km s −1 , λ 0 is the mean wavelength, g eff is the effective Landé factor and c the speed of light in kms −1 .
The B time series of Gl 205 is listed in Table A.2 of the Appendix A. The mean values of the B time series is 1.6 G, the standard deviation is 2.9 G and the mean of the error bars is 1.0 G. It is expected that the longitudinal magnetic field is modulated by the rotational period of the star, since the largescale magnetic topology at the surface of the star is expected to evolve on a longer time scale. To measure the stellar rotation period, we first computed the generalized Lomb-Scargle (GLS) periodogram (Lomb 1976;Scargle 1982;Zechmeister & Kürster 2009) implemented in the astropy (Astropy Collaboration et al. 2013, 2018) Python package. The GLS periodogram of the B time series shows a strong peak of periodicity at 32.7 d with false alarm probability (FAP) below 1% (see Figure 10). However, it has been shown that stellar activity follows a quasi-periodic behavior (e.g., Haywood et al. 2014;Angus et al. 2018) rather than a single sinusoidal signal, as is the case of the periodicity searched in a GLS periodogram. Moreover, the magnetic field of Gl 205 is known to evolve on a timescale of a few rotation cycles (Hébrard et al. 2016). Thus, we take advantage of the flexibility of the GPs to constrain the rotational period of the star using the B time series. For this purpose, it is important to highlight that the three well defined observational seasons of the B (see Figure 4) are long enough to cover a few stellar rotation cycles.
The kernel of the GP regression is set to be the quasi-periodic as defined in Roberts et al. (2012): where x i and x j are two observation dates, A is the amplitude of the covariance, l is the decay time, β is the smoothing factor, P rot is the stellar rotation period, and σ is the uncorrelated white noise also known as the jitter term. The posterior distributions of the hyper-parameters were sampled from a Markov-chain Monte Carlo (MCMC) routine using the package emcee (Foreman-Mackey et al. 2013). We set up 50 walkers and 5000 steps after a burn-in phase of 500 steps. The priors used and the final posterior distributions are listed in Table 3 and in Figure B.1 is displayed the corner plot of the posterior distributions. The best-fitting model of the GP regression is illustrated in Figure 4 which has a reduced χ 2 of 0.92. As a sanity check, we plot the GLS periodogram of the GP model residuals in Figure 5 where we see that there is no periodic signal left.
We measured a stellar rotation period of P rot = 34.4 ± 0.5 d and a decay time of l = 62 +15 −12 d, which implies that the active features could evolve relatively fast, on a time scale of about two rotation cycles, as the GP decay time has been proved as a good indicator of the average time scale evolution of the active features (Nicholson & Aigrain 2022). With this new estimation of the rotation period, we can derive the rotational velocity v sin i. Assuming a stellar radius of 0.556 ± 0.033M and inclination of 60 • ± 10 • (see Table 1), we obtained a v sin i of 0.7 ± 0.1 km s −1 . (Hébrard et al. 2016) obtained a v sin i of 1.0 ± 0.5 km s −1 from their ZDI analysis, which is in agreement with our results.

Zeeman-Doppler Imaging
We use Zeeman-Doppler imaging (Semel 1989;Brown et al. 1991;Donati & Brown 1997) to invert the observed Stokes V LSD profiles into distributions of the large-scale magnetic field at the surface of Gl 205. As described in Donati et al. (2006), ZDI decomposes the large-scale field vector into its poloidal and toroidal components, both expressed as weighted sums of spherical harmonics. For a given field distribution, local Stokes V profiles are computed for each resolution element of the visible hemisphere of the star using analytical expressions from the Notes. The symbol U(a, b) defines an uniform prior with a and b the minimum and maximum limits, respectively.
Unno-Rachkovski's solution to the radiative transfer equation in a plane-parallel Milne-Eddington atmosphere (Unno 1956). These profiles are then (i) shifted to the local projected rotational velocity, (ii) weighted according to stellar inclination and limbdarkening law (assumed linear with a coefficient of 0.3 Claret & Bloemen 2011), and (iii) combined into global profiles at the times of the observations. ZDI uses a conjugate gradient algorithm to iteratively compare the synthetic profiles to the observed Stokes V LSD profiles down to a given reduced χ 2 . The degeneracy between multiple magnetic maps is lifted by imposing a maximum-entropy regularization condition to the fit, assuming that the map with the minimum amount of information is the most reliable (Skilling & Bryan 1984) The intrinsic evolution of the magnetic field is not yet included in our ZDI code, despite notable progress over the last few years (Yu et al. 2019;Finociety & Donati 2022). To prevent the code from focusing on the intrinsic evolution of the field rather than on its rotational modulation, we divide our observations into four subsets of data, namely S 1 , S 2 , S 3 , S 4 , containing respectively 36, 31, 19 and 31 observations. The time windows spanned by these four data sets are indicated in Table 4 and depicted in Figure 4. We then apply ZDI independently to each of these data sets. Note that these four seasons were defined so that the longitudinal field remains roughly periodic on time scales larger than a rotation period, ensuring that the magnetic topology does not dramatically evolve during each season and that the variation of the Stokes V profiles reflects primarily the stellar modulation rather than the intrinsic evolution of the field.
Our ZDI reconstruction includes the modelling of latitudinal differential rotation (DR) shearing the large-scale field at the surface of Gl 205 (Donati et al. 2000;Petit et al. 2002). The stellar rotation rate Ω is assumed to vary as a function of the colatitude θ such that where Ω eq and dΩ stand respectively for the rotation rate at the stellar equator and the difference in rotation rate between the equator and the pole. The best DR parameters explaining the observations are estimated using the method described in Donati et al. (2000). For a wide range of DR parameters, we apply ZDI to a fixed level of entropy. The best parameters with error bars are estimated by fitting a 2D paraboloid to the resulting χ 2 distribution in the Ω eq , dΩ space. The best-fitting maps of the large-scale field vector for the four subsets of data are shown in Figure 6, and the associated magnetic properties are listed in Table 4. Note that the best-fits to the observed Stokes V LSD profiles are shown in Appendix C for the four seasons. Data sets S 1 to S 4 have been fitted to reduced χ 2 of 1.0, 1.08, 1.1 and 1.05, starting from reduced χ 2 of 2.4, 1.5, 4.1 and 1.4. As expected from the time series of longitudinal magnetic field shown in Figure 4, we observe a sig- nificant fluctuation in the field topology throughout our observations. In Seasons S 1 and S 3 , the field is found to be a dipole of 13-15 G (consistent with the topology found in Hébrard et al. 2016), respectively tilted at 36 • and 56 • to the rotation axis towards phases 0.78 and 0.84. A weaker and more complex field is found in Season S 2 . Though dominantly poloidal, the magnetic energy budget is much more spread into the dipolar (45%), quadripolar (20%) and octupolar (31%) than in Seasons S 1 and S 3 (both dipolar at more than 90%). Between Seasons S 3 and S 4 , the field quickly evolves, on a 2-month time scale, from a poloidal dipole to a dominantly toroidal topology 8 . Notes. Total time span of each subset of data ∆T , average large-scale magnetic strength < B >, fraction of poloidal field f pol , fraction of axisymmetric field f axi , best fitting DR parameters Ω eq and dΩ, equatorial and polar rotation periods, P eq and P pol .
parameters is strongly limited by the relatively low latitudinal precision of our reconstruction, due to the very low v sin i of the star. As a consequence, no conclusion can be drawn on a potential evolution of the DR throughout the observations. We report an inverse-variance weighted average of the DR values from Season S 1 to S 4 of P eq = 32.0 ± 1.2 d for the rotation period at the stellar equator, and P pol = 45.5 ± 0.3 d for the rotation period at the poles.

Periodogram analysis
The almost simultaneous acquisition of SOPHIE and SPIRou RVs allows us to quantify the stellar activity jitter in the optical and near-infrared domains. We computed the GLS periodogram to look for periodicities in the RVs. The whole RVs time series are shown in Figure 1 along with its GLS periodogram. We identified a highly significant (FAP << 0.1%) period at 34.3 d and a second peak with also low FAP at 37.9 d. There is also a third peak with low FAP at ∼ 0.5 d, which is an alias of the 1-d periodicity due to the daily sampling. The SOPHIE and SPIRou data sets do not show the same periodicities. In the SOPHIE RVs, there is no significant periodicity found by the GLS periodogram, however, the strongest peak occurs close to the P rot /2 harmonic at ∼ 17 d (see the topright panel of Figure 9). The SPIRou RVs on the contrary, show a clear peak of periodicity at 34.4 d (see the top-left panel of Figure 10) which is the expected stellar rotation period. In terms of the scatter of the RVs, the data sets have a root-mean-square (RMS) of 4.7 m s −1 for SPIRou and 4.8 m s −1 for SOPHIE.

Global modeling
In order to obtain a first approximation of the RV jitter magnitude, we applied a GP with a quasi-periodic kernel in the SO-PHIE and SPIRou RV time series, following Section 5.1. The MCMC procedure to obtain the posterior distribution is set up with 50 walkers and 5000 steps after a burn-in phase of 500 steps. The priors applied in the hyper-parameters are listed in Table 5. Since we measured the stellar rotation period from the spectropolarimetric data, we applied a more constraint prior to the GP period of the radial velocities.
The best-fit model of the GP regressions are shown in Figure  8 and the final posterior distribution of the GP hyper-parameters are listed in Table 5. The period found in the SOPHIE and SPIRou RVs of 34.2 +2.8 −0.9 d and 39.2 ± 2.4 d, respectively, are consistent at a 3σ level with the period of the longitudinal magnetic field of 34.4±0.5 d (see Section 5.1. The decay time found for the SOPHIE RVs is larger than for SPIRou and consistent with more than two stellar rotations (l = 96 +41 −47 d). In the case of SPIRou, however, the decay time of 38 +4 −2 d is comparable with the rotation period of 39.2 ± 2.4 d. This effect is problematic for the quasi-periodicity of the signal. When there is a short evolution time scale, l P rot , the periodicity of the signal is meaningless (see e.g., Rajpaul et al. 2015;Barragán et al. 2022a).
We tested applying a different prior in the decay time for the SPIRou RVs, keeping it uniform but with limits between 70 to 150 d, to ensure at least two rotation cycles. As results, we obtained consistent posterior distributions, within error bars, with the values from Table 5, except for the decay time. With a different prior, we obtained a decay time of 73 +5 −2 d. However, the log-likelihood of the first model (log-L = -374) is slightly higher than the model with a new prior in the decay time (log-L = -384), meaning that the first model fits better the data. Notes. The symbol U(a, b) defines an uniform prior with a and b the minimum and maximum limits, respectively.

Seasonal analysis
The RVs observation time-span is clearly divided in three seasons (see Figure 1): the first epoch from 58738 BJD to 58922 BJD (September 2019 -March 2020), the second epoch from 59088 BJD to 59297 BJD (August 2020 -March 2021), and the third epoch from 59440 BJD to 59607 BJD (August 2021 -January 2022). These subsets are longer than the subsets defined in Section 5.2 since in order to look for periodic variability it is required a longer time-span than for the ZDI analysis. The details of the RV subsets such as the start and ending dates as well as the number of data points included are listed in Table 6. Note that these RV subset are defined as S RV 1 , S RV 2 , and S RV 3 in order to differentiate them from the seasons defined in Section 5.2. These RV subsets are depicted in Figure 1.
Even though the seasons S RV 1 , S RV 2 overlaps at some extent with S 1 and S 2 , we can not directly compare S 3 with the RV data since only three SOPHIE observations are within this sea-  Fig. 7: Distribution of reduced χ 2 as a function of Ω eq and dΩ extracted from the Stokes V LSD profiles in Season S 1 to S 4 , defined in Table 4. In each map the 1 and 3 σ contours are indicated by the black solid lines.
son. Therefore we defined a fourth season of RVs observations S RV 4 , from 59500 BJD to 59607 BJD (October 2021 -January 2022) that overlaps with S 4 and it will allow us to measure the differences between seasons S 3 and S 4 of the B in the RVs data sets.
To have some insight about the variability of the RV signal, we define the amplitude A as the peak-to-valley difference and Notes. Number of data points N, the amplitude of the RV variation is defined as the peak-to-valley difference of the GP model. The goodness of the fit is described with the RMS of the residuals alongside the reduced χ 2 .
we listed it alongside the RMS of the model residuals and the reduced χ 2 in Table 6, for all the data set and each RV season. It is clearly seen in Figure 8 that the best-fitting GP model for the SOPHIE RVs is more stable in time than the one for the SPIRou RVs, and it is similar to a double sinusoidal model at P rot and its first harmonic. As seen in the middle and bottom panels of Figure 8, the signal in the SOPHIE RVs remains more or less consistent between S RV 1 and S RV 2 , with similar shapes and amplitudes of 10.8 ± 2.9 and 9.8 ± 3.0 m s −1 , respectively. The goodness of the model is similar for the season S RV 1 through S RV 4 , with RMS between 2.3 and 2.8 m s −1 . However, during S RV 3 and S RV 4 the reduced χ 2 gets slightly worse than the other seasons. This may be due to the presence of outliers in the sub-data set.
On the contrary, the SPIRou RV signal is not consistent between one season to the other (see bottom panels of Figure 8), exhibiting high variability. The amplitude of the RV signal goes slightly up between S RV 1 and S RV 4 , from 13.7 ± 2.3 to 15.2 ± 2.2 m s −1 , but it not significant and the difference is within the error bars. As in the SOPHIE RVs, the there no important differences in the RMS or reduced χ 2 of each season, however, the goodness of the model gets worse for S RV 3 . As for the SOPHIE RVs, this may be due to the presence of outliers. If these outliers are removed, the RMS improves to 2.4 m s −1 .
At this point, we can not tell if there are significant differences between S RV 3 and S RV 4 of the SOPHIE and SPIRou RVs, in order to measure a possible impact in the RVs due to the changes in the magnetic field topology (see Section 5.2). As listed in Table 5, there is little discrepancy between these two seasons in the amplitudes, RMS, and reduced χ 2 . Despite the fact that the B shows high variability at this epochs, the radial velocities do not increase their scatter. This may mean that the variability in the longitudinal magnetic field is not directly expressed in high RV dispersion.

Optical activity indicators
Using the SOPHIE data we computed standard activity indicators that quantify distortions in the CCF profile, such as the BIS, FWHM, and contrast. The bisector inverse slope (BIS) ) is defined as the velocity span between the top (55% Fig. 8: Time series of the SOPHIE and SPIRou radial velocities. In blue and red colors are the best fitting GP models using a quasi-periodic kernel in the SOPHIE and SPIRou data, respectively. The light-blue and light-red colors depict the 3σ level of the uncertainties. The residuals of this model have an scatter of 2.6 m s −1 for SOPHIE, and 2.4 m s −1 for SPIRou and they are shown below the time series (top). Zoom in of the SOPHIE and SPIRou radial velocities divided in the seasons defined in Table 6 with their best fitting GP models (middle and bottom).
< CCF depth < 80%) and the bottom (20% < CCF depth < 40%) of the CCF. The FWHM and the contrast are direct measurements of the width and the depth of the CCF, respectively.
The time series of the BIS, FWHM, and contrast of the CCF from the SOPHIE data are shown in Figure 9, with their corresponding GLS periodograms. No significant periodicity close to the expected stellar rotation period of P rot = 34.4 ± 0.5 d is found in their periodograms.
Within the SOPHIE spectral domain is located the wellknown Hα line (Kürster et al. 2003;Bonfils et al. 2007;Boisse et al. 2009), and CaII H&K lines that are measured to compute the Mt. Wilson S index (Wilson 1968;Baliunas et al. 1995). The so-called S index is a measure of the emission at the core of the CaII H&K lines, located at 3922Å and 3968Å and for the Hα is the flux at 6562Å. To compute these indices we follow Boisse et al. (2009), where it is defined as the ratio between the flux in a defined range of continuum and the core of the absorption lines.
In Figure 9 are shown the time series of the Hα and S indices of our SOPHIE data along with their GLS periodograms. The time series of Hα shows a peak of periodicity at 33.7 d and in the case of the S index, long-term trends dominate the periodogram.
Usually the correlation or anti-correlation between the RV and the activity indicators is a hint of the stellar origin of the signal (e.g., Queloz et al. 2001;Boisse et al. 2011). However, the lack of correlation can be due to phase shifts between the RVs and the activity indicators (e.g., Bonfils et al. 2007;Dumusque et al. 2014b). For example, Collier Cameron et al. (2019) measured a temporal lag of 1 and 3 days between the maxima in the RVs and the maxima of the FWHM and BIS, respectively. In the present work we do not explore the possibility of the phase lags in our data set but we rather warn the reader about it.
The SOPHIE BIS, FWHM, and contrast are not correlated or anti-correlated with the RVS, and the S index and Hα are correlated with the RVs with Pearson's coefficients of 0.4. We computed the p-value associated to this correlation in order to prove the significance. The p-value is defined as the probability that the observed correlation is a false positive, under a true nullhypothesis. In this case, the null-hypothesis is that there is no correlation between the variables. The correlation between the RVs and the Hα and the S index has a p-value of 8 · 10 −4 and 9 · 10 −4 , respectively. This means that the found correlations are statistically significant.
We applied a GP regression following the same procedure as in Section 5.1 to study the periodicity of the signal, since we have seen how the longitudinal magnetic field and the radial velocities follow a quasi-periodic trend. This behavior could have an effect in the classical GLS periodogram and hide the real period of the signal. Since the SOPHIE DRS do not have implemented yet the error bars derivation for the FWHM and the contrast, we assumed equal uncertainties for all the data points of 0.1 %CCF for the contrast, and 0.01 km s −1 for the FWHM.
We used the same priors as in Section 6 which are listed in Table 5. To obtain the posterior distribution from the MCMC we 50 walkers and 5000 steps after a burn-in phase of 500 steps. The best-fitted GP model for the SOPHIE activity indicators are shown in Figure E In the case of the SOPHIE activity indicators, the quasiperiodic GP fits a model with a periodicity in agreement with the expected rotation period of P rot = 34.4±0.5 d . This is in particular interesting for the BIS, FWHM, and CCF contrast since their GLS periodogram do not exhibit significant peaks of periodicity at the rotation period. The decay time of the GP model of the FWHM, BIS, and Hα are consistent with an average time scale decay of at least two rotation cycle, as also seen in the longitudinal magnetic field B .

Near-infrared activity indicators
Within the SPIRou's LBL framework, we can obtain the differential line width (dLW, Zechmeister et al. 2018) and the chromatic velocity slope. The differential line width corresponds to the second derivative of the spectral profile and can be expressed in units of FWHM. We follow the definition by Artigau et al. (2022) using this quantity as FWHM LBL . The chromatic velocity slope is defined as the RV gradient as a function of wavelength (Zechmeister et al. 2018).
The time series and the GLS periodograms of FWHM LBL and the chromatic velocity slope are shown in Figure 10, along with their phase-folded and correlation with RVs plots. When phase-folding FWHM LBL by the stellar rotation period of 34.4 d, it seems to be modulated in time. Moreover, FWHM LBL is slightly correlated with the RVs with a Pearson's coefficient of 0.3 and a p-value of 4 · 10 −5 , which means that the found correlation is significant. The highest peak of the FWHM LBL GLS periodogram is at 34.4 d but with FAP slight above 1%, which corresponds to the stellar rotation period. The chromatic velocity slope indicator is not correlated or anti-correlated with the RVs, however the most significant peak of periodicity in its periodogram is located at 31.3 d, close to the rotation period.
As for the SOPHIE activity indicators, we also computed a GP model in the SPIRou activity indicators including the FWHM, chromatic velocity slope, and pEW of the near-infrared spectral lines. The best-fit GP model for the FWHM and the chromatic velocity slope are shown in Figure E Similar to the activity indicators in the optical, the measured decay time of the signal are consistent with two stellar rotation cycles. The results of the near-infrared spectral lines are discussed in the next subsection.

Diagnostics on near-infrared spectral lines
Observations in the infrared domain introduce two challenges. On the one hand, we lack well-identified and characterized spectral lines in this domain that can be used as stellar activity tracers. On the other hand, M dwarf spectra do not always show a clear flux continuum within the whole near-infrared wavelength range, since their spectra have a large number of absorption lines and hamper the proper measurement of equivalent widths.
We selected near-infrared absorption lines of chemical species that have been proved as activity indicators in optical or near-infrared wavelengths for Sun-like stars or M dwarfs, to test their potential for near-infrared M dwarfs data. These absorption lines are the titanium (Ti I) line at 10499Å (Spina et al. 2020 Table 7. The equivalent width (EW) of spectral lines can be used as a proxy for the strength of the chromospheric activity and it is defined as following Gray (2008): where F λ is the flux of the spectral line and F 0 is the continuum flux surrounding the spectral line. As the continuum in the M dwarfs spectra is unknown, we used a pseudo-continuum as in Schöfer et al. (2019), and therefore we measured the pseudo-equivalent width (pEW). To this, we first stacked the four consecutive exposures of each SPIRou observation to obtain one single high signal-to-noise spectra per epoch. Then, we used a Python re-implementation of the continuum fitting routine from IRAF 9 to obtain the pseudo-continuum. This pseudo-continuum is defined as the local continuum of a spectral zone of 2 to 5 Å where the spectral line is at the center. The details of the wavelength range that we used to compute the pseudo-continuum are listed in Table 7.
To each spectrum per epoch, we fitted a Voigt profile in the absorption lines within an MCMC framework using the emcee . We used the posterior distributions of the fitted parameters to compute the uncertainties of the pEW using the bootstrap method. To look for periodicities in the time series of the tested spectral lines, we computed the GLS periodogram. We then considered as significant a peak with its FAP below 1%. In Table 7 are shown the highest peaks of periodicities in the GLS periodograms for each 9 Image Reduction and Analysis Facility, https:// iraf-community.github.io Article number, page 13 of 41 A&A proofs: manuscript no. aanda line within a range of 1 d to 100 d, to exclude periodicities coming from the 1-d alias and long-term variability. In Figure 10 are displayed the pEW time series of the Al I, Ti I, K I, Fe I, and He I lines, along with their GLS periodograms, phase-folded plot, and correlation plot with RVs. The highest peak of periodicity of each line is listed in Table 7, for all the data and for each of the RV seasons. Only Al I exhibits peaks of periodicities with a FAP below 1% at 35 d , close to the expected stellar rotation of 34.4 d. Moreover, Al I shows several peaks between 30 to 40 d that may be related to the differential rotation. The lines best correlated with the RVs are Al I and Fe I with a Pearson's coefficient of 0.3 and 0.2, respectively. However, the correlation between the RVs and Al I is more significant with a p-value of 4 · 10 −4 versus the p-value of 0.01 in the correlation with the Fe I.
Excluding Al I, the rest of the spectral lines tested Fe I, Ti I, K I, and He I do not seem to be modulated by the stellar rotation and show small or none correlation with the RVs, at least from the analysis of their periodograms. The Fe I line in particular, shows a peak of periodicity at 44.7 d which is similar to the expected stellar rotation at the poles (P pol = 45.5 d). This line could be tracing active surface features at latitudes close to the stellar pole.
The best-fitting GP models of the tested spectral lines are shown in Figure E.2 and the final posterior distribution of the hyper-parameters are listed in Table E.1. Most of the spectral lines have periodicities longer than the measured rotation period of 34.4 ± 0.5 , except for the He I. However, as we have seen in previous results within this work, it seems that the He I line is not sensitive to stellar activity. Moreover, it is clear that the GP do not fully modeled the variability in the test spectral lines, and we observed high values of the reduced χ 2 . The underestimation of the uncertainties of the pEW computation could play a role in this, nevertheless, we will investigate it further in future works.
Thanks to the analysis of the GLS periodograms and the GP regression, we can see that the periodicity of the Al I and Fe I lines are consistent between these two techniques, however, the decay time is shorter than expected and do not covers two complete cycles.  Figure 9 for SPIRou data.

Seasonal analysis
Since stellar activity has a quasi-periodic behavior, the activity signals should be consistent in short-time scales and therefore, their periodicities could be measured with the periodograms in a seasonal analysis. We observed that the strongest peaks of the GLS periodograms of the optical activity indicators are not consistent through the three seasons of data previously defined as S RV 1 , S RV 2 , and S RV 3 . The GLS periodograms for each one of the three subsets are displayed in Figure D  Notes. Wavelength values are in vacuum. In bold are the peaks of the GLS periodograms with FAP<1%. We searched for periodicities related to the stellar rotation period in a range between 1 d and 100 d and the BIS, the periodicity close to the rotation period gains significance towards the second subset but remains with a high FAP. On the other hand, the peak at the stellar rotation period decreases in significance towards the 2021/2022 observations. During S RV 1 , the Hα time series shows an important power excess at a period slightly longer than the expected rotation period. During S RV 2 and S RV 3 , Hα keeps some power excess around the rotation period but is not significant.
The SPIRou activity indicators, the FWHM LBL , and the chromatic velocity slope, do not show the same season behavior. As shown in Figure D.2, the FWHM LBL have a peak of periodicity not related to the stellar rotation in S RV 1 , during S RV 2 and S RV 3 we can see some power excess close to the rotation period but with FAP above 10%. On the other hand, the chromatic velocity slope has a high-significant peak of periodicity slightly shorter than the rotation period in S RV 3 , and nothing during S RV 1 or S RV 3 . The periodicity seen in S RV 3 may be affected by the window function of that particular season (see Figure D.2).
The tested near-infrared spectral lines also do not keep consistent periodicities across the three subsets of RVs observations S RV 1 , S RV 2 , and S RV 3 . The strongest peaks of the GLS periodograms of Al I, Ti I, K I, and Fe I change from season to season and they are close to the rotation period at least during one season (see Table 7. However, this periodicity observed during at least one season is ∼ 39 d instead of the expected P rot . This may be related to active features in the stellar surface at mid-latitudes where the rotation should the larger than at the equator. The Fe I line in particular, even though when considering the whole data set its periodicity is larger than P rot , the periodicities of each RV season is between 30 to 40 d which could be coming from the stellar rotation modulation. This variability in the periodicities observed for the spectral lines may be a hint of the high temporal variability in the stellar activity manifest as highly variable active features. Although we can see some hints of seasonal behaviors in the optical and near-infrared activity indicators in this work, most of their periodicities are not significant and therefore, we can not describe trends of the stellar activity during each season. For example, Hα is clearly modulated by the stellar rotation during S RV 1 which could be due to high chromospheric activity levels. However, we do not see this behavior in other activity indicators. Rajpaul et al. (2015) describes a multi-dimensional GP framework on which the stellar activity in the RVs time series can be modeled simultaneously with the Keplerian signal using the information from activity indicators. In summary, this framework assumes that the stellar activity signals and the RVs can be modeled by the same latent function, namely G(t), and its time derivative,Ġ(t). G(t) is related to the area of the visible stellar disc covered by active regions andĠ(t) describes the evolution in time of these active regions. Therefore, the RV variations induced by activity can be expressed as a linear combination of G(t) andĠ(t). Physically, this linear combination will account for the convective blueshift suppression and the evolution of spots on the stellar surface.

Stellar activity modeling
In principle, the ancillary time series, such as the activity indicators, should be also defined as linear combinations of G(t) andĠ(t) if they are affected by both processes: the covered area by active regions and its time evolution in the stellar surface (e.g., BIS). However, some activity indicators, such as logR HK and S HK , can be described only by G(t) as they do not account for the time evolution of spots (see Rajpaul et al. 2015 and Barragán et al. 2022a for more details).
We used the Python code pyaneti 10 (Barragán et al. 2019; Barragán et al. 2022a) to apply the multi-dimensional GP framework in our SOPHIE and SPIRou data. As the code was built for planet RV and transit modeling and there is no planet signal detected in this system, we set up a planet signal with an amplitude of K = 0 m/s then only the stellar activity signal is considered in the modeling. The kernel of the GP regression is set to be the same as in Section 5, Equation 2. However, the amplitude A in Equation 2 in the multi-dimensional framework is replaced by the amplitudes C i as the following: . . .
where Θ 1,...,N are the activity indicators. In this way it is guaranteed that the hyper-parameters of the kernel will be shared except for the amplitudes C i . We follow the procedure as in Section 5 to obtain the posterior distributions. We modelled the stellar activity of the SOPHIE RVs using as ancillary time series the Hα index and the BIS as the following: We chose the Hα instead of the S index because the former shows a peak of periodicity at the stellar rotation period and is clearly correlated with the RVs (see Figure 9). The details about the computation of the activity indicators are in Section 7.
The model obtained from the multi-dimensional GP regression is shown in Figure 11 for the RVs, the Hα, and the BIS time series. The priors and posteriors are listed in Table 8 and in Figure F.1 is the corner plot of the posterior distributions. The RV jitter is 2.7 ± 0.6 m/s and the rotation period is determined at 35.4 ± 1.0 d. The residuals of the RVs have a scatter of 2.9 m/s and do not show any significant periodicity in the GLS periodogram.
In the case of the SPIRou data set, we used as ancillary time series the FWHM LBL as it is the activity indicator that is best correlated to the RVs and shows a significant peak at the stellar rotation period (see Figure 10). The model is the following: We only used FWHM LBL because there is no analogous to the BIS or any other activity indicator that could trace the temporal evolution of the active regions in the SPIRou data set. The model obtained for the RVs and the FWHM LBL is shown in We tested the B as ancillary time series together with the FWHM LBL , since it has been shown how it traces the magnetic activity, we could filter more activity signal in the RVs. The stellar rotation period obtained from this test is in better agreement with the value obtained in Section 5 than the one from only using FWHM LBL as ancillary time series, with a value of P rot = 34.6±0.8 d. However, the scatter of the residuals is higher with a value of 3.7 m s −1 .
In Figure 11 we see the complexity in the structures describing the SOPHIE RVs which is due to the high harmonic complexity of the signal. We can probe this with the smoothing factor β value from the GP model. The smoothing factor β is higher for the SOPHIE RVs (β = 0.81 +0.29 −0.20 ) than for SPIRou (β = 0.62 +0.31 −0.18 ). The high harmonic complexity seen in the optical can be empirically proven with the amplitudes of G(t) andĠ(t) (Barragán et al. 2022b). The functionĠ(t) describes the area covered by active regions on the stellar surface as a function of time (Aigrain et al. 2012;Rajpaul et al. 2015). We see that the amplitude of G(t), C 0 = 3.6 +1.6 −1.0 is smaller than the amplitude ofĠ(t), C 1 = 24.2 +12.3 −8.9 , which explains the observed complexity. However, we see a different result in the near-infrared. The amplitudes of G(t) andĠ(t) are similar with values of C 0 = 3.8 +1.5 −0.8 and C 1 = 3.2 +4.7 −2.3 . This could be an indication that the functionĠ(t) is not fully describing the origin of the RV variations and might be evidence of a different process dominating the stellar activity in the near-infrared, such as the Zeeman effect.

Detection of RV planet's signal
Up to date, no planet has been confirmed orbiting Gl 205. From our RV analysis of Section 6, we do not observe periodicities that could be attributable to Keplerian signals. Considering the total time span of our SOPHIE and SPIRou observations of almost 900 days, we can discard long-period planets with orbital periods longer than 450 days. In particular, the detection of long-period planets would be highly affected due to the two gaps of obser-vations of ∼100 days each (see Figure 1). Further observation campaigns of Gl 205 may help on the discovery of long-period planets around this star.
Each well-defined season of RVs observations in Figure 1 last 184, 209, and 167 days, therefore our highest detection sensitivity is given for planets with periods of less than 100 days and with amplitudes greater than 4.8 m s −1 , which corresponds to the scatter in the RVs. Moreover, we do not observe periodicity peaks in the GLS periodograms of the SOPHIE and SPIRou RVs that are not related to the stellar rotation period.
The observed peaks of periodicity in the RVs GLS periodogram are not consistent during the three seasons of observations and they are always correlated with one or more activity indicators. Furthermore, the procedure to filter the stellar activity described in Section 8 allows us to discard remnant Keplerian periodic signals on the RVs. In both cases, for the SOPHIE and SPIRou data sets, the multi-dimensional GP absorbs most of the activity-induced RV signal leaving an RMS of the residuals of ∼3 m s −1 . The GLS periodogram of these RVs residuals does not exhibit any periodicity left with low FAP that could be attributed to a Keplerian signal.
Furthermore, we tested the coherence and consistency of the periodicities using the Stacked Bayesian generalized Lomb-Scargle (Stacked-BGLS) periodogram from Mortier & Collier Cameron (2017). The main idea behind this algorithm to disentangle signals is that stellar activity will produce short-lived incoherent signals while a Keplerian is a long-lived consistent signal.
The Stacked-BGLS periodogram for the RVs from SOPHIE and SPIRou, together and independently, are shown in Figure 13. We observed that there are no strong signals in the periodograms at periods between 0 to 100 days. However, the activity close to 34 d is accompanied by other signals at close periods and show  Notes. The symbol U(a, b) defines an uniform prior with a and b the minimum and maximum limits, respectively. some level of variable probability, as expected for stellar activity signals.

Discussion and Conclusions
In this work, we present the long-term stellar activity and magnetic field characterization of the early, moderately-active M dwarf Gl 205 using quasi-simultaneous optical and nearinfrared high-resolution spectra from the SOPHIE/OHP and SPIRou/CFHT spectrographs. We used the SPIRou circularly polarized spectra to measure the Zeeman signature produced by the presence of the magnetic field. The longitudinal magnetic field B is modulated by the stellar rotation with a pe-riod of P rot = 34.4 ± 0.5 d , which is in agreement with previous works (Kiraga & Stepien 2007;Bonfils et al. 2013;Hébrard et al. 2016). The analysis of the periodicities found for the B and activity indicators reinforces the efficiency and accuracy of the B to constraint the P rot , over the standard activity indicators for early M dwarfs. We applied the Zeeman Doppler imaging technique to reconstruct the large-scale magnetic field map of the star over three seasons of observations. We observe a temporal evolution of the magnetic field topology while the strength of the large-scale field remains mostly constant. Moreover, we confirmed the differential rotation over the stellar surface of Gl 205 which could explain the disagreement between the stellar rotation period measured by previous studies and from different activity proxies.
We derived the radial velocities using the CCF technique for the SOPHIE data and the line-by-line method for the SPIRou data. Both RVs data sets are comparable in amplitude and scatter with variations due to stellar activity. We applied a quasiperiodic GP regression in the SOPHIE and SPIRou radial velocities. Their periodicities are compatible with the result of the longitudinal magnetic field B .
We filtered the activity-induced RV variations using a multidimensional Gaussian Process regression framework. For the optical RVs, we used the Hα index and the BIS as ancillary time series and, for the near-infrared, the FWHM LBL . The obtained models fit the RVs down to the noise level with 2.7 m s −1 of scattering in the residuals for both instruments. Since we did not detect periodic signals left, we can rule out a priori the presence of Keplerian signatures in the given data set. Moreover, we used TESS photometry to discard the presence of transit events in the two available sectors. Nevertheless, we can not rule out completely the presence of planets around Gl 205 due to the flexibility of the GPs. At certain periods, in particular close to the stellar rotation period, the GP could still absorb the Keplerian signature in the RVs Rajpaul et al. (2015).

Magnetic field temporal evolution
The long-term follow-up of more than two years of data presented in this work allows us to characterize the evolution of the large-scale magnetic field of Gl 205. The topology evolves significantly from a poloidal dipolar field with a strength of <B>=12 G in 2019 to a dominantly toroidal field in 2022 with a similar field strength of <B>=12.3 G. This change in the topology occurred quickly within two months during our last set of SPIRou observations. Previous studies of moderately-active early M dwarfs samples have shown the diversity of magnetic field topologies Hébrard et al. 2016;Martioli et al. 2022) proposing that there is no unique topology pattern for these stars. This diversity is especially seen in partly-convective, slowly rotating M dwarfs. Long-term follow-up campaigns of the early M dwarfs, such as for Gl 410 and Gl 846 Hébrard et al. 2016), revealed a temporal evolution of the topology of the magnetic field. The variability of the magnetic field seen in early M dwarfs is at least partly attributable to their differential rotation over the stellar surface Morin et al. 2008;Hébrard et al. 2016).
Although most of the seasons observed in this study and Hébrard et al. (2016) exhibit poloidal fields for the case of Gl 205, the fact that at least during one of our seasons the field is dominantly toroidal is unexpected, since previous studies in partly-convective stars See et al. 2016) found that only fast rotators could generate toroidal dominated fields.
Our results of the large-scale magnetic field during the four seasons of B (see Table 4) are mostly consistent with the previous magnetic field study by Hébrard et al. 2016, however, in this work we have been able to characterize the long-term evolution of the large-scale field. The long-term SPIRou monitoring was key to revealing the true nature of the magnetic field evolution and its intrinsic variability.

Diagnostics on optical and near-infrared activity indicators
Previous works on photometric and spectroscopic/spectropolarimetric data of Gl 205 have measured similar rotation period values (P rot ∼ 33 d). Nevertheless, the rotational modulated variations are not always clear in all the activity indicators time series. Bonfils et al. (2013) showed that the strongest peaks at P rot of the Hα and S index periodograms are seasonal, meaning that they become significant when the data set is restricted to one observation season of ∼ 300 d. Hébrard et al. (2016) measured the periodicity of B , RV, FWHM, and Hα finding values between 33.46 to 41.9 d, where the best result was given by the B with a rotation period of 33.63 ± 0.37 d. This discrepancy between activity indicators was explained due to the presence of differential rotation over the stellar surface but the HARPS-Pol and NARVAL data did not allow for confirming it.
We studied stellar activity indicators of Gl 205 in the optical and near-infrared domains. In the optical domain, the Hα index from SOPHIE exhibits a good correlation with the RVs and is modulated by the stellar rotation period as seen by Bonfils et al. (2013) in HARPS data. This is expected since the Hα index is a well-studied activity indicator for M dwarfs and its correlation with the stellar rotation is clear in large samples (e.g., Delfosse et al. 1998;Newton et al. 2017;Jeffers et al. 2018). In particular, the Carmencita sample of Jeffers et al. (2018) shows that Hα inactive (pEW(Hα) > -0.5Å) early M dwarfs, which is the case of Gl 205, tend to exhibit modulations at larger periods from 10 to 100 d.
The other optical indicators included in this work do not show clear hints of tracing activity. Even though most of them tend to have incipient peaks of periodicities close to the stellar rotation in their GLS periodograms, these peaks are not significant or they do not persist for longer than one season. For example, the BIS becomes significantly modulated by the stellar rotation during the season S RV 2 . However, the periodicity is slightly longer than the rotation period suggesting that the traced active features were closer to the poles. It is not surprising that the BIS is not anti-correlated with the RVs (Desort et al. 2007) in the case of Gl 205 since the SOPHIE resolution is larger than the vsini of 0.7 km s −1 .
In the near-infrared domain, we found that the FWHM LBL is the best tracer of activity in our data set as it is correlated with the RV with a Pearson's coefficient of 0.4, and its periodogram has the strongest peak at the stellar rotation period. Previous studies have found the same trend for other M dwarfs (e.g., Klein et al. 2021;Zicher et al. 2022). Moreover, we tested the performance of near-infrared spectral lines as possible activity tracers. The He I triplet at 10833Å has been extensively studied in the CARMENES sample of M dwarfs (Fuhrmeister et al. 2019(Fuhrmeister et al. , 2020 showing that its variability is correlated with Hα only for active cases. In our work, we obtained high error bars from the fitting of the He I line due to the high amount of absorption lines around it making it difficult to obtain a good Voigt profile. The index we derived does not show to be modulated by the stellar rotation period. The K line at 12435Å is studied in detail by Fuhrmeister et al. (2022). The authors find that the K I is rarely correlated or anti-correlated with Hα since this line is less sensitive to chromospheric variability than Hα. Our results agree with the findings of Fuhrmeister et al. (2022) seeing that the K I line is not correlated with the RV variations and it is not modulated by P rot .
Only the Al I line at 13154Å shows several peaks of periodicity within the range of the expected differential rotation suggesting that this line could be tracing several active features across the stellar latitudes. Even though the Ti I at 10499Å and Fe I at 11693Å lines are not modulated by the stellar rotation, they appear to be affected by the magnetic field since they follow more or less the same changes in phase. Al I and Ti I have not been previously studied as activity tracers for Sun-like or M dwarf stars, further analysis in a larger sample would be of great benefit to prove their capacity of activity tracer as they are located in relatively clear parts of the spectrum with a small presence of telluric lines.
Interestingly, the analysis of several activity indicators over time reveals changes in the periodicities over different seasons of observations which may indicate an evolution in the distribution of active features over the stellar surface of Gl 205. All the activity indicators studied in this work reveal different information as may be tracing various activity processes in the stellar surface and disclose the complexity of the stellar activity of Gl 205.
As explained before, the lack of correlation or anticorrelation between the RVs and the activity indicators can be a result of time lags between the time series. For that reason, we can not completely discard as activity tracers the activity indicators that are not correlated with the RVs. Nevertheless, the existence of the correlation is a hint of the stellar nature of the signal.
On the other hand, the GLS periodogram is the standard tool to detect periodicities in time series, however, its efficiency depends on the number of data points, time span, and sampling, among other characteristics. If a significant peak of periodicity is not found by the periodogram, it does not directly imply that the indicator is not rotationally modulated. In this work, we compared the periodicities found in the GLS periodograms versus the period of the quasi-periodic GP applied in all the activity indicators. In some cases where the GLS periodogram does not find periodicities in the dataset, the GP succeeded in constraining a periodicity close to the expected rotation period. However, in this particular case, we know beforehand the expected stellar rotation and then, we can constraint properly the period of the GP. In blind searches of the stellar rotation period, the results may differ.
The periods found in the GP models are all within the ranges of the differential rotation of the star, between 32 d to 45.5 d, except for He I. In particular, these periods tend to be longer than the period measured in the longitudinal magnetic field. This may be a geometrical effect due to the inclination of Gl 205 which is 60 deg. From the observer's point of view, there is one pole hidden producing that we observe preferably active regions at longer latitudes whose rotation periods are longer.
Finding near-infrared spectral lines that could work as stellar activity tracers remains an open challenge for stellar characterization and exoplanet searches in this domain. This is particularly important for less active M dwarfs since chromospheric lines may be better tracers of stellar activity than CCF-based activity indicators (Lafarga et al. 2021). A complementary route to face this challenge consists in identifying subsets of lines that are more or less sensitive to the activity jitter ).

Optical and near-infrared activity-induced RV jitter
In this study, we confirm that the observed RV variations of Gl 205 in the optical and near-infrared are due to stellar activity as previously stated. The RV periodicities over the three RVs sub-datasets and during the whole time series are always linked to one or more activity indicators, in both domains. Hébrard et al. 2016 defined the total RV jitter as the sum of the jitter due to the rotational modulation and the jitter that comes from random components (e.g., short-lived spots). Even though we are close to reaching the noise level of the data set with the multi-dimensional GP modeling, the remaining jitter of 2.7 ± 0.3 m/s for SPIRou and 2.7 ± 0.6 m/s for SOPHIE is comparable with the random component contribution proposed for Gl 205 of the order of J r = 2.7 m/s. This jitter contribution is greater than the average error bars of 1.9 m/s.
In the GP models applied to the radial velocities, we observe that the model of the optical RVs is consistent over time and resembles a sinusoidal with its first harmonic. The nearinfrared RVs, on the other hand, show a more complex behavior with a rapidly evolving signal. The amplitudes measured from the GP model show a slightly larger value for the near-infrared RVs.Suárez Mascareño et al. (2018)  We have found that the optical and near-infrared RVs of Gl 205 are similar in amplitude and jitter levels for the whole time series, meaning that there is at least a little gain in observing at near-infrared wavelengths which is expected for early, moderately, or low active, M dwarfs. Reiners et al. (2010) showed that the gain in precision towards the near-infrared is seen especially for M4 and later spectral types. In the case of early M dwarfs, the best precision is obtained in the V band even though the highest S/N is reached in the J band. Moreover, this effect may not be exclusively dependent on the spectral type only but on levels of activity too. Robertson et al. 2020 analyzed a small sample of fastrotating M dwarfs to compare their optical and near-infrared RVs. The late M dwarf Gl 3959 exhibits comparable optical and near-infrared RVs from the HIRES and HPF spectrographs, while some other targets showed comparable RVs only during one season of observations. Active M dwarfs such as Gl 388 or AU Mic, show stronger RV jitter in the optical than in the near-infrared while less-active stars, such as Gl 205, do not. The strength of the magnetic field may play a key role in this phenomenon since it likely impacts the brightness contrast between magnetic and non-magnetic features. We have shown that Gl 205 has a much weaker field than those on active M dwarfs.
We plan to extend the work of quasi-simultaneous observations with SOPHIE and SPIRou for more M dwarfs to help understand the multi-wavelength stellar activity contribution in RVs studies.
Acknowledgements. PCZ thanks Oscar Barragán for the helpful discussions about pyaneti and the multi-dimensional GPs. This work is based on observations collected with the SOPHIE spectrograph on the 1.93 m telescope at the Observatoire de Haute-Provence (CNRS), France. We thank the staff of the Observatoire de Haute-Provence for their support at the 1.93 m telescope and on SOPHIE. This work is 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. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of MaunaKea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. Based on observations obtained with SPIRou, an international project led by Institut de Recherche en Astrophysique et Planétologie, Toulouse, France. This paper includes data collected by the TESS mission that are publicly available from the Mikulski Archive for Space Telescopes (MAST). We acknowledge funding from the French National Research Agency (ANR) under contract number ANR-18-CE31-0019 (SPlaSH). PCZ thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant #1829740, the Brinson Foundation, and the Moore Foundation; her participation in the program has benefited this work. BK acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 865624,GPRV). AAK acknowledges that this work was supported by FCT -Fundação para a Ciência e a Tecnologia through national funds and by FEDER through COMPETE2020 -Programa Operacional Competitividade e Internacionalização by these grants:      Table 4) are shown on the left-hand panel whereas the profiles observed between August and October 2020 (Season S 2 in Table 4) are shown in the right-hand panel. The stellar rotation cycle, computed using the first observation (BJD = 2 458 738.128) as a reference time and assuming a rotation period of 34.4 d (see Table 3), are given on the right-hand side of each observation. The ±1σ error bars are indicated on the left-hand side of each profile.