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 ± 0.5 days 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 3 m s⁻¹ for both instruments while using Hα 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.