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 Prot = 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 Peq = 32.0 ± 1.8 days at the stellar equator and Ppol = 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.