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

The stellar activity of M dwarfs is the main limitation for discovering and characterizing exoplanets orbiting them since it induces quasi-periodic RV variations. We aim to characterize the magnetic field and stellar activity of the early, moderately active, M dwarf Gl205 in the optical and nIR domains. 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 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 GP regression. 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 GP regression with activity indicators as ancillary time series. The optical and nIR RVs have 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$\pm$0.5 d in the longitudinal magnetic field. Using ZDI we measure the amount of latitudinal differential rotation (DR) shearing the stellar surface yielding rotation periods of Peq=32.0$\pm$1.8 d at the stellar equator and Ppol=45.5$\pm$0.3 d at the poles. We observed inconsistencies in the activity indicators' periodicities that could be explained by these DR values. The multi-dimensional GP modeling yields an RMS of the RV residuals down to the noise level of 3 m/s for both instruments, using as ancillary time series H$\alpha$ and the BIS in the optical, and the FWHM in the nIR.