Magnetized winds of M-type stars and star-planet magnetic interactions: uncertainties and modeling strategy

M-type stars are the most common stars in the universe. They are ideal hosts for the search of exoplanets in the habitable zone (HZ), as their small size and low temperature make the HZ much closer in than their solar twins. Harboring very deep convective layers, they also usually exhibit very intense magnetic fields. Understanding their environment, in particular their coronal and wind properties, is thus very important, as they might be very different from what is observed in the solar system. The mass loss rate of M-type stars is poorly known observationally, and recent attempts to estimate it for some of them (TRAPPIST-1, Proxima Cen) can vary by an order of magnitude. In this work, we revisit the stellar wind properties of M-dwarfs in the light of the latest estimates of $\dot{M}$ through Lyman-$\alpha$ absorption at the astropause and slingshot prominences. We outline a modeling strategy to estimate the mass loss rate, radiative loss and wind speed, with uncertainties, based on an Alfv\'en wave driven stellar wind model. We find that it is very likely that several TRAPPIST-1 planets lie within the Alfv\'en surface, which imply that these planets experience star-planet magnetic interactions (SPMI). We also find that SPMI between Proxima Cen b and its host star could be the reason of recently observed radio emissions.