From Stellar Coron{\ae} to Gyrochronology: a theoretical and observational exploration

Context. Stellar spin-down is the result of a complex process involving rotation, dynamo, wind and magnetism. Multi-wavelength surveys of solar-like stars have revealed the likely existence of relationships between their rotation, X-ray luminosity, mass-losses and magnetism. Those impose strong constraints on the corona and wind of cool stars. Aims. We aim to provide power-law prescriptions of the mass-loss of stars, of their magnetic field, and of their base coronal density and temperature that are compatible with their observationally-constrained spin-down. Methods. We link the magnetic field and the mass-loss rate from a wind torque formulation in agreement with the distribution of stellar rotation periods in open clusters and the Skumanich law. Given a wind model and an expression of the X-ray luminosity from radiative losses, we constrain the coronal properties by assuming different physical scenarii linking closed loops to coronal holes. Results. We find that the magnetic field and the mass loss are involved in a one-to-one correspondence constrained from spin-down considerations. We show that a magnetic field depending on both the Rossby number and the stellar mass is required to keep a consistent spin-down model. The estimates of the magnetic field and the mass-loss rate obtained from our formalism are consistent with statistical studies as well as individual observations and give new leads to constrain the magnetic field-rotation relation. The set of scaling-laws we derived can be broadly applied to cool stars from the PMS to the end of the MS, and allow for a stellar wind modelling consistent with all the observational constraints available to date.