Heating and ionization of stellar chromospheres by nonthermal proton beams: Implications for impulsive phase, redshifted Lyman-alpha radiation in stellar flares

We investigate the physical basis for the timescale of impulsive-phase, redshifted Lyman-alpha emission in stellar flares on the assumption that it is determined by energy losses in a nonthermal proton beam that is penetrating the chromosphere from above. The temporal evolution of ionization and heating in representative model chromospheres subjected to such beams is calculated. The treatment of 'stopping' of beam protons takes into account their interactions with (1) electrons bound in neutral hydrogen, (2) nuclei of neutral hydrogen, (3) free electrons, and (4) ambient thermal protons. We find that, for constant incident beam flux, the system attains an equilibrium with the beam energy input to the chromosphere balanced by radiative losses. In equilibrium, the beam penetration depth is constant, and erosion of the chromosphere ceases. If the redshifted, impulsive-phase stellar flare Lyman-alpha emission is produced by downstreaming hydrogen formed through charge exchange between beam protons and ambient hydrogen, then the emission should end when the beam no longer reaches neutral hydrogen. The durations of representative emission events calculated on this assumption range from 0.1 to 14 s. The stronger the beam, the shorter the timescale over which the redshifted Lyman-alpha emission can be observed.