STOCHASTIC FERMI ACCELERATION IN SOLAR-FLARES

We have calculated proton spectra, valid from non- to ultra-relativistic energies, resulting from stochastic Fermi acceleration in solar flares. These spectra were obtained by numerically solving the Fokker-Planck equation, in which the escape of the particles from the acceleration region is characterized by an energy-independent escape time. In addition to equilibrium spectra, we also present time-dependent energy spectra showing the approach to equilibrium. We compare these numerical equilibrium spectra with previous results which were obtained either by Monte Carlo simulations or approximate analytical treatments. There are no analytic solutions valid in the transrelativistic regime, which is very important for the production of pions and neutrons in solar flares. We relate the acceleration efficiency to physical parameters, in particular the energy density in either magnetosonic or Alfvén waves, and place a lower limit on either of these energy densities from acceleration times implied by gamma-ray observations. We also discuss the physical interpretation of the escape time.