Minimum Mach number for impulsive proton acceleration by parallel collisionless shock waves in solar flares

A lower limit to the minimum Alfven Mach number M(A) is determined for which a parallel shock could accelerate protons to 10 MeV in 2 s for low coronal conditions in which the density is 9.5 x 10 to the 10th/cu cm, the magnetic field is 56 G, and the temperature is 1.6 x 10 to the 6th K. A lower limit to the time to generate the Alfven waves necessary to scatter the protons back to the shock where they can be accelerated by the shock compression is derived, and the upstream and downstream mean free paths are determined. The actual acceleration time after the Alfven waves are established. The lower limit to M(A) is 2.0. 16 references