Pitch-Angle Dependence of First-Order Fermi Acceleration at Shock Fronts

A model of first-order Fermi acceleration at shock fronts, developed in a previous paper, is extended to consider the angular distribution of the accelerated particles. In the model, particles trapped between a hydromagnetic shock front and a moving magnetic mirror are accelerated by a Fermi process. Scattering of the particles in pitch angle during acceleration was assumed to offset the decrease in pitch angle due to reflection from the moving mirror. In the previous paper, the trapped particle distribution was averaged over pitch angle and the scattering was assumed sufficient to keep the distribution nearly isotropic. Here we consider the limit in which scattering is assumed weak, although it is still strong enough to offset the effect of the mirror. An approximate energy spectrum is obtained, and it is found that acceleration may be very efficient when the scattering is small. In particular, a mirror ratio BM/Bo 2 may suffice to produce the observed energetic electron spectra observed near the Earth's bow shock.