Fermi acceleration at shock waves and the galactic cosmic ray spectrum: numerical simulations

A hybrid simulation scheme is applied to treat first-order Fermi acceleration in the neighborhood of shock waves. A steady-state hydrodynamic approximation that describes the shock transition is used to evaluate the evolution in position, phase, and pitch angle of relativistic test particles. The relativistic test particles are followed to obtain the distribution function for a specific time after injection. The hydrodynamic model has four components: the main background plasma, a minor relativistic particle population, an oblique averaged magnetic field, and self-generated MHD waves. Various physical parameters are analyzed as a function of time for different approximations to the diffusion coefficients and spectral indexes for the power law MHD turbulent spectrum (normalized to the wave pressure obtained from the hydrodynamic approximation). Results are applied to a proposed scenario of the main global acceleration mechanism of galactic cosmic rays from a few GeVs to 10⁵-10⁶ GeV