Acceleration time scale at ultrarelativistic shock waves

Abstract

The first-order cosmic ray acceleration at ultrarelativistic shocks is investigated using the Monte Carlo method. We apply a method of discrete particle momentum scattering as a model of particle pitch angle diffusion to reproduce highly anisotropic conditions at the shock wave. Shocks with Lorentz factors $γ$ up to 320 and varying magnetic field inclinations $ψ$ are considered. Values of diffusion coefficients upstream in the point where energy spectral indices stabilize to the limit 2.2 were calculated. The obtained acceleration time does not depend on shock conditions.