Acceleration time scale for the first - order Fermi acceleration in relativistic shock waves

The acceleration time scale for the process of first-order Fermi acceleration in rela-tivistic shock waves with oblique magnetic field configurations is investigated by the method of Monte Carlo particle simulations. We discuss the differences in derivation of the cosmic ray acceleration time scale for non-relativistic and relativistic shocks. We demonstrate the presence of correlation between the particle energy gain at interaction with the shock and the respective time elapsed since the previous interaction. Because of that any derivation of the acceleration time scale can not use the distribution of energy gains and the distribution of times separately. The time scale discussed in the present paper, T (c) acc , is the one describing the rate of change of the particle spectrum cutoff energy in the time dependent evolution. It is derived using a simplified method involving small amplitude particle momentum scattering and intended to model the situations with anisotropic cosmic ray distributions. We consider shocks with parallel, as well as oblique, sub-and super-luminal magnetic field configurations with finite amplitude perturbations, δB. At parallel shocks T (c) acc diminishes with the growing perturbation amplitude and shock velocity U 1. Another feature discovered in oblique shocks are non-monotonic changes of T (c) acc with δB. The effect arises due to the particle cross-field diffusion. The acceleration process leading to power-law spectra is possible in super-luminal shocks only in the presence of large amplitude turbulence. Then, T (c) acc always increases with increasing δB. In some of the considered shocks the acceleration time scale can be shorter than the particle gyroperiod upstream the shock. We also indicate the relation existing for relativistic shocks between the acceleration time scale and the particle spectral index. A short discussion of the numerical approach modelling the pitch angle diffusion versus the large angle momentum scattering is given. We stress the importance of the proper evaluation of the effective magnetic field (including the perturbed component) in simulations involving discrete particle momentum scattering.