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

Abstract

The acceleration time-scale for the process of first-order Fermi acceleration in relativistic 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 a correlation between the particle energy gain at interaction with the shock andthe respective time elapsed since the previous interaction. Because of this, any derivation of the acceleration time-scale cannot treat the distribution of energy gains and the distribution of times separately. The time-scale discussed in the present paper, T_acc^(c), is the one describing the rate of change of the particle spectrum cut-off energy in the time-dependent evolution. It is derived using a simplified method involving small-amplitude particle momentum scattering, and is intended to model 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_acc^(c) diminishes with increasing perturbation amplitude and shock velocity U₁. Another feature discovered in oblique shocks is non-monotonic changes of T_acc^(c) with δB. This effect is 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_acc^(c) always increases with increasing δB. In some of the shocks considered the acceleration time-scale can be shorter than the particle gyroperiod upstream of 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 to 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.