THE ACCELERATION TIME-SCALE FOR FIRST-ORDER FERMI ACCELERATION IN RELATIVISTIC SHOCK-WAVES

The acceleration time-scale for the process of first-order Fermi accel eration in relativistic shock waves with oblique magnetic field config urations is investigated by the method of Monte Carlo particle simulat ions. We discuss the differences in derivation of the cosmic ray accel eration time-scale for non-relativistic and relativistic shocks. We de monstrate the presence of a correlation between the particle energy ga in at interaction with the shock and the respective time elapsed since the previous interaction. Because of this, any derivation of the acce leration 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 par ticle momentum scattering, and is intended to model situations with an isotropic cosmic ray distributions. We consider shocks with parallel, as well as oblique, sub- and super-luminal magnetic field configuratio ns with finite-amplitude perturbations, delta B. At parallel shocks T- acc((c)) diminishes with increasing perturbation amplitude and shock v elocity U-1. Another feature discovered in oblique shocks is non-monot onic changes of T-acc((c)) with delta B. This effect is due to the par ticle 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 i ncreasing delta B. In some of the shocks considered the acceleration t ime-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 ang le diffusion versus the large-angle momentum scattering is given. We s tress the importance of the proper evaluation of the effective magneti c field (including the perturbed component) in simulations involving d iscrete particle momentum scattering.