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The first-order Fermi acceleration process at an ultra-relativistic shock wave is expected to create a particle spectrum with the unique asymptotic spectral index sigma_{gamma >> 1} approximately 2.2. Below, we discuss this result and differences in its various derivations, which -- explicitly or implicitly -- always require highly turbulent conditions downstream of the shock. In the presence of medium amplitude turbulence the generated particle spectrum can be much steeper than the above asymptotic one. We also note problems with application of the pitch angle diffusion model for particle transport near the ultra-relativistic shocks., Comment: Substantially modified and shorted version, accepted to A&A

The process of cosmic ray acceleration up to energies in excess of $10^{20}$ eV at relativistic shock waves with large Lorentz factors, $\Gamma \gg 1$ requires $\sim \Gamma^2$ particle energy gains at single reflections from the shock (cf. Gallant & Achterberg 1999). In the present comment, applying numerical simulations we address an efficiency problem arising for such models. The actual efficiency of the acceleration process is expected to be substantially lower than the estimates of previous authors., Comment: 3 pages, 3 figures, accepted for publication in MNRAS (Letters)

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 $\gamma$ up to 320 and varying magnetic field inclinations $\psi$ 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., Comment: 4 pages, 2 figures, to appear in proceedings of 16th ECRS

Energy spectra of particles accelerated by the first-order Fermi mechanism are investigated at ultrarelativistic shock waves, outside the range of Lorentz factors considered previously. For particle transport near the shock a numerical method involving small amplitude pitch-angle scattering is applied for flows with Lorentz factors $\gamma$ from 3 to 243. For large $\gamma$ shocks a convergence of derived energy spectral indices up to the value $\sigma_\infty \approx 2.2$ is observed for all considered turbulence amplitudes and magnetic field configurations. Recently the same index was derived for $\gamma$-ray bursts by Waxman [Astrophys. J. Lett. 485, L5 (1997)]., Comment: 4 pages, 6 figures

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 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, $T_{acc}^{(c)}$, 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 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, $\delta B$. We got some interesting results like non-monotonic changes of $T_{acc}^{(c)}$ with $\delta B$, which arises due to the particle cross-field diffusion., Comment: 10 pages, TeX type, 11 Encapsulated PostScript figures, psbox.tex included, uses mn.sty (MN.sty - 11 pages, [hbt] above figures, footer on each page), accepted for publication in MNRAS (July 1996)