Energy Spectra of Cosmic Rays Accelerated at Ultrarelativistic Shock Waves

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 $\gamma -\mathrm{ray}$ bursts by Waxman [Astrophys. J. Lett. 485, L5 (1997)].