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Cosmic Ray Acceleration at Ultrarelativistic Shock Waves: Effects of a “Realistic” Magnetic Field Structure

First-order Fermi acceleration processes at ultrarelativistic (γ ~ 5-30) shock waves are studied with Monte Carlo simulations. The accelerated particle spectra are derived by integrating the exact particle trajectories in a turbulent magnetic field near the shock. The magnetic field model upstream of the shock assumes finite-amplitude perturbations within a wide wavevector range and with a predefined wave power spectrum, imposed on the mean field component inclined at some angle to the shock normal. The downstream field structure is obtained as the compressed upstream field. We show that the main acceleration process at superluminal shocks is the particle compression at the shock. Formation of energetic spectral tails is possible in a limited energy range only for highly perturbed magnetic fields. Cutoffs in the spectra occur at low energies within the resonance energy range considered. These spectral features result from the anisotropic character of particle transport in the magnetic field downstream of the shock, where field compression produces effectively two-dimensional perturbations. We also present results for parallel shocks. Because of the turbulent field compression at the shock, the acceleration process becomes inefficient for larger turbulence amplitudes, and features observed in oblique shocks are recovered in this case. For small-amplitude perturbations, particle spectra are formed in the wide energy range, and modifications of the acceleration process due to the existence of long-wave perturbations are observed, as reported previously for mildly relativistic shocks. The critical turbulence amplitude required for efficient acceleration at parallel shocks decreases with increasing shock Lorentz factor γ. In both subluminal and superluminal shocks, an increase of γ leads to steeper spectra with lower cutoff energies. The spectra obtained for the "realistic" background conditions assumed in our simulations do not converge to the "universal" spectral index claimed in the literature. Thus, the role of the first-order Fermi acceleration in astrophysical sources hosting relativistic shocks requires serious reanalysis.