Cosmic‐Ray Acceleration at Relativistic Shock Waves with a “Realistic” Magnetic Field Structure

The process of cosmic ray first-order Fermi acceleration at relativistic shock waves is studied with the method of Monte Carlo simulations. The simulations are based on numerical integration of particle equations of motion in a turbulent magnetic field near the shock. In comparison to earlier studies, a few "realistic" features of the magnetic field structure are included. The upstream field consists of a mean field component inclined at some angle to the shock normal with finite-amplitude sinusoidal perturbations imposed upon it. The perturbations are assumed to be static in the local plasma rest frame. Their flat or Kolmogorov spectra are constructed with randomly drawn wave vectors from a wide range $(k_{min}, k_{max})$. The downstream field structure is derived from the upstream one as compressed at the shock. We present particle spectra and angular distributions obtained at mildly relativistic sub- and superluminal shocks and also parallel shocks. We show that particle spectra diverge from a simple power-law, the exact shape of the spectrum depends on both the amplitude of the magnetic field perturbations and the wave power spectrum. Features such as spectrum hardening before the cut-off at oblique subluminal shocks and formation of power-law tails at superluminal ones are presented and discussed. At parallel shocks, the presence of finite-amplitude magnetic field perturbations leads to the formation of locally oblique field configurations at the shock and the respective magnetic field compressions. This results in the modification of the particle acceleration process, introducing some features present in oblique shocks, e.g., particle reflections from the shock. We demonstrate for parallel shocks a (nonmonotonic) variation of the particle spectral index with the turbulence amplitude.