In paper Meli et al. (2005) the particle shock acceleration at super-luminal shocks is discussed and evaluated by performing Monte Carlo calculations. Relativistic beaming in many relativistic astrophysical sources suggests the appearance of super-luminal shocks. The range of magnetic field orientations for which a shock is super-luminal increases as the upstream plasma flow speed increases. In order to study the sub-luminal case, it is possible to find a relativistic transformation to the frame of reference (so called De Hoffmann-Teller frame; introduced in paper De Hoffmann and Teller, 1950), in which the shock front is stationary and the electric field is zero (E = 0) in both upstream and downstream regions. However, superluminal shock fronts do not admit a transformation to such a frame, the particle 'diffusive' approximation cannot be applied and the particles are more likely to gain energy as their gyro-center makes a single crossing of the shock front from upstream to downstream or else as doing 'drifts' parallel or antiparallel to the present electric field. Meli et al. (2005) have established a numerical Monte Carlo method to study the CR acceleration properties (spectral shapes, energy gain, scattering model dependence, magnetic field dependence, etc) in highly relativistic super-luminal shocks by following the helix-trajectory of the particle in the region of a shock front.
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