The Cosmic-Ray Shock Structure Problem for Relativistic Shocks

The time asymptotic behaviour of a relativistic (parallel) shock wave significantly modified by the diffusive acceleration of cosmic-rays is investigated by means of relativistic hydrodynamical equations for both the cosmic-rays and thermal gas. The form of the shock structure equation and the dispersion relation for both long and short wavelength waves in the system are obtained. The dependence of the shock acceleration efficiency on the upstream fluid spped, long wavelength Mach number and the ratio N = P_co/cP sub co+P_go)(Psub co and P_go are the upstream cosmic-ray and thermal gas pressures respectively) are studied.