On the theory of cosmic-ray-mediated shocks with variable compression ratio

Cosmic-ray-mediated shocks may accelerate enough cosmic rays to high enough energies that they escape the shock, carrying an appreciable amount of energy before being convected to downstream infinity. Under such conditions, it is noted, the overall compression ratio cannot be determined from the conservation equations as in conventional hydrodynamic treatments, and the standard equations for shock acceleration admit arbitrarily high compression ratios. A procedure is outlined for obtaining the structure of high Mach number, cosmic-ray-mediated shocks, including their overall compresion ratio, around a low Mach number viscous subshock. Analytic solutions are obtained by quardrature for an energy-dependent diffusion coefficient in the limit of extreme sensitivity to energy, which, unlike previous solutions, include the finite thermal pressure of the preshock gas.

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