Time-dependent evolution of cosmic-ray-mediated shocks in the two-fluid model

We present the results of an extensive series of time-dependent numerical simulations of cosmic-ray-mediated shocks based on the two-fluid model. We examine the time evolution of both plane parallel, piston-driven shocks and spherical adiabatic blast waves. We include shocks that sweep up ambient cosmic rays as well as those that inject the cosmic rays directly. Starting from a state with no upstream precursor, the plane shocks eventually reach equilibrium states given by earlier analytic treatments. The time required to reach those states is controlled by the development of the shock precursor. It is typically two orders of magnitude greater than the cosmic-ray diffusion time, and coincidentally similar to the mean time to accelerate the cosmic rays to high energy. It depends, however, upon the degree of shock restructuring that is necessary to achieve equilibrium. Spherical shocks do not reach equilibrium conditions in these simulations. Postshock cosmic-ray pressures are much smaller than for equilibrium plane shocks of comparable strength and similar upstream and/or injection conditions. We find that supernova shocks are capable of transferring ∼10% of the blast energy through diffusive processes into cosmic rays at the end of the adiabatic, Sedov-Taylor phase. However, the efficiency for doing this is considerably less than indicated in onion-skin or similar models which inject particles and assume instantaneous equilibrium for the shocks.