RENORMALIZED 2-FLUID HYDRODYNAMICS OF COSMIC-RAY-MODIFIED SHOCKS

A simple two-fluid model of diffusive shock acceleration, introduced b y Axford, Leer, & Skadron and Drury & Volk, is revisited. This theory became a chief instrument in the studies of shock modification due to particle acceleration. Unfortunately its most intriguing steady state prediction about a significant enhancement of the shock compression an d a corresponding increase of the cosmic-ray production violates assum ptions which are critical for the derivation of this theory. In partic ular, for strong shocks the spectral flattening makes a cutoff-indepen dent definition of pressure and energy density impossible and therefor e causes an additional closure problem. Confining ourselves for simpli city to the case of plane shocks, assuming reacceleration of a preexis ting cosmic-ray population, we argue that also under these circumstanc es the kinetic solution has a rather simple form. It can be characteri zed by only a few parameters, in the simplest case by the slope and th e magnitude of the momentum distribution at the upper momentum cutoff. We relate these parameters to standard hydrodynamic quantities like t he overall shock compression ratio and the downstream cosmic-ray press ure. The two-fluid theory produced in this way has the traditional for m but renormalized closure parameters. By solving the renormalized Ran kine-Hugoniot equations, we show that for the efficient stationary sol ution, most significant for cosmic-ray acceleration, the renormalizati on is needed in the whole parameter range of astrophysical interest.