Exact Expressions for the Critical Mach Numbers in the Two-Fluid Model of Cosmic-Ray Modified Shocks

The acceleration of relativistic particles due to repeated scattering across a shock wave remains the most attractive model for the production of energetic cosmic rays. This process has been analyzed extensively during the past two decades using the ``two-fluid'' model of diffusive shock acceleration. It is well known that 1, 2, or 3 distinct solutions for the flow structure can be found depending on the upstream parameters. The precise nature of the critical conditions delineating the number and character of shock transitions has remained unclear, mainly due to the inappropriate choice of parameters used in the determination of the upstream boundary conditions. We derive the exact critical conditions by reformulating the upstream boundary conditions in terms of two individual Mach numbers defined with respect to the cosmic-ray and gas sound speeds, respectively. The gas and cosmic-ray adiabatic indices are assumed to remain constant throughout the flow, although they may have arbitrary, independent values. Our results provide for the first time a complete, analytical classification of the parameter space of shock transitions in the two-fluid model. When multiple solutions are possible, we propose using the associated entropy distributions as a means for indentifying the most stable configuration.