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 acros s a shock wave remains the most attractive model for the production of ener getic cosmic rays. This process has been analyzed extensively during the pa st two decades using the "two-fluid" model of diffusive shock acceleration. It is well known that one, two, or three distinct solutions for the flow s tructure can be found depending on the upstream parameters. Interestingly, in certain cases both smooth and discontinuous transitions exist for the sa me values of the upstream parameters. However, despite the fact that such m ultiple solutions to the shock structure were known to exist, the precise n ature of the critical conditions delineating the number and character of sh ock transitions has remained unclear, mainly due to the inappropriate choic e of parameters used in the determination of the upstream boundary conditio ns. In this paper we derive the exact critical conditions by reformulating the upstream boundary conditions in terms of two individual Mach numbers de fined with respect to the cosmic-ray and gas sound speeds, respectively. Th e gas and cosmic-ray adiabatic indices are assumed to remain constant throu ghout the flow, although they may have arbitrary, independent values. Our r esults provide for the first time a complete, analytical classification of the parameter space of shock transitions in the two-fluid model. We use our formalism to analyze the possible shock structures for various values of t he cosmic-ray and gas adiabatic indices. When multiple solutions are possib le, we propose using the associated entropy distributions as a means for id entifying the most stable configuration.