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Bifurcation, Efficiency, and the Role of Injection in Shock Acceleration with the Bohm Diffusion

The efficiency and uniqueness of diffusive shock acceleration is studied on the basis of novel kinetic solutions. These solutions, obtained earlier (Paper I), self-consistently describe a strong coupling of cosmic rays with the gas flow. They show that the dependence of the acceleration efficiency on physical parameters is critical in nature. In this paper, we investigate a steady acceleration in the parameter space formed by the injection rate ν, the upper cutoff momentum p1, and the Mach number M, while the flow compression R serves as an order parameter. We determine a manifold of all possible solutions in this parameter space. To elucidate the differences between the present kinetic results and the well-known two-fluid predictions, we particularly focus on the ν → 0, M → ∞ limit, where the two-fluid model suffers from particularly serious closure problems and displays an "unphysical" behavior. We show that in contrast to the two-fluid model, three different solutions occur for arbitrarily large M provided that p1 is sufficiently high. The three solutions appear together only if the injection rate ν lies between two critical values, ν1 < ν < ν2. For ν < ν1(M, p1), only an inefficient solution is possible. For ν > ν2(M, p1), an efficient solution only occurs with a very high cosmic ray production rate. On the basis of the obtained bifurcation surface R(ν, p1), we consider the limit p1 → ∞, ν → 0, which completely uncovers the long-debated anomalies of the two-fluid model. The construction of a steady state manifold that is at least partially an attractor of a time-dependent system allows us to speculate on the nonstationary acceleration.