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

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Abstract
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.
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