7 v 1 1 2 A ug 1 99 7 Bifurcation , efficiency , and the role of injection in shock acceleration with the Bohm diffusion

Received ; accepted – 2 – ABSTRACT The efficiency and uniqueness of the diffusive shock acceleration is studied on the basis of the novel kinetic solutions. These solutions obtained earlier (paper I) selfconsistently describe a strong coupling of cosmic rays with the gas flow. They show that the dependence of the acceleration efficiency upon 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 p 1 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 especially serious closure problems and displays an 'unphysical' behavior. We show that in contrast to the two-fluid model three different solutions occurs also for arbitrarily large M provided that p 1 is sufficiently high. The three solutions appear together only if the injection rate ν lies between two critical values, ν 1 < ν < ν 2. For ν < ν 1 (M, p 1) only the inefficient solution is possible. For ν > ν 2 (M, p 1) only the efficient solution with a very high cosmic ray production rate occurs. On the basis of the obtained bifurcation surface R(ν, p 1) we consider the limit p 1 → ∞, ν → 0 which completely uncovers the long debating anomalies of the two-fluid model. The constructed steady state manifold that, at least partially is an attractor of a time dependent system, allows us to speculate on the nonstationary acceleration.