On the role of injection in kinetic approaches to nonlinear particle acceleration at non relativistic shocks

The dynamical reaction of the particles accelerated at a shock front by the first order Fermi process can be determined within kinetic models that account for both the hydrodynamics of the shocked fluid and the transport of the accelerated particles. These models predict the appearance of multiple solutions, all physically allowed. We discuss here the role of injection in selecting the real solution, in the framework of a simple phenomenological recipe, which is a variation of what is sometimes referred to as thermal leakage. In this context we show that multiple solutions basically disappear and when they are present they are limited to rather peculiar values of the parameters. Diffusive shock acceleration is thought to be responsible for acceleration of cosmic rays in several astrophysical environments. Despite the success of this theory, some issues are still subjects of much debate, for the theoretical and phenomenological implications that they may have. One of the most important of these is the reaction of the accelerated particles on the shock: the violation of the test particle approximation occurs when the acceleration process becomes sufficiently efficient that the pressure of the accelerated particles is comparable with the incoming gas kinetic pressure. Both the spectrum of the particles and the structure of the shock are changed by this phenomenon, which is therefore intrinsically nonlinear (Ellison, these proceedings). Nonlinear effects in shock acceleration of thermal particles result in the appearance of multiple solutions in certain regions of the parameter space. This phenomenon is very general and was found in both the two-fluid [4] and kinetic models [7, 2, 3]. Here we investigate the phenomenon of multiple solutions and show that the appearance of these solutions is dramatically reduced if a self consistent model for injection is adopted. A SEMI–ANALYTICAL APPROACH TO THE PROBLEM Following the approach presented in [1, 2, 3], we solve the steady-state transport equation for the cosmic ray distribution function f (x, p) at a plane shock wave: