Time dependent cosmic–ray shock acceleration with self–consistent injection

One of the key questions to understanding the efficiency of diffusive shock acceleration of the cosmic rays (CRs) is the injection process from thermal particles. A self–consistent injection model based on the interactions of the suprathermal particles with self–generated magneto–hydrodynamic waves has been developed recently by Malkov (1998). By adopting this analytic solution, a numerical treatment of the plasma–physical injection model at a strong quasi–parallel shock has been devised and incorporated into the combined gas dynamics and the CR diffusion–convection code. In order to investigate self–consistently the injection and acceleration efficiencies, we have applied this code to the CR modified shocks of both high and low Mach numbers (M = 30 and M = 2.24) with a Bohm type diffusion model. Both simulations have been carried out until the maximum momentum (pmax/mpc) ∼ 1 is achieved to illustrate early evolution of a Bohm type diffusion. We find the injection process is self–regulated in such a way that the injection rate reaches and stays at a nearly stable value after quick initial adjustment. For both shocks about 10 of the incoming thermal particles are injected into the CRs. For the weak shock, the shock has reached a steady state within our integration time and ∼ 10% of the total available shock energy is transfered into the CR energy density. The strong shock has achieved a higher acceleration efficiency of ∼ 20% by the end of our simulation, but has not yet reached a steady–state. With such efficiencies shocks do not become CR–dominated or smoothed completely during the early stages when the particles are only mildly relativistic. Later, as the CR pressure becomes dominated by highly relativistic particles that situation should change, but is difficult to compute, since the maximum CR momentum increases approximately linearly with time for this model. In the near future we intend to extend such shock simulations as these to include much higher CR moSend offprint requests to: [email protected] ⋆ Present address: Universität Siegen, Fachbereich Physik, 57068 Siegen, Germany menta using an adaptive mesh refinement technique currently under development.