We have developed a numerical method to include self-consistently the injection of the supra-thermal particles into the cosmic-ray population at quasi-parallel shocks according to the analytic solution of Malkov (1998). Toward this end, we have adopted the "transparency function" which expresses the probability that supra-thermal particles at a given velocity can leak upstream through the magnetic waves, based on non-linear particle interactions with self-generated waves. We have incorporated the transparency function into the existing numerical code which solves the cosmic-ray transport equation along with the gas dynamics equations. In order to investigate the interaction of high energy particles, accelerated by the Fermi process, with the underlying plasma flow without using a free parameter for the injection efficiency, we have applied our code with the new injection scheme to both strong () and weak () parallel shocks.
The main conclusions from the simulation results are as follows:
While the weak shock model of reaches a steady-state, the strong shock model of has not reached a steady-state by the end of our simulation. We expect for the strong shock that the CR pressure continues to increase and the shock becomes CR dominated, leading to the greater total velocity jump and more efficient acceleration. In realistic shocks such as SNRs, however, escaping particles due to non-planar geometry or lack of scattering at high momentum are likely to become important. To resolve this non-linear evolution, much longer physical time scales have to be simulated, until CRs reach energies where escape is likely to be important. The key problem here is the range in configuration and momentum space that has to be computed. Our method uses a grid with uniform cells in configuration space, chosen fine enough to capture the evolution of at near-thermal momenta where the diffusion coefficient is proportional to (Bohm diffusion). This leads to a computationally extremely expensive calculation, especially because the grid has to be large enough to contain the diffusion length scale of the highest momentum CRs. The problem can be solved on a much larger time scale by using an adaptive mesh refinement (AMR) code with the shock tracking techniques (Kang & Jones 1999). In the near future we plan to incorporate the injection model presented here into the powerful shock tracking AMR-code, to calculate the evolution of the phase-space distribution of the plasma during different phases of SNRs. This would allow us to investigate with a plasma-physical based injection model how the slowly growing cosmic-ray pressure at a strong shock eventually modifies the shock structure. A strong modification will cause the velocity jump across the subshock to decrease and the distribution function of the suprathermal particles to steepens. This might have further back reaction on the injection efficiency. Also the CR distribution will deviate from a simple power-law. For a calculation up to the highest energy CRs, also the spherical geometry of a SNR should be taken into account. Such an approach could lead to a consistent calculation of the complete phase-space distribution at quasi-parallel shocks, and should be a promising step towards a calculation of the overall efficiency of SNRs in producing CRs during their evolution.
For oblique shocks, the injection efficiencies calculated here for a parallel shock should define an upper limit, because the statistical probability of a particle to cross the shock from downstream to upstream decreases with the intersection velocity of magnetic field and shock front. This kinematical effect was investigated by Baring et al. (1993) with the use of Monte-Carlo simulations. However, in the model we have incorporated here, the injection is already suppressed strongly (compared to the purely kinematical model) by the reduced transparency of the plasma due to the high amplitude Alfvén waves. We point out, that for oblique shocks, the filtering due to Alfvén waves may be reduced due to the decreased downstream amplification of the wave amplitude. This would allow lower energy particles to be injected, and the kinematical effect could be partly balanced. As a result, we speculate that the dependence on the obliquity might be significantly weaker than calculated by Baring et al. (1993). Resolution of that important question must await more complete understanding of the injection physics.
In summary, we have shown that the process of particle acceleration under consideration of a plasma physical injection model underlies a rather effective self-regulation. Apart from the direct particle-wave interaction described by the injection model itself, also the energetic feedback of the energy transfer between thermal plasma and cosmic-rays keeps the fraction of particles in the non-thermal distribution at roughly of the particles swept through the shock. These self-regulation mechanisms lead to a quite stable injection efficiency, which depends weakly on the initial conditions.
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001