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Astron. Astrophys. 364, 911-922 (2000)

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6. Conclusions

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" [FORMULA] 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 ([FORMULA]) and weak ([FORMULA]) parallel shocks.

The main conclusions from the simulation results are as follows:

  1. The injection process is regulated by the overlap of the population of supra-thermal particles in the injection pool and the function [FORMULA]. As being in the high energy tail of the Maxwell velocity distribution, the population in the injection pool depends strongly on the gas temperature and the particle momentum. The function [FORMULA] behaves like a delta-function defined near a narrow injection pool. As the postshock gas cools due to high initial injection, the Maxwell distribution shifts to lower momenta. But the transparency function also shifts to lower momenta, as well, due to its dependence on the postshock flow velocity. As a result, the injection rate reaches and stays at a stable value after a quick initial adjustment, and also depends only weakly on the initial conditions. This self-regulated injection may imply a broad application of our simulation methods.

  2. The fraction of the background particles that are accelerated to form the non-thermal part of the distribution turns out to be in the range [FORMULA] for the range of initial wave-amplitudes [FORMULA] at a [FORMULA] shock. For a [FORMULA] shock, a slightly higher injection is achieved at [FORMULA], but this could be a lower limit. Such values for the particle injection efficiency have been used as a parameter for spherically expanding SNRs by several authors (Dorfi 1990; Jones & Kang 1992; Berezhko et al. 1995; Berezhko & Völk 2000). These values are well above the "critical injection rate" of [FORMULA] above which spherical shocks of this Mach number are CR dominated according to Berezhko et al. (1995).

  3. Due to computational limitations of using a Bohm type diffusion model, we have integrated our models until the maximum momentum reaches about [FORMULA]. For the [FORMULA] shock model, the energy flux in the total CR distribution was about [FORMULA] of the energy flux in the thermal plasma and shocks didn't become CR dominated and smoothed completely by the end of our simulations. For the [FORMULA] shock model, the acceleration efficiency is lower by a factor of two compared to the high Mach shock because of the smaller velocity jump across the shock.

  4. Just above the injection pool, the distribution function changes sharply from a Maxwell distribution to an approximate power-law whose index is close to the test-particle slope. We estimated this critical momentum as [FORMULA] where [FORMULA]. This determines the number of particles in the injection pool by [FORMULA]. For strong shocks this translates into a distribution function at injection energies of [FORMULA].

While the weak shock model of [FORMULA] reaches a steady-state, the strong shock model of [FORMULA] 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 [FORMULA] at near-thermal momenta where the diffusion coefficient is proportional to [FORMULA] (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 [FORMULA] 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.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001