3. Monte-Carlo simulations
We present results for test-particle acceleration at multiple quasi-parallel and oblique shocks. We have used a Monte-Carlo code which was described by Gieseler et al. (1999), and extended it to multiple shocks. We refer to this previous work for a more detailed description of the code, and summarize here only the main features. A particle is described by three coordinates: the distance from the shock, the magnitude of the momentum , and the pitch angle between particle momentum and magnetic field . Upstream and downstream the particle momentum is conserved, and the small scale irregularities lead to pitch-angle scattering. We do not consider cross field diffusion. The magnetic moment is always conserved in the non-relativistic shocks with velocity , which we consider here. When a particle crosses the shock, the momentum and pitch angle are transformed to the new relevant rest frame. Depending on the pitch angle µ and inclination angle , particles can be reflected upstream of the shock. If a particle reaches a distance of a few diffusion length scales downstream of the shock where the density distribution has reached its constant downstream value, it is considered as escaped from the single shock. Between shocks, decompression of the plasma leads to a shift in the momentum distribution, as described in Sect. 2.
For exactly planar shocks, every escaping particle will reach the next shock, and can be further accelerated. To account approximately for the more realistic situation that some particles found in the downstream region of the Nth shock have been accelerated only at shocks, we use the fixed escape probability . This gives the fraction of particles of the downstream distribution of each shock that will not be further accelerated. These particles remain in the system but bypass the following shocks and therefore contribute directly to the downstream distribution of the Nth shock. Exactly planar shocks would be described by , whereas reveals a spectrum produced by one single shock.
The distinction of particles which are able to diffuse (and for which the shock is a discontinuity) from the thermal particles, which are not subject to the first order acceleration process, is based on the gyro radius, and therefore on the momentum of the particles. We do not include a self consistent injection model here. Instead we inject particles with a momentum , for which they have an already relativistic velocity, and can be accelerated immediately. However, when a particle has a momentum , we consider this particle as re-thermalised and suppress further acceleration, even though the momentum does not correspond to a mean thermal momentum.
3.1. Multiple shocks with diffusive acceleration
In order to compare our Monte-Carlo simulations with analytical calculations, we first consider identical planar quasi-parallel shocks. Here, all particles are transported through the subsequent shocks with decompression between them. This corresponds to for the escape probability. We also do not include effects of re-thermalisation. Therefore we can compare the results of the Monte-Carlo simulations directly to the analytical results described in Sect. 2. To retrieve the spectral index from the momentum distribution, we use the linear interpolation between two neighboring momentum bins in the region . The statistical fluctuation is then expressed solely by the scatter of the result in this region. For higher momenta, where the distribution deviates only slightly from a pure power law, we fit a power law over four equidistant momentum bins in . The result for particles is presented in Fig. 1. The open crosses represent particles which are injected at the first shock with . The final distribution is measured downstream of the 5th shock before decompression. Eq. (2) is shown by the dashed line. The filled dots show the spectral slope in the case when particles are injected at all shocks. The corresponding analytical result from a derivation of Eq. (3) is shown by the solid line in Fig. 1. In both cases, the agreement is quite good.
In Fig. 2 we have included effects of finite extent of the acceleration region by choosing the free parameter , as described above. This leads essentially to a reduction of the number of shocks at which the particles in the final downstream distribution have been accelerated, and therefore the momentum distribution is steeper. In addition we remove particles from the acceleration mechanism with momentum , which correspond to re-thermalisation. This produces a cutoff of the test-particle distribution at . To show the effect of the escape, we have included in Fig. 2 the analytical results shown in Fig. 1 as lines. The discrete symbols represent the Monte-Carlo results. The number of initial particles here is . The symbols and the fit procedure used are the same as described above. Note that the distribution of particles injected at the first shock is a subset of the particles injected at all shocks. We do not perform independent runs of the code to measure these two momentum distributions. With increasing momentum, the distribution of particles injected at all shocks has an increasing fraction of particles which were injected at the first shock, because these have the highest probability to gain momentum. Therefore, the statistical fluctuations of the Monte-Carlo distributions shown in Fig. 2 (and also in Fig. 1 and Fig. 3) are correlated at high momentum.
3.2. Multiple oblique shocks
By increasing the inclination angle , the ratio of the particle velocity to the intersection velocity of magnetic field and shock becomes larger, and the acceleration can no longer be described by the diffusion approximation. Then, the calculations of Sect. 2 break down.
We consider here strong shocks with speed , and . The resulting velocity of the intersection point is . Such a shock produces a spectrum with canonical index (Kirk & Heavens 1989). The realisation of the multiplicity we investigate here is one in which the compression by the shock and the decompression of the plasma described in Sect. 2 are both effective only along the shock normal (which has the same direction for all subsequent shocks). Therefore, the decompression will restore the magnetic field to the initial upstream value and orientation. This realisation can be described by multiple identical oblique shocks of the same inclination angle . This situation reveals most clearly the direct effect of the shock multiplicity, together with re-thermalisation and escape (for which we chose ). The results are presented in Fig. 3, where the discrete symbols represent the Monte-Carlo calculations in the same way as described above, except that the linear interpolation of two adjacent bins is used up to , and an initial number of independent particles is simulated. The dashed line indicates the canonical spectral index for a single shock of this type. Although the escape between shocks leads to a steepening of the spectrum towards the canonical result, the slope at multiple oblique shocks is still flatter than for a single shock. The relative hardening due to the multiplicity of the shock is in the high energy part (from about three orders of magnitude above the injection momentum) quite similar to the diffusive case, shown in Fig. 2. However, the main effect of producing a hard spectrum is due to the obliquity ( compared to ), and this is independent of momentum for relativistic particle velocities, as long no loss mechanisms are included.
© European Southern Observatory (ESO) 2000
Online publication: June 5, 2000