## 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 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 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 accelerationIn order to compare our Monte-Carlo simulations with analytical
calculations, we first consider identical planar quasi-parallel
shocks. Here,
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 shocksBy 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 © European Southern Observatory (ESO) 2000 Online publication: June 5, 2000 |