Method

We treat the effects of DSA at a single shock, adiabatic decompression and synchrotron losses in terms of operators, , , , respectively, operating on an initial distribution function to produce the final distribution function. DSA is described by (e.g., Melrose 1986, p. 249)
 

with b=3r/(r-1), where r (1<r<4) is the compression ratio for the shock. One has b>4 with for , corresponding to the strongest possible shock in a nonrelativistic plasma. Decompression is required to reduce the magnetic field, B, from its compressed value (which applies strictly only to a perpendicular shock) just behind the shock to its ambient value, , before the arrival of the next shock. Then implies
 

Synchrotron losses are described by , , where is the classical radius of the electron, and is the mass of the electron. Let p' be the solution of at t that produces p at , so that one has . Then, if the synchrotron losses are allowed to operate for a time , Liouville's theorem implies
 

The dependence of the synchrotron loss rate on the strength of the shock, , implies that for a strong shock () the losses are most rapid in the compressed-B region just downstream of the shock. Assuming that the losses are important only in this region, may be identified as the time spent by the electron there. However, for our numerical calculations only the combination need be specified, and this is a free parameter.

In our calculations, we inject an initial -function distribution, , and subject it sequentially to DSA, synchrotron losses and decompression. In multiple DSA, the resulting output distribution is the input into a second identical sequence, and this sequence is repeated as many times as desired. We consider both the case where there is a single initial injection and the case where there is an injection at each shock. In the absence of synchrotron losses it is necessary to extend the calculations to very much higher p-values than are ultimately of interest to avoid rounding errors. A rounding error at after the first shock propagates down to after N shocks, and for large N this leads to errors in conservation of particles. We chose a sufficiently high such that particles are conserved to an accuracy of % for N=50.

Another feature of the model is the neglect of the spatial coordinates, which assumption is usually made in this context. The underlying idea is that spatial inhomogeneities are taken into account through implicit coupling between separate regions of space. Here such couplings are implicit in the operators themselves. For example, couples the downstream to the upstream region, and couples the region immediately behind the shock to the decompressed region further downstream.

© Copyright Astronomical Society of Australia 1997