2. Analytical considerations
In the diffusion approximation, the problem of subsequent shocks can be solved analytically. For an upstream phase-space distribution , the downstream distribution without additional injection is given by
During transport to the next shock, due to conservation of the phase space volume , we have to consider the effect of the decompression of the plasma on the momentum of the particles. Expansion of the plasma by the compression ratio r leads to the shift of the downstream momentum to the new upstream momentum at the next shock (Schneider 1993). Applying Eq. (1) to N subsequent identical shocks with adiabatic decompression between them, we get the spectral index downstream of the Nth shock for a delta-function injection distribution at only at the first shock (see Melrose & Pope 1993):
where the nth contribution is calculated by subsequent application of Eq. (1), and considering decompression between the shocks, as shown by Melrose & Pope (1993). The spectral index 3 is given by , which can be easily calculated for shocks, which we consider here. For an infinite number of shocks, the result asymptotes to (e.g., White 1985).
© European Southern Observatory (ESO) 2000
Online publication: June 5, 2000