*Astron. Astrophys. 357, 1133-1136 (2000)*
## 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 *N*th shock for a delta-function
injection distribution at only at the
first shock (see Melrose & Pope 1993):
For exactly identical shocks, we have to consider injection at all
shocks. The downstream distribution at the last shock is then given by
a sum of the distributions injected with
at each shock:
where the *n*th 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
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