SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 357, 1133-1136 (2000)

Previous Section Next Section Title Page Table of Contents

2. Analytical considerations

In the diffusion approximation, the problem of subsequent shocks can be solved analytically. For an upstream phase-space distribution [FORMULA], the downstream distribution without additional injection is given by

[EQUATION]

During transport to the next shock, due to conservation of the phase space volume [FORMULA], 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 [FORMULA] to the new upstream momentum [FORMULA] 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 [FORMULA] only at the first shock (see Melrose & Pope 1993):

[EQUATION]

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 [FORMULA] at each shock:

[EQUATION]

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 [FORMULA], which can be easily calculated for [FORMULA] shocks, which we consider here. For an infinite number of shocks, the result asymptotes to [FORMULA] (e.g., White 1985).

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
helpdesk.link@springer.de