Our Monte-Carlo simulation of first order Fermi acceleration, using the shock-drift process and diffusively (or statistically) particle transport reproduced the analytical steady state test-particle spectrum at multiple shocks, for which the diffusion approximation does apply. We introduced a phenomenological re-thermalisation effect, which accounts for the fact, that below the injection momentum no acceleration can occur. Furthermore, with regard to the finite extension of the shock, we allowed for escape of particles between subsequent shocks. This reduces the ability of multiple shocks to flatten the spectrum which would be produced at one single shock.
As a more realistic setup of multiple shock acceleration at fast shocks we considered oblique shocks, and included a finite escape probability between shocks. We found very hard spectral distributions at strong shocks () with velocity and obliquity . The main effect here is produced by the obliquity, because the canonical spectral index is . In addition, the spectrum becomes even harder with at at the relatively low number of these moderately oblique shocks considered here (Fig. 3).
An example geometrical situation where multiple oblique shocks with (about) the same inclination angle are likely is a jet with helical magnetic field and along which a number of shocks exist. We assume, that the shocks are propagating along the jet axis and do not modify the jet geometry. The compression by the shock and the decompression of the plasma described in Sect. 2 are then both effective only along the shock normal. Therefore, the decompression will restore the initial upstream magnetic field orientation. This situation can be described by multiple identical oblique shocks. If the magnetic field would not be restored to the upstream direction, the obliquity for subsequent plan-parallel shocks would increase, leading to even harder spectra than described above.
In regions where geometrically uncorrelated shocks exist, like central regions of AGN, a large number of subsequent shocks, through which a single distribution is processed, might be a too strong idealisation. Here, we would have to average over a low number of shocks with different inclination angles . If we consider electrons, in addition effects of losses and magnetic field strength would generally have to be included. However, our results indicate, that very flat synchrotron spectra with index can be produced even at a low number () of oblique shocks.
© European Southern Observatory (ESO) 2000
Online publication: June 5, 2000