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

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1. Introduction

The theory of first order Fermi acceleration is commonly used to understand non thermal particle spectra or source distributions of synchrotron radiation in various astrophysical objects (for reviews see Drury 1983; Blandford & Eichler 1987; Kirk et al. 1994). In the diffusion approximation the steady-state spectral index of accelerated test particles is solely a function of the compression ratio of the shocked plasma flow. For strong shocks, the compression ratio is given by [FORMULA], where [FORMULA] and [FORMULA] are the downstream and upstream densities respectively. This leads to a phase space distribution of accelerated particles [FORMULA] with the canonical spectral index [FORMULA]. The synchrotron emission of such an (e.g.) electron distribution as a function of frequency is given by [FORMULA], where [FORMULA]. Unless the injection into the acceleration process is very effective, (leading to a strongly modified shock structure), this is the hardest spectrum which can be produced through first order Fermi acceleration by a single shock for which the diffusion approximation is valid. In complex large scale structures like the galaxy or active galactic nuclei (AGN) particles may encounter several shocks. If the transport time between shocks is larger than the acceleration time at a single shock, the problem can be described by subsequent shocks, where (in the case of planar geometry) the downstream distribution is transported (and decompressed) upstream of the next shock.

It is well known from the theory of diffusive shock acceleration, that the supply of an upstream power law distribution with the canonical spectral index leads to an amplification of the distribution, and the spectral index is not changed. However, the increased number of high energy particles is accompanied by a decrease of the number density at the low energy cutoff, leading to a flattening of the distribution at intermediate energies. This can be seen by calculating the spectrum as it is processed through a number of shocks, suppressing new injection, and considering adiabatic decompression, as shown by Melrose & Pope (1993). At sufficiently high energies, a power law with the canonical spectral index is always revealed. This applies for multiple identical shocks without losses, which are subject of this work. 1

In the limit of an infinite number of subsequent shocks with injection at each shock, the flattening of the spectrum (compared to a single shock) extends even to the highest energy particles, with a momentum dependence of [FORMULA] (White 1985; Achterberg 1990).

Since the modification of the particle spectrum evolves from the low energy part of the spectrum due to further acceleration and adiabatic decompression, effects of injection can be very important. Including a theory of the injection process is well beyond the scope of this work. However, given a momentum [FORMULA] at which particles are injected, we assume, that this is the momentum that divides particles which are able to diffuse across the shock from the thermal pool. Adiabatic decompression can shift the momentum of some particles to [FORMULA]. These particles are considered to be `re-thermalised' and no longer take part in the acceleration process and the resulting spectrum.

In order to compare our numerical work with analytical treatments we first consider subsequent identical shocks with a quasi-parallel magnetic field, leading to diffusive acceleration. The relative orientation of magnetic field and shock normal is, however, very likely to be oblique. Then, the diffusion approximation may no longer apply. To investigate the principal effect of a low number of multiple oblique shocks, we use for each shock the same inclination angle [FORMULA] between magnetic field and shock normal in the upstream rest frame. We use a fixed escape probability [FORMULA] for particles during the propagation between subsequent planar shocks to account for effects of finite shock extension. 2 The acceleration process is described by the shock-drift mechanism. We take into account that multiple shock encounters are possible. For highly oblique geometry, repeated reflections off the compressed downstream field are the most important acceleration process. The transport upstream and downstream is of diffusive character, but the phase space distribution at the shock can be highly anisotropic, and the standard diffusion approximation of the acceleration problem (namely, [FORMULA] is only to second order anisotropic) does not apply.

Even if we consider shocks with different inclination angles [FORMULA], each shock will produce locally a characteristic pitch-angle distribution, and spectral index. Only during the transport from one shock to the next, the pitch-angle distribution is isotropised. Anastasiadis & Vlahos (1993) used a random pitch angle before every electron shock interaction. This isotropisation produces steeper spectra as opposed to pitch-angle scattering, as shown by Naito & Takahara (1995).

After referring to some analytical results in Sect. 2, we compare our Monte-Carlo results for multiple identical planar shocks to analytical calculations in Sect. 3.1, where we also include effects of finite spatial extend of the shocks, and re-thermalisation. In Sect. 3.2 we present results for multiple oblique shocks.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000