Introduction

Diffusive shock acceleration (DSA) is the favored acceleration mechanism for relativistic electrons in most synchrotron sources (e.g., the reviews by Drury 1983; Blandford & Eichler 1987). DSA results naturally in a power law electron energy or momentum distribution, and hence a power law synchrotron spectrum. In terms of the momentum distribution function, a power law distribution is of the form , where p is the momentum and b is the power law index. A power law synchrotron spectrum of the form , where is the frequency, corresponds to electrons with . DSA at a single shock gives a power law distribution with b=3r/(r-1), where r is the compression ratio of the shock, which has a maximum value r=4 for strong shocks in a nonrelativistic gas with ratio of specfic heats 5/3. Hence the flattest distribution that can be produced by a single shock has b=4 and hence .

Flat radio spectra () in some extragalactic sources are usually attributed to self absorption (e.g., Kellermann & Pauliny-Toth 1969). For the self-absorption to produce a flat spectrum over at least a decade in frequency (as often observed) requires specific geometric properties in the source (e.g., Marscher 1977), leading to description of the self-absorption interpretation as a `cosmic conspiracy' model (Cotton et al. 1980). There are nonthermal sources with flat spectra in the Galactic Centre region (e.g., Yusef-Zadeh 1989) which are not plausibly self-absorbed. Thus, at least for the galactic sources with flat synchrotron spectra, a more plausible explanation is that the the electron distribution is flatter than b=4 (e.g., Melrose 1996). It is known that DSA at a sequence of shocks (``multiple DSA'') tends to a distribution after an arbitrarily large number of shocks (e.g., White 1985; Achterberg 1990; Schneider 1993; Melrose & Pope 1993; Pope & Melrose 1994), and this does imply a flat synchrotron spectrum, , as required. However, an explanation of flat spectra in terms of multiple DSA in this way has some unsatisfactory features: the approach to the asymptotic distribution is slow and requires that a large number of shocks propagate across the acceleration region; the particles must escape from the system before they have time to cool due to synchroton losses; and multiple DSA cannot account for even weakly inverted spectra, . However, the relation between the actual particle distribution (which in inhomogeneous due to the shocks, and only approaches asymptotically) and the synchrotron spectrum involves an integral over the entire source, and this needs to be modelled in detail to determine the actual synchrotron spectrum.

In this paper we show that synchrotron losses combined with multiple DSA can be efficient in forming flat and inverted synchrotron spectra. The underlying idea is threefold. First, it is known that the effect of synchrotron losses on a power law distribution is to steepen a distribution with b>4, and to cause a pile up (an integrable divergence) for b<4, as we show explicitly in §5 below. Second, multiple DSA is known to form a curved distribution with the flatter portion (b<4) moving to higher energies as the number of shocks increases. Third, synchrotron losses, which are most important in the compressed-B region just downstream of the shock, limit the maximum p to which the particles can be accelerated. We refer to this maximum p as the synchrotron cutoff, which is defined in a more formal manner below. Combining these ideas, when multiple DSA produces a slope b<4 just below the synchrotron cutoff energy, synchrotron losses tend to flatten the distribution even further. Our objective in this paper is to describe this flattening in detail using a simple numerical model. In this model all processes (DSA, synchrotron losses and adiabatic decompression here) are treated as independent, described by relatively simple operators, and a combination of processes is treated by applying the appropriate operators sequentially.

The model is described in §2. For the combination of DSA and synchrotron losses the assumption that the processes may be treated sequentially is justified in §3. Our results are presented in §4, and their interpretation is discussed in §5. Our conclusions are summarized in §6.

© Copyright Astronomical Society of Australia 1997