Interpretation

The foregoing results show three notable effects of synchrotron losses on multiple DSA: (a) it provides a high-p synchrotron cutoff (denoted ) beyond which no particle can be accelerated by DSA; (b) for a single initial injection, a plateau distribution, , develops at ; (c) the cumulative effect of injection at every shock leads to a distribution for ; and (d) the distribution in (c) has a slope that rises gradually to a peak (with at in Figureá 3).

á 
Figure 6: The effect of synchrotron losses on initial power law distributions with b=3 (upper), b=4 (middle) and b=5 (lower).

In the following discussion an important (and long-known) effect of synchrotron losses plays a central role: synchrotron losses tend to steepen a distribution with b>4 and to cause a turn-up in a distribution with b<4. This is illustrated in Figureá 6 where the initial distribution is a power law () that extends to . After a time t the particles initially with have , which is the synchrotron cutoff in this case. A distribution with b>4 initially becomes steeper (both with increasing p and increasing t) with for . The distribution with b=4 does not change in shape and cuts off abruptly at . A distributions with b<4 initially develops a pile up with for .

The formation of a plateau distribution for a single initial injection subjected to many shocks can be understood in terms of two effects. One effect is that multiple DSA tends to flatten the distribution towards the asymptotic distribution . Thus, although DSA at a single shock (in a plasma with ratio of specific heats 5/3) cannot produce a distribution flatter than b=4, and b=4 only for the strongest possible shock with r=4, multiple DSA can produce a distribution with b<4. The other effect is that once a distribution with b<4 forms, synchrotron losses tend to cause electrons to pile up just below the synchrotron cutoff, cf. Figureá 6. Together these effects account for the distributions in Figureá 3-5, with b close to 3 well below and a peak in the slope just below the cutoff at .

Schlickeiser (1984) showed that the combination of (second-order) Fermi acceleration and synchrotron losses causes a `pile up' just below synchrotron cutoff, and our result is related to Schlickeiser's result. The combination of DSA and decompression should lead to a Fermi-like acceleration mechanism, in the sense that the combination may be described by a diffusion equation in momentum space. Hence, the asymptotic solution for multiple DSA should approach the asymptotic solution for Fermi acceleration: for constant injection at this is a plateau (b=0) for and is b=3 for . The synchrotron losses provide a high-p barrier that prevents particles from diffusing to very high p, and this may be regarded as a reflecting boundary in momentum space. This reflection acts like a source of particles at the synchrotron cutoff so that one expects the asymptotic spectrum to approach a plateau just below the cutoff. The tendency to form a plateau distribution for a single initial injection, cf. Figureá 1, is a manifestation of this effect. When expressed in terms of the energy spectrum of the electrons, , with for highly relativistic particles, and hence
á 

a plateau momentum distribution implies an energy spectrum . Thus our results show that DSA combined with synchrotron losses produces a pile up similar to that found by Schlickeiser (1984) for Fermi acceleration combined with synchrotron losses. However, in the more realistic case where there is injection at each shock (which would be simulated by constant injection in Fermi acceleration) the asymptotic distribution is or , becoming somewhat flatter just below the synchrotron cutoff. This portion of the distribution with b<3 implies that it is possible in principle for the model to account for weakly inverted spectra (), but only at relatively high frequencies, corresponding to emission by electrons with momenta just below (around according to Figureá 3).

á© Copyright Astronomical Society of Australia 1997