Results

Our numerical results are illustrated in Figures 1-5. In these figures the logarithm of the distribution function is plotted as a function of , so that a power law distribution corresponds to a straight line. The absolute values of f(p) and of p are unimportant. The synchrotron cutoff momentum, , is a free parameter and is chosen to be either three () or six () orders of magnitude above the injection momentum. All the shocks have the same strength, specified by the value of r, and the calculations are performed both for strong shocks with r=3.8 and for shocks with r=2.0. The adiabatic decompression after each shock moves the curve to the left (by ), without changing its shape, so that after N shocks the lowest energy particle in the distribution has .

In Figure 1 the evolution of a single distribution injected at the first shock is shown after 1, 5, 25 and 50 shocks, all with r=3.8 and with the synchrotron cutoff . The sharply-peaked distribution is immediately after the first shock, before adiabatic decompression, and the other curves peaking to the left of the first are after 5, 25 and 50 shocks, respectively. The formation of a plateau (the nearly horizontal portion of the curve) is evident after 50 shocks.

 
Figure 3: The cumulative effect after many shocks with injection at each shock; (a) the distribution, (b) the slope of the distribution for : r=3.8, , N=1, 10, 30, 50.

In Figure 2 the distribution shown in Figure 1 for N=50 is compared for the same case without synchrotron losses. In effect all the particles that would be above the synchrotron cutoff in the absence of synchrotron losses (dashed curve) appear in a hump just below the synchrotron cutoff when synchrotron losses are included (solid curve). This hump is a manifestation of the `pile up' effect due to synchrotron losses, as discussed in §5 below.

 
Figure 4: As for Figure 3, except for r=2.0 and N=1, 20, 80, 200.

 
Figure 5: As for Figure 3, except for .

In the theory of DSA it is usually assumed that there is injection at every shock. Hence the cases shown in Figure 1 and 2, where there is only a single initial injection with this distribution subjected to many shocks, is not realistic in practice. One expects the distribution after N shocks to consist of the sum over the distribution injected at the first shock subjected to N shocks, the distribution injected at the second shock subjected to N-1 shocks, and so on to the distribution injected at the Nth shock subjected to only one shock. This sum is performed in evaluating the distributions shown in Figure 3. In Figure 3a the curves correspond to the sums after N=1, 10, 30, 50, with the innermost distribution being for N=1 (which is the same as the N=1 case in Figure 1) and the outermost curves being for N=50. An inflection develops in the distribution just below the synchrotron cutoff. To illustrate this more clearly, the slope of the distributions are plotted in Figure 3b. The lowermost curve is for N=1; at low it shows a power law with index b=4.07, corresponding to b=3r/(r-1) with r=3.8, cf. (1); the slope steepens as the synchrotron cutoff is approached. As N is increased the slope decreases monotonically and approaches b=3 at low , in accord with theoretical predictions (White 1985; Achterberg 1990; Schneider 1993; Melrose & Pope 1993; Pope & Melrose 1994). Nearer the synchrotron cutoff, after about 10 shocks, a peak in the slope starts to develop and becomes increasingly prominent with increasing N. This peak may be attributed to the contribution from the plateau-like portions of the distributions resulting from injection at the earliest shocks.

In order to illustrate that the effects shown in Figure 3 are not unique to very strong shocks, we performed calculations for weaker shocks with r=2.0. The results are shown in Figure 4. As in Figure 3, an inflection in the distribution develops just below the synchrotron cutoff after many shocks. The development of this feature is slower for weaker shocks, and more shocks (up to N=200) are included in Figure 4 than in Figure 3 (up to N=50). The slope of the distribution for N=1 is b=6.0 at , corresponding to b=3r/(r-1) with r=2.0, and it increases towards b=3 with increasing N. The peak in the slope (Figure 4b) just below the synchrotron cutoff is somewhat broader, with the peak at a somewhat lower momentum, than for the stronger shocks. We also performed calculations for 100 shocks of random strength, with r chosen as a random variable between 1.5 and 4.0, and the results are similar to those shown in Figure 2-4.

To study the effect of increasing the range between the injection and the synchrotron cutoff, we repeated the calculations in Figure 3 for . The results are plotted in Figure 5. Compared with Figure 3, Figure 5 shows that the portion of the distribution with (corresponding to a flat synchrotron spectrum) extends over most of the wider range , with the peak in the slope remaining essentially unchanged at .

© Copyright Astronomical Society of Australia 1997