Alternative treatment of DSA with synchrotron losses

Our sequential procedure for treating synchrotron losses involves neglecting the synchrotron losses so that DSA forms a distribution that extends well beyond the synchrotron cutoff, and then allowing synchrotron losses to modify this distribution. In order to check the validity of this sequential procedure we treat DSA in another manner that allows one to include the acceleration and the synchrotron losses at the same time, rather than sequentially. Our numerical results show that the two procedures produce indistinguishable results. Here we summarize the alternative procedure and outline an analytic proof that the two procedures are equivalent.

The alternative method is an adaptation of that explained by Achterberg (1990). An individual electron gains in one cycle time, , where a cycle involves crossing the shock from upstream to downstream and back to upstream again. There is a probability, P(p), of an electron escaping downstream in each cycle. For a highly relativistic particle, the theory implies
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where is the shock speed; depends on the mean free paths of the electrons in the upstream and downstream regions. Then the distribution of electrons escaping downstream is
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where the normalization of f(p) depends on an arbitrary injection rate. The integral in (5) with (4) is elementary, and the result reproduces (1) for an initial distribution .

Synchrotron losses may be included in (4) and (5) by including the synchrotron losses in the calculation of in each cycle. The average acceleration rate over one cycle, , is then modified to
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where the additional term describes the synchrotron losses. The synchrotron cutoff corresponds to , and no electron can be accelerated to beyond this cutoff (f(p)=0 for ). The resulting distribution of electrons then follows from (5) with reinterpreted in this way, that is, with the replacement , where is the synchrotron loss in one cycle. The integral can be performed analytically and the result is the same as is obtained using the sequential procedure (3). The effective synchrotron loss time, in (3), found by equating the two results, is
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Thus, except for very weak shocks (), (7) implies and the change over any one cycle is small, justifying our use of the discrete change in one cycle to determine a continuous changes over many cycles.

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Figure 1: The effect of multiple DSA and synchrotron losses is illustrated for the distribution resulting from a single initial injection: r=3.8, , N=1, 5, 25, 50.

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Figure 2: Comparison of the distributions with (solid curve) and without (dashed curve) synchrotron losses for injection only at the initial shock: r=3.8, , N=50.

The foregoing proof of equivalence of the two treatments of DSA and synchrotron losses applies for an initial distribution . The generalization to an arbitrary initial distribution, say, follows simply by applying the operation . (In effect the solution for an initial -function distribution is a Green's function for the general case.) This establishes the equivalence of the two procedures: the sequential procedure is exact within the framework of this alternative treatment of DSA. Thus we are well justified in using the sequential procedure in our numerical calculations.

б© Copyright Astronomical Society of Australia 1997