The first term in Eq. 4.21.86 is an exponential in n times a Poisson probability of > n acceleration events, and the second term, corresponding to a finite probability of residence time T = t, is a Poisson distribution at (n) = rat. Usually re << ra so the result (in terms of momentum) is a power law spectrum with a hump and subsequent cutoff after ~ rat acceleration events. A more complicated analytic expression can be derived for the more realistic case in which ra and re depend on n (and particle momentum).

The following system of differential equations can be shown to be equivalent to the above approach, and is more convenient for computations. Ruffolo and Channok (2003) express P(n, t) as the sum of E (n, t) and A(n, t), the fraction of particles escaping and remaining, respectively, after n acceleration events at time t. Then dAT = -(,n + re,n K + ^n-1 An-1; ^ = ^-1 An-1 (4.21.87)

dt dt with the initial condition A(0,0) = 1 and all other A, E equal to zero at t = 0.

For a general shock angle, we may use ra,n and re,n which depend on the particle velocity vn (following Drury, 1983):

u1 u2

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