If the loss rate is p = -op (the generalization to different loss rates upstream and downstream is trivial) the basic equation becomes d
(4np2Lf)+ ^(o - 4np2/ (p)ol)= Q - ^np2/(p)- 4napAf (p)dL2 . (4.22.7) at dp dp
In the steady state and away from the source region this gives immediately the remarkably simple result for the logarithmic slope of the spectrum, d ln f = - 3(u1 - 4opL -op2 (dL1/dp)) (4 22 8)
The denominator goes to zero at the critical momentum pcr = (( - u2 )(3aL), (4.22.9)
where the losses exactly balance the acceleration. If the numerator at this point is negative, the slope goes to - ^ and there is no pile-up. However, the slope goes to + ^ and a pile-up occurs if u1 -4u2 + 3op2(dL1/dp)> 0 at p = pcr . (4.22.10)
In early analytic work (Webb et al., 1984; Bregman et al., 1981) the diffusion coefficient was taken to be constant, so dL1/ dp = 0 and this condition reduces to u > 4u2 . However, if, as in the work of Protheroe and Stanev (1998), the diffusion coefficient is an increasing function of energy or momentum the condition becomes less restrictive. For a power-law dependence of the form K ^ p0 the condition for a pile-up to occur reduces to u1 - 4u2 + 0(u1 - u2)—L— > 0. (4.22.11)
Drury et al. (1999) note that the equivalent criterion for the model used by Protheroe and Stanev (1998) is slightly different, namely, u1 - 4u2 + 0(u1 - u2 )> 0 (4.22.12)
because of their neglect of the additional loss process. For the case in which L\l L2 = u^ u1 and with 0 = 1 this condition predicts that shocks with compression ratios greater than about r = 3.45 will produce pile-ups whilst weaker shocks will not.
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