The effect of finite width of shock wave front

According to Berezhko et al. (M1988) the effect of a finite width l of the shock wave's front can be accounted for approximately by changing the position of the injection source: to put it behind the front at some distance x = l instead of x = 0 as was assumed in the previous Sections 4.21.3-4.21.5. The physical meaning of this shifting is that for a real shock wave the heating of plasma occurs not at the point x = 0 but over the some length l, which may be considered as effective width of shock wave front. It means that the source function for Eq. 4.21.12 will now be

Let us suppose that this source is mono-energetic (i.e. Qo (p) is described by Eq. 4.21.19), and other sources are absent, i.e. /ยป(p) = 0. In this case the solution before the front is described by the same Eq. 4.21.16, but for the region behind the front the solution will be

4npQ

From Eq. 4.2i.24 it can be seen that the taking into account of the finite width of the shock wave's front leads to the appearing of a modulating factor exp(- ^l/K2 ) which sufficiently decreases the flux of accelerated particles if u2l > K2 . Berezhko et al. (Mi988) came to the conclusion that the effective shock acceleration may be realized only for particles for which the diffusion length is bigger than the width l of the shock wave front, i.e.,

This means that for small energy particles with too small diffusion coefficient the Eq. 4.21.25 will not be satisfied and the acceleration of such particles will be not effective.

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