Let us consider, following Berezhko et al. (M1988), some elementary model of particle acceleration by a spherical standing shock wave in the case of accretion. The structure of accreting plasma in the approximation of spherical symmetry is characterized by the increase of plasma velocity u directed towards the gravitational center from zero on infinite distance to some value ui on the distance r = ro of shock wave transition and then have a constant value u2 = ui/a at r < ro
[- U2 at r < ro, where the sign - means that the plasma flux is directed to the center of coordinates. In this case the solution of Eq. 4.21.48 with the boundary condition Eq. 4.21.49, and with injection sources described by Eq. 4.21.50 and Eq. 4.21.51, will be for the outer region (before the shock front at r > ro):
For the region behind the shock front at r < ro the solution can be obtained very easily for a small diffusion coefficient, when K2 << u2ro or g2 << 1 (it can be actually caused by high plasma excitation in this region). This means that in Eq. 4.21.48 can be neglected by the diffusion term in comparison with terms describing convection and adiabatic heating, and in the stationary case for the region r < ro Eq. 4.21.48 can be rewritten as f vu f 0
dr 3 dr
The solution of this equation with boundary condition f (ro, p ) = fro (p) is as follows
The factor exp
in Eq. 4.21.63 reflects the adiabatic increasing of particle momentum in compressing plasma ( Vu < 0).
The equation for the momentum spectrum on the shock front fro (p ) can be found on the basis of the boundary condition described by Eq. 4.21.49 for the mono-energetic injection source (Eq. 4.21.50-4.21.51) and taking into account Eq. 4.21.60 and Eq. 4.21.63:
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