V 'o reflects the adiabatic changing of the particle momentum s(t) is the solution of the equation ds/dt = u(s,t) with the boundary conditions s[fVo )= ro (tro ), s(t) = r . This means that neglecting the particle diffusion leads to the situation in which accelerated particles in the region 2 are moving together with the plasma fluxes.

For the region 1 at r > ro, where u = 0, let us suppose that the diffusion coefficient K\(r ) = const and injection of particles into the regime of acceleration is only at the shock front. In this case we obtain the transport equation f = K 1. ( r 2 f dt r2 dr V dr

the solution of which for the boundary conditions

where the function ¿up) is the solution of the integral equation

V ^tToP^ expi - (pp ^ (f )u(f)dt +Ui = ro (()foo (t). (4.21.72)

For the case g\ >> 1 the approximate solution of Eq. 4.21.72 gives

By substituting Eq. 4.21.73 in Eq. 4.21.71 and integrating over t we obtain fi(r,p,t)= L (p,t^exp<[-gi^

where the parameter vdetermines the character of the expansion of the shock wave: v = dlnr,(t)/dlnt; ^(t) = o(t/to)v ; o = r,(to). (4.21.76)

Eq. 4.21.75 shows that the width L of the region before the shock front occupied by accelerated particles at g1 >> 1 is very narrow in comparison with ro (t): L ~ ro/g1 = k/U1. With decreasing of modulation parameter g1 the width L increases up to radius of shock wave ro . The Eq. 4.21.68 and Eq. 4.21.75 described solutions for both regions behind and before shock front. The function fro (p, t)

which enter into these equations can be determined by sewing together the solutions described by Eq. 4.21.68 and Eq. 4.21.75 on the basis of the boundary conditions on the shock front. Therefore we obtain

Ui where

gl g2

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