## Acceleration by spherical running shock wave

Let us consider, following Berezhko et al. (M1988), the particle acceleration by spherical running shock wave. The spherically symmetric running shock wave is determined by the low of its extending ro p). The field of plasma velocities in the case in which the undisturbed plasma is in the rest will be u(r, t ) =

\ru(r, t Vr if r < r° (t), [0 if r > r° (().

The plasma leaks on the shock front with the velocity u = dro/dt and flows behind the front with velocity U2 = u^ j, where o is the compressibility of plasma. The particle acceleration by this running shock wave is a sufficiently non-stationary problem, described by Eq. 4.21.48 for the field velocities of matter Eq. 4.21.66. But the main peculiarities of this acceleration investigated in Krymsky and Petukhov (1980), Prishchep and Ptuskin (1981), Berezhko and Krymsky (1982), Drury (1983) can be, according to Berezhko et al. (1984, M1988), analyzed on the basis of approximate solutions of Eq. 4.21.48. The matter of this method is that solutions before the shock front f1(r, p, t) and behind the front f (r, P, t) are expressed through the momentum spectrum on the front fVo (p, t) = f1po,p,t) = f (p,p, t)

which then can be found from the boundary conditions described by Eq. 4.21.49-4.21.50 (as it was made in Section 4.21.12 for the problem of particle acceleration by the standing shock wave in the case of accretion). This method can be applied for any type of shock wave extending. The simple solutions can be obtained more easy for very small parameters of modulation g1 2 << 1 and very big parameters of modulation g12 >> 1.

Let us consider first the problem for the region behind the front r < ro p). If we suppose that the diffusion coefficient here is very small (as in Section 4.21.12), the transport Eq. 4.21.48 can be transformed into

dt 3 dr dr

the solution of which satisfying the boundary condition f (ro, p, t) = f^ (p, t) will be f2 (r, P,t) = frr