Acceleration by spherical standing shock wave in the solar or stellar wind

According to Parker (M1963) and Baranov and Krasnobaev (M1977) the dependence of plasma velocity in its dependence on the distance to the star at the presence of the standing shock wave at the distance r = ro can be described as follows:

where = u\j a. As was shown in Webb et al. (1981), Forman et al. (1981), Webb et al. (1983), the analytical solution of Eq. 4.21.48 can be obtained if the diffusion coefficient has the form

\K1(r/ro ) if r < ro• [k2 (r/ro ) if r > ro•

where K1 and k are constants.

By using Fourier's transformation, for the stationary case and injection sources described by Eq. 4.21.50 and Eq. 4.21.51 the following solution was obtained, following to Berezhko et al. (M1988):

g2ro

2r 2

g2ro

2r 2

where

From Eq. 4.21.54 and Eq. 4.21.55 it follows that in the case considered there is important adiabatic deceleration of particles in the region 2 at r < ro and the finite dimension of the shock wave which leads to additional runaway particles in region 1 at r > ro from the neighborhood of the shock wave front compared with the case

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