The expression for the flux density of particles (Eq. 2.13.16) can be written in the form (Parker, 1965; Jokipii, 1971):


n(r, Ek, * )dEk is the so called Compton-Getting's factor which was called after the paper (Compton and Getting, 1935).

2.14. Spectral representations of Green's function of non-stationary equation of CR diffusion

2.14.1. Formulation of the problem

In studying CR propagation in interplanetary space, the model of isotropic diffusion including convection and adiabatic deceleration of particles is successfully applied. In recent years a substantial result was achieved in this direction, and first of all in this Section we should notice the solution used above (see Section 2.7) of the equation for the Green's function of the stationary equation of isotropic transfer with arbitrary dependence of the diffusion coefficient on the particle momentum and with the power dependence on the distance (Toptygin, 1973a,b). As to solutions of non-stationary problems which are of primary importance in studying a great number of aspects of the propagation theory (propagation of the solar CR, Forbush effect, the 11-year variation including the hysteresis phenomena, etc), the situation is more complicated in this case. Mathematical difficulties arising in the solution of non-stationary problems are very substantial and it is extremely difficult to obtain closed expressions for a solution of non-stationary equations describing the actual physical situations. Having no claim on a complete solution of the problem described, Dorman and Katz (1977a,b,c) considered some simplest models of non-stationary propagation of CR in a medium with the constant diffusion coefficient. As will be shown below, it is possible in these models to find a spectral expression for the Green's function of the equation of a transfer of CR. The assumption of constancy of the diffusion coefficient is, of course, an idealization, but we hope that in future it will be possible to generalize the considered class of non-stationary solutions for the Green's function taking into account a dependence of the diffusion coefficient on a distance and from particle momentum.

2.14.2. Determining of the radial Green's function for a non-stationary diffusion including convection

Consider initially a non-stationary diffusion of CR including their convective transfer by radially expanding plasma of solar wind. If we neglect the process of adiabatic variation of particle energy, the particle density n(r, t) satisfies the equation of a non-stationary diffusion including convection:

where k is the coefficient of particle diffusion, uo is the solar wind velocity which is assumed to be directed along a radius away from the Sun, i.e. uo = uo r/r ,

Qo (r, t) is the source function. Including the radial dependence of solar wind velocity uo, we write the Eq. 2.14.1 in the spherical coordinates dn (A d i

where the notations are used. In Eq. 2.14.2

V r2 dr' dr' A0<P r2sin039^" 3d' d<2 r2sin^' represents the radial and angular parts of Laplacian. Writing Eq. 2.14.2 in the form n + Q1, (2.14.5)

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