S

Here 0)^(kz) = kzV^. The connection between MHD waves and CR propagation is determined by the effective frequency of particle scattering on waves:

v± = 2n2oHkresW±(kres), where kres = ZeH(pc\1, oH = ZeHv(pc)-1 .(3.12.5)

Let us consider the isotropic part of the CR distribution function fo = J f (p,^)dQ/4n. For times At >> v-1 and distances Ar >> vv-1 the transport equation in the diffusion approximation will be do dt

CR particles penetrating into the Heliosphere generate MHD waves along the spiral magnetic field in direction to the Sun (kz < 0); therefore we will use sign '-', i.e.

v~, W-. Because we shell consider only this case, we do not use the index 'z' and the sign '-' any further. Let us introduce the function F (k ) = kW (k). Then we obtain:

- r_2 9 j&H^ 1 - ^ (( ^)-1 f + u f - ^ f = o ,(3.12.8)

dr 4n2Ze r 20 dr 'dr 3r 'dp dE 3uF 2nVAr1 ZeH1 7 , 2 f ZeH f dfo n ^^^ u— +----1 J dpp c 1--= 0. (3.12.9)

For approximate calculations we replace in the integral of Eq. 3.12.8 F (ZeH/pc\ by F ZZeH/pc) and introduce functions F (p ) = F (k = ZeH/pc) and r(p )=r(( = ZeH/pc) . Then from Eq. 3.12.8 and Eq. 3.12.9 we obtain

dr 6n Ze rF(p)sin0 dr dr 3r dp u 'dF{p) + 3uF(p)-4nVAvp\ f = o. (3.12.11)

dr r 3r sin0 dr

For the boundary condition f0(r = r0,p) = fe((), where fe(p) is the CR distribution function out of the Heliosphere, the approximate solution of Eq. 3.12.10 and Eq. 3.12.11 will be fo(r,p) =-( f ))( 3 3); F(p) = 4nVAp4vfo(r,p(3-12.12)

3cr12H1

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