General theory of CR propagation and acceleration in space plasma with two types of scatters moving with different velocities

According to Dorman and Shogenov (1985, 1990) the CR distribution function in the presence of two types of scatters will be described by the well known Boltzmann equation with collision term a^jpO + v^jpO + Ze [v - u1,B(r,,)]]pil = Sf , (4.20.1) dt dr c dp where v and p are the velocity and momentum of the CR particle, B(r, t) is the frozen magnetic field in the background plasma,

Stf = n(r,t)v - u2|a(jv - u2\,a)(r,p',t)- f (r,p,t)}', (4.20.2)

and the distribution function of second type of scatters is p(r, U2, t) = «(r,t)^(u2 - (r, t)). (4.20.3)

Let us suppose that B(r,t) = Bo (r, t)+ Bi(r, t), where Bo (r,t) = (B(r, t)) and (Bi (r, t)) = 0. Then f (r, p, t) = F(r, p, t)+ f1(r,p, t), where F(r, p, t) = (f (r, p, t)) and (f1(r,p,t)) = 0 . According to Eq. 4.20.1 and Eq. 4.20.2 we obtain aF(pi) + v apW) + Ze [v - u1,ffl(r,t^fcpO = StF + st1F , (4.20.4) dt dr c dp where ®(r, t) = ZecBo (r, t)/E . Here Ze and E are electrical charge and total energy of CR particle and

StF = «(r, t )J| v - u2^jv - U2\,a) (r, p', t)-F (r, p, t)}', (4.20.5)

where

Dik=

is the diffusion coefficient in the momentum space and

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