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In Eq. 2.4.28 it was included that the first argument in Bad describes a slow variation of mean square of a stochastic field with a distance (see Eq. 2.4.23), and therefore one can neglect the action of the operator exp(- L0t) on this argument. At stochastic velocities ui = 0 Eq. 2.4.27-2.4.29 transform into the equations which have been derived for the first time by Dolginov and Toptygin (1966a) using the diagrams technique.

Let us elucidate the character of the approximations which were made in deducing the kinetic equation. Deducing Eq. 2.4.26-2.4.29, we have closed the chain of equations postulating F2 = 0 . This assumption holds true in the case in which the corrections to the distribution function F connected with including the subsequent terms of the functional series Eq. 2.4.15 are small. In the case under consideration, however, it is not necessary to calculate the subsequent vanishing approximation but is possible to use the known quantum mechanical analogy (see, for example, Bonch-Bruevich and Tyablikov, M1961) according to which the approximation based on the assumption F2 = 0 corresponds to Bom's approximation for perturbation theory. If a magnetic field is stationary in time the conditions of applicability of Born's approximation in the case under consideration mean that variations of a particle momentum S in a stochastic field H1 are small compared to the particle momentum p. The ratio of these quantities

P cP rL1

determines the condition of applicability of the considered approximation. As is seen, the condition reduces to a small value of the ratio of the correlation radius of a stochastic field to a particle Larmor radius rLi = cpjJ in the stochastic field. This condition means that a particle is scattered at a small angle 9 ~ lc/rn by every inhomogeneity. Thus an application of the diagram technique with a Gaussian distribution of inhomogeneities (Dolginov and Toptygin, 1966a, 1968a,b) and approximation F2 = 0 in the functional method (Dorman and Katz, 1972a,b; 1974a,b) resulted in the kinetic equation with a collisional term which is determined by the correlation tensor of the second rank of the stochastic magnetic field. As was noted, in this case particles are scattered at a small angle by every inhomogeneity. To consider the cases when particles are scattered at large angles, one should take into account the correlators of the higher ranks in the kinetic equation. According to (Katz, 1973) the functional method makes it possible to exceed the limits of Born's approximation and to take into account the triple correlation, i.e. to consider the cases when a particle in interaction with a separate inhomogeneity of magnetic field is scattered at large angles.

2.5. Kinetic equation for propagation of CR including electric fields in plasma

In actual conditions, a turbulent motion is not obliged to be a set of some separate inhomogeneities of a magnetic field. Very often it can be represented as a set of weakly interacting collective oscillations of a medium. These oscillations in the solar wind have the most frequently a form of AlfVén waves. One should take into account the presence of the random electric fields of oscillations in plasma in addition to the turbulent magnetic fields, considering the interaction of charged particles of cosmic radiation with the magnetized plasma of the solar wind (Dorman and Katz, 1972b). In this case the kinetic equation has the following form:

^ + L0jF(r,p,t) = [Cj _jd_pT^exp(-L0T)TaÀ(r,p;r'F(r,P,i)) ,(2.5.1) where

TaÀ = eafiYeÀvv/3v'ju{H1yH 'v) + c£a/3yvp{H1rE'à)

and E are the random electric fields of oscillations and the dashed symbols correspond to the field components depending on r'. If the regular magnetic field can be considered as homogeneous in space at distances of the order of the correlation radius of a stochastic field, the variations of the radius-vector and of the momentum of a particle are determined by the expressions:

Ar(r) = R(0,r,r;p) = r + (hv)vr + [[jiv]]]^ - [hv]1-^^ , (2.5.3)

mL mL

Ap(r) = P(0,r, p) = (hp)h - [h[hp]]cosffi>LT + [hp]sinffi»LT, (2.5.4)

where h = H0/H0 and a>L = ecH0/E is the Larmor frequency of a particle with total energy E. In this case Eq. 2.5.1 takes a form d

The Eq. 2.5.1-2.5.5 at the given components of corresponding correlation tensors, completely describe the process of a propagation of CR in the random electromagnetic fields. The collisional terms of these equations include both elastic and inelastic particle interactions with magnetic field inhomogeneities (or with turbulent pulsations of electromagnetic fields of oscillations). These equations give the most complete description of the spatial and angular distribution and also of variation of energy spectrum of galactic and solar CR in their interaction with space plasma.

2.6. Kinetic equation for the propagation of CR in the presence of a strong regular field in low-turbulence magnetized plasma in which the Alfven waves are excited

2.6.1. Formulation of the problem and deduction of the basic equation

Experimental studies of CR of low energies (i-i0 MeV) showed that the free path of these particles in interplanetary space exceeds i AU (see, for example, Vernov et al., i968a). As a result of this fact there was found a presence of the pronounced anisotropy in the angular distribution of particles moving from the Sun. To study a propagation of such particles one should directly use the kinetic equation. The first theoretical treatment of the processes of propagation of low energy CR were carried out by Tverskoy (i967b, i969) who paid his greatest attention to an analysis of the effects of particle acceleration in the interplanetary space. The formulation of the problem proposed by B. A. Tverskoy was used, however, as the basis in a majority of the subsequent studies where a propagation of charged particles in cosmic conditions was investigated. The most detailed consideration of the process of multiple scattering of low energy charged particles by stochastic inhomogeneities of a magnetic field has been carried out by Galperin et al. (i97i) and Toptygin (i973a,b). In this Section, basing on the kinetic equation we consider a motion of low energy charged particles through weakly turbulent magnetized solar wind plasma in which Alfven oscillations are induced. Note that the presence of Alfven waves in the solar wind plasma is confirmed by direct measurements (see, for example, Belcher and Davis, i969, i97i). When analyzing the motion of charged particles, together with particle scattering on the turbulent pulsations of the magnetic field we consider also the energy exchange between turbulent pulsations and charged particles owed to particle interaction with stochastic electric fields of Alfven waves.

A propagation of CR in low turbulent magnetized plasma in which the Alfven waves are excited has been studied in Toptygin (i97i), Dorman and Katz (i972b). The process was considered of a propagation of particles the Larmor radius of which is far less than the correlation radius of a random magnetic field.

The particles of relatively small energy satisfy this condition and the magnetic field should be strong enough. The particles with energies up to ~ i000 MeV satisfy this condition (cp/eHo << lc ) when CR propagate in the interplanetary field. The collision term of the kinetic equation in the case under consideration is determined by the right hand side of Eq. 2.5.4:

StF = e —— JdT/p; R(0,t,0; p), P(0,t, p)] — , (2.6.i)

where R and P are determined by the Eq. 2.5.3 and Eq. 2.5.4 and Taj is determined by Eq. 2.5.2. If the Alfvén waves are excited with the frequency o(k) = vakz - Y(k), (2.6.2)

where va is the Alfvén velocity, /(k) is the fading coefficient of the Alfvén waves with the wave vector k, then the electric and magnetic fields of the oscillation in these wave are connected by the relation

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