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-exp

Therefore an additional factor (x0/G)exp(v/ (v- l) ) will appear in the expression for the free path A in Eq. 2.7.20. Basing on the obtained relations and using observational data on the spectrum of magnetic field inhomogeneities, one can estimate a free path of low energy particles in the interplanetary space. However, the experimental data obtained by various authors in different time, are considerably different. Using, for example, the data from (Jokipii and Coleman, 1968) and estimating the collision width of the Cherenkov resonance, we find x0 = 0.9 for protons with the energy ~ 1 MeV. This means that a weakening of scattering at x ~ x0 is small. The estimate of a free path according to Eq. 2.7.4 gives the value of the order of 1 AU This value of A// is in agreement with the observational data presented in (Vernov et al., 1968a). On the other hand, a free path calculated from the data of (Sari and Ness, 1969, 1970) appears to be more than 1 AU. It should be emphasized that if the main contribution in the observed spectrum of magnetic field inhomogeneities is caused by hydromagnetic discontinuities (as was assumed by Sari and Ness, 1969), the developed theory can appear to be in applicable since a particle can be scattered at a large angle immediately when passing through a discontinuity.

2.8. Kinetics of CR in a large scale magnetic field

2.8.1. The kinetic equation deriving on the basis of the functional method

The problem of propagation of CR in a large scale magnetic field has been discussed in Toptygin (1973a, M1983) on the basis of the drift kinetic equation. The quasi-linear approximation, which permits describing in unique way processes of CR scattering and their diffusion across the lines of force of a regular magnetic field, was used by Toptygin (1973a, M1983) to derive the kinetic equation for the average distribution function of CR. The problem of the diffusion of CR in a large scale field has been discussed by other methods in Ptuskin (1985) and Zybin and Istomin (1985). A kinetic equation describing the propagation of CR in a large scale field was derived by Dorman, Katz and Stehlik (1988) on the basis of the functional method (Klyatskin, M1975; Rytov et al., M1977; see also above, Section 2.2).

In order to describing the motion of CR particles in a strong magnetic field we shall use the equations of motion in the drift approximation (Sivukhin, 1963)

where R(t) is the radius vector of the guiding center, P± and P// are the transverse and longitudinal components of the particle's momentum with respect to the direction of the magnetic field H(r,t), h = H/H, and V± and V// are the components of the particle's velocity perpendicular and parallel to the magnetic field. The large scale field H(r,t) has regular Ho and random H1(r, t) components:

If the random component is much larger than the regular one, then expanding Eq. 2.8.1 and Eq. 2.8.2 into series in powers of the random field to within the accuracy of second-order terms and averaging the equations obtained over the directions of the particle's momentum in the plane perpendicular to the regular field Ho , we obtain:

where

Ofr = -npAar(n)-(1l2)naAPy(n\ Aay(n)—Say- nanr, n — Ho/\Ho\, H1a± — Aa/(n)H/ — Aa»^, (2.8.7)

and r(t) is the radius vector of the guiding center of the particle averaged over the directions of the particle's momentum vector in the plane perpendicular to the field Ho, which is assumed to be uniform in space and constant in time, and pL and p,, are the components of the particle's momentum vector across and along the direction of the magnetic field Ho . The random field H1(r,t) in Eq. 2.8.4-2.8.7 is measured below in units of Ho .

The system of Eq. 2.8.4-2.8.6 satisfies the condition (Sivukhin, 1963)

dt dpL dt dp n dt a consequence of which is the Liouville theorem f — f VAf + ¿pLf + ±1Lf — 0, (2.8.9)

dt dt dt dpL dt dp// dt i.e., the equality to zero of the total derivative of the distribution function f (r,pl,p//,t), calculated along the drift trajectory of motion of the particles. If particles moving in a large scale field interact with very small scale inhomogeneities of the magnetic field, it is necessary to supplement the Eq. 2.8.9 with a collision integral, which determines the variations of the distribution function as a consequence of the scattering of particles in the very small scale random magnetic field:

It is necessary to average Eq. 2.8.10 over the ensemble of realizations of the large scale random field:

where F = (/) is the average distribution function. We shall make use of the functional method of Klyatskin (M1975) and Rytov et al. (M1977) to carry out the averaging. We shall write the average which figures in the second term in Eq. 2.8.11 in the form

(df/) = V""aF + V//(Hai(r,t)/) + tHy(r,t)/) . (2.8.12)

For the averaging of the second and third terms on the right hand side of Eq. 2.8.12 we shall make use of the well known formula (Klyatskin, M1975; Rytov et al., M1977):

(H^(r, t)/ [H1] = 0 dt1 J dr^r, t;r1, t1 ))' (2'8'13)

which is valid for any Gaussian field H1(r, t) with zero average value and correlation tensor

Bp/u(r, t;r1, t1 ) = (H1^(r, t )^1^(r1, t1)) . (2.8.14)

It is necessary for performing the averaging to calculate the functional derivative 8/ [H1 ]] t1). We shall make use for its calculation of Eq. 2.8.10

= -V//8(t -11 ){Vir/l8(r - r1) - (v±^8(r -11 )Opi)}[H1 ], (2.8.15)

The formal solution of Eq. 2.8.15 is of the form

= -JdP1V1 //{{(r,t;r1,t1 )vi^ - (virMGW1 (r,t;r1,t1 X°Pi)}[H1,rbPl,hi (2816)

where GppL (r,t;rbti) is the Green's function of Eq. 2.8.15, p = {,p//}, and dp = p±dp± . Taking Eq. 2.8.16 into account we obtain from Eq. 2.8.13:

(H1ß(r, t )f [H1] = - J dt1 J dr1<ip\V1 //Gpp1 (r, ^1, ^)

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