Here J0 (x) is a modified Bessel function. We returned to the variables z, 0 in Eq. 2.7.7 and Eq. 2.7.8.
At 00 = 0 the expression for the Green's function Eq. 2.7.7 transforms to the expression derived for the first time in (Galperin et al., 1971) where it was used to explain the cases of anisotropic propagation of particles with the energy 1-5 MeV observed by direct measurements in the interplanetary space (see below, Section 2.10).
The Green's function Eq. 2.7.7 describes a distribution of particles ejected by a point source. When considering the concrete cases one should know the actual source function (it is possible that the conditions in a process of particle propagation which are constrained by conservation of the adiabatic invariant sin2 o0h result in an insignificant of the character of the angular distribution of particles in a source). We should note that it is necessary to know Green's function of the kinetic equation when we analyze the finer questions of a kinetic of CR pertaining to fluctuation effects arising in their motion in the interplanetary space (see below, Section 2.8).
The Green's function of the kinetic equation can be derived in a non-stationary case as well as in the case when the particles propagate in a diffusion way across the regular magnetic field. We now write the resulting expression
G = G// (z, 0; zo ,0o )±(x, y, z; Xo, yo, zo), (2.7.9)
where xo, yo, zo are the source coordinates, G// is the Green's function of the field-aligned particle motion determined by Eq. 2.7.7, and
z z o o is the Green's function of the transversal motion of particles, A± is the free path of a particle across the lines of force of regular field (Toptygin, 1973a,b).
Now we consider particle scattering in the range of angles in which the inequality |cos0| = x<< 1 is satisfied (Galperin et al., 1971). The coefficient of diffusion D00 in angular space as a function of 0 is somewhat decreased as 0
grows of owing to a decreasing contribution of the cyclotron resonances of higher orders. At the same time the presence of the second summand 3 (see Eq. 2.7.2) including Eq. 2.6.22-2.6.23 related to Cherenkov resonance starting from some 6 value in D66 results in the increase of D66. This behavior of the diffusion coefficient is repeated at 6>n/ 2, so that D66(6) = D66(n-6). Thus there are two maxima on the curve of the dependence of D66(6) at x = xo = |cos6o| on the location and depth, depending on the relation between Cherenkov's and cyclotron summands. It can be shown (Toptygin, 1973a,b) that the value of xo is determined by the expression
It appeared that the values of free path and isotropization time are substantially different for the cases in which xo = 1 and when xo << 1. In the former of these cases the minimum is not deep or is even absent and it does not affect considerably the scattering of particles. As in the region of Cherenkov resonance the scattering is rapid, the isotropization time is generally determined by the range of angles 0 < 6 < 1, i.e. by the range where the expression for D1 determined by Eq. 2.6.23 is applicable (or, in other words, the field-aligned free path is determined by the Eq. 2.7.4). In this case the isotropization time is determined by the relation t = a///v. We shall emphasize the characteristic dependence of a free path on the momentum of a particle (Galperin et al.,1971). At v> 2 the free path is decreased with the growth of a particle momentum. This is related to the fact that with a growth of Larmor radius, the particles will be scattered by inhomogeneities of a more scale of increasing number. At v = 2 the free path stops being dependent on the momentum. This circumstance was emphasized by Dorman and Miroshnichenko (1965) when analyzing the data on a propagation of CR from solar flares.
In the case of a narrow Cherenkov resonance, the pitch-angle scattering of particles in the range cos6 = xo is abruptly weakened; this results in a considerable increase of isotropization time and free path of particles. The analytical solution of the Eq. 2.7.1 in this case can be obtained with the conditions xo < x << 1, Ho = const:
The signs ± corresponds to cosd > 0 or cosd < 0 , respectively, and the quantity l =
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