where a1 and a2 are determined by Eq. 2.12.6. One should keep in mind that according to Eq. 2.12.16, Y is a function of r,ds,ps, and Ko = vA/3 is a function of r, d, p, and of some parameter of solar activity with definite time delay S(t - r/u).
Let us now determine the drift velocities udr of CR particles. The drift can be caused by many factors: electromagnetic drift caused by the moving of solar wind plasma with frozen in magnetic fields, density drift in magnetic field, curvature drift. Here we will consider electromagnetic drift caused the convection of CR particles. When plasma with magnetic field H moves with a velocity u = ur/r an electric field E arises in the Sun-stars coordinate system. Charged particles will drift under action of E and H in this system with the velocity
H2 r rH2
Consider first the drift velocity under action of regular spiral interplanetary magnetic field. Since the angle between r and H is determined by Eq. 2.12.16, we shall obtain basing on Eq. 2.12.20 (irrespective of the magnetic field direction to the Sun or away from it):
(udr,r ) = 2 + Q2 2 • 20' W ) = 0 (udr,V j = 2 + Q2 2 • 20 ' (21221)
Consider now the second extreme case. Let us assume that the interplanetary field is only a set of inhomogeneities in which the direction of the magnetic field H may be arbitrary with equal probability (a turbulent field). Let the angle between r and H be £. Then averaging Eq. 2.12.20 over all possible directions of H we obtain
(udr,r )tob = fu, (Udr,0 )tob = )tob = 0 . (2.12.22)
In the presence of the field inhomogeneities on the background of a regular magnetic field, including Eq. 2.12.21 and Eq. 2.I2.22 and designating a relative share of the large-scale field by
udr,r = ~2-Tl-2— +-^-' udr,0 = udr,m = ~2-T2-2— .(2.12.24)
It is easy to see that with ¡3^ 1 Eq. 2.I2.24 transforms to Eq. 2.12.21, and with ¡3^ 0 it transforms to Eq. 2.12.22.
2.12.4. On rotation of CR gas in the interplanetary space
The assumption is of common use in the literature that a gas of CR co-rotates with the Sun with the same angular velocity Q. In particular, a mean solar anisotropy of galactic CR is usually explained by this phenomenon. It should be emphasized that this concept holds, to a certain degree, true for a region near the Earth's orbit, whilst this concept is completely wrong with removal from the Sun. First of all, Eq. 2.12.21 and Eq. 2.12.22 show that the rotation of CR gas ( udr^
component) is provided by a drift in a regular field with spiral force lines, whilst a turbulent field component gives a sufficient contribution to udr,r . Since the radial component of drift velocity is balanced mainly by a diffusion flow, the real average motion of the CR gas is provided by udr^ .
Let us suppose that ¡^ 1 (the field is purely regular). Consider the region near the plane of the helio-equator (d~n2). Since the average solar wind velocity u ~ 400 km/sec, we have u ~ QrE, where rE = 1.5 x1013 cm is the distance from the Earth to the Sun. The Eq. 2.12.21 results in (udr,r ~Qr(r/rE)<< u and
(udr^)reg ~Qr at r << rE . Near the Earth's orbit (r ~ rE), we have (udr r )rg ~ u/2, (udr,p)reg ~ ^rE/2 (about half of the Sun angle rotation). Far behind the Earth's orbit at r >>rE we have (udr,r)g = u, ('udr^)r ~Q,r(rE/r)2 <<Qr (much smaller than the Sun angle rotation).
Thus, the radial component of the drift velocity near the Sun where the field is radial, is sufficiently less than the wind's velocity; however, CR gas in this region must co-rotate synchronously together with the Sun. Near the Earth's orbit the radial drift increases to one half of the wind velocity and the rotation of CR gas is decelerated to half of the velocity of solar rotation. Finally, at distances more than several AU, where the field is practically azimuthally, the rotation of CR gas is ^ (r/rE )-2 from the solar rotation. The above considerations result in there being a differentional rotation of CR gas in the interplanetary space and its angular velocity near the Sun coincides with the solar rotation, and this velocity is sharply decreased with the distance from the Sun.
2.12.5. Temporal variations and spatial anisotropy of CR in the interplanetary space
Substituting Eq. 2.12.19 and Eq. 2.12.24 in Eq. 2.12.13 we obtain the equation determining the spatial-temporal variation of CR density. Using the boundary and initial conditions it is possible to determine the sought function n (r,0,p,R,Z,t) by means of this equation. Substituting the n(r,0,p,R,Z,t) obtained and Eq. 2.12.19 and Eq. 2.12.24 in Eq. 2.12.9 we shall then find the spatial particle fluxes which determine CR anisotropy: in the radial direction it is the so called 12- and 24-hour anisotropy ( Jr/nv ); in the direction normal to the helio-equatorial plane it is the so called North-South asymmetry ( Jg/nv ); in the direction along the Earth's orbit it is 6- and 18-hour anisotropy ( J(nv ). When comparing theoretical results with experimental data it is necessary to keep in mind that Eq. 2.12.13 gives the results, which are related to the coordinate system connected with the helio-equator.
2.12.6. The region where the CR anisotropic diffusion approximation is applicable
Let us now discuss the question concerning the region where the Eq. 2.12.1 is applicable. It is known that the anisotropic diffusion approximation is the are better applicable to a description of a process the slower are the processes of variations of density and fluxes within the distances of the order of free path. For the estimates one can use the following criteria:
n dxa J dxa
In all cases of galactic CR modulation, which are of practical interest, the criteria determined by Eq. 2.12.25 hold true. In fact, after the measurements Neher and Anderson (1964) the value of the relative radial intensity gradient of CR was in
1962 — — = (12 ± 4)%/AU in the interplanetary space near the Earth's orbit for n dr particles with the energy > 10 MeV. However, at this distance from the Sun for such soft particles, according to the study of solar CR propagation (Dorman and
Miroshnichenko, M1968), A = 1012cm; therefore in this case ——A < 0.01.
For the energetic particles A increases approximately «R but (according to Dorman, M1963a) the relative gradient should decrease « R-1, so that the above estimate should not in practice be strongly dependent on particle energy. In the maximum of solar activity, the value of relative gradient should be increased several times (this results from the experimental data on the 11-year CR variation) but in this case A slightly decreases so that the criterion 2.12.25 is still satisfied.
It is easy to show that the criterion 2.12.25 is equivalent to the condition JI nv << 1. But according to numerous investigations of diurnal variations, J/nv < 0.01 in the whole range of the studied rigidities except for some occasional periods of large Forbush effects when J/nv reaches 0.03-0.05.
However, in all cases when it is necessary to determine the distribution function of CR (numerous problems of solar CR propagation, the effects of CR interaction with interplanetary shock waves, a distortion of the external anisotropy of CR in the interplanetary space and so on), one carries out the study based on the kinetic equation.
2.13. On a relation between different forms of the equation of anisotropic diffusion of CR
In the theory of CR propagation different forms of the equation of anisotropic diffusion are used, depending on the choice of variables: the momentum p (Dolginov and Toptygin, 1966), rigidity R (Dorman, 1965), total energy E and kinetic energy Ek (Parker, 1965; Jokipii, 1971). When applying these equations the necessity arises of inter-relating the quantities included in these equations. In the papers (Dorman, Katz and Shakhov, 1976, 1977) the various forms of the anisotropic diffusion equations were analyzed, their identity was proved and the relation was found between the phase density of particles and the flux density of particles, expressed in different variables.
We shall start from the anisotropic diffusion equation for the phase density n(r, p,t) and the expression for the flux density J(r, p,t) of particles (which were obtained in Dolginov and Toptygin, 1966a,b):
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