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at X = 0°; the values of Rcj are 1.7 GV and 0.11 GV at 0 = + 90°; Rcj = 6.6 GV and 0.41 GV at 0 = - 90° respective for r = 0.25 and 1 AU.

Cutoff Rigidity
Fig. 2.11.1. Variation of interplanetary magnetic cut-off rigidity with latitude and directions of incidence at two distances from the Sun. Curves a, b, c correspond to the 0, 90, and - 90° respectively. According to Gall et al. (1977).

From Fig. 2.11.1 can be seen the presence of the East-West asymmetry in Rcj, which results in appearance of CR anisotropy with the direction to the maximum at 18 hours of the local solar time (i.e. with the same phase as it results from the convection-diffusion theory); in this case the amplitude of anisotropy should be increased approaching the Sun. As Rcj is substantially dependent on r (^ r"3 according to Eq. 2.11.47), this will result in the appearance of an additional positive gradient, i.e. again of the same sign as expected from diffusion theory. The importance was emphasized of the experimental test of the predicted effects the relative significance of which should pronouncedly increase approaching the Sun.

With the assumption that the sector structure of the interplanetary magnetic field is a separating boundary between the magnetic field of the opposite polarities in the northern and southern hemispheres of the Sun, Svalgaard and Wilcox (1976) studied a connection between the extent of these field and the 11-year variation of CR. The sector magnetic field in the photosphere with the intensity 0.5 Gauss near the minimum of the solar activity is extended in the latitude range ± 40° and its extent at the distance of 1 AU is only ± 15°. This field compression may be caused of an excess magnetic pressure in the polar regions of the Sun. Near the maximum of solar activity when the sign of the general field changes, the field intensity in the polar region is decreased and a compression of the structure of the equatorial field will also be decreased. An increase of the volume occupied by the sector field with a complicated structure should result in an increase of scattering of galactic CR in their diffusion into the solar system. This geometrical effect may be the main cause of the 11-year variation of CR. To test this hypothesis, Svalgaard and Wilcox (1976) determined the 11-year variation of helio-latitude extent of sector structure of magnetic field. It was shown that this parameter is in a good correlation with the inverse wave of the 11-year variation of CR intensity measured in the stratosphere above Murmansk and Mirny (Antarctica).

Humble and Pelechaty (1977) have made calculations of trajectories of CR with a rigidity from 150 to 9000 GV in interplanetary space, including the sector structure of the magnetic field at low helio-latitudes, and that the field is unidirectional in the high latitude region. The calculations were carried out for particles coming to Hobart (Australia) in various seasons of a year. A possibility of change the field polarity in the high latitude region owed to inversion of the solar general magnetic field has also been taken into account.

Krainev and Stozhkov (1977) reported their theoretical model of a magnetic field in interplanetary space based on the data on large scale photospheric magnetic field which have a dipole character. The presence of general magnetic field of the sun and their variations with the 22-year period will result in the corresponding variations of intensity and anisotropy of galactic CR. The model was developed in the paper (Krainev, 1978) in which the equation of CR diffusion in interplanetary field was solved including a dipole character of the high latitude magnetic field of the Sun stretched out by the solar wind. To simplify the calculations, variations of particle energy in the process of CR propagation in interplanetary space were neglected. it was found that the depth of modulation depends substantially on a direction of the solar magnetic dipole Ms and on the sign of the charge of CR

particles: near the ecliptic plane the depth of modulation is considerable (by 2-4 times) larger at sign (H±,q) < 0 than at sign (H±,q) > 0 (here q is a charge of particles, H ± is the interplanetary magnetic field component normal to the ecliptic plane); when moving away from the ecliptic plane both the depth of modulation and the ratio value described are decreased. It is assumed that Ms changes its direction to the opposite near the epoch of the maximum of solar activity every ~ 11 years.

The spectrum over total energy E per nucleon of the type of « E~2 6 was chose as the non-modulated interstellar spectrum. The considered model results in the appearance of 22-year harmonic in the variation of intensity of galactic CR in interplanetary space which is superposed in the 11-year harmonics caused by the 11-year cycle of the solar activity.

2.11.6. Derivation of the Fokker-Planck CR transport equation from variational principle

Burgoa (2003) proposed a Lagrangian density for obtaining the Fokker-Planck CR transport equation and determining the energy-momentum tensor and

CR currents of a single CR source by applying the Noether's theorem (see in Sokolov et al., M1989).

According to Burgoa (2003), the Fokker-Planck CR transport field is possible define by y = y(X) and the complex conjugate ^ (X) with parameters X = x0, x1,

2 3 X , X , x where x = ct ; x , x , x = x, y, z and x = bp, where p is the momentum modulus, b a dimensional constant and x, y, z the position coordinates. So here, the Latin index i, j from 1 to 3 and the Greek index f, v, p from 0 to 4 in analogy of the relativity theory. In this model the diffusion tensor Kvfl{x', x4) is defined by:

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