It should be noted that in the initial formulation of the problem of CR propagation (Krymsky, 1964; Dorman, 1965; Parker, 1965), a form of Fokker-Planck equation was postulated, basing on the concept of a systematic energy losses of particles in their interaction with the radially-divergent inhomogeneities of magnetic field (in contrast to the paper Dolginov and Toptygin (1966a,b) where a consistent deduction of this equation was carried out for the first time). In this case, the exact expression was used for the particle flux j(r, p,t) in space and the particle flux in the momentum space was determined by the expression where the kinetic coefficient (dp/dt) has the meaning of variation of particle momentum per unit time, and for calculation of this coefficient some intuitive considerations were involved using the assumption of systematic losses of particle energy. In spite of the fact that the equation (obtained from these not completely exact assumption) having been a quite correct equation of the transfer, a canonical form ascribed to it does not correspond to reality2 and based on the interpretation of physical phenomena taking place in the CR propagation in interplanetary space appears to be incorrect. Moreover, in a consistent phenomenological treatment there does not arise the problem of calculation of the kinetic coefficient (3p/3t), but, as seen from Eq. 2.16.10, it is necessary to determine the crossed coefficient of diffusion Dpa characterizing the process of energy exchange between CR and magnetic inhomogeneities which is caused by spatial inhomogeneity of the particle distribution function, in accordance with the general conclusion resulting from the Eq. 2.16.3. Therefore, the concept of adiabatic deceleration of particles does not have a global character and the process of energy exchange in the system CR-solar wind is determined by a concrete form of the distribution function of particles. In this case, galactic CR when propagating in the solar wind, appear to be in an acceleration regime, accumulating energy in the process of scattering on the radially moving inhomogeneities of the magnetic field.
In the conclusion of this problem we present the relations resulting from Eq. 2.l6.10, which determine a variation of momentum and energy of a particle per unit time:
2 In a later publication (see, for example, Jokipii, 1971) the correct expression for the flux J of particles was used (see Eq. 2.16.5) corresponding to the expression obtained in (Dolginov and Toptygin, 1966a,b).
and m is a particle rest mass.
These relations show as well that a variation of particle momentum and energy is determined by a sign of the radial gradient of CR, and with the positive radial gradient, an increase of particle energy takes place. The specific feature of the quoted relation consists in the fact that mean variation of particle energy being determined by the value of the relative gradient of CR, which is the parameter characterizing the collective properties of the particle ensemble under consideration. Similar results was obtained some later also by Gleeson and Webb (1978).
2.17. The second order pitch-angle approximation for the CR Fokker-Planck kinetic equation
The study of multiple charged particle scattering in magnetic field with random inhomogeneities as scattering centers is important in turbulent plasma theory (e.g., Shkarofsky et al., M1966), in problems of cosmic ray particle propagation through cosmic media (e.g., Jokipii, 1966; Dorman and Katz, 1977), and many other problems of particle transport (e.g., Case and Zweifel, M1967). If the magnetic field is sufficiently strong that the Larmor radius of particle rL << X (X being the particle mean free path with respect to its scattering by inhomogeneities of the magnetic field), the averaging over a particle's spiral motion around the magnetic field can be performed, and one can restrict oneself to a simple rectilinear system.
The paper of Dorman, Shakhov and Stehlik (2003) deals with solution of the equation for the particle distribution function in the second order approximation in the pitch-angle. The exact analytical solution is obtained in an integral form. The well known solution in the first order pitch-angle approximation can be restored by performing the small time limit in the result. Unlike the first order solution the solution obtained in the second approximation rightly shows that the pitch-angle diffusion is closely connected with the particle transport along the mean magnetic field.
The diffusive particle propagation and its angular scattering along the mean magnetic field is governed by kinetic equation of the Fokker - Planck form, and
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