CR particle motion in magnetic fields scattering by magnetic inhomogeneities

1.8.1. CR particle motion in the regular magnetic fields frozen into moving plasma formations

The magnetized plasma moving at high velocities is an important constituent of the space. The electric fields are known to be practically absent under the conditions of a high conductivity (the time of their neutralization « c"1 ,where c> 1012 ^1014 CGSE units), so that only magnetic fields H may exist in a coordinate system related to the moving plasma (and, correspondingly, to the moving frozen magnetic field). The magnetic and induced electric fields in a coordinate system with respect to which a plasma formation moves at velocity u (here the case where |u| << c is of practical interest) are determined on the basis of Lorentz transformations by the relations

When affected by these fields the motion of a particle with momentum p, velocity v and charge Ze will be defined by the equation

Since v >> u the motion in homogeneous field may be treated as that along a spiral line with the curvature (Larmor) radius rL = cpJZeH . (1.8.3)

In this case the energy of a particle traversing a region of size l in the direction perpendicular to u and H changes by

If the size of the region l is such that l > rL, such a region will significantly change the particle motion direction. If rL << l the particle will move practically along magnetic force lines and, if the force lines are closed such plasma formation may be a magnetic trap for particles with given rL (see below, Section 1.10). Many publications have been devoted to the problems of charged particle motion in regular magnetic fields; we shall dwell on these problems below in connection with specific problems of CR propagation in space plasma (interplanetary space, the Earth's and other planets magnetospheres, interstellar and intergalactic space). Here we shall only refer the reader to the monographs (Boguslavsky, M1929; Alfven, M1950; Stormer, M1955; Spitzer, M1956; Pikelner, M1961) and to the general works (Hayakava and Obayashi, 1963a,b), in which these problems are formulated and solved to various approximations (analytical finding of trajectories, the drift approximation, numerical integration of the motion equations).

1.8.2. CR particle moving in essentially inhomogeneous magnetized plasma

Strictly speaking, the propagation of the CR particles in essentially inhomogeneous magnetized plasma should be studied by solving the Bolzman kinetic equation for the function of CR distribution f ((, r,p):

dt or c op where Acol is the term reflecting the role of elastic and inelastic collisions of CR particles with plasma particles (this term also reflects the fragmentation and energy losses according to Sections 1.1-1.4). The structure of the real fields is, however, fairly complex, and therefore the Eq. 1.8.5 can be successfully solved as yet only for the simplest cases and by reducing to the diffusion approximation. However, to understand the basic features of CR interactions with magnetic fields the elementary approach is sufficient in many cases. In this approach at first the features of an isolated particle interaction with various types of magnetic field inhomogeneities is considered and then one or another statistical set of inhomogeneities is treated and the transport scattering path of particles and the diffusion coefficient are estimated.

1.8.3. Two-dimensional model of CR particle scattering by magnetic inhomogeneities of type H = (0,0, H )

Dorman and Nosov (1965) studied the scattering properties of the magnetic fields of the simplest configurations in a plane perpendicular to magnetic field H = (0,0, H ). For the two-dimensional case the differential effective cross section is da = (drjdë)de, (1.8.6)

where r is the impact parameter, 6 is the scattering angle (in the two-dimensional case a is of dimensionality of length).

1.8.4. Scattering by cylindrical fibers with a homogeneous field

If ro is the radius of a cylinder inside which H = const, then

where rL = cp/ZeH is the curvature radius of particle inside the cylinder (see Figs 1.8.1 and 1.8.2).

Fig 1.8.1. Geometry of scattering, and the notations
Fig. 1.8.2. The cases of scattering for various values of a = rLjr0

After calculating dr/dd by Eq. 1.8.7 and substituting in Eq. 1.8.6, we find that r da = rro-2

2a sin

where we have denoted:

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