If the distance between inhomogeneities is /, the mean time between two collisions iS T = / 3A ' v±, and the diffusion coefficient of the curvature center is

3t 3l3 HO

o and the transport scattering path

i.e. it should be practically independent of a particle's rigidity (it will be reminded that this result is valid subject that R± > AHo ). If, however, R± < AHo , or R±<Ah , then g = R±/Ho at Ho < h and g = R±/h at Ho > h which may approximately be written in the form

i.e. k_ x R_l and Ax ^ R_. The Eq. 1.9.46 and Eq. 1.9.48 for Ax may be combined to within a factor of ~2:


Let 8 << 1 (the field in inhomogeneities are little different from the basic field); then jAt -382

and at Ri > 88HoA it will be Ai ^ A4l "3 8 2, whereas at Rl < SHoA it will be

i.e. A±=A4l "3 at R±>> hA and A± = RjA2/(l3h2) at R± << hA . Thus, the measurements of the transport scattering path on the basis of data on CR variations are of extreme interest since this parameter is a sensitive characteristic of the magnetic inhomogeneities in space.

1.9.10. The transport path for scattering with anisotropic distribution of magnetic inhomogeneities in space

Many of the diffusion models of a propagation of CR in interplanetary space are based on the concep of the scattering centers in solar wind. These scattering centers are magnetic inhomogeneities frozen in interplanetary plasma which radially move away from the Sun together with solar wind (Belov and Dorman, 1972). In a region remote from the Sun at the distance r, let the average distance between inhomogeneities along the radius is lr (r) and transverse to the radius (over 9 and 9, where 9 is polar, 9 is azimuthal angles) is le(r) and l^(r). Suppose that at some distance ro from the Sun lr(ro) = le(ro) = l^(ro), i.e. the scattering centers are isotropically distributed. If a diffusion picture does not vary with time, then lr (r) = lr (ro ) = lo, le(r ) = l> ) = . (1.9.53)

Then at r ^ ro the isotropy in a distribution of inhomogeneities is conserved only in the plane, normal to the radius. The natural question arises of whether the anisotropic distribution of scattering centers will result in anisotropic diffusion. To verify this assumption let us consider the following spatial structure in the location of inhomogeneities: let the neighboring inhomogeneities be located at the same distances which are equal to a in radial direction, and in the directions, perpendicular (over 9 and 9) to a radius, they equal b. Let us consider as well that every scattering occurs isotropically and all of inhomogeneities has the same dimension A. A probability that a particle, moving initially along the radius, will be scattered at the distance a is A2/b2 ;the probability (a2/b2)(l-A2/b2) corresponds to the distance 2a, etc. A probability of a free path ka in the radial direction will be

(a2/b2)-A2/b2) 1. Summing with the respective weight all possible free paths, we obtain the average free path



where N = l/ (ab2) is a density of inhomogeneities; <r = A is the differential cross-section of scattering. Thus in the absence of a regular magnetic field the diffusion remains isotropic, in spite of anisotropy in scattering centers.

Consider now a large-scale regular magnetic field. For particles with Larmor radius rL >> max{a,b} the isotropy of diffusion is not violated, but for particles with Larmor radius rL << min{a,b} diffusion will be essentially anisotropic. These particles will be bound to the lines of force of a regular magnetic field, and a free path in the direction of the field will be A/ = dr2f AA , where d is the average distance between inhomogeneities along the lines of force. In particular, if the regular field has a spiral structure then d = ao


where u is the solar wind velocity, and H is the angular velocity of solar rotation. We see that the distance d is essentially dependent on the distance from the Sun. Near the Sun d ~ ao, and at great distances (near the ecliptic plane) d ~ aor/ro . A

free path A varies together with a distance r. Thus in the presence of a regular magnetic field an additional radial dependence of the diffusion coefficient arises in interplanetary space which is related to a divergent character of a motion of scattering centers (resulting in an anisotropic distribution in space). A similar situation may occur in expanding shells of Supernovae, in spiral branches of galaxies, in Metagalaxy.

1.10. Magnetic traps of CR in space

1.10.1. Types of CR magnetic traps and main properties

CR in space are essentially confined within magnetic traps of one or another scale and fail to propagate freely (excluding for CR y-quanta and neutrinos which cannot be trapped by magnetic fields). Enormous CR traps are very diverse in their properties and the charged particle behavior in such traps is essentially energy-dependent. The trap in the Earth's environment formed by the near-dipolar magnetic field exhibits a high degree of stability and considerable lifetime of particles in the trap. At the same time the traps in the vicinities of chromospheric flares or in solar corpuscular streams of magnetized plasma are much more transparent for particles and the mode of particle ejection from such traps resembles the diffusion in irregular magnetic fields. Traps of various kinds are also formed in the vicinities of normal stars, in particular in the solar system and supernova shells. On the other hand, the Galaxy (the galactic disc and halo) also forms a peculiar trap of dimensions of many thousand of parsecs which can safely (with a ~ 107 years life time) retain the particles of moderate and high energies and is very transparent for the super-high energy particles. It is quite possible that the clusters of galaxies form even more enormous traps of the super-high energy particles.

What formation in the space should be considered as a magnetic trap? Perhaps they are the formation with regular magnetic fields of peculiar configuration where the charged particle lifetime is very great and the accumulation effect is significant, or should the term be much extended? It seems to be expedient from the viewpoint of the study of the general regularities of the temporal variations of CR intensity to consider the CR traps as any magnetic formations in which the motion and time of residence of charged particles is substantially different from those in the free space of the same volume (it should be emphasized that the properties of the traps are essentially dependent on particle energy and that the same magnetic formation may be an excellent trap for particles with energies lower than some critical energy and, at the same time, may be practically transparent for particles of higher energies). The CR intensity inside a trap is determined by the powers of both internal and external source of particles, the absorption, nuclear conversions, loss owed interactions with magnetic fields (the latter is of importance to electrons) inside a trap, and the extent of the exchange with the outer space particles. The temporal variations of the above said factors will in their turn result in the temporal variations of the trapped radiation. Such an approach will permit the diverse types of CR to be uniquely considered and understood. The cosmic traps are characterized, first of all, by the structure of the magnetic fields that determine the charged particle's motion, the exchange with the outer space (ejection from a trap and the possibility of being trapped) and, to a great extent, by the particle absorption inside a trap. An extremely important characteristic of the traps is their dynamism; the traps can be static, moving, expanding, compressed, and, besides that, can exhibit their internal dynamics.

Many works devoted to the theoretical development and experimental study of traps with regular magnetic fields for containing hot plasma appeared in connection with the recent research into controllable thermonuclear reactions (see, for example, in Artsymovich, M1961). These problems have been also sufficiently elaborated in connection with the development of space electrodynamics (Pikelner, M1961;

Spitzer, M1956) and intensive exploration of the magnetic trap and the properties of trapped radiation in the Earth's environments.

1.10.2. Traps of cylindrical geometry with a homogeneous field

The simplest traps of cylindrical type with regular magnetic fields resembling homogeneous fields may be formed, for example, in the galactic arms, in solar corpuscular streams, and in extended magnetic formations in interplanetary space. In this case CR particles move along a spiral with radius of curvature rL = cp^/(ZeH), where p± = psin#; 6 is the angle between the particle motion direction and a magnetic force line. The particles will move along the field at a velocity v// = p//c2 /E = pc2 cosd/E where E = (2c2 + m^c4) is the total energy of particle. Thus if the width of the regular field region is L, the particles with rL << L cannot be ejected through the side walls across the field, excluding for the surface layer of thickness 2rL . If rL ~ L such a region will only scatter the particle and change the direction of its motion by an angle determined by Eq. 1.8.7 (see Section 1.8.4).

1.10.3. Traps with strength-less structure of the field

Willis (1966) has studied the motion of an individual charged particle in a relatively simple strength-less field H = (0, Ho sin ax, Ho cos ax), where Ho and a are constants. In such field the equations of particle motion can be exactly solved without assuming a leading center. One of the components of the equation of particle motion is formally reduced to the general equation of pendulum motion. It has been shown that under some conditions a particle may move predominantly across the lines of force and thus be ejected from magnetic trap. The criterion of realization of such possibility is the condition L < rL/2, where L is the characteristic scale of the field; rL is the maximum gyration radius of particle in homogeneous magnetic field Ho . It should be noted that examination of the traps of such kind is of great interest in analyzing the possibility of solar CR trapping by sunspot magnetic fields and in studying the propagation of galactic CR through the interstellar and interplanetary space.

1.10.4. The effect of magnetic field inhomogeneities

If the trap field comprises magnetic field irregularities which can scatter the particles, the diffusion across the field will also take place. Let the field inhomogeneities be characterized by size A and the mean distance between them l; let also the field in the inhomogeneities differ from the regular part by, on the average, ±AH and the field direction in inhomogeneities be the same as in the main part of the trap. Within the time of particle motion through an inhomogeneity, the curvature center will shift by

Since within 1 sec a particle will encounter vA2l 3 inhomogeneities, the mean velocity of the displacement of the center of curvature will be u±_A± vA2l "3, (1.10.2)

whence, including Eq. 1.10.1, we shall obtain for the diffusion coefficient of the curvature center across the magnetic field:

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