From Eq. 4.8.6 follows that the injection energy Eki x Z . It means that for heavy nuclei the injection energy will be sufficiently bigger than for protons, and in practice according to Eq. 4.8.8 the ratio of accelerated heavy nuclei number to contents in sources is expected to be much smaller than for protons. So the contribution of heavy nuclei to the total flux of accelerated particles expected to be negligible. However, the real situation is the opposite: in galactic CR the content of heavy nuclei relative to abundances in sources is much bigger than for protons (see review in Ginzburg and Syrovatsky, M1963; Dorman, M1963a, M1972, M1975a; Berezinsky et al., M1990; Schlickeiser, M2001; see also in Section 1.4 in Dorman, M2004, and very short information above, in Section 1.2). This serious contradiction was widely discussed in the literature after the first paper of Fermi (1949) on the statistical acceleration mechanism, where this problem was for the first time noted.
To solve this problem Korchak and Syrovatsky (1959) take into account that if the temperature of background plasma is not too high, heavy ions are only one or two times ionized, so really the acceleration starts from the effective charge of heavy ion Z* = 1 or 2, and in this case, according to Eq. 4.8.4 and Eq. 4.8.5, the energy of injection will be about the same as for protons. With energy increasing the effective charge of particle will also be increased, but during all processes of acceleration the energy gain will be bigger than ionization losses.
Moreover, according to Korchak and Syrovatsky (1959), if the acceleration starts from velocities v < ve, where ve is the velocity of electrons of the background plasma, the value of the injection energy does not have a sense and practically all particles with these velocities will be involved in the acceleration process. Therefore from the condition
may be found the critical value of the acceleration parameter acr, higher of which the injection threshold absent and particles are accelerated independence from their initial energy. In this case acr (A, Z )= acr (p )(Z *)/A , (4.8.10)
where acr (p) is the critical value of the acceleration parameter for protons. Let us account the loss of electrons (increasing Z*) by ion with increasing energy during its acceleration (Bore's formula):
where v is the velocity of accelerated ion, and vo = e2jmero (here ro is the classical radius of electron). After substituting Eq. 4.8.11 into Eq. 4.8.10 we obtain acr (A, Z) = acr (p )ve/vo )(Z *)3/A . (4.8.12)
Because heavy ions of not very hot background plasma starts to accelerate from only single or double ionized (Z* = 1 or 2), for them acr becomes smaller than for protons. Therefore, at the same initial ionization of background plasma is possible such parameter of acceleration a, that ions with big A will be accelerated independence from their initial energy, but the acceleration of protons will be depressed because of the high threshold of injection.
4.9. Statistical acceleration in the turbulent plasma confined within a constant magnetic field
As a rule the space plasma is turbulent and confined within a more or less regular magnetic field. Important studies of the turbulent acceleration mechanisms were carried out in Lucke (1962), Tsytovich (1969), Schatzman (1969), Hall (1969) and others. The detail analysis of this problem was given by Tsytovich (I966c).
4.9.1. The magnetic field effect on plasma turbulence
The problem of the effect of large scale magnetic field on turbulent motion in an electro-conducting medium seems to be highly important to the problems of particle acceleration in such a medium and CR propagation. Without going into details of this complicated problem, we shall only note the work (Rädler, 1974) which considers the turbulent motions in homogeneous incompressible electro-conducting medium in the presence of a magnetic field which is on the average homogeneous and stationary. Adopting the model in which the turbulence is owed to stochastic volume force, and assuming weak interaction between the motion and the magnetic field, Rädler (1974) develops a method for calculating the paired correlation tensor of the velocity field. Calculated as an example is such a tensor for homogeneous stationary turbulence that is isotropic and mirror-symmetric at a beam magnetic field. It has been found that: first, the field suppresses the turbulence, namely, the component parallel to the field that is smaller than the perpendicular component; second, the correlation length parallel to the field tends to exceed the perpendicular length. The probability is considered of particular situations in which the turbulent velocities are enhanced by the field and the anisotropy of the velocity components and correlation lengths is opposite to that indicated above.
The magnetic field gives rise to the change of the spectrum of the quasi-longitudinal plasma fluctuations determined by the equation at small kV^fc. In the case of a weak spatial dispersion the effect of systematically change in the particle velocity for vz >> is
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