where the dielectric constant
where the plasmons numbers dNo , (o)+vo)Te)
dkz z vz
In Eq. 4.9.3 and 4.9.4 are used notations
Important application of CR charged particle's acceleration by plasma turbulence in the interstellar space pointed out by Jokipii (1977). It was shown that the age of CR particles in the Galaxy disk is enough to increase particle's energy several times from the turbulence energy of magnetized space plasma in the disk. The possibility of CR particle acceleration by frozen in magnetic turbulence of solar and stellar winds (in the case in which the CR density gradient is directed from the star outside) was shown in papers of Dorman et al. (1980, 1987).
4.9.3. Acceleration by magneto-sound and Alfven waves
The phase velocities of the AlfVen and magneto-sound waves may be very high if va0 = H2/^nNjmj >> 1. In this case, the Alfven velocity may approach the speed of light,
This circumstance is of importance because the acceleration effect appears at vz < va , where vz is the field-aligned velocity of particle. According to Tsytovich (1966c) we obtain dE„ 2e1a)^iva — .„ ^
dt 2^, ao where Na is the mean number of Alfven waves in unit of volume. The works of (Toptygin, 1972; 1973) treat the acceleration of fast particles by the Alfven and magnetosonic waves of small amplitudes existing in a medium with a strong homogeneous magnetic field. The contribution from the small-scale (harmonics with k > , where rg is the Larmor radius of particles) and large scale (k < rL-1)
random field to the acceleration is included. The energy dependence of the particle diffusion coefficient in momentum space has been calculated. The effect of the moe anisotropy of the particle distribution function on the acceleration has been considered. It has been shown that when the particles are accelerated by the Alfven waves whose spectral function has the spectrum exponent v > 2 and the amplitude is sufficiently small, an energy dependence of the diffusion coefficient, which is stronger than that in the Fermi acceleration mechanism, may arise. In the case of magneto-sonic waves at v > 2 the energy dependence of the diffusion coefficient is the same as that in the acceleration by the Fermi mechanism, whilst at v < 2 the energy dependence is weaker.
4.9.4. Cyclotron acceleration of ions by plasma waves
The existence of the magnetic field facilitates the injection in the case of acceleration by high-frequency fluctuations since it is known that very slow waves of high frequencies may move perpendicularly to the magnetic field. The cyclotron acceleration of ions may be owed to the absorption and emission of the plasma waves with low phase velocities which propagate perpendicularly to the magnetic field. In this case (Tsytovich, 1963a):
= — +l n2(al,u)Jiv±n(ojLl,M)1 - ) )N(aI) - )2 j-12 dt mev± i -1 J
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