3cAdE 8u 2 E
we shall obtain for the accelerated particle spectrum
It is of importance to emphasize that in the relativistic case considered the particle energy gain and the generation of the spectrum are also mainly accounted for not by the difference in the frequency of the head-on and overtaking collisions (this concept is widely used in the literature) but by the effect of the systematic difference in the particle energy gain and loss in each particle collision with clouds, namely, the relative importance of the former phenomenon as compared with the latter is determined by a factor of 1:3.
4.5. Statistical acceleration of particles during the variations in the acceleration mechanism parameters as particles gain energy
4.5.1. The expected variations of the acceleration mechanism parameters as a particles gain energy
It was assumed above that the parameters A, u, and t are the constants independent of time. This is, however, is not the fact of reality, even in the stationary case. The fact is that, as the particle energy increases, the properties of the magnetic inhomogeneities (size, magnetic field intensity, velocity of motion) involved effectively in scattering and energy change of the particle vary; therefore, the effective values of A and u will vary with changing of E. As it follows from the results presented in Chapter 1 (Section 1.9), it should be expected that over a wide energy range where A, is the transport path at particle energy of injection E,, and it is most probable that 0<3< 1. Only at very high E, when the Larmor radius of the accelerated particles exceeds the largest scale of inhomogeneities, ¡^ 2.
Generally speaking, the specific value of the parameter 3 is a function of the magnetic inhomogeneity spectrum and the nature of the fields in the inhomogeneities (see Section 1.9). As to the inhomogeneity velocity u, it should increase with increasing A (for example, in case of developed turbulence u - A23). Since according to Eq. 4.5.1 A is a power function of E, we shall assume that where ui is the velocity of movement of the inhomogeneities ensuring the effective scattering of the particles with energy Ei (in the case of developed turbulence of Kolmogorov type 8 = Ifi/3 ). If t in Eq. 4.2.7 is determined by diffusive escaping from the acceleration region with effective size L, then
where t = ¿¡Iv^ is the mean lifetime of particles with energy Ej and velocity Vj in their source. It follows from Eq. 4.5.3 that the effective time of particle acceleration in the source decreases with increasing the particle energy E.
4.5.2. The mode of particle energy change and formation of the spectrum in the non-relativistic range for the statistical acceleration mechanism including the dependence of A and u on energy.
Let it be assumed that the dependences determined by Eq. 4.5.1, 4.5.2, and 4.5.3 are also valid for Ek, i.e. in the non-relativistic energy range where these dependences may be written in the form
¿ = ^(Ek!Ekl u = ut (Ek/Ekl T = Tl(Ek/EM)-(P+1/2),
where , Uj, and t are respectively the transport scattering path, the velocity of scattering inhomogeneities, and the mean time living of particles in the acceleration source at kinetic energy of ejection En . Then instead of Eq. 4.4.18 we shall obtain at Ek >> macu212:
Let us examine the following cases.
(1) The case 28 - 1 = 0. Including the initial condition — Eki at t — 0, and 5 +1/2, we shall then obtain
5Ek v ki y
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