l we shall obtain, after tedious transformations and neglecting the terms of higher orders than u2/c2 , the expression coinciding with Eq. 4.7.26. 4.7.5. The mode of particle energy change in time

The mean change in the particle energy with time dE/dt for scattering through some angles 9 is determined not only by the value of AE/E in each collision event but also by the collision frequency v. Since v^(d3/12)- then dE/dt ^ (d3/12sin2(9/2). On the other hand, in the models of magnetic clouds or inhomogeneities of type j = 1, 2, 3 the transport scattering path X = (d V l2 )//2 at /<<1. Thus at small scattering angles we shall obtain the expressions of Sections 4.2-4.5 for dE/dt, in which X should be meant not as the path before collisions with magnetic clouds but as the particle transport scattering path. Since at large scattering angles the path before scattering with magnetic clouds coincides with the transport scattering path, we can draw the following important conclusion: the results of Sections 4.2-4.5 (and hence of Section 4.6, the statistical acceleration in particle rigidities) are valid not only for mirror reflections but also for statistical scatterings through any angles; it is necessary only to understand X in the expressions of Sections 4.2-4.5 and 4.6 as the transport scattering path of particles.

4.8. Injection energy and the portion of the accelerated particles in the statistical mechanism

The initial acceleration process takes place most frequently in the non-relativistic energy range where the energy loss is of significance. It is the initial acceleration process that determines so called injection energy, i.e. the minimum energy from which the particle acceleration becomes possible. Detailed examination of the acceleration in the non-relativistic energy range makes it possible also to find the portion of the accelerated particles.

4.8.1. Injection energy in the statistical acceleration mechanism

Since the statistical acceleration mechanisms are characterized by a comparatively slow rate of energy gain, the various kinds of energy loss by the accelerated particles are very essential, especially in the first stages of acceleration. Let us consider the initial stage of acceleration (Dorman, 1959). The Eq. 4.3.5 in the non-relativistic energy range for the energy gain rate and analogues expressions in Sections 4.4-4.7, may be generalized in the form where mac and Ek are the rest mass and the kinetic energy of the accelerated will be assumed in accordance with Section 4.5 that in the non-relativistic energy range where aj is the acceleration parameter at Ek = kT (where T is the plasma temperature). On the other hand, the energy loss for collisions (the ionization loss) is

particle; a(Ek) is the acceleration parameter including its dependence on Ek . It

1 =- 4nZ2e4NeIijm~j2Ek/>

1 =- 4nZ2e4NeIijm~j2Ek/>

ion m e

where the logarithmic term Li = 20 for the characteristic space conditions, Ze is the charge of the accelerated particle, Ne is the electron concentration in the medium; me is the electron mass. It is obvious that the particle acceleration is possible only if

dt Jac V dt Jlon

Substituting Eq. 4.8.2 In Eq. 4.8.1 and comparing with Eq. 4.8.3, we find that the condition described by Eq. 4.8.4 is satisfied at > En , where

It can be easily seen that at 28 _ 3 = 0 and at a constant acceleration parameter we obtain the conventionally used expression for particle injection

meoc oc

If the initial energy of a particle Eko < Eki, such particle cannot be accelerated. The acceleration will occur only for the particles with Eko > Eki.

4.8.2. The injection from background plasma: conditions for acceleration of all particles

Let us consider different cases.

The case 28-3 = - 1- It can be seen from the comparison between Eq. 4.8.1 and Eq. 4.8.3 that in this case (dEk/dt)ac and (dE^/dt) vary with Ek in a similar manner. Therefore, if

2nZ2e4NL , Z2Ne _i

meckT ckT

then Eki ^ 0 and all the plasma particles should be accelerated. If, however, the Eq. 4.8.7 is not satisfied then (dEk/dt )ac < (dEk/dt) over the entire range of nonrelativistic energies and the acceleration proves to be impossible at 28_3 = _ 1.

The case 28-3 > _ 1- If in this case Eq. 4.8.7 is satisfied then Eki - kT, which is also equivalent to the acceleration of all the plasma particles since in this case the particle acceleration condition Eq. 4.8.4 is satisfied from thermal energies. Since the overall heating of the plasma takes place in this case, T increases abruptly and the acceleration conditions change; such a process, which is essentially non-stationary, takes place at high values of aj .

4.8.3. The injection from background plasma: quasi-stationary acceleration of a small part of the particles

If the parameter aT (the acceleration parameter a at Ek = kT ) is sufficiently small then Eki >> kT and only small portion of plasma particles are accelerated. In this case the ion distribution function varies little in time and the acceleration is close to a quasi-stationary process. In this case, as was shown by Gurevich (i960), the rate of generation of run-away particles dn/dt from background plasma, including in the acceleration process, will be (in cm-3 sec-1):

where N is the concentration of ions (cm-3 ) and vo is the frequency of collisions

(sec-1) in the background plasma. The Eq. 4.8.8 is valid in the case when the distribution function of background plasma particles is close to the equilibrium (i.e. the part of accelerated particles for the average time of acceleration is much smaller than the total number of particles in the background plasma.

4.8.4. The problem of injection and acceleration of heavy nuclei from background plasma

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