The Eq. 4.15.74 can be written in a more compact form if we include that in the case under consideration (when a particle's motion across the magnetic field is not included), the quantities o2k"2-K2 and 0.(21 HNK"i — Kj represent the diffusion coefficient in the normal to the front direction, and u^ = u - U2 is the jump of velocity on the front. As a result Eq. 4.15.74 comes to the form
The boundary condition reflects the fact of particle acceleration at the crossing of the shock front. A part of this acceleration is caused by a difference of the magnetic inhomogeneity velocities before and behind the front and represents the Fermi acceleration. The value 1 + ¡o = 1 in the right hand of Eq. 4.15.75 should correspond to only this process. Another part of the acceleration is owed to a jump of the regular magnetic field on the wave front. The latter part is included by the summand ¡o(Au/3)p(9N/dp) on the right-hand of Eq. 4.15.75. While the Fermi component of the acceleration can be written basing on a general structure of the transfer equation, the inclusion of an additional acceleration require a consideration of the elementary processes on the front and their including into the boundary conditions; this has been carried out in (Vasilyev et al., 1978).
4.16. Acceleration of particles in case of magnetic collapse and compression
It was shown above that in some cases the injection energy for the statistical acceleration mechanism could prove to be very high (see Section 4.8). In such a case an essential particle injector might be the first-order Fermi mechanism of particle acceleration in a magnetic trap between two mutually approaching magnetic mirrors considered in (Fermi, 1954; Spitzer, M1956) for the case in which the initial particle velocity vo >> u (u is the speed of magnetic mirror motion to meet each other).
However, the acceleration stage when vo < u is of greatest interest from the viewpoint of the problem of injection. Such case was examined in (Dorman, 1959a) on the assumption of particle injection from the space of the magnetic mirrors. If, however, the particle's thermal velocity is much below the speed of mutual approach of the mirrors, such case is little effective. The case is of great interest where the particles are injected from the space between magnetic clouds (Dorman,
4.16.1. Non-relativistic case of particle acceleration during magnetic collapse
Let the semi-space 1 (the left of plane A) and semi-space 2 (the right of plane B) contain homogeneous magnetic fields of intensity H (assumed, for the sake of simplicity, to be the same in either semi-space). The fields are parallel to the planes A and B, but, generally speaking, are not parallel to each other. The field between the planes is zero. In semi-space 1 a particle will be affected by the magnetic field H
and the electric field E = - [uH]c. The particle affected by these fields will move in a rest coordinate system along a trochoid and in a coordinate system relative to semi-space 1 along a circle at a velocity vo + u and frequency m^ = ZeH/macc . In this case a particle, when affected by the electric field E during its motion in the magnetized semi-space, will gain velocity 2u if the particles are non-relativistic (see Section 4.3). Let a particle with velocity vo be injected at the moment t = — tin
I2 = —nmacc/ZeH to the semi-space 1 from the space between the planes A and B. At moment t = 0 the particle will be ejected from semi-space 1 at velocity vi = vo + 2u and, after that, from semi-space 2 at velocity V2 = vo + 4u . At that moment, the distance between the planes will be
Extending the above examination further, we shall find that at the k-th ejection of the particle from any semi-space the particle velocity will be vk
and the moment tk/to of the k-th ejection will be
= t± = 2(k - l)+^l(kk -1)(( + a1 + l) k to 2k + «1 -1
where to =
l1 ZeHlo - 2nucma
where 0<t< 1. After expressing k from Eq. 4.16.3 in terms of Tk , substitution in Eq. 4.16.2, and considering Eq. 4.16.4, we obtain
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