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Let us return to a general Eq. 4.15.55 and consider it at a >> u^v . Neglecting the small terms we shall have:

macvui

for the condition ¡u~ ^ considered, so 1 1 - H2/ H . At the transition from the region 2 into region 1 the corresponding increment is given by the equation:

macVU1

a1 v

"A

where -1 < u < -ui /a^v . A transition through the front is possible only with «1 > ui/v . If 1 >> «i >> ui/v , then

AEk21 =

macvu1 «1

In all the cases the maximum increment of energy at a reflection or at a transition through the front is several times higher than the particle's initial energy, and is a small part of it at a >> ui/v .

Since a magnetic field has a random component the above formulae are valid if the angles a and a are weakly dependent on the distance which a particle covers during a time of acceleration. A simple estimate shows that this condition will be satisfied if the correlation length Lo fits the inequalities Lo >> rgvjui (H2/Hi) at

«1 < ui /v and Lo >> (rg ja\ )(H2/Hi) at a > u^v . Since in interplanetary space the transport path A // in the direction of the magnetic field is usually larger than the correlation length Lo , the above inequalities also provide the small value of a particle pitch angle variation owed to scattering by a small scale field during the period of acceleration.

Let us now find, according to the paper (Vasilyev et al., 1978), the boundary conditions on the shock wave front. We shall consider the particle distribution function as sufficiently isotropic one both before a shock wave front and behind it in order to use the diffusion approximation. Furthermore suppose that a >> ui/v which results in that AE^ is small as compared to a particle energy . This condition makes it possible to expand the distribution function over the small additions to the energy. Otherwise the boundary conditions will have the form of the equations in the finite differences over the energy. A deduction of the boundary conditions is based on a calculation of the particle fluxes through the wave front in the z-axis direction and in the opposite one.

The particle flux J2 (Ek) through the plain, which is on the distance from the front of about one Larmor radius, in the direction of the z-axis (along the normal to the front) can be written by means of the distribution function F2 (Ek ¡):

where an integration is made within the limits from 0 to n/ 2 + U2/ a^y 2 over a pitch angle and from 0 to 2n over the angle of cyclotron rotation of particles. Since a reflection of particles moving from the medium 2 with the condition under consideration is absent, the flux J2 (Ek) can be formed only by the particles moving from the region 1 and crossing the shock wave front. Let us designate the flux of these particles as J'p2 (Ek):

J\2(Ek) = I(nv)p12(¡¡FX(Ek -AEk12(¡¡,t) . (4.15.63)

The argument of the distribution function includes the energy increment on the front; the factor p12(¡) (see Eq. 4.15.45 ) represents the probability of a particle transition through the front without reflection. The first boundary condition will have a form:

The second boundary condition is obtained from the consideration of the particles crossing the same plane but in the direction opposite to the z-axis. It has the form:

where J1E) is the flux through the plane expressed by the distribution function in the medium 1; JH (Ek) is the flux of particles moving from the medium 2 and crossing the wave front; Jj^ (Ek) is the flux of particles incident from the medium 1

and reflecting from the front. For all of these fluxes it is easy to write the expressions similar to Eq. 4.15.62 and 4.15.63.

The distribution function in the system of rest plasma in the diffusion approximation has the form

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