It is of importance to emphasize that, according to Dorman and Freidman (1959), the energy increase is dependent on neither the particle's charge, nor the shade wave speed, nor the magnetic field intensity (this is associated with the fact that the increases of the intensity of field resulting in a more intensive drift and in a more rapid ejection of particles from the acceleration region, so that the total effect remains the same). The particle acceleration effect is eventually determined by the parameter a = H2/H1 = u^U2 , i.e., the degree of enhancement of the transverse magnetic field and compression of medium during shock wave movement.

It follows from Eq. 4.15.7 in the non-relativistic case, when Eo ~ macc , that

and in the ultra-relativistic case when it may be assumed that cpo ~ Eo we get

4.15.2. Acceleration in a single passage of a transversely incident particle (the shock front is limited)

If the shock wave front is limited and its size is L, the particle may be ejected from the zone of shock wave acceleration even earlier than it can drift from region 1 to region 2 shown in Fig. 4.15.1. In this case the maximum energy that the particle may acquire will be

c where u is the movement velocity of the front, H is the field intensity in the front. Thus in case of a limited shock front the particle energy gain will be determined either according to Eq. 4.15.7 if AE < AEmax, or otherwise by Eq. 4.15.10.

4.15.3. Exact integration of the particle motion equations for an oblique incidence of a non-relativistic particle onto a shock front

The above approximate estimates are concordant with the results obtained by Shabansky (1961) for trajectory calculations for a particle with the initial momentum po which is incident at some angle po to a front normal. It was found that if po > 0 the ratio p/po is not high (< 1.5); p/po increases rapidly and approaches the value ~ 4 at po =-n/4. The maximum acceleration corresponding to p/po ~ 5.23 is realized at po = -n/2 .

4.15.4. Particle acceleration by a transverse shock wave at v >> u in general case (including oblique incidence of particles)

Such a problem was investigated by Shabansky (1966). Let us consider a plane hydromagnetic shock wave propagating in the direction normal to the magnetic field (a transverse shock wave). Let all the quantities with index '1' be related to those before the front, and after behind the front they have index '2'. An energetic particle with a velocity which is assumed to be far more than the front velocity passes from the medium 1 to the medium 2. During this transition a particle occurs alternatively in the regions 1 and 2 moving along the arcs of circles in the coordinate systems which are motionless relative to the media 1 and 2, respectively. Because in this case the velocity parallel to the magnetic field does not vary, we limit a priori our consideration by the motion of an energetic particle only in the plane normal to the field. In the general case this is equivalent to a consideration in the coordinate system moving with the velocity v// along the field.

At the successive passages of the regions 1 and 2, the arc of a particle's trajectory in the region 1 will be decreased and in the region 2 it will be increased until it will be equal to 2n and a particle will be always in the region 2 behind the wave front. If the angle between a particle velocity v and the normal to the front (directed from the medium 1 to 2) in the moment of transition is 9, the central angle 6 of an arc of a particle motion in the medium 2 will be related to 9 as follows: 6 = n + 29. In this case the angles 9 and 6 very within the limits -n/2 <9<n/2, 0 <6< 2n. The front's displacement relative to the medium 2 during the time of a particle's motion in the medium 2 is, on the one hand, Ax2 = u2At2 = u2 (62/mL2) and, on the other hand, Ax2 = rL2A92cos92 , where u2 is the front velocity relative to the medium 2, rLi = pc/ZeE, (i = 1, 2) is the Larmor radius, mLi = ZeEilmacc is the frequency of a particle Larmor rotation. Then the change of the angle 9 and the front displacement relative to the medium 2 during a particle passage through the medium 2 is equal to

Similarly to this, during a particle's passage through the medium 1 the front displacement relative to the medium 1 Ax'1 = u\At\ = u1 (6 / (og\) = u1 (n + 291 )/®g!

and on the other hand, Ax'1 = rg1A91 cos 91. The front displacement relative to the medium 2 will be Ax1 = Ax'1 ((21 u), where u1 is the front velocity in the medium 1 and 91 is the angle between the front normal (directed from the medium 2 to medium 1) and a particle velocity. Since 99 =-9, the angle variation A91 and the front displacement relative to the medium 2 during a particle passage through the medium 1 are

v cosp v

Together with the angle variation described by Eq. 4.15.11 and Eq. 4.15.12 owed to finiteness of particle motion in the media 2 and 1, the angle will be increased by Ap which is related to a change of a particle's momentum at the reflection from the medium 2. Normal to the front plane component of a momentum, p± = pcosp, will be increased by Ap^ = 2(2 -U1 )e/c2 and the longitudinal component p// = p sinp will have Ap// = 0 . Here E and p are the total energy and the momentum of a particle, c is the velocity of light. From the equations

Ap sinp + pApo cosp = 0, Ap cosp-pApo sinp = 2( - U2 ))/c2 , (4.15.13)

which have been obtained by differentiation of these expressions, we shall find Apo and the momentum variation Ap :

Apo = -2(( - U2))sinppc2 = -2(( -U2)sinp/v , (4.15.14)

The total increment of the angle during a circle is composed by the increments described by Eq. 4.15.11, 4.15.12 and 4.15.14 Ap = Ap1 + Ap2 + Apo and is equal to v cosp

sinp cosp

Let us now determine a displacement of the instantaneous center of rotation during one revolution. As is seen from the expression for the Larmor radius

/ZeH2 )[pH] a displacement in the direction of the wave propagation

{cj ZeH2 )[ApH2 ] x is equal to zero at the transition through the front (since Ap is normal to the front).

One can neglect the angle variation Ap and a change of position of the front itself during a revolution when determining a displacement of the instantaneous rg = c

rotation center in the front plane in the direction normal to the field. A displacement of the instantaneous center is

Dividing Eq. 4.15.15 by Eq. 4.15.16 and passing to the limit Ap/Ap^dp/dp we obtain the equation for a particle momentum where f (ç)=

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