This shows that with |e| ~ Hn the particles will be accelerated to relativistic energies. For |e| << Hn the value AEk = 2up'z, where p'z ~ p// (p// is a particle's momentum along the magnetic field) for a small a', and therefore a discontinuity with a tangential electric field E accelerates the particles similar to a magnetic mirror moving along the discontinuity with the velocity u = c|E|/Hn .
4.15.7. Particle acceleration at a multiple reflection from a shock wave front
In the presence of inhomogeneities of the magnetic field ahead of the shock front there will be particle scattering both by undisturbed medium and by a shock wave front. As a result a certain share of particles will undergo a multiple acceleration on the shock wave front and their energy can be far increased. A realization of such a mechanism of acceleration with the certain concrete condition will be considered below (see Sections 4.21-4.24). Here we shall present the solution obtained by Vasilyev et al. (1978) for the problem of energetic particles acceleration by a shock wave propagating in a turbulent medium. The paper of Vasilyev et al. (1978) is founded upon the following assumptions: a) a particle's Larmor radius rg is large compared to the shock front's thickness; b) the shock waves are propagated in the medium with a regular magnetic field Ho and a random field H , and H < Ho ; c) the spectrum of the random field falls with the increase of the wave number k so that the large scale component of the field (the scales are more than a particle Larmor radius rg) prevails over the small scale component; the main turbulence scale Lo
(correlation length) satisfies the condition Lo >> rg . If these conditions are satisfied, a particle's transport path will be large compared to the Larmor radius in the resultant field composed of a regular magnetic field H o and by a large scale of a random field H . A small scale field within each of the Larmor revolutions will produce only a small distortion of a trajectory. Therefore near the front within several Larmor revolutions one can use the leading center approximation, neglecting a small scale component. In a system of coordinates with a shock front at rest let the plasma before the shock moves normally to it. The large-scale field before and behind the front is denoted by H1 and H2, respectively, and the small random component is neglected. If the angles a1, a2 (related by the expression U1ctga1 = U2ctga2 owed to the boundary conditions) satisfy the inequality u;ctga;- < c (i = 1, 2), the coordinate system K' will exist in which u'IIH' and an electric field E' is equal to zero in the whole space. The K' system moves opposite to the x-axis with the velocity U = ufitgai (Landau and Lifshitz, M1957).
In this system the energy of particles does not change and their trajectories in the regions 1 and 2 have the form of spiral segments. At arbitrary values a the coefficients of particle reflection from the front f (m) and those of particle's passage through the front p(M) as functions of m = cos 6 (the pitch angle cosine), averaged over a period of cyclotron rotation, can be found by a numerical calculation. However, if the condition a^ << 1 is valid the magnetic flux through the orbit (the transverse adiabatic invariant) will be conserved in the K' system when a particle crosses the wave front. Furthermore, we shall restrict our consideration to this case and for simplicity consider a non-relativistic case: u/c << a << 1. The differences of a' from a and of H' from H as well as tangential velocity component behind the front will then be negligibly small. From a comparison of the transverse adiabatic invariant before and behind the front, sin2 (6\/ H\ ) = sin2 (2/ H2), we find the boundary value o = V 1 - H1/H2 , (4.15.44)
separating the particles moving from the region 1 and reflecting from the front, from those passing through the front. Thus, the coefficient of passage is
. x fi at 1 >M > M o, P2 (M) = j0 t 0 < , < (4.15.45)
and the reflection coefficient fa (/') = 1-P12 (/'). F°r the particles incident upon the front from the region 2,
021^') = 1, /21l"') = 0 at -1 <M'< 0. (4.15.46)
A relation of pitch-angles in the stroked and initial systems is given by the expression
cos 6' = (cos 6 + u1l a1v )[1 ■^(u1/a1v )cos6 + (u1/a1v )2 ] ,(4.15.47)
which results in all values 6'-1 < cos 6' < 1 being possible with a^ < 1. For u1/a1v > 1 only positive values -y/1 -(a1v/u1) < cos6< 1 are possible. Since the reflection condition is cos 6'< y] 1 - H^H2 both latter inequalities will be valid simultaneously, but having the condition a >(u1/ v )(H1/ H2 ))2. For a,1 < (1/vH2))2 a particle with an arbitrary cos6 coming to the front from the region 1 will pass through it.
The dependence of cos 6 on cos 6' is double-valued:
cos6 = ±cos6^1 -((/a1v)2sin2 6 - (a1v)sin2 6 . (4.15.48)
In the case a^ < 1 one should take only the sign +. For a^ > 1, one should take into account both of the signs. The Eq. 4.15.44-4.15.48 give the values of ^ limiting the reflection region and the region of passage for the particles coming from the medium 1 into medium 2. They are different for three regions of values of «1:
region 1 a > u^ v ; reflection at - u^ a\v </</0+ , passage at /0+</< 1, where
/ = ±7(1 - H|/H2)(1 - (1/«v)2(HJH2))-U1HJavH2 ; (4.15.49
region 2 (1/v \H1H2 )2 < «1 < U1Iv ; reflection at /0- < / < /0+, passage at / + < / < 1 and - 1 < / < /0 - ;
region 3 (1/c)<< «1 < (1/H2 )2 ; only the passage through the front is possible.
For the particles in the medium 2 incidence onto the front is possible at -1 </<-U1I«v providing by the condition u^«v < 1; all these particles will pass through the front; there is no reflection. At u^«v > 1 a hit onto the front from the medium 2 is impossible. The coefficients of reflection f and of passage p, which are considered as the functions of are equal to 0 or 1 depending on in what range ¡n is located. When calculating an increment of energy of a particle reflecting from the shock front, let us include that in the system K', the energy variation of a particle does not take place, and the longitudinal momentum is changed to the reversal momentum:
Here we have neglected a difference between p// = pcos 9 and px as a result of ai angle being small. The momentum along Hj and a particle energy in the initial system before a reflection are designated by p// and Ek, and p' '// and E' \ correspond to the same quantities in the system K ' after reflection. Furthermore, a particle was considered to be non-relativistic: Ek = macv2 ¡2 .
passing again to the initial coordinate system, we obtain the energy variation at a reflection:
AEk = 2macVUl a
Remember that particle reflection is possible only with a >(M1/v H2 ))2 ;
substituting the boundary value «1 = (1/v H2 )2 at which the single value cosO = ¡o =-(1/H2 ))2 is possible, we shall obtain the maximum possible energy of the reflected particles ( for a given ration H^H2 ):
At the characteristic value H2/ H1 = 3, typical for interplanetary shock waves, the energy of a particle increases no more than 9 times in a reflection.
In the opposite limiting case m^a1v << 1, an energy increment at the reflection is a small portion of the initial particle energy:
When passing through a shock front the energy and momentum of a particle varies in the coordinate system K' in a following way:
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