r where rg is determined by Eq. 4.13.2, Ei is the initial average energy, and r¡i is the initial rate of charged particles injection into acceleration process. The obtained slope 5/2 in the spectrum is in good agreement with observations of CR in high energy range (about 2.6 - 2.7).
Let us note that the interaction of charged particles with strong low-frequency electromagnetic field generated in case of rotation of inclined magnetic dipole has been studied analytically and numerically by Grewing and Heintzmann (1973a,b,c,d). It has been shown that, when reasonable initial conditions for rotating magnetized neutron star are properly selected, the particle may acquire an energy comparable with the highest observed CR energy.
4.14. Particle acceleration by moving magnetic piston
As a result of the ejection of magnetized plasma (for example, during the chromosphere flares, coronal ejections, during the explosions of Novae and Supernovae) the particles will be reflected from such a condensation (let us call it a magnetic piston) and the particle energy will increase or decrease depending on the type of an acting collision: head-on or overtaking collision.
4.14.1. Acceleration and deceleration at a single interaction of particles with magnetic piston
Let a magnetic piston have a thickness l, the intensity of a magnetic field in it be H, and let it move with a velocity u (u << c). If the angle between the velocity v of a particle's motion and the piston velocity u is 9, then the particle will be reflected from the piston having a Larmor radius inside the piston:
Since a particle's velocity in the coordinate system related to the piston is v-u, then rL =■
The Eq. 4.14.1 and Eq. 4.14.2 result in that the piston not reflecting the particles lV2
which have the velocities v > vcr, where
At a reflection when v < vcr, a relative energy variation of a particle, according to Sections 2.2-2.5, will be
If v > vcr, the particle will pass through a magnetic piston. In this case, it will be scattering at the angle 9 which is determined by the equation:
where rL is determined according to Eq. 4.14.2. In the case of scattering at the small angles 6 << 1, the equation results in
0 ~ 2arcsin l cosp 2rT
The energy variation in this case is determined according to Sections 4.3-4.5:
It should be noted that Eq. 4.14.4 and Eq. 4.14.6 hold true also at large values of u. For non-relativistic particle energies the relative variation of kinetic energy hEk/Ek will be considerable even in a single reflection:
The Eq. 4.14.7 results in AE^/E^ being able to be very large at v ~ u. In the range of super-relativistic energies
at the particle reflection from a magnetic piston and
when a particle crosses it. The Eq. 4.14.9 and Eq. 4.14.10 show that the relative energy variation is not large, of the order of u/c, in the case of relativistic energies.
4.14.2. Acceleration and deceleration of particles at the multiple interactions with magnetic piston
In the presence of scattering medium behind or/and before a magnetic piston, a multiple interaction of particles with a piston will take place and particle energy l c c c c u v variation will be very significant. In this case a share of particles interacting with a piston will be pronouncedly decreased with increase of the multiplicity of interaction. Therefore the differential energy spectrum of accelerated particles will fall down with a growth of particle energy; the detailed form of a spectrum will be determined by the probability dependence for a given multiplicity of particle interaction with a piston, The same conclusion can also be drawn for a dependence of the relative share of decelerated particles An/n on the particle energy - it should also be decreased with a growth of particle energy. Some examples of particle acceleration and deceleration in the process of multiple interactions with a magnetic piston will be discussed below.
4.15. Mechanisms of particle acceleration by shock waves and other moving magneto-hydrodynamic discontinuities during a single interaction
The particle acceleration by the moving magneto-hydrodynamic discontinuities is probably one of the accelerated processes which are most frequent in the space (in the solar atmosphere, in interplanetary space and in the magnetospheres of the planets, in the Galaxy, etc.). The mechanism of acceleration by the transverse and oblique shock fronts for normal and oblique incidence of particles, including the scattering by magnetic inhomogeneities of medium has been most comprehensively developed (Dorman and Freidman, 1959; Shabansky, 196l, I966; Schatzman, I963; Korobeinikov and Lomnev, 1964; Alekseev and Kropotkin, 1970; Vasilyev at al., 1978; and others).
4.15.1. Acceleration for single passage of a laterally incident particle (the shock front is unlimited)
Consider a shock wave in a medium with a frozen magnetic field parallel to the shock front plane. Let the shock front move at velocity U1. In undisturbed space 1 the field intensity is H1, in disturbed space 2 moving at velocity u - U2 relative to the rest system, the field intensity is H2 (see Fig. 4.15.1). A particle moving in undisturbed space 1 will collide with the magnetized shock front, be reflected from the front, and gain an additional momentum as in a head-on collision with mirror. Then the particle will again collide with the front, etc. After a while, however, the process will stop due to the particle drift to undisturbed space 2 behind the front. In addition to that, the drift along the front takes place owing to the difference between H1 and H2 . In this case if the front is limited, the acceleration may stop even earlier, before the particle is completely transferred to space 2. The calculations carried out by Dorman and Freidman (1959) for the case of an infinite front show that such a mechanism may give a considerable increase of the particle's energy. In Dorman and Freidman (1959) the particle acceleration was estimated, on the assumption of normal particle incidence onto the front.
In the coordinate system relative to the shock front (as in Fig. 4.15.1, in which the shock front is in plane yz, the magnetic field along z axis), the particle motion will be as follows. In space 1 a charged particle affected by magnetic field H and electric field E1 =-—x Hx will drift at velocity u1 towards the front. Near the c front the particle is affected by the difference in H1 and H2, and will drift also along the front towards the Y axis (i.e. along the electric field Ej = E2); on traversing the shock front plane the particle will drift in disturbed space at a velocity u2 .
Let the particle move in undisturbed space at velocity vo perpendicular to the magnetic field. In a fixed coordinate system the particle will then move along a spiral with curvature radius n = cpo/ZeH1 and frequency (( = ZecH1/Eo . Here Ze is the particle charge, po is the initial momentum, Eo is the total initial energy of the particle. When the front approaches the particle, the drift will be toward the Y axis and the particle energy will change by
where l is the drift along the y axis. Let us estimate l. The shift during a single cycle will be
where a = H2IH1 = u^u2 is the degree of the compression of transverse magnetic field in the shock wave. We assume here that the particle energy is almost invariable during a single cycle. The time of a single cycle will be
n n nEo
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