and Ek max = 32 and 100 MeV for electrons at lo = 10 cm and lo = 10 cm, respectively. However, it is necessary that the injection condition should also be satisfied for the above possibility of particle acceleration to be realized.
4.16.4. The particle injection conditions for acceleration in a magnetic trap
It is evident that the most crucial moment for the particle's acceleration in a magnetic trap is the initial stage of acceleration, i.e. the first collisions of particles with magnetized clouds. In a magnetized cloud the particle's velocity in the first collision will be v = u + vo. According to Spitzer (M1956) the period between the collisions with the thermal particles of cloud plasma, i.e. the time during which our test particle is gradually deflected by 90° as a result of multiple small deflections, is determined as
where N is the concentration of the medium; m, is the mass of the plasma particles;
T is the temperature in the space plasma; A = (3/2Ze3 )k 3T VnN ^ . The functions O and G and the values of ln A are presented in Chapter 5 of the monograph Spitzer (M1956). If vtd > nrg, i.e. exceeds the path in a magnetic cloud, the particle may be ejected from the cloud without colliding with the particles of cloud plasma. Later on the particle will move between the clouds, and if the condition vtd > l proves to be satisfied the particle can enter another cloud. Since the particle velocity increased as a result of the first and second collisions, the condition vtd > nrg will be explicitly satisfied in the second cloud, and, moreover, this condition will be satisfied during the subsequent motion of the particle through the space between the clouds.
When the particle velocity exceeds the electron velocity in space, the energy loss for dynamical friction is more significant. In this case, according to Spitzer (M1956), dv/dt = -v/ts , (4.16.24)
23 macv memacv /„^„^ ts =-—-ac-, , —r-=-- (4.16.25)
8nNZ e (1 + macjme )(me/2kT ) G((m,/2kT )ln A 4nNZ2e4ln A (the latter is valid at v >> ^2kT/me and mac >> me).
It can be seen from the comparison between Eq. 4.16.23 and Eq. 4.16.24 that under the above conditions ts/td ~ me/mac, i.e. the validity of the conditions vts > 7Lrg and vts > l becomes more critical for heavy particles at high velocities.
These conditions may be more accurately obtained by integrating Eq. 4.16.24 along the path of motion. The resultant path is
8ne4 NZ2 ln A
where vjn and vfm are the particle's initial and final velocities. The appropriate conditions will be L > 7Lrg and L > l. When considering electron acceleration, me _ mac and the factors memac in Eq. 4.16.25 and. Eq. 4.16.26 and mac in Eq.
4.16.23 will turn out to be me . In this case td ~ ts since O - G ~ 1 in the cases of interest to us.
4.16.5. Diffusive compression acceleration of charged particles
Jokipii et al. (2003) consider the acceleration of fast charged particles by smooth compressions and expansions in a collisionless fluid by using the diffusion approximation. If the diffusion length k/u is of the order of the fluid scale or larger, efficient acceleration occurs which has similarities with both 2nd-order Fermi acceleration and diffusive shock acceleration, but is different from both. A simple one-dimensional sinusoidal flow is analyzed in Jokipii et al. (2003). It was shown that the acceleration dominates, even with equal amounts of compression and expansion. The acceleration time is = tcju2 . They suggest that this mechanism may be an important accelerator in regions where there are large-scale compressive disturbances, but few shocks. It may contribute to the acceleration of CR elsewhere in the Heliosphere and the Galaxy. It was suggest the name 'diffusive compression acceleration' for this mechanism.
Now a number of general acceleration mechanisms have been suggested. The most successful of these has been the acceleration by collisionless shocks (Krymsky, 1977; Axford et al., 1977; Bell, 1978; Blandford and Ostriker, 1978; Drury, 1983; Jones and Ellison, 1991; see also Sections 4.14, 4.15, 4.21-4.24). However, there are important situations in which energetic particles are accelerated with no shocks present. One recent and particularly clear example consists of the energetic ions observed in interplanetary co-rotating interaction regions near 1 AU, well inside the radius at which the associated co-rotating shocks form (Mason, 2000). Giacalone et al (2002) found that compression acceleration provided a natural and compelling interpretation of the observations.
Jokipii et al. (2003) consider the transport of CR in the diffusion approximation, in which the (nearly-isotropic) distribution function f(r,p,t) as a function of position r, momentum magnitude p and time t satisfies the Parker equation (Parker, 1965):
f dt dxi
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