Analytical considerations

The main assumption in diffusive shock acceleration is that the pitch-angle distribution is nearly isotropic. By requiring the diffusive streaming anisotropy to be small, one can readily derive an expression for the injection velocity, winj (c.f.,

Giacalone and Jokipii, 1999). The most general expression is given by:

winj = 3u1

where k_|_, and K// are the components of the diffusion tensor perpendicular and parallel to the mean magnetic field, respectively, and the anti-symmetric component of the diffusion tensor is Kg = vrgj 3 (rg is the Larmor radius of accelerated particles in the mean magnetic field). For the case in which the correlation scale of the turbulent magnetic field is much larger than the gyro-radius of the particles of interest, it has been shown from numerical simulations that Kj_/k// is independent of energy (Giacalone and Jokipii, 1999). Thus, taking £ = k__/k// << 1 and n = A//rgj3, where A// is the parallel mean-free path and rg is the Larmor radius, Eq. 4.26.1 can be rewritten as:

winj - winj,ll

1 + (l/rjf sin2 eBn + sin2 0Bn cos2 dBn (s sin2 9Bn + cos2 0Bn F

where winj- // = 3ui is the injection velocity for a parallel shock.

Shown in Fig. 4.26.1 is the solution to Eq. 4.26.2 for n = 100 and £ = 0.02. The dashed curve is sec 0Bn , which is the scatter-free approximation which is clearly invalid for the case of a turbulent magnetic field. Note that at low-energies, the injection velocity at a perpendicular shock approaches 3u1 , which is the same as that obtained for a parallel shock (Giacalone, 2003).

Thus, it can be conclude that enhanced motion normal to mean field by field-line random walk significantly decreases the injection velocity threshold for acceleration. Thus, the theory predicts that there should not be an injection problem at nearly perpendicular shocks. The acceleration rate, vac, in diffusive shock acceleration is given by

inj.i

Fig. 4.26.1. The injection velocity derived from the diffusive streaming anisotropy for the case of field-line random walk (solid line) normalized to that at a parallel shock. The dashed curve assumes the scatter-free approximation. According to Giacalone and Jokipii (2005).

Thus, taking vac // as the acceleration rate at a parallel shock (dgn = 0), and £ = (as before), we obtain

Eq. 4.26.4 is plotted as a function of &Bn for the case of £ = 0.02 in the Fig. 4.26.2. From Fig. 4.26.2 clearly can be seen that the acceleration rate is a maximum at perpendicular shocks. Therefore, it can be conclude that perpendicular shocks are both efficient and rapid accelerators of charged particles are most important in producing high-energy CR in a wide variety of astrophysical plasmas.

Fig. 4.26.2. The acceleration rate, normalized to that at a parallel shock, as a function of dBn . According to Giacalone and Jokipii (2005).

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