It can be seen from Eq. 4.11.28, Eq. 4.11.29, and Eq. 4.11.32 that at P = 0 the expected spectra of the accelerated particles are of power form. In the opposite case, at P > 0, the power exponent in the accelerated particle spectrum increases with increasing the energy or rigidity of the particles.
4.11.5. Formation of the particle spectrum in the magnetic pumping mechanism including absorption in the source
Fälthammer (1963) studied this acceleration mechanism in detail, examined the injection conditions, and derived the following form of the kinetic equation for the function of particle distribution in the space of coordinates and moments for a stationary case:
dP \ t J Tab where / is the sought distribution function, p is the particle momentum, k is the diffusion coefficient, t is the effective duration of acceleration, and Tab is the effective duration of absorption. Assuming that the particle injection is determined by the condition p//t = noS(r) at p = po, Fälthammer (1963) finds the solution of the equation presented above (subject to k, t, and Tab are independent of the spatial coordinates r of the leading center of particles but are functions of only momentum p) for three-dimensional case in the form:
T dp Tab P
In the two-dimensional case the resultant solutions are the same; the only
difference is that the denominator contains nRp instead of [nRpj . Fälthammer
(1963) has shown that Alfven's partial solution (Alfven, 1959) can be obtained from the reduced general solution if the following dependences on p for k, t and Tab are selected:
K=Ko(p/po)1-^, <p = const; t = t0(p/po)a, a = const; Tab = const.(4.11.36) In this case the resultant spectrum in the range of large momentum is of a power form. In the small-momentum range at small distances from the injection region, the resultant spectrum is also of a power form, whereas at great distances the spectrum should have a significant fall at low energies. These results are qualitatively in a good agreement with the experimental data. In order to obtain the exponent y = 2.5 in the differential spectrum at high energies, it is necessary to the two-dimensional
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