Above we do not take into account the energy lost by the particle during its regular acceleration by the shock wave. In real space plasma accelerated particles lose energy in different types of particle interactions with matter, magnetic fields and photons (see detailed description above, in Chapter 1). According to Bulanov and Dogiel (1979) this account can be made on the basis of the Fokker-Planck equation which in the one-dimensional approximation will be (Berezhko et al., M1988):
where the term pu/ dx )p/3) describes the adiabatic energy change caused by the change of plasma velocity, F = (Ap/At) is the averaged temp of momentum changing owing to energy lose, and D = p/2)Ap2/At^ is the diffusion coefficient in momentum space. Berezhko et al. (M1988) for simplicity considered the case in which in the regions i = 1, 2 the energy lose can be presented in the form
where the characteristic times Ti of energy lose do not depend on the momentum p. It is supposed also that the parameter a■ = 4kJ tu2 is the same in both regions, i.e., a1 = a2 = a. In this case Eq. 4.21.31 can be solved and its solutions for regions i = 1, 2 are as follows:
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