One can see from the Eq. 2.17.6 that both processes of the pitch-angle diffusion and the particle transport are connected to each other here. Unlike the first approximation f there is not 'free' propagation in the small time limit solution as well as in the second approximation f2 . So the first approximation is suitable only in a very small time interval, t << 1.
2.17.4. Peculiarities of the second pitch-angle approximation
Unlike the first approximation, the function f2(6) describes the initially anisotropic stream during a longer time after the particle injection, and one has a low level at y ~t especially for y >> 1. The space distribution f2(y) has been depicted in Fig. 2.17.1, where the pitch- angle 6 is fixed. The picture is similar for non-zero 6. The space distribution possess rather wide 'tail' behind the front of the first particles at y = t. Its width increase with increasing time, and the maximum decreases in amplitude and becomes later with increasing time. Temporal development of f2(y,T) is shown in Fig. 2.17.2. We conclude that unlike the first approximation in pitch-angle the derived expressions for the particle distribution function in the second approximation as well as the particle density gives a more realistic picture of the pitch-angle distribution after an immediately unidirectional particle injection.
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