F4 fWMl 2I71

Note that cross-field transport (i.e. perpendicular diffusion and drift, energy change, or adiabatic focusing) is not included into the model.

2.17.2. The first order approximation

Using the Fourier transform in the space variable y, and the Laplace transform in the time t , Eq. 2.17.1 gives the ordinary differential equation that does not lead to known special functions (Komarov et al., M1976); therefore some approximation of Eq. 2.17.1 is necessary. The simplest approximation corresponds to very small pitch-angle 6, when one can put sinO ^ 6 and cosO ^ 1. In this case the function f1 (y,T,0,0o) in the first order approximation is well known (e.g., Dorman and Katz, 1977d):

where lo (x) is the zero order Bessel function with an imaginary argument.

2.17.3. The second order approximation

In the second order pitch-angle approximation one must also hold the term of

O(d1). This means that sin0^0 and cos0^ 1 -02ยก2 and Eq. 2.17.1 forf2 in the second order approximation reads f+ dT

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