On the Balance between Radial and Pitch Angle Diffusion

Abstract

The various attempts to mathematically describe the observed populations of trapped particles can be summarized by the following set of equations expressing conservation of particle number f and energy:

$$\frac{\partial }{{\partial t}}f({x^\mu },t) = \frac{\partial }{{\partial {x^\mu }}}\left( {{D^{\mu v}}\frac{{\partial f}}{{\partial {x^v}}}} \right) - L({x^\mu }) + S({x^\mu })$$

(1)

and conservation of electromagnetic wave energy e:

$$\frac{\partial }{{\partial t}}\varepsilon (\omega ,t) = - {\text{div(}}\varepsilon {{\text{v}}_g}{\text{) + 2(}}\gamma {\text{ - }}l{\text{)}}\varepsilon$$

(1)

. The variables x ^μ represent the adiabatic invariants which undergo small fluctuations by either interaction with the electromagnetic wave field or collisions. D ^μv is the diffusion tensor which so far has been considered mostly for L and pitch angle diffusion. S represents any internal particle source like the neutron albedo decay and L losses due to collisions.