Pitch Angle Diffusion With a Partially Filled Loss Cone: Exactly Solvable Models and General Approximations

Analytical solutions for the steady state pitch angle diffusion equation are reconsidered, both outside and inside the loss cone. Kennel and Petschek (1966, https://doi.org/10.1029/JZ071i001p00001 ) obtained expressions, still in widespread use, by matching different approximations on either side of the loss cone, but the derivation relied on a simple, artificial form of the pitch angle diffusion coefficient, . Schulz and Lanzerotti (1974) presented a solution using a somewhat more realistic and more flexible expression for , assuming a perfectly empty loss cone. Here, these approaches are combined, matching the latter results outside the loss cone to the earlier expressions inside the loss cone. The expressions show explicitly that while lifetimes scale inversely with the diffusion coefficient when is small, can be several times the scaled empty‐loss‐cone value when is large. The approach is also developed for constant , requiring Legendre functions, and variants of are constructed for pitch angle distributions given by several other “special functions.” Matching to the analytical solution inside the loss cone is also implemented for arbitrary forms of , extending a previously presented “shooting” algorithm for exactly determining to allow for a partially filled loss cone. Finally, this approach is used to augment a simple integral expression for approximating for any to also allow for a partially filled loss cone.