Closed-form solution for the general phase-resolved boundary-layer problem

Derive a closed-form solution for the phase-resolved boundary-layer problem defined by the two-dimensional Fokker–Planck equation with inclination diffusion and an inclination-dependent loss surface for arbitrary loss-cone fullness parameter q.

Background

The paper formulates a phase-resolved Fokker–Planck problem for angular-momentum magnitude and inclination at fixed energy, with the loss process implemented through a pericenter reset and with q measuring the relative importance of orbital-period depletion and angular-momentum diffusion. The general intermediate-fullness regime requires retaining both radial phase and inclination transport.

The authors state that only the empty-cone and full-cone limits admit analytic treatment. A closed-form solution for arbitrary q would provide an analytic description of the boundary-layer problem across the intermediate regime and would clarify how inclination diffusion modifies relativistic loss fluxes beyond the asymptotic limits.