Semi-analytic Orbits: A Practical Implementation of Lynden-Bell's Planar Orbits and Extension to Vertical Oscillations

We present a practical implementation of the perturbation theory derived by Lynden-Bell (2015) for describing, to arbitrary precision, the orbit of a particle in an arbitrary spherically-symmetric potential. Our implementation corrects minor but important errors in the initial derivation, and extends the formalism in two ways. First, a numerical method is developed to efficiently and precisely solve the analogue to the Kepler problem, and second, a method is introduced to track the particle's vertical oscillations about an axisymmetric disk, even when the vertical oscillation frequency varies with radius. While not as flexible as numerical integration, this method guarantees conservation of energy, angular momentum, and related quantities, and may be used to evaluate a particle's position and velocity in constant time. Our implementation is written in Python and is pip installable as the package lbparticles.