Generalized true- and eccentric-anomaly parametrizations for the perturbed Kepler motion

TL;DR

This paper extends classical Kepler orbit parametrizations to quasiperiodic orbits affected by perturbations, providing methods for averaging observables and evaluating integrals relevant to gravitational radiation.

Contribution

It introduces generalized true- and eccentric-anomaly parametrizations for perturbed Kepler motion and develops a toolbox for averaging and integral evaluation in these systems.

Findings

Generalized parametrizations apply to quasiperiodic orbits.

Residue theorem effectively evaluates integrals over motion periods.

Pole analysis reveals conditions for additional poles in integrals.

Abstract

The true- and eccentric-anomaly parametrizations of the Kepler motion are generalized to quasiperiodic orbits by considering perturbations of the radial part of kinetic energy as a series in the negative powers of the orbital radius. A toolbox of methods for averaging observables in terms of the energy $E$ and angular momentum $L$ is developed. A broad range of systems governed by the generic Brumberg force, as well as recent applications of the theory of gravitational radiation involve integrals over a period of motion. These integrals are evaluated by using the residue theorem. It is shown that the pole of the integrand is located in the origin and that under certain circumstances an additional pole emerges.