The true and eccentric anomaly parametrizations of the perturbed Kepler motion

TL;DR

This paper extends true and eccentric anomaly parametrizations to quasiperiodic perturbed Kepler orbits, develops averaging methods for observables, and investigates the existence and pole structure of these parametrizations.

Contribution

It generalizes anomaly parametrizations to perturbed orbits and analyzes the conditions for their existence and pole locations using complex analysis techniques.

Findings

Anomaly parametrizations exist under specific conditions.

Residue theorem is used to evaluate integrals of orbital functions.

Conditions for poles at the origin are identified.

Abstract

The true and eccentric anomaly parametrizations of the Kepler motion are generalized to quasiperiodic orbits, by considering perturbations of the radial part of the kinetic energy in a form of a series of negative powers of the orbital radius. A toolchest of methods for averaging observables as functions of the energy $E$ and angular momentum $L$ is developed. A broad range of systems governed by the generic Brumberg force and recent applications in the theory of gravitational radiation involve integrals of these functions over a period of motion. These integrals are evaluated by using the residue theorem. In the course of this work two important questions emerge: (1) When does the true and eccentric anomaly parameter exist? (2) Under what circumstances and why are the poles in the origin? The purpose of this paper is to find the answer to these queries.