Analytic Solution for Perturbed Keplerian Motion Under Small Acceleration Using Averaging Theory

TL;DR

This paper introduces an improved analytic method for orbit propagation under small perturbations, achieving higher accuracy than existing techniques by combining averaging theory and linearization of orbital equations.

Contribution

It develops a novel asymptotic expansion approach that enhances orbit prediction accuracy for perturbed Keplerian motion using linear systems and averaging theory.

Findings

Positional error reduced by an order of magnitude compared to state-of-the-art methods.

Achieves tens of meters accuracy in LEO orbits after five periods.

Effective for small tangential accelerations like 1e-5.

Abstract

A novel approach is developed for analytic orbit propagation based on asymptotic expansion with respect to a small perturbative acceleration. The method improves upon existing first order asymptotic expansions by leveraging on linear systems and averaging theories. The solution starts with the linearization of Gauss planetary equations with respect to both the small perturbation and the six orbital elements. Then, an approximate solution is obtained in terms of secular and short period components. The method is tested on a low-thrust maneuver scenario consisting of a Keplerian orbit perturbed by a constant tangential acceleration, for which a solution can be obtained in terms of elliptic integrals. Results show that the positional propagation error is about one order of magnitude smaller with respect to state-of-the-art methods. The position accuracy for a LEO orbit, apart from pathological cases, is typically in the range of tens of meters for a dimensionless tangential acceleration of 1e-5 after 5 orbital periods propagation.