Approximate Analytical Solution to the Zonal Harmonics Problem Using Koopman Operator Theory

TL;DR

This paper presents a novel method using Koopman operator theory to derive approximate analytical solutions for satellite orbit evolution affected by zonal harmonics, applicable to various orbit types with adjustable accuracy.

Contribution

It introduces a new approach leveraging Koopman operator theory for analytical orbit solutions, including a modified orbital element set for rapid convergence.

Findings

Applicable to all orbit types including hyperbolic

Provides automated solutions with adjustable accuracy

Demonstrates effectiveness through multiple examples

Abstract

This work introduces the use of the Koopman operator theory to generate approximate analytical solutions for the zonal harmonics problem of a satellite orbiting a non-spherical celestial body. Particularly, the solution proposed directly provides the osculating evolution of the system under the effects of any order of the zonal harmonics, and can be automated to obtain any level of accuracy in the approximated solution. Moreover, this paper defines a modified set of orbital elements that can be applied to any kind of orbit and that allows the Koopman operator to have a fast convergence. In that regard, several examples of application are included, showing that the proposed methodology can be used in any kind of orbit, including circular, elliptic, parabolic and hyperbolic orbits.