# Exploring the long-term dynamics of perturbed Keplerian motion in high degree potential fields

## TL;DR

This paper investigates the long-term behavior of perturbed Keplerian orbits using high-fidelity potential models, emphasizing symbolic algebra and recursive methods to improve computational efficiency over brute-force approaches.

## Contribution

It demonstrates how Kaula's recursion formulas can efficiently replace expanded expressions and brute-force methods in analyzing perturbed orbital dynamics with high-degree potential fields.

## Key findings

- Kaula's recursion formulas outperform expanded expressions.
- Symbolic algebra systems handle complex potential models effectively.
- Recursive methods simplify long-term orbital analysis.

## Abstract

The long-term dynamics of perturbed Keplerian motion is usually analyzed in simplified models as part of the preliminary design of artificial satellites missions. It is commonly approached by averaging procedures that deal with literal expressions in expanded form. However, there are cases in which the correct description of the dynamics may require full, contrary to simplified, potential models, as is, for instance, the case of low-altitude, high-inclination lunar orbits. In these cases, dealing with literal expressions is yet possible with the help of modern symbolic algebra systems, for which memory handling is no longer an issue. Still, the efficient evaluation of the averaged expressions related to a high fidelity potential is often jeopardized for the expanded character of the output of the automatic algebraic process, which unavoidably provides huge expressions that commonly comprise tens of thousands of literal terms. Rearrangement of the output to generate an efficient numerical code may solve the problem, but automatization of this kind of post-processing is a non trivial task due to the ad-hoc heuristic simplification procedures involved in the optimization process. However, in those cases in which the coupling of different perturbations is not of relevance for the analysis, the averaging procedure may preserve the main features of the structure of the potential model, thus avoiding the need of the typical blind computer-based brut force perturbation approach. Indeed, we show how standard recursions in the literature may be used to efficiently replace the brut force approach, in this way avoiding the need of further simplification to improve performance evaluation. In particular, Kaula's seminal recursion formulas for the gravity potential reveal clearly superior to the use of both expanded expressions and other recursions more recently proposed in the literature.