Higher-Order Gravitational Models: A Tutorial on Spherical Harmonics and the Newtonian Model

TL;DR

This paper provides a tutorial on spherical harmonic gravity models, explaining their theoretical basis and importance for accurately predicting spacecraft trajectories affected by non-spherical celestial bodies.

Contribution

It offers a comprehensive introduction to higher-order gravitational models and demonstrates their practical impact on orbital dynamics around irregular bodies.

Findings

Higher-order terms significantly affect orbit predictions.

Spherical harmonics improve accuracy over point-mass models.

Examples include LEO satellites and asteroid missions.

Abstract

Accurate modeling of gravitational interactions is fundamental to the analysis, prediction, and control of space systems. While the Newtonian point-mass approximation suffices for many preliminary studies, real celestial bodies exhibit deviations from spherical symmetry, including oblateness, localized mass concentrations, and higher-order shape irregularities. These features can significantly perturb spacecraft trajectories, especially in low-altitude or long-duration missions, leading to cumulative orbit prediction errors and increased control demands. This article presents a tutorial introduction to spherical harmonic gravity models, outlining their theoretical foundations and underlying assumptions. Higher-order gravitational fields are derived as solutions to the Laplace equation, providing a systematic framework to capture the effects of non-uniform mass distributions. The impact of these higher-order terms on orbital dynamics is illustrated through examples involving Low Earth Orbit satellites and spacecraft near irregularly shaped asteroids, highlighting the practical significance of moving beyond the point-mass approximation.