Higher-Order Gravitational Models: A Tutorial on Spherical Harmonics and the Newtonian Model
Felipe Arenas-Uribe
TL;DR
This work addresses the need for accurate gravity modeling beyond the point-mass approximation by deriving and implementing a spherical-harmonics expansion of the exterior gravitational potential. It shows that, outside a sphere of radius $R$, the potential satisfies $\nabla^2 U = 0$, allowing a separable solution that yields $U(r,\theta,\varphi)=\frac{GM}{r}+\sum_{\ell\ge1}\sum_{m=-\ell}^{\ell}\frac{B_{\ell m}}{r^{\ell+1}}Y_\ell^m(\theta,\varphi)$ and its Brillouin-sphere normalized form $U(r,\theta,\varphi)=\frac{GM}{r}\left[1+\sum_{\ell\ge1}\sum_{m=-\ell}^{\ell} \bar B_{\ell m}(R/r)^{\ell}Y_\ell^m(\theta,\varphi)\right]$. The article details computational strategies, including radial and angular decompositions, Legendre recursions, and Cartesian solid-harmonic recursions, plus approaches to estimate harmonic coefficients from data or shape models. It introduces a practical library, ARC-Grav, to assemble multi-sub-body gravity fields and demonstrates its use with Low Earth Orbit perturbations and irregular asteroid modeling. The results highlight that higher-order terms can induce measurable perturbations, especially in polar or irregular-body contexts, underscoring their importance for precise orbit prediction and mission design in perturbed environments.
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.