Potential series expansion method applied in Analytical Modeling of Gravitational field of Irregularly Shaped Celestial Bodies

TL;DR

This paper introduces a series expansion method for efficiently modeling the gravitational field of irregularly shaped celestial bodies, achieving high accuracy with reduced computation time, useful for space mission planning.

Contribution

The study applies the Series Potential Expansion Method (PSEM) to polyhedral models of celestial bodies, providing a faster and sufficiently accurate alternative to classical methods.

Findings

Relative errors below 0.1% at higher orders (11-12)

Significant reduction in computational time

Effective for stability analysis and trajectory planning

Abstract

This study aims to establish an analytical model that reproduces the gravitational field around non-spherical bodies with constant density. Due to the non-spherical geometry of such bodies, their gravitational potential is disturbed relative to a central field. By considering the body as a polyhedron and decomposing it into tetrahedral elements, we use the Series Potential Expansion Method (PSEM) to approximate the total potential by summing the potentials of each tetrahedron. While this model does not offer higher accuracy than the classical polyhedral approach, it achieves relative errors below 0.1\% for points outside the body when developed to higher orders (e.g., orders 11 and 12), and significantly reduces execution time. To validate this approach, we apply our model to asteroids (87) Sylvia, (101955) Bennu, (99942) Apophis, and (25143) Itokawa. We determine equilibrium points, analyze stability, investigate zero-velocity planes, and calculate the relative errors between the gravitational field modeled by PSEM and the results obtained using both the classical polyhedral method by Tsoulis and Petrovic and the mass concentration method. Our results highlight the computational efficiency of PSEM in modeling the gravitational potential of irregularly shaped bodies. This efficiency stems from expressing the gravitational potential through a homogeneous analytical function that is easy to manipulate algebraically, enabling explicit determination of the acceleration vector. Our model provides a robust framework for more complex analyses, such as studying periodic orbits around non-spherical celestial bodies, assessing their stability, and planning the smooth landing trajectories of spacecraft.