Closed Form Expressions for the Potentials and Accelerations of Generalized Ring Models

TL;DR

This paper derives closed-form mathematical expressions for gravitational potentials and accelerations of various ring and arc mass distributions, utilizing elliptic functions, to enhance modeling efficiency in celestial mechanics.

Contribution

It introduces new closed-form formulas for potentials and accelerations of rings and arcs, including time-averaged cases, expressed through elliptic functions, improving computational modeling.

Findings

Expressions often simplified to real forms in certain limits.

Formulas applicable to arbitrary eccentricities and time-averaged scenarios.

Potential for faster, more accurate celestial mechanics modeling.

Abstract

We present several closed-form expressions of useful mass distributions. These include the potentials and accelerations of circular rings and arcs, the potentials of uniform density rings and arcs at arbitrary eccentricities, and the potentials and accelerations of rings and arcs when the mass is time-averaged over a Kepler orbit. We show that these expressions can be expressed, often simply, in terms of elliptic functions of complex arguments. We show that in a few limiting cases, the expressions are entirely real. We expect that these expressions will allow for more rapid modeling in many areas of celestial mechanics.