The planar two-body problem for spheroids and disks

TL;DR

This paper presents a new exact numerical method for solving the two-body problem involving spheroids and disks by integrating gravitational potential over surfaces, avoiding series truncation errors, and demonstrating its effectiveness through simulations.

Contribution

It introduces a surface integral-based method for the two-body problem with spheroids and disks, providing exact solutions without series truncation errors.

Findings

The method accurately models non-Keplerian precession patterns.

Conservation of energy and angular momentum is verified.

The approach smoothly transitions to point mass and disk cases.

Abstract

We outline a new method suggested by Conway (2016) for solving the two-body problem for solid bodies of spheroidal or ellipsoidal shape. The method is based on integrating the gravitational potential of one body over the surface of the other body. When the gravitational potential can be analytically expressed (as for spheroids or ellipsoids), the gravitational force and mutual gravitational potential can be formulated as a surface integral instead of a volume integral, and solved numerically. If the two bodies are infinitely thin disks, the surface integral has an analytical solution. The method is exact as the force and mutual potential appear in closed-form expressions, and does not involve series expansions with subsequent truncation errors. In order to test the method, we solve the equations of motion in an inertial frame, and run simulations with two spheroids and two infinitely thin disks, restricted to torque-free planar motion. The resulting trajectories display precession patterns typical for non-Keplerian potentials. We follow the conservation of energy and orbital angular momentum, and also investigate how the spheroid model approaches the two cases where the surface integral can be solved analytically, i.e. for point masses and infinitely thin disks.