The accuracy of mutual potential approximations in simulations of binary asteroids

TL;DR

This paper compares an exact surface integral method for calculating mutual gravitational potentials in binary asteroid simulations with traditional series expansion methods, highlighting accuracy issues and computational efficiency.

Contribution

It introduces a surface integral approach for exact potential calculations and evaluates its accuracy and efficiency against truncated series expansions.

Findings

Differences in forces and torques are largest when bodies are nearly touching.

Approximate solutions can differ by over 1000% for elongated primary bodies.

Long-term dynamics show over 100% difference when bodies are initially close.

Abstract

Simulations of asteroid binaries commonly use mutual gravitational potentials approximated by series expansions, leading to truncation errors, and also preventing correct computations of the forces and torques when the bodies are close. We make of a recently developed method where the mutual potential is calculated with the use of surface integrals and is exact for bodies of ellipsoidal shapes. The solutions produced by the surface integration method are compared with an approach that expands the mutual potential, truncated at second and fourth order. The approximate solutions are generated with the ``General Use Binary Asteroid Simulator'' (gubas). We find that the differences in the forces and torques are the largest when the bodies are nearly touching. These differences can exceed 1000% if the shape of the primary is highly elongated. Long term simulations show more than 100% difference in the dynamics if the bodies are initially close, while the differences are negligible if the bodies are initially far apart. For simulations with two triaxial ellipsoids, the computational efficiency of the surface integral method is comparable to fourth order approximations with gubas, and superior to potentials truncated to order eight or higher.