Stability of the Euler Resting N-Body Relative Equlilbria

TL;DR

This paper investigates the stability of linear arrangements of equal-sized gravitating spheres in rotation, revealing that such configurations are only stable for up to five bodies, thus limiting the aspect ratio of rubble pile shapes.

Contribution

It generalizes the Euler Resting configuration stability analysis to N bodies, establishing a maximum of five for stability and implications for asteroid shapes.

Findings

Euler Resting configuration stable only up to 5 bodies

Configurations with 6 or more bodies are inherently unstable

Limits the aspect ratio of rubble pile bodies to 5:1

Abstract

The stability of a system of $N$ equal sized mutually gravitating spheres resting on each other in a straight line and rotating in inertial space is considered. This is a generalization of the "Euler Resting" configurations previously analyzed in the finite density 3 and 4 body problems. Specific questions for the general case are how rapidly the system must spin for the configuration to stabilize, how rapidly it can spin before the components separate from each other, and how these results change as a function of $N$. This paper shows that the Euler Resting configuration can only be stable for up to 5 bodies, and that for 6 or more bodies the configuration can never be stable. This places an ideal limit of 5:1 on the aspect ratio of a rubble pile body's shape.