Stability and Motion around Equilibrium Points in the Rotating Plane-Symmetric Potential Field

TL;DR

This paper analyzes the stability and motion near equilibrium points in a rotating plane-symmetric potential field, with applications to planetary science and asteroid dynamics.

Contribution

It classifies equilibrium points, establishes stability conditions, and applies the theory to rotating cubes and asteroids, revealing phenomena like equilibrium point annihilation.

Findings

Non-degenerate equilibrium points are classified into twelve cases.

Resonant equilibrium points are identified as Hopf bifurcation points.

Equilibrium points can collide and annihilate as rotation speed varies.

Abstract

This study presents a study of equilibrium points, periodic orbits, stabilities, and manifolds in a rotating plane symmetric potential field. It has been found that the dynamical behaviour near equilibrium points is completely determined by the structure of the submanifolds and subspaces. The non-degenerate equilibrium points are classified into twelve cases. The necessary and sufficient conditions for linearly stable, non resonant unstable and resonant equilibrium points are established. Furthermore, the results show that a resonant equilibrium point is a Hopf bifurcation point. In addition, if the rotating speed changes, two non degenerate equilibria may collide and annihilate each other. The theory developed here is lastly applied to two particular cases, motions around a rotating, homogeneous cube and the asteroid 1620 Geographos. We found that the mutual annihilation of equilibrium points occurs as the rotating speed increases, and then the first surface shedding begins near the intersection point of the x axis and the surface. The results can be applied to planetary science, including the birth and evolution of the minor bodies in the Solar system, the rotational breakup and surface mass shedding of asteroids, etc.