Collision and Annihilation of Relative Equilibrium Points Around Asteroids with a Changing Parameter

TL;DR

This paper analyzes how the number and types of relative equilibrium points around asteroids change with parameters, revealing bifurcations and a conserved quantity that restricts equilibria in gravitational fields.

Contribution

It introduces a theorem on a conserved quantity that limits the number of equilibria and details various bifurcation types in asteroid gravitational potential.

Findings

Number of equilibria varies in pairs and is odd.

Multiple bifurcation types identified, including saddle-node and pitchfork bifurcations.

For asteroid 216 Kleopatra, equilibria decrease from 7 to 1 as rotation period varies.

Abstract

In this work, we investigate the bifurcations of relative equilibria in the gravitational potential of asteroids. A theorem concerning a conserved quantity, which is about the eigenvalues and number of relative equilibria, is presented and proved. The conserved quantity can restrict the number of non-degenerate equilibria in the gravitational potential of an asteroid. It is concluded that the number of non-degenerate equilibria in the gravitational field of an asteroid varies in pairs and is an odd number. In addition, the conserved quantity can also restrict the kinds of bifurcations of relative equilibria in the gravitational potential of an asteroid when the parameter varies. Furthermore, studies have shown that there exist transcritical bifurcations, quasi-transcritical bifurcations, saddle-node bifurcations, saddle-saddle bifurcations, binary saddle-node bifurcations, supercritical pitchfork bifurcations, and subcritical pitchfork bifurcations for the relative equilibria in the gravitational potential of asteroids. It is found that for the asteroid 216 Kleopatra, when the rotation period varies as a parameter, the number of relative equilibria changes from 7 to 5 to 3 to 1, and the bifurcations for the relative equilibria are saddle-node bifurcations and saddle-saddle bifurcations.