On the circular Sitnikov problem: the alternation of stability and instability in the family of vertical motions

TL;DR

This paper investigates the stability patterns of vertical oscillations in a special three-body problem, revealing how stability alternates as the amplitude of motion changes continuously.

Contribution

It provides a detailed analysis of the stability and instability alternation in the family of vertical motions within the circular Sitnikov problem.

Findings

Stability alternates periodically with amplitude changes.

Linear stability analysis confirms the pattern of stability and instability.

Results enhance understanding of dynamical behavior in restricted three-body systems.

Abstract

We consider the special case of the restricted circular three-body problem, when the two primaries are of equal mass, while the third body of negligible mass performs oscillations along a straight line perpendicular to the plane of the primaries (so called periodic vertical motions). The main goal of our investigation is to study the stability of these periodic motions in the linear approximation. A special attention is given to the alternation of stability and instability within the family of periodic vertical motions, whenever their amplitude is varied in a continuous monotone manner.