A generalized Sitnikov problem

TL;DR

This paper analyzes a generalized Sitnikov problem involving multiple primary bodies and a massless particle, classifying possible motions and exploring conditions for periodicity related to pyramidal central configurations.

Contribution

It provides a classification of the massless particle's motions and establishes conditions for periodicity in a generalized gravitational setup.

Findings

Classified all motions of the massless particle in rigid primary configurations.

Derived conditions for the existence of periodic motions with the same period as primaries.

Linked periodicity to the existence of pyramidal central configurations.

Abstract

In this paper we address a $n+1$-body gravitational problem governed by the Newton's laws, where $n$ primary bodies orbit on a plane $\Pi$ and an additional massless particle moves on the perpendicular line to $\Pi$ passing through the center of mass of the primary bodies. We find a condition for that the configuration described be possible. In the case that the primaries are in a rigid motion we classify all the motions of the massless particle. We study the situation when the massless particle has a periodic motion with the same minimal period than primary bodies. We show that this fact is related with the existence of certain pyramidal central configuration.