Non-collinear equilibrium points in the perturbed restricted three-body problem with unstable elongated primary

Abstract

In this paper, we have studied the motion around the non-collinear equilibrium points in the perturbed circular restricted three-body problem. The bigger primary is considered a point mass, and the smaller primary is an unstable elongated primary. The unstable elongated primary refers to the elongated primary rotates about its centre point, i.e. at an instant, the line joining the two ends of the elongated primary and the x-axis makes an angle, \(\theta \in [0,360^\circ )\). Computations of the non-collinear equilibrium points \(L_{4,5}\), and their linear stability, are investigated, and the results are applied to the Jupiter-Amalthea, Saturn-Prometheus and Pluto-Hydra systems. It is found that the positions of the non-collinear equilibrium points \(L_{4,5}\) remain unchanged with variation in the angle, \(\theta \), but small variation in the critical mass parameter, \(\mu _c\) is observed. Variations of the critical mass, \(\mu _c\), with different segment-length and its rotation are studied. Stable solutions around \(L_4\) are obtained in the systems of Jupiter-Amalthea, Saturn-Prometheus, and Pluto-Hydra using the established results.