On the spatial collinear restricted four-body problem with non-spherical primaries

TL;DR

This paper systematically analyzes the spatial collinear restricted four-body problem with non-spherical primaries, exploring libration points, their stability, regions of motion, and convergence basins, revealing stability conditions and orbital dynamics.

Contribution

It provides a comprehensive numerical study of libration points, stability, and basins of convergence in a non-spherical four-body problem, including parametric effects.

Findings

Libration points' positions vary with oblateness and prolateness.

Only collinear libration points are linearly stable under certain conditions.

Non-collinear libration points are linearly unstable and thus nonlinearly unstable.

Abstract

In the present work a systematic study has been presented in the context of the existence of libration points, their linear stability, the regions of motion where the third particle can orbit and the domain of basins of convergence linked to libration points in the spatial configuration of the collinear restricted four-body problem with non-spherical primaries (i.e., the primaries are oblate or prolate spheroid). The parametric evolution of the positions of the libration points as function of the oblateness and prolateness parameters of the primaries and the stability of these points in linear sense are illustrated numerically. Moreover, the numerical investigation shows that the only libration points which lie on either of the axes are linearly stable for several combinations of the oblateness parameter and mass parameter whereas the non-collinear libration points are found linearly unstable, consequently unstable in nonlinear sense also, for studied value of mass parameter and oblateness parameter. Moreover, the regions of possible motion are also depicted, where the infinitesimal mass is free to orbit, as function of Jacobian constant. In addition, the basins of convergence (BoC) linked to the libration points are illustrated by using the multivariate version of the Newton-Raphson (NR) iterative scheme.