Investigating the basins of convergence in the circular Sitnikov three-body problem with non-spherical primaries

TL;DR

This study numerically analyzes how the oblateness of primaries influences the convergence regions of equilibrium points in the Sitnikov three-body problem, revealing the geometric and iterative properties of these basins.

Contribution

It provides a detailed numerical exploration of the Newton-Raphson basins of convergence in the non-spherical Sitnikov problem, highlighting the impact of oblateness on convergence regions.

Findings

Oblateness significantly alters the shape of convergence basins.

The number of iterations varies with oblateness and region.

Probability distributions of convergence are characterized.

Abstract

In this work we numerically explore the Newton-Raphson basins of convergence, related to the equilibrium points, in the Sitnikov three-body problem with non-spherical primaries. The evolution of the position of the roots is determined, as a function of the value of the oblateness coefficient. The attracting regions, on several types of two dimensional planes, are revealed by using the classical Newton-Raphson iterative method. We perform a systematic and thorough investigation in an attempt to understand how the oblateness coefficient affects the geometry as well as the overall properties of the convergence regions. The basins of convergence are also related with the required number of iterations and also with the corresponding probability distributions.