On the Newton-Raphson basins of convergence of the out-of-plane equilibrium points in the Copenhagen problem with oblate primaries

TL;DR

This paper investigates how the oblateness of primary bodies in the Copenhagen three-body problem influences the Newton-Raphson basins of convergence around out-of-plane equilibrium points, revealing complex fractal structures.

Contribution

It provides a systematic analysis of the impact of oblateness on convergence basins and their fractal properties in the restricted three-body problem.

Findings

Oblateness significantly alters the geometry of convergence basins.

Fractal dimension and basin entropy quantify the complexity of the basins.

The number of iterations correlates with basin structure and probability distributions.

Abstract

The Copenhagen case of the circular restricted three-body problem with oblate primary bodies is numerically investigated by exploring the Newton-Raphson basins of convergence, related to the out-of-plane equilibrium points. The evolution of the position of the libration points 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 multivariate Newton-Raphson iterative method. We perform a systematic and thorough investigation in an attempt to understand how the oblateness coefficient affects the geometry of the basins of convergence. The convergence regions are also related with the required number of iterations and also with the corresponding probability distributions. The degree of the fractality is also determined by calculating the fractal dimension and the basin entropy of the convergence planes.