Investigating the Newton-Raphson basins of attraction in the restricted three-body problem with modified Newtonian gravity

TL;DR

This study explores how the basins of attraction in the restricted three-body problem change with varying gravitational potential power, revealing the significant influence of this parameter on the system's convergence behavior.

Contribution

It introduces a systematic numerical analysis of the Newton-Raphson basins of attraction in a modified three-body problem with variable gravity power, highlighting the impact of parameter p.

Findings

Basins of attraction are highly sensitive to the power p of the gravitational potential.

The shape and geometry of attraction basins vary significantly with changes in p.

The number of iterations correlates with basin convergence properties.

Abstract

The planar circular restricted three-body problem with modified Newtonian gravity is used in order to determine the Newton-Raphson basins of attraction associated with the equilibrium points. The evolution of the position of the five Lagrange points is monitored when the value of the power $p$ of the gravitational potential of the second primary varies in predefined intervals. The regions on the configuration $(x,y)$ plane occupied by the basins of attraction are revealed using the multivariate version of the Newton-Raphson iterative scheme. The correlations between the basins of convergence of the equilibrium points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We conduct a thorough and systematic numerical investigation by demonstrating how the dynamical quantity $p$ influences the shape as well as the geometry of the basins of attractions. Our results strongly suggest that the power $p$ is indeed a very influential parameter in both cases of weaker or stronger Newtonian gravity.