Generalization of Levi-Civita regularization in the restricted three-body problem

TL;DR

This paper generalizes the Levi-Civita regularization method in the restricted three-body problem using polynomial functions of degrees 2 to 8, establishing analytical relationships and demonstrating numerical trajectories.

Contribution

It introduces a family of polynomial regularization functions extending Levi-Civita's method, with analytical and numerical validation in the three-body problem.

Findings

Polynomial functions of degrees 2 to 8 are used for regularization.

Analytical relationships between physical and regularized planes are derived.

Numerical trajectories demonstrate the effectiveness of the generalized method.

Abstract

A family of polynomial coupled function of $n$ degree is proposed, in order to generalize the Levi-Civita regularization method, in the restricted three-body problem. Analytical relationship between polar radii in the physical plane and in the regularized plane are established; similar for polar angles. As a numerical application, trajectories of the test particle using polynomial functions of $2, 3,..., 8$ degree are obtained. For the polynomial of second degree, the Levi-Civita regularization method is found.