A family of zero-velocity curves in the restricted three-body problem

TL;DR

This paper analyzes zero-velocity curves and equilibrium points in the regularized circular restricted three-body problem, using polynomial transformations to explore properties of Roche varieties and their asymptotic behavior.

Contribution

It introduces a generalized Levi-Civita transformation with polynomial functions to analyze Roche varieties in the three-body problem, extending previous work.

Findings

Identified five families of equilibrium points in the parametric plane.

Plotted Roche varieties for polynomial degrees n=1 to 6.

Derived and analyzed the asymptotic variety and its shape.

Abstract

The equilibrium points and the curves of zero-velocity (Roche varieties) are analysed in the frame of the regularized circular restricted three-body problem. The coordinate transformation is done with Levi-Civita generalized method, using polynomial functions of n degree. In the parametric plane, five families of equilibrium points are identified. These families of points correspond to the five equilibrium points in the physical plane L1, L2, ..., L5. The zero-velocity curves from the physical plane are transformed in Roche varieties in the parametric plane. The properties of these varieties are analysed and the Roche varieties for n = {1,2,...,6} are plotted. The equation of the asymptotic variety is obtained and its shape is analysed. The slope of the Roche variety in L11 point is obtained. For n = 1 the slope obtained by Plavec and Kratochvil (1964) in the physical plane was found.