The dynamics around the collinear points of the elliptic three-body problem: A normal form approach

TL;DR

This paper analyzes the dynamics near collinear points in the elliptic three-body problem using a normal form approach, enabling the construction of various orbits and comparison with numerical simulations.

Contribution

It introduces a normal form method for the elliptic three-body problem to approximate and analyze the dynamics around collinear points, including orbit construction.

Findings

Normal form provides accurate approximate solutions for orbit trajectories.

Constructed planar, vertical Lyapunov, and halo orbits.

Analytical results closely match numerical simulations.

Abstract

We study the dynamics of the collinear points in the planar, restricted three-body problem, assuming that the primaries move on an elliptic orbit around a common barycenter. The equations of motion can be conveniently written in a rotating pulsating barycentric frame, taking the true anomaly as independent variable. We consider the Hamiltonian modeling this problem in the extended phase space and we imple ment a normal form to make a center manifold reduction. The normal form provides an approximate solution for the Cartesian coordinates, which allows us to construct several kinds of orbits, most notably planar and vertical Lyapunov orbits, and halo orbits. We compare the analytical results with a numerical simulation, which requires special care in the selection of the initial conditions.