The rectilinear three body problem as a basis for studying highly-eccentric systems

TL;DR

This paper investigates the rectilinear restricted three-body problem, identifying stable periodic orbits and their resonances, and applies these findings to analyze the stability of the highly eccentric planetary system HD7449.

Contribution

It extends previous work by finding and analyzing stable periodic orbits in the rectilinear TBP and demonstrates their relevance to real highly eccentric planetary systems.

Findings

Identified eight linearly stable periodic orbits in the rectilinear TBP.

Stable orbits are associated with specific mean motion resonances.

FLI maps show stable regions around these periodic orbits in phase space.

Abstract

The rectilinear elliptic restricted Three Body Problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity $e'=1$, but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter $\mu=0.5$ (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke's computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to $\mu$ and $e'<1$. Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449.