Dynamics of a small planetoid in Newtonian gravity field of Lagrangian configuration of three primaries

TL;DR

This paper introduces a semi-analytical method to approximate the trajectory of a small planetoid near a Lagrangian configuration of three primaries in a restricted 4-body problem, focusing on quasi-planar motion.

Contribution

It develops a novel semi-analytical approach to model the dynamics of a planetoid in a four-body system with primaries in Lagrangian configuration, including equations for planar and out-of-plane motion.

Findings

Derived coupled differential equations for planetoid coordinates

Obtained an expression for out-of-plane oscillations

Provided approximate trajectories for specific mass configurations

Abstract

Novel method for semi-analytical solving of equations of a trapped dynamics for a planetoid m4 close to the plane of mutual motion of main bodies around each other (in case of a special type of Bi-Elliptic Restricted 4-Bodies Problem) is presented. We consider here three primaries m1, m2, m3 orbiting around their center of mass on elliptic orbits which are permanently forming Lagrangian configuration of an equilateral triangle. Our aim is to obtain appoximate coordinates of quasi planar trajectory of the infinitesimal planetoid m4, when the primaries have masses equal to 1/3. Results are as follows: 1) equations for coordinates {x, y} are described by system of coupled 2-nd order ordinary differential equations with respect to true anomaly f, 2) expression for z stems from solving second order Riccati ordinary differential equation that determines the quasi-periodical oscillations of planetoid m4 not far from invariant plane {x, y, 0}.