A complete and explicit solution to the three-dimensional problem of two fixed centres

TL;DR

This paper provides the first explicit, exact, and comprehensive closed-form solution to the three-dimensional two fixed centres problem, applicable to all initial conditions and parameters, including unbounded and repulsive cases.

Contribution

It introduces a novel, complete solution using Weierstrass elliptic functions, extending previous partial or numerical approaches to an exact analytical framework.

Findings

Solution valid for all initial conditions and parameters

Analysis of quasi-periodic and periodic orbits

Identification of regions of motion and equilibrium points

Abstract

We present for the first time an explicit, complete and closed-form solution to the three-dimensional problem of two fixed centres, based on Weierstrass elliptic and related functions. With respect to previous treatments of the problem, our solution is exact, valid for all initial conditions and physical parameters of the system (including unbounded orbits and repulsive forces), and expressed via a unique set of formulae. Various properties of the three-dimensional problem of two fixed centres are investigated and analysed, with a particular emphasis on quasi-periodic and periodic orbits, regions of motion and equilibrium points.