A closed-form solution for the three-body problem and Sun-Earth-Moon orbit solution

Abstract

Abstract The study of the motions of planets, satellites, and other celestial bodies is one of the important problems in basic physics and astronomy. The solution to the two-body problem enables astronomers to predict the orbits of the Moon, satellites, and spaceships around the Earth. The general analytic solution for the three-body problem stands unsolved except in some special cases, such as the Sun-Earth-Moon problem, in which the mass of the Moon is neglected. This reduces the problem to a two-body problem. In this work, the author presents a general solution to the problem in a closed form in terms of two basic particle-particle vectors. The position vector of each particle is expressed in terms of the center of mass and the two basic particle-particle vectors. This solution is used for studying the three-body problem with gravitational interaction without imposing the non-zero-total angular-momentum condition nor ignoring any masses. The Sun-Earth-Moon problem was solved in the general case and showed an expected orbital motion while a perturbation in Earth-Sun orbit due to the revolution of the Moon. This solution is the key for future studies for n-body problem solutions.