Kepler’s problem of a two-body system perturbed by a third body

Abstract

One of the most important problems in basic physics and astronomy is studying the motion of planets, satellites, and other celestial bodies. 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. This reduces the problem to a two-body problem. In this work, the authors present a closed-form approach to the three-body problem theoretically and numerically based on particle–particle vector analysis. The theoretical approach, which is based on the real Moon–Sun–Earth problem information, illustrates the perturbation of the Moon in the Sun–Earth problem and shows an expected orbital motion with a perturbation in the Sun–Earth orbit due to the revolution of the Moon. The numerical investigation uses the same information to study the same problem and calculate the angular momentums of each pair of objects. The two solutions show good agreement with the well-known Earth-Moon and Sun–Earth momentums. The Moon–Sun orbit is close to an elliptic shape with angular momentum of about 3.27 × 1038 J.s. This approach is the key to future studies for n-body problem solutions.