The Three-Body Problem Earth-Moon-Spaceship / Das Dreikörperproblem Erde-Mond-Weltraumschiff / Le problème des trois corps Terre-Lune-Astronef

Abstract

With reference to the method discussed on the occasion of the IXth International Astronautical Congress, Amsterdam, the solution of the astronomical n-body problem using Lie series is discussed and the known algebraic integrals (conservation of momentum, conservation of angular momentum, conservation of energy) are reproduced. In order to prepare the solution of the three-body problem, the two-body problem is solved by using the new method. Furthermore, the anomaly λ is introduced as an independent variable in order to obtain a parametric representation r(t), φ(t). In doing this, a generalization of Kepler’s equation was found. The solutions of the two-body problem are discussed. Next, the solution of the plane three-body problem is given in which case a decomposition of the Lie operator is useful. After a thorough discussion of the initial data, the closed solution of the three-body problem is given, and two different methods for numerical computation are given in such detail that immediate programming is possible. The voluminous calculations with respect to the spatial three-body problem are not described in extenso due to lack of space and time.