Symplectic Integrators for Hill’s Lunar Problem

Abstract

Hill’s lunar problem is one of the simplest, yet realistic, gravitational three-body problems. It models, e.g., the motion of the Earth’s moon or the motion of Saturn’s coorbital satellites. It still has a great importance since it is being used as a reference problem in certain more accurate perturbation models of the moon’s motion. In regularized coordinates Hill’s lunar problem may be described by a polynomial Hamiltonian of degree 6.

Here we will use an additive splitting of this Hamiltonian in order to implement symplectic composition integrators for Hill’s lunar problem. Experiments indicate that these integrators are very accurate for orbits consisting of many revolutions. E.g., in integrations of quasiperiodic or homoclinic orbits the Hamiltonian remains nearly constant until the orbit escapes. However, a comparison with high-order Taylor series methods shows that these methods are considerably faster and therefore quite competitive, although the Hamiltonian seems to grow linearly with time.

In an appendix we discuss new algorithms for evaluating the Stumpff functions since these functions are needed for constructing our symplectic integrators.