The flight path of a spacecraft in the Earth-Moon system

Consider the problem of calculating a flight path from the Earth to the Moon and back. We are required to find some initial conditions in the vicinity of Earth such that the spacecraft will be swung round by the gravity of the Moon and return to the vicinity of Earth without further thrust by its rocket motors. The astronauts of Apollo 13, whose engine failed on the flight to the Moon, had to follow such a flight path back to Earth.

We assume that Earth and Moon both revolve in circular orbits about their common centre-of-mass, unaffected by the low mass spacecraft. The effects of the sun and other planets are neglected. This then is an example of the restricted 3-body problem. Assume that the Earth-Moon distance is always and that the Earth's mass . Allow for an adjustable Earth/Moon mass ratio mr so that the Moon mass is . Thus we find from Newtonian gravity that the period of rotation of the Earth-Moon system is . (The actual value of mr is 81).

Compared with Newton's equations of motion, Hamilton's equations have two advantages:

1. They are differential equations of the first order and so directly integrable by Runge-Kutta methods.
2. Hamilton's equations are easy to write down in any coordinate system appropriate for the problem. In this case we choose plane polar coordinates.