THE RESTRICTED HILL 4-BODY PROBLEM WITH APPLICATIONS TO THE EARTH-MOON-SUN SYSTEM

Starting from the four-body problem a generalization of both the restr icted three-body problem and the Hill three-body problem is derived. T he model is time periodic and contains two parameters: the mass ratio upsilon of the restricted three-body problem and the period parameter m of the Hill Variation orbit. In the proper coordinate frames the res tricted three-body problem is recovered as m --> 0 and the classical H ill three-body problem is recovered as upsilon --> 0. This model also predicts motions described by earlier researchers using specific model s of the Earth-Moon-Sun system. An application of the current model to the motion of a spacecraft in the Sun perturbed Earth-Moon system is made using Hill's Variation orbit for the motion of the Earth-Moon sys tem. The model is general enough to apply to the motion of an infinite simal mass under the influence of any two primaries which orbit a larg er mass. Using the model, numerical investigations of the structure of motions around the geometric position of the triangular Lagrange poin ts an performed. Values of the parameter upsilon range in the neighbor hood of the Earth-Moon value as the parameter m increases from 0 to 0. 195, at which point the Hill Variation orbit becomes unstable. Two fam ilies of planar periodic orbits are studied in detail as the parameter s m and upsilon vary. These families contain stable and unstable membe rs in the plane and all have the out-of-plane stability. The stable an d unstable manifolds of the unstable periodic orbits are computed and found to be trapped in a geometric area of phase space over long perio ds of time for ranges of the parameter values including the Earth-Moon -Sun system. This model is derived from the general four-body problem by rigorous application of the Hill and restricted approximations. The validity of the Hill approximation is discussed in light of the actua l geometry of the Earth-Moon-Sun system.