Astronomy and Astrophysics – Astronomy
Scientific paper
Feb 1998
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998cemda..70...75s&link_type=abstract
Celestial Mechanics and Dynamical Astronomy, v. 70, Issue 2, p. 75-98 (1998).
Astronomy and Astrophysics
Astronomy
8
Hill Four-Body Problem, Earth-Moon-Sun System, Hill Four-Body Problem, Earth&Ndash, Moon&Ndash, Sun System
Scientific paper
Starting from the four-body problem a generalization of both the restricted three-body problem and the Hill three-body problem is derived. The model is time periodic and contains two parameters: the mass ratio ν of the restricted three-body problem and the period parameter m of the Hill Variation orbit. In the proper coordinate frames the restricted three-body problem is recovered as m → 0 and the classical Hill three-body problem is recovered as ν → 0. This model also predicts motions described by earlier researchers using specific models 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 system. The model is general enough to apply to the motion of an infinitesimal mass under the influence of any two primaries which orbit a larger mass. Using the model, numerical investigations of the structure of motions around the geometric position of the triangular Lagrange points are performed. Values of the parameter ν range in the neighborhood 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 families of planar periodic orbits are studied in detail as the parameters m and ν vary. These families contain stable and unstable members in the plane and all have the out-of-plane stability. The stable and unstable manifolds of the unstable periodic orbits are computed and found to be trapped in a geometric area of phase space over long periods 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 actual geometry of the Earth-Moon-Sun system.
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