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  • A class of high energy, close approach trajectories within the three-body problem

    Paper ID



    • James L. Kamm
    • Odus R. Burggraf


    Ohio State University,






    In the past several years there has been a significant increase in interest in the three-body problem due to the application by Lagerstrom and Kevorkian [1] of matched asymptotic expansion techniques to obtain uniformly valid three-body trajectories. The problem solved by Lagerstrom and Kevorkian and extended by several others [2—5] is the classical restricted three-body perturbation problem with the ratio of the primary masses considered to be the perturbation parameter. Their combined analysis provides a rather extensive coverage of the problem for ranges of the initial conditions typical of earth-moon and moon-earth transfers. This classical free-fall perturbation problem is a very realistic one for trajectories within our solar system. Nevertheless, it is of interest to investigate the situation where the primaries have about the same mass, and indeed when the mass of the third body is also of the same order of magnitude as the primaries. Such a problem could be attempted by finding another perturbation scheme. This paper deals with a class of planar free-fall three-body problems such that the ratio of two characteristic lengths of the problem is small and can serve as a legitimate perturbation parameter. That is, we seek a solution in which the perturbation scheme is provided by the initial conditions and not by the physical structure of the system. If the problem to be considered is one in which two bodies are initially rotating about each other, one obvious characteristic length is the initial distance, d, between the bounded bodies, hereafter called the “primaries”. Interestingly, by suitable normalization of variables, d can be eliminated from the equations; however, it cannot simultaneously be eliminated from the initial conditions.