A comparison of three perturbation methods for earth-moon-spaceship problem

The limiting case of the restricted three-body problem, in which the mass of one of the finite bodies is much smaller than the mass of the other, is of the singular perturbation type. The first-order perturbation solution has a logarithmic singularity at the position of the smaller body, and higher approximations are increasingly more singular. Three methods of treating singular perturbation problems are compared as applied to this problem. Uniformly valid first-order solutions are obtained for the problem of a two-fixed force-center by using the generalized method. It is shown that the generalized method gives better approximations than the method of inner and outer expansions and thus can be used for wider ranges of the small parameter. Furthermore, the Poincare-Lighthill-Kuo (PLK) method is shown to give incorrect results for the trajectory.