Bifurcations of the Hill's Region in the Three Body Problem

In the spatial three body problem, the topology of the integral manifolds M(c; h) (i.e. the level sets of energy h and angular momentum c, as well as center of mass and linear mo-mementum) and the Hill's regions H(c; h) (the projection of the integral manifold onto position coordinates) depends only on the quantity = hjcj 2 : In 1] and 3], it was established that, for h < 0 and c 6 = 0, there are exactly eight bifurcation values for at which the topology of the integral manifold changes. It was also shown that each of these values, the topology of the Hill's region changes as well. In this work, it is shown that there are no other values of for which the topology of the Hill's region changes. That is, a bifurcation of the Hill's region occurs if and only if a bifurcation of the integral manifold occurs. 1. Introduction The spatial three-body problem admits the ten integrals of energy, linear momentum and angular momentum. Fixing these integrals de-nes an eight dimensional algebraic set called the integral manifold, M(c; h), which depends on the energy level h and the magnitude c of the angular momentum vector. The seven dimensional reduced integral manifold, M R (c; h), is the quotient space M(c; h)=SO 2 where the SO 2 action is rotation about the angular momentum vector. The Hill's region H(c; h) and the reduced Hill's region H R (c; h) are the projection onto position coordinates of M(c; h) and M R (c; h) respectively. In 3], the dependence of the topology of these spaces on the parameters c and h was investigated. It is a classical result that the topology of