THE PLANAR ISOSCELES PROBLEM FOR MANEFF GRAVITATIONAL LAW

Maneff's gravitational law explains, with a very good approximation, t he perihelion advance of the inner planets as well as the orbit of the Moon. Here the invariant set of planar isosceles solutions of the thr ee-body problem for Maneff's model is studied. The application of Mane ff's law in atomic physics provides, in the case of the isosceles prob lem, a model with relativistic correction for the helium atom. It is s hown that every solution leads to a collision singularity and conseque ntly has no periodic orbits. Using McGehee's technique the triple-coll ision singularity is blown up and the binary-collision solutions are r egularized. The flow on the collision manifold is shown to be nongradi entlike and the set of collision/ejection solutions is described. The center manifold and the block-regularization problems are analyzed. Th e network of homoclinic and heteroclinic orbits is further discussed. Finally an anisotropic model having the property that the flow on the collision manifold changes drastically when the mass parameter is vari ed is studied, giving rise to a subcritical pitchfork bifurcation of t he equilibria.