The invariant manifolds at infinity of the RTBP and the boundaries of
bounded motion

Regina Mart\'{\i}nez$^{(1)}$ and Carles Sim\'o$^{(2)}$

$^{(1)}$ Departament de Matem\`atiques, Universitat Aut\`onoma de
Barcelona, Bellaterra, Barcelona, Spain 
$^{(2)}$ Departament de Matem\`atica Aplicada i An\`alisi, Universitat de
Barcelona, Barcelona, Spain 
reginamb@mat.uab.cat, carles@maia.ub.es

Abstract

The invariant manifolds of the periodic orbit at infinity in the planar circular
RTBP are studied. To this end we consider the intersection of the manifolds
with the passage through the barycentric pericentre. The intersections of the
stable and unstable manifolds have a common even part, which can be seen as a
displaced version of the two-body problem, and an odd part which gives rise to
a splitting. The theoretical formulas, obtained for Jacobi constant $C$ large
enough, are compared to direct numerical computations, showing improved
agreement when $C$ increases. A return map to the pericentre passage is derived
and using an approximation by standard-like maps one can make a prediction of
the location of the boundaries of bounded motion. This result is compared to
numerical estimates, again improving for $C$ increasing. Several anomalous
phenomena are described.