Geometric chaos indicators and computations of the spherical hypertube manifolds of the spatial circular restricted three-body problem

The circular restricted three-body problem has five relative equilibria L1,L2, ...,L5. The invariant stable-unstable manifolds of the center manifolds originating at the partially hyperbolic equilibria L1,L2 have been identified as the separatrices for the motions which transit between the regions of the phase-space which are internal or external with respect to the two massive bodies. While the stable and unstable manifolds of the planar problem have been extensively studied both theoretically and numerically, the spatial case has not been as deeply investigated. This paper is devoted to the global computation of these manifolds in the spatial case with a suitable finite time chaos indicator. The definition of the chaos indicator is not trivial, since the mandatory use of the regularizing Kustaanheimo-Stiefel variables may introduce discontinuities in the finite time chaos indicators. From the study of such discontinuities, we define geometric chaos indicators which are globally defined and smooth, and whose ridges sharply approximate the stable and unstable manifolds of the center manifolds of L1,L2. We illustrate the method for the Sun-Jupiter mass ratio, and represent the topology of the asymptotic manifolds using sections and three-dimensional representations.