Orbital Element Distribution Of Invariant Manifolds Associated With Lyapunov Family Of Periodic Orbits Around L-1 And L-2

This study explores the initial configurations that lead to an eventual approach to a given planet, particularly Jupiter, using the invariant manifold of Lyapunov orbits around Lagrangian points L-1 or L-2. Reachability to the vicinity of planets would provide information on developing a process for capturing irregular satellites, which are small bodies orbiting around a giant planet with a high eccentricity that are considered to have been captured by the mother planet, rather than formed in situ. A region several times the Hill radius is often used for determining reachability, combined with checking the velocity of the planetesimal with respect to the mother planet. This kind of direct computation is inapplicable when the Hill sphere is widely open and its boundary cannot be clearly recognized. Here, we thus employ Lyapunov periodic orbits (LOs) as a formal definition of the vicinity of Jupiter and numerically track the orbital distribution of the invariant manifold of an LO while varying the Jacobi constant, C-J. Numerical tracking of the manifold is carried out directly via repeated Poincare mapping of points initially allocated densely on a fragment of the manifold near the fixed points, with the assistance of multi-precision arithmetic using the Multiple Precision Floating-Point Reliable Library. The numerical computations show that the distribution of the semi-major axis of points on the manifolds is quite heavily tailed and that its median spans roughly 1-2 times the Jovian orbital radius. The invariant manifold provides a distribution profile of a that is similar to that obtained using a direct method.