Invariant manifolds and transport in a Sun-perturbed Earth-Moon system

Abstract

This dissertation is devoted to the analysis of the motion of small bodies, like asteroids, in the neighbourhood of the Earth-Moon system from a celestial mechanics approach. This is an extensive area of research where probably, the most extended simplified mathematical model is the well-known autonomous Hamiltonian system the Restricted Three-Body Problem (RTBP). Many modifications to this model have been proposed, looking for a more accurate description of the system. One of the simplest ways of introducing additional physical effects is through time-periodic perturbations, such that such that the new non-autonomous system is close to the autonomous one, and it has many periodic or quasi-periodic solutions. If these solutions are hyperbolic, they have stable/unstable invariant manifolds, such that stable manifolds approach the quasi-periodic solutions forward in time, meanwhile unstable manifolds do it backward in time, constituting the skeleton for the dynamical transport phenomena we are interested in. Notice that one dimension can be reduced by defining a suitable temporal Poincar´e map. Therefore, our aim is to compute the quasi-periodic solutions and their manifolds in this map. Most of the effort of this dissertation is addressed to the Bicircular Problem (BCP), in which the Earth and Moon are treated as the primaries in the RTBP and the gravitational field of the Sun is introduced as a time-periodic forcing of the RTBP. In particular, we have extensively analysed the horizontal family of two dimensional quasi-periodic solutions in the neighbourhood of the collinear unstable equilibrium point L3. We found that diverse trajectories connecting the Earth, the Moon and the outside Earth-Moon system are governed by L3 dynamics. Big attention is paid to the trajectories coming from the Moon towards the Earth, since they may give an insight of the travel that lunar meteorites perform before landing in our planet. These results have been translated and compared with those of a realistic model based on JPL (Jet Propulsion Laboratory) ephemeris, showing a good agreement between the results obtained. We also have proposed and carried out a strategy for capturing a Near Earth Asteroid (NEA) using the stable invariant manifolds of the horizontal family of quasi-periodic orbits around L3 in the BCP. To this aim the high order parametrization of the stable/unstable invariant manifolds is introduced, for which computation we have employed the jet transport technique. Finally, the strategy is applied to the NEA 2006 RH120. The contributions to the BCP presented in this dissertation include two other applications. The first one is devoted to the study of the unstable behaviour near the triangular points, meanwhile the second is devoted to a family of stable invariant curves around the Moon that are close to a resonance, promoting the appearance of chaotic motion. The last part of the dissertation is focused on the effective computation of the high or- der parametrization of the stable and unstable invariant manifolds associated with reducible invariant tori of any high dimension. To this aim, we resort on the reducible system, that offers a high degree of parallelization of the computations. Besides, we explain how to com- bine the presented methods with multiple shooting techniques to accurately compute highly unstable invariant objects. Finally, we apply the developed algorithms to compute the high order parametrization of the manifolds associated to L1 and L2 in an Earth-Moon system that includes five time-periodic forcings regarded to four physical features of the system, besides the solar gravitational field.

Esta tesis analiza el movimiento de pequeños cuerpos, como asteroides, en el sistema Tierra­ Luna desde el marco de la mecánica celeste. El modelo que hemos empleado en mayor profundidad es el Problema Bicircular (PBC), el cual se puede entender como una perturbación periódica en el tiempo del conocido Problema Restringido de Tres Cuerpos (PRTC), dado que en el PBC se incluye el campo gravitatorio de un tercer cuerpo masivo que rota en movimiento circular alrededor de la configuración del PRTC. El cuerpo que causa la perturbación es para nosotros el Sol de tal forma que los objetos invariantes del PRTC adquieren una dimensión angular debida a la frecuencia del movimiento relativo entre el Sol y el baricentro Tierra-Luna. En el marco del PBC hemos analizado los fenómenos de transporte gobernados por la familia horizontal de soluciones cuasi-periódicas dos dimensionales (toros 2D) alrededor punto inestable colinear L3. Estas soluciones tienen asociadas variedades invariantes estables e inestables que constituyen el esqueleto de los fenómenos que queremos estudiar. Las trayectorias encontradas conectan la Tierra y la Luna y también el exterior/interior del sistema Tierra-Luna. Hemos prestado especial atención a las trayectorias que van de la Luna a la Tierra ya que podrían explicar el viaje que realizan los meteoritos lunares encontrados en nuestro planeta. Estos resultados han sido testeados en un modelo más realista basado en las efemérides del JPL (Jet Propulsion Laboratory). Otra de las aplicaciones propuestas es la de capturar un asteroide cercano a la Tierra usando la parametrización a orden alto de las variedades invariantes asociadas a los toros 2D alrededor de L3. La parte final trata del desarrollo de algoritmos para el cálculo preciso de la parametrización a orden alto de variedades invariantes estables/inestables asociadas a toros reducibles de cualquier dimensión alta. Además, se explica cómo combinar dichos algoritmos con métodos de tiro múltiple para aquellos objetos invariantes que sean muy inestables. Finalmente, aplicamos la metodología al cálculo de las variedades asociadas a L1 y L2 de un sistema Tierra-Luna que incluye cinco perturbaciones periódicas en el tiempo.