Contributions to Libration Orbit Mission Design using Hyperbolic Invariant Manifolds Elisabet Canalias Vila Departament de Matem`atica Aplicada 1 { Universitat Polit`ecnica de Catalunya Programa de Doctorat en Ci`encia i Tecnologia Aeroespacial Mem`oria per aspirar al grau de Doctor per la Universitat Polit`ecnica de Catalunya Certifico que aquesta tesi ha estat realitzada per l'Elisabet Canalias Vila i dirigida per mi. Barcelona, 11 de maig del 2007. Josep Joaquim Masdemont Soler Al meu avi, una de les persones de qui m´es coses he apr`es. Contents Abstract v Acknowledgments vii Preface ix 1 Motivation and State of the Art 1 2 Introduction 7 2.1 The restricted three body problem . 7 2.2 The phase space around the libration points . 11 2.2.1 Stability of the Li points . 11 2.2.2 Types of libration orbits . 13 2.3 Lindstedt-Poincar´e procedures . 17 2.4 JPL Solar System Ephemeris . 20 3 Eclipse avoidance and impulsive transfers in Lissajous orbits 21 3.1 Introduction . 21 3.2 Linear approximation to Lissajous orbits. 22 3.3 Non escape maneuvers . 23 3.3.1 In plane maneuvers . 24 3.3.2 Out of plane maneuvers . 29 3.4 The effective phases plane (EPP) . 30 3.5 Eclipse avoidance . 32 3.5.1 Exclusion zones . 32 3.5.2 Eclipse avoidance strategy . 35 3.5.3 Results . 41 3.5.4 Comments on eclipse avoidance for non-square Lissajous . 47 3.5.5 Alternatives for short term spatial missions . 48 3.6 Impulsive transfers between Lissajous orbits of different amplitudes . 50 3.6.1 Combined maneuvers . 52 3.6.2 Increasing the size of a Lissajous orbit using a combined maneuver . 52 3.6.3 Reducing the size of a Lissajous orbit using a combined maneuver . 54 3.6.4 Eclipse avoidance in combined maneuvers to change the amplitude . 55 3.7 Rendez-vous . 61 i 3.7.1 Rendez-vous maintaining the amplitudes . 61 3.7.2 Rendez-vous with amplitude change . 68 4 Homoclinic and heteroclinic connections between planar Lyapunov orbits 75 4.1 Introduction . 75 4.2 Methodology . 76 4.2.1 Lyapunov orbits . 76 4.2.2 Fixed energy surfaces . 77 4.2.3 KS-Regularisation . 79 4.2.4 Homoclinic and heteroclinic phenomena . 82 4.2.5 Details on the numerical methodology . 85 4.3 Families of connections: Sun-Earth and Earth-Moon systems. 87 4.3.1 Homoclinic connecting trajectories . 88 4.3.2 Heteroclinic connecting trajectories . 94 5 Transfer trajectories between the Sun-Earth and Earth-Moon L2 regions 101 5.1 Introduction . 101 5.2 Connecting trajectories between planar Lyapunov orbits . 102 5.2.1 Lyapunov orbits and their hyperbolic invariant manifolds . 102 5.2.2 Poincar´e section . 104 5.2.3 Intersections on the Poincar´e section . 104 5.2.4 Connecting trajectories . 109 5.2.5 Preliminary explorations . 110 5.2.6 Families of connecting trajectories . 116 5.2.7 Zero cost connecting trajectories . 120 5.2.8 Families of zero cost connecting trajectories . 122 5.3 Connecting trajectories between Lissajous orbits . 127 5.3.1 Introduction . 127 5.3.2 Lissajous orbits and their hyperbolic manifolds . 127 5.3.3 Coupling between the two Restricted Three Body Problems . 128 5.3.4 Poincar´e section . 129 5.3.5 Intersections on the Poincar´e section . 131 5.3.6 Computation of connecting trajectories between Lissajous orbits. 132 5.3.7 Preliminary explorations . 139 5.3.8 Results. 140 5.4 Refinement to JPL coordinates . 146 5.4.1 Multiple shooting method . 147 5.4.2 First approximation to real ephemeris connecting trajectories . 148 5.4.3 Zero cost connecting trajectories in JPL coordinates . 153 5.4.4 Results . 154 Conclusions and Future work 167 A. Guide to the attached DVD 169 ii B. Resum 179 Bibliography 181 iii Abstract This doctoral thesis lies within the framework of astrodynamics. In particular, it deals with mission design near libration point orbits. The starting point of the studies contained in the present dissertation is dynamical systems theory, which provides an accurate description of the dynamics governing libration regions. However, this work is aimed at real applications, and therefore it makes use of this theoretical description as a means to provide solutions to problems that have been identified in mission design. The restricted three body problem (RTBP) is a well known model to study the motion of an infinitesimal mass under the gravitational attraction of two massive bodies. Its 5 equilibrium points have been thoroughly studied since the last century. The results contained in the present dissertation refer to two of these equilibrium points: L1 and L2, which lie on both sides of the smallest of the massive bodies of the system and are the ones on which more practical interest has been focused in the last decades ( for missions such as SOHO, Genesis, Hershel-Planck. ). Instability is a basic property of the aforementioned equilibrium points, which is inherited by the orbits surrounding them and accounts for the existence of stable and unstable directions at each point of these orbits. The union of these directions or, more precisely, of the asymptotic orbits arising from the periodic and quasi-periodic motions around L1 and L2, forms an invariant object either approaching (stable directions) or leaving (unstable directions) the vicinity of libration points. These invariant objects are the hyperbolic manifolds of libration point orbits. A proper knowledge and description of such manifolds is extremely useful for mission design, as they are the key to understanding the dynamics of the system. The first problem that has been tackled in our work is the eclipse avoidance in Lissajous orbits. Generically, a spatial probe placed in an orbit around the solar libration point L2 is affected by occultations due to the shadow of the Earth, unless eclipse avoidance maneuvers are planned. If the orbit surrounds L1, eclipses due to the strong solar electromagnetic influence occur. On the other hand, Lissajous-type orbits are a kind of libration motion resulting from the combination of two perpendicular oscillations. Their main advantage over other kinds of orbits, such as the elongated Halo orbits, is that the amplitudes of each one of the oscillations can be chosen independently, and this fact makes them suitable for certain mission requirements. This work uses the linear approximation to the analytical description of Lissajous orbits in order to compute the so-called non escape direction which allows for transfers between different orbits by changing either the amplitudes or the phases (or both at the same time) while avoiding the unstable part of the movement. Furthermore, another interesting problem in space mission design is the rendez-vous, under- stood in our work as the strategy to make two different satellites meet at a certain orbit or to approach each other to a given small distance. The tools developed for eclipse avoidance in Lis- sajous orbits also allow us to plan simple rendez-vous strategies, which can be used either for preliminary mission analysis or as a contingency plan. On the other hand, there exist low cost channels between the libration points L1 and L2 of a given system, like the ones used in the Genesis mission. These channels provide a natural way of transfering between the libration regions and they can be found by intersecting stable and unstable manifolds of orbits around L1 and L2. Remember that stable manifolds tend to an invariant object in forwards time. Unstable manifolds do so backwards in time. Therefore, when an intersection v is found between a stable manifold and an unstable one, it provides a path that goes away from a libration orbit and approaches another one. Connections between planar Lyapunov orbits, which are planar periodic motions around L1 and L2 are studied in this dissertation, being specifically computed for the Sun-Earth and Earth-Moon systems. Moreover, the idea of intersecting stable and unstable manifolds in order to find low cost connecting trajectories can also be applied in the search for low cost paths from the lunar libration regions to the solar libration orbits. It is well known that the stable manifolds of orbits around libration points of the Earth-Moon problem do not come close enough to the Earth as to provide a direct transfer to the Moon. On the contrary, stable and unstable manifolds of some libration orbits around L2 in the Sun-Earth problem do come to a close approach with the Earth. Therefore, if a natural path between the solar libration orbits and the lunar ones could be found, this would result in a cheap way of reaching the Moon. And the other way round, a path from the lunar libration regions to the solar ones would allow for the placement of a gateway station at the vicinity of a lunar libration point aimed at providing services to solar libration missions, for instance. This is the idea that drives the last part of the dissertation. With the goal of joining the lunar libration orbits and the solar ones by using invariant manifolds, the four body problem Sun-Earth-Moon- spacecraft is decoupled in two restricted three body problems. Then, we can search for intersections between manifolds of libration orbits belonging to both problems. At first, connecting trajectories from the planar Lyapunov orbits around L2 in the Earth-Moon system to planar Lyapunov orbits around the solar L2 point are computed. Afterwards, the search is conducted in the 3-dimensional case, between Lissajous type orbits around the aforementioned libration points of both problems. The computation of connecting trajectories in the spatial case is much more complicated, as the dimension of the state space in which we look for intersections increases with respect to the planar case.