Gerard Gómez

Connecting orbits and invariant manifolds in the spatial restricted three-body problem

The invariant manifold structures of the collinear libration points for the restricted three-body problem provide the framework for understanding transport phenomena from a geometrical point of view. In particular, the stable and unstable invariant manifold tubes associated with libration point orbits are the phase space conduits transporting material between primary bodies for separate three-body systems. These tubes can be used to construct new spacecraft trajectories, such as a ‘Petit Grand Tour’ of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. This work extends the results to the three-dimensional case. Besides providing a full description of different kinds of libration motions in a large vicinity of these points, this paper numerically demonstrates the existence of heteroclinic connections between pairs of libration orbits, one around the libration point L_1 and the other around L_2. Since these connections are asymptotic orbits, no manoeuvre is needed to perform the transfer from one libration point orbit to the other. A knowledge of these orbits can be very useful in the design of missions such as the Genesis Discovery Mission, and may provide the backbone for other interesting orbits in the future.

Study of the transfer from the Earth to a halo orbit around the equilibrium pointL 1

The purpose of this paper is to study a transfer strategy from the vicinity of the Earth to a halo orbit around the equilibrium pointL1 of the Earth-Sun system. The study is done in the real solar system (we use the DE-118 JPL ephemeris in the simulations of motion) although some simplified models, such as the restricted three body problem (RTBP) and the bicircular problem, have been also used in order to clarify the geometrical aspects of the problem. The approach used in the paper makes use of the hyperbolic character of the halo orbits under consideration. The invariant stable manifold of these orbits enables the transfer to be achieved with, theoretically, only one manoeuvre: the one of insertion into the stable manifold. For the total Δv required, the figures obtained are similar to the ones given by the standard procedures of optimization.

The Bicircular Model Near the Triangular Libration Points of the RTBP

We present a study of a simplified model of the Restricted Four Body Problem consisting of Earth, Moon, Sun and a massless particle, as a model of the dynamics of a spacecraft. The region where we look for the motion is a vicinity of the triangular libration points of the Restricted Three Body Problem. The model we discuss here is the so called Bicircular Problem. The main question is the existence of zones where the motion has good stability properties. The answer is positive, but the stable motions can not be confined to a small distance of the ecliptic plane. Both numerical simulations and analytical results are presented. Some tentative explanations offer a possible way to study many other kinds of problems. Some applications to space missions are mentioned.

STUDY OF THE TRANSFER BETWEEN HALO ORBITS

Abstract Two methods of transfer between halo orbits of the same family are developed making use of the geometry of the phase space around these solutions and the Floquet theory for periodic orbits. The study is done for the halo orbits around the libration point L1 of the Earth–Sun system, in the framework of the restricted three body problem. The optimal performance of the method is given. In a second step one of the methods is adapted to the real situation taking into account the influence of all the bodies of the solar system. The results obtained are compared with other strategies studied by Hiday and Howell 13 , 14 .

The invariant manifold structure of the spatial Hill's problem

The paper studies the invariant manifolds of the spatial Hills problem associated to the two liberation points. A combination of analytical and numerical tools allow the normalization of the Hamiltonian and the computation of periodic and quasi-periodic (invariant tori) orbits. With these tools, it is possible to give a complete description of the center manifolds, association to the liberation points, for a large set of energy values. A systematic exploration of the homoclinic and heteroclinic connections between the center manifolds of the liberation points is also given.

Homoclinic and heteroclinic solutions in the restricted three-body problem

In this work we have performed a systematic computation of the homoclinic and heteroclinic orbits associated with the triangular equilibrium points of the restricted three-body problem. Some analytical results are given, related to their number when the mass ratio varies.

LIBRATION POINT ORBITS: A SURVEY FROM THE DYNAMICAL POINT OF VIEW

The aim of this paper is to provide the state of the art on libration point orbits. We will focus in the Dynamical Systems approach to the problem, since we believe that it provides the most global picture and, at the same time, allows to do the best choice of both strategy and parameters in several mission analysis aspects. I. Dynamics and phase space around the Libration Points 1 Equations of motion and Libration Points 1.1 The Restricted Three Body Problem and its perturbations It is well known that several very simple models, such as the Two Body Problem or the Restricted Three Body Problem (RTBP), are suitable for spacecraft mission design, since they give good insight of the dynamics in many real situations. In this section we will review some of the most relevant restricted models for the analysis of the motion in the vicinity of the libration points. Most of the well known restricted problems take as starting point the circular RTBP, that models the motion of a massless particle under the gravitational attraction of two punctual primaries revolving in circular orbits around their center of mass. In a suitable coordinate system and with adequate units, the Hamiltonian of the RTBP is (Szebehely [70]) H(x, y, z, px, py, pz) = 1 2 (px + p 2 y + p 2 z) + ypx − xpy − 1− μ ((x− μ)2 + y2 + z2)1/2 − μ ((x− μ+ 1)2 + y2 + z2)1/2 , being μ = m2/(m1 + m2), where m1 > m2 are the masses of the primaries. In order to get closer to more realistic situations, or simplifications, this model is modified in different ways. For instance, 1. Hill’s problem. Is useful for the analysis of the motion aroundm2. Can be obtained setting the origin at m2, rescaling coordinates by a factor μ1/3 and keeping only the dominant terms of the expanded Hamiltonian in powers of μ1/3. The Hamiltonian function is H = 1 2 (px + p 2 y + p 2 z) + ypx − xpy − 1 (x2 + y2 + z2)1/2 − x + 1 2 (y + z). This Hamiltonian corresponds to a Kepler problem perturbed by the Coriolis force and the action of the Sun up to zeroth-order in μ1/3. Hill’s model is the first approximation to Departament de Matematica Aplicada i Analisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, E.T.S.E.I.B., Diagonal 647, 08028 Barcelona, Spain Departament de Matematica Aplicada i Analisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

Invariant Manifolds, Lagrangian Trajectories and Space Mission Design

The last 30 years have produced an explosion in the capabilities of designing and managing libration point missions. The starting point was the ground-breaking mission of the third International Sun-Earth Explorer spacecraft (ISEE–3). The ISEE-3 was launched August 12, 1978 to pursue studies of the Earth–Sun interactions, in a first step of what now is known as Space Weather. After a direct transfer of the ISEE-3 to the vicinity of the Sun-Earth Lagrange point, it was inserted into a nearly-periodic halo orbit, in order to monitor the solar wind about 1 h before it reached the Earth’s magneto-sphere as well as the ISEE–1 and 2 spacecraft (which where in an elliptical orbit around the Earth).

Solar system models with a selected set of frequencies

The purpose of this paper is to develop a methodology to generate simplified models suitable for the analysis of the motion of a small particle, such as a spacecraft or an asteroid, in the Solar System. The procedure is based on applying refined Fourier analysis methods to the time-dependent functions that appear in the dierential equations of the problem. The equations of the models obtained are quasi-periodic perturbations of the Restricted Three Body Problem that depend explicitly on natural frequencies of the Solar System. Some examples of these new models are given and compared with other ones found in the literature. For one of these new models, close to the Earth-Moon system, we have computed the dynamical substitutes of the collinear libration points. The methodology developed in this paper can also be used for the analytical construction of simplified models of systems governed by dierential equations which have a quasi-periodic (in time) external excitation and such that the form of the equations is rather cumbersome.

Spatial p-q Resonant Orbits of the RTBP

The purpose of this paper is to extend the study of the so called p-q resonant orbits of the planar restricted three-body problem to the spatial case. The p-q resonant orbits are solutions of the restricted three-body problem which have consecutive close encounters with the smaller primary. If E, M and P denote the larger primary, the smaller one and the infinitesimal body, respectively, then p and q are the number of revolutions that P gives around M and M around E, respectively, between two consecutive close approaches. For fixed values of p and q and suitable initial conditions on a sphere of radius μα around the smaller primary, we will derive expressions for the final position and velocity on this sphere for the orbits under consideration.