Fast numerical computation of Lissajous and quasi-halo libration point trajectories

In this paper we present a methodology for the automatic generation of quasi–periodic libration point trajectories (Lissajous and quasi–halo) of the Spatial, Circular Restricted Three–Body Problem. This methodology is based on the computation of a mesh of orbits which, using interpolation strategies, gives an accurate quantitative representation of the full set of libration point orbits. This representation, when combined with the one obtained using Poincar´e maps, provides a useful tool for the design of missions to libration points fulfilling specific requirements. The same methodology applies to stable and unstable manifolds as well. This paper extends and improves results presented in [10].Postprint (published version

A Hopf variables view on the libration points dynamics

The dynamics about the libration points of the Hill problem is investigated analytically. In particular, the use of Lissajous variables and perturbation theory allows to reduce the problem to a one degree of freedom Hamiltonian depending on two physical parameters. The invariant manifolds structure of the Hill problem is then disclosed, yet accurate computations are limited to energy values close to that of the libration points

Control of Lagrange point orbits using solar sail propulsion

Several missions have utilised halo orbits around the L1 and L2 previous termLagrangenext term points of the Earth-Sun system. Due to the instability of these orbits, station-keeping techniques are required to prevent escape after orbit insertion. This paper considers using solar sail propulsion to provide station-keeping at quasi-periodic orbits around L1 and L2. Stable manifolds will be identified which provide near-Earth insertion to a quasi-periodic trajectory around the libration point. The possible control techniques investigated include solar sail area variation and solar sail pitch and yaw angle variation. Hill's equations are used to model the dynamics of the problem and optimal control laws are developed to minimise the control requirements. The constant thrust available using solar sails can be used to generate artificial libration points Sunwards of L1 or Earthwards of L2. A possible mission to position a science payload Sunward of L1 will be investigated. After insertion to a halo orbit at L1, gradual solar sail deployment can be performed to spiral Sunwards along the Sun-Earth axis. Insertion -V requirements and area variation control requirements will be examined. This mission could provide advance warning of Earthbound coronal mass ejections (CMEs) responsible for magnetic storms

Higher Order Approximation to the Hill Problem Dynamics about the Libration Points

An analytical solution to the Hill problem Hamiltonian expanded about the libration points has been obtained by means of perturbation techniques. In order to compute the higher orders of the perturbation solution that are needed to capture all the relevant periodic orbits originated from the libration points within a reasonable accuracy, the normalization is approached in complex variables. The validity of the solution extends to energy values considerably far away from that of the libration points and, therefore, can be used in the computation of Halo orbits as an alternative to the classical Lindstedt-Poincar\'e approach. Furthermore, the theory correctly predicts the existence of the two-lane bridge of periodic orbits linking the families of planar and vertical Lyapunov orbits.Comment: 28 pages, 8 figure

A Motivating Exploration on Lunar Craters and Low-Energy Dynamics in the Earth -- Moon System

It is known that most of the craters on the surface of the Moon were created by the collision of minor bodies of the Solar System. Main Belt Asteroids, which can approach the terrestrial planets as a consequence of different types of resonance, are actually the main responsible for this phenomenon. Our aim is to investigate the impact distributions on the lunar surface that low-energy dynamics can provide. As a first approximation, we exploit the hyberbolic invariant manifolds associated with the central invariant manifold around the equilibrium point L_2 of the Earth - Moon system within the framework of the Circular Restricted Three - Body Problem. Taking transit trajectories at several energy levels, we look for orbits intersecting the surface of the Moon and we attempt to define a relationship between longitude and latitude of arrival and lunar craters density. Then, we add the gravitational effect of the Sun by considering the Bicircular Restricted Four - Body Problem. As further exploration, we assume an uniform density of impact on the lunar surface, looking for the regions in the Earth - Moon neighbourhood these colliding trajectories have to come from. It turns out that low-energy ejecta originated from high-energy impacts are also responsible of the phenomenon we are considering.Comment: The paper is being published in Celestial Mechanics and Dynamical Astronomy, vol. 107 (2010

A note on the dynamics around the L1,2 Lagrange points of the Earth-Moon system in a complete solar system model

The purpose of this paper is the study of the phase space around the collinear libration points L1 and L2 of the Earth–Moon system when the gravitational effects of the remaining bodies of the Solar System are taken into account. In the simplified model defined by the Circular Restricted Three Body Problem (CR3BP), the description of the phase space around these two points has already been done in the past, either using semi-analytical techniques1 or numerical ones.2 When using more realistic models of motion, the refinement of the different kinds of libration point orbits around both points (see G´omez et al.3) has some problems when the time interval used is large. These problems are more evident for the Earth–Moon L2 point due to a 1:2 resonance between the natural frequency of some halo orbits (!h) and the external frequency due to the perturbation of the Sun (!s); in fact !h ' 2!s for some orbits of the halo family of periodic orbits around this point. To analyse more closely this fact, Andreu4 introduced an intermediate model between the CR3BP and the restricted n-body problem, which is the so called Quasi-Bicircular Problem. It is a restricted four body problem in which the three primaries move following a “true” solution of the three body problem along orbits close to circular (the Earth and the Moon around their barycenter, and the Earth-Moon barycenter around the Sun). The description of the different types of orbits in the neighbourhood of L2 is also done in Reference 4 by means of the reduction of the Hamiltonian of the problem to the central manifold. The reduced Hamiltonian is then studied by means of Poincar´e maps at different energy levels.Postprint (published version

Transfer orbits in the Earth-Moon system and tefinement to JPL ephemerides

We describe how to determine transfers between libration point orbits and either the surface of the Moon or a Low Earth Orbit within the Circular Restricted Three – Body Problem (CR3BP) assumptions. Moreover, we explain how to refine such trajectories to ones verifing more comprehensive equations of motion. We are interested in seeing how the geometry of the nominal target orbits and of the associated stable manifolds drives the connections and also how much reliable the CR3BP is. The main tools we take advantage of are the Lindstedt–Poincaré semi-analytical method, differential correction procedures and an optimizer.Postprint (published version

Computation of libration point orbits and manifolds using collocation methods

This thesis contains a methodology whose aim is to compute trajectories describing natural motion of the phase space in a neighborhood of Libtation points and stable/unstable manifolds which correspond to these orbits in the Restricted Three Body Problem. There are two models the Circular Restricted Three Body Problem and Elliptic Restricted Three Body Problem which are special cases of RTBP . In this paper we pay attention to CRTBP which is autonomous (depending on time). The CRTBP is the most easily understood and well-analysed in a coordinate system rotating with two large bodies. The method is based on the collocation method implemented in AUTO - 07p software and must provide an isolated periodic solution. The paper includes explanation of the collocation method, its application in case of CRTBP, numerical and graphical results of its implementation

Uncontrolled spacecraft formations on two-dimensional invariant tori

Within the class of natural motions near libration point regions quasi-periodic trajectories evolving on invariant tori are studied. Those orbits prove beneficial for relative spacecraft configurations with large distances among satellites. In this study properties of invariant tori are outlined, and non-resonant and resonant tori around the Sun/Earth libration point L1 are computed. A numerical approach to obtain the frequency base and to parametrize a torus in angular phase space is introduced. Initial states for spacecraft formations on the torus’ surface are defined. The formation naturally evolve along its surface such that the relative positions within a formation are unaltered and the relative distances and the orientation are closely bounded. An in-plane coordinate frame together with a modified torus motion is introduced and the inner and outer behaviour of the formation’s geometry is investigated