Earth-Moon Lagrange Point Research Articles

Launch and deployment of distributed small satellite systems

The rise in launch and use of small satellites in the past decade, a result of improved functionality through technology miniaturisation and alternative design philosophies, has spawned interest in the development of distributed systems or constellations of small satellites. However, whilst a variety of missions based on constellations of small satellites have been proposed, issues relating to the launch and deployment of these distributed systems mean that few have actually been realised. A number of strategies have been proposed which enable multiple small satellites comprising a constellation to be launched together and efficiently separated on-orbit, thus reducing the total cost of launch. In this paper, two such strategies which have the potential to significantly increase the viability of small satellite constellations in Earth orbit are investigated. Deployment using natural Earth perturbations to indirectly achieve plane separations is analysed using a developed method and compared to deployment utilising the Earth–Moon Lagrange point L1 as a staging area prior to return to LEO. The analysis of three example missions indicates that these two strategies can facilitate the successful establishment of small satellite constellations in Earth orbit whilst also reducing propulsive requirements, system complexity, and/or cost. The study also found that the method of nodal precession is sensitive to the effects of orbital decay due to drag and can result in long deployment times, and the use of Lunar L1 is more suitable for constellation configurations where several satellites are present in each orbital plane.

Control of chaos in the vicinity of the Earth‐Moon L5 Lagrangian point to keep a spacecraft in orbit

AbstractThe concept of Space Manifold Dynamics is a new method of space research. We have applied it along with the basic idea of the method of Ott, Grebogi, and York (OGY method) to stabilize the motion of a spacecraft around the triangular Lagrange point L5 of the Earth‐Moon system. We have determined the escape rate of the trajectories in the general three‐ and four‐body problem and estimated the average lifetime of the particles. Integrating the two models we mapped in detail the phase space around the L5 point of the Earth‐Moon system. Using the phase space portrait our next goal was to apply a modified OGY method to keep a spacecraft close to the vicinity of L5. We modified the equation of motions with the addition of a time dependent force to the motion of the spacecraft. In our orbit‐keeping procedure there are three free parameters: (i) the magnitude of the thrust, (ii) the start time, and (iii) the length of the control. Based on our numerical experiments we were able to determine possible values for these parameters and successfully apply a control phase to a spacecraft to keep it on orbit around L5. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Spacecraft Rendezvous Utilizing Invariant Manifolds for a Halo Orbit

For spacecraft to rendezvous in a halo orbit around Earth Moon Lagrange Point 2 with an amplitude of several ten-thousand kilometers, we propose selecting a chaser's rendezvous trajectory from two different types, depending on the phase difference to the target. In the first trajectory, the chaser approaches the target from behind along the orbit, similarly to a rendezvous in a low Earth orbit. The second trajectory utilizes the homoclinic intersection of invariant manifolds of the halo orbit extended toward the Moon, where the chaser's trajectory is controlled so that it first departs from the halo orbit along an unstable manifold, is connected to a stable manifold through the intersection, and then returns to the halo orbit. We showed that this detour can adjust the time of arrival to the halo orbit with low fuel usage and the total delta-v for the rendezvous can be significantly reduced in comparison to the first trajectory if the initial phase difference is large. The strategy employed can significantly increase the flexibility of the flight plan, increase the launch window of the visiting vehicle to the target and enhance the tolerance against failure compared to the application of a traditional phasing method.

Optimal interior Earth–Moon Lagrange point transfer trajectories using mixed impulsive and continuous thrust

Optimal interior Earth–Moon Lagrange point transfer trajectories results with fixed flight time using mixed impulsive and continuous thrust propulsion in the framework of the planar Circular Restricted Three-Body Problem are presented. A virtual spacecraft departs geosynchronous or low-Earth orbit and enters a specified Lyapunov orbit around the interior Earth–Moon Lagrange point. For these transfer trajectory designs, a direct transcription and collocation method is employed to reformulate the continuous dynamic optimization problem into a discrete optimization problem, which then is solved using nonlinear programming software. As the design parameters, flight time and relative weighting factor between impulsive and continuous thrust are adjusted in the transfer trajectory design procedure. For practically implementable transfer trajectory designs, the constraints for coast arc and bounded control thrust are applied in the problem formulation as equality and inequality constraints, respectively. In addition, both quadratic performance index and minimum fuel performance index are used to show the relative comparison and analysis in the transfer trajectory design results. According to the relative weighting factor, the transfer trajectories are classified into direct and spiral departure trajectories whose virtual spacecraft directly departs the Earth parking orbit and departs after generating spirals around the Earth, respectively. The convergence of these transfer trajectory design methods are numerically verified by generating varying numbers of nodes and comparing their design results. Especially, the primer vector theory is utilized to analyze the transfer trajectory design results using minimum fuel performance index. Finally, the progressive homotopy continuation method is applied to design low-Earth departure transfer trajectories. These transfer trajectories show that they are numerically and theoretically valid results with the benefit of mixed impulsive and continuous thrust.

Alternative transfer to the Earth–Moon Lagrangian points L4 and L5 using lunar gravity assist

Lagrangian points L4 and L5 lie at 60° ahead of and behind the Moon in its orbit with respect to the Earth. Each one of them is a third point of an equilateral triangle with the base of the line defined by those two bodies. These Lagrangian points are stable for the Earth–Moon mass ratio. As so, these Lagrangian points represent remarkable positions to host astronomical observatories or space stations. However, this same distance characteristic may be a challenge for periodic servicing mission. This paper studies elliptic trajectories from an Earth circular parking orbit to reach the Moon’s sphere of influence and apply a swing-by maneuver in order to re-direct the path of a spacecraft to a vicinity of the Lagrangian points L4 and L5. Once the geocentric transfer orbit and the initial impulsive thrust have been determined, the goal is to establish the angle at which the geocentric trajectory crosses the lunar sphere of influence in such a way that when the spacecraft leaves the Moon’s gravitational field, its trajectory and velocity with respect to the Earth change in order to the spacecraft arrives at L4 and L5. In this work, the planar Circular Restricted Three Body Problem approximation is used and in order to avoid solving a two boundary problem, the patched-conic approximation is considered.

Optimal station-keeping near Earth–Moon collinear libration points using continuous and impulsive maneuvers

In this study the gravitational perturbations of the Sun and other planets are modeled on the dynamics near the Earth–Moon Lagrange points and optimal continuous and discrete station-keeping maneuvers are found to maintain spacecraft about these points. The most critical perturbation effect near the L1 and L2 Lagrange points of the Earth–Moon is the ellipticity of the Moon’s orbit and the Sun’s gravity, respectively. These perturbations deviate the spacecraft from its nominal orbit and have been modeled through a restricted five-body problem (R5BP) formulation compatible with circular restricted three-body problem (CR3BP). The continuous control or impulsive maneuvers can compensate the deviation and keep the spacecraft on the closed orbit about the Lagrange point. The continuous control has been computed using linear quadratic regulator (LQR) and is compared with nonlinear programming (NP). The multiple shooting (MS) has been used for the computation of impulsive maneuvers to keep the trajectory closed and subsequently an optimized MS (OMS) method and multiple impulses optimization (MIO) method have been introduced, which minimize the summation of multiple impulses. In these two methods the spacecraft is allowed to deviate from the nominal orbit; however, the spacecraft trajectory should close itself. In this manner, some closed or nearly closed trajectories around the Earth–Moon Lagrange points are found that need almost zero station-keeping maneuver.

A methodology to support strategic decisions in future human space exploration: From scenario definition to building blocks assessment

The human exploration of multiple deep space destinations (e.g. Cis-Lunar, NEAs), in view of the final challenge of sending astronauts to Mars, represents a current and consistent study domain especially in terms of its possible scenarios and mission architectures assessments, as proved by the numerous on-going activities about this topic and moreover by the global exploration roadmap. After exploring and analysing different possible solutions to identify the most flexible path, a detailed characterisation of several Design Reference Missions (DRMs) represents a necessity in order to evaluate the feasibility and affordability of deep space exploration missions, specifically in terms of enabling technological capabilities.The study presented in this paper was aimed at defining an evolutionary scenario for deep space exploration in the next 30 years with the final goal of sending astronauts on the surface of Mars by the end of 2030 decade. Different destinations were considered as targets to build the human exploration scenario, with particular attention to Earth–Moon Lagrangian points, NEA and Moon. For all the destinations selected as part of the exploration scenario, the assessment and characterisation of the relative Design Reference Missions were performed. Specifically they were defined in terms of strategies, architectures and mission elements. All the analyses were based on a pure technical approach with the objective of evaluating the feasibility of a long term strategy for capabilities achievement and technological development to enable future space exploration.This paper describes the process that was followed within the study, focusing on the adopted methodology, and reports the major obtained results, in terms of scenario and mission analysis.

Attitude Dynamics of Rigid Bodies in the Vicinity of the Lagrangian Points

I N THE last 30 years, several sun-monitoring spacecraft have orbited around the sun–Earth L1 point, and several space observatories will be placed near the sun–EarthL2 in the next decade. Although a large body of literature exists on the orbital dynamics of spacecraft in the restricted three-body problem, very few have explored the attitude dynamics of a rigid Lagrangian-point spacecraft. Kane and Marsh [1] considered the attitude dynamics of an axial symmetric satellite that is rotating about its axis of symmetry, which is normal to the primary bodies’ orbital plane. Robinson [2] first studied the attitude dynamics of a dumbbell satellite located at a triangular point. The same author [3] later determined the equilibrium attitudes of an arbitrary shaped satellite located at a collinear point or at a triangular point and constructed a linear stability diagram about one of the equilibrium configurations. Misra andBellerose [4] studied the librational dynamics of a tethered satellite located at the Earth–moon Lagrangian points and obtained the libration frequencies. The rigid spacecraft is assumed to be held at the Lagrangian points in all of these studies. In practice, however, collinear point satellites are not located directly at the Lagrangian point but in periodic orbits around the point. This note extends the work done by Robinson [3] and Misra and Bellerose [4] and addresses the question of how the translational motions of the spacecraft affect its attitude dynamics. The attitude dynamics of the satellite are studied while it is in a planar Lyapunov orbit and its three-dimensional counterpart. A triangular point spacecraft was previously demonstrated to have two equilibrium configurations, and linear stability diagrams about both equilibrium configurations for a L4 spacecraft are also presented. II. Problem Formulation

Planetary Defense From Space: Part 1-Keplerian Theory

A system of two space bases housing missiles is proposed to achieve the Planetary Defense of the Earth against dangerous asteroids and comets. We show that the layout of the Earth–Moon system with the five relevant Lagrangian (or libration) points in space leads naturally to only one, unmistakable location of these two space bases within the sphere of influence of the Earth. These locations are at the two Lagrangian points L1 (in between the Earth and the Moon) and L3 (in the direction opposite to the Moon from the Earth). We show that placing bases of missiles at L1 and L3 would cause those missiles to deflect the trajectory of asteroids by hitting them orthogonally to their impact trajectory toward the Earth, so as to maximize their deflection. We show that the confocal conics are the best class of trajectories fulfilling this orthogonal deflection requirement. An additional remark is that the theory developed in this paper is just a beginning of a larger set of future research work. In fact, while in this paper we only develop the Keplerian analytical theory of the Optimal Planetary Defense achievable from the Earth–Moon Lagrangian points L1 and L3, much more sophisticated analytical refinements would be needed to: 1. Take into account many perturbation forces of all kinds acting on both the asteroids and missiles shot from L1 and L3; 2. add more (non-optimal) trajectories of missiles shot from either the Lagrangian points L4 and L5 of the Earth–Moon system or from the surface of the Moon itself; 3. encompass the full range of missiles currently available to the US (and possibly other countries) so as to really see “which asteroids could be diverted by which missiles”, even in the very simplified scheme outlined here. Outlined for the first time in February 2002, our Confocal Planetary Defense concept is a Keplerian Theory that proved simple enough to catch the attention of scholars, representatives of the US Military and popular writers. These developments could possibly mark the beginning of an “all embracing” mathematical vision of Planetary Defense beyond all learned activities, dramatic movies and unknown military plans covered by secret.

Optimal trajectories from the Earth-Moon L1 and L3 points to deflect hazardous asteroids and comets.

Software code named asteroff was recently created by the author to simulate the deflection of hazardous asteroids off of their collision course with the Earth. This code was both copyrighted and patented to avoid unauthorized use of ideas that could possibly be vital to construct a planetary defense system in the vicinity of the Earth. Having so said, the basic ideas and equations underlying the asteroff simulation code are openly described in this paper. A system of two space bases housing missiles is proposed to achieve the planetary defense of the Earth against dangerous asteroids and comets, collectively called impactors herein. We show that the layout of the Earth-Moon system with the five relevant Lagrangian (or libration) points in space leads naturally to only one, unmistakable location of these two space bases within the sphere of influence of the Earth. These locations are at the two Lagrangian points L(1) (between the Earth and the Moon) and L(3) (in the direction opposite to the Moon from the Earth). We show that placing missile bases at L(1) and L(3) would enable those missiles to deflect the trajectory of impactors by hitting them orthogonally to their impact trajectory toward the Earth, so as to maximize their deflection. We show that confocal conics are the best class of trajectories fulfilling this orthogonal deflection requirement. One additional remark is that the theory developed in this paper is just a beginning for a wider set of future research. In fact, we only develop the Keplerian analytical theory for the optimal planetary defense achievable from the Earth-Moon Lagrangian points L(1) and L(3). Much more sophisticated analytical refinements would be needed to: (1) take into account many perturbation forces of all kinds acting on both the impactors and missiles shot from L(1) and L(3); (2) add more (non-optimal) trajectories of missiles shot from either the Lagrangian points L(4) and L(5) of the Earth-Moon System or from the surface of the Moon itself; and (3) encompass the full range of missiles currently available to the US (and possibly other countries) so as to really see which impactors could be diverted by which missiles, even in the very simplified scheme outlined here. Published for the first time in February 2002, our Keplerian planetary defense theory has proved, in just one year, to be simple enough to catch the attention of scholars, in addition to popular writers, and even of someone from the US Military. These recent developments might possibly mark the beginning of an all embracing vision in planetary defense beyond all learned congressional activities, dramatic movies, and unknown military plans covered by secrecy.

Materials trade study for lunar/gateway missions

The National Aeronautics and Space Administration (NASA) administrator has identified protection from radiation hazards as one of the two biggest problems of the agency with respect to human deep space missions. The intensity and strength of cosmic radiation in deep space makes this a ‘must solve’ problem for space missions. The Moon and two Earth-Moon Lagrange points near Moon are being proposed as hubs for deep space missions. The focus of this study is to identify approaches to protecting astronauts and habitats from adverse effects from space radiation both for single missions and multiple missions for career astronauts to these destinations. As the great cost of added radiation shielding is a potential limiting factor in deep space missions, reduction of mass, without compromising safety, is of paramount importance. The choice of material and selection of the crew profile play major roles m design and mission operations. Material trade studies in shield design over multi-segmented missions involving multiple work and living areas in the transport and duty phase of space mission's to two Earth-Moon co-linear Lagrange points (L 1) between Earth and the Moon and (L 2) on back side of the moon as seen from Earth, and to the Moon have been studied. It is found that, for single missions, current state-of-the-art knowledge of material provides adequate shielding. On the other hand, the choice of shield material is absolutely critical for career astronauts and revolutionary materials need to be developed for these missions. This study also provides a guide to the effectiveness of multifunctional materials in preparation for more detailed geometry studies in progress.

A search for objects near the Earth-Moon Lagrangian points

A photographic search of the five Earth-Moon Lagrangian positions included the solar-synchronized positions in the stable L4/L5 libration orbits, the potentially stable nonplanar orbits near L1/L2, Earth-Moon L3, and also L2 in the Sun-Earth system. Observations using the 61-cm Burrell Schmidt telescope at the Warner and Swasey Observatory, Kitt Peak Station, spanned 60° along the lunar orbital plane × 5° around Earth-Moon L5, 48° × 5° around L4, 25° × 13° around L3, 15° × 24° around the Moon (L1/L2), and 14° × 14° around Sun-Earth L2. Limiting magnitude for the detection of libration objects near L3, L4, and L5 was 17–19th magnitude, 10–18th magnitude for L1/L2 plates, and 14–16th magnitude for Sun-Earth L2. No natural or artificial objects were found. An automated search of selected priority plates was attempted using the Faint Object Classification and Analysis System (FOCAS) software package.