Earth-Moon Lagrange Point Research Articles

A Relativistic Framework to Estimate Clock Rates on the Moon

As humanity aspires to explore the solar system and investigate distant worlds such as the Moon, Mars, and beyond, there is a growing need to estimate and model the rate of clocks on these celestial bodies and compare them with the rate of standard clocks on Earth. According to Einstein’s theory of relativity, the rate of a standard clock is influenced by the gravitational potential at its location and its relative motion. A convenient choice of local reference frames allows for the comparison of local time variations of clocks due to gravitational and kinematic effects. We estimate the rate of clocks on the Moon using a locally freely falling reference frame coincident with the center of mass of the Earth–Moon system. A clock near the Moon’s selenoid ticks faster than one near the Earth’s geoid, accumulating an extra 56.02 μs day−1 over the duration of a lunar orbit. This formalism is then used to compute the clock rates at Earth–Moon Lagrange points. Accurate estimation of the rate differences of coordinate times across celestial bodies and their intercomparisons using clocks on board orbiters at Lagrange points as time transfer links is crucial for establishing reliable communications infrastructure. This understanding also underpins precise navigation in cislunar space and on celestial bodies’ surfaces, thus playing a pivotal role in ensuring the interoperability of various position, navigation, and timing systems spanning from Earth to the Moon and to the farthest regions of the inner solar system.

The Lunar Space Elevator: A Key Technology for Realising the Greater Earth Lunar Power Station (GEO-LPS)

The Greater Earth Lunar Power Station (GE -LPS) is a multi-purpose concept that aims to solve several critical issues related to lunar development and terrestrial energy production. As the GE -LPS concept and its energy production functions may be scaled to any dimension, larger versions could be positioned in Earth orbit to provide clean solar energy for terrestrial purposes and thereby reduce the mass needed to be launched from Earth to build SPS units by 80% or more. The construction of the GE -LPS from mostly lunar materials requires the establishment of industrial-scale automated mining and manufacturing processes on the Moon. A key technology is a Lunar Space Elevator (LSE) deployed as a transportation system to move SPS components from an anchor point on the surface of the Moon to a docking and assembly station at Earth-Moon Lagrange point 1 (EM-L1). Significantly, a LSE could be built today with existing tether materials such as Dyneema or Zylon which are already commercially available. Additionally, lunar sourced basalt fibre may be sufficient for reinforcing and extending the LSE once it becomes operational. An Earth-pointing LSE could become a valuable cislunar infrastructure asset – “the Suez Canal of cislunar space” – linking the Earth and Moon economies. Note: The ‘O’ symbol is an ancient European symbol for planet Earth: it is used here to mean “Greater Earth”, a region defined by the Earth’s gravitational field, which includes the Moon. Keywords: Solar Power Satellite, Lunar ISRU, Lunar Space Elevator, Energy, Europe

Fully Autonomous Orbit Determination and Synchronization for Satellite Navigation and Communication Systems in Halo Orbits

This paper presents a solution for autonomous orbit determination and time synchronization of spacecraft in Halo orbits around Lagrange points using inter-satellite links. Lagrange points are stable positions in the gravitational field of two large bodies that allow for a sustained presence of a spacecraft in a specific region. However, a challenge in operating at these points is the lack of fixed landmarks for orbit determination. The proposed solution involves using inter-satellite links to perform range and range-rate measurements, allowing for accurate computation of the spacecraft’s orbit parameters without the need for any facilities on Earth. Simulations using a fleet of three satellites in Near Rectilinear Halo Orbits around the Earth–Moon Lagrange point, proposed for the Lunar Gateway stations, were conducted to demonstrate the feasibility of the approach. The results show that inter-satellite links can provide reliable and accurate solutions for orbit determination with a DRMS error lower than one meter (90th percentile) and synchronization errors of around one nanosecond. This solution paves the way for a fully autonomous fleet of spacecraft that can be used for observation, telecommunication, and navigation missions.

Debris Collision Risk Analysis Following Simulated Cislunar Spacecraft Explosions

This research simulates spacecraft explosions in cislunar space and analyzes the resulting risk to other notional spacecraft. Three case studies are considered, which include simulations of spacecraft explosions during Earth–moon Hohmann transfers, along the manifold of the Earth–moon Lagrange point, and at the stable and Earth–moon Lagrange points. Debris from the explosions is propagated using a four-body dynamical model that incorporates the gravitational forces of the Earth, moon, and sun. Finally, the risk to nearby cislunar spacecraft is quantified using two different models for the vulnerability of spacecraft to debris, which include a logistic vulnerability model that incorporates only debris particle mass and a ballistic limit vulnerability model that incorporates both particle mass and velocity. The results from each model are compared to evaluate their application to cislunar debris scenarios. The results of this study showed potential debris hazards to spacecraft near Earth from a mishap during an Earth–moon transfer and hazard probabilities to nearby cislunar spacecraft on the order of from Lagrange point mishaps. Overall, research into cislunar debris propagation enhances operational planning outside the traditional near-Earth paradigm of spacecraft mission operations and increases understanding of the debris-related consequences of mishaps within this orbital regime.

In‐Orbit Performance Analysis of Solar Array for Chang'e‐4 Relay Satellite in an Earth–Moon Lagrangian Point 2 Halo Orbit

Herein, the in‐orbit performance of the solar array for Chang'e‐4 Relay Satellite in an Earth–Moon Lagrangian point 2 (EML2) Halo orbit is analyzed. The daily degradation coefficient of the photovoltaic current is calculated to be 0.99989 independent of whether the relay antenna outspreads, indicating an annual degradation coefficient of 3.94%. However, due to the shading effect of the antenna and the electrical performance degradation, the output voltage of the solar array dissatisfies the requirement of the power controller on around the 700th day, resulting in an abrupt decrease of the photovoltaic current. In addition, the in‐orbit working temperature of the solar array for Chang'e‐4 Relay Satellite is also studied. The working temperature mainly ranges from 52.3 to 76.8 °C, while the working temperature decreases to the range of 0–25 °C once the relay antenna outspreads. The present work is helpful to a deepen the understanding of EML2 Halo orbit and provide reference to the solar array design for a deep space exploration aircraft.

Spacecraft survivability in the natural debris environment near the stable Earth-Moon Lagrange points

The theoretical analysis of the motion of natural space debris near the stable Earth-Moon Lagrange Points, L4 and L5, is presented with a focus on the potential debris risks to spacecraft operating near these points. Specifically, the research formulates a debris propagation model using four-body dynamics, then applies candidate probabilistic survivability models to a notional spacecraft operating at the L4 and L5 Lagrange points to quantify the collision risks to the spacecraft from natural debris particles. Of the survivability models implemented, the natural debris collision risks to spacecraft survivability are found to be incredibly low, but mitigation strategies to reduce the risk further are identified in this study. Overall, research into stable Lagrange point natural debris propagation improves understanding of the collision risks posed by the naturally occurring Kordylewski clouds and enhances operational planning for Lagrange point space missions.

Orbital Guidance Using Higher-Order State Transition Tensors

This paper derives the equations necessary to use state transition tensors (STTs) in a spacecraft guidance problem. The derivation includes all necessary equations up to the fourth-order expansions in the dynamics and the controls. These are first derived for the general case for a finite burn beginning at the epoch time; simplifications are made for when the target is the spacecraft state at a desired final time, and the controls at the epoch time are impulsive ’s. An application is presented comparing the STT method with a linearized numerical predictor–corrector for computing an impulsive station-keeping maneuver in an unstable halo orbit around the Earth–moon Lagrange point. The STT method is shown to perform significantly faster, while also converging on accurate solutions for long-horizon targets where the predictor–corrector method fails. The formulation is able to accommodate any number of perturbations in the dynamics, rendering it suitable for onboard operational usage. This approach provides promise for fast and efficient determination of the desired controls to achieve a nonlinear target constraint surface in a highly nonlinear dynamic environment.

Mission to Earth–Moon Lagrange Point by a 6U CubeSat: EQUULEUS

EQUULEUS (EQUilibriUm Lunar-Earth point 6U Spacecraft) will be the world's smallest spacecraft to explore the Earth-Moon Lagrange point. It is being jointly developed by JAXA (Japan Aerospace Exploration Agency) and the University of Tokyo, and will be launched by NASA's Space Launch System Exploration Mission-1. The spacecraft will fly to a libration orbit around the Earth-Moon L2 point (EML2) and will demonstrate low-energy trajectory-control techniques within the Sun-Earth-Moon region for the first time by a nano-class spacecraft. EQUULEUS also carries three scientific observation missions: imaging of Earth's plasmasphere by extreme ultraviolet wavelength, lunar impact flash observation on the far side of the moon, and micrometeoroid flux measurements in the cis-lunar region. While all these missions have their own scientific objectives, they will also contribute to future human activity and/or infrastructure development in the cis-lunar region. Most parts of the spacecraft system use commercial off-the-shelf components, or are designed based on the experiences of various past space missions, with the exception of the newly developed water resistojet propulsion system. EQUULEUS uses X-band frequency for deep space telecommunication. Japanese deep space antennas (64-m and 34-m) will be nominally used for spacecraft operation, and support from the deep space network of JPL (Jet Propulsion Laboratory) is also being planned, especially for the initial phase of operation. The spacecraft will fly to EML2 in less than one year, and will remain there for scientific observations until shortly before the depletion of the onboard propellant, when the spacecraft will leave the orbit for space-debris compliance.

Fireworks Algorithm Applied to Trajectory Design for Earth to Lunar Halo Orbits

A simple and efficient method using the fireworks algorithm (FWA) is presented for pruning the search space of an impulsive maneuver transfer from a user-defined low Earth orbit (LEO) to a desired lunar halo orbit around Earth–moon Lagrange point 2. The dynamical framework adopted is the circular restricted three-body problem. The FWA method is used to optimize both the two-burn direct impulsive transfer and the impulsive transfer that uses stable manifolds. Sample LEO to lunar halo trajectories, along with their required costs and times of flight (TOFs), are provided and compared to known numerical techniques. The results obtained are close in both and TOFs to what the existing literature reports, which demonstrates the validity of the proposed method.

Rendezvous Strategies in the Vicinity of Earth-Moon Lagrangian Points

In the context of Human Spaceflight exploration mission scenario, with the Lunar Orbital Platform- Gateway (LOP-G) orbiting about Earth-Moon Lagrangian Point (EML), Rendezvous and Docking (RVD) operational activities are mandatory and critical for the deployment and utilization of the LOP-G (station assembly, crew rotations, cargo delivery, lunar sample return). There is extensive experience with RVD in the two-body problem: in Low Earth Orbit (LEO) to various space stations, or around quasi-circular Low Lunar Orbits (LLO), the latter by Apollo by means of manual RVD. However, the RVD problem in non-Keplerian environments has rarely been addressed and no RVD has been performed to this date in the vicinity of Lagrangian points (LP) where Keplerian dynamics are no longer applicable. Dynamics in such regions are more complex, but multi-body dynamics also come with strong advantages that need to be further researched by the work proposed here. The aim of this paper is to present methods and results of investigations conducted to first set up strategies for far and close rendezvous between a target (the LOP-G, for example) and a chaser (cargo, crew vehicle, ascent and descent vehicle, station modules, etc.) depending on target and chaser orbit. Semi-analytical tools have been developed to compute and model families of orbits about the Lagrangian points in the Circular Restricted Three Body Problem (CR3BP) like NRHO, DRO, Lyapunov, Halo and Lissajous orbits. As far as close rendezvous is concerned, implementation of different linear and non-linear models used to describe cis-lunar relative motion will be discussed and compared, in particular for NRHO and DRO.

Erratum: Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth–Moon Lagrange point L5 – I. Three-dimensional celestial mechanical modelling of dust cloud formation

Erratum: Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth–Moon Lagrange point L5 – I. Three-dimensional celestial mechanical modelling of dust cloud formation

Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth–Moon Lagrange point L5 – Part II. Imaging polarimetric observation: new evidence for the existence of Kordylewski dust cloud

Telescopes mounted with polarizers can study the neutral points of the Earths atmosphere, the solar corona, the surface of planets/moons of the Solar System, distant stars, galaxies and nebulae. These examples demonstrate well that polarimetry is a useful technique to gather astronomical information from spatially extended phenomena. There are two enigmatic celestial objects that can also effectively be studied with imaging polarimetry, namely the Kordylewski dust clouds (KDCs) positioned around the L4 and L5 triangular Lagrangian libration points of the Earth-Moon system. Although in 1961 the Polish astronomer, Kazimierz Kordylewski had observed two bright patches near the L5 point with photography, many astronomers assume that these dust clouds do not exist, because the gravitational perturbation of the Sun, solar wind and other planets may disrupt the stabilizing effect of the L4 and L5 Lagrange points of the Earth and Moon. Using ground-born imaging polarimetry, we present here new observational evidence for the existence of the KDC around the L5 point of the Earth-Moon system. Excluding artefacts induced by the telescope, cirrus clouds or condensation trails of airplanes, the only explanation remains the polarized scattering of sunlight on the particles collected around the L5 point. By our polarimetric detection of the KDC we think it is appropriate to reconsider the pioneering photometric observation of Kordylewski. Our polarimetric evidence is supported by the results of simulation of dust cloud formation in the L5 point of the Earth-Moon system presented in the first part of this paper.

Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth–Moon Lagrange point L5 – I. Three-dimensional celestial mechanical modelling of dust cloud formation

Since the discovery in 1772 of the triangular Lagrange points L4 and L5 in the gravitational field of two bodies moving under the sole influence of mutual gravitational forces, astronomers found a large number of minor celestial bodies around these points of the Sun-Jupiter, Sun-Earth, Sun-Mars and Sun-Neptune systems. The L4 and L5 points of the Earth and Moon may be empty due to the gravitational perturbation of the Sun. However, in 1961 Kordylewski found two bright patches near the L5 point, which may refer to an accumulation of interplanetary particles. Since that time this formation is called the Kordylewski dust cloud (KDC). Until now only a very few computer simulations studied the formation and characteristics of the KDC. To fill this gap, we investigated a three-dimensional four-body problem consisting of the Sun, Earth, Moon and one test particle, 1860000 times separately. We mapped the size and shape of the conglomeratum of particles not escaped from the system sooner than an integration time of 3650 days around L5. Polarimetric observations of a possible KDC around L5 are presented in the second part of this paper.

Determining the necessity of fault tolerance techniques in FPGA devices for space missions

Functionality of electronic components in space is strongly influenced by the impact of radiation induced errors which may interfere with the proper operation of the equipment. In space missions, FPGA implementations are generally protected using computationally expensive radiation-error mitigation techniques such as error co rrecting codes (ECC) and triple modular redundancy (TMR). For high-performance systems, such fault tolerance techniques can prove problematic due to both the added computational requirements and their resulting power overhead. As such it is important to make a proper assessment of the expected error rates to make a proper selection of mitigation techniques. This paper provides an extensive overview of the techniques used for determining the necessity of such mitigation techniques in space missions and other situations where a large radiation dose will be encountered. Given the presented study and radiation analysis, in this paper an experimental example is presented in the form of a case study on the Digital Receiver System (DRS) in the Netherlands–China Low-frequency Explorer (NCLE) mission, which is implemented using a Xilinx Kintex-7 SRAM FPGA. Fault rates are estimated for a five-year mission to the second Earth-Moon Lagrange point (L2) and the chosen fault mitigation strategy as implemented in NCLE–DRS is presented. The effect of potential upsets on the functionality of DRS has been taken into account in order to make error estimations more precise. Thus, two test-benches are developed and presented to experimentally evaluate the effect of upsets in FPGA configuration memory and the data on the DRS final outputs. The approach provided in this paper should generalize well to other space missions, as long as a general estimate of the expected radiation environment is available.

Enhancing Station-Keeping Control With the Use of Extended State Observers

Recently there has been a resurgence of interest in missions to the moon and a major challenge of such missions is to provide a continuous communication between the Earth and the Moon's far side. Orbits around the $L_2$ Earth-Moon Lagrange point have been a topic of interest in this field due to their potential for constant communication with both the Earth and the Moon, however the Lagrange point orbits are innately unstable and so station-keeping control is required to maintain them. Station-keeping problems are highly nonlinear and a traditional approach to control design is first to linearize the nonlinear system. However, this first-order approximation introduces errors if there are large injection errors. This paper demonstrates how a simple Extended State Observer (ESO) can be used to improve the convergence time of spacecraft to the reference orbit given with large injection errors. Additionally, solar radiation pressure (SRP) a dominant disturbance in deep-space, can lead to inefficient station-keeping if it is not taken into account in the reference orbit design. New reference orbits can be designed that exploit the SRP perturbation but this assumes that it is known apriori. Here we show how an ESO could provide an in-orbit measurement of the SRP which could be used to modify the reference trajectory to a more fuel efficient one. Finally, it is shown how an ESO can be used to estimate, not only the disturbance, but simultaneously the velocity of the spacecraft meaning that only the position of the spacecraft is required.

Moon-based planetary defense campaign

The Moon is an ideal location to launch intercepting missions to life-threatening and catastrophic asteroids. The effectiveness of the interception greatly depends on the weight of the spacecraft. Unfortunately, interceptors launched from the Earth lose more than 98% of their weight by burning the majority of their onboard fuel and by jettisoning their lower stage structures before entering a heliocentric orbit. However, if interceptors are launched from the Moon by a lunar surface accelerator, they can enter a heliocentric orbit without consuming any onboard fuel or jettisoning any part of the spacecraft. A 5-ton construction package, which consists of robots and industrial production equipment, would enable mining on the moon and construction of a 3.5 km-long, 5,000-ton accelerator.Large asteroid impacts have and will inevitably occur, and it is important to be prepared to avoid catastrophes, but they may not happen immediately or even within the next fifty years. The future planetary defense system must be a dual-use system, which continuously provides a secondary benefit to justify its operation and maintenance costs. When it is not defending the planet, the Lunar Electromagnetic Interceptor Launch System (LEILAS) can send over a thousand tons of construction material and fuel annually to the Low Earth Orbit (LEO) or Earth-Moon Lagrange Point Two (EML-2) to build space stations and to construct large spacecraft for deep space missions.

Effect of orbital shadow at an Earth-Moon Lagrange point on relay communication mission

The shadow effect is an important constraint to be considered during the implementation of exploration missions. In this paper, for the Earth-Moon Lagrange point L2 relay communication mission, shadow effect issues on a periodic orbit about L2 are investigated. A systematic analysis based on the time domain and phase space is performed including the distribution, duration, and frequency of shadows. First, the Lindstedt-Poincare and second-order differential correction methods are used in conjunction with the DE421 planetary ephemeris to achieve a mission trajectory family in a high-precision ephemeris model. Next, on the basis of a conical shadow model, the influence of different orbital phases and amplitudes on the shadow is analyzed. The distribution of the shadow is investigated as well. Finally, the configuration of the shadow and its characteristics are studied. This study provides an important reference and basis for mission orbit design and shadow avoidance for relay satellites at an Earth-Moon Lagrange point.

A Chang’e-4 mission concept and vision of future Chinese lunar exploration activities

A novel concept for Chinese Chang'e-4 lunar exploration mission is presented in this paper at first. After the success of Chang'e-3, its backup probe, Chang'e-4 lander/rover combination, would be upgraded and land on the unexplored lunar farside by the aid of a relay satellite near the second Earth-Moon Lagrange point. Mineralogical and geochemical surveys on the farside to study the formation and evolution of lunar crust and observations at low radio frequencies to track the signals of the Universe's Dark Ages are priorities. Follow-up Chinese lunar exploration activities before 2030 are envisioned as building a robotic lunar science station by three to five missions. Finally several methods of international cooperation are proposed.

Erratum: Earth-moon Lagrangian points as a test bed for general relativity and effective field theories of gravity [Phys. Rev. D92, 064045 (2015)

Erratum: Earth-moon Lagrangian points as a test bed for general relativity and effective field theories of gravity [Phys. Rev. D<b>92</b>, 064045 (2015)

Earth-moon Lagrangian points as a test bed for general relativity and effective field theories of gravity

In the restricted four-body problem consisting of the Earth, the Moon and the Sun as the primaries and a spacecraft as the planetoid, we take into account the solar perturbation in the description of the motion of a spacecraft in the vicinity of the stable Earth-Moon libration points L4 and L5 both in the classical regime and in the context of effective field theories of gravity. We then evaluate the location of all Lagrangian points in the Earth-Moon system within the framework of general relativity. For the points L4 and L5, the corrections of coordinates are of order a few millimeters. After that, we set up a scheme where the theory which is quantum corrected has as its classical counterpart the Einstein theory, instead of the Newtonian one. By virtue of the effective-gravity correction to the longdistance form of the potential among two point masses, all terms involving the ratio between the gravitational radius of the primary and its separation from the planetoid get modified. Within this framework, for the Lagrangian points of stable equilibrium, we find quantum corrections of order two millimeters, whereas for Lagrangian points of unstable equilibrium we find quantum corrections below a millimeter. Finally, general relativity corrections to Newtonian position of collinear Lagrangian points turn out to be below the millimiter, whereas on stable equilibrium points they are of order of a few millimiters.