Gravitational lunar capture based on bicircular model in restricted four body problem

Gravitational capture is a useful phenomenon in the design of the low energy transfer (LET) orbit for a space mission. In this paper, gravitational lunar capture based on the Sun–Earth–Moon bicircular model (BCM) in the restricted four body problem is studied. By the mechanical analysis in the space near the Moon, we first propose a new parameter \(k\) , the corrected ratio of the radial force, to investigate the influence of the radial force on the capture eccentricity in the BCM. Then, a parametric analysis is performed to detect the influences on the corrected ratio \(k\) . Considering the restriction of time-of-flight and corrected ratio, we investigate, respectively, the minimum capture eccentricity and the corrected minimum capture eccentricity. Via numerical analysis, we discover two special regions on the sphere of capture, in which the capture point possesses the global minimum capture eccentricity and corrected capture eccentricity. They denote the optimal capture regions in terms of minimizing the fuel consumption of the maneuver. According to the results obtained, some suggestions on the design of the LET orbit are given.     

Direct and indirect capture of near-Earth asteroids in the Earth–Moon system

Near-Earth asteroids have attracted attention for both scientific and commercial mission applications. Due to the fact that the Earth–Moon \(\hbox {L}_{1}\) and \(\hbox {L}_{2}\) points are candidates for gateway stations for lunar exploration, and an ideal location for space science, capturing asteroids and inserting them into periodic orbits around these points is of significant interest for the future. In this paper, we define a new type of lunar asteroid capture, termed direct capture. In this capture strategy, the candidate asteroid leaves its heliocentric orbit after an initial impulse, with its dynamics modeled using the Sun–Earth–Moon restricted four-body problem until its insertion, with a second impulse, onto the \(\hbox {L}_{2}\) stable manifold in the Earth–Moon circular restricted three-body problem. A Lambert arc in the Sun-asteroid two-body problem is used as an initial guess and a differential corrector used to generate the transfer trajectory from the asteroid’s initial obit to the stable manifold associated with Earth–Moon \(\hbox {L}_{2}\) point. Results show that the direct asteroid capture strategy needs a shorter flight time compared to an indirect asteroid capture, which couples capture in the Sun–Earth circular restricted three-body problem and subsequent transfer to the Earth–Moon circular restricted three-body problem. Finally, the direct and indirect asteroid capture strategies are also applied to consider capture of asteroids at the triangular libration points in the Earth–Moon system.     

A Hopf variables view on the libration points dynamics

This study analyzes a recently discovered class of exterior transfers to the Moon. These transfers terminate in retrograde ballistic capture orbits, i.e., orbits with negative Keplerian energy and angular momentum with respect to the Moon. Yet, their Jacobi constant is relatively low, for which no forbidden regions exist, and the trajectories do not appear to mimic the dynamics of the invariant manifolds of the Lagrange points. This paper shows that these orbits shadow instead lunar collision orbits. We investigate the dynamics of singular, lunar collision orbits in the Earth–Moon planar circular restricted three-body problem, and reveal their rich phase space structure in the medium-energy regime, where invariant manifolds of the Lagrange point orbits break up. We show that lunar retrograde ballistic capture trajectories lie inside the tube structure of collision orbits. We also develop a method to compute medium-energy transfers by patching together orbits inside the collision tube and those whose apogees are located in the appropriate quadrant in the Sun–Earth system. The method yields the novel family of transfers as well as those ending in direct capture orbits, under particular energetic and geometrical conditions.     

On the history of the lunar orbit

A frequency-dependent model of tidal friction is used in the determination of the time rate of change of the lunar orbital elements and the angular velocity of the Earth. The variational equations consider eccentricity, the solar tide on the Earth, Earth oblateness, and higher-order terms in the Earth's tidal potential. A linearized solution of the equations governing the precission of the Earth's rotational angular momentum and the lunar ascending node is found. This allows the analytical averaging of the variational equations over the period of relative precession which, though large, is necessarily small in comparison to the time step of the numerical integrator that yields the system history over geological time. Results for this history are presented and are identified as consistent with origin of the Moon by capture. This model may be applied to any planet-satellite system where evolution under tidal friction is of interest.     

On the capture of the Moon

This paper studies the possibility of lunar capture depending on variations of the solar mass under certain well specified conditions and assumptions regarding the behaviour of the three-body dynamical system formed by the Sun, Earth and Moon. It is found that a large amount of decrease in the solar mass (approximately 37%) would be required to allow capture if the model of the planar restricted problem of three bodies is assumed, if the masses of the Earth and Moon did not change and if the angular momentum of the Sun-Earth system did not change. Such large mass-changes of the Sun can not be associated with radiation mass losses only with catastrophic events, such as stellar close approaches.     

Formation of Embryos of the Earth and the Moon from a Common Rarefied Condensation and Their Subsequent Growth

Embryos of the Moon and the Earth may have formed as a result of contraction of a common parental rarefied condensation. The required angular momentum of this condensation could largely be acquired in a collision of two rarefied condensations producing the parental condensation. With the subsequent growth of embryos of the Moon and the Earth taken into account, the total mass of as-formed embryos needed to reach the current angular momentum of the Earth–Moon system could be below 0.01 of the Earth mass. For the low lunar iron abundance to be reproduced with the growth of originally iron-depleted embryos of the Moon and the Earth just by the accretion of planetesimals, the mass of the lunar embryo should have increased by a factor of 1.3 at the most. The maximum increase in the mass of the Earth embryo due to the accumulation of planetesimals in a gas-free medium is then threefold, and the current terrestrial iron abundance is not attained. If the embryos are assumed to have grown just by accumulating solid planetesimals (without the ejection of matter from the embryos), it is hard to reproduce the current lunar and terrestrial iron abundances at any initial abundance in the embryos. For the current lunar iron abundance to be reproduced, the amount of matter ejected from the Earth embryo and infalling onto the Moon embryo should have been an order of magnitude larger than the sum of the overall mass of planetesimals infalling directly on the Moon embryo and the initial mass of the Moon embryo, which had formed from the parental condensation, if the original embryo had the same iron abundance as the planetesimals. The greater part of matter incorporated into the Moon embryo could be ejected from the Earth in its multiple collisions with planetesimals (and smaller bodies).     

The Flux of Lunar Meteorites onto the Earth

Numerous new finds of lunar meteorites in Oman allow detailed constraints to be obtained on the intensity of the transfer of lunar matter to the Earth. Our estimates show that the annual flux of lunar meteorites in the mass interval from 10 to 1000 g to the entire Earth's surface should not be less than several tenths of a kilogram and is more likely equal to tens or even a few hundred kilograms, i.e., a few percent of the total meteorite flux. This corresponds to several hundred or few thousand falls of lunar meteorites on all of Earth per year. Even small impact events, which produce smaller than craters on the Moon smaller than 10 km in diameter, are capable of transferring lunar matter to the Earth. In this case, the Earth may capture between 10 to 100% of the mass of high-velocity crater ejecta leaving the Moon. Our estimates for the lunar flux imply rather optimistic prospects for the discovery of new lunar meteorites and, consequently, for the analyses of the lunar crust composition. However, the meteorite-driven flux of lunar matter did not play any significant role in the formation of the material composition of the Earth's crust, even during the stage of intense meteorite bombardment.     

Comparison of low-energy lunar transfer trajectories to invariant manifolds

In this study, transfer trajectories from the Earth to the Moon that encounter the Moon at various flight path angles are examined, and lunar approach trajectories are compared to the invariant manifolds of selected unstable orbits in the circular restricted three-body problem. Previous work focused on lunar impact and landing trajectories encountering the Moon normal to the surface, and this research extends the problem with different flight path angles in three dimensions. The lunar landing geometry for a range of Jacobi constants is computed, and approaches to the Moon via invariant manifolds from unstable orbits are analyzed for different energy levels.     

Tether orientation control for lunar elevator

This study focuses on spatial motion of the lunar elevator which is studied in the framework of elliptical restricted three-body problem. Analysis of dynamics of a spacecraft anchored to the Moon by a tether is done assuming that the tether’s length can be changed according to a prescribed law. The goal is to find the control laws that allow one to compensate for the eccentricity of the orbits, i.e., to maintain the pendulum at a fixed angle with respect to the Earth–Moon direction. The results have shown that the fixed orientation of the tether can be kept for several configurations of the system; some of these configurations are found to be stable. The obtained results can be applied to study the properties and possible configurations of the lunar elevator, as well as applications for small planets and asteroids.     

Design and analysis of Weak Stability Boundary trajectories to Moon

An algorithm is developed to find Weak Stability Boundary transfer trajectories to Moon in high fidelity force model using forward propagation. The trajectory starts from an Earth Parking Orbit (circular or elliptical). The algorithm varies the control parameters at Earth Parking Orbit and on the way to Moon to arrive at a ballistic capture trajectory at Moon. Forward propagation helps to satisfy launch vehicle’s maximum payload constraints. Using this algorithm, a number of test cases are evaluated and detailed analysis of capture orbits is presented.     

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. In the former case, as main outcome, we observe a more relevant bombardment at the apex of the lunar surface, and a percentage of impact which is almost constant and whose value depends on the assumed Earth–Moon distance d_EM. In the latter, it seems that the Earth–Moon and Earth–Moon–Sun relative distances and the initial phase of the Sun θ ₀ play a crucial role on the impact distribution. The leading side focusing becomes more and more evident as d_EM decreases and there seems to exist values of θ ₀ more favorable to produce impacts with the Moon. Moreover, the presence of the Sun makes some trajectories to collide with the Earth. The corresponding quantity floats between 1 and 5 percent. 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.     

On Choosing a Rational Flight Trajectory to the Moon

The algorithm for choosing a trajectory of spacecraft flight to the Moon is discussed. The characteristic velocity values needed for correcting the flight trajectory and a braking maneuver are estimated using the Monte Carlo method. The profile of insertion and flight to a near-circular polar orbit with an altitude of ~100 km of an artificial lunar satellite (ALS) is given. The case of two corrections applied during the flight and braking phases is considered. The flight to an ALS orbit is modeled in the geocentric geoequatorial nonrotating coordinate system with the influence of perturbations from the Earth, the Sun, and the Moon factored in. The characteristic correction costs corresponding to corrections performed at different time points are examined. Insertion phase errors, the errors of performing the needed corrections, and the errors of determining the flight trajectory parameters are taken into account.     

Origin of the moon by tidal capture and some geophysical consequences

Of the many proposed modes of origin of the Moon, some violate physical laws; many are in conflict with observations; all are improbable. Perhaps the least improbable - based on recent tidal theory calculations and on the interpretation of lunar rock data - is capture of the Moon as it passed near the Earth in adirect (prograde) orbit, shortly after the formation of Moon and Earth, about 4.5 billion years ago. (Capture of the Moon from an initiallyretrograde orbit which had been proposed some years ago, leads to physically unacceptable consequences.) The effects of capture on the Earth would have been cataclysmic, leading to intensive heating of its interior, to volcanism, and to the immediate formation of an atmosphere and hydrosphere. Thus capture of a Moon may have given rise to the unique properties of the Earth (in the Solar System) and to the early evolution of life, about 3.5 billion years ago.Presented at the NATO Advanced Study Institute on Lunar Studies in Patras, Greece, September, 1971.     

Optimization of lunar occultations for determining the positions of point-like X-ray sources

In view of the scheduled satellite mission EXOSAT (European X-Ray Observatory Satellite) of ESA (European Space Agency) the lunar occultation technique to determine the position of point-like X-ray sources is investigated. An error analysis for the source coordinates resulting from this technique is presented and an occultation strategy is proposed to achieve optimum lunar occultations. The analysis takes into account the errors of the space coordinates of the satellite and the Moon, the unevenness of the lunar surface, the intensities of source and background, the apparent angular velocity of the Moon as seen from the satellite, the finite sizes of the preoccultation position error boxes of the X-ray sources and the inaccuracies in the satellite orbit correction manoeuvres necessary to achieve the occultations.     

Analytical theory of the libration of the Moon

This paper presents a new theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field. This field is represented through its third degree harmonics for the torque due to the Earth (second degree for the torque due to the Sun).The orbital motion of the Moon is described by the ELP 2000 solution (Chapront-Touzé, 1980) of the main problem of lunar theory.the physical libration variables are obtained as Poisson series and comparisons with the results of Eckhardt (Eckhardt, 1981) and Migus (Migus, 1980) are presented.     

Fast Earth–Moon transfers with ballistic capture

This contribution deals with fast Earth–Moon transfers with ballistic capture in the patched three-body model. We compute ensembles of preliminary solutions using a model that takes into account the relative inclination of the orbital planes of the primaries. The ballistic capture orbits around the Moon are obtained relying on the hyperbolic invariant structures associated to the collinear Lagrangian points of the Earth–Moon system, and the Sun–Earth system portion of the transfers are quasi-periodic orbits obtained by a genetic algorithm. The trajectories are designed to be good initial guesses to search optimal cost-efficient short-time Earth–Moon transfers with ballistic capture in more realistic models.     

Study of lunar gravity assist orbits in the restricted four-body problem

In this paper, the lunar gravity assist (LGA) orbits starting from the Earth are investigated in the Sun–Earth–Moon–spacecraft restricted four-body problem (RFBP). First of all, the sphere of influence of the Earth–Moon system (SOIEM) is derived. Numerical calculation displays that inside the SOIEM, the effect of the Sun on the LGA orbits is quite small, but outside the SOIEM, the Sun perturbation can remarkably influence the trend of the LGA orbit. To analyze the effect of the Sun, the RFBP outside the SOIEM is approximately replaced by a planar circular restricted three-body problem, where, in the latter case, the Sun and the Earth–Moon barycenter act as primaries. The stable manifolds associated with the libration point orbit and their Poincaré sections on the SOIEM are applied to investigating the LGA orbit. According to our research, the patched LGA orbits on the Poincaré sections can efficiently distinguish the transit LGA orbits from the non-transit LGA orbits under the RFBP. The former orbits can pass through the region around libration point away from the SOIEM, but the latter orbits will bounce back to the SOIEM. Besides, the stable transit probability is defined and analyzed. According to the variant requirement of the space mission, the results obtained can help us select the LGA orbit and the launch window.     

Fission and geosynchronous release of the Moon

The problem of the origin of the Moon has led to various hypotheses: simultaneous accretion, fission, capture, etc. These theories were based primarily on global mechanical considerations. New geological data (Turcotteet al., 1974; Kahn and Pompea, 1978) have led to fresh approaches and new versions of these theories.As suggested by Wise (1969) and O'Keefe (1972), the initial Earth may have taken unstable forms when radial segregation sped up the rotation. The Moon may have been created as the small part of the pyroid of Poincaré.Fission theory was mainly discarded, in the past, on the basis of energy considerations. We are now arriving at the conclusion that these considerations are void if the fission was followed by a very long period of geostationary rotation of the Moon at a distance of about 3 Earth radius (i.e., out of the Roche limit). Indeed the large amount of energy of the initial system could have been released slowly and therefore evacuated by losses of material and radiation.The accretion of the Earth and the radial segregation of heavy chemicals toward the center has led to a differential rotation of the different layers with a faster rotation at the center. During the geostationary period the Moon was synchronous with respect to the surface layer. That Earth-Moon system has both a correct angular momentum and a large stability provided that the viscosity of intermediate layers was small enough, which is in concordance with its high temperature.Even with a very hot system, a superficial cold layer appears because of its low conductivity and the radiation equilibrium with outer space. This implies a slow loss of energy: the geosynchronous Moon receded extremely slowly.During the geostationary period lithophile elements were extracted with water by the radial segregation and were deposited in the area facing the Moon. One massive continent was formed, as suggested by Grjebine (1978).As the continent became thicker and sank into the mantle, convection currents appeared and speeded up the cooling of the Earth. The viscosity increased and the synchronization between the Moon and the surface of the Earth became more difficult to maintain. When synchronism was broken important lunar tides transferred energy and momentum from the Earth to the Moon which receded toward its present position and the modification of its equilibrium shape explains the formation of lunar maria in the near side.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

Accretional heating as the major cause of compositional differences among meteorite parent bodies,the Moon,and Earth

Accretional energy can be retained with sufficient efficiency in the outer layers of the Moon due to the considerable amount of debris falling back into large craters.Heating of meteorite parent bodies occurs mainly after their accretion, by destructive collisions. The heating was generally not sufficient to differentiate the parent bodies completely so that iron meteorites would originate from the mantle, rather than from the core of a meteorite parent body. Assuming that the Earth and Moon accreted from material of similar chemical composition, we suggest that only from the outer lunar shell is there a loss of gases and volatiles due to accretional melting. The Earth melted completely and degassing was efficient for the whole mass of the Earth leading to its ≈20% higher uncompressed mean density in comparison to the Moon. Because of its lower gravitational field, gases and volatiles escaped much more easily from the lunar atmosphere than from the terrestrial one, leading to the observed depletion in volatiles of the outer parts of the Moon.     

Optimisation of seismic network design: Application to a geophysical international lunar network

Fundamental scientific questions concerning the internal structure and dynamics of the Moon, and their implications on the Earth-Moon System, are driving the deployment of a new broadband seismological network on the surface of the Moon. Informations about lunar seismicity and seismic subsurface models from the Apollo missions are used as a priori information in this study to optimise the geometry of future lunar seismic networks in order to best resolve the seismic interior structure of the Moon. Deep moonquake events and simulated meteoroid impacts are the assumed seismic sources. Synthetic P and S wave arrivals computed in a radial seismic model of the Moon are the assumed seismic data. The linearised estimates of resolution and covariance of radial seismic velocity perturbations can be computed for a particular seismic network geometry. The non-linear inverse problem relating the seismic station positions to the linearised estimates of covariance and resolution of radial seismic velocity perturbations is written and solved by the Neighbourhood Algorithm. This optimisation study favours near side seismic station positions at southern latitudes in order to constrain the deep mantle structure from deep moonquake data at large epicentral distances. The addition of a far side station allows to divide by two the size of the error bar on the seismic velocity model. The monitoring of lunar impact flashes from the Earth allows to improve the radial seismic model in the top of the mantle by adding much more meteor impact data at short epicentral distances due to the high accuracy of the space/time location of these seismic sources. Such meteor impact detections may be necessary to investigate the 3D structure of the lunar crust.