Study of the transfer between libration point orbits and lunar orbits in Earth–Moon system

This paper is devoted to the study of the transfer problem from a libration point orbit of the Earth–Moon system to an orbit around the Moon. The transfer procedure analysed has two legs: the first one is an orbit of the unstable manifold of the libration orbit and the second one is a transfer orbit between a certain point on the manifold and the final lunar orbit. There are only two manoeuvres involved in the method and they are applied at the beginning and at the end of the second leg. Although the numerical results given in this paper correspond to transfers between halo orbits around the \(L_1\) point (of several amplitudes) and lunar polar orbits with altitudes varying between 100 and 500 km, the procedure we develop can be applied to any kind of lunar orbits, libration orbits around the \(L_1\) or \(L_2\) points of the Earth–Moon system, or to other similar cases with different values of the mass ratio.     

Cylindrical isomorphic mapping applied to invariant manifold dynamics for Earth–Moon Missions

Several families of periodic orbits exist in the context of the circular restricted three-body problem. This work studies orbital motion of a spacecraft among these periodic orbits in the Earth–Moon system, using the planar circular restricted three-body problem model. A new cylindrical representation of the spacecraft phase space (i.e., position and velocity) is described, and allows representing periodic orbits and the related invariant manifolds. In the proximity of the libration points, the manifolds form a four-fold surface, if the cylindrical coordinates are employed. Orbits departing from the Earth and transiting toward the Moon correspond to the trajectories located inside this four-fold surface. The isomorphic mapping under consideration is also useful for describing the topology of the invariant manifolds, which exhibit a complex geometrical stretch-and-folding behavior as the associated trajectories reach increasing distances from the libration orbit. Moreover, the cylindrical representation reveals extremely useful for detecting periodic orbits around the primaries and the libration points, as well as the possible existence of heteroclinic connections. These are asymptotic trajectories that are ideally traveled at zero-propellant cost. This circumstance implies the possibility of performing concretely a variety of complex Earth–Moon missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining a suitable, convenient end-of-life strategy for spacecraft placed in any of the unstable orbits. The final disposal orbit is an externally confined trajectory, never approaching the Earth or the Moon, and can be entered by means of a single velocity impulse (of modest magnitude) along the right unstable manifold that emanates from the Lyapunov orbit at \(L_2\) .     

Quasi-periodic orbits about the translunar libration point

Analytical solutions for quasi-periodic orbits about the translunar libration point are obtained by using the method of Lindstedt-Poincaré and computerized algebraic manipulations. The solutions include the effects of nonlinearities, lunar orbital eccentricity, and the Sun's gravitational field. For a small-amplitude orbit, the orbital path as viewed from the Earth traces out a Lissajous figure. This is due to a small difference in the fundamental frequencies of the in-plane and out-of-plane oscillations. However, when the amplitude of the in-plane oscillation is greater than 32 379 km, there is a corresponding value of the out-of-plane amplitude that will produce a path where the fundamental frequencies are equal. This synchronized trajectory describes a halo orbit of the Moon.     

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 note on the dynamics around the Lagrange collinear points of the Earth–Moon system in a complete Solar System model

In this paper we study the dynamics of a massless particle around the L _1,2 libration points of the Earth–Moon system in a full Solar System gravitational model. The study is based on the analysis of the quasi-periodic solutions around the two collinear equilibrium points. For the analysis and computation of the quasi-periodic orbits, a new iterative algorithm is introduced which is a combination of a multiple shooting method with a refined Fourier analysis of the orbits computed with the multiple shooting. Using as initial seeds for the algorithm the libration point orbits of Circular Restricted Three Body Problem, determined by Lindstedt-Poincaré methods, the procedure is able to refine them in the Solar System force-field model for large time-spans, that cover most of the relevant Sun–Earth–Moon periods.     

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.     

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.     

On quasi-periodic motions around the collinear libration points in the real Earth–Moon system

Due to various perturbations, the collinear libration points of the real Earth–Moon system are not equilibrium points anymore. Under the assumption that the Moon’s motion is quasi-periodic, special quasi-periodic orbits called dynamical substitutes exist. These dynamical substitutes replace the geometrical collinear libration points as time-varying equilibrium points. In the paper, the dynamical substitutes of the three collinear libration points in the real Earth–Moon system are computed. For the points L ₁ and L ₂, linearized motions around the dynamical substitutes are described, and the variational equations of the dynamical substitutes are reduced to a form with a near constant coefficient matrix. Then higher order analytical formulae of the central manifolds are constructed. Using these analytical solutions as initial seeds, Lissajous orbits and halo orbits are computed with numerical algorithms.     

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.     

Sun–Earth L_{1} and L_{2} to Moon transfers exploiting natural dynamics

This paper examines the design of transfers from the Sun–Earth libration orbits, at the \(L_{1}\) and \(L_{2}\) points, towards the Moon using natural dynamics in order to assess the feasibility of future disposal or lifetime extension operations. With an eye to the probably small quantity of propellant left when its operational life has ended, the spacecraft leaves the libration point orbit on an unstable invariant manifold to bring itself closer to the Earth and Moon. The total trajectory is modeled in the coupled circular restricted three-body problem, and some preliminary study of the use of solar radiation pressure is also provided. The concept of survivability and event maps is introduced to obtain suitable conditions that can be targeted such that the spacecraft impacts, or is weakly captured by, the Moon. Weak capture at the Moon is studied by method of these maps. Some results for planar Lyapunov orbits at \(L_{1}\) and \(L_{2}\) are given, as well as some results for the operational orbit of SOHO.     

Spacecraft trajectories to the L<Subscript>3</Subscript> point of the Sun–Earth three-body problem

Of the three collinear libration points of the Sun–Earth Circular Restricted Three-Body Problem (CR3BP), L₃ is that located opposite to the Earth with respect to the Sun and approximately at the same heliocentric distance. Whereas several space missions have been launched to the other two collinear equilibrium points, i.e., L₁ and L₂, taking advantage of their dynamical and geometrical characteristics, the region around L₃ is so far unexploited. This is essentially due to the severe communication limitations caused by the distant and permanent opposition to the Earth, and by the gravitational perturbations mainly induced by Jupiter and the close passages of Venus, whose effects are more important than those due to the Earth. However, the adoption of a suitable periodic orbit around L₃ to ensure the necessary communication links with the Earth, or the connection with one or more relay satellites located at L₄ or L₅, and the simultaneous design of an appropriate station keeping-strategy, would make it possible to perform valuable fundamental physics and astrophysics investigations from this location. Such an opportunity leads to the need of studying the ways to transfer a spacecraft (s/c) from the Earth’s vicinity to L₃. In this contribution, we investigate several trajectory design methods to accomplish such a transfer, i.e., various types of two-burn impulsive trajectories in a Sun-s/c two-body model, a patched conics strategy exploiting the gravity assist of the nearby planets, an approach based on traveling on invariant manifolds of periodic orbits in the Sun–Earth CR3BP, and finally a low-thrust transfer. We examine advantages and drawbacks, and we estimate the propellant budget and time of flight requirements of each.     

Strategies for plane change of Earth orbits using lunar gravity and derived trajectories of family G

The dynamics of the circular restricted three-body Earth-Moon-particle problem predicts the existence of the retrograde periodic orbits around the Lagrangian equilibrium point L1. Such orbits belong to the so-called family G (Broucke, Periodic orbits in the restricted three-body problem with Earth-Moon masses, JPL Technical Report 32–1168, 1968) and starting from them it is possible to define a set of trajectories that form round trip links between the Earth and the Moon. These links occur even with more complex dynamical systems as the complete Sun-Earth-Moon-particle problem. One of the most remarkable properties of these trajectories, observed for the four-body problem, is a meaningful inclination gain when they penetrate into the lunar sphere of influence and accomplish a swing-by with the Moon. This way, when one of these trajectories returns to the proximities of the Earth, it will be in a different orbital plane from its initial Earth orbit. In this work, we present studies that show the possibility of using this property mainly to accomplish transfer maneuvers between two Earth orbits with different altitudes and inclinations, with low cost, taking into account the dynamics of the four-body problem and of the swing-by as well. The results show that it is possible to design a set of nominal transfer trajectories that require ΔV _Total less than conventional methods like Hohmann, bi-elliptic and bi-parabolic transfer with plane change.     

Impulsive time-free transfers between halo orbits

A methodology is developed to design optimal time-free impulsive transfers between three-dimensional halo orbits in the vicinity of the interior L ₁ libration point of the Sun-Earth/Moon barycenter system. The transfer trajectories are optimal in the sense that the total characteristic velocity required to implement the transfer exhibits a local minimum. Criteria are established whereby the implementation of a coast in the initial orbit, a coast in the final orbit, or dual coasts accomplishes a reduction in fuel expenditure. The optimality of a reference two-impulse transfer can be determined by examining the slope at the endpoints of a plot of the magnitude of the primer vector on the reference trajectory. If the initial and final slopes of the primer magnitude are zero, the transfer trajectory is optimal; otherwise, the execution of coasts is warranted. The optimal time of flight on the time-free transfer, and consequently, the departure and arrival locations on the halo orbits are determined by the unconstrained minimization of a function of two variables using a multivariable search technique. Results indicate that the cost can be substantially diminished by the allowance for coasts in the initial and final libration-point orbits.An earlier version was presented as Paper AIAA 92-4580 at the AIAA/AAS Astrodynamics Conference, Hilton Head Island, SC, U.S.A., August 10–12, 1992.     

Planetary perturbations on the Libration of the Moon

A theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field, was recently built (Moons, 1982). The Lie transforms method was used to reduce the Hamiltonian of the main problem of the libration of the Moon and to produce the usual libration series p₁, p₂ and . This main problem takes into account the perturbations due to the Sun and the Earth on the rotation of a rigid Moon about its center of mass. In complement to this theory, we have now computed the planetary effects on the libration, the planetary terms being added to the mean Hamiltonian of the main problem before a last elimination of the angles. For the main problem, as well as for the planetary perturbations, the motion of the center of mass of the Moon is described by the ELP 2000 solution (Chapront and Chapront-Touze, 1983).     

Using space manifold dynamics to deploy a small satellite constellation around the Moon

The possibility of communicating with the far side of the Moon is essential for keeping a continuous radio link with lunar orbiting spacecraft, as well as with manned or unmanned surface facilities in locations characterized by poor coverage from Earth. If the exploration and the exploitation of the Moon must be sustainable in the medium/long term, we need to develop the capability to realize and service such facilities at an affordable cost. Minimizing the spacecraft mass and the number of launches is a driving parameter to this end. The aim of this study is to show how Space Manifold Dynamics can be profitably applied in order to launch three small spacecraft onboard the same launch vehicle and send them to different orbits around the Moon with no significant difference in the Delta-V budgets. Internal manifold transfers are considered to minimize also the transfer time. The approach used is the following: we used the linearized solution of the equations of motion in the Circular Restricted Three Body Problem to determine a first–guess state vector associated with the Weak Stability Boundary regions, either around the collinear Lagrangian point L1 or around the Moon. The resulting vector is then used as initial state in a numerical backward-integration sequence that outputs a trajectory on a manifold. The dynamical model used in the numerical integration is four-body and non-circular, i.e. the perturbations of the Sun and the lunar orbital eccentricity are accounted for. The trajectory found in this way is used as the principal segment of the lunar transfer. After separation, with minor maneuvers each satellite is injected into different orbits that lead to ballistic capture around the Moon. Finally, one or more circularization maneuvers are needed in order to achieve the final circular orbits. The whole mission profile, from launch to insertion into the final lunar orbits, is modeled numerically with the commercial software STK.     

Mars interplanetary trajectory design via Lagrangian points

With the increase in complexities of interplanetary missions, the main focus has shifted to reducing the total delta-V for the entire mission and hence increasing the payload capacity of the spacecraft. This paper develops a trajectory to Mars using the Lagrangian points of the Sun-Earth system and the Sun-Mars system. The whole trajectory can be broadly divided into three stages: (1) Trajectory from a near-Earth circular parking orbit to a halo orbit around Sun-Earth Lagrangian point L₂. (2) Trajectory from Sun-Earth L₂ halo orbit to Sun-Mars L₁ halo orbit. (3) Sun-Mars L₁ halo orbit to a circular orbit around Mars. The stable and unstable manifolds of the halo orbits are used for halo orbit insertion. The intermediate transfer arcs are designed using two-body Lambert’s problem. The total delta-V for the whole trajectory is computed and found to be lesser than that for the conventional trajectories. For a 480 km Earth parking orbit, the total delta-V is found to be 4.6203 km/s. Another advantage in the present approach is that delta-V does not depend upon the synodic period of Earth with respect to Mars.     

Lissajous trajectories for lunar global positioning and communication systems

This work proposes a Lunar Global Positioning System (LGPS) and a Lunar Global Communication System (LGCS) using two constellations of satellites on Lissajous trajectories around the collinear L ₁ and L ₂ libration points in the Earth–Moon system. This solution is compared against a Walker constellation around the Moon similar to the one used for the Global Positioning System (GPS) on the Earth to evaluate the main differences between the two cases and the advantages of adopting the Lissajous constellations. The problem is first studied using the Circular Restricted Three Body Problem to find out its main features. The study is then repeated with higher fidelity using a four-body model and higher-order reference trajectories to simulate the Earth-Moon-spacecraft dynamics more accurately. The LGPS performance is evaluated for both on-ground and in-flight users, and a visibility study for the LGCS is used to check that communication between opposite sides of the Moon is possible. The total ΔV required for the transfer trajectories from the Earth to the constellations and the trajectory control is calculated. Finally, the estimated propellant consumption and the total number of satellites for the Walker constellation and the Lissajous constellations is used as a performance index to compare the two proposed solutions.     

Effect of the Earth’s compression on the physical libration of the Moon

The effect of the Earth??s compression on the physical libration of the Moon is studied using a new vector method. The moment of gravitational forces exerted on the Moon by the oblate Earth is derived considering second order harmonics. The terms in the expression for this moment are arranged according to their order of magnitude. The contribution due to a spherically symmetric Earth proves to be greater by a factor of 1.34 × 10⁶ than a typical term allowing for the oblateness. A linearized Euler system of equations to describe the Moon??s rotation with allowance for external gravitational forces is given. A full solution of the differential equation describing the Moon??s libration in longitude is derived. This solution includes both arbitrary and forced oscillation harmonics that we studied earlier (perturbations due to a spherically symmetric Earth and the Sun) and new harmonics due to the Earth??s compression. We posed and solved the problem of spinorbital motion considering the orientation of the Earth??s rotation axis with regard to the axes of inertia of the Moon when it is at a random point in its orbit. The rotation axes of the Earth and the Moon are shown to become coplanar with each other when the orbiting Moon has an ecliptic longitude of L _? = 90° or L _? = 270°. The famous Cassini??s laws describing the motion of the Moon are supplemented by the rule for coplanarity when proper rotations in the Earth-Moon system are taken into account. When we consider the effect of the Earth??s compression on the Moon??s libration in longitude, a harmonic with an amplitude of 0.03?? and period of T ₈ = 9.300 Julian years appears. This amplitude exceeds the most noticeable harmonic due to the Sun by a factor of nearly 2.7. The effect of the Earth??s compression on the variation in spin angular velocity of the Moon proves to be negligible.     

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.     

Optimal low-thrust transfers between libration point orbits

Over the past three decades, ballistic and impulsive trajectories between libration point orbits (LPOs) in the Sun–Earth–Moon system have been investigated to a large extent. It is known that coupling invariant manifolds of LPOs of two different circular restricted three-body problems (i.e., the Sun–Earth and the Earth–Moon systems) can lead to significant mass savings in specific transfers, such as from a low Earth orbit to the Moon’s vicinity. Previous investigations on this issue mainly considered the use of impulsive maneuvers along the trajectory. Here we investigate the dynamical effects of replacing impulsive ΔV’s with low-thrust trajectory arcs to connect LPOs using invariant manifold dynamics. Our investigation shows that the use of low-thrust propulsion in a particular phase of the transfer and the adoption of a more realistic Sun–Earth–Moon four-body model can provide better and more propellant-efficient solution. For this purpose, methods have been developed to compute the invariant tori and their manifolds in this dynamical model.