Collinear Libration Points Research Articles

Lissajous Orbits at the Collinear Libration Points in the Restricted Three-Body Problem with Oblateness

In the present work, the collinear equilibrium points of the restricted three-body problem are studied under the effect of oblateness of the bigger primary using an analytical and numerical approach. The periodic orbits around these points are investigated for the Earth-Moon system. The Lissajous orbits and the phase spaces are obtained under the effect of oblateness.

Periodic motion analysis around the libration points by polynomial expansion method in planar circular restricted three-body problem

We propose a polynomial expansion method to investigate periodic motions around both collinear and triangular libration points in the planar circular restricted three-body problem. The polynomial expansion method focuses on the approximate nonlinear polynomial relations between the two directions in the motion plane, which provide an alternative way to inspect the periodic motions. Based on the nonlinear polynomial relations, the planar two-dimensional system is decoupled. As an example, the $$3\mathrm{rd}$$ order analytical solutions for the periodic motion are determined by solving the decoupled system. The efficiency of the polynomial expansion method has been validated by numerical method.

Mirror Trajectories in Space Mission Analysis

The theorem of mirror trajectories was proven almost six decades ago by Miele, and states that for a given path in the restricted problem of three bodies (with primaries in mutual circular orbits) there exists a mirror trajectory (in two dimensions) and three mirror paths (in three dimensions). This theorem regards feasible trajectories and proved extremely useful for investigating the natural spacecraft dynamics in the circular restricted problem of three bodies. Several trajectories of crucial importance for mission analysis and design can be identified by using the theorem of mirror trajectories, i.e. (a) free return paths, (b) periodic orbits, and (c) homoclinic connections. Free return paths have been designed and flown in the Apollo missions, because they allow safe ballistic return toward the Earth in case of failure of the main propulsive system. These trajectories belong to the class of mirror paths with a single orthogonal crossing of the axis that connects the Earth and the Moon in the synodic reference system. Instead, if two orthogonal crossings of the same axis exist, then the resulting path is a periodic orbit. A variety of such orbits can be found, encircling either both primaries, a single celestial body, or a collinear libration point. In all cases, periodic orbits represent repeating trajectories that can be traveled indefinitely, (ideally) without any propellant expenditure. Homoclinic connections are special paths that belong to both the stable and the unstable manifold associated with a periodic orbit. These trajectories depart asymptotically from a periodic orbit and can encircle both primaries (even with close approaches) before converging asymptotically toward the initial periodic orbit. After almost six decades, the theorem of mirror trajectories, which clarified the fundamental symmetry properties related to the motion of a third body (or, more concretely, a space vehicle), still excerpts a considerable influence in space mission analysis and design.

A note on the full two-body problem and related restricted full three-body problem

Truncating at the second order of the mutual potential between two rigid bodies, time-explicit first order solutions to the rotations and the orbital motion of the two bodies in the planar full two-body problem (F2BP) are constructed. Based on this analytical solution, equations of motion (EOMs) for the related restricted full three-body problem are given. In the case of the synchronous or double synchronous states for the full two-body problem, EOMs for the related restricted full three-body problem (RF3BP) are also given. At last, one example—the “collinear libration point” in the binary asteroid system—is given.

A Study of Libration Points in Modified CR3BP Under Albedo Effect when Smaller Primary is an Ellipsoid

As we know that the Sun is a source of radiation in our solar system, the other planets or asteroids absorb some of the radiations incident on it and some reflected back into the space, these reflected radiations are called Albedo. The spacecraft is affected by both radiations i.e direct radiations as well as albedo. In this paper this is investigated how albedo perturbed the libration points and its stability in restricted three-body problem when less massive primary is an ellipsoid? It is found that there exist five libration points, three collinear and two non-collinear, the non-collinear libration points are stable for a critical value of mass parameter μ≤μ c , where μ c= 0.0385208965 …− (0.00891747 + 0.222579k) α– 0.02206859 σ 1 – 0.04071097 σ 2 but collinear libration points are still unstable. Also, an example of Sun-Earth system is taken in the last as a real application.

Design of the stabilizing control of the orbital motion using the analytical representation of an invariant manifold in the vicinity of a collinear libration point

Design of the stabilizing control of the orbital motion using the analytical representation of an invariant manifold in the vicinity of a collinear libration point

Modeling of controlled coupled attitude-orbit motion in the neighborhood of collinear libration point L1

Modeling of controlled coupled attitude-orbit motion in the neighborhood of collinear libration point L1

The optimal stabilization of orbital motion in a neighborhood of collinear libration point

In this paper we consider the special problem of stabilization of controllable orbital motion in a neighborhood of collinear libration point $L_2$ of Sun-Earth system. The modification of circular three-body problem -nonlinear Hill's equations, which describe orbital motion in a neighborhood of libration point is used as a mathematical model. Also, we used the linearized equations of motion. We investigate the problem of spacecraft arrival on the unstable invariant manifold. When a spacecraft reaches this manifold, it does not leave the neighborhood of $L_2$ by long time. The distance to the unstable invariant manifold is described by a special function of phase variables, so-called ''hazard function'. The control action directed along Sun-Earth line.

A study of libration points in CR3BP under albedo effect

In this paper this is investigated how albedo perturbed the libration points from its original position? It is found that there exist five libration points, three collinear and two non-collinear and all the libration points are affected by Albedo. The non-collinear libration points are stable for a critical value of mass parameter µ ≤ µc, where µc = µo − (0.00891747 + 0.222579k)α (µo is the critical mass parameter for classical case) but collinear libration points are still unstable.

Invariant manifold connections via polyhedral representation

In recent years, manifold dynamics has assumed an increasing relevance for analysis and design of low-energy missions, both in the Earth-Moon system and in alternative multibody environments, and several space missions have already taken advantage of the results of the related studies. Recent efforts have been devoted to developing a suitable representation for the manifolds, which would be extremely useful for mission analysis and optimization. This work describes and uses a recently-introduced, intuitive polyhedral interpolative approach for each state component associated with manifold trajectories, both in two and in three dimensions. A grid of data, coming from the numerical propagation of a finite number of manifold trajectories, is used. This representation is employed for some invariant manifolds, associated with the two planar Lyapunov orbits at the collinear libration points located in the proximity of the Moon, and with a three-dimensional Halo orbit. Accuracy is evaluated, and is proven to be satisfactory, with the exclusion of limited regions of the manifolds. This paper describes the use of this polyhedral representation for the detection of homoclinic and heteroclinic connections. In particular, a variety of homoclinic trajectories connected with two Lyapunov orbits are detected. Then, the polyhedral interpolating technique is successfully employed for the determination of heteroclinic connections between the manifolds associated with the Lyapunov orbits. Lastly, near-homoclinic connections between the manifolds emanating from a Halo orbit are detected. The results achieved in this paper prove utility and effectiveness of the polyhedral interpolative technique for detecting manifold connections in the circular restricted three-body problem, and represent the premise for its application to space mission analysis involving invariant manifold dynamics.

Suitable configurations for triangular formation flying about collinear libration points under the circular and elliptic restricted three-body problems

The design of formations of spacecraft in a three-body environment represents one of the most promising challenges for future space missions. Two or more cooperating spacecraft can greatly answer some very complex mission goals, not achievable by a single spacecraft. The dynamical properties of a low acceleration environment such as the vicinity of libration points associated to a three-body system, can be effectively exploited to design spacecraft configurations able of satisfying tight relative position and velocity requirements. This work studies the evolution of an uncontrolled formation orbiting in the proximity of periodic orbits about collinear libration points under the Circular and Elliptic Restricted Three-Body Problems. A three spacecraft triangularly-shaped formation is assumed as a representative geometry to be investigated. The study identifies initial configurations that provide good performance in terms of formation keeping, and investigates key parameters that control the relative dynamics between the spacecraft within the three-body system. Formation keeping performance is quantified by monitoring shape and size changes of the triangular formation. The analysis has been performed under five degrees of freedom to define the geometry, the orientation and the location of the triangle in the synodic rotating frame.

Arnold’s mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument

We consider the spatial circular restricted three-body problem, on the motion of an infinitesimal body under the gravity of Sun and Earth. This can be described by a 3-degree of freedom Hamiltonian system. We fix an energy level close to that of the collinear libration point L1, located between Sun and Earth. Near L1 there exists a normally hyperbolic invariant manifold, diffeomorphic to a 3-sphere. For an orbit confined to this 3-sphere, the amplitude of the motion relative to the ecliptic (the plane of the orbits of Sun and Earth) can vary only slightly.We show that we can obtain new orbits whose amplitude of motion relative to the ecliptic changes significantly, by following orbits of the flow restricted to the 3-sphere alternatively with homoclinic orbits that turn around the Earth. We provide an abstract theorem for the existence of such ‘diffusing’ orbits, and numerical evidence that the premises of the theorem are satisfied in the three-body problem considered here. We provide an explicit construction of diffusing orbits.The geometric mechanism underlying this construction is reminiscent of the Arnold diffusion problem for Hamiltonian systems. Our argument, however, does not involve transition chains of tori as in the classical example of Arnold. We exploit mostly the ‘outer dynamics’ along homoclinic orbits, and use very little information on the ‘inner dynamics’ restricted to the 3-sphere.As a possible application to astrodynamics, diffusing orbits as above can be used to design low cost maneuvers to change the inclination of an orbit of a satellite near L1 from a nearly-planar orbit to a tilted orbit with respect to the ecliptic. We explore different energy levels, and estimate the largest orbital inclination that can be achieved through our construction.

Restricted three-body problem with stokes drag effect when less massive primary is an ellipsoid

The present paper deals with the effect of Stokes drag force on the existence and stability of collinear and non-collinear libration points in circular restricted three-body problem when less massive primary is an ellipsoid. During the investigation, it is found that there exist five libration points Li (i = 1, 2… 5) out of which three are collinear and two are non-collinear. We observed that the Stokes drag force does not affect the collinear libration points while non-collinear libration points are affected by it and all the libration points either collinear or non-collinear are unstable in Lyapunov sense for the given range of dissipative constant k and mass parameter µ.

A general method for the generation and extension of collinear libration point orbits

This paper is devoted to the study on applying numerical techniques to accurately compute and robustly extend the libration point orbits (LPOs). A new methodology is proposed exploiting the hyperbolic dynamics of the collinear libration points. Numerical tools are developed to facilitate the efficient computation process, which are applicable to realistic force models and inherently parallelizable. Extensive numerical explorations in the Earth–Moon system are carried out, revealing the delicate structures of nested island chains and bounded chaotic motions on the center manifold. Numerical results confirm that the proposed approach can handle the computations of various types of LPOs in a unified manner and is operational over a wide range of energy levels. LPOs obtained with this approach offer a broad range of future mission possibilities in an extended vicinity of the collinear libration points.

Periodic orbits around the collinear libration points

Periodic orbits around the collinear libration points

Optimization of low-thrust transfers to libration point orbits

Purpose – The purpose of this paper is to develop a superconvergent trajectory optimization method for the design of low-thrust transfer trajectories from parking orbit to libration point orbits near the collinear libration points. Design/methodology/approach – The optimization method is developed by merging the concept of Lyapunov feedback control law with the manifold dynamics. First, the whole transfer trajectory is divided into two segments, raising segment and coast segment. Then, the trajectories of each segment are described in different coordinate frames, and designed with Lyapunov feedback control law and the manifold dynamics, respectively. The Poincare section is used as an effective tool to search the compatible patching point between these two segments on the manifold. Finally, the transfer trajectory is optimized using sequential quadratic programming. Findings – In the numerical simulation, the proposed optimization method does not have any convergence problem. It is a fast and effective me...

Birth of periodic and artificial halo orbits in the restricted three-body problem

We investigate the bifurcation of artificial halo orbits from the Lyapunov planar family of periodic orbits around the collinear libration points of the circular, spatial, restricted three-body problem. Beside the gravitational forces, our model includes also the effect of the Solar Radiation Pressure (SRP) and this motivates the use of the term ‘artificial’ halo orbits. Indeed, as a typical problem, one may think of a solar sail, which is characterized by a performance parameter measuring the strength of the effect of the SRP on the spacecraft.To settle the model, we determine the position of the collinear points as a function of the mass and performance parameters and the energy values at which Hill׳s surfaces allow for transit orbits between the primaries. To analyze the dynamics we use a consolidated procedure which consists in the computation of a resonant normal form, allowing the reduction to the center manifold and providing an integrable approximation of the Hamiltonian dynamical system. Finally, we compute the bifurcation thresholds of the 1:1 resonant periodic orbit families (which have the standard ‘halo’ orbits as their first member) as a function of the performance and mass parameters.The results show that SRP is indeed a relevant ingredient for new dynamical features and must definitely be considered when planning a mission of a solar sail with trajectories in the neighborhoods of collinear points.

Combined effects of Finite Straight Segment and Oblateness on the Libration Points in the Restricted-Three Body Problem

This paper deals with the existence and stability of the libration points in the restricted-three body problem when the smaller primary is a finite straight segment and bigger primary is an oblate spheroid. We have observed that five libration points do exist in this problem, out of which three are collinear and two are non-collinear with the primaries. The collinear libration points are unstable for all values of mass parameter /u and the non-collinear libration points are stable if jU < /Jc, where jU is a critical value of mass parameter /u. The range of stability depends length of the finite straight segment and oblateness factor.

Stationary solutions in a model three-body problem

Two visco-elastic bodies (deformable spheres) are considered which interact with each other and move in quasi-circular orbits in the attractive force field of a fixed centre – a heavy point mass. Their axes of rotation are perpendicular to their orbital plane. Stationary solutions of the evolutionary equations of motion are found. In one particular case, they extend solutions of the restricted circular three-body problem corresponding to two collinear libration points. All three bodies are located along a straight line. This implies synchronization of motion of the barycentre of the two visco-elastic bodies relative to the attracting centre with their orbital motion relative to the barycentre in a 1:1 resonance. The rotation of the two bodies relative to their own centres of mass takes place in such a way that the bodies “view” the attracting centre and each other from the same side, i.e., they are synchronized in a 1:1 resonance with their orbital motion. Instability of stationary solutions is analytically proven.

A note on transfers from LEOs to GEOs visiting libration points of the Sun–Earth CRTBP

The objective of this work is to explore the use the invariant manifold dynamics of the Circular Restricted Three Body Problem to construct transfer trajectories from a Low Earth Orbit (LEO) to a Geostationary Earth Orbit (GEO). The underlying idea is to determine orbits that shadow the stable and unstable manifolds of the central manifold of the collinear libration points of the Sun–Earth system, and connect both kinds of orbits around the Earth. The resulting transfer orbits have two legs connected at the libration point region. After a first maneuver performed in the LEO (Δv≈3.1km/s) the spacecraft reaches the neighborhood of the equilibrium point, L1 or L2, driven by the stable manifold of the central manifold of the point. With a small impulse (Δv≈100m/s), the spacecraft is transferred back to a GEO shadowing an orbit of the unstable manifold of a libration point orbit. Once the GEO is reached, an insertion manoeuvre must be done (Δv≈1.2km/s). In the paper the total Δv, together with the total time of flight, are computed considering initial LEOs with different altitudes and several inclinations and final GEOs with inclinations of ±5° around the equator.