Collinear Libration Points Research Articles

Prospects for the use of space robots in the neighbourhood of the libration points

This paper is devoted to the navigation tasks for space robots operating in the neighborhood of unstable libration points of the Earth-Sun system in the near-Earth space. The motion of robot in space is examined under a circular restricted three-body problem. The possibility of a space robot maneuvering in the near-Earth space with return to the neighborhood of a collinear libration point and stabilizing in the neighborhood of this point are shown.

Existence and stability of the non-collinear libration points in the restricted three body problem when both the primaries are ellipsoid

This paper deals with the existence and stability of the non-collinear libration points in the restricted three-body problem when both the primaries are ellipsoid with equal mass and identical in shape. We have determined the equations of motion of the infinitesimal mass which involves elliptic integrals and then we have investigated the existence and stability of the non-collinear libration points. This is observed that the non-collinear libration points exist only in the interval 52∘<φ<90∘ and form an isosceles triangle with the primaries. Further we observed that the non collinear libration points are unstable in 52∘<φ<90∘.

On motion around the collinear libration points in the elliptic R3BP with a bigger triaxial primary

We investigate the motion of a test particle in the vicinity of a binary made of a triaxial primary and a spherical companion moving along elliptic orbits about their common barycenter in the neighborhood of collinear libration points. Their positions and stability are found to be affected by the triaxiality of the bigger primary, and by the semi-major axis and the eccentricity of the binary’s orbits as well. The analytic results obtained are applied to binary neutron stars consisting of a bigger triaxial primary and a spherical companion. A numerical analysis shows that the positions of the collinear points of PSR J1518+4904, PSR B1534+12, PSR B1914+16, and PSR B2127+11c are affected by the triaxiality of the bigger primary, and by the eccentricity of the orbits. The stability behavior however remains unchanged: the collinear points remain unstable in the Lyapunov sense.

High-order solutions of invariant manifolds associated with libration point orbits in the elliptic restricted three-body system

High-order analytical solutions of invariant manifolds, associated with Lissajous and halo orbits in the elliptic restricted three-body problem (ERTBP), are constructed in this paper. The equations of motion of ERTBP in the pulsating synodic coordinate system have five equilibrium points, and the three collinear libration points as well as the associated center manifolds are unstable. In our calculation, the general solutions of the invariant manifolds associated with Lissajous and halo orbits around collinear libration points are expressed as power series of five parameters: the orbital eccentricity, two amplitudes corresponding to the hyperbolic manifolds, and two amplitudes corresponding to the center manifolds. The analytical solutions up to arbitrary order are constructed by means of Lindstedt–Poincare method, and then the center and invariant manifolds, transit and non-transit trajectories in ERTBP are all parameterized. Since the circular restricted three-body problem (CRTBP) is a particular case of ERTBP when the eccentricity is zero, the general solutions constructed in this paper can be reduced to describe the dynamics around the collinear libration points in CRTBP naturally. In order to check the validity of the series expansions constructed, the practical convergence of the series expansions up to different orders is studied.

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

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

Restricted three-body problem when one of the primaries is an ellipsoid

This paper deals with the existence and the stability of the libration points in the restricted three-body problem when the smaller primary is an ellipsoid. We have determined the equations of motion of the infinitesimal mass which involves elliptic integrals and then we have investigated the collinear and non collinear libration points and their stability. This is observed that there exist five collinear libration points and the non collinear libration points are lying on the arc of the unit circle whose centre is the bigger primary. Further observed that the libration points either collinear or non-collinear all are unstable.

Trajectory design strategies applied to temporary comet capture including Poincaré maps and invariant manifolds

Temporary satellite capture (TSC) of Jupiter-family comets has been a focus of investigation within the astronomy community for decades. More recently, TSC has been approached from the perspective of dynamical systems theory, within the context of the circular restricted three-body problem (CR3BP). Thus, this problem serves as a testbed for exploring techniques that support trajectory design in similar dynamical regimes. In particular, an association between the invariant manifolds of libration point orbits and the paths of comets that experience TSC has been explored. In this investigation, TSC is further examined from the perspective of transit, that is, transition through the gateways associated with the collinear libration points, in the three-body problem. Periapsis Poincare maps, previously employed for trajectory design in several investigations, are used to deliver insight into the nature of transit trajectories for energy levels near those associated with several Jupiter-family comets. The evolution of transit trajectories with increasing energy is explored, and the existence of solutions with similar characteristics to the paths of comets P/1996 R2, 82P/Gehrels 3, and 147P/Kushida–Muramatsu is demonstrated within the context of the planar CR3BP using planar periapsis maps. During TSC, the path of comet 111P/Helin–Roman–Crockett is highly inclined with respect to Jupiter; the motion of this comet is examined relative to invariant manifolds in the spatial CR3BP. A method to display the information contained in higher-dimensional Poincare maps is also demonstrated, and is employed to locate a trajectory possessing the same qualitative characteristics as the path of 111P/Helin–Roman–Crockett.

Bifurcating families around collinear libration points

The planar and the vertical Lyapunov families are two basic periodic families around the collinear libration points. The stability curves of these two families are given first, and then periodic families bifurcating from them are explored in detail. Several properties of these bifurcating families are found. This study follows a series of the authors’ publications on periodic families around the libration points in the restricted three-body problem.

Low-energy Earth–Moon transfers involving manifolds through isomorphic mapping

Analysis and design of low-energy transfers to the Moon has been a subject of great interest for decades. Exterior and interior transfers, based on the transit through the regions where the collinear libration points are located, have been studied for a long time and some space missions have already taken advantage of the results of these studies. This paper is concerned with a geometrical approach for low-energy Earth-to-Moon mission analysis, based on isomorphic mapping. The isomorphic mapping of trajectories allows a visual, intuitive representation of periodic orbits and of the related invariant manifolds, which correspond to tubes that emanate from the curve associated with the periodic orbit. Two types of Earth-to-Moon missions are considered. The first mission is composed of the following arcs: (i) transfer trajectory from a circular low Earth orbit to the stable invariant manifold associated with the Lyapunov orbit at L1 (corresponding to a specified energy level) and (ii) transfer trajectory along the unstable manifold associated with the Lyapunov orbit at L1, with final injection in a periodic orbit around the Moon. The second mission is composed of the following arcs: (i) transfer trajectory from a circular low Earth orbit to the stable invariant manifold associated with the Lyapunov orbit at L1 (corresponding to a specified energy level) and (ii) transfer trajectory along the unstable manifold associated with the Lyapunov orbit at L1, with final injection in a capture (non-periodic) orbit around the Moon. In both cases three velocity impulses are needed to perform the transfer: the first at an unknown initial point along the low Earth orbit, the second at injection on the stable manifold, the third at injection in the final (periodic or capture) orbit. The final goal is in finding the optimization parameters, which are represented by the locations, directions, and magnitudes of the velocity impulses such that the overall delta-v of the transfer is minimized. This work proves how isomorphic mapping (in two distinct forms) can be profitably employed to optimize such transfers, by determining in a geometrical fashion the desired optimization parameters that minimize the delta-v budget required to perform the transfer.

Control of spacecraft groupings near sun-earth collinear libration points by means of solar sails

The adjustment and maintenance of spacecraft configurations by means of solar sails is considered.

On the control of linear time-varying systems of a special form

Some classes of linear time-varying control systems of the first and second order that can be reduced to a time-invariant system by nonsingular transformation of the same order as the original system are considered. A description of the classes of systems that admit such a reduction by extending the space of states is given. By way of example, the problem of a spacecraft control in the vicinity of the collinear libration point of a planar restricted circular three-body problem is discussed.

On two kinds of intermediate orbits for asteroid explorations

In the paper, two kinds of intermediate orbits for asteroid explorations are proposed. One is around the collinear libration points of the Sun-asteroid restricted three-body problem. The other is around the asteroid itself. The first kind of intermediate orbit is applicable to asteroids with known masses, while the second is suitable for asteroids with unknown or negligible masses. Analytical solutions of these two intermediate orbits in the simplified models are introduced first, and then numerical algorithms are used to refine them to obtain the true orbits in the real force model. At last, the problem of station-keeping is addressed. The linear optimal feedback control law is used, and numerical simulations are made to both kinds of intermediate orbits. The results show that both kinds of orbits are feasible. The cost is reasonable and mainly depends on the initial insertion error.

Natural Regions Near the Collinear Libration Points Ideal for Space Observations with Large Formations

This investigation explores regions near libration points that might prove suitable for space observations with large formations. Recent analyses have considered occulters located at relatively large distances from the telescope near the L 2 Sun-Earth libration point for detection of exoplanets. During the science mode, the telescope-occulter distance, as well as the pointing direction toward the star, are typically fixed. Quasi-periodic Lissajous trajectories are employed as a tool to determine regions near the telescope orbit where the large formation can be maintained. By placing the occulter in these locations, the control required to maintain the line-of-sight is reduced.

Ballistic Deployment of Science Packages on Binary Asteroids

A simple, adaptable, and robust method for ballistic landings on the surface of a binary asteroid system is presented. The desire is to deploy a lander, from a mother spacecraft passing in the vicinity of one of the collinear libration point of the system, on an unpowered trajectory, meaning that active guidance is not needed. From the concepts of the circular restricted three-body problem, suitably adapted for the objective, the features that make such a strategy feasible for any asteroid binary system are detailed. Proving that this strategy is effective for a wide range of actual binary systems, the requirements on the deployment are found to be very robust to operational accuracy.

Autonomous optical navigation for orbits around Earth–Moon collinear libration points

The analysis of optical navigation in an Earth–Moon libration point orbit is examined. Missions to libration points have been winning momentum during the last decades. Its unique characteristics make it suitable for a number of operational and scientific goals. Literature aimed to study dynamics, guidance and control of unstable orbits around collinear libration points is vast. In particular, several papers deal with the optimisation of the Δv budget associated to the station-keeping of these orbits. One of the results obtained in literature establishes the critical character of the Moon–Earth system in this aspect. The reason for this behaviour is twofold: high Δv cost and short optimal manoeuvre spacing. Optical autonomous navigation can address the issue of allowing a more flexible manoeuvre design. This technology has been selected to overcome similar difficulties in other critical scenarios. This paper analyses in detail this solution. A whole GNC system is defined to meet the requirements imposed by the unstable dynamic environment. Finally, a real simulation of a spacecraft following a halo orbit of the L2 Moon–Earth system is carried out to assess the actual capabilities of the optical navigation in this scenario.

Periodic orbits in the concentric circular restricted four-body problem and their invariant manifolds

We give numerical calculations of periodic orbits in the planar concentric restricted four-body problem. It is assumed that the motion of a massless body is governed by three primaries m1, m2 and m3. We suppose that m1≫m2,m3 and that, in an m1 centered inertial reference frame, m2 and m3 move in different circles about m1 and m1 is fixed. Although the motion of the primaries m1,m2,m3 does not satisfy Newton’s equations of motion, this approximation is a good to model, for instance, the Jupiter–Europa–Ganymede–spacecraft system. We compute periodic orbits in both the m1–m2 and m1–m3 rotating frames. Periodic orbits that orbit around one of the primaries are found. Using a method that is based on the well-known Laplace resonance we also find unstable periodic orbits about the collinear libration points near m2 and m3. Since the periodic orbits near the collinear libration points are unstable they have stable/unstable manifolds, which we compute. We observe a lack of intersection of the stable and unstable manifolds of different periodic orbits.

Disturbance compensating control of orbit radially aligned two-craft Coulomb formation

This paper investigates the orbit radial stabilization of a two-craft virtual Coulomb structure about circular orbits and at Earth–Moon libration points. A generic Lyapunov feedback controller is designed for asymptotically stabilizing an orbit radial configuration about circular orbits and collinear libration points. The new feedback controller at the libration points is provided as a generic control law in which circular Earth orbit control form a special case. This control law can withstand differential solar perturbation effects on the two-craft formation. Electrostatic Coulomb forces acting in the longitudinal direction control the relative distance between the two satellites and inertial electric propulsion thrusting acting in the transverse directions control the in-plane and out-of-plane attitude motions. The electrostatic virtual tether between the two craft is capable of both tensile and compressive forces. Using the Lyapunov’s second method the feedback control law guarantees closed loop stability. Numerical simulations using the non-linear control law are presented for circular orbits and at an Earth–Moon collinear libration point.

Computer Assisted Existence Proofs of Lyapunov Orbits at $L_2$ and Transversal Intersections of Invariant Manifolds in the Jupiter--Sun PCR3BP

We present a computer assisted proof of existence of a family of Lyapunov orbits which stretches from $L_2$ (the collinear libration point between the primaries) up to half the distance to the smaller primary in the Jupiter--Sun planar circular restricted three body problem. We then focus on a small family of Lyapunov orbits with energies close to comet Oterma and show that their associated invariant manifolds intersect transversally. Our computer assisted proof provides explicit bounds on the location and on the angle of intersection.

Stability of the collinear libration points of the photogravitational three-body problem with an internal fourth order resonance

The problem of the stability of the collinear libration points of the photogravitational three-body problem is considered. It is shown that these points are stable in a Lyapunov sense in the case of a fourth-order resonance.

Hamiltonian normal forms in the neighbourhoods of collinear libration points in the restricted circular three body problem

The fourth-degree Hamiltonian normal forms in the neighbourhoods of the collinear libration points in the restricted plane circular three-body problem are found using an invariant normalization algorithm. Exact relations between the coefficients of the normal forms and the ratio of the masses of the heavy bodies are obtained. The subspaces of the phase variables, the initial values of which determine the periodic trajectories in the neighbourhoods of the libration points, are found.