Collinear Lagrangian Points Research Articles

Orbital transfers in an asteroid system considering the solar radiation pressure

The present paper studies the effects of the radiation pressure in the trajectories of a spacecraft in transfers between the collinear Lagrange points of a double asteroid system. The system considered is this paper is formed by the double asteroid 1996FG3 and the maneuvers are always assumed to be bi-impulsive. In a system formed by asteroids, the solar radiation pressure has a significant influence in the transfers paths. This occurs because the gravitational forces in these systems are smaller if compared with systems formed by larger bodies. Solutions with lower and higher fuel consumption can be found by adding the solar radiation pressure. The radiation pressure was not used as a control but its effects over the transfers were measured. For a small system of primaries such as an asteroid system, it is very important to take into account this force to make sure that the spacecraft will reach the desired point.

Distant retrograde orbits and the asteroid hazard

Distant Retrograde Orbits (DROs) gained a novel wave of fame in space mission design because of their numerous advantages within the framework of the US plans for bringing a large asteroid sample in the vicinity of the Earth as the next target for human exploration. DROs are stable solutions of the three-body problem that can be used whenever an object, whether of natural or artificial nature, is required to remain in the neighborhood of a celestial body without being gravitationally captured by it. As such, they represent an alternative option to Halo orbits around the collinear Lagrangian points L1 and L2. Also known under other names (e.g., quasi-satellite orbits, cis-lunar orbits, family-f orbits) these orbital configurations found interesting applications in several mission profiles, like that of a spacecraft orbiting around the small irregularly shaped satellite of Mars Phobos or the large Jovian moon Europa. In this paper a basic explanation of the DRO dynamics is presented in order to clarify some geometrical properties that characterize them. Their accessibility is then discussed from the point of view of mission analysis under different assumptions. Finally, their relevance within the framework of the present asteroid hazard protection programs is shown, stressing the significant increase in warning time they would provide in the prediction of impactors coming from the direction of the Sun.

Preliminary results on the dynamics of large and flexible space structures in Halo orbits

The global exploration roadmap suggests, among other ambitious future space programmes, a possible manned outpost in lunar vicinity, to support surface operations and further astronaut training for longer and deeper space missions and transfers. In particular, a Lagrangian point orbit location - in the Earth- Moon system - is suggested for a manned cis-lunar infrastructure; proposal which opens an interesting field of study from the astrodynamics perspective. Literature offers a wide set of scientific research done on orbital dynamics under the Three-Body Problem modelling approach, while less of it includes the attitude dynamics modelling as well. However, whenever a large space structure (ISS-like) is considered, not only the coupled orbit-attitude dynamics should be modelled to run more accurate analyses, but the structural flexibility should be included too. The paper, starting from the well-known Circular Restricted Three-Body Problem formulation, presents some preliminary results obtained by adding a coupled orbit-attitude dynamical model and the effects due to the large structure flexibility. In addition, the most relevant perturbing phenomena, such as the Solar Radiation Pressure (SRP) and the fourth-body (Sun) gravity, are included in the model as well. A multi-body approach has been preferred to represent possible configurations of the large cis-lunar infrastructure: interconnected simple structural elements - such as beams, rods or lumped masses linked by springs - build up the space segment. To better investigate the relevance of the flexibility effects, the lumped parameters approach is compared with a distributed parameters semi-analytical technique. A sensitivity analysis of system dynamics, with respect to different configurations and mechanical properties of the extended structure, is also presented, in order to highlight drivers for the lunar outpost design. Furthermore, a case study for a large and flexible space structure in Halo orbits around one of the Earth-Moon collinear Lagrangian points, L1 or L2, is discussed to point out some relevant outcomes for the potential implementation of such a mission.

The effects of oblateness and solar radiation pressure on halo orbits in the photogravitational Sun-Earth system

In this paper, we construct a third-order analytic approximate solution using the Lindstedt-Poincare method in the photogravitational circular restricted three body problem considering the Sun as a radiating source and the Earth as an oblate spheroid for computing halo orbits around the collinear Lagrangian points L1 and L2. Further, the well-known differential correction and continuation schemes are used to compute halo orbits and their families numerically. The effects of solar radiation pressure and oblateness on the orbit are studied around both Lagrangian points. From the study, it is noticed that time period of the halo orbit increases around L1 and L2 accounting oblateness of the Earth and solar radiation pressure of the Sun. It is also found that stability of halo orbits is a weak function of the out-of-plane amplitude and mass reduction factor.

Testing the strong equivalence principle with spacecraft ranging towards the nearby Lagrangian points

General relativity is supported by great experimental evidence. Yet there is a lot of interest in precisely setting its limits with on going and future experiments. A question to answer is about the validity of the Strong Equivalence Principle. Ground experiments and Lunar Laser Ranging have provided the best upper limit on the Nordtvedt parameter $\sigma[\eta]=4.4\times 10^{-4}$. With the future planetary mission BepiColombo, this parameter will be further improved by at least an order of magnitude. In this paper we envisage yet another possible testing environment with spacecraft ranging towards the nearby Sun-Earth collinear Lagrangian points. Neglecting errors in planetary masses and ephemerides, we forecast $\sigma[\eta]=6.4\,(2.0)\times10^{-4}$ (5 yr integration time) via ranging towards $L_1$ in a realistic (optimistic) scenario depending on current (future) range capabilities and knowledge of the Earth's ephemerides. A combined measurement, $L_1$+$L_2$, gives instead $4.8\,(1.7)\times10^{-4}$. In the optimistic scenario a single measurement of one year would be enough to reach $\approx3\times10^{-4}$. All figures are comparable with Lunar Laser Ranging, but worse than BepiColombo. Performances could be much improved if data were integrated over time and over the number of satellites flying around either of the two Lagrangian points. We point out that some systematics (gravitational perturbations of other planets or figure effects) are much more in control compared to other experiments. We do not advocate a specific mission to constrain the Strong Equivalence Principle, but we do suggest analysing ranging data of present and future spacecrafts flying around $L_1$/$L_2$ (one key mission is, for instance, LISA Pathfinder). This spacecraft ranging would be a new and complementary probe to constrain the Strong Equivalence Principle in space.

Control of asteroid retrieval trajectories to libration point orbits

The fascinating idea of shepherding asteroids for science and resource utilization is being considered as a credible concept in a not too distant future. Past studies identified asteroids which could be efficiently injected into manifolds which wind onto periodic orbits around collinear Lagrangian points of the Sun-Earth system. However, the trajectories are unstable, and errors in the capture maneuver would lead to complete mission failure, with potential danger of collision with the Earth, if uncontrolled. This paper investigates the controllability of some asteroids along the transfers and the periodic orbits, assuming the use of a solar-electric low-thrust system shepherding the asteroid. Firstly, an analytical approach is introduced to estimate the stability of the trajectories from a dynamical point of view; then, a numerical control scheme based on a linear quadratic regulator is proposed, where the gains are optimized for each trajectory through a genetic algorithm. A stochastic simulation with a Monte Carlo approach is used to account for different perturbed initial conditions and the epistemic uncertainty on the asteroid mass. Results show that only a small subset of the considered combinations of trajectories/asteroids are reliably controllable, and therefore controllability must be taken into account in the selection of potential targets.

Solar Sail Equilibrium Orbits in the Circular Restricted Three-Body Problem with Oblateness

We investigate solar sail in the circular restricted three-body problem, where the larger primary is a source of radiation and the smaller primary is an oblate spheroid in the system. Firstly, the differential equations of motion for solar sail in the system combined effects of radiation and oblateness of celestial bodies are built. Then the positions of the solar sail collinear Lagrange points are calculated as mass ratio or oblateness changes in certain extent. Linearization near the collinear equilibria of the system is applied. A linear quadratic regulator is used to stabilize the nonlinear system. Three different cases of solar sail equilibrium orbits are studied each with different choices for the weight matrices. The simulations reveal that solar sail equilibrium orbits can be stable under active control by changing three angles, incident angle, cone angle and clock angle of the solar sail.

Fugitive stars in active galaxies

We investigate in detail the escape dynamics in an analytical gravitational model which describes the motion of stars in a quasar galaxy with a disk and a massive nucleus. We conduct a thorough numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. In order to distinguish safely and with certainty between ordered and chaotic motion we apply the Smaller ALingment Index (SALI) method. It is of particular interest to locate the escape basins through the openings around the collinear Lagrangian points $L_1$ and $L_2$ and relate them with the corresponding spatial distribution of the escape times of the orbits. Our exploration takes place both in the configuration $(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape process as well as the overall orbital properties of the galactic system. Our numerical analysis reveals the strong dependence of the properties of the considered escape basins with the total orbital energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. We hope our outcomes to be useful for a further understanding of the escape mechanism in active galaxy models.

Escape dynamics in a binary system of interacting galaxies

The escape dynamics in an analytical gravitational model which describes the motion of stars in a binary system of interacting dwarf spheroidal galaxies is investigated in detail. We conduct a numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. In order to distinguish safely and with certainty between ordered and chaotic motion, we apply the Smaller ALingment Index (SALI) method. It is of particular interest to locate the escape basins through the openings around the collinear Lagrangian points L1 and L2 and relate them with the corresponding spatial distribution of the escape times of the orbits. Our exploration takes place both in the configuration (x, y) and in the phase (x,x˙) space in order to elucidate the escape process as well as the overall orbital properties of the galactic system. Our numerical analysis reveals the strong dependence of the properties of the considered escape basins with the total orbital energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. It was also observed that for energy levels close to the critical escape energy the escape rates of orbits are large, while for much higher values of energy most of the orbits have low escape periods or they escape immediately to infinity. We hope our outcomes to be useful for a further understanding of the escape mechanism in binary galaxy models.

Escape dynamics and fractal basin boundaries in Seyfert galaxies

The escape dynamics in a simple analytical gravitational model which describes the motion of stars in a Seyfert galaxy is investigated in detail. We conduct a thorough numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. In order to distinguish safely and with certainty between ordered and chaotic motion, we apply the Smaller ALingment Index (SALI) method. It is of particular interest to locate the escape basins through the openings around the collinear Lagrangian points $L_1$ and $L_2$ and relate them with the corresponding spatial distribution of the escape times of the orbits. Our exploration takes place both in the physical $(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape process as well as the overall orbital properties of the galactic system. Our numerical analysis reveals the strong dependence of the properties of the considered escape basins with the total orbital energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. It was also observed, that for energy levels close to the critical escape energy the escape rates of orbits are large, while for much higher values of energy most of the orbits have low escape periods or they escape immediately to infinity. We also present evidence obtained through numerical simulations that our model can describe the formation and the evolution of the observed spiral structure in Seyfert galaxies. We hope our outcomes to be useful for a further understanding of the escape mechanism in galaxies with active nuclei.

Lissajous and Halo Orbits in the Restricted Three-Body Problem

We study the dynamics near the collinear Lagrangian points of the spatial, circular, restricted three-body problem. Following a standard procedure, we reduce the system to the center manifold and we analyze the Lissajous orbits as well as the halo orbits, the latter ones arising from bifurcations of the planar Lyapunov family of periodic orbits. To obtain the Lissajous orbits, we perform a classical perturbation theory and we provide a formal approximate solution under suitable non-degeneracy and non-resonance conditions. As for the halo orbits, we construct a normal form adapted to the synchronous resonance: introducing a detuning, measuring the displacement from the resonance, and expanding the energy in series of the detuning, we are able to evaluate the energy level at which the bifurcation takes place. Except for a particular case, the analytical values obtained after a second order resonant perturbation theory are in very good agreement (in some cases up to the fourth decimal digit) with the numerical values found in the literature.

Qualitative and analytical results of the bifurcation thresholds to halo orbits

We study the dynamics in the neighborhood of the collinear Lagrangian points in the spatial, circular, restricted three--body problem. We consider the case in which one of the primaries is a radiating body and the other is oblate (although the latter is a minor effect). Beside having an intrinsic mathematical interest, this model is particularly suited for the description of a mission of a spacecraft (e.g., a solar sail) to an asteroid. The aim of our study is to investigate the occurrence of bifurcations to halo orbits, which take place as the energy level is varied. The estimate of the bifurcation thresholds is performed by analytical and numerical methods: we find a remarkable agreement between the two approaches. As a side result, we also evaluate the influence of the different parameters, most notably the solar radiation pressure coefficient, on the dynamical behavior of the model. To perform the analytical and numerical computations, we start by implementing a center manifold reduction. Next, we estimate the bifurcation values using qualitative techniques (e.g. Poincar\'e surfaces, frequency analysis, FLIs). Concerning the analytical approach, following \cite{CPS} we implement a resonant normal form, we transform to suitable action-angle variables and we introduce a detuning parameter measuring the displacement from the synchronous resonance. The bifurcation thresholds are then determined as series expansions in the detuning. Three concrete examples are considered and we find in all cases a very good agreement between the analytical and numerical results.

Asteroid Observation and Landing Trajectories Using Invariant Manifolds

In this paper, a study of the equilibrium points of a rotating nonspherical asteroid is performed with special emphasis on the equilibria aligned with the longest axis of the body. These equilibrium points have the same spectral behavior as the collinear Lagrange points of the restricted three-body problem, saddle centers, and therefore unstable and stable invariant manifolds can be computed. The invariant manifolds of the equilibrium point or periodic orbits around it, which are fuel-free trajectories, can approach the surface of the asteroid, orbit around it for different amounts of time, and then impact on it. This paper studies the dependence of the existence of fuel-free trajectories to the surface of the asteroid from the equilibrium point on the shape and rotation rate of the body. A possible maneuver to orbit the asteroid to observe it and later achieve vertical landing is proposed. The theory developed is then applied to asteroid 4660 Nereus, for which an approach, observation phase, and landing maneuver are designed.

Orbit control strategy for Lagrange point orbits based on an analytical method

The periodic or quasi-periodic orbits around collinear Lagrange points present many properties that are advantageous for space missions. These Lagrange point orbits are exponentially unstable. On the basis of an analytical method, an orbit control strategy that is designed to eliminate the dominant unstable components of Lagrange point orbits is developed. The proposed strategy enables the derivation of the analytical expression of nonlinear control force. The control parameter of this strategy can be arbitrarily selected provided that the parameter is considerably lower than the negative eigenvalue of motion equations, and that the energy required keeps the same order of magnitude. The periodic or quasi-periodic orbit of controlled equations remains near the periodic or quasi-periodic orbit of uncontrolled equations.

Analytic approach on geometric structure of invariant manifolds of the collinear Lagrange points

An analytical method is proposed to find geometric structures of stable, unstable and center manifolds of the collinear Lagrange points. In a transformed space, where the linearized equations are in Jordan canonical form, these invariant manifolds can be approximated arbitrarily closely as Taylor series around Lagrange points. These invariant manifolds are represented by algebraic equations containing the state variables only without the help of time. Thus the so-called geometric structure of these invariant manifolds is obtained. The stable, unstable and center manifolds are tangent to the stable, unstable and center eigenspaces, respectively. As an example of applicability, the invariant manifolds of L1 point of the Sun-Earth system are considered. The stable and unstable manifolds are symmetric about the line from the Sun to the Earth, and they both reach near the Earth, so that the low energy transfer trajectory can be found based on the stable and unstable manifolds. The periodic or quasi-periodic orbits, which are chosen as nominal arrival orbits, can be obtained based on the center manifold.

Dynamic stabilization of L2 periodic orbits using attitude-orbit coupling effects

Numerical explorations show how the known periodic solutions of the Hill problem are modified in the case of the attitude-orbit coupling that may occur for large satellite structures. We focus on the case in which the elongation is the dominant satellite?s characteristic and find that a rotating structure may remain with its largest dimension in a plane parallel to the plane of the primaries. In this case, the effect produced by the non-negligible physical dimension is dynamically equivalent to the perturbation produced by an oblate central body on a masspoint satellite. Based on this, it is demonstrated that the attitude-orbital coupling of a long enough body may change the dynamical characteristics of a periodic orbit about the collinear Lagrangian points.

Periodical Motion of a Tethered Satellite System around Collinear Libration Points

This paper studies the motion of a rotational tethered satellite system flying around collinear Lagrangian points of the Earth-Moon system. Equations of spatial motion of the mass center and the tether libration are deduced, but only the planar motion with constant rotation rate is discussed. First, the equation of the mass center is simplified and solved analytically for the scenario where the system is moving in the neighborhood of the libration point. The analytical solution indicates that both a bounded solution and periodical solution exist. A fully nonlinear dynamics analysis is then performed. A numerical method combining continuation and piecewise differential corrections is applied to obtain the periodical solution. Tether tension is also investigated. Simulation results validate the effectiveness of the proposed numerical method and the existence of the periodical orbits. The influence of parameters, such as spin rate, mass ratio of end satellites, and tether length are analyzed.

Propellantless Stationkeeping at Enceladus via the Electromagnetic Lorentz Force

An introduction to the dynamics of an electrostatically charged spacecraft in two- and three-body regimes is presented, with particular attention to a promising application at Enceladus. Equilibrium solutions to the equations of motion are found, and the stability of the orbits assessed. The perturbative Lorentz force, produced by interaction of the spacecraft with a planetary magnetic field, creates trajectories that are not possible with Newtonian gravity alone. A heuristic, analytical control law with single-variable scalar feedback allows a charged spacecraft to remain near the collinear Lagrange points on either side of Enceladus. The mission feasibility depends on the charge requirement, which is primarily affected by the navigational accuracy. For example, an insertion error into the cis-Enceladus Lagrange point of 10 km in position and 1 m/s in velocity requires a specific charge of approximately 0.05 C/kg. Other significant gravitating bodies in the Saturnian system do not have a substantive effect on the charge requirements or on the stability. Given a sufficiently accurate insertion, a charged spacecraft could maintain station near Enceladus without propellant.

Launch windows for libration point missions

The four upcoming ESA missions to the collinear Lagrange points in the sun–Earth system, Herschel and Planck, GAIA, and LISA Pathfinder, will be launched from French Guiana by different Rockets, namely Ariane 5, Soyuz, and VEGA. Herschel and Planck will be launched together in 2008 on an Ariane 5. The launcher injects Herschel directly into the stable manifold of a large amplitude quasi-Halo orbit at L 2 . Departing from this launch orbit, Planck performs a three manoeuvre transfer to a Lissajous orbit, on which the sun–spacecraft–Earth angle remains below 15 ∘ . The daily and seasonal launch window for Herschel/Planck will be constrained by sun aspect angle conditions, by eclipse conditions, and by the propellant allocation on the Planck satellite. GAIA will be launched by Soyuz/Fregat end 2011. The target orbit at L 2 is similar to that of Planck, with a constrained sun–spacecraft–Earth angle. Initially, to reach a full year launch window, a launch scenario was preferred using a nearly circular parking orbit at 19 ∘ inclination, before igniting the Fregat upper stage to enter into the transfer. Recently alternative launch scenarios have been introduced, to guarantee a controlled re-entry of the Soyuz third stage, but leading to a reduction of the launch window. LISA Pathfinder will be delivered 2010 by the VEGA launcher to a low inclination elliptical orbit, with 1624 km apogee height. From there an apogee raising manoeuvre sequence, using a propulsion module connected to the spacecraft, reaches L 1 transfer conditions. Launch window calculations for these projects combine the choice of lift-off time and launch orbit conditions, with the selection of the target orbit at the Lagrange points and the minimisation of propellant to reach that orbit. For Herschel and LISA Pathfinder the target orbits have been chosen such that a transfer without deterministic manoeuvres is possible. Planck and GAIA both use a multiple manoeuvre fast transfer strategy reaching the Lissajous orbit at L 2 within 50 days.

Planar dynamics of variable length multi-tethered spacecraft near collinear Lagrangian points

This paper examines the planar dynamics of a wheel-and-spoke configured multi-spacecraft system, connected together by variable length tethers, near the second Sun–Earth Lagrangian point. The closed form solutions of the system under some simple tether length functions are determined and numerical results for the tether pitch librations under more complex tether length functions are obtained, along with the control effort required to maintain the desired tether librations.