Collinear Lagrangian Points Research Articles

Exploring the collinear Lagrangian points of exoplanet systems with P–R drag and oblateness

Exploring the collinear Lagrangian points of exoplanet systems with P–R drag and oblateness

Analysis of the dynamics of a spacecraft in the vicinity of an asteroid binary system with equal masses

In this work, we performed a dynamical analysis of a spacecraft around a nearly equal-mass binary near-Earth asteroid with application to the asteroid 2017 YE5, which is also a possible dormant Jupiter-family comet. Thus, we investigated the motion of a particle around this binary system using the circular restricted three-body problem. We calculated the locations of the Lagrangian points of the system and their Jacobi constant. Through numerical simulations, using the Poincaré Surface of Sections, it was possible to find several prograde and retrograde periodic orbits around each binary system's primary, some exhibiting significantly-sized higher-order behavior. We also calculated the stability of these orbits. After finding the periodic orbits, we investigated the influence of solar radiation pressure on these orbits. For this analysis, we considered that the area-to-mass ratio equals 0.01 and 0.1. We also performed a spacecraft lifetime analysis considering the physical and orbital characteristics of the 2017YE5 system and investigated the behavior of a spacecraft in the vicinity of this system. We analyzed direct and retrograde orbits for different values of Jacobi's constant. This study investigated orbits that survive for at least six months, not colliding or escaping the system during that time. We also analyze the initial conditions that cause the spacecraft to collide with M1 or M2, or escape from the system. In this work, we take into account the gravitational forces of the binary asteroid system and the solar radiation pressure (SRP). Finally, we calculated optimal bi-impulsive orbital maneuvers between the collinear Lagrangian points. We found a family of possible orbital transfers considering times of flight between 0.1 and 1 day.

Analysis of Resonant Periodic Orbits in the Framework of the Perturbed Restricted Three Bodies Problem

In this work, the perturbed equations of motion of the infinitesimal body are constructed in the framework of the circular restricted three-body problem when the main two bodies are oblate and radiating. Under the perturbations effects of the oblateness and the radiation pressure the positions of collinear Lagrange points are evaluated, the interior and exterior first-order resonant periodic orbits are also studied. In addition, the initial positions of the periodic orbits and the size of loops have been estimated under these effects. Thus, the characteristics of periodic orbits have been studied under the combine effects of two, three and four perturbations for all the possible combinations of the perturbed parameters. The different order of resonant periodic orbits have been also analysed under the effects of Jacobi constant, mass factor, order of resonance and number of loops.

Families of periodic orbits about Lagrangian points L1, L2 and L3 with continuation method

This study deals the impact of radiation pressure and albedo from the respective primaries in the context of regularized periodic motion of infinitesimal mass under the frame of restricted three body problem with the help of continuation method. The effect of these perturbing parameters on the orbital properties about the collinear Lagrangian points and their stability property are investigated. An expansion in the structure of Lyapunov orbits are found due to radiation pressure effect and albedo whereas crossing of stability bounds (±1) occurs in the continuation process of Lyapunov orbits indicating the presence of halo bifurcation. This occurrence of halo bifurcation leads to the origin of halo periodic orbits from Lyapunov periodic orbits at different values of radiation parameter ( q ) and albedo ( Q A ). From these bifurcating orbits, families of halo periodic orbits are determined using continuation in radiation parameter and albedo, respectively. Due to the existence of one of the stability index ( ν ), which is either greater than 1 or less than ​− ​1, periodic orbits become unstable. • Regularized restricted three body problem with radiation pressure and albedo are discussed. • Numerical computation of periodic orbits about collinear equilibrium points are discussed. • The parametric evolution of periodic orbit as the function of radiation and albedo parameter are presented. • Families of periodic orbits are found with the help of continuation in radiation and albedo parameter.

Geometry of transit orbits in the periodically-perturbed restricted three-body problem

In the circular restricted three-body problem, low energy transit orbits are revealed by linearizing the governing differential equations about the collinear Lagrange points. This procedure fails when time-periodic perturbations are considered, such as perturbation due to the sun (i.e., the bicircular problem) or orbital eccentricity of the primaries. For the case of a time-periodic perturbation, the Lagrange point is replaced by a periodic orbit, equivalently viewed as a hyperbolic-elliptic fixed point of a symplectic map (the stroboscopic Poincaré map). Transit and non-transit orbits can be identified in the discrete map about the fixed point, in analogy with the geometric construction of Conley and McGehee about the index-1 saddle equilibrium point in the continuous dynamical system. Furthermore, though the continuous time system does not conserve the Hamiltonian energy (which is time-varying), the linearized map locally conserves a time-independent effective Hamiltonian function. We demonstrate that the phase space geometry of transit and non-transit orbits is preserved in going from the unperturbed to a periodically-perturbed situation, which carries over to the full nonlinear equations.

Periodic orbit in the photo-gravitational restricted three body problem around the collinear Lagrangian points when more massive primary is an oblate spheroid and source of radiation

The circular Restricted Three Body Problem is considered with the more massive primary as an oblate spheroid and source of radiation. A new mean motion expression given by n2=1+6A is used in the present study, when the secular effect of the oblateness on the mean motion, argument of perigee and right ascension of the ascending node is considered. The locations of the collinear Lagrangian points are found. It is found to have some variation from the previous study conducted on the same because of the new mean motion that is considered in this study. The variations of the location of the Lagrangian points due to the unperturbed as well as the perturbed problem due to oblateness and radiation pressure are studied. A study on the eccentricity e and angular frequency s at L1, L2 and L3 is carried out. It is observed how the change in effect of oblateness and radiation pressure has affected the location, angular frequency and eccentricity at L3, though there are only small changes noticed in the case of L1 and L2. Â

Networks and Bifurcations of Eccentric Orbits in Exoplanetary Systems

A systematic study of families of planar symmetric periodic orbits of the elliptic restricted three-body problem is presented, in exoplanetary systems. We find families of periodic orbits that surround only one of the primaries (Satellite-Type), that are moving around both primaries (Planet-Type), and also moving about the collinear Lagrange points. The linear stability of every periodic orbit is calculated, and the families are interpreted through stability diagrams. We focus on quasi-satellite motions of test particles that are associated with the known family [Formula: see text] that consists of 1:1 resonant retrograde Satellite-Type orbits. Over the last years, quasi-satellite orbits are of special interest due to the many applications in the design of spacecraft missions around moons and asteroids. We find the critical simple (1:1 resonant) periodic orbits of the basic families of the circular problem from which we calculate new families of the elliptic problem. Additionally, families of the elliptic problem which bifurcate from the main family [Formula: see text], for various resonances, are also presented and discussed. Hundreds of critical orbits (bifurcation points), from which families of the elliptic problem of higher multiplicity emerge, are found and the corresponding resonances are identified.

Euler's three-body problem

In physics and astronomy, Euler's three-body problem is to solve for the motion of a body that is acted upon by the gravitational field of two other bodies. This problem is named after Leonhard Euler (1707-1783), who discussed it in memoirs published in the 1760s. In these publications, Euler found that the parameter that controls the relative distances among three collinear bodies is given by a quintic equation. Later on, in 1772, Lagrange dealt with the same problem, and demonstrated that for any three masses with circular orbits, there are two special constant-pattern solutions, one where the three bodies remain collinear, and the other where the bodies occupy the vertices of two equilateral triangles. Because of their importance, these five points became known as Lagrange points. The quintic equation found by Euler for the relative distances among the collinear bodies was also found later by Lagrange, and because of that, Euler has also been given credit for the discovery of the three collinear Lagrange points. A practical application of the collinear points for satellite location is also presented.

Comparison of material sources and customer locations for commercial space resource utilization

Space resource utilization (SRU) can be a catalyst for a growing space economy. Significant reductions in cost, launch mass and risk for human and robotic activities beyond Earth can be achieved. Commercial entities will have the opportunity to generate revenue through economic development, allowing for a decreased dependency on governmental agencies for the exploration of space. A wide variety of materials can be found on planetary and small solar system bodies, which are potentially of great use to customers in space. This paper conducts a comparison of the economic and commercial potential of SRU on the surface of the Moon, Mars, a representative near-Earth asteroid and a representative main-belt asteroid. In particular, mining of water will be considered which will be electrolysed into hydrogen and oxygen using solar power. Coupled economic and trajectory optimization then results in the minimum specific cost to deliver propellant (LOX/LH2) from these bodies to a number of customer locations. Customers on the surface of or in orbit around the Earth, Moon, and Mars are investigated, along with Psyche, a large metallic main-belt asteroid. Using a cost model that considers technology maturation to allow current aviation-like costs to be used for development and manufacturing, missions are optimized. Results show that for customers beyond low-Earth orbit, SRU should be used instead of launching water-derived resource directly from the Earth. In particular, near-Earth asteroids are shown to be in an attractive location for mining, processing and delivering a large quantity of LOX/LH2 at a low specific cost, for customers in the vicinity of the Earth and the Moon, including nearby collinear Lagrange points. Exceptions are customers on the surface of the Earth and the Moon, who, together with customers on the surface of Mars, are better off mining and processing on that same surface. A sensitivity analysis shows that, for all investigated customers except Psyche, there are near-Earth asteroids available which can be used to deliver water-derived resources for a lower specific cost than the most ideal main-belt asteroid. Only once these near-Earth asteroids have been depleted, mining main-belt asteroids and transporting resources back to a customer in the vicinity of the Earth is worth considering.

Computation of Periodic Orbits around L1 and L2 using PSO Technique

In this work, an optimization problem related to a non-linear function is tackled mathematically by the use of particle swarm optimization (PSO) method in the case of circular restricted three body problem (CRTBP). This method is applied to find the family of periodic orbits around the collinear Lagrangian points $${{L}_{1}}$$ and $${{L}_{2}}$$ in Sun–Earth system. To deal with the chosen like Sun–Earth-spacecraft system, which is constructed as CRTBP problem, the Lagrangian points associated with this system are located by the application of the Newton–Raphson method. Moreover, the initial conditions for periodic orbit of the dynamical system are determined with the same proposed method. The application of the PSO method to this problem is motivated by its higher numerical accuracy.

Minimum Fuel-Consumption Stabilization of a Spacecraft at the Lagrangian Points

We consider the motion of a spacecraft described by the differential equations of the three-body problem in the Earth-Moon system. The goal is to stabilize the spacecraft in the neighborhood of the collinear Lagrangian points (which are know to be unstable equilibria) via use of minimum fuel-consumption control. The adopted approach is based on l1-optimization of linearized and discretized equations with terminal conditions being the target Lagrangian point. Therefore, the problem reduces to a linear program, and its solution defines pulse controls for the original three-body equations. Upon reaching the desired neighborhood, the spacecraft performs control-free flight until its deviation from the Lagrangian point exceeds certain prespecified threshold. The correction is then applied repeatedly, so that the spacecraft is kept within a small neighborhood of the unstable equilibrium point.

Linking low- to high-energy dynamics of invariant manifolds, transit orbits, and singular collision orbits in the planar circular restricted three-body problem

The first part of the present paper reveals interplays among invariant manifolds emanating from planar Lyapunov orbits around the three collinear Lagrange points \(L_1, L_2\), and \(L_3\) for high energies. Once the energetically forbidden region vanishes, the invariant manifolds together form closed separatrices bounding transit orbits in the phase space, deviating from the low-energy picture of invariant manifold tubes. Though the qualitatively different behavior of invariant manifolds emerges for high energies, associated transit orbits possess a common feature generalized from that of low-energy transit orbits. The second part extends our previous proposal of using singular collision orbits associated with the secondary to find trajectories reaching the vicinity of the secondary to low energies. Statistical analyses indicate that singular collision orbits are useful to find such transfer trajectories except for the very low-energy regime. These results are numerically obtained in the Earth–Moon and Sun–Jupiter planar circular restricted three-body problems.

Monitoring Venus and communications relay from Lagrange Points

Orbits around the two close collinear Sun-Earth Lagrange points have been utilized in recent decades for many solar and astronomical missions to exploit the observational advantages. DSCOVR is the first satellite to observe Earth continuously from the vicinity of the L1 point and is filing a crucial gap in Earth observations with its vast collection of polar, geosynchronous and some retrograde low inclination orbits. Spacecraft orbiting Venus in the last four decades have provided us with a wealth of information and many unanswered questions, which cannot be addressed adequately by observing from polar or near equatorial orbits around the planet. The Sun-Venus collinear Lagrange points L1 (sunward) and L2 (behind the planet from the Sun) points are key vantage points located about a million km away from the planet along the direction to the Sun which enable continuous monitoring of the planet's day and night hemispheres. As spacecraft positions at L1 and L2 points are unstable, they can be inserted in orbits around them to observe Venus over a small range of phase angles unlike any Venus orbiter observations which cover 0–180° solar phase angle twice each orbit for a very long time with minimal station keeping costs. To help better understand Venus, we propose that monitoring Venus continuously at nearly the same phase angle from the vicinity of L1 and L2 Lagrange points is critical. Such Lagrange orbiters around Sun-Venus L1 and L2 points can provide crucial information continuously about the evolution and variability of: (i) reflectance of the global cloud cover, (ii) the night side cloud cover opacity, (iii) surface activity, and (iv) interaction of planet's atmosphere with the solar wind, and loss of atmosphere. In addition, the two Lagrange orbiters can provide a crucial continuous communications capability for relaying data from in-situ atmospheric or surface platforms. Well instrumented missions to L1 and L2 points of Venus would significantly improve our understanding Earth's perplexing neighbor by obtaining continuous record of data on the day and night hemispheres not available from Venus orbiting missions.

Annotated Translations of Three of the Euler’s Papers on Celestial Mechanics

Annotated translations from Latin of three of the Euler’s papers on celestial mechanics are presented, which fall into the category of three-body problems. The first translation deals with an exact solution of three bodies that move around the common center of mass and always line up. This is considered the first work from which the three collinear Lagrange points could be obtained. The second translation deals with motions of Sun, Earth and Moon in syzygy and Moon libration as well, where, for the first time, Euler introduces an archaic form of a Fourier sine series expansion to describe the Moon’s wagging motion. The last translation relates to a paper that was written with the goal of alleviating astronomical computations of the perturbed motion of the Moon around the Earth by the Sun, ending up with eight coupled differential equations for resolving the perturbed motion of this celestial body. Despite showing great analytical skills, Euler gave no indications on how this system of equations could be solved, which renders his efforts practically useless in the determination of the variations of the nodal line and inclination of the Moon’s orbit.

Perturbed locations and linear stability of collinear Lagrangian points in the elliptic restricted three body problem with triaxial primaries

Abstract This paper aims to study the effect of the triaxiality and the oblateness as a special case of primaries on the locations and stability of the collinear equilibrium points of the elliptic restricted three body problem (in brief ERTBP). The locations of the perturbed collinear equilibrium points are first determined in terms of mass ratio of the problem (the smallest mass divided by the total mass of the system) and different concerned perturbing factors. The difference between the locations of collinear points in the classical case of circular restricted three body problem and those in the perturbed case is represented versus mass ratio over its range. The linear stability of the collinear points is discussed. It is observed that the stability regions for our model depend mainly on the eccentricity of the orbits in addition to the considered perturbations.

High-Fidelity Trajectory Optimization with Application to Saddle-Point Transfers

A method to optimize space trajectories subject to impulsive controls is presented. The method employs a high-fidelity model and a multiple-shooting technique. The model accounts for an arbitrary number of gravitational attractions, their corrections due to celestial bodies oblateness, and solar radiation pressure. The peculiarity of this paradigm is that the equations of motion are written in a rotopulsating frame, where two primaries are at rest despite the fact that their motion and the one of the perturbers are given by a real ephemeris model. Direct transcription of the dynamics coupled with a multiple-shooting technique and an efficient computation of the Jacobian of the defects are used to optimize trajectories subject to a finite number of impulsive controls. The method has been applied to find a multitude of solutions to the problem of transferring a spacecraft from the sun–Earth collinear Lagrange points to the sun–Earth gravitational saddle point, where a theoretical zero background acceleration allows testing possible deviations from general relativity. The problem of targeting the sun–Earth saddle point with high accuracy represents a major flight dynamics challenge. The applicative scenario encompasses the possible mission extension option for LISA Pathfinder as special case.

Fast Earth–Moon transfers with ballistic capture

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

Looking for a new test of general relativity in the solar system

This paper discusses three matter-of-principle methods for measuring the general relativity correction to the Newtonian values of the position of collinear Lagrangian points L1 and L2 of the Sun–Earth-satellite system. All approaches are based on time measurements. The first approach exploits a pulsar emitting signals and two receiving antennas located at L1 and L2, respectively. The second method is based on a relativistic positioning system based on the Lagrangian points themselves. These first two methods depend crucially on the synchronization of clocks at L1 and L2. The third method combines a pulsar and an artificial emitter at the stable points L4 or L5 forming a basis for the positioning of the collinear points L1 and L2. Further possibilities are mentioned and the feasibility of the measurements is considered.

Computation of three-dimensional periodic orbits in the sun-earth system

In this paper, a third-order analytic approximation is described for computing the three-dimensional periodic halo orbits near the collinear L1 and L2 Lagrangian points for the photo gravitational circular restricted three-body problem in the Sun-Earth system. The constructed third-order approximation is chosen as a starting initial guess for the numerical computation using the differential correction method. The effect of the solar radiation pressure on the location of two collinear Lagrangian points and on the shape of the halo orbits is discussed. It is found that the time period of the halo orbit increases whereas the Jacobi constant decreases around both the collinear points taking into account the solar radiation pressure of the Sun for the fixed out-of-plane amplitude.

Halo orbit transfer trajectory design using invariant manifold in the Sun-Earth system accounting radiation pressure and oblateness

In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth $L_{1}$ and $L_{2}$ -halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at a possible intersection between the Earth’s parking orbit of the spacecraft and the manifold.