The effect of eccentricity and inclination on the motion near the Lagrangian points L4 and L5

We study the effect of eccentricity and inclination on small amplitude librations around the equilibrium points L ₄ and L ₅ in the circular restricted three-body problem. We show that the effective libration centres can be displaced appreciably from the equilateral configuration. The secular extrema of the eccentricity as a function of the argument of pericentre are shifted by ∼25 ° This revised version was published online in July 2006 with corrections to the Cover Date.     

A note on a Lagrangian formulation for motion about the collinear points

A lagrangian formulation for the three-dimensional motion of a satellite in the vicinity of the collinear points of the circular-restricted problem is reconsidered. It is shown that the influence of the primaries can be expressed in the form of two third-body disturbing functions. By use of this approach, the equations for the Lagrangian and for the motion itself are readily developed into highly compact expressions. All orders of the non-linear developments are shown to be easily obtainable using well-known recursive relationships. The resulting forms for these equations are well suited for use in the initial phase of canonical or non-canonical investigations.     

Three-body problem, its Lagrangian points and how to exploit them using an alternative transfer to L4 and L5

An alternative transfer strategy to send spacecraft to stable orbits around the Lagrangian equilibrium points L4 and L5 based in trajectories derived from the periodic orbits around L1 is presented in this work. The trajectories derived, called Trajectories G, are described and studied in terms of the initial generation requirements and their energy variations relative to the Earth through the passage by the lunar sphere of influence. Missions for insertion of spacecraft in elliptic orbits around L4 and L5 are analysed considering the restricted three-body problem Earth–Moon-particle and the results are discussed starting from the thrust, time of flight and energy variation relative to the Earth.     

A possible long-lived asteroid population at the equilateral Lagrangian points of Saturn

The Lagrangian equilateral points of a planetary orbit are points of equilibrium that trail at 60°, ahead (L4) or behind (L5), the trajectory of a planet. Jupiter is the only major planet in our Solar system harbouring a known population of asteroids at those locations. Here we report the existence of orbits close to the Lagrangian points of Saturn, stable at time-scales comparable to the age of the Solar system. By scaling with respect to the Trojan population we have estimated the number of objects that would populate the regions, which gives a significant figure. Moreover, mutual physical collisions over the age of the Solar system would be very rare, so the evaporation rate of this swarm arising from mutual interactions would be very low. A population of asteroids not self-collisionally evolved after their formation stage would be the first to be observed in our planetary system. Our present estimations are based on the assumption that the capture efficiency at Saturn's equilateral points is comparable with the one corresponding to Jupiter, thus our figures may be taken as upper limits. In any case, observational constraints on their number would provide fundamental clues to our understanding of the history of the outer Solar system. If they existed, the surface properties and size distribution of those objects would represent unusually valuable fossil records of our early planetary system.     

Solar and planetary destabilization of the Earth-Moon triangular Lagrangian points

In the restricted circular three-body problem, two massive bodies travel on circular orbits about their mutual center of mass and gravitationally perturb the motion of a massless particle. The triangular Lagrange points, L₄ and L₅, form equilateral triangles with the two massive bodies and lie in their orbital plane. Provided the primary is at least 27 times as massive as the secondary, orbits near L₄ and L₅ can remain close to these locations indefinitely. More than 2200 cataloged asteroids librate about the L₄ and L₅ points of the Sun-Jupiter system, and five bodies have been discovered around the L₄ point of the Sun-Neptune system. Small satellites have also been found librating about the L₄ and L₅ points of two of Saturn's moons. However, no objects have been discovered around the Earth-Moon L₄ and L₅ points. Using numerical integrations, we show that orbits near the Earth-Moon L₄ and L₅ points can survive for over a billion years even when solar perturbations are included, but the further addition of the far smaller perturbations from other planets destabilize these orbits within several million years. Thus, the lack of observed objects in these regions cannot be used as a constraint on Solar System formation, nor on the tidal evolution of the Moon's orbit.     

Asymptotic solutions of the restricted problem near the equilateral Lagrangian points

The restricted problem in the vicinity of the Lagrangian point L₄ is studied by finding a convergent binomial expansion of the disturbing function. Using a Hamiltonian formulation in Delaunay variables and removing the short-period terms a resonance problem (already considered by Giacaglia (1970) in an attempt of enlarging the Ideal Resonance) is obtained. It is shown that this extension is reducible to Garfinkel's ideal resonance in the libration region.     

The stability of the Lagrangian point L4

The Hamiltonian function of the restricted problem of three bodies near the triangular Lagrangian point is normalized through sixth order terms with the help of MACSYMA. The same calculations were done previously with an algebraic processor in order to establish the stability at a critical value of the mass ratio.     

The stability of the triangular Lagrangian points for commensurability of order two

The two variable expansion method is used to show that the triangular equilibrium points in the planar restricted problem of three bodies are unstable at µ=µ₂=0.024293.... It is also shown that even though the equilibrium points are stable for µµ₂ finite motions in the neighborhood of the equilibrium points may be unstable.     

The effect of solar radiation pressure on the Lagrangian points in the elliptic restricted three-body problem

In this paper the effect of solar radiation pressure on the location and stability of the five Lagrangian points is studied, within the frame of elliptic restricted three-body problem, where the primaries are the Sun and Jupiter acting on a particle of negligible mass. We found that the radiation pressure plays the rule of slightly reducing the effective mass of the Sun and changes the location of the Lagrangian points. New formulas for the location of the collinear libration points were derived. For large values of the force ratio β, we found that at β=0.12, the collinear point L₃ is stable and some families of periodic orbits can be drawn around it.     

Three-dimensional periodic oscillations generating from plane periodic ones around the collinear Lagrangian points

The three families of three-dimensional periodic oscillations which include the infinitesimal periodic oscillations about the Lagrangian equilibrium pointsL ₁,L ₂ andL ₃ are computed for the value =0.00095 (Sun-Jupiter case) of the mass parameter. From the first two vertically critical (|a _v |=1) members of the familiesa, b andc, six families of periodic orbits in three dimensions are found to bifurcate. These families are presented here together with their stability characteristics. The orbits of the nine families computed are of all types of symmetryA, B andC. Finally, examples of bifurcations between families of three-dimensional periodic solutions of different type of symmetry are given.     

Analytical investigation of non-linear stability of the Lagrangian pointL 4 around the commensurability 1:2

The problem of stability of the Lagrangian pointL ₄ in the circular restricted problem of three bodies is investigated close to the 1 : 2 commensurability of the long and short period libration. By stability we define boundedness of the solution for a given initial finite displacement from the equilibrium point as function of the mass parameter close to the commensurability. A rigorous treatment close to the resonance condition is possible using a transformation that diagonalizes the matrix related to the linear part of the equations of motion. The so obtained equations are further transformed to action angle type variables. Then using an isolated resonance approach, only the slowly varying terms are kept in the equations and two independent isolating first integrals can be found. These integrals finally enable us to solve the stability problem in an exact way. The so obtained results are compared to numeric integration of the equations of motion and are found to be in perfect agreement.     

Physical significance and the role of Lagrangian points and the Oort clouds of planets in the Solar System

Here, the role played by Lagrangian points and the Oort clouds of planets in the evolution and structure of the Solar System has been discussed. It is revealed that the Lagrangian points are not mere isolated points in space associated with the orbit of a planet at which the resultant gravitational force of the system of three bodies is zero as thought previously to be, but their existence has much deeper physical significance as regards the origin of the Solar System and those of satellite systems of planets.     

The Uppsala‐DLR Trojan Survey of L4, the preceding Lagrangian cloud of Jupiter

We have used the ESO Schmidt telescope during the apparitions 1996, 1997 and 1998 to study the preceding Lagrangian cloud of Jupiter, L₄. In our observed field of view, about 700 square degrees, during 1996 we found 399 moving objects that we classified as Trojans. From this we conclude that down to absolute magnitude H ≤ 13.0 there are ≈ 1100 Trojans in the L₄ cloud. Proper elements of 696 numbered and multiopposition Trojans were calculated and one of these in the L₅ cloud were found to be a non Trojan. A discussion of the Trojan proper elements is also presented.     

A new orbit for comet C/1858 L1 (Donati)

A new orbit for comet C/1858 L1 (Donati), based on 1036 observations in α and 971 in σ made between 7 June 1858 and 5 March 1859, is calculated using iteratively reweighted least squares. Residuals were weighted by the Welsch weighting function. The orbit represents a high eccentricity ellipse, e = 0.996265, with large semi‐major axis, a = 154.8612 AU, and long period, P = 1927.22 yr. The residuals are relatively random, a 10.7% chance of being random, but with a slight indication of possible nongravitational forces influencing the motion. The comet will not return until the year 3759, when it will pass 0.8442 AU from the Earth. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equilibrium points and hill’s regions in the 1+2 and 2+2 problem

For the 1+2 and 2+2 problems, we obtain the equilibrium points, their stability, and the topology of the constant energy manifold.     

Gravitational deflection of light at 1 2 1 PPN order

We compute in parametrized form the effect of source-velocity terms in the gravitational deflection of light. Internal rotation corresponds to sources of order ³ where . We find for rigidly rotating models of the Sunand of Jupiter that, in each case, this effect is of order 10^–6 arc sec. A very high accuracy that may be achievable by future space-bourne instruments is thus necessary for its detection.