Existence and Stability of L4,5 In The Relativistic Restricted Three-Body Problem

The existence and stability of triangular libration points in the relativistic restricted three-body problem has been studied. It is found that L_4,5 are unstable in the whole range 0 ≤ μ ≤ 1/2 in contrast to the classical restricted three-body problem where they are stable for 0 < μ < μ₀, where μ is the mass parameter and μ₀ = 0.03852.... This revised version was published online in July 2006 with corrections to the Cover Date.     

Counterglow from the Earth-Moon libration points

Results from the OSO-6 Rutgers Zodiacal Light Analyzer experiment show photometric perturbations above the background in the anti-Sun line of sight. Sixteen successive lunations were examined, and the accumulated perturbations show a maximum value in the direction of the L₄ and L₅ Earth-Moon libration points. This is interpreted as a counterglow from a cloud of particles at the libration points. The average brightness of these libration clouds is 20 S₁₀ Vis. The average angular size of the libration clouds is approximately 6 degrees. Their position varies from one lunation to the next, within an ellipsoidal zone centered on the libration point direction, with its semi-major axis, of approximately 6 degrees, nominally in the ecliptic and its semi-minor axis, of approximately 2 degrees perpendicular to the ecliptic. The position of these clouds with respect to the Lagrangian L₄ and L₅ points, is towards the Moon in the northern summer and away from the Moon in the northern winter.     

Effect of Stellar Wind and Poynting–Robertson Drag on Photogravitational Elliptic Restricted Three Body Problem

The existence and linear stability of the planar equilibrium points for photogravitational elliptical restricted three body problem is investigated in this paper. Assuming that the primaries, one of which is radiating are rotating in an elliptical orbit around their common center of mass. The effect of the radiation pressure, forces due to stellar wind and Poynting–Robertson drag on the dust particles are considered. The location of the five equilibrium points are found using analytical methods. It is observed that the collinear equilibrium points L₁, L₂ and L₃ do not lie on the line joining the primaries but are shifted along the y-coordinate. The instability of the libration points due to the presence of the drag forces is demonstrated by Lyapunov’s first method of stability.     

On the triangular libration points in photogravitational restricted three-body problem with variable mass

This paper investigates the triangular libration points in the photogravitational restricted three-body problem of variable mass, in which both the attracting bodies are radiating as well and the infinitesimal body vary its mass with time according to Jeans’ law. Firstly, applying the space-time transformation of Meshcherskii in the special case when q=1/2, k=0, n=1, the differential equations of motion of the problem are given. Secondly, in analogy to corresponding problem with constant mass, the positions of analogous triangular libration points are obtained, and the fact that these triangular libration points cease to be classical ones when α≠0, but turn to classical L ₄ and L ₅ naturally when α=0 is pointed out. Lastly, introducing the space-time inverse transformation of Meshcherskii, the linear stability of triangular libration points is tested when α>0. It is seen that the motion around the triangular libration points become unstable in general when the problem with constant mass evolves into the problem with decreasing mass.     

Orbits of the Tethys Lagrangian bodies

The 1980 observations of the Saturn system have revealed objects at both the preceding (L₄) and following (L₅) triangular libration points of Tethys (S4). The observations indicate a small (～2°) libration amplitude for the L₄ body while the data on the L₅ object are insufficient to define its libration amplitude.     

Revealing the existence and stability of equilibrium points in the circular autonomous restricted four-body problem with variable mass

We have numerically investigated the circular autonomous restricted four-body problem where the fourth particle of variable mass is moving under the gravitational influence of three bodies known as primaries. Moreover, these primaries move in circular orbit around their common center of mass in such a way that their configuration remains an equilateral triangle configuration. The effect of the parameter α on the existence as well as on the locations of the libration points are investigated. The parametric variation of the positions of the libration points and zero velocity curves are also revealed when the parameter α (which occurs in Jeans’ law) increases. Moreover, the Newton–Raphson basins of convergence corresponding to the libration points are unveiled numerically when the parameter α increases. The obtained results strongly suggest that the study of the evolution of the attracting domains of the proposed dynamical system is worth studying in spite of their complexity.     

Trojan-type orbits in the system of two gravitational centers with variable masses and separation

Trojan type orbits in the system of two gravitational centers with variable separation are studied within the framework of the restricted problem of three bodies. The backward numerical integration of the equations of motion of the bodies starting in the triangular libration pointsL ₄ andL ₅ (reverse problem) finds the breakdown of librations as the separation decreases because of the mass gain of the smaller component and an approach of the body of negligible, mass to the latter up to the distance below its sphere of action with a relative velocity approximately equal to the escape one on this sphere. The breakdown of librations aboutL ₅ occurs under the mass gain of the smaller component considerably larger than in the case ofL ₄ and implications are made for the asymmetry of the number of librators aboutL ₄ andL ₅ in the solar system (Greeks and Trojans). Other parameters of the libration motion near 1/1 commensurability are obtained, namely, the asymmetry of the libration amplitudes about the triangular points as well as the values of periods and amplitudes within the limits of those for real Trojan asteroids. Trojans could be supposedly, formed inside the Proto-jupiter and escape during its intensive mass loss.     

Effect of the Earth’s compression on the physical libration of the Moon

The effect of the Earth??s compression on the physical libration of the Moon is studied using a new vector method. The moment of gravitational forces exerted on the Moon by the oblate Earth is derived considering second order harmonics. The terms in the expression for this moment are arranged according to their order of magnitude. The contribution due to a spherically symmetric Earth proves to be greater by a factor of 1.34 × 10⁶ than a typical term allowing for the oblateness. A linearized Euler system of equations to describe the Moon??s rotation with allowance for external gravitational forces is given. A full solution of the differential equation describing the Moon??s libration in longitude is derived. This solution includes both arbitrary and forced oscillation harmonics that we studied earlier (perturbations due to a spherically symmetric Earth and the Sun) and new harmonics due to the Earth??s compression. We posed and solved the problem of spinorbital motion considering the orientation of the Earth??s rotation axis with regard to the axes of inertia of the Moon when it is at a random point in its orbit. The rotation axes of the Earth and the Moon are shown to become coplanar with each other when the orbiting Moon has an ecliptic longitude of L _? = 90° or L _? = 270°. The famous Cassini??s laws describing the motion of the Moon are supplemented by the rule for coplanarity when proper rotations in the Earth-Moon system are taken into account. When we consider the effect of the Earth??s compression on the Moon??s libration in longitude, a harmonic with an amplitude of 0.03?? and period of T ₈ = 9.300 Julian years appears. This amplitude exceeds the most noticeable harmonic due to the Sun by a factor of nearly 2.7. The effect of the Earth??s compression on the variation in spin angular velocity of the Moon proves to be negligible.     

Numerical Exploration of Chermnykh’s Problem

We study the equilibrium points and the zero-velocity curves of Chermnykh’s problem when the angular velocity ω varies continuously and the value of the mass parameter is fixed. The planar symmetric simple-periodic orbits are determined numerically and they are presented for three values of the parameter ω. The stability of the periodic orbits of all the families is computed. Particularly, we explore the network of the families when the angular velocity has the critical value ω = 2√2 at which the triangular equilibria disappear by coalescing with the collinear equilibrium point L₁. The analytic determination of the initial conditions of the family which emanate from the Lagrangian libration point L₁ in this case, is given. Non-periodic orbits, as points on a surface of section, providing an outlook of the stability regions, chaotic and escape motions as well as multiple-periodic orbits, are also computed. Non-linear stability zones of the triangular Lagrangian points are computed numerically for the Earth–Moon and Sun–Jupiter mass distribution when the angular velocity varies.     

Effect of peturbations on the location of equilibrium points in the restricted problem of three bodies with variable mass

The effect of small perturbations in the coriolis and the centrifugal forces on the location of equilibrium points in the restricted problems of three bodies with variable mass has been studied. It is found that the points L₄ and L₅ form nearly equilateral triangles with the primaries and the points L₁, L₂, L₃ remain collinear and lie on the line joining the primaries.     

Comment on the paper “On the triangular libration points in photogravitational restricted three-body problem with variable mass” by Zhang, M.J., et al.

In a recent paper, published in Astrophys. Space Sci. (337:107, 2012) (hereafter paper ZZX) and entitled “On the triangular libration points in photogravitational restricted three-body problem with variable mass”, the authors study the location and stability of the generalized Lagrange libration points L ₄ and L ₅. However their study is flawed in two aspects. First they fail to write correctly the equations of motion of the variable mass problem. Second they attribute a variable mass to the third body of the restricted three-body model, a fact that is not compatible with the assumptions used in deriving the mathematical formulation of this model.     

The influence of triaxiality and oblateness on the triangular points of double pulsars in the ER3BP

We consider the motion of a test particle around a triaxial primary and an oblate companion orbiting each other in elliptic orbits about their common barycenter in the neighborhood of triangular libration points. The positions and stability of these points are influenced by the triaxiality and oblateness of the primary and secondary, and by the semi-major axis and eccentricity of the orbits. The triangular points are stable for 0<μ<μ _c ; where μ is the mass ratio (μ≤1/2) and μ _c is the critical mass value influenced by the eccentricity, oblateness, semi major axis and triaxiality factors. The size of the region of stability increases with decreasing values of triaxiality and oblateness. An application of the results obtain to double neutron star binaries results show that the positions and stability of the triangular points of PSR J1518+4904, PSR B1534+12, PSR B1914+16 and PSR B2127+11c are affected by the parameters in the systems’ dynamics.     

Existence and Stability of Libration Points in the Restricted Three-Body Problem When the Primaries are Triaxial Rigid Bodies

This paper deals with the stationary solutions of the planar restricted three-body problem when the primaries are triaxial rigid bodies with one of the axes as the axis of symmetry and its equatorial plane coinciding with the plane of motion. It is seen that there are five libration points, two triangular and three collinear. It is further observed that the collinear points are unstable, while the triangular points are stable for the mass parameter 0 < _crit(the critical mass parameter). It is further seen that the triangular points have long or short periodic elliptical orbits in the same range of .This revised version was published online in October 2005 with corrections to the Cover Date.     

Stability of L4 for radiated rigid primaries in resonant cases

The non-linear stability of the triangular libration point L₄ of the restricted three-body problem is studied under the presence of third- and fourth-order resonances, when the more massive primary is a triaxial rigid body and source of radiation. In this study, Markeev's theorems are applied with the help of Moser's theorem. It is found that the stability of the triangular libration point is unstable in the third-order resonance case and in the fourth-order resonance case, this is stable or unstable depending on A₁ and A₂, and a source of radiation parameter α, where A₁, A₂ depend upon the lengths of the semi-axes of the triaxial rigid body.     

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.     

On the theoretical possibility of the libration cloud

In the present paper, the problem of whether the interplanetary matter has a tendency to accumulate around the Lagrangian libration pointsL ₄ andL ₅, is examined statistically. It is concluded that: (1) If the particles are initially assumed to be distributed uniformly, they keep the uniformity ever after around the libration points. (2) If the particles receive random stochastic perturbations, their distribution tends to become uniform even if initially they have non-uniform distributions. (3) If the particles mutually collide inelastically, they have a tendency to avoid the regions near the libration points. Therefore, the interplanetary matter will not tend to accumulate near the libration points. Even if the observations of the libration cloud so far reported are confirmed, the clouds are likely to be but temporary objects.     

On the theoretical possibility of the libration cloud

We show within the framework of the restricted three body problem that

1. Only in the immediate neighbourhood of the Lagrangian pointsL ₄ andL ₅ the distribution of a cloud of particles tends to become uniform under the influence of random stochastic perturbations. No consequences can be derived from this fact for a tendency of dispersion of clouds librating at arbitrary distances around the Lagrangian points.
2. From an elementary inspection of the Jacobi integral we cannot conclude that mutual completely inelastic collisions tend to drive the particles away from the vicinity of the libration points.

Finally the motion of a particle within the libration cloud, approximated as a resisting medium, is briefly examined.     

Analytical model of the long-period forced longitude librations of Mercury

The shaking of Mercury’s orbit by the planets forces librations in longitude in addition to those at harmonics of the orbital period that have been used to detect Mercury’s molten core. We extend the analytical formulation of Peale et al. (Peale, S.J., Margot, J.L., Yseboodt, M. [2009]. Icarus 199, 1-8) in order to provide a convenient means of determining the amplitudes and phases of the forced librations without resorting to numerical calculations. We derive an explicit relation between the amplitude of each forced libration and the moment of inertia parameter (B-A)/C_m. Far from resonance with the free libration period, the libration amplitudes are directly proportional to (B-A)/C_m. Librations with periods close to the free libration period of ∼12 years may have measurable (∼arcsec) amplitudes. If the free libration period is sufficiently close to Jupiter’s orbital period of 11.86 years, the amplitude of the forced libration at Jupiter’s period could exceed the 35 arcsec amplitude of the 88-day forced libration. We also show that the planetary perturbations of the mean anomaly and the longitude of pericenter of Mercury’s orbit completely determine the libration amplitudes.While these signatures do not affect spin rate at a detectable level (as currently measured by Earth-based radar), they have a much larger impact on rotational phase (affecting imaging, altimetry, and gravity sensors). Therefore, it may be important to consider planetary perturbations when interpreting future spacecraft observations of the librations.     

On quasi-periodic motions around the collinear libration points in the real Earth–Moon system

Due to various perturbations, the collinear libration points of the real Earth–Moon system are not equilibrium points anymore. Under the assumption that the Moon’s motion is quasi-periodic, special quasi-periodic orbits called dynamical substitutes exist. These dynamical substitutes replace the geometrical collinear libration points as time-varying equilibrium points. In the paper, the dynamical substitutes of the three collinear libration points in the real Earth–Moon system are computed. For the points L ₁ and L ₂, linearized motions around the dynamical substitutes are described, and the variational equations of the dynamical substitutes are reduced to a form with a near constant coefficient matrix. Then higher order analytical formulae of the central manifolds are constructed. Using these analytical solutions as initial seeds, Lissajous orbits and halo orbits are computed with numerical algorithms.