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.     

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.     

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.     

Central-body square configuration of restricted six-body problem

This paper deals with the existence and stability of libration points in linear sense in the central-body square configuration of restricted six-body problem. It is found that there exist twelve libration points, four collinear and eight non-collinear. All libration points lie on the concentric circles C₁, C₂ and C₃ centered at origin. The libration points L_1,3,5,7 lie on circle C₁, L_9,10,11,12 on C₂ and L_2,4,6,8 on C₃. This is also observed that the eight libration points are on the axes and four are off the axes, i.e., L_1,2,3,4 are on x-axis, L_5,6,7,8 on y-axis and rest are off the axes. The libration points on circles C₁ and C₃ are unstable for all values of mass parameter µ while the libration points on circle C₂ are stable for the critical mass parameter µ_c = 0.00910065.     

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.     

A note on the dynamics around the Lagrange collinear points of the Earth–Moon system in a complete Solar System model

In this paper we study the dynamics of a massless particle around the L _1,2 libration points of the Earth–Moon system in a full Solar System gravitational model. The study is based on the analysis of the quasi-periodic solutions around the two collinear equilibrium points. For the analysis and computation of the quasi-periodic orbits, a new iterative algorithm is introduced which is a combination of a multiple shooting method with a refined Fourier analysis of the orbits computed with the multiple shooting. Using as initial seeds for the algorithm the libration point orbits of Circular Restricted Three Body Problem, determined by Lindstedt-Poincaré methods, the procedure is able to refine them in the Solar System force-field model for large time-spans, that cover most of the relevant Sun–Earth–Moon periods.     

The effect of the Earth’s oblateness on the Moon’s physical libration in latitude

The Moon’s physical libration in latitude generated by gravitational forces caused by the Earth’s oblateness has been examined by a vector analytical method. Libration oscillations are described by a close set of five linear inhomogeneous differential equations, the dispersion equation has five roots, one of which is zero. A complete solution is obtained. It is revealed that the Earth’s oblateness: a) has little effect on the instantaneous axis of Moon’s rotation, but causes an oscillatory rotation of the body of the Moon with an amplitude of 0.072″ and pulsation period of 16.88 Julian years; b) causes small nutations of poles of the orbit and of the ecliptic along tight spirals, which occupy a disk with a cut in a center and with radius of 0.072″. Perturbations caused by the spherical Earth generate: a) physical librations in latitude with an amplitude of 34.275″; b) nutational motion for centers of small spiral nutations of orbit (ecliptic) pole over ellipses with semi-major axes of 113.850″ (85.158″) and the first pole rotates round the second one along a circle with radius of 28.691″; c) nutation of the Moon’s celestial pole over an ellipse with a semi-major axis of 45.04″ and with an axes ratio of about 0.004 with a period of T = 27.212 days. The principal ellipse’s axis is directed tangentially with respect to the precession circumference, along which the celestial pole moves nonuniformly nearly in one dimension. In contrast to the accepted concept, the latitude does not change while the Moon’s poles of rotation move. The dynamical reason for the inclination of the Moon’s mean equator with respect to the ecliptic is oblateness of the body of the Moon.     

Planetary perturbations on the Libration of the Moon

A theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field, was recently built (Moons, 1982). The Lie transforms method was used to reduce the Hamiltonian of the main problem of the libration of the Moon and to produce the usual libration series p₁, p₂ and . This main problem takes into account the perturbations due to the Sun and the Earth on the rotation of a rigid Moon about its center of mass. In complement to this theory, we have now computed the planetary effects on the libration, the planetary terms being added to the mean Hamiltonian of the main problem before a last elimination of the angles. For the main problem, as well as for the planetary perturbations, the motion of the center of mass of the Moon is described by the ELP 2000 solution (Chapront and Chapront-Touze, 1983).     

On the construction of low-energy cislunar and translunar transfers based on the libration points

There exist cislunar and translunar libration points near the Moon, which are referred to as the LL ₁ and LL ₂ points, respectively. They can generate the different types of low-energy trajectories transferring from Earth to Moon. The time-dependent analytic model including the gravitational forces from the Sun, Earth, and Moon is employed to investigate the energy-minimal and practical transfer trajectories. However, different from the circular restricted three-body problem, the equivalent gravitational equilibria are defined according to the geometry of the instantaneous Hill boundary due to the gravitational perturbation from the Sun. The relationship between the altitudes of periapsis and eccentricities is achieved from the Poincaré mapping for all the captured lunar trajectories, which presents the statistical feature of the fuel cost and captured orbital elements rather than generating a specified Moon-captured segment. The minimum energy required by the captured trajectory on a lunar circular orbit is deduced in the spatial bi-circular model. The idea is presented that the asymptotical behaviors of invariant manifolds approaching to/traveling from the libration points or halo orbits are destroyed by the solar perturbation. In fact, the energy-minimal cislunar transfer trajectory is acquired by transiting the LL ₁ point, while the energy-minimal translunar transfer trajectory is obtained by transiting the LL ₂ point. Finally, the transfer opportunities for the practical trajectories that have escaped from the Earth and have been captured by the Moon are yielded by the transiting halo orbits near the LL ₁ and LL ₂ points, which can be used to generate the whole of the trajectories.     

The vector approach to the problem of physical libration of the Moon. II. The nonlinear problem

By the new vector method in a nonlinear setting, a physical libration of the Moon is studied. Using the decomposition method on small parameters we derive the closed system of nine differential equations with terms of the first and second order of smallness. The conclusion is drawn that in the nonlinear case a connection between the librations in a longitude and latitude, though feeble, nevertheless exists; therefore, the physical libration already is impossible to subdivide into independent from each other forms of oscillations, as usually can be done. In the linear approach, ten characteristic frequencies and two special invariants of the problem are found. It is proved that, taking into account nonlinear terms, the invariants are periodic functions of time. Therefore, the stationary solution with zero frequency, formally supposing in the linear theory a resonance, in the nonlinear approach gains only small (proportional to e) periodic oscillations. Near to zero frequency of a resonance there is no, and solution of the nonlinear equations of physical libration is stable. The given nonlinear solution slightly modifies the previously unknown conical precession of the Moon’s spin axis. The character of nonlinear solutions near the basic forcing frequency Ω₁, where in the linear approach there are beats, is carefully studied. The average method on fast variables is obtained by the linear system of differential equations with almost periodic coefficients, which describe the evolution of these coefficients in a nonlinear problem. From this follows that the nonlinear components only slightly modify the specified beats; the interior period T ≈ 16.53 days appears 411 times less than the exterior one T ≈ 18.61 Julian years. In particular, with such a period the angle between ecliptic plane and Moon orbit plane also varies. Resonances, on which other researches earlier insisted, are not discovered. As a whole, the nonlinear analysis essentially improves and supplements a linear picture of the physical libration.     

High-Precision Ephemerides of Planets—EPM and Determination of Some Astronomical Constants

The latest version of the planetary part of the numerical ephemerides EPM (Ephemerides of Planets and the Moon) developed at the Institute of Applied Astronomy of the Russian Academy of Sciences is presented. The ephemerides of planets and the Moon were constructed by numerical integration in the post-Newtonian metric over a 140-year interval (from 1880 to 2020). The dynamical model of EPM2004 ephemerides includes the mutual perturbations from major planets and the Moon computed in terms of General Relativity with allowance for effects due to lunar physical libration, perturbations from 301 big asteroids, and dynamic perturbations due to the solar oblateness and the massive asteroid ring with uniform mass distribution in the plane of the ecliptic. The EPM2004 ephemerides resulted from a least-squares adjustment to more than 317000 position observations (1913–2003) of various types, including radiometric measurements of planets and spacecraft, CCD astrometric observations of the outer planets and their satellites, and meridian and photographic observations. The high-precision ephemerides constructed made it possible to determine, from modern radiometric measurements, a wide range of astrometric constants, including the astronomical unit AU = (149597870.6960 ± 0.0001) km, parameters of the rotation of Mars, the masses of the biggest asteroids, the solar quadrupole moment J ₂ = (1.9 ± 0.3) × 10⁻⁷, and the parameters of the PPN formalism β and γ. Also given is a brief summary of the available state-of-the-art ephemerides with the same precision: various versions of EPM and DE ephemerides from the Jet Propulsion Laboratory (JPL) (USA) and the recent versions of these ephemerides—EPM2004 and DE410—are compared. EPM2004 ephemerides are available via FTP at ftp://qua-sar.ipa.nw.ru/incoming/EPM2004.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 3, 2005, pp. 202–213.Original Russian Text Copyright © 2005 by Pitjeva.     

A dynamical equivalent to the equilateral libration points of the earth-moon system

Consider the Earth-Moon-particle system as a Restricted Three Body Problem. There are two equilateral libration points. In the actual world system, those points are no longer relative equilibrium points mainly due to the effect of the Sun and to the noncircular motion of the Moon around the Earth. In this paper we present the problem as a perturbation of the RTBP and we look for the dynamical equivalent of L _4,5. It turns out to be a quasiperiodic orbit. It is obtained for a simplified model but the procedure to obtain it is general and can be carried out with an additional computational effort.     

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.     

Effects of axis-symmetric body,zonal harmonic and potential from a belt on the stability of triangular libration points in the CR3BP

Within the frame work of the circular restricted three-body problem (CR3BP) we have examined the effect of axis-symmetric of the bigger primary, oblateness up to the zonal harmonic J ₄ of the smaller primary and gravitational potential from a belt (circular cluster of material points) on the linear stability of the triangular libration points. It is found that the positions of triangular libration points and their linear stability are affected by axis-symmetric of the bigger primary, oblateness up to J ₄ of the smaller primary and the potential created by the belt. The axis-symmetric of the bigger primary and the coefficient J ₂ of the smaller primary have destabilizing tendency, while the coefficient J ₄ of the smaller primary and the potential from the belt have stabilizing tendency. The overall effect of these perturbations has destabilizing tendency. This study can be useful in the investigation of motion of a particle near axis-symmetric—oblate bodies surrounded by a belt.     

Effect of perturbed potentials on the stability of libration points in the restricted problem

The location and the stability in the linear sense of the libration points in the restricted problem have been studied when there are perturbations in the potentials between the bodies. It is seen that if the perturbing functions satisfy certain conditions, there are five libration points, two triangular and three collinear. It is further observed that the collinear points are unstable and for the triangular points, the range of stability increases or decreases depending upon whetherP> or <0 wherep depends upon the perturbing functions. The theory is verified in the following four cases:

1. There are no perturbations in the potentials (classical problem).
2. Only the bigger primary is an oblate spheroid whose axis of symmetry is perpendicular to the plane of relative motion (circular) of the primaries.
3. Both the primaries are oblate spheroids whose axes of symmetry are perpendicular to the plane of relative motion (circular) of the primaries.
4. The primaries are spherical in shape and the bigger is a source of radiation.

     

The three-dimensional motion of Trojan asteroids

The problem is considered within the framework of the elliptic restricted three-body problem. The asymptotic solution is derived by a three-variable expansion procedure. The variables of the expansion represent three time-scales of the asteroids: the revolution around the Sun, the libration around the triangular Lagrangian pointsL ₄,L ₅, and the motion of the perihelion. The solution is obtained completely in the first order and partly in the second order. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter. As an example for the perturbations of the orbital elements the main perturbations of the eccentricity, the perihelion longitude and the longitude of the ascending node are given. Conditions for the libration of the perihelion are also discussed.     

Lissajous trajectories for lunar global positioning and communication systems

This work proposes a Lunar Global Positioning System (LGPS) and a Lunar Global Communication System (LGCS) using two constellations of satellites on Lissajous trajectories around the collinear L ₁ and L ₂ libration points in the Earth–Moon system. This solution is compared against a Walker constellation around the Moon similar to the one used for the Global Positioning System (GPS) on the Earth to evaluate the main differences between the two cases and the advantages of adopting the Lissajous constellations. The problem is first studied using the Circular Restricted Three Body Problem to find out its main features. The study is then repeated with higher fidelity using a four-body model and higher-order reference trajectories to simulate the Earth-Moon-spacecraft dynamics more accurately. The LGPS performance is evaluated for both on-ground and in-flight users, and a visibility study for the LGCS is used to check that communication between opposite sides of the Moon is possible. The total ΔV required for the transfer trajectories from the Earth to the constellations and the trajectory control is calculated. Finally, the estimated propellant consumption and the total number of satellites for the Walker constellation and the Lissajous constellations is used as a performance index to compare the two proposed solutions.     

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.     

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.