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.     

Cylindrical isomorphic mapping applied to invariant manifold dynamics for Earth–Moon Missions

Several families of periodic orbits exist in the context of the circular restricted three-body problem. This work studies orbital motion of a spacecraft among these periodic orbits in the Earth–Moon system, using the planar circular restricted three-body problem model. A new cylindrical representation of the spacecraft phase space (i.e., position and velocity) is described, and allows representing periodic orbits and the related invariant manifolds. In the proximity of the libration points, the manifolds form a four-fold surface, if the cylindrical coordinates are employed. Orbits departing from the Earth and transiting toward the Moon correspond to the trajectories located inside this four-fold surface. The isomorphic mapping under consideration is also useful for describing the topology of the invariant manifolds, which exhibit a complex geometrical stretch-and-folding behavior as the associated trajectories reach increasing distances from the libration orbit. Moreover, the cylindrical representation reveals extremely useful for detecting periodic orbits around the primaries and the libration points, as well as the possible existence of heteroclinic connections. These are asymptotic trajectories that are ideally traveled at zero-propellant cost. This circumstance implies the possibility of performing concretely a variety of complex Earth–Moon missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining a suitable, convenient end-of-life strategy for spacecraft placed in any of the unstable orbits. The final disposal orbit is an externally confined trajectory, never approaching the Earth or the Moon, and can be entered by means of a single velocity impulse (of modest magnitude) along the right unstable manifold that emanates from the Lyapunov orbit at \(L_2\) .     

Rotational dynamics of celestial deformable bodies

In a previous paper of this series (Tokis, 1974b), we have discussed the solution of the Eulerian equation which governs the axial rotation, applied to the effects of viscous friction exhibited in binary systems which consist of a close pair of fluid bodies of arbitrary structure. The aim of the present paper will be to give an application of those results to the Earth-Moon system. It is shown that synchronism between the axial rotation of the Earth and the revolution of the Moon will occur at the value of 650 h, in a time scale which depends strongly on the value of the mean viscosity of the Earth (regarded as spherical or spheroidal). In particular, the variation of rotational angular velocity of the Earth over the next ten centuries commencing from 1900 A.D., depends sensitively on the value of viscosity. On the other hand, the time for synchronism of axial rotation of the Moon is not affected by the viscosity for values between 10²⁴g cm^?1 s^?1 and 10²⁷g cm^?1 s^?1.     

Magnetization of celestial bodies with special application to the primeval Earth and Moon

The magnetic fields of celestial bodies are usually supposed to be due to a ‘hydromagnetic dynamo’. This term refers to a number of rather speculative processes which are supposed to take place in the liquid core of a celestial body. In this paper we shall follow another approach which is more closely connected with hydromagnetic processes well-known from the laboratory, and hence basically less speculative. The paper should be regarded as part of a general program to connect cosmical phenomena with phenomena studied in the laboratory. As has been demonstrated by laboratory experiments, a poloidal magnetic field may be increased by the transfer of energy from a toroidal magnetic field through kink instability of the current system. This mechanism can be applied to the fluid core of a celestial body. Any differential rotation will produce a toroidal field from an existing poloidal field, and the kink instability will feed toroidal energy back to the poloidal field, and hence amplify it. In the Earth-Moon system the tidal braking of the Earth's mantle acts to produce a differential angular velocity between core and mantle. The braking will be transferred to the core by hydromagnetic forces which at the same time give rise to a strong magnetic field. The strength of the field will be determined by the rate of tidal braking. It is suggested that the magnetization of lunar rocks from the period ?4 to ?3 Gyears derives from the Earth's magnetic field. As the interior of the Moon immediately after accretion probably was too cool to be melted, the Moon could not produce a magnetic field by hydromagnetic effects in its core. The observed lunar magnetization could be produced by such an amplified Earth field even if the Moon never came closer than 10 or 20 Earth's radii. This hypothesis might be checked by magnetic measurements on the Earth during the same period.     

Effects of a physical librations of the moon caused by a liquid core,and determination of the fourth mode of a free libration

An analytical theory of lunar physical librations based on its two-layer model consisting of a non-spherical solid mantle and ellipsoidal liquid core is developed. The Moon moves on a high-precision orbit in the gravitational field of the Earth and other celestial bodies. The defined fourth mode of a free libration is caused by the influence of the liquid core, with a long period of 205.7 yr, with amplitude S = 0″0395 and with an initial phase Π₀ = ?134° (for the initial epoch 2000.0). Estimates of dynamic (meridional) oblatenesses of a liquid core of the Moon have been estimated: ?_D = 4.42 × 10^?4, μ_D = 2.83 × 10^?4 (?_D + μ_D = 7.24 × 10^?4). These results have been obtained as a result of comparison of the developed analytical theory of physical librations of the Moon with the empirical theory of librations of the Moon constructed on the basis of laser observations.     

Indirect transfer to the Earth–Moon L1 libration point

The Earth–Moon L1 libration point is proposed as a human gateway for space transportation system of the future. This paper studies indirect transfer using the perturbed stable manifold and lunar flyby to the Earth–Moon L1 libration point. Although traditional studies indicate that indirect transfer to the Earth–Moon L1 libration point does not save much fuel, this study shows that energy efficient indirect transfer using the perturbed stable manifold and lunar flyby could be constructed in an elegant way. The design process is given to construct indirect transfer to the Earth–Moon L1 libration point. Simulation results show that indirect transfer to the Earth–Moon L1 libration point saves about 420 m/s maneuver velocity compared to direct transfer, although the flight time is about 20 days longer.     

Luni-solar effects of geosynchronous orbits at the critical inclination

The luni-solar effects of a geosynchronous artificial satellite orbiting near the critical inclination is investigated. To tackle this four-degrees-of-freedom problem, a preliminary exploration separately analyzing each harmonic formed by a combination of the satellite longitude of the node and the Moon longitude of the node is opportune. This study demonstrates that the dynamics induced by these harmonics does not show resonance phenomena. In a second approach, the number of degrees of freedom is halved by averaging the total Hamiltonian over the two non-resonant angular variables. A semi-numerical method can now be applied as was done when considering solely the inhomogeneity of the geopotential (see Delhaise et Henrard, 1992). Approximate surfaces of section are constructed in the plane of the inclination and argument of perigee. The main effects of the Sun and Moon attractions compared to the terrestrial attraction alone are a strong increase in the amplitude of libration in inclination (from 0.6° to 3.2°) and a decrease of the corresponding libration period (from the order of 200 years to the order of 20 years).Research Assistant for the Belgian National Fund for Scientific Research     

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).     

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.     

Determining parameters of Moon’s orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models

The aim of this work is to combine the model of orbital and rotational motion of the Moon developed for DE430 with up-to-date astronomical, geodynamical, and geo- and selenophysical models. The parameters of the orbit and physical libration are determined in this work from lunar laser ranging (LLR) observations made at different observatories in 1970–2013. Parameters of other models are taken from solutions that were obtained independently from LLR. A new implementation of the DE430 lunar model, including the liquid core equations, was done within the EPM ephemeris. The postfit residuals of LLR observations make evident that the terrestrial models and solutions recommended by the IERS Conventions are compatible with the lunar theory. That includes: EGM2008 gravitational potential with conventional corrections and variations from solid and ocean tides; displacement of stations due to solid and ocean loading tides; and precession-nutation model. Usage of these models in the solution for LLR observations has allowed us to reduce the number of parameters to be fit. The fixed model of tidal variations of the geopotential has resulted in a lesser value of Moon’s extra eccentricity rate, as compared to the original DE430 model with two fit parameters. A mixed model of lunar gravitational potential was used, with some coefficients determined from LLR observations, and other taken from the GL660b solution obtained from the GRAIL spacecraft mission. Solutions obtain accurate positions for the ranging stations and the five retroreflectors. Station motion is derived for sites with long data spans. Dissipation is detected at the lunar fluid core-solid mantle boundary demonstrating that a fluid core is present. Tidal dissipation is strong at both Earth and Moon. Consequently, the lunar semimajor axis is expanding by 38.20 mm/yr, the tidal acceleration in mean longitude is \(-25.90 {{}^{\prime \prime }}/\mathrm{cy}^2\), and the eccentricity is increasing by \(1.48\times 10^{-11}\) each year.     

Disturbance compensating control of orbit radially aligned two-craft Coulomb formation

This paper investigates the orbit radial stabilization of a two-craft virtual Coulomb structure about circular orbits and at Earth–Moon libration points. A generic Lyapunov feedback controller is designed for asymptotically stabilizing an orbit radial configuration about circular orbits and collinear libration points. The new feedback controller at the libration points is provided as a generic control law in which circular Earth orbit control form a special case. This control law can withstand differential solar perturbation effects on the two-craft formation. Electrostatic Coulomb forces acting in the longitudinal direction control the relative distance between the two satellites and inertial electric propulsion thrusting acting in the transverse directions control the in-plane and out-of-plane attitude motions. The electrostatic virtual tether between the two craft is capable of both tensile and compressive forces. Using the Lyapunov’s second method the feedback control law guarantees closed loop stability. Numerical simulations using the non-linear control law are presented for circular orbits and at an Earth–Moon collinear libration point.     

New Trends in the Development of the Lunar Physical Libration Theory

A review of the modern state of the lunar libration theory is presented. A significant progress in the lunar investigation is achieved due to the simultaneous processing of results of the satellite Doppler tracing and of the lunar laser ranging. The data evidencing existence of a small iron core in the Moon are discussed. In this connection, the further development of the theory of rotation of the Moon presents the study of internal structure and dynamics of a lunar body. A model of a two-layer Moon can have a very advanced application to explain some observed phenomena and to be as a first approach in the modelling of internal processes determining the lunar rotation.     

Solitary waves of vortices within a rotating,fluid shell

We consider a spherical, solid planet surrounded by a thin layer of an incompressible, inviscid fluid. The planet rotates with constant angular velocity about a fixed axis. The motion imparted by this planetary rotation upon the fluid particles of the ocean has been assumed to be governed by a linear version of the Navier-Stokes equation.We study the vortex motion within this rotating ocean and establish that the propagation of vortices depends on a third-order partial differential equation for the stream function. We prove that, in the most general case, this vorticity equation cannot generate any solitary waves; however, should the vertical component of vorticity satisfy a certain functional relationship, then we have obtained a family of solitary waves of permanent form.Retired, U.S. Naval Research Laboratory, Washington, D.C., U.S.A.     

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.     

Analytical theory of the libration of the Moon

This paper presents a new theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field. This field is represented through its third degree harmonics for the torque due to the Earth (second degree for the torque due to the Sun).The orbital motion of the Moon is described by the ELP 2000 solution (Chapront-Touzé, 1980) of the main problem of lunar theory.the physical libration variables are obtained as Poisson series and comparisons with the results of Eckhardt (Eckhardt, 1981) and Migus (Migus, 1980) are presented.     

A nonlinear analysis of the moon's physical libration in longitude

The Euler equations for the forced physical librations of the Moon have already been solved by using a digital computer to perform the semi-literal mathematical manipulations. Very near resonance, the computer solution for the physical libration in longitude is complemented by the solution of the appropriate Duffing equation with a dissipation term. Because of its apparent proximity to a resonant frequency, the term whose argument is 2 - twice the mean angular distance of the Moon's perigee from the ascending node of its orbit - is especially important. Its phase, which soon should be measurable, is related to the Moon's anelasticity. The term's frequency, in units of the sidereal month, increases as the semi-major axis of the Moon's orbit about the Earth increases. Using the Moon's mechanical ellipticity of Koziel and the rate of increase of the semi-major axis of MacDonald, it is estimated that the 2 term will cross the resonant frequency in 130 million years and, if the rate of energy dissipation is sufficiently low, a transient libration will be induced.     

Librations of the Galilean satellites: The influence of global internal liquid layers

The four Galilean satellites are thought to harbor one or even two global internal liquid layers beneath their surface layer. The iron core of Io and Ganymede is most likely (partially) liquid and also the core of Europa may be liquid. Furthermore, there are strong indications for the existence of a subsurface ocean in Europa, Ganymede, and Callisto. Here, we investigate whether libration observations can be used to prove the existence of these liquid layers and to constrain the thickness of the overlying solid layers. For Io, the presence of a small liquid core increases the libration of the mantle by a few percent with respect to an entirely solid Io and mantle libration observations could be used to determine the mantle thickness with a precision of several tens of kilometers given that the libration amplitude can be measured with a precision of 1 m. For Europa, Ganymede, and Callisto, the presence of a water ocean close to the surface increases by at least an order of magnitude the ice shell libration amplitude with respect to an entirely solid satellite. The shell libration depends essentially on the shell thickness and to a minor extent on the density difference between the ocean and the ice shell. The possible presence of a liquid core inside Europa and Ganymede has no noticeable influence on their shell libration. For a precision of several meters on the libration measurements, in agreement with the expected accuracy with the NASA/ESA EJSM orbiter mission to Europa and Ganymede, an error on the shell thickness of a few tens kilometers is expected. Therefore, libration measurements can be used to detect liquid layers such as Io’s core or water subsurface oceans in Europa, Ganymede, and Callisto and to constrain the thickness of the overlying solid surface layers.     

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.     

Solitary waves of vortices

We study the propagation of solitary waves of vortices within a spherical shell which constitutes the uppermost layer of a solid planet. This solid-liquid configuration rotates with constant angular velocity about an axis which is fixed with respect to the solid surface. The fluid within the shell is inviscid, incompressible, and of constant density. The motion imparted by the planetary rotation upon this fluid mass is governed by the Laplace tidal equation from which the potential of the extraplanetary forces has been deleted. Consistent with this ocean model, we establish that the stream function of a solitary wave of vortices must satisfy a third-order partial differential equation. We obtain solutions to this wave equation by imposing the condition that the vertical component of vorticity be functionally related to the stream function. We find that this dependence must necessarily be of the exponential type and that the solution to the wave equation then reduces to a quadrature depending on some arbitrary parameters. We prove that we can always choose the values of these parameters in order to approximate the integral in question by means of an analytic function: we reach a representation of the stream function of a solitary wave of vortices in terms of hyperbolic functions of time and position.This paper is dedicated to the memory of Professor Zdenek Kopal.