Free Libration Research Articles

Complements to Moons’ Lunar Libration Theory

AbstractAnalytical complements have been brought to Moons’ lunar libration theory concerning tidal effects, direct perturbations due to the Earth’s figure, and indirect non periodic perturbations. Comparisons to JPL numerical integrations DE245 and DE403 have been performed and the residuals treated by frequency analysis, allowing the determination of fitted free libration parameters and numerical complements.

Determination of Some Physical Parameters of the Moon with Lunar Laser Ranging Data

AbstractInformation about the structure of lunar interior and evolution could be obtained from measurements of lunar free librations, gravitational field, dissipation etc. In this paper the precision of determining free librations from Lunar Laser Ranging (LLR) data are estimated. Using the observing data from four telescopes for eighteen years, the amplitudes and phases of free librations, and ratios of the moments of inertia of the Moon were determined.

Estimation of the Lunar Physical Librations

AbstractThe recent long-term integration of JPL ephemeris DE403/LE403 yielded lunar physical librations covering 6000 years. A Fourier analysis of a 718-year subset of this span produced estimates of the component frequencies of the forced and free librations. A subsequent iterative least-squares estimation procedure provided precise values for phases and for time-varying amplitudes and frequencies. Two free libration modes were found; presence of a third is possible but close to the noise.

Estimation of the lunar physical librations

The recent long-term integration of JPL ephemeris DE403/LE403 yielded lunar physical librations covering 6000 years. A Fourier analysis of a 718-year subset of this span produced estimates of the component frequencies of the forced and free librations. A subsequent iterative least-squares estimation procedure provided precise values for phases and for time-varying amplitudes and frequencies. Two free libration modes were found; presence of a third is possible but close to the noise.

Determination of some physical parameters of the Moon with Lunar Laser Ranging data

Information about the structure of lunar interior and evolution could be obtained from measurements of lunar free librations, gravitational field, dissipation etc.. In this paper the precision of determining free librations with Lunar Laser Ranging (LLR) data are estimated. Using the observing data from four telescopes for eighteen years the amplitudes and phases of free librations, the moments of inertia ratio of The Moon were determined.

Passing through resonance: The excitation and dissipation of the lunar free libration in longitude

The acceleration of the mean lunar longitude has a small effect on the periods of most terms in a Fourier expansion of the longitude. There are several planetary perturbation terms that have small amplitudes, but whose periods are close to the resonant period of the lunar libration in longitude. Some of these terms are moving toward resonance, some are moving away from resonance, and the periods of those terms that do not include the Delaunay variables in their arguments are not moving. Because of its acceleration of longitude, the Moon is receding from the Earth, so the magnitude of the restoring torque that the Earth exerts on the rotating Moon is gradually attenuating; thus resonance itself is moving, but at a much slower rate than the periods of the accelerating planetary perturbations. There are five planetary perturbation terms from the ELP-2000 Ephemeris (with amplitudes of 0”.00001 or greater) that have passed through resonance in the past two million years. One of them is of special interest because it appears to be the excitation source of a supposed free libration in longitude that has been detected by the lunar laser ranging experiment. The amplitude of the term is only 0”.00021 but it could be the source of the ~ 1” amplitude free libration term if the viscoelastic properties of the Moon are similar to those of the Earth.

Constants of the Moon's free libration on the basis of heliometric observations from the years 1841–1945

The results of simultaneous and homogeneous reduction of 8 libration series are presented. On this basis all 6 free libration constants have been estimated using some new formulae.

The problem of the Eulerian oscillations: A weakness of numerical versus analytical methods

The application of computer technology has permitted more and more problems of dynamical astronomy to be solved more easily, quickly, and accurately. In this area, numerical integration is often very efficient, and sometimes essential. There is often, however, a temptation to choose numerical integration simply because it is the easiest way to attack the problem. Sometimes this works to the detriment of a satisfactory understanding of the physics of the problem under study. It is particularly the case for the "free", or Eulerian, oscillations. The forces that create such a motion are not of gravitational origin and are not even conservative. The theory can only specify the frequencies of oscillation, not their amplitudes nor phases. The case is complicated when the free oscillations interact with gravitationally-forced oscillations, a situation that is almost inevitable, since nothing is isolated in the Universe. The first author has particularly studied this problem in the case of the rotation of the Moon, and published the first credible determinations of the lunar free libration. In this kind of problem, the observations have to be used and care must to be taken to create no spurious free librations in the results by using numerical integrations to describe the other related motions. A differential correction of the starting conditions to fit the observations does not necessarily give any valid information on the real free oscillation contained in the data. An analytical model is necessary, if the goal of the research is to understand the origins and characteristics of an Eulerian oscillation in such a system.

Free librations of the moon from lunar laser ranging

Free librations of the moon from lunar laser ranging

Galilean satellites: Evolutionary paths in deep resonance

The Laplace resonance among the inner three Galilean satellites (mean motions n 1 − 3 n 2 + 2 n 3 = 0) has stable configurations in “deep resonance,” i.e., where mean motions taken by pairs are in ratios very close to 2:1. The present satellite configuration, with the resonance variable φ ≡ λ 1 − 3 λ 2 + 2 λ 3 stable at 180°, is unstable near this exact commensurability. But there is a continuous path of stable conditions branching from φ = 180° to higher and lower values of φ and toward very deep resonance, according to a theory extended to third order in orbital eccentricity. This path provides a track for tidal evolution of the system. Thus, scenarios involving evolution (probably episodic) from deep resonance are viable, and eliminate the requirement by the alternative equilibrium hypothesis for rapid tidal dissipation in Jupiter. Evolution out from deep resonance is consistent with the free eccentricity of Ganymede, the free libration of φ, and observational constraints on Io's secular acceleration. Also, the relatively large forced eccentricities in deep resonance may have controlled geophysical processes in the satellites by much greater tidal heating and global stress than at present.

High‐Precision Earth Rotation and Earth—Moon Dynamics: Lunar Distances and Related Observations

The last decade or so has seen the practical development of a number of high‐precision, space‐related geodetic techniques, specifically, lunar laser ranging (LLR), satellite ranging, and very long baseline interferometry (VLBI). One consequence has been an enlarged and improved data base available for studies of lunar motion and earth rotation.The impact on lunar studies has been particularly striking. The vast improvement in lunar positioning data provided by the LLR experiment has revived interest in the previously lethargic business of modeling the lunar orbit and librations (librations are rotational displacements of the moon about its center of mass). Several numerical and analytical models of the orbit and librations have been developed over the past few years. Numerical models, constructed by numerically solving appropriate differential equations, are generally more accurate than existing analytical models and are, in principal, better suited to removing the effects of lunar motion from the LLR residuals. On the other hand, results from numerical models are less amenable to physical interpretation, and so analytical models continue to receive attention. Analytical models are particularly useful in trying to understand the lunar librations, both because of the possibility of excited free librations and because the dynamical behavior of the moon during the forced librations is not completely understood.

The free librations of a dissipative Moon

Dissipation in the Moon produces a small offset, ca. 0.23", of the Moon's rotation axis from the plane defined by the ecliptic and lunar orbit normals. Both solid body tidal friction and viscous fluid friction at a core—mantle interface are plausible mechanisms. In this paper, I discuss the merits of each and find that solid friction requires a low lunar tidal Q , ca . 28, while turbulent fluid friction requires a core of radius 330 km to cause the signature observed by lunar laser ranging. Large ( ca . 0.4—8.0") free librations of the lunar figure have also been detected by laser ranging. Both a very recent impact on the Moon and fluid turbulence in the lunar core are plausible mechanisms for generating these free librations.

Planetary and Earth Figure Perturbations in the Librations of the Moon

AbstractComparisons of numeric, semi-analytic and analytic lunar libration models indicate that the planetary and Earth figure perturbations used to supplement the semi-analytic and analytic models are inadequate. Using the ELP-2000 of Chapront and Chapront-Touzé, an improved solution of these perturbations is developed. For the libration in longitude there are numerous terms with periods near that of the free libration; their amplitudes remain ill-defined.

Lunar Crater Giordano Bruno: A.D. 1178 Impact Observations Consistent with Laser Ranging Results

The hypothesis of Hartung, that the impact formation of lunar crater Giordano Bruno (103 degrees east, 36 degrees north) was observed and recorded 800 years ago, is considered in the context of data from the Luna 24 mission and laser range observations. It is concluded that (i) the event would certainly have been visible, and (ii) current determinations of the free libration in longitude in the moon's rotation are consistent with the hypothesis. Such a study cannot prove Hartung's interpretation, but it is nonetheless supportive of it.

The dynamical evolution and origin of the Martian moons

The satellites of Mars are found today on relatively circular, un-inclined orbits that lie close to their primary. Assuming that the secular acceleration of Phobos observed today is caused by solid body tides, the orbital evolution equations can be integrated backwards in time. Doing this indicates that the original orbits of Phobos and Deimos were both close to the synchronous orbit and were of low inclination; the eccentricity of Phobos' orbit may have been moderate in the past but Deimos' was always near zero. Integrating forward shows that Phobos will crash onto Mars in less than 10 8 years. Spacecraft measurements of low densities and albedos for Phobos and Deimos, as well as Phobos' reflectance spectrum suggest that the martian moons are carbonaceous chondritic in composition, that is, derived from material presumably formed much further away from the Sun than Mars. Two origin scenarios are described that produce satellites, composed of chondritic material, in fairly regular orbits. The first is the usual accretion theory, except that the composition of the satellites is determined by the last material brought into the system and that matter is not compacted since the central pressures are low. The second suggests that the martian satellites are the remnants of an interplanetary meteoroid which was captured and fragmented upon its collision with a primordial circumplanetary nebula surrounding Mars; further interaction with the nebula regularized the orbits. This mechanism appears to be successful in also explaining the irregular satellites found in the outer solar system. The rotations of Phobos and Deimos are tidally locked to be synchronous with their orbital periods; the longest axes of the satellites point approximately along the Mars-to-satellite lines while the shortest are normal to the satellite orbital planes. These configurations are a natural outcome of the action of tides on the satellites; their adoptions take place with characteristic timescales of ∼10 6 years for Phobos and ∼10 8 years for Deimos. Phobos is found to exhibit free librations of ∼5° about its synchronous position due to a near-resonance between its natural libration frequency and the frequency at which it is being forced. The current rotation states of the martian moons say nothing about their origin.

Hamiltonian Theory of the Libration of the Moon

AbstractThe feasibility of applying the Lie transform method to the problem of the physical libration of the Moon is investigated. By a succession of canonical transformations, the Hamiltonian of the problem is brought under a form suitable for perturbation technique. The mean value of the inclination of the angular momentum upon the ecliptic and the frequencies of the free libration are computed.

Theory of Motion of Jupiter’s Galilean Satellites

AbstractThe final results for the theory enabling one to calculate the positions of the Galilean satellites and their partial derivatives are presented, following the techniques outlined in earlier papers. Extensive use of algebraic manipulation software on a digital computer is employed to generate the final expressions. The new theory is, in effect, a revitalization of Sampson’s theory in which we (a) remove algebraic and mathematical errors existing in Sampson’s work, (b) introduce some neglected effects due to solar interactions and the 3-7 commensurability, (c) allow for non-zero amplitude and phase of the free libration, (d) express the final results as analytic functions of variations in 49 arbitrary constants of integration and physical parameters, (e) construct the theory in a manner which readily allows for future revision, and (f) provide analytic expressions for the partial derivatives with respect to the 49 parameters.

The molecular motion in the solid phases of t-butyl chloride by cold neutron scattering

Cold neutron scattering experiments have been performed on tertiary butyl chloride at the temperature 173 °K in phase III, 205 °K in phase II, and 235 °K in phase I. The crystal is plastic in phases I and II and ordinary crystalline in phase III. The experimental data are interpreted by fitting of computer calculated cross sections, based on various physical models. The main aim was to study the rotational motion. By use of the data from the phase III a reasonalbe Debye form for the translational spectrum is obtained and used also in phases II and I. Similarly the rather well separated torsional spectrum of the methyl groups is simply approximated. The rotational spectra dominate the elastic and near elastic scattering. Models ranging from free rotation to undamped libration, including isotropic Langevin rotational diffusion as an intermediate case when the rotational relaxation times are short, are tested. It is found, that no extreme model corresponding either to almost free rotation, or to almost undamped libration, or to uniaxial rotational diffusion can describe the data. On the other hand, either isotropic Langevin rotational diffusion or a stochastic two-step rotational diffusion model involving consecutive free rotations and librations fit the data well. It is found that the time a molecule needs for moving from one potential well to another is about 30% of the time it stays in a well. It is also found that both times are short, of order 0.2<τ<2.10−12 sec. It is therefore concluded that the Frenkel model type is not applicable to the rotational motion of t-butyl chloride in plastic phases I and II.

Excitation and relaxation of the wobble, precession, and libration of the Moon

The rate of meteorite-impact excitation of the free wobble, free precession, and free libration of the moon above a given amplitude is estimated for two crater-size scaling laws and is compared with the rate of damping by tidal and rotational distortion as well as by a possible core-mantle interaction. Criteria for the probable existence and the probable nonexistence of observable amplitudes of the free motions are developed in terms of upper bounds on the damping factor for the various motions and in terms of ranges for the kinematic viscosity of a possible core. It is shown that although observable amplitudes are compatible with reasonable values of the damping factor and kinematic viscosity, other reasonable values could keep the amplitudes below the observable level most of the time. It is found that the free libration is the least likely motion to be observed and that a lunar core with earthlike properties might keep some of the free motions damped, but not all of them. Uncertainties in some of the assumptions used in the analysis are discussed.

Free librations of the moon determined by an analysis of laser range measurements

Among the lunar laser range measurements obtained during the past six years at McDonald Observatory, those available cover the period August 1969-November 1974, being 1377 normal observations made on the three Apollo reflectors and that of Lunokhod II. The fit of these data led to a rms residual of 55 cm. In this study, a large number of parameters have been resolved, including the geocentric coordinates of the telescope, the selenocentric coordinates of each of the reflectors, as well as orbital elements of the Moon. In addition, the interest has been directed more specially towards the study of the rotational motion of the Moon and particularly the problem of its free librations. The performed resolutions give the evidence of the three modes of free oscillations. The determined amplitudes arise to 1″.7 in longitude and 0″.5 and 8″.7 in latitude, with the respective periods of 2.9 years, 27.3 days and 75 years. In connection with these parameters, the fittings determined also the most of part of elements of lunar gravitational field: the moment of inertia parametersβ andγ, and a number of the third degree harmonics. These new results should now permit a research on the implications of these oscillations effects, concerning the impact history of the Moon and the properties of its internal structure. On the basis of the amplitudes determined here, one can already estimate an order of the magnitude for theQ dissipation coefficient comparable with that determined from seismic studies of the Moon.