Free Libration Research Articles

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.

The rotation of Janus and Epimetheus

Epimetheus, a small moon of Saturn, has a rotational libration (an oscillation about synchronous rotation) of 5.9 ° ± 1.2 ° , placing Epimetheus in the company of Earth’s Moon and Mars’ Phobos as the only natural satellites for which forced rotational libration has been detected. The forced libration is caused by the satellite’s slightly eccentric orbit and non-spherical shape. Detection of a moon’s forced libration allows us to probe its interior by comparing the measured amplitude to that predicted by a shape model assuming constant density. A discrepancy between the two would indicate internal density asymmetries. For Epimetheus, the uncertainties in the shape model are large enough to account for the measured libration amplitude. For Janus, on the other hand, although we cannot rule out synchronous rotation, a permanent offset of several degrees between Janus’ minimum moment of inertia (long axis) and the equilibrium sub-Saturn point may indicate that Janus does have modest internal density asymmetries. The rotation states of Janus and Epimetheus experience a perturbation every 4 years, as the two moons “swap” orbits. The sudden change in the orbital periods produces a free libration about synchronous rotation that is subsequently damped by internal friction. We calculate that this free libration is small in amplitude (<0.1°) and decays quickly (a few weeks, at most), and is thus below the current limits for detection using Cassini images.

Latitudinal librations of Mercury with a fluid core

Abstract In the framework of the space missions to Mercury, an accurate model of rotation is needed. Librations around the 3:2 spin-orbit resonance as well as latitudinal librations have to be predicted with the best possible accuracy. In this paper, we use a Hamiltonian analysis and numerical integrations to study the librations of Mercury, both in longitude and latitude. Due to the proximity of the period of the free libration in longitude to the orbital period of Jupiter, the 88-day and 11.86-year contributions dominate Mercury’s libration in longitude (with the Hermean parameters chosen). The amplitude of the libration in latitude is much smaller (under 1 arcsec) and should not be detected by the space missions. Nevertheless, we point out that this amplitude could be much larger (up to several tens of arcsec) if the free period related to the libration in latitude approaches the period of the Jupiter–Saturn Great Inequality (883 years). Given the large uncertainties on the planetary parameters, this new resonant forcing on Mercury’s libration in latitude should be borne in mind.

Resonant forcing of Mercury's libration in longitude

The period of free libration of Mercury's longitude about the position it would have had if it were rotating uniformly at 1.5 times its orbital mean motion is close to resonance with Jupiter's orbital period. The Jupiter perturbations of Mercury's orbit thereby lead to amplitudes of libration at the 11.86 year period that may exceed the amplitude of the 88 day forced libration determined by radar. Mercury's libration in longitude may be thus dominated by only two periods of 88 days and 11.86 years, where other periods from the planetary perturbations of the orbit have much smaller amplitudes.

Titan’s rotational state

In Noyelles et al. (Astron. Astrophys. 478, 959–970 (2008)), a resonance involving the wobble of Titan is hinted at. This paper studies this scenario and its consequences. The first step is to build an accurate analytical model that would help to find the likely resonances in the rotation of every synchronous body. In this model, I take the orbital eccentricity of the body into account, as well as its variable inclination with respect to Saturn’s equator. Then an analytical study using the second fundamental model of the resonance is performed to study the resonance of interest. Finally, I study the dissipative consequences of this resonance. I find that this resonance may have increased the wobble of Titan by several degrees. For instance, if Titan’s polar momentum C is equal to 0.355MR 2 (M and R T being, respectively, Titan’s mass and radius), the wobble might be forced to 41 degrees. Thanks to an original formula, I find that the dissipation associated with the forced wobble might not be negligible compared to the contribution of the eccentricity. I also suspect that, due to the forced wobble, Titan’s period of rotation may be somewhat underestimated by observers. Finally, I use the analytical model presented in this paper to compute the periods of the free librations of the four Galilean satellites as well as the Saturnian satellite Rhea. For Io and Europa, the results are consistent with previous studies. For the other satellites, the periods of the free librations are, respectively, 186.37 d, 23.38 y and 30.08 y for Ganymede, 2.44 y, 209.32 y and 356.54 y for Callisto, and 51.84 d, 2.60 y and 3.59 y for Rhea.

Free librations of the two-layer Moon and the possibilities of their detection

Based on the forthcoming second stage of the Japanese Lunar mission ILOM (2013), when an optical telescope will be set on the surface near one of the Lunar poles, the possibility to detect free Lunar modes (Chandler-like wobble and free-core nutation) is considered. The difference between the Lunar Eulerian and Chandler-like wobble is explained. The terms “arbitrary libration” and “free libration” are discussed. The geometrical and physical interpretations of the free polar motion over the Lunar surface are considered from the viewpoints of Lunar surface-based observations and the Lunar Navigation Almanac. The dependencies of the free libration period on the core’s radius, density, and ellipticity are modelled and discussed.

Titan's rotation

Aims. We study the forced rotation of Titan seen as a rigid body at the equilibrium Cassini state, involving the spin-orbit synchronization. Methods. We used both the analytical and the numerical ways. We analytically determined the equilibrium positions and the frequencies of the 3 free librations around it, while a numerical integration associated to frequency analysis gave us a more synthetic, complete theory, where the free solution split from the forced one. Results. We find a mean obliquity of 2.2 arcmin and the fundamental frequencies of the free librations of about 2.0977, 167.4883, and 306.3360 years. Moreover, we bring out the main role played by Titan’s inclination on its rotation, and we suspect a likely resonance involving Titan’s wobble.

Tidal evolution of Mimas, Enceladus, and Dione

The tidal evolution through several resonances involving Mimas, Enceladus, and/or Dione is studied numerically with an averaged resonance model. We find that, in the Enceladus–Dione 2:1 e-Enceladus type resonance, Enceladus evolves chaotically in the future for some values of k 2 / Q . Past evolution of the system is marked by temporary capture into the Enceladus–Dione 4:2 e e ′ -mixed resonance. We find that the free libration of the Enceladus–Dione 2:1 e-Enceladus resonance angle of 1.5° can be explained by a recent passage of the system through a secondary resonance. In simulations with passage through the secondary resonance, the system enters the current Enceladus–Dione resonance close to tidal equilibrium and thus the equilibrium value of tidal heating of 1.1 ( 18 , 000 / Q S ) GW applies. We find that the current anomalously large eccentricity of Mimas can be explained by passage through several past resonances. In all cases, escape from the resonance occurs by unstable growth of the libration angle, sometimes with the help of a secondary resonance. Explanation of the current eccentricity of Mimas by evolution through these resonances implies that the Q of Saturn is below 100,000. Though the eccentricity of Enceladus can be excited to moderate values by capture in the Mimas–Enceladus 3:2 e-Enceladus resonance, the libration amplitude damps and the system does not escape. Thus past occupancy of this resonance and consequent tidal heating of Enceladus is excluded. The construction of a coherent history places constraints on the allowed values of k 2 / Q for the satellites.

Project “The Moon – 2012+”: Spin-orbital evolution, geophysics and selenodesy of the Moon

The Russian scientific project “The Moon – 2012+” is directed at solving fundamental problems of celestial mechanics, selenodesy and geophysics of the Moon through the pursuance of theoretical research and computer simulations of the following fields. 1. Spin-orbital longtime evolution and physical librations of the multilayered Moon: (a) development of the analytical theory of rotation of the two- /three-layer Moon and construction of the physical libration’s tables for processing accurate observations and for constructing a lunar annual book; and (b) analysis of the spin-orbital evolution of the early Moon, an estimation of internal energy dissipation, and modeling of the long-term mechanism maintaining the free librations of the Moon. 2. Geodynamics of a lunar core: analysis of differentiation of a lunar core, detailed elaboration of plume-tectonics of mantle and a core of the early Moon, evolution of a boundary layer of a core–mantle boundary, reconstruction of the gravitational and viscous–mechanical interactions of a lunar core and mantle, research on resonant dissipation of internal energy, and calculation of free and forced nutations of a lunar core and of free fluctuations of a core–mantle system. 3. Selenodesy of lunar far-side: solution of an inverse problem in lunar gravimetry, construction of a geodynamic model of the lunar crust and of a Moho’s boundary, reconstruction of initial mascons on the far-side of the Moon, and creation of accurate topographical and gravitational models of the Moon on the basis of modern observations.

Long-period forcing of Mercury's libration in longitude

Planetary perturbations of Mercury's orbit lead to forced librations in longitude in addition to the 88-day forced libration induced by Mercury's orbital motion. The forced librations are a combination of many periods, but 5.93 and 5.66 years dominate. These two periods result from the perturbations by Jupiter and Venus respectively, and they lead to a 125-year modulation of the libration amplitude corresponding to the beat frequency. Other periods are also identified with Jupiter and Venus perturbations as well as with those of the Earth, and these and other periods in the perturbations cause several arc second fluctuations in the libration extremes. The maxima of these extremes are about 30″ above and the minima about 7″ above the superposed ∼ 60 ″ , 88-day libration during the 125-year modulation. Knowledge of the nature of the long-period forced librations is important for the interpretation of the details of Mercury's rotation state to be obtained from radar and spacecraft observations. We show that the measurement of the 88-day libration amplitude for the purposes of determining Mercury's core properties is not compromised by the additional librations, because of the latter's small amplitude and long period. If the free libration in longitude has an amplitude that is large compared with that of the forced libration, its ∼10-year period will dominate the libration spectrum with the 88-day forced libration and the long-period librations from the orbital perturbations superposed. If the free libration has an amplitude that is comparable to those of the long-period forced libration, it will be revealed by erratic amplitude, period and phase on the likely time span of a series of observations. However, a significant free libration component is not expected because of relatively rapid damping.

Nonharmonic transverse vibration of the H-bonded molecules and the THz spectra in ice and water

A nonlinear equation of motion is found for the dimer comprising two charged H 2O molecules. The THz dielectric response to nonharmonic vibration of a nonrigid dipole, forming the hydrogen bond (HB), is found in the direction transverse to this bond. An explicit expression is derived for the autocorrelator that governs the spectrum generated by transverse vibration (TV) of such a dipole. This expression is obtained by analytical solution of the truncated set of recurrence equations. The far infrared (FIR) spectra of ice at the temperature − 7 °C are calculated. The wideband, in the wavenumber (frequency) ν range 0…1000 cm − 1 , spectra are obtained for liquid water at room temperature and for supercooled water at − 5.6 °C. All spectra are represented in terms of the complex permittivity ε( ν) and the absorption coefficient α( ν). The obtained analytical formula for ε comprises the term ε ⊥ pertinent to the studied TV mechanism with three additional terms Δ ε q , Δ ε μ , and ε or arising, respectively, from: elastic harmonic vibration of charged molecules along the H-bond; elastic reorientation of HB permanent dipoles; and rather free libration of permanent dipoles in ‘defects’ of water/ice structure. The suggested TV-dielectric relaxation mechanism allows us: (a) to remove the THz ‘deficit’ of loss ε″ inherent in previous theoretical studies; (b) to explain the THz loss and absorption spectra in supercooled (SC) water; and (c) to describe, in agreement with the experiment, the low- and high-frequency tails of the two bands of ice H 2O located in the range 10…300 cm − 1 . Specific THz dielectric properties of SC water are ascribed to association of water molecules, revealed in our study by transverse vibration of HB charged molecules.

The rotation of Io

The paper develops, in the framework of Hamiltonian mechanics, a theory of the rotation of Io, considered as a rigid body. The theory includes the perturbation due to Jupiter (considered as an oblate body) and the indirect perturbations due to the other Galilean satellites. In order to describe the orbit of Io around Jupiter, we use the synthetic theory of Lainey [2002, PhD dissertation, Observatoire de Paris], the result of a frequency analysis of a numerically integrated jovian system. The direct effects of the other Galilean satellites are found to be negligible, but their indirect effects are important. Our theory is consistent with the rigid body model and with Lainey's description of the orbit of Io, at least down to 10 −6 rad (0.2 arc-second). Of course the effects of the nonrigidity of Io and of a probable liquid core should be considered. We find a mean obliquity of 7.619 × 10 −4 rad (157 arc-second) and the period of the three free librations to be 13.25 days (free libration in longitude), 159.39 days (free libration in latitude), and 229.85 days (free wobble). Fourier series are produced describing, in the body frame, the motion of the polar axis of Jupiter, the motion of the unit vector pointing toward Jupiter, and the “motion of the pole” (the motion of the angular momentum with respect to the axis of largest inertia). Free librations (depending on three arbitrary parameters) are also computed.

The free precession and libration of Mercury

An analysis based on the direct torque equations including tidal dissipation and a viscous core–mantle coupling is used to determine the damping time scales of O ( 10 5 ) years for free precession of the spin about the Cassini state and free libration in longitude for Mercury. The core–mantle coupling dominates the damping over the tides by one to two orders of magnitude for the plausible parameters chosen. The short damping times compared with the age of the Solar System means we must find recent or on-going excitation mechanisms if such free motions are found by the current radar experiments or the future measurement by the MESSENGER and BepiColombo spacecraft that will orbit Mercury. We also show that the average precession rate is increased by about 30% over that obtained from the traditional precession constant because of a spin–orbit resonance induced contribution by the C 22 term in the expansion of the gravitational field. The C 22 contribution also causes the path of the spin during the precession to be slightly elliptical with a variation in the precession rate that is a maximum when the obliquity is a minimum. An observable free precession will compromise the determination of obliquity of the Cassini state and hence of C / M M R 2 for Mercury, but a detected free libration will not compromise the determination of the forced libration amplitude and thus the verification of a liquid core.

The Spin-Orbit Resonant Rotation of Mercury: A Two Degree of Freedom Hamiltonian Model

The paper develops a hamiltonian formulation describing the coupled orbital and spin motions of a rigid Mercury rotation about its axis of maximum moment of inertia in the frame of a 3:2 spin orbit resonance; the (ecliptic) obliquity is not constant, the gravitational potential of mercury is developed up to the second degree terms (the only ones for which an approximate numerical value can be given) and is reduced to a two degree of freedom model in the absence of planetary perturbations. Four equilibria can be calculated, corresponding to four different values of the (ecliptic) obliquity. The present situation of Mercury corresponds to one of them, which is proved to be stable. We introduce action-angle variables in the neighborhood of this stable equilibrium, by several successive canonical transformations, so to get two constant frequencies, the first one for the free spin-orbit libration, the other one for the 1:1 resonant precession of both nodes (orbital and rotational) on the ecliptic plane. The numerical values obtained by this simplified model are in perfect agreement with those obtained by Rambaux and Bois [Astron. Astrophys. 413, 381–393].

“Sloshing” Librations in Lopsided Disk Galaxies

A combined particle-mesh N-body/gridded hydrodynamics/stellar evolution code is employed to model the response of a galaxy disk to an impulsive perturbation. The resulting galaxy is examined for asymmetry in morphology and dynamics to answer the question of whether free librations in the plane of the disk can explain the relatively large proportion of asymmetric field spirals. The simulation produced a perturbed disk containing streamers of gas and stars, gas shocks, and m = 1 and m = 2 spiral structure. Strong kinematic peculiarities were generated as well. However, although the resulting disk reproduced many of the kinematic features observed in H I synthesis maps, the effects are relatively short-lived, lasting only on the order of a dynamical timescale. The similarities between the model and the tidally deformed galaxy NGC 5474, however, do leave open the possibility of forced sloshing modes playing a role in galaxies undergoing tidal interactions.

Numerical Simulations of the Orbits of the Galilean Satellites

Long-term numerical simulations of the orbits of the Galilean satellites are carried out using a realistic physical model. The free libration of the Laplace angle, which plays a central role in the classical description of the tidal evolution of the Laplace resonance, is isolated from the simulated orbits. The amplitude and the period of the libration agree well with the values obtained by J. Lieske. In addition, we point out that tidal dissipation models need to take into account the large variations in Io's eccentricity due to Jupiter's oblateness.

Lunar Laser Ranging: Glorious Past And A Bright Future

Lunar Laser Ranging (LLR), a part of the NASA Apollo program, has beenon-going for more than 30 years. It provides the grist for a multi-disciplinarydata analysis mill. Results exist for solid Earth sciences, geodesy and geodynamics,solar system ephemerides, terrestrial and celestial reference frames, lunar physics,general relativity and gravitational theory. Combined with other data, it treatsprecession of the Earth's spin axis, lunar induced nutation, polar motion/Earthrotation, Earth orbit obliquity to the ecliptic, intersection of the celestial equatorwith the ecliptic, luni-solar solid body tides, lunar tidal deceleration, lunar physicaland free librations, structure of the moon and energy dissipation in the lunar interior.LLR provides input to lunar surface cartography and surveying, Earth station and lunar retroreflector location and motion, mass of the Earth-moon system, lunar and terrestrial gravity harmonics and Love numbers, relativistic geodesic precession, and the equivalence principle of general relativity. With the passive nature of the reflectors and steady improvement in observing equipment and data analysis, LLR continues to provide state-of-the-art results. Gains are steady as the data-base expands. After more than 30 years, LLR remains the only active Apollo experiment. It is important to recognize examples of efficient and cost effective progress of research. LLR is just such an example.

Connaissance de la libration lunaire à l'ère de la télémétrie laser–Lune

A renovated knowledge of the lunar librations is exposed in the present paper. This updating derives both from progress of observational techniques of the Moon's motions and from theoretical methods of celestial dynamics. First is presented a summary of classical libration families such as the apparent librations, the physical librations, or the called free librations. Ambiguities and redundancies of this cutting out are discussed. On the one hand, lunar librations are strongly overlapped and datum of their branching is not immediate. On the other hand, new librations have been made into evidence. Starting from a causal analysis of libration nature, the necessary rearrangement of the lunar librations is first based on the modern knowledge of dynamical systems. It takes then into account the post-Newtonian gravitation acquisitions in use of reference systems required for dealing with rotation motions combined with translation ones. By the way of the new families of lunar librations, namely the potential and kinetical librations, the paper gives the libration amplitudes discovered at the joint epoch of the LLR experiment and elaborated methods of numerical integrations of the coupled spin–orbit motion of the Moon. The paper presents also a description of the lunar tides. It takes stock on the relation between the accurate study of the Moon's librations and the knowledge of its internal structure. Finally, it takes into account the lunar spin–orbit interaction whereby the mechanism of the secular acceleration of the Moon is matched by a secular slowing-down of its rotation.

Advanced Observations of Lunar Physical Librations and Gravitational Fields in Japanese Lunar Missions in the Near Future

AbstractIn the SELenological and ENgineering Explorer (SELENE) project which is the Japanese lunar program to be launched in 2003 by the Institute of Space and Astronautical Science (ISAS) and National Space Development Agency of Japan (NASDA), we measure angular distance between a radio transmitter on a relay satellite, that on the Moon and quasars by differential VLBI and determine amplitudes of the physical librations, gravitational harmonic coefficients of the Moon and lunar ephemeris with an accuracy one or two orders higher than before in cooperation with 4-way Doppler measurements and two-way Doppler and ranging measurements using the lunar orbiter and the relay satellite. We are proposing another selenodetic mission, In situ Lunar Orientation Measurement (ILOM) to study lunar rotational dynamics by direct observations of the lunar physical libration from the lunar surface with an accuracy of 1 milliarcsecond in a post-SELENE project which will be launched about three years after SELENE. Year-long trajectories of the stars provide information on various components of the physical librations and we will also try to detect the lunar free librations in order to investigate the lunar mantle and the liquid core.

Complements to Moons'1 Lunar Libration Theory

Analytical 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.