Free Libration Research Articles

Chaos over order: mapping 3D rotation of triaxial asteroids and minor planets

ABSTRACT Celestial bodies approximated with rigid triaxial ellipsoids in a two-body system can rotate chaotically due to the time-varying gravitational torque from the central mass. At small orbital eccentricity values, rotation is short-term orderly and predictable within the commensurate spin–orbit resonances, while at eccentricity approaching unity, chaos completely takes over. Here, we present the full three-dimensional rotational equations of motion around all three principal axes for triaxial minor planets and two independent methods of numerical solution based on Euler rotations and quaternion algebra. The domains of chaotic rotation are numerically investigated over the entire range of eccentricity with a combination of trial integrations of Euler’s equations of motion and the GALI(k) (Generalized Alignment Index) method. We quantify the dependence of the order–chaos boundaries on shape by changing a prolateness parameter, and find that the main 1:1 spin–orbit resonance disappears for specific moderately prolate shapes already at eccentricities as low as 0.3. The island of short-term stability around the main 1:1 resonance shrinks with increasing eccentricity at a fixed low degree of prolateness and completely vanishes at approximately 0.8. This island is also encroached by chaos on longer time-scales, indicating longer Lyapunov exponents. Trajectories in the close vicinity of the 3:2 spin–orbit resonance become chaotic at smaller eccentricities, but separated enclaves of orderly rotation emerge at eccentricities as high as 0.8. Initial perturbations of rotational velocity in latitude away from the exact equilibrium result in a spectrum of free libration, nutation, and polar wander, which is not well matched by the linearized analysis omitting the inertial terms.

Librations of a body composed of a deformable mantle and a fluid core

We present fully three-dimensional equations to describe the rotations of a body made of a deformable mantle and a fluid core. The model in its essence is similar to that used by INPOP (Integration Plan\'{e}taire de l'Observatoire de Paris), e.g. Viswanathan et al. (2019), and by JPL (Jet Propulsion Laboratory), e.g. Folkner et al. (2014), to represent the Moon. The intended advantages of our model are: straightforward use of any linear-viscoelastic model for the rheology of the mantle; easy numerical implementation in time-domain (no time lags are necessary); all parameters, including those related to the "permanent deformation", have a physical interpretation. The paper also contains: 1) A physical model to explain the usual lack of hydrostaticity of the mantle (permanent deformation). 2) Formulas for free librations of bodies in and out-of spin-orbit resonance that are valid for any linear viscoelastic rheology of the mantle. 3) Formulas for the offset between the mantle and the idealized rigid-body motion (Peale's Cassini states). 4) Applications to the librations of Moon, Earth, and Mercury that are used for model validation.

A benchmarking and sensitivity study of the full two-body gravitational dynamics of the DART mission target, binary asteroid 65803 Didymos

NASA's Double Asteroid Redirection Test (DART) is designed to be the first demonstration of a kinetic impactor for planetary defense against a small-body impact hazard. The target is the smaller component of the binary asteroid 65,803 Didymos. We have conducted high-fidelity rigid full two-body simulations of the mutual dynamics of this system in a broad benchmarking exercise to find the best simulation methodologies, and to understand the sensitivity of the system to initial conditions. Due to the non-spherical shapes of the components and their close proximity, the components cannot be treated as point masses and so the dynamics differ significantly from a simple Keplerian orbit, necessitating the use of numerical simulations to fully capture the system's dynamics. We find that the orbit phase (angular position or true anomaly) of the secondary is highly sensitive to the initial rotation phase of the primary, making prediction of the secondary's location from numerical simulation challenging. Finally, we show that the DART impact should induce significant free and forced librations on the secondary. If this libration can be measured by ESA's recently approved follow-up spacecraft, Hera, it may be possible to constrain properties of the secondary's interior structure.

On the secondary’s rotation in a synchronous binary asteroid

ABSTRACT This article studies the secondary’s rotation in a synchronous binary asteroid system in which the secondary enters the 1:1 spin-orbit resonance. The model used is the planar full two-body problem, composed of a spherical primary plus a triaxial ellipsoid secondary. Compared with classical spin-orbit work, there are two differences: (1) influence of the secondary’s rotation on the mutual orbit is considered and (2) instead of the Hamiltonian approach, the approach of periodic orbits is adopted. Our studies find the following. (1) The genealogy of the two families of periodic orbits is the same as that of the families around triangular libration points in the restricted three-body problem. That is, the long-period family terminates on to a short-period orbit travelling N times. (2) In the limiting case where the secondary’s mass is negligible, our results can be reduced to classical spin-orbit theory, by equating the long-period orbit with free libration and the short-period orbit with the forced libration caused by orbit eccentricity. However, the two models show obvious differences when the secondary’s mass is non-negligible. (3) By studying the stability of periodic orbits for a specific binary asteroid system, we are able to obtain the maximum libration amplitude of the secondary (which is usually less than 90°) and the maximum mutual orbit eccentricity that does not break the secondary’s synchronous state. We also find an anti-correlation between the secondary’s libration amplitude and the orbit eccentricity. The (65803) Didymos system is taken as an example to show the results.

Estimation of the lunar free libration modes based on the recent ephemerides

The accuracy of lunar laser ranging (LLR) experiment has already increased to about 1 cm in recent years. When the LLR data joined to the lunar dynamical model updated from DE430, three independent numerically integrated ephemerides called DE436, EPM2017 and INPOP17a were released in 2017. It prompts us to reanalysis the lunar rotation parameters of the ephemerides, and also, especially estimation of the lunar free libration modes. To this end, we performed a Fourier analysis and least-square fit with the data from these three ephemerides to estimate lunar free librations. The results show that the free librations for each ephemeris are consistent with each other and with the previously published DE421 and DE430 free libration values; minor differences in detail may result from differences in the physical parameters contained in each ephemeris. The Euler angles integrated in each ephemeris however, show a significant difference before 1800, relative to the difference after 1970. This indicates that special attention must be paid to backward integration when building ephemerides.

Cassini's motions of the Moon and Mercury and possible excitations of free librations

On the basis of conditionally-periodic solutions of Hamiltonian systems at resonance of main frequencies Cassini's motions, their stability, Cassini's angle and periods of free librations of the Moon and Mercury have been recently studied and determined. The generalized formulations of Cassini's laws for the motion of the Moon and Mercury, that are considered as absolutely rigid non-spherical bodies, have been determined. The study of the second approximation equations of the desired quasi-periodic solutions in the case of the Moon allows us to determine the constant components of the first order for six Andoyer variables and the constant component of the second order for the angular velocity of the Moon. These effects are caused by the influence of the third harmonic of selenopotential. In this paper, these effects are described by analytical formulas, the dynamic and geometric interpretations are given, and a new interpretation of Mercury's motion under the generalized Cassini's laws has been proposed. Predictions of the existence of free librations of significant amplitude in the Mercury longitude, that are confirmed by the radar measurements data of the Mercury angular velocity, and in its pole motion in the body and in space have been made. The mechanism describing free librations of celestial bodies and their pole oscillations has been proposed due to the forced relative oscillations and wobble of the core-mantle system of celestial bodies (Moon, Mercury, Earth and other bodies in the solar system) under gravitational action of the external celestial bodies. The paper shows that the ascending node of equator of Mercury (and the intermediate plane orthogonal to the angular momentum) of epoch 2000.0 on the ecliptic does not coincide with the ascending node of orbital plane of Mercury on the same plane, and is ahead of it at an angle 23º4’. Angular momentum vector of the rotational motion of Mercury forms a constant angle ρG=4'1±1'1 with normal to the moveable plane of its orbit. The observed inclination of the angular velocityρω=2'1±0'1, can be considered as a possible evidence of a significant amplitude of the poles free motion of the Mercury rotation axis (c amplitude of about 2′- 4′).

Determination of the free lunar libration modes from ephemeris DE430

The Moon’s physical librations have been extensively studied, and elaborate researches have been developed for the purpose of deriving accurate modes of free librations. Our motivation comes from the Planetary and Lunar Ephemeris DE430 by JPL/NASA, which was created in April 2013, and is reported to be the most accurate lunar ephemeris today using the data from Gravity Recovery and Interior Laboratory (GRAIL). Therefore, the residuals after fitting the model have reduced owing to improvement in the libration models, and the free librations embedded in the Euler angles have also improved. We use Fourier analysis to extract the approximate frequencies from DE430 and then a quadratic interpolation method is used to determine higher accuracy frequencies. With the frequencies, the linear least-squares fitting method is employed to fit the lunar physical librations to DE430. From this analysis we identified the three modes of free physical librations, and estimated the amplitudes as in longitude, in latitude and for the wobble, with the respective periods of 1056.16, 8806.9 and 27262.99 d. Since the free librations damp with time, they require recent excitation or a continuous stimulating mechanism in order to sustain.

The analytical and numerical approaches to the theory of the Moon’s librations: Modern analysis and results

Observing physical librations of celestial bodies and the Moon represents one of the astronomical methods of remotely assessing the internal structure of a celestial body without conducting expensive space experiments. The paper contains a review of recent advances in studying the Moon’s structure using various methods of obtaining and applying the lunar physical librations (LPhL) data.In this article LPhL simulation methods of assessing viscoelastic and dissipative properties of the lunar body and lunar core parameters, whose existence has been recently confirmed during the seismic data reprocessing of “Apollo” space mission, are described. Much attention is paid to physical interpretation of the free librations phenomenon and the methods for its determination.In the paper the practical application of the most accurate analytical LPhL tables (Rambaux and Williams, 2011) is discussed. The tables were built on the basis of complex analytical processing of the residual differences obtained when comparing long-term series of laser observations with the numerical ephemeris DE421.In the paper an efficiency analysis of two approaches to LPhL theory is conducted: the numerical and the analytical ones. It has been shown that in lunar investigation both approaches complement each other in various aspects: the numerical approach provides high accuracy of the theory, which is required for the proper processing of modern observations, the analytical approach allows to comprehend the essence of the phenomena in the lunar rotation, predict and interpret new effects in the observations of lunar body and lunar core parameters.

Tides in a body librating about a spin–orbit resonance: generalisation of the Darwin–Kaula theory

The Darwin-Kaula theory of bodily tides is intended for celestial bodies rotating without libration. We demonstrate that this theory, in its customary form, is inapplicable to a librating body. Specifically, in the presence of libration in longitude, the actual spectrum of Fourier tidal modes differs from the conventional spectrum rendered by the Darwin-Kaula theory for a non-librating celestial object. This necessitates derivation of formulae for the tidal torque and the tidal heating rate, that are applicable under libration. We derive the tidal spectrum for longitudinal forced libration with one and two main frequencies, generalisation to more main frequencies being straightforward. (By main frequencies we understand those emerging due to the triaxiality of the librating body.) Separately, we consider a case of free libration at one frequency (once again, generalisation to more frequencies being straightforward). We also calculate the tidal torque. This torque provides correction to the triaxiality-caused physical libration. Our theory is not self-consistent: we assume that the tidal torque is much smaller than the permanent-triaxiality-caused torque; so the additional libration due to tides is much weaker than the main libration due to the permanent triaxiality. Finally, we calculate the tidal dissipation rate in a body experiencing forced libration at the main mode, or free libration at one frequency, or superimposed forced and free librations.

Determining the amplitude of Mercury's long period librations with the BepiColombo radio science experiment⋆

The Mercury Orbiter Radio science Experiment (MORE) is one of the experiments on-board the ESA/JAXA BepiColombo mission to Mercury. A crucial goal of MORE is to determine the gravity field and rotational state of Mercury in order to enable a better understanding of the planet's geophysics. The authors have recently reported on the results of a set of simulations of the MORE gravimetry and rotation experiments, carried out with the dedicated ORBIT14 software. Since that time, the launch date has been postponed twice, leading to a shift of more than one year in the orbital phase of the mission. Actually, the updated schedule results in a more suitable planetary configuration to determine the amplitude of the forced librations in longitude induced by Jupiter. In fact, the amplitude can be considerably enhanced due to a near-resonance with the free librations period, a key parameter to constrain the interior structure of Mercury. We show that the newest launch date allows the measurement of the long period librations amplitude forced by Jupiter with an accuracy of some tenth of arcseconds, a significant improvement with respect to the results with the previous mission schedule.

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: ɛD = 4.42 × 10−4, μD = 2.83 × 10−4 (ɛD + μD = 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.

Influence of an inner core on the long-period forced librations of Mercury

The planetary perturbations on Mercury’s orbit lead to long-period forced librations of Mercury’s mantle. These librations have previously been studied for a planet with two layers: a mantle and a liquid core. Here, we calculate how the presence of a solid inner core in the liquid outer core influences the long-period forced librations. Mantle–inner core coupling affects the long-period libration dynamics mainly by changing the free libration: first, it lengthens the period of the free libration of the mantle, and second, it adds a second free libration, closely related to the free gravitational oscillation between the mantle and inner core. The two free librations have periods between 2.5 and 18y depending on the internal structure. We show that large amplitude long-period librations of a few tens of arcsec are generated when the period of a planetary forcing approaches one of the two free libration periods. These amplitudes are sufficiently large to be detectable by spacecraft measurements of the libration of Mercury. The amplitudes of the angular velocity of Mercury’s mantle at planetary forcing periods are also amplified by the resonances, but remain much smaller than the current precision of Earth-based radar observations unless the period is very close to a free libration period. The inclusion of mantle–inner core coupling in the rotation model does not significantly improve the fit to the radar observations. This implies that it is not yet possible to determine the size of the inner core of Mercury on the basis of available observations of Mercury’s rotation rate. Future observations of the long-period librations may be used to constrain the interior structure of Mercury, including the size of its inner core.

The role of Mercury’s core density structure on its longitudinal librations

As Mercury orbits the Sun, gravitational torques on its ellipsoidal figure give rise to a longitudinal libration. Earth-based radar observations indicate that the outermost part of the core is fluid and only the mantle participates in the libration. If a solid inner core is present, it can alter the libration of the mantle through gravitational coupling. Previous studies have shown that the influence of the inner core on libration remains small, though not negligible if its radius is larger than 1000km. However, these studies considered a simplified model of the libration dynamics, where the density in both the inner core and fluid core were assumed uniform. Here, we build a model of Mercury’s libration that takes into account the radially varying density profile within the core. Our results indicate that the amplitude of the 88-day mantle libration is slightly larger than that predicted on the basis of the uniform density model, though the difference is well below the current error bars in observations. We also show that the period of the free libration of the combined inner core and mantle is largely unaffected by the details of the core density structure. However, the period of the second free mode of libration – broadly described by the gravitational oscillation between the mantle and inner core – is significantly longer when a more realistic core density profile is used. We discuss the implication of this result, and we also show how the role of the inner core can be taken into account when observations of gravity and rotation are combined to form constraints on the interior structure of Mercury.

Internal forcing of Mercury’s long period free librations

Observations of Mercury’s spin rate do not rule out that, in addition to the forced 88-day mantle libration induced by solar torques, a decadal timescale libration may also be present. It has been proposed that this signal represents an amplified forced libration caused by the periodic 11.86yr perturbation on Mercury’s orbit by Jupiter. Here, we investigate the possibility that a decadal libration may be produced by a forcing internal to the planet. Our mechanism is based on the presence of time-dependent zonal flows generated by turbulent convection in Mercury’s fluid core. Through electromagnetic coupling, these flows entrain longitudinal displacements of the inner core, which then entrain mantle librations by gravitational coupling. We construct a simple model to capture this exchange of angular momentum. Although core zonal flows are expected to have a broad frequency spectrum, we show that the resulting mantle libration is dominated by the amplification that occurs at the period of the free decadal libration of the combined mantle and inner core. Our results suggest that for plausible values of Mercury’s internal magnetic field, if the inner core of Mercury is large (⩾1000km), decadal mantle libration amplitudes of the order of 1arcsec can be generated by zonal flows of the order of 1kmyr−1. Conversely, if future observations can robustly determine an upper bound on the amplitude of Mercury’s decadal librations, our results can be used as constraints on the convective dynamics in Mercury’s core.

The effect of tides and an inner core on the forced longitudinal libration of Mercury

Mercury's longitudinal libration at a period of 88days depends on the moment of inertia of the planet's silicate outer part and as such contains information on the interior structure and composition of Mercury. Secondary effects on the libration may affect the interpretation of libration observations in terms of interior structure properties and may also provide further information on Mercury's interior. Here we assess the importance of the effects of tides and the existence of a solid inner core on the librations of Mercury with particular focus on the 88days libration amplitude. Tides affect the librations of Mercury by changing the gravitational torque exerted by the Sun on Mercury and by changing the polar moment of inertia. We show that they slightly decrease the 88days libration amplitude by about 1–2m, which is below the current and future expected observational precision. We further demonstrate that the effect of an inner core on Mercury's 88days libration amplitude is also below the observational precision for small inner cores, but could be observed if Mercury's inner core is larger than at least 1000km. An inner core also changes considerably the free libration period by up to 25% if the inner core is very large. Besides giving information on the moment of inertia of the silicate shell, observations of Mercury's libration can therefore also yield information on the inner core.

Theory of longitude librations of mercury based on the ground radar observations

On the basis of the analytic theory of longitude libration of Mercury (in the elliptical orbit) and the data of determination of values of the angular velocity of Mercury rotation obtained using high-precision complex method of ground radar tracking, the amplitude, phase, and period of free libration and amplitudes of five fundamental harmonics (annual, semi-annual, third annual, …) of forced librations in longitude of the planet are determined.

The free librations of Mercury and the size of its inner core

[1] The predicted period of the free libration of Mercury's mantle, if fully decoupled from the core, is approximately 12 yr. If the innermost part of the core is solid, this period is altered and a second free mode is possible. The period of these two modes depend on the mean solar torque acting on the elliptical figures of the mantle and inner core, and also on the gravitational torque they exert on one another. We show that one mode is well approximated by the free gravitational oscillation between the mantle and inner core that would occur in the absence of a solar torque; its period is as small as 4 yr depending on inner core size and other model parameters. The second free mode consists of an in-phase libration of the mantle and inner core. When the inner core is small, the period of this mode coincides with that of the decoupled free mantle libration of 12 yr. However, the period lengthens considerably when the inner core radius is larger than 1000 km, and can be as high as approximately 19 yr if Mercury's core is almost fully solidified. These modes should affect Mercury's libration dynamics at decadal timescale. This suggests that, in principle, constraints on the size of Mercury's inner core may be extracted from its long term libration motion, though in practice this may be difficult given data limitations.

The Moon’s physical librations and determination of their free modes

The Lunar Laser Ranging experiment has been active since 1969 when Apollo astronauts placed the first retroreflector on the Moon. The data accuracy of a few centimeters over recent decades, joined to a new numerically integrated ephemeris, DE421, encourages a new analysis of the lunar physical librations of that ephemeris, and especially the detection of three modes of free physical librations (longitude, latitude, and wobble modes). This analysis was performed by iterating a frequency analysis and linear least-squares fit of the wide spectrum of DE421 lunar physical librations. From this analysis we identified and estimated about 130–140 terms in the angular series of latitude librations and polar coordinates, and 89 terms in the longitude angle. In this determination, we found the non-negligible amplitudes of the three modes of free physical libration. The determined amplitudes reach 1.296′′ in longitude (after correction of two close forcing terms), 0.032′′ in latitude and 8.183′′ × 3.306′′ for the wobble, with the respective periods of 1056.13 days, 8822.88 days (referred to the moving node), and 27257.27 days. The presence of such terms despite damping suggests the existence of some source of stimulation acting in geologically recent times.

Theory of the rotation of Janus and Epimetheus

The saturnian coorbital satellites Janus and Epimetheus present a unique dynamical configuration in the Solar System, because of high-amplitude horseshoe orbits, due to a mass ratio of order unity. As a consequence, they swap their orbits every 4 years, while their orbital periods is about 0.695 days. Recently, Tiscareno et al. (Tiscareno, M.S., Thomas, P.C., Burns, J.A. [2009]. Icarus 204, 254–261) got observational informations on the shapes and the rotational states of these satellites. In particular, they detected an offset in the expected equilibrium position of Janus, and a large libration of Epimetheus. We here propose to give a three-dimensional theory of the rotation of these satellites in using these observed data, and to compare it to the observed rotations. We consider the two satellites as triaxial rigid bodies, and we perform numerical integrations of the system in assuming the free librations as damped. The periods of the three free librations we get, associated with the three dimensions, are respectively 1.267, 2.179 and 2.098 days for Janus, and 0.747, 1.804 and 5.542 days for Epimetheus. The proximity of 0.747 days to the orbital period causes a high sensitivity of the librations of Epimetheus to the moments of inertia. Our theory explains the amplitude of the librations of Janus and the error bars of the librations of Epimetheus, but not an observed offset in the orientation of Janus.

In Situ Measurements of Lunar Rotation for the Study of the Interior of the Moon

Measurements of physical and free librations of lunar rotation are important because they provide information of the physical state of the lunar interior. For example, we can discuss if the lunar core is molten from the amplitudes of libration terms. Previously only passive LLR (Lunar Laser Ranging) using CCR (corner cube reflectors) has been applied for the detailed study of lunar librations. As for candidate instruments for SELENE-II (forthcoming lunar landing mission by JAXA), we propose detailed measurements of lunar rotation by ILOM (In-situ Lunar Orientation Measurement) and IVLBI (Inverse-VLBI) in addition to LLR.