Free Wobble Research Articles

Detection of the Chandler Wobble of Mars From Orbiting Spacecraft

AbstractFor the first time for any planetary body other than the Earth, the free wobble of the pole called the Chandler wobble has been detected for Mars with a period of 206.9 ± 0.5 days and amplitude of 10 cm from radio tracking observations of Mars Odyssey, Mars Reconnaissance Orbiter (MRO), and Mars Global Surveyor (MGS), in order of decreasing sensitivity. The motion of the rotation pole location on the surface of Mars, or polar motion, is observed using two different approaches: (1) joint global estimates of Mars' orientation and its gravity field and (2) time series solutions of C21 and S21. For Mars interior models, the Chandler wobble period is combined with other measurements including the moments of inertia from our estimated precession rate and tidal Love number k2 = 0.169 ± 0.006. The Chandler wobble period constrains the rheology of the Martian mantle and in particular its long‐term frequency dependence.

Chandler Wobble and Free Core Nutation: Theory and Features

Being a torque free motion of the rotating Earth, Chandler wobble is the major component in the Earth’s polar motion with amplitude about 0.05-0.2 arcsec and period about 430-435 days. Free core nutation, also called nearly diurnal free wobble, exists due to the elliptical core-mantle boundary in the Earth and takes almost the whole part of un-modelled variation of the Earth’s pole in the celestial sphere beside precession and nutation. We hereby present a brief summary of their theories and report their recent features acquired from updated datasets (EOP C04 and ECMWF) by using Fourier transform, modelling, and wavelet analysis. Our new findings include (1) period-instability of free core nutation between 420 and 450 days as well as its large amplitude-variation, (2) re-determined Chandler period and its quality factor, (3) fast decrease in Chandler amplitude after 2010.

New solution for the Earth’s free wobble and its geophysical implications

In this paper, the theory of the free wobble of the triaxial Earth is developed and new conclusions are drawn: the Euler period should be actually expressed by the first kind of complete elliptic integral; the trace of the free polar motion is elliptic and the orientations of its semi-minor and major axes are approximately parallel to the Earth’s principal axes A and B, respectively. In addition, the present theory shows that there is a mechanism of frequency-amplitude modulation in the Chandler wobble, which might be a candidate for explaining the correlation between the amplitude and period of the Chandler wobble.

On mechanical Q parameters of the lower mantle inferred from data on the Earth’s free oscillations and nutation

A new method for the interpretation of recent tidal and astrometry data based on integral relations connecting the mantle Q parameters to variations in frequencies of the main Earth’s spheroidal oscillation, nearly diurnal free nutation, Chandler wobble, and Love numbers is proposed in this paper. The normalized weighting functions in the aforementioned integral relations are shown to be very close to each other in a wide range of oscillation periods (from 54 minutes to 14 months) for the actual Earth’s model and therefore the detection of the dependence of the mantle Q parameters averaged with these weighting functions on frequency does not need to be solved for the inverse problem of the Q parameters as a function of the oscillation frequency and depth: instead, it is sufficient to determine the dependence of the adequate integral characteristics of the mantle’s inelasticity only on frequency. This makes it possible to significantly simplify the solution of the inverse problem and improve the rheologic models of the lower mantle.

Free Wobble of the Triaxial Earth: Theory and Comparisons with International Earth Rotation Service (IERS) Data

Earth’s free wobble is often referred to as the Euler wobble (for the rigid case) or the Chandler wobble for the real case. In this study, we investigate the theory of the free wobble of the triaxial Earth and demonstrate that: (1) the Euler period should actually be expressed by the complete elliptic integral of first kind, and (2) the trace of the free polar motion is elliptic, with the orientations of its semi-minor and major axes being approximately parallel to the Earth’s principal axes A and B, respectively. Numerical calculations show that, due to the triaxiality of the Earth, the spin rate ω 3 fluctuates with the semi-Euler/Chandler period, although its amplitude (about 10−19 rad/s) is rather small and beyond the present measurement accuracy; the tilt of the instantaneous spin axis (or the amplitude of the free wobble), θ, has a fluctuation whose amplitude is around 0.34 milli-arcsecond (mas), which could be detected by present observations. Thus, we conclude that the Earth’s triaxial nature has little impact on ω 3, but has an influence on the polar motion which should not be ignored. On the other hand, our study shows that there is a mechanism of frequency–amplitude modulation in the Chandler wobble which might be a candidate to explain the correlation between the amplitude and period of the Chandler wobble. We compare the theoretical polar parameters (m 1, m 2) with the observed values for the Chandler components obtained from the data EOP (IERS) C 04, and find that they coincide with each other quite well, especially for recent years. In addition, a polar wander towards 76.7°W, which is in agreement with previous results given by other scientists, is also obtained.

Determination of the Earth’s free core nutation parameters by using tidal gravity data

The Earth’s free core nutation (FCN) is a retrograde eigenmode which is attributed to the interaction between the solid mantle and the liquid core of the rotational elliptical Earth. This mode appears as an eigenmode of nearly diurnal free wobble (NDFW) in a terrestrial reference frame with a period of about one day (XU et al, 2001). Therefore, the NDFW will lead to an obvious resonance enhancement in the diurnal tidal gravity observations, especially those of the tidal waves with frequencies closed to its eigenfrequency such as P1, K1, Ψ1 and Φ1. The FCN resonance parameters can be retrieved accurately by high-precision tidal gravity observations, especially those recorded with the superconducting gravimeters (SG). The Global Geodynamics Project (GGP) organized by IUGG took it as an important content for determining the FCN resonance parameters by using gravity data. However, the results are affected by many factors such as station location, background noise, the selection of the tide-generating potential tables, ocean tide models, data processing techniques and so on. In our study, the FCN parameters will be retrieved by using the SG observations at Wuhan, and the effects of the choices of various tide-generating potential tables, oceanic models and weight functions on the estimation of the FCN parameters will be discussed in detail.

A global solution for the Mars static and seasonal gravity, Mars orientation, Phobos and Deimos masses, and Mars ephemeris

With the collection of six years of MGS tracking data and three years of Mars Odyssey tracking data, there has been a continual improvement in the JPL Mars gravity field determination. This includes the measurement of the seasonal changes in the gravity coefficients (e.g., J ¯ 2 , J ¯ 3 , C ¯ 21 , S ¯ 21 , C ¯ 31 , S ¯ 31 ) caused by the mass exchange between the polar ice caps and atmosphere. This paper describes the latest gravity field MGS95J to degree and order 95. The improvement comes from additional tracking data and the adoption of a more complete Mars orientation model with nutation, instead of the IAU 2000 model. Free wobble of the Mars' spin axis, i.e. polar motion, has been constrained to be less than 10 mas by looking at the temporal history of C ¯ 21 and S ¯ 21 . A strong annual signature is observed in C ¯ 21 , and this is a mixture of polar motion and ice mass redistribution. The Love number solution with a subset of Odyssey tracking data is consistent with the previous liquid outer core determination from MGS tracking data [Yoder et al., 2003. Science 300, 299–303], giving a combined solution of k 2 = 0.152 ± 0.009 using MGS and Odyssey tracking data. The solutions for the masses of the Mars' moons show consistency between MGS, Odyssey, and Viking data sets; Phobos GM = ( 7.16 ± 0.005 ) × 10 −4 km 3 / s 2 and Deimos GM = ( 0.98 ± 0.07 ) × 10 −4 km 3 / s 2 . Average MGS orbit errors, determined from differences in the overlaps of orbit solutions, have been reduced to 10-cm in the radial direction and 1.5 m along the spacecraft velocity and normal to the orbit plane. Hence, the ranging to the MGS and Odyssey spacecraft has resulted in position measurements of the Mars system center-of-mass relative to the Earth to an accuracy of one meter, greatly reducing the Mars ephemeris errors by several orders of magnitude, and providing mass estimates for Asteroids 1 Ceres, 2 Pallas, 3 Juno, 4 Vesta, and 324 Bamberga.

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.

Normal modes of synchronous rotation

The dynamics of synchronous rotation and physical librations are revisited in order to establish a conceptually simple and general theoretical framework applicable to a variety of problems. Our motivation comes from disagreements between the results of numerical simulations and those of previous theoretical studies, and also because different theoretical studies disagree on basic features of the dynamics. We approach the problem by decomposing the orientation matrix of the body into perfectly synchronous rotation and deviation from the equilibrium state. The normal modes of the linearized equations are computed in the case of a circular satellite orbit, yielding both the periods and the eigenspaces of three librations. Libration in longitude decouples from the other two, vertical modes. There is a fast vertical mode with a period very close to the average rotational period. It corresponds to tilting the body around a horizontal axis while retaining nearly principal-axis rotation. In the inertial frame, this mode appears as nutation and free precession. The other vertical mode, a slow one, is the free wobble. The effects of the nodal precession of the orbit are investigated from the point of view of Cassini states. We test our theory using numerical simulations of the full equations of the dynamics and discuss the disagreements among our study and previous ones. The numerical simulations also reveal that in the case of eccentric orbits large departures from principal-axis rotation are possible due to a resonance between free precession and wobble. We also revisit the history of the Moon's rotational state and show that it switched from one Cassini state to another when it was at 46.2 Earth radii. This number disagrees with the value 34.2 derived in a previous study.

Parameters of the fluid core resonance inferred from superconducting gravimeter data

Abstract We have estimated the parameters of fluid core resonance (FCR) due to the nearly diurnal free wobble of the Earth's core based on the superconducting gravimeter (SG) data obtained at the following four observation sites; Esashi and Matsushiro in Japan, Canberra in Australia and Membach in Belgium. By fitting the tidal admittances normalized with the O1 wave at each site to a model of the damped harmonic oscillator, we obtained values of 429.66 ± 1.43 sidereal days, 9350–10,835, −4.828E−4 ± 3.4E−6, −3.0E−5 ± 4.5E−6 for the eigenperiod, the Q-value and the real and imaginary parts of the resonance strength, respectively. Our values obtained from only using the gravity data are very consistent with those inferred from the VLBI nutation data. Our study strongly indicates that the systematic difference between two estimations from the gravity and the nutation in particular for the Q-value, which has been shown in previous works, is mainly caused by the inaccurate correction for the ocean tide effects. The error in the ocean tide correction is discussed based on the comparison among four global ocean tide models; Schwiderski model (1980), NAO.99b (Matsumoto et al., 2000), CSR4.0 (Eanes and Bettadpur, 1994) and GOT99.2b (Ray, 1999).

Decadal free polar motion of triaxial Earth

SUMMARY By using the elliptic function solution, the exact rotation solution of triaxial Earth is provided as motion of couple periods with an image decadal period longitudinally and Chandler wobble as transverse wave. To obtain the evident function result, we had to treat with Euler rotation equations with three times separated polar motions. The image period means transverse wave modulation that appears as phase modulation and the phase time series of the free wobble show decadal variation. The decadal wobble causes frequency modulation for free polar motion. The decadal wobble has 14.6 yr period that is almost 73/6 times or about 12 times of 1.191 yr period for Chandler wobble so that subharmonic resonance takes place. Applying subharmonic resonance theory in non-linear oscillations, the amplitude of polar motions excited by the decadal additional wobble is provided as about 8.2 times magnifying the excitation amplitude. The decadal wobble of amplitude about 13 mas excites the polar motion on the average amplitude of 132 mas in addition of an about 108 mas amplitude modulation, range in (24, 240). This amplitude variation is almost identical to the observed data series.

Consistent atmospheric and oceanic excitation of the Earth's free polar motion

SUMMARY Earth orientation parameters as observed by space-geodetic techniques show a broad spectrum of frequencies. Many efforts have been made to relate spectral peaks to geophysical processes, such as atmospheric or oceanic variations. However, the mechanisms of excitation of some typical oscillations are still unclear. In order to gain insight into the rotational dynamics of the Earth, the non-linear gyroscopic Dynamic Model for Earth Rotation and Gravity (DyMEG) has been developed at DGFI (Deutsches Geodatisches Forschungsinstitut). The present paper studies the superposed effect of atmospheric and oceanic excitation on the free rotation of a viscoelastic gyro. In contrast to former investigations, period and damping of the Earth's free wobble (Chandler wobble) are generated by the gyro from geometrical and rheological parameters. Numerical results for polar motion in the time domain demonstrate that DyMEG creates a damped oscillation in absence of geophysical and gravitational excitations. Damping vanishes when atmospheric and oceanic angular momentum is regarded. Simulated polar mo- tion for the period from 1975 to 1994 forced by consistent atmospheric and oceanic angular momentum shows significant correlation with geodetic observations, which indicates that the free gyroscopic model DyMEG is able to reproduce realistic variations of the Earth's rotation. Spectral analyses of both atmospheric and oceanic excitations give no hint for increased power in the Chandler frequency band as it was stated for the maintenance of the Chandler wobble in recent research studies. Thus, stochastic signals in the climate dynamics as caused by both the weather and oceanic mass redistributions are found to be a sufficient source to maintain the amplitude of the Earth's free wobble by resonant interaction.

Hydromagnetic and Gravitomagnetic Crust‐Core Coupling in a Precessing Neutron Star

We consider two types of mechanical coupling between the crust and the core of a precessing neutron star. First, we find that a hydromagnetic (MHD) coupling between the crust and the core strongly modifies the star's precessional modes when $t_a\le\sim (T_s\times T_p)^{1/2}$; here $t_a$ is the Alfven crossing timescale, and $T_s$ and $T_p$ are the star's spin and precession periods, respectively. We argue that in a precessing pulsar PSR B1828-11 the restoring MHD stress prevents a free wobble of the crust relative to the non-precessing core. Instead, the crust and the proton-electron plasma in the core must precess in unison, and their combined ellipticity determines the period of precession. Link has recently shown that the neutron superfluid vortices in the core of PSR B1828-11 cannot be pinned to the plasma; he has also argued that this lack of pinning is expected if the proton Fermi liquid in the core is type-I superconductor. In this case, the neutron superfluid is dynamically decoupled from the precessing motion. The pulsar's precession decays due to the mutual friction between the neutron superfluid and the plasma in the core. The decay is expected to occur over tens to hundreds of precession periods and may be measurable over a human lifetime. Such a measurement would provide information about the strong n-p interaction in the neutron-star core. Second, we consider the effect of gravitomagnetic coupling between the neutron superfluid in the core and the rest of the star and show that this coupling changes the rate of precession by about 10%. The general formalism developed in this paper may be useful for other applications.

Distributions of fluctuations in motion of the Earth’s pole

1. According to observation data and measurements made by the International Earth Rotation Service in the last 15‐20 years, motion of the Earth’s pole involves a principal component (free nutation and Chandlers wobble) with an amplitude of 0.20  —0.25  and a period of 433 ∠ 2 sidereal days [1], a regular annual component (365 sidereal days) with an amplitude of about 0.07  — 0.08  , and a comparatively slow irregular drift (trend) of the axis of the Earth’s figure. The annual wobbles of the Earth’s axis are caused by the moment of gravitational forces of the Sun, the Earth’s revolution around the Sun, and the diurnal tides of the mantle [2, 3]. The causes and mechanism of exciting the annual wobbles are usually attributed to seasonal geophysical phenomena [4, 5]. Measurements of motion of the Earth’s pole were statistically analyzed in many works (see, e.g., [6]). Analytical stochastic spectrally correlation models of motion of the Earth’s pole were developed in [7‐9] on the basis of celestial mechanics. In this paper continuing studies [7‐9], we consider one-dimensional distributions of fluctuations in motion of the deformable Earth.

Experimental earth tidal models in considering nearly diurnal free wobble of the Earth's liquid core

Based on the 28 series of the high precision and high minute sampling tidal gravity observations at 20 stations in Global Geodynamics Project (GGP) network, the resonant parameters of the Earth's nearly diurnal free wobble (including the eigenperiods, resonant strengths and quality factots) are precisely determined. The discrepancy of the eigenperiod between observed and theoretical values is studied, the important conclusion that the real dynamic ellipticity of the liquid core is about 5% larger than the one under the static equilibrium assumption is approved by using our gravity technique. The experimental Earth's tidal gravity models with considering the nearly diurnal free wobble of the Earth's liquid core are constructed in this study. The numerical results show that the difference among three experimental models is less than 0.1%, and the largest discrepancy compared to those widely used nowdays given by Dehant (1999) and Mathews (2001) is only about 0.4%. It can provide with the most recent real experimental tidal gravity models for the global study of the Earth's tides, geodesy and space techniques and so on.

Experimental earth tidal models in considering nearly diurnal free wobble of the Earth?s liquid core

Experimental earth tidal models in considering nearly diurnal free wobble of the Earth?s liquid core

Oceanic excitation of the chandler wobble

We estimate the oceanic contribution to the excitation of the Chandler wobble using an 11-year time series of ocean angular momentum. The time domain comparison of this series with the non-atmospheric excitation inferred from the polar motion and atmospheric angular momentum data shows a high correlation over most of the period of 1985 to 1996 when they overlap. The frequency domain comparison yields a high coherence in the vicinity of the resonant Chandler frequency. In terms of the free wobble excitation power, we compute the ocean contribution to be 11.9 mas 2/cpy, which is approximately the same amount as supplied by the atmosphere. The combined action of the atmosphere and oceans provides about 80% of the power needed to maintain the wobble observed during 1985 to 1996, with the remaining deficit being at the level of one standard deviation of the excitation power estimated from a 100-year record of polar motion data.

Euler, Jacobi, and missions to comets and asteroids

Whenever a freely spinning body is found in a complex rotational state, this means that either the body experienced some interaction within its relaxation-time span, or that it was recently “prepared” in a non-principal state. Both options are encountered in astronomy where a wobbling rotator is either a recent victim of an impact or a tidal interaction, or is a fragment of a disrupted progenitor. Another factor (relevant for comets) is outgassing. By now, the optical and radar observational programmes have disclosed that complex rotation is hardly a rare phenomenon among the small bodies. Due to impacts, tidal forces and outgassing, the asteroidal and cometary precession must be a generic phenomenon: while some rotators are in the state of visible tumbling, a much larger amount of objects must be performing narrow-cone precession not so easily observable from the Earth. The internal dissipation in a freely precessing top leads to relaxation (gradual damping of the precession) and sometimes to spontaneous changes in the rotation axis. Recently developed theory of dissipative precession of a rigid body reveals that this is a highly nonlinear process: while the body is precessing at an angular rate ω, the precession-caused stresses and strains in the body contain components oscillating at other frequencies. Dependent upon the spin state, those frequencies may be higher or, most remarkably, lower than the precession rate. In many states dissipation at the harmonics is comparable to or even exceeds that at the principal frequency. For this and other reasons, in many spin states the damping of asteroidal and cometary wobble happens faster, by several orders, than believed previously. This makes it possible to measure the precession-damping rate. The narrowing of the precession cone through the period of about a year can be registered by the currently available spacecraft-based observational means. We propose an appropriate observational scheme that could be accomplished by comet and asteroid-aimed missions. Improved understanding of damping of excited rotation will directly enhance understanding of the current distribution of small-body spin states. It also will constrain the structure and composition of excited rotators. However, in the near-separatrix spin states a precessing rotator can considerably slow down its relaxation. This lingering effect is similar to the one discovered in 1968 by Russian spacecraft engineers who studied free wobble of a tank with viscous fuel.

Relative wobble of the Earth's inner core derived from polar motion and associated gravity variations

SUMMARY In this paper an explanation of the observed decadal variations of the polar motion of the Earth is presented. Recent investigations show that the contribution of surface processes is too small to excite the observed magnitudes of the decadal variations of polar motion. After removing the known effects of atmospheric variations from the observed polar motion, we obtain significant residuals that obviously can only be explained by processes in the core. In this paper, we investigate the effect of a relative inner-core rotation. In particular, we assume that the orientation of the figure axis of the inner core changes with respect to the outer core and mantle. Due to the flattening of the inner core and the density difference between the inner and outer core, these changes contribute to variations of the Earth's polar motion due to internal mass redistributions. The objective of this study is to determine those changes in the orientation of the figure axis of the inner core that produce mass redistributions necessary to excite the decadal variations of the observed polar motion minus the estimated atmospheric influence. Using polar motion data and the atmospheric excitation function, it is possible to determine the excitation function of the internal process superimposed on the free wobble of the Earth by linear approximation of the Liouville equation. On the other hand, provided that only the contribution of the mass redistributions is significant, we can express the time function obtained in terms of the orientation of the figure axis of the inner core. The final expression then contains the corresponding orientation angles as unknowns. Using this expression, we calculate the orientation angles from the numerically determined values of the excitation function. The calculation results in a mean tilt of 1°, and a mean eastward drift of 0.7° yr−1 of the figure axis of the inner core and quasi-periodic decadal variations of its orientation angles. The associated changes of the mass geometry in the core due to these variations of the figure axis of the inner core are then used to estimate their influence on the Earth's outer gravity field. Finally, we compare the magnitude of the resulting gravity field variations with the accuracy of recent and future gravity field models.

The Akp-Btk value method and the results for the retrieval of the parameters of the Earth’s free core nutation

The rotation of the Earth’s liquid core creates the Nearly Diurnal Free Wobble (NDFW). It is one of the problems of researching the Earth’s liquid core for us to retrieve the parameters of the Earth’s Free Core Nutation (FCN), from the Earth’s liquid core resonance of gravity tide waves on the diurnal frequency band. Since 1987, some scientists in many different countries have come to calculate the parameters of FCN by using the observational data of gravity tide waves on the diurnal frequency band. They basically followed the Stacking method, which needed five diurnal waves for the resolution. In this paper, authors introduced the Akp-Btk value method with clearly geometrical meaning as anew method, which only requires three very high signal-noise-ratio waves O1, K1 and P1 for the resolution. Authors chose the observational data of the three superconducting gravimeter stations respectively located in Cantley of Canada, Wuhan of China and Brussels of Belgium, to compute the parameters of FCN. It was the first time that the observational results of the parameters of FCN obtained from gravitational tide were in accord with the parameters of FCN gained from VLB1 since 1987.