Rigid Earth Model Research Articles

Forced Nutation for Rigid Earth Model with Different Theories

Where Earth is not strictly rigid body but can responds to any effects that tend to its rotation and shape, we will explain, in the present paper, the goal which is to define the forced nutation for a rigid Earth model using two different theories. We will formulate a first order Hamiltonian of a deformable Earth for its rotational motion around the Sun through the contribution of triaxial symmetry of the Earth. The formulation of the theory will be formed twice times. Firstly, deduce the tidal affect’s forces by Luni - Solar attraction coupling with the Earth’s geopotential force. Secondly, through the formulation, we will neglect the coupling between the different effects (the geopotential Earth force effect and the Luni - Solar attraction force), so, we will find the transform of the Hamiltonian for each force separately. The analytical solution for the formulated Hamiltonian will be derived for the two cases by using perturbation technique of Lie - Hori series. Once can get the analytical solution by getting the generation function, we will derive the nutation series analytically and numerically for each case and conclude over the results.

GENERAL THEORY OF THE ROTATION OF THE NON-RIGID EARTH AT THE SECOND ORDER. I. THE RIGID MODEL IN ANDOYER VARIABLES

This paper is the first part of an investigation where we will present an analytical general theory of the rotation of the non-rigid Earth at the second order, which considers the effects of the interaction of the rotation of the Earth with itself, also named as the spin-spin coupling. Here, and as a necessary step in the development of that theory, we derive complete, explicit, analytical formulae of the rigid Earth rotation that account for the second-order rotation-rotation interaction. These expressions are not provided in this form by any current rigid Earth model. Working within the Hamiltonian framework established by Kinoshita, we study the second-order effects arising from the interaction of the main term in the Earth geopotential expansion with itself, and with the complementary term arising when referring the rotational motion to the moving ecliptic. To this aim, we apply a canonical perturbation method to solve analytically the canonical equations at the second order, determining the expressions that provide the nutation-precession, the polar motion, and the length of day. In the case of the motion of the equatorial plane, nutation-precession, we compare our general approach with the particular study for this motion developed by Souchay et al., showing the existence of new terms whose numerical values are within the truncation level of 0.1 μas adopted by those authors. These terms emerge as a consequence of not assuming in this work the same restrictive simplifications taken by Souchay et al. The importance of these additional contributions is that, as the analytical formulae show, they depend on the Earth model considered, in such a way that the fluid core resonance could amplify them significatively when extending this theory to the non-rigid Earth models.

Models and nomenclature in Earth rotation

AbstractThe celestial Earth's orientation is required for many applications in fundamental astronomy and geodesy; it is currently determined with sub-milliarcsecond accuracy by astro-geodetic observations. Models for that orientation rely on solutions for the rotation of a rigid Earth model and on the geophysical representation of non-rigid Earth effects. Important IAU 2000/2006 resolutions on reference systems have been passed (and endorsed by the IUGG) that recommend a new paradigm and high accuracy models to be used in the transformation from terrestrial to celestial systems. This paper reviews the consequences of these resolutions on the adopted Earth orientation parameters, IAU precession-nutation models and associated nomenclature. It summarizes the fundamental aspects of the current IAU precession-nutation models and reports on the consideration of General Relativity (GR) in the solutions. This shows that the current definitions and nomenclature for Earth's rotation are compliant with GR and that the IAU precession-nutation is compliant with the IAU 2000 definition of the geocentric celestial reference system in the GR framework; however, the underlying Earth's rotation models basically are Newtonian.

Comparison of Nutation Theories

AbstractAnalyses of residuals between VLBI observations and combinations of nutation series show that the MHB 2000 nonrigid-Earth nutation model applied to the REN 2000 rigid-Earth model results in the best fit, and that amplitudes of any possible periodic terms remaining in the observed corrections to the MHB2000 theory could be expected to be less than 0.1 mas.

Rigid-Earth Nutation Models

AbstractDespite the fact that the main causes of the differences between the observed Earth nutation and that derived from analytical calculations come from geophysical effects associated with nonrigidity (core flattening, core-mantle interactions, oceans, etc…), efforts have been made recently to compute the nutation of the Earth when it is considered to be a rigid body, giving birth to several “rigid Earth nutation models.” The reason for these efforts is that any coefficient of nutation for a realistic Earth (including effects due to nonrigidity) is calculated starting from a coefficient for a rigid-Earth model, using a frequency-dependent transfer function. Therefore it is important to achieve high quality in the determination of rigid-Earth nutation coefficients, in order to isolate the nonrigid effects still not well-modeled.After reviewing various rigid-Earth nutation models which have been established recently and their relative improvement with respect to older ones, we discuss their specifics and their degree of agreement.

Corrections and new developments in rigid earth nutation theory

We present here the new tables REN-2000 of the nutation for a rigid Earth model, starting from Hamiltonian theory, with a level of truncature at 0.1 mu as for individual coefficients instead of 5 mu as (Kinoshita & Souchay 1990). For this presentation to be achieved we first carry out the calculations of the second-order effects due to crossed-nutations and spin-orbit coupling, at the same level of truncation as above. This paper is the third and last one in the frame of the complete reconstruction of the theory of the rigid Earth nutation. It is the complementary part to previous studies concerning the luni-solar nutation involving indirect planetary effects (Souchay & Kinoshita 1996), and the influence of the second-order geopotential (J(3), J(4)) and of the direct planetary effect (Souchay & Kinoshita 1997). Quasi-diurnal and sub-diurnal nutations coming from the harmonics of degree 2, 3 and 4 of the geopotential are also included in REN-2000, their values being taken from Folgueira et al. (1998a,b). A presentation of the series REN-2000 is done at the end of the paper, with separated informations for each contribution.

The geophysical approach towards the nutation of a rigid Earth

This paper presents the new and complete se- ries named H96NUT for the nutation of a rigid Earth model. In contrast to the traditional computation method, the nutation is calculated here indirectly via the new and highly accurate tidal potential development recently given by Hartmann & Wenzel (1995a,b), (HW95). All contribu- tions up to 0.45 as were taken into account leading to 699 nutation terms separated after origin (or 607 terms summed according to the same arguments). These are: the main terms due to the Moon, the Sun and the plan- ets, indirect planetary eects, eects due to the triaxi- ality of the Earth, eects due to the J3 and J4 geopo- tential, the geodesic nutation and second order terms. Nutation series for the angular momentum axis, the gure axis and the rotation axis are derived; they are in good agreement with those given in Roosbeek & Dehant (1997) with exception of several terms whose cause is discussed. Modern constants (like the precession constant) have been used throughout. The value for the dynamical elliptic- ity Hdyn, which is compatible with these computations, is Hdyn =3 :273 792 489 10 3 . In addition, some other val- ues related with precession are also deduced from the tidal potential. One main advantage of this work is that a com- pletely independent analytical computation method has been established which is extensively described here (see also Hartmann (1996) for more details) and which pro- vides { besides a numerical calculation { a useful valida- tion of existing nutation series.

Comparisons of the REN-2000 Tables with Numerical Integration and Other Recent Analytic Tables

We present a comparison of the nutation given by the new REN-2000 analytic tables of Souchay et al. (for a rigid-Earth model) with that obtained from a numerical integration. We show that the standard deviation in the residuals are about 5 μas for both Δψ sin 0 and Δ, given a 134 yr time span. We also compare the nutation given by the REN-2000 series with that obtained by Roosbeek & Dehant (RDAN97) and Bretagnon et al. (BRS97). The agreement with Bretagnon et al. concerning the nutation for the figure axis is very satisfactory, with no difference exceeding 1 μas for individual coefficients of nutation, both in Δψ sin and Δ, except for the 18.6 yr component and the month component, for which the difference is a few microarcseconds. We show that the only substantial difference between REN-2000 and BRS97, concerning the 18.6 yr component, is a result of the different choice of the value of the dynamical ellipticity Hd. Moreover, comparison of the REN-2000 series with the RDAN97 series yields larger residuals than comparison with the BRS97 series. On the other hand, these residuals are noticeably smaller for the global contribution of the Oppolzer terms.

The Theory of the Nutation for a Rigid Earth Model: Current State of the Situation

Whereas no particular attention was paid to the theory of the nutation for a rigid Earth model, for more than a decade after the adoption by the International Astronomical Union (IAU) of coefficients as calculated and listed by Kinoshita (1977), an increasing number of studies have been done in the recent years aiming to improve this theory. The improvement became necessary mainly because of the big parallel improvement of the VLBI observations itself, which leads to present determinations of some coefficients of nutation at the level of a few 10μas. Therefore the amelioration of the theory of the nutation for a rigid Earth model can be divided in two aspects: one is to consider a smaller level of truncation of the coefficients of nutation; the other is to evaluate in the best way the coefficients already taken into account, particularly the leading coefficients which are typically those subject to the largest absolute differences.

Effects On the Nutation of the Non-Zonal Harmonics of Third Degree

In this paper, we calculate the coefficients of the nutation for a rigid Earth model due to the C-3m and S-3m (m not equal 0) harmonics of the geopotential, starting from the Hamiltonian theory as developped by Kinoshita (1977). We show that these coefficients are far from bring negligible as given the level of truncation of 0.1 mu as which is necessary in the reconstruction of the tables of nutation, and also that their value is very close to that given by Bretagnon st al. (1997).

A New Nutation Series for a Rigid Earth Model

AbstractA new nutation series for a rigid Earth model was derived from a new and highly accurate tidal potential series. A new second order theory for the nutational amplitudes of rotation-, figure- and angular momentum-axis based on tidal amplitudes is formulated. Amplitudes larger than 0.45μas are taken into account leading to a series with 699 terms. The new series may serve as test of other ones that were recently published in the literature.

New series of rigid and non rigid Earth nutation. Comparison with observations

After analysing the recent developments of the theory of the nutation for a simplified rigid Earth model (Kinoshita and Souchay, 1990) with new corrections and new contributions (Williams, 1994; Souchay and Kinoshita, 1996), we will study the effect of these developments on the calculation of the coefficients of nutation for a non rigid Earth model, based on the transfer function given by Wahr (1979). The relative improvements characterized by the residuals with the observations are explained in the following.

Recent Results of the Nutation Theory for a Rigid Earth

AbstractAlthough the main questions remaining in the frame of the study of the nutation-precession phenomena concern principally the effects due to physical interactions, recent works have shown that even in the simplified case of a rigid Earth model of nutation, important changes have to be taken into account. Here we present the various steps having led to a more complete and more accurate rigid Earth nutation theory, since the pionnier work of Woolard (1953), insisting on the most recent improvements.

The nutation of a rigid earth model: Direct influences of the planets

view Abstract Citations (15) References (10) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Nutation of a Rigid Earth Model: Direct Influences of the Planet Hartmann, Torsten ; Soffel, Michael Abstract This paper presents the nutation of the rigid earth due to the direct influence of the planets, which can reach up to 0.5 milliarcsec (mas) in longitude and 0.2 mas in obliquity. In contrast to the direct computation method of Kinoshita & Souchay, it is here calculated indirectly via the new planetary tidal potential development recently given by Hartmann & Wenzel (1994). The results are (except for two terms) equal within the stated truncation threshold of 0.005 mas. Since we lowered this level to 0.0025 mas, another eleven terms are added. Values for the precession due to the planets are also deduced from the tidal potential. Publication: The Astronomical Journal Pub Date: September 1994 DOI: 10.1086/117139 Bibcode: 1994AJ....108.1115H Keywords: Earth (Planet); Earth Rotation; Lunar Effects; Nutation; Precession; Solar System; Accuracy; Earth Axis; Earth Tides; Graphs (Charts); Legendre Functions; Poisson Equation; Tables (Data); Very Long Base Interferometry; Astrophysics; CELESTIAL MECHANICS; EARTH full text sources ADS |

The motion of Mars' pole. I - Rigid body precession and nutation

Abstract : Differences between the observed forced nutation component amplitudes of the Earth and predicted nutation amplitudes for a rigid Earth are a result of differences between the theoretical rigid structure of the Earth used in older models and the actual elastic Earth with a liquid core. However, except for the period of the Chandler wobble, the observations of the motion of the Earth's pole were not accurate enough to observe the effects of the elastic, liquid core Earth until the last 30 years. In more recent works, such as those by Wahr (198la,b), the nutation resulting from an elastic, liquid core Earth are modeled as perturbations of the rigid Earth model nutation. This is the approach adopted for the 1980 IAU Theory of Nutation to determine the amplitude of the various nutational elements (Kaplan 1981). These perturbations result in modifications to the nutation amplitudes for the Earth from about 1% to 0.01% of the theoretical rigid nutation amplitudes or about O.019 for the largest term in the series. Since the l950s the improvement in the measurement of the motion of the Earth's pole have made its observation a powerful probe of the structure of the Earth.

Forced nutations of the Earth: Influence of inner core dynamics: 1. Theory

Gravitational and pressure couplings between the solid inner core and the rest of the Earth give rise to torques through which the inner core influences the nutational motions of the Earth. In view of the very small magnitude of the moment of inertia of the inner core relative to that of the the Earth as a whole, one expects from physical considerations that inclusion of the inner core in the dynamics should lead to a new nutational normal mode with a natural frequency not too far from that of the free core nutation, and to an associated weak resonance in the amplitude of forced nutations. We present here a treatment of the nutation problem for an oceanless, elastic, spheroidally stratified Earth, with the dynamical role of the inner core explicitly included in the formulation. As a preliminary to the setting‐up of dynamical equations, we devote some attention to a careful definition of a suitable coordinate system and of certain basic dynamical variables. We use the approach of Sasao et al. (1980), with their system of dynamical equations enlarged by the inclusion of two additional equations which are needed to describe the rotational motion of the inner core. An extension and sharpening of a line of reasoning employed by them enables us to derive expressions for the torques which couple the mantle and the fluid outer core to the solid inner core. Solving the enlarged system of equations, we show that a new nearly diurnal eigenfrequency does emerge; a rough estimate places it not very far from the prograde annual tidal excitation frequency, so that possible resonance effects on nutation amplitudes need careful consideration. Another eigenfrequency, attributable to a wobble of the inner core, is also found; its value is estimated to be a few times smaller than the wobble frequency that the inner core would have in the absence of couplings to the rest of the Earth. Considering an expansion, in terms of resonance contributions, of the amplitude of forced nutations normalized relative to that for a corresponding rigid Earth model, we indicate how the coefficients in the expansion are related to those in expansions of the type used by Wahr (1981b). Finally, we discuss the problem of comparing observed nutation amplitudes with those computed on the basis of Earth models generated from seismological data, with special reference to the fact that the dynamical ellipticity of the Earth, as computed from published Earth models which assume the condition of hydrostatic equilibrium, differs significantly from that determined from the precession constant. Numerical results, corrections for unmodeled effects, and comparison with observational results will be dealt with in accompanying papers.

Forced nutations of the Earth: Influence of inner core dynamics: 2. Numerical results and comparisons

We apply the theory developed in Paper 1 (Mathews et al., this issue), which includes the solid inner core explicitly in the dynamical equations, to obtain the eigenfrequencies and other characteristics of the Earth's nutational normal modes as well as the amplitudes of forced nutations at various tidal frequencies, for two commonly used Earth models, 1066A and the Preliminary Reference Earth Model (PREM). We also make an evaluation of various procedures for taking account of known deviations of the Earth from models, notably in the dynamical ellipticity e for which the two models yield values which are over 1% smaller than the value e* deduced from the precession constant. On adopting the procedure of simply replacing e by e* in the equations of our theory, the values obtained for some of the nutation amplitudes for model 1066A differ significantly from the corresponding results of Wahr (1981b). The largest of the differences, which occur in the prograde semiannual, retrograde 18.6 year, and retrograde annual nutation terms, amount to −0.59, 0.35, and −0.25 milliarcseconds (mas), respectively, while the standard errors in the very long baseline interferometry (VLBI) determinations are now only about 0.04 mas except in the long period terms. The difference in the procedures used to take account of the discrepancy between e and e* contributes −0.56, 0.81, and −0.17 mas, respectively, to the above‐noted differences. For the purpose of comparison with VLBI‐observed data, we use the results for a “modified PREM,” defined by a set of Earth model parameters which differ from those of PREM only in having e* for the dynamical ellipticity of the Earth as a whole and a modified value for the dynamical ellipticity eƒ of the fluid outer core. The amplitudes computed for this model, with corrections applied for the effects of ocean tides and mantle anelasticity, are in generally satisfactory agreement with observed values, when the modified eƒ is determined by matching the theoretical and observed values for the retrograde annual term. (The modified eƒ is 0.002665, about 4.6% higher than in PREM, equivalent to an increase, relative to PREM, of about 430 meters in the difference between the equatorial and the polar radii of the core‐mantle boundary. We find that contributions from inner‐core dynamics to the prograde semiannual and annual, and the retrograde 18.6 year and annual terms, recomputed for modified PREM, amount to −0.09, 0.03, −0.36, and −0.09 mas, respectively.) The largest residual remaining, other than in the long‐period terms which still have an uncertainty of about 1 mas, is −0.25 mas in the prograde fortnightly amplitude. Consideration of possible sources of the discrepancies is facilitated by a resonance expansion of the amplitude of forced nutations, as a function of frequency, normalized relative to that for a rigid Earth model. We also provide tables which exhibit the sensitivities of various relevant quantities (the eigenfrequencies and the coefficients which appear in the resonance expansion, as well as the nutation amplitudes at important tidal frequencies) to possible errors in the Earth parameters which enter our theory. Reconciliation of theoretical and experimental values for the prograde fortnightly term, for instance, could be accomplished, without affecting significantly the comparison for other nutation terms, by a decrease of about 10% in the value of the compliance parameter k that represents, in effect, the deformability of the Earth as a whole in response to perturbations of its rotation; but this change in k would have to be produced by some mechanism which does not affect the values of the other compliances.

Comparison of new nutation series with numerical integration

To check the results given by the new tables of the nutation for the rigid Earth model in (Kinoshita and Souchay, 1990), we perform a calculation of this nutation by numerical integration, using the basic equations of Kinoshita (1977). Results show very small long period residuals, with a discrepancy of 0.06 mas sin Ω in longitude, -0.04 mas cos Ω in obliquity, without out-of-phase components. Moreover, the total residuals of short period do not exceed 0.1 mas in longitude, 0.06 mas in obliquity. All these results are considerably better than those found by previous numerical integrations, thus confirming the validity of the new tables.

The theory of the nutation for the rigid earth model at the second order

We perform a complete reconstruction of the series of the nutation for a rigid Earth model with the use of the very accurate theories ELP2000 and VSOP82 for the motion of the Moon and the planets respectively, in such a way that all the individual contributions up to 0.005 mas should be taken. This implies the introduction of the planetary effects, of the influence of second-order parts of the potential of the Earth (J3, triaxiality), and some improvements due to an extension of the theory at the second order. All this increase notably the number of coefficients to be taken in account, and modifies also in a significant way the value of some of them.

The Results of the Reconstruction of the Coefficients of the Nutation for the Rigid Earth Model and Their Comparison With Numerical Integration

In view of the present accuracy of the astrometric observations and of the development of the theory of the nutation, it became necessary to make a complete revision of this theory for a rigid Earth model. We present the results of our recent one (Kinoshita and Souchay, 1989), which includes planetary effects and second order effects no considered in the previous tables (Kinoshita, 1977). We analyze the difference between these tables and the new ones providing from the revision above and the comparison between the theory and numerical integrations recently performed (Kubo and Fukushima, 1987; Shastok et al., 1987; Shastok et al., 1989). The results of this comparison are much better after revision than before.