Secular variation of the obliquity of the ecliptic from a core-mantle coupling

The Earth is assumed to consist of a rigid core and a rigid mantle coupled by a non-isotropic friction. It is shown that if the core rotates slightly faster than the mantle, then the observed secular change of the obliquity of the ecliptic can be accounted for without giving rise to an unacceptably high rate of spin-down of the Earth.     

Kinematical modeling of the Earth rotation,focusing on the Oppolzer terms in a rigid Earth and the Oppolzer-like terms in an elastic Earth

Under perturbations from outer bodies, the Earth experiences changes of its angular momentum axis, figure axis and rotational axis. In the theory of the rigid Earth, in addition to the precession and nutation of the angular momentum axis given by the Poisson terms, both the figure axis and the rotational axis suffer forced deviation from the angular momentum axis. This deviation is expressed by the so-called Oppolzer terms describing separation of the averaged figure axis, called CIP (Celestial Intermediate Pole) or CEP (Celestial Ephemeris Pole), and the mathematically defined rotational axis, from the angular momentum axis. The CIP is the rotational axis in a frame subject to both precession and nutation, while the mathematical rotational axis is that in the inertial (non-rotating) frame. We investigate, kinematically, the origin of the separation between these two axes—both for the rigid Earth and an elastic Earth. In the case of an elastic Earth perturbed by the same outer bodies, there appear further deviations of the figure and rotational axes from the angular momentum axis. These deviations, though similar to the Oppolzer terms in the rigid Earth, are produced by quite a different physical mechanism. Analysing this mechanism, we derive an expression for the Oppolzer-like terms in an elastic Earth. From this expression we demonstrate that, under a certain approximation (in neglect of the motion of the perturbing outer bodies), the sum of the direct and convective perturbations of the spin axis coincides with the direct perturbation of the figure axis. This equality, which is approximate, gets violated when the motion of the outer bodies is taken into account.     

A Hamiltonian theory for an elastic earth: First order analytical integration

In this article an approximate analytical integration is performed of the Hamiltonian corresponding to the rotational motion of an Earth whose elastic mantle is deformed by rotation and lunisolar attraction, using Deprit's perturbation method for the first order. Besides the usual terms, this Hamiltonian includes the perturbation of the kinetic energy and the elastic energy produced with the deformation, as well as their causes, the tidal and the centrifugal potential; these new terms have already been studied for the tidal deformation in previous articles (Getino and Ferrándiz, 1990a, 1990b). The effects of the deformation due to the centrifugal potential are studied in this article, following the same method as that used for the tidal deformation. Numerical tables of the periodic perturbations corresponding to the nutation in obliquity and longitude are obtained. As for the secular effects, a theoretical value of 457 days is obtained for Chandler's period.     

Tides and nutation of the Earth: II. Realistic models of the liquid core

In the first part of this study (Molodensky, 2004; hereinafter, paper I), a survey of the theory of tides and nutation of the Earth was given with the inclusion of the dynamical effects of the elastic mantle and liquid core in the framework of a very simple model of a homogeneous, incompressible core. Here, effects of the inhomogeneity, compressibility, and viscosity of the liquid core are considered, along with those of electromagnetic coupling of the liquid core with the mantle and solid inner core. Errors in the present-day measurements of the amplitudes of forced nutation (of the order of 20 arc s) are well below not only the dynamical effects of the Earths liquid core but also the effects of the inelasticity of the mantle, the dynamical effects of the solid inner core, and the possible effects of electromagnetic coupling between the liquid core, solid inner core, and mantle. This opens up new avenues for astrometric studies of the mechanical properties and electrical conductivity of the mantle and solid inner core at very low frequencies. The modern data on the amplitudes and phases of the Earths forced nutation cannot be interpreted entirely unambiguously, since the following factors remain unknown: (i) the role of the core-mantle electromagnetic coupling compared to the coupling due to core viscosity near the core-mantle boundary and (ii) the frequency dependence of the creep function of the mantle at low frequencies. In large measure, the effects of electromagnetic and viscous coupling can be separated if high-precision data on the tidal variations in the gravitational force at periods of about a day are invoked and allowances are made for the differences between the effects of viscous and electromagnetic coupling on the amplitudes and phases of forced nutation and on the tidal variations in the gravitational force. Here, ranges of possible values of the creep function are constructed for periods from one hour to one day; for these calculations, values consistent with the entire set of data on the forced nutation of the Earth are assumed for the effective dynamical flattening of the liquid core-mantle and liquid core-solid inner core boundaries (these values describe the ellipticity of the boundaries of the inner core, outer core, and mantle, as well as the electromagnetic coupling between the liquid core, mantle, and solid inner core).Translated from Astronomicheskii Vestnik, Vol. 39, No. 1, 2005, pp. 61–80.Original Russian Text Copyright © 2005 by Molodensky.     

On the effect of the mantle elasticity on the earth's rotation

Starting from the Hamiltonian model for a solid Earth with an elastic mantle previously developped by the authors, analytical expressions are derived which give the nutation series corresponding to the plane perpendicular to the angular momentum vector, to the plane perpendicular to the rotational axis and to the equator of figure, as well as the series that give the polar motion. The effects of the different perturbations — solid Earth, centrifugal and tidal potentials — are calculated separately. The corrections due to the elasticity of the mantle, which mostly correspond to the Oppolzer terms, are calculated with an accuracy of 10^–6 arc sec., given that the intrinsic observational accuracy has reached 0.01 mas.     

Empiric models of the Earth’s free core nutation

Free core nutation (FCN) is the main factor that limits the accuracy of the modeling of the motion of Earth’s rotational axis in the celestial coordinate system. Several FCN models have been proposed. A comparative analysis is made of the known models including the model proposed by the author. The use of the FCN model is shown to substantially increase the accuracy of the modeling of Earth’s rotation. Furthermore, the FCN component extracted from the observed motion of Earth’s rotational axis is an important source for the study of the shape and rotation of the Earth’s core. A comparison of different FCN models has shown that the proposed model is better than other models if used to extract the geophysical signal (the amplitude and phase of FCN) from observational data.     

弹性地球各种几何轴和物理轴的定义

基于经典的弹性地球自转动力学理论,建立了极移和章动的联合动力学方程。由此给出了弹性地球各种几何轴和物理轴(Tisserand轴、自转轴、瞬时形状轴、角动量轴、CEP和CIP轴)的极移、岁差章动的动力学方程,明确了各种轴的定义及其之间的理论关系。理论研究表明,联合动力学方程要比经典动力学方程综合性强易于理解,可同时求解极移和章动,特别是在文[1]理论中出现的倾斜模(TOM),在此只是作为了一个特解而存在。     

Accurate Analytical Calculation of Effects of Rorations of the Central Planet on a Staellite's Orbit

Satellite orbital perturbations due to many rotations of the planet-fixed reference frame are calculated by a general analytical method. For the International Terrestrial Reference Frame (ITRF) the effects of the Earth irregular rotation, precession, nutation, and polar motion are considered. Gravity coefficients of the Earth potential expansion are expressed in an inertial Celestial Reference Frame (CRF) as functions of the set of standard constant coefficients derived in the ITRF and of the rotation angles between the CRF and ITRF. The analytical motion theory uses time dependent gravity coefficients, and the Lagrange motion equations are integrated in the CRF, as it is done by numerical methods. Comparison of the proposed analytical method with a numerical one is presented. Motion of the ETALON-1 geodetic satellite perturbed by the geopotential (36*36) and by the full effects of the Earth irregular rotation, precession, nutation and polar motion is predicted. The r.m.s. difference between the satellite's coordinates calculated by both methods over a year interval is 2 cm. This revised version was published online in July 2006 with corrections to the Cover Date.     

RDAN97: An Analytical Development of Rigid Earth Nutation Series Using the Torque Approach

New series of rigid Earth nutations for the angular momemtum axis, the rotation axis and the figure axis, named RDAN97, are computed using the torque approach. Besides the classical J2 terms coming from the Moon and the Sun, we also consider several additional effects: terms coming from J3 and J4 in the case of the Moon, direct and indirect planetary effects, lunar inequality, J2 tilt, planetary‐tilt, effects of the precession and nutations on the nutations, secular variations of the amplitudes, effects due to the triaxiality of the Earth, new additional out‐of‐phase terms coming from second order effect and relativistic effects. Finally, we obtain rigid Earth nutation series of 1529 terms in longitude and 984 terms in obliquity with a truncation level of 0.1 μ (microarcsecond) and 8 significant digits. The value of the dynamical flattening used in this theory is HD=(C-A)/C=0.0032737674 computed from the initial value pa=50′.2877/yr for the precession rate. These new rigid Earth nutation series are then compared with the most recent models (Hartmann et al., 1998; Souchay and Kinoshita, 1996, 1997; Bretagnon et al., 1997, 1998. We also compute a benchmark series (RDNN97) from the numerical ephemerides DE403/LE403 (Standish et al., 1995) in order to test our model. The comparison between our model (RDAN97) and the benchmark series (RDNN97) shows a maximum difference, in the time domain, of 69 μas in longitude and 29 μas in obliquity. In the frequency domain, the maximum differences are 6 μas in longitude and 4 μ as in obliquity which is below the level of precision of the most recent observations (0.2 mas in time domain (temporal resolution of 1 day) and 0.02 mas in frequency domain). This revised version was published online in July 2006 with corrections to the Cover Date.     

The outer core rotation about an inclined axis from geomagnetic secular variations (Mitteilungen des Zentralinstituts für Physik der Erde,Nr. 1832)

Non-axisymmetric motions of the outer core of the Earth are important for the dynamo problem and the excitation of the decade variations of the polar motion. The components of the vector of a rigid rotation of the outer core about an inclined axis were estimated by a first-order approximation of the frozen- field theory of the geomagnetic secular variation from 1903.5 to 1975.5. The trends and quasi-periodic constituents of these quantities were computed. It was shown that the position and time behaviour of the rotational pole of the outer core differ considerably from the well-know co-ordinates of the dipole axis. Some periods of the equatorial components of the rotational vector are comparable with those of the axial component previously derived for a pure axial rotation. Additionally, the time behaviour of the pole path shows events like the well-known Markowitz wobble but naturally with other extent. These and other results suggested that the investigations are worth to be continued in future by some physical interpretations.     

用Hamilton方法计算弹性地球的章动序列

本文利用Hamilton方法研究弹性地球自转运动，采用地球模型PREM参数，给出了形状轴的章动序列．结果表明我们的方法是可行的，计算是可靠的．弹性地幔对地球章动的影响仅在毫角秒量级上，它相对液核对地球竟动的影响要小得多．     

国内天文地球动力学中的潮汐研究

简要说明了天文地球动力学范畴内所研究的潮汐现象，包括由日月引潮力引起的固体潮、海洋潮、大气潮和由于地球自转轴的极移引起的极潮，以及这些潮汐对地球自转和地球自转的测量产生的效应。重点阐述中国天文学界在这一领域里的研究成果。这些研究涉及潮汐影响地球自转的机制，也就是各种潮汐效应与极移、自转速率变化和章动的关系，包括构建这类关系的理论模型，分析潮汐对它们的影响，利用中国古代丰富的天象记录计算地球自转的长期减慢，计算弹性或滞弹地球的洛夫数，依据某一地球模型计算潮汐效应或章动序列等等。研究也涉及在测量地球自转参数的不同技术中各种潮汐效应对测量结果产生的影响及其改正，并涉及与潮汐有关的观测方法的优化和数据处理过程的改进。最后介绍了中国学者所发现的脉冲星的周期和周期变率测量中的潮汐效应，尽管它们的量级甚微，但不容忽视。     

关于章动序列计算中的地球动力学扁率的取值

刚体地球章动序列和非刚体地球章动的转换函数都和地球动力学扁率有关。ＩＡＵ１９８０章动理论中采用了一个不一致的地球动力学扁率值,从而影响了章动振幅的计算。本文介绍了章动序列计算中地球动力学扁率的取值。由地球模型１０６６Ａ或ＰＲＥＭ得到的地球动力学扁率值比由岁差观测得到的约小１％,并且不可靠。当考虑体静力学平衡被破坏时新的地球物理模型,可得到与岁差常数相一致的地球动力学扁率值。地球动力学扁率值Ｈ＝０．     

Atmospheric tides and the rotation of Venus II. Spin evolution

Tides in the atmosphere of Venus may help to stabilize its slow retrograde rotation. The frequency dependence of the body tides also affects its rotational stability. However, the obliquity is probably maintained near 180° by friction between the core and mantle of Venus. In any case, it appears most likely that Venus originated with an obliquity greater than 90°.     

Considerations concerning the non-rigid Earth nutation theory

This paper presents the reflections of the Working Group of which the tasks were to examine the non-rigid Earth nutation theory. To this aim, six different levels have been identified: Level 1 concerns the input model (giving profiles of the Earth's density and theological properties) for the calculation of the Earth's transfer function of Level 2; Level 2 concerns the integration inside the Earth in order to obtain the Earth's transfer function for the nutations at different frequencies; Level 3 concerns the rigid Earth nutations; Level 4 examines the convolution (products in the frequency domain) between the Earth's nutation transfer function obtained in Level 2, and the rigid Earth nutation (obtained in Level 3). This is for an Earth without ocean and atmosphere; Level 5 concerns the effects of the atmosphere and the oceans on the precession, obliquity rate, and nutations; Level 6 concerns the comparison with the VLBI observations, of the theoretical results obtained in Level 4, corrected for the effects obtained in Level 5.Each level is discussed at the state of the art of the developments.     

地球动力学扁率及其与岁差章动的关系

由岁差常数求得的日月岁差是天文学的重要参数之一,它和地球动力学扁率相联系。地球动力学扁率在章动理论的计算中也是一个重要的物理量。介绍了由不同的观测方法和模型给出的地球动力扁率值,并讨论了它也岁差的关系和对章动计算的影响。在刚体地球章动振幅的计算中,地球动力学扁率值起着尺度因子的作用,要改善刚体地球章动振幅的计算,需要修改目前的黄经总岁差值。非刚体地球章动的转换函数中所采用的简正模和常数都直接或间接地依赖地球动力学扁率值。在IAU1980章动理论中,计算刚体地球章动振幅所使用的地球动力学扁率值计算转换函数中简正模频率和常数所使用的地球动力学扁率值并不一致。随着观测和计算精度的提高,地球动力学扁率值的不一致将影响章动振幅的计算。在建立刚体地球章地动理论中,如何解释地球动力学扁率值的差异,如何选取地球动力学扁率值,还有待进一步的研究。     

Possible relations between the rotational axis of the Earth's inner core and the magnetic dipole axis

The geomagnetic field is maintained by amagnetohydrodynamic dynamo process within the liquid outer core. The distribution of the associated electric currents is modified if the outer core is bounded by electrically conducting material. Then, eddy currents and the related magnetic fields are generated within these regions. In particular, the relative rigid rotation of the inner core produces a secondary magnetic field, which is superimposed on the dynamo field. The angle between the dipole axis of the total field and the rotational axis of the inner core is an important quantity needed for the theory of polar motion of the Earth. This angle is investigated for a broad spectrum of angular velocities of the inner core. To simplify the mathematical procedure, we model the dynamo field using an axisymmetric field generated by a system of electric currents within the outer core. The conductivity of the mantle is neglected. We find that the position of the dipole axis depends on the angular velocity of the inner core as well as on the distribution of the current system within the outer core. Coincidence of both axes can be reached if the angular velocity is high enough and if the current system is concentrated within a thin sheet near the outer core-inner core boundary.     

Averaged rotational dynamics of an asteroid in tumbling rotation under the YORP torque

The secular effect of YORP torque on the rotational dynamics of an asteroid in non-principal axis rotation is studied. The general rotational equations of motion are derived and approximated with an illumination function expanded up to second order. The resulting equations of motion can be averaged over the fast rotation angles to yield secular equations for the angular momentum, dynamic inertia and obliquity. We study the properties of these secular equations and compare results to previous research. Finally, an application to several real asteroid shapes is made, in particular we study the predicted rotational dynamics of the asteroid Toutatis, which is known to be in a non-principal axis state.     

Behavior of nearby synchronous rotations of a Poincaré–Hough satellite at low eccentricity

This paper presents a study of the Poincaré–Hough model of rotation of the synchronous natural satellites, in which these bodies are assumed to be composed of a rigid mantle and a triaxial cavity filled with inviscid fluid of constant uniform density and vorticity. In considering an Io-like body on a low eccentricity orbit, we describe the different possible behaviors of the system, depending on the size, polar flattening and shape of the core. We use for that the numerical tool. We propagate numerically the Hamilton equations of the system, before expressing the resulting variables under a quasi-periodic representation. This expression is obtained numerically by frequency analysis. This allows us to characterise the equilibria of the system, and to distinguish the causes of their time variations. We show that, even without orbital eccentricity, the system can have complex behaviors, in particular when the core is highly flattened. In such a case, the polar motion is forced by several degrees and longitudinal librations appear. This is due to splitting of the equilibrium position of the polar motion. We also get a shift of the obliquity when the polar flattening of the core is small.