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.     

Corrections stemming from the non-osculating character of the Andoyer variables used in the description of rotation of the elastic Earth

We explore the evolution of the angular velocity of an elastic Earth model, within the Hamiltonian formalism. The evolution of the rotation state of the Earth is caused by the tidal deformation exerted by the Moon and the Sun. It can be demonstrated that the tidal perturbation to spin depends not only upon the instantaneous orientation of the Earth, but also upon its instantaneous angular velocity. Parameterizing the orientation of the Earth figure axis with the three Euler angles, and introducing the canonical momenta conjugated to these, one can then show that the tidal perturbation depends both upon the angles and the momenta. This circumstance complicates the integration of the rotational motion. Specifically, when the integration is carried out in terms of the canonical Andoyer variables (which are the rotational analogues to the orbital Delaunay variables), one should keep in mind the following subtlety: under the said kind of perturbations, the functional dependence of the angular velocity upon the Andoyer elements differs from the unperturbed dependence (Efroimsky in Proceedings of Journées 2004: Systèmes de référence spatio-temporels. l’Observatoire de Paris, pp 74–81, 2005; Efroimsky and Escapa in Celest. Mech. Dyn. Astron. 98:251–283, 2007). This happens because, under angular velocity dependent perturbations, the requirement for the Andoyer elements to be canonical comes into a contradiction with the requirement for these elements to be osculating, a situation that parallels a similar antinomy in orbital dynamics. Under the said perturbations, the expression for the angular velocity acquires an additional contribution, the so called convective term. Hence, the time variation induced on the angular velocity by the tidal deformation contains two parts. The first one comes from the direct terms, caused by the action of the elastic perturbation on the torque-free expressions of the angular velocity. The second one arises from the convective terms. We compute the variations of the angular velocity through the approach developed in Getino and Ferrándiz (Celest. Mech. Dyn. Astron. 61:117–180, 1995), but considering the contribution of the convective terms. Specifically, we derive analytical formulas that determine the elastic perturbations of the directional angles of the angular velocity with respect to a non-rotating reference system, and also of its Cartesian components relative to the Tisserand reference system of the Earth. The perturbation of the directional angles of the angular velocity turns out to be different from the evolution law found in Kubo (Celest. Mech. Dyn. Astron. 105:261–274, 2009), where it was stated that the evolution of the angular velocity vector mimics that of the figure axis. We investigate comprehensively the source of this discrepancy, concluding that the difference between our results and those obtained in Ibid. stems from an oversimplification made by Kubo when computing the direct terms. Namely, in his computations Kubo disregarded the motion of the tide raising bodies with respect to a non-rotating reference system when compared with the Earth rotational motion. We demonstrate that, from a numerical perspective, the convective part provides the principal contribution to the variation of the directional angles and of length of day. In the case of the x and y components in the Tisserand system, the convective contribution is of the same order of magnitude as the direct one. Finally, we show that the approximation employed in Kubo (Ibid.) leads to significant numerical differences at the level of a hundred micro-arcsecond.     

The kinematical mechanism for the perturbation of the rotational axis in the rotation of the elastic Earth

Kubo (Celest Mech Dyn Astron 110:143–168, 2011) investigated the kinematical structure of the perturbation in the rotation of the elastic Earth due to the deformation caused by the outer bodies. In that paper, while the mechanism for the perturbation of the figure axis was made clear, that for the rotational axis was not shown explicitly. In the present study, following the same method, the structure of the perturbation of the rotational axis is investigated. This perturbation consists of the direct perturbation and the convective perturbation. First the direct perturbation is shown to be (A − C)/A times as large as that of the figure axis, coinciding with the analytical expressions obtained in preceding studies by other authors. As for the convective perturbation, which appears only in the perturbation of the rotational axis but not in that of the figure axis, it is shown to be (A − C)/A times the angular separation between the original figure axis and the induced figure axis produced by the elastic deformation, A and C being the principal moments of inertia of the Earth. If the perturbing bodies are motionless, the conclusion of Kubo (Celest Mech Dyn Astron 105:261–274, 2009) holds strictly, i.e. the sum of the direct and the convective perturbations of the rotational axis coincides with the perturbation of the figure axis.     

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。     

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.     

非刚体地球的受迫章动奥波策项和倾斜模

讨论了非刚体地球受迫章动奥波策项与简正模表达式中倾斜模的关系。结果表明天球历书极章动中倾斜振项对应于角动量极的章动,在球历书极章动与角动量极的章动奥波策项之和。同时还给出了岁差速率与自转极的章动奥波策项间的数学关系。     

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．     

A precise modeling of Eros 433 rotation

Thanks to the recent data obtained from the NEAR space probe, we calculate in this paper, with a precision never reached so far for an asteroid, the precession and the nutation of Eros 433. In a preliminary step, we show that Eros obliquity has a remarkable value of 89.0° which tends to align its figure axis along the orbital plane. This very specific obliquity has some consequences on the motion of the axis of figure: one is the very small amplitude of the precession in longitude, for which we get the value . Moreover, we calculate Eros nutation for the figure axis due to the Sun, after developing the perturbing potential at the 4th order of the eccentricity. We show that the figure axis undergoes very large oscillations in the direction perpendicular to Eros orbital plane, due to the nutation in obliquity. Peak to peak, these oscillations reach 55″, which is far larger than the amplitudes of the nutations of the Earth due to the Sun (of the order of 2″). Moreover, we give the analytical developments of Δψ and Δε, both for the axis of angular momentum, and the axis of figure.     

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.     

Remarks on dynamical ellipticity at the Earth's

In this paper, two factors — the redistribution of the density and the variation in the angular velocity of the Earth rotation, that affect the adopted value of the flattening for equidensity surface within the Earth, are discussed. The computational results show that the contribution of the redistribution of the density in the Earth interior (especially in the core) on the change of the flattening at the core-mantle boundary (CMB) is marginal, and that the calculated value of the flattening at the CMB can be in good agreement with the VLBI observed value so long as the fact that the angular velocity of the Earth rotation has undergone the tidal evolution is taken into account. As a result, this paper presents a set of recommended values of the dynamical parameters of the Earth (see Table III) for computing Earth's forced nutation series.     

Theory of the rotation of the rigid earth

An analytical theory is developed for planes normal to the angular-momentum axis, to the figure axis, and to the rotational axis of the triaxial rigid Earth. One of the purposes of this paper is to determine the effect on nutation and precession of Eckertet al.'s improvement to Brown's tables of the Moon and to check Woolard's theory from a different point of view. The present theory is characterized by the use of Andoyer variables, a moving reference plane, and Hori's averaging perturbation method. A comparison with Woolard's results shows that (1) the maximum difference in nutation for the plane normal to the angular-momentum axis, calculated from the same constants as Woolard adopted, reaches 0.0017, (2) the discrepancy in Oppolzer terms is large compared with the discrepancy in nutation for the plane normal to the angular-momentum axis, and (3) the present theory does not include some of the secular terms that are incorporated in Woolard's theory and that have an effect on the establishment of a reference system. The nutation coefficients 0.0001 for the three above-mentioned planes are calculated by using the numerical values recommended at the working meeting of the International Astronomical Union held in Washington in September 1974. The effects on precession and nutation due to the higher geopotential (n3) are also investigated. Any future revision of the lunar theory will not alter the values of the coefficients of the nutational terms derived here.     

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。     

Dynamics of the Earth’s core from VLBI observations

The Earth’s rotation is accompanied by free circadian oscillations of its liquid core in the inner cavity of the lower mantle, which perturb the angular momentum of the entire Earth and produce an additional free nutation of the celestial pole called free core nutation (FCN). Since this nutation causes resonances in the diurnal tides and in the expansions of luni—solar nutation, its study, especially an improvement of the FCN period, is of fundamental importance for the theory of the Earth’s rotation. We have determined the FCN parameters from a joint analysis of equidistant series of coordinates of the celestial pole obtained from the combined processing of VLBI observations on global networks of stations for the interval 1984.0–2008.4 by IERS (International Earth Rotation and Reference System Service, Paris, France) and NEOS (National Earth Orientation Service, Washington, USA). Applying a moving least-squares filter (MLSF) to these data has shown that the FCN period averaged over this time interval differs significantly from the theoretical one and its phase varies over a wide range. Using the mean quadratic collocation (MQC) method, we have obtained a new, more accurate stochastic FCN model. Its analysis by the envelope method has revealed long-term linear phase trends, calling into question not only the adopted FCN period but also its stability and, hence, the stability of the resonant effects in the Earth’s luni—solar nutation.     

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。     

Solution to the rotation of the elastic earth by method of rigid dynamics

Hamiltonian mechanics is applied to the problem of the rotation of the elastic Earth. We first show the process for the formulation of the Hamiltonian for rotation of a deformable body and the derivation of the equations of motion from it. Then, based on a simple model of deformation, the solution is given for the period of Euler motion, UT1 and the nutation of the elastic Earth. In particular it is shown that the elasticity of the Earth acts on the nutation so as to decrease the Oppolzer terms of the nutation of the rigid Earth by about 30 per cent. The solution is in good agreement with results which have been obtained by other, different approaches.     

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．     

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.     

On the possibility of creating a radio-astronomical inertial coordinate system based on the measurement of arcs between QSOs with the aid of VLBI

The aim of this paper has been to discuss a possibility of the formation of an inertial system of coordinates, precise within 0″.001, with the aid of the measures of mutual angular distances of quasars by the VLBI technique used in radio-astronomy. Such a system depends on neither the rotation, nor revolution of the Earth around the Sun — a fact which permits a separation of the astronomical and geophysical aspects of the problem. The proposed system should permit us to resolve some general astrometric problems — such as of the precession and nutation of the Earth as well as of the motion of the terrestrial axis of rotation within the Earth, or fluctuations in the angular veoocity of terrestrial rotation.     

地球自转变化的重力效应

极移对重力测量的影响在前人的研究和工作中已得到了充分的考虑并有不同形式的改正公式。但是由于地轴转动速率变化的重力效应的量级很小,且限于以前的测量精度以及资料处理方法, 因而常常被忽略。随着重力测量精度的提高以及资料处理方法的改进, 有必要重新讨论该项影响。本文从推导地球瞬时自转极运动对地面台站重力测量值影响的公式入手, 将地球自转角速率变化的重力效应和极移的重力效应分离开来, 并对它们的量级分别作了估计。本文着重讨论自转角速率变化对重力测量的影响。用 Ｉ Ｅ Ｒ Ｓ 的日长变化序列和极移序列分别具体地计算了１９８２ ～１９９６ 年期间对武汉的重力测量值的影响, 结果表明, 自转角速率变化对武汉地区重力值的影响最大可达０ ．２５ ｍｉｃｒｏｇａｌ（ｐｅａｋ －ｔｏ － ｐｅａｋ） , 这在高精度的绝对重力测量和相对重力测量中不应再被忽略