Polar motion for an elastic Earth model with a homogeneous liquid core using a canonical theory

The rotational motion for an elastic Earth model with a homogeneous liquid core has been obtained using Hamilton's equations. From the canonical equations for the precessional and nutational motions in an inertial frame, the corresponding equations in an Earth fixed frame are deduced. The linearized equations obtained for polar motion and liquid core motion are equivalent to the Sasao-Okubo-Saito equations.     

Polar motion for an elastic earth model using a canonical theory

The purpose of this paper is to develop a canonical formulation of the rotational motion for an elastic Earth model. We have obtained the canonical equations for the precession and nutation motion in an inertial frame, and from this we have deduced the equations in an Earth-fixed frame. The linearized equations deduced for polar motion are equivalent to those obtained using Liouville's equations.     

同时顾及章动和极移的地球自转方程

通过引进章动坐标系相对惯性参照系的转动角速度随时间的变化 ,导出了一个可同时解出章动和极移的地球自转方程 ,用这个方程可同时研究地球的强迫和自由转动。与现行研究地球自转的惯用方法相比 ,该方法综合性强 ,易于理解。     

Analytical versus semi-analytical determinations of the Oppolzer terms for a non-rigid Earth

Estimations of the Oppolzer terms for the angular momentum and rotation axes of a non-rigid Earth are obtained from two different approaches and compared. The first approach is an analytical method which relies on the solutions of the Liouville equations for a two-layer Earth model. The Oppolzer terms are evaluated from analytical expressions. The results are then compared to those calculated from Wahr's theory of nutation for a non-rigid Earth, which is the second approach used. Results are obtained for the main nutation frequencies and for the precession case. The differences between the two solutions are generally quite small (the relative error is most of the time under 8%) and are, for a large part, due to successive approximations and truncation effects during their determination. Departures of the results from the two methods are significantly larger for frequencies near the Free Core Nutation (FCN) resonance. This is particularly true for the Oppolzer terms of the angular momentum axis. The Earth model adopted is a little bit different in each case: for the Liouville system solution, we have limited the model to a homogeneous elastic mantle and a homogeneous liquid core. Another source of some of the small differences in the results is the presence of a solid inner core in Wahr's theory. We confirm through the analytical calculation the strong effect of the core on the Oppolzer terms of the angular momentum axis for a non-rigid Earth at the precession frequency. Finally, an application is given in the determination of the axes' position at J2000 for a non-rigid Earth. Received: 23 February 1998 /Accepted: 18 November 1998     

EARTH ROTATION EQUATIONS CONTAINING BOTH NUTATION AND POLAR MOTION

1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｍｏｄｅｒｎＥａｒｔｈｒｏｔａｔｉｏｎｔｈｅｏｒｙ ,ｔｈｒｅｅｒｅｆｅｒｅｎｃｅｆｒａｍｅｓａｒｅｕｓｕａｌｌｙｕｓｅｄ ,ｉ.ｅ .,ｔｈｅｓｐａｔｉａｌ (ｉｎｅｒｔｉａｌ)ｆｒａｍｅＯξ1 ξ2 ξ3,ｔｈｅｔｅｒｒｅｓｔｒｉａｌｆｒａｍｅＯｘ1 ｘ2 ｘ3ａｎｄｔｈｅｎｕｔａｔｉｏｎｆｒａｍｅＯｘ01 ｘ02 ｘ03.Ｔｈｅｎｕｔａｔｉｏｎｆｒａｍｅｄｅｆｉｎｅｓａｃｅｌｅｓｔｉａｌｅｐｈｅｍｅｒｉｓｐｏｌｅ .Ｔｈｅｍｏｔｉｏｎｏｆｔｈｉｓｃｅｌｅｓｔｉａｌｅｐｈｅｍｅｒｉｓｐｏｌｅｗ…     

液核地球极移和章动理论模型的改进

以经典的液核弹性地球自转动力学理论为基础,通过引入章动坐标系相对惯性空间的运动,建立液核地球极移和章动的联合动力学方程。由此定义液核地球的CIP轴,并对其进行解算。研究表明新方法克服了传统方法不能解决的一些问题(例如CEP和CIP的定义),简化了传统方法求解的复杂性。值得注意的是新方法可同时求解极移和章动,特别是在Smith M.L.(1977)理论中出现的倾斜模(TOM),在此只是作为一个特解而存在。     

将微椭对称体的弹性运动方程化算到球域上的一种新方法

研究微椭对称弹性地球模型运动方程的简化问题。将它们化算到了球域上，得到的方程等于球对称弹性地球模型的运动方程加上扁率改正项。     

Four dimensional studies in Earth space

Summary A system of reference which is directly related to observations, is proposed for four dimensional studies in Earth space. The requisite data is used to define both global control network and also polar wandering. The determination of variations of the Earth’s gravitational field with time also forms part of such a system. Techniques are outlined for the unique definition of the motion of the geocenter, and the changes in the location of the axis of rotation of an instantaneous Earth model, in relation to values at some epoch of reference. The instantaneous system referred to is directly related to a fundamental equation in geodynamics. The reference system defined would provide an unambiguous frame for long period studies in Earth space, provided the scale of the space were specified. Presented at the I.A.G./A.G.U. Symposium on Earth Gravity Models related problems, St Louis, Missouri, U.S.A., 16–18 August 1972.     

用切贝谢夫配点法消除地球自振常微分方程组在地心处的奇异性

用切贝谢夫配点法求解地球自振常微分方程组，无需进一步改化即可消除这组方程在地心处的奇异性，并能获得高精度的结果     

Observation of the Earth’s nutation by the VLBI: how accurate is the geophysical signal

We compare nutation time series determined by several International VLBI Service for geodesy and astrometry (IVS) analysis centers. These series were made available through the International Earth Rotation and Reference Systems Service (IERS). We adjust the amplitudes of the main nutations, including the free motion associated with the free core nutation (FCN). Then, we discuss the results in terms of physics of the Earth’s interior. We find consistent FCN signals in all of the time series, and we provide corrections to IAU 2000A series for a number of nutation terms with realistic errors. It appears that the analysis configuration or the software packages used by each analysis center introduce an error comparable to the amplitude of the prominent corrections. We show that the inconsistencies between series have significant consequences on our understanding of the Earth’s deep interior, especially for the free inner core resonance: they induce an uncertainty on the FCN period of about 0.5 day, and on the free inner core nutation (FICN) period of more than 1000 days, comparable to the estimated period itself. Though the FCN parameters are not so much affected, a 100 % error shows up for the FICN parameters and prevents from geophysical conclusions.     

三轴弹性地球自转的动力学方程

整体地球自转动力学理论一般假设地球是旋转对称的,但实际上地球是一个非对称的旋转椭球体。因此,三轴地球自转的研究是符合现实的。本文在弹性地球自转Liouville方程的基础上,在推导过程中所有量保留到极移平方和椭率乘积量级而忽略其更小量级的情况下,给出了适用于地球自转研究的三轴弹性地球自转的动力学方程。同时也列出了求解三轴弹性地球自转自由摆动的两种方法,即椭圆积分方法和椭圆函数方法。最后指出,如果在推导过程中保留更小量级,则弹性地球模型的自转动力学方程无解析解;旋转对称地球自转的线性解是三轴地球模型在极移量级下的一种特例,且三轴弹性地球模型不可能出现第二自由摆动。     

The inertial positioning problem in Hamiltonian mechanics

The general inertial positioning problem—to determine the actual coordinates of a vehicle, moving with respect to the inertial frame, from its measured inertial accelerations—is discussed in Hamiltonian mechanics. The use of the Hamiltonian formalism allows the avoidance of sophisticated geometrical considerations. Starting from the canonical equations of motion for the general problem, the navigation equations of some special problems (motion on a rotating sphere, motion on a rotating ellipsoid) are derived in linearized form. Using the simplicity of the linearized form, the solutions can be easily computed in analytical form. This is done for the above problems.     

On the dynamical effects of an inhomogeneous liquid core in the theory of nutation

As shown in previous work, dynamical effects of a realistic model of a heterogeneous, compressible, stably stratified liquid core may be obtained by means of a simple analysis of the generalized two-dimensional Laplace tidal equation which describes tidal flows of an incompressible and non-gravitating fluid in a thin spherical layer with mobile boundaries. The solution was presented in the form of expansions in powers of a small parameter κ being the ratio of nutational motion frequency in space to the frequency of the Earth's diurnal rotation. Whereas in an earlier paper only first-order terms were taken into account, our present approach includes not only main second-order terms in the spherical harmonic expansions of the solutions, but also the terms of higher orders. These effects are calculated numerically for realistic models of the Earth's outer liquid core, solid inner core and anelastic mantle (PREM model). All tables are found in electronic version at http://www.tu-darmstadt.de/fb/vw/ipg/Welcome2.html Received: 12 June 1997 / Accepted: 11 December 1997     

Geocenter variations derived from a combined processing of LEO- and ground-based GPS observations

GNSS observations provided by the global tracking network of the International GNSS Service (IGS, Dow et al. in J Geod 83(3):191–198, 2009) play an important role in the realization of a unique terrestrial reference frame that is accurate enough to allow a detailed monitoring of the Earth’s system. Combining these ground-based data with GPS observations tracked by high-quality dual-frequency receivers on-board low earth orbiters (LEOs) is a promising way to further improve the realization of the terrestrial reference frame and the estimation of geocenter coordinates, GPS satellite orbits and Earth rotation parameters. To assess the scope of the improvement on the geocenter coordinates, we processed a network of 53 globally distributed and stable IGS stations together with four LEOs (GRACE-A, GRACE-B, OSTM/Jason-2 and GOCE) over a time interval of 3 years (2010–2012). To ensure fully consistent solutions, the zero-difference phase observations of the ground stations and LEOs were processed in a common least-squares adjustment, estimating all the relevant parameters such as GPS and LEO orbits, station coordinates, Earth rotation parameters and geocenter motion. We present the significant impact of the individual LEO and a combination of all four LEOs on the geocenter coordinates. The formal errors are reduced by around 20% due to the inclusion of one LEO into the ground-only solution, while in a solution with four LEOs LEO-specific characteristics are significantly reduced. We compare the derived geocenter coordinates w.r.t. LAGEOS results and external solutions based on GPS and SLR data. We found good agreement in the amplitudes of all components; however, the phases in x- and z-direction do not agree well.     

三轴刚性地球自由旋转欧拉角变化数值模拟研究

假定地球是一个三轴刚性体,在Euclid空间中做自由旋转.在设定主惯性矩A小于B小于C的情况下,求解欧拉运动方程,得到数值解.计算结果表明:地球在除了自转和自由进动之外,同时还存在着自由章动.章动角会随着时间做周期性变化.重点讨论的是章动角的变化.     

On calculating the Earth's C21 and S21 gravity coefficients in the IERS terrestrial reference frame

Modern models of the Earth's gravity field are developed in the IERS (International Earth Rotation Service) terrestrial reference frame. In this frame the mean values for gravity coefficients of the second degree and first order, C ₂₁(IERS) and S ₂₁(IERS), by the current IERS Conventions are recommended to be calculated by using the observed polar motion parameters. Here, it is proved that the formulae presently employed by the IERS Conventions to obtain these coefficients are insufficient to ensure their values as given by the same source. The relevant error of the normalized mean values for C ₂₁(IERS) and S ₂₁(IERS) is 3×10⁻¹², far above the adopted cutoff (10⁻¹³) for variations of these coefficients. Such an error in C ₂₁ and S ₂₁ can produce non-modeled perturbations in motion prediction of certain artificial Earth satellites of a magnitude comparable to the accuracy of current tracking measurements. Received: 14 September 1998 / Accepted: 20 May 1999     

The gravity field variation caused by inner core super rotation

Due to the super rotation of the Earth’s inner core, the tilted figure axis of the inner core would progress with respect to the mantle and thus cause the variation of the Earth’s external gravity field. This paper improves the present model of the gravity field variation caused by the inner core super rotation. Under the assumption that the inner core is a stratifying ellipsoid whose density function is fitted out from PREM and the super rotation rate is 0.27～0.53°/yr, calculations show that the global temporal variations on the Earth’s surface have a maximum value of about 0.79～1.54×10?3 μGal and a global average intensity of about 0.45～0.89×10?3 μGal in the whole year of 2007, which is beyond the accuracy of the present gravimetry and even the super conducting gravimeter data. However, both the gravity variations at Beijing and Wuhan vary like sine variables with maximal variations around 0.33 μGal and 0.29 μGal, respectively, in one cycle. Thus, continuous gravity measurements for one or two decades might be able to detect the differential motion of the inner core.     

Prediction of Earth orientation parameters by artificial neural networks

 Earth orientation parameters (EOPs) [polar motion and length of day (LOD), or UT1–UTC] were predicted by artificial neural networks. EOP series from various sources, e.g. the C04 series from the International Earth Rotation Service and the re-analysis optical astrometry series based on the HIPPARCOS frame, served for training the neural network for both short-term and long-term predictions. At first, all effects which can be described by functional models, e.g. effects of the solid Earth tides and the ocean tides or seasonal atmospheric variations of the EOPs, were removed. Only the differences between the modeled and the observed EOPs, i.e. the quasi-periodic and irregular variations, were used for training and prediction. The Stuttgart neural network simulator, which is a very powerful software tool developed at the University of Stuttgart, was applied to construct and to validate different types of neural networks in order to find the optimal topology of the net, the most economical learning algorithm and the best procedure to feed the net with data patterns. The results of the prediction were analyzed and compared with those obtained by other methods. The accuracy of the prediction is equal to or even better than that by other prediction methods. Received: 6 February 2001 / Accepted: 23 October 2001     

Contributions to reference systems from Lunar Laser Ranging using the IfE analysis model

Lunar Laser Ranging (LLR) provides various quantities related to reference frames like Earth orientation parameters, coordinates and velocities of ground stations in the Earth-fixed frame and selenocentric coordinates of the lunar retro-reflectors. This paper presents the recent results from LLR data analysis at the Institut für Erdmessung, Leibniz Universität Hannover, based on all LLR data up to the end of 2016. The estimates of long-periodic nutation coefficients with periods between 13.6 days and 18.6 years are obtained with an accuracy in the order of 0.05–0.7 milliarcseconds (mas). Estimations of the Earth rotation phase \(\Delta \)UT are accurate at the level of 0.032 ms if more than 14 normal points per night are included. The tie between the dynamical ephemeris frame to the kinematic celestial frame is estimated from pure LLR observations by two angles and their rates with an accuracy of 0.25 and 0.02 mas per year. The estimated station coordinates and velocities are compared to the ITRF2014 solution and the geometry of the retro-reflector network with the DE430 solution. The given accuracies represent 3 times formal errors of the parameter fit. The accuracy for \(\Delta \)UT is based on the standard deviation of the estimates with respect to the reference C04 solution.