## 1. Introduction

[2] In the Euler's dynamical equations of the rotating Earth in the body-fixed reference frame, the angular momentum of the nonrigid Earth is expressed as **H** = **I**(*t*)** ω**(

*t*) +

**h**(

*t*), where

**I**(

*t*) is the time-varying second-degree symmetric inertia tensor,

**(**

*ω**t*) is the angular velocity vector of the body-fixed axes,

**h**(

*t*) is the relative angular momentum. The time-independent parts of the diagonal components of

**I**(

*t*) correspond to the principal axes of inertia for an undeformed Earth. The figure axis of the Earth is the axis of maximum inertia for the deformed (oblate) Earth. Neglecting the second-order terms, the unit vector of the figure axis can be defined by the products of inertia, (

*I*

_{xz},

*I*

_{yz},

*C*−

*A*)/(

*C*−

*A*), where

*C*and

*A*are the Earth's polar and equatorial moments of inertial, respectively [

*Moritz and Mueller*, 1987]. The Earth's external gravitational potential is conveniently expressed in body fixed coordinates through a spherical harmonic expansion as

where *r* is the geocentric distance, ϕ is the latitude, and *λ* is the longitude of the point at which *U* is determined. *G* is the gravitational constant, *M* is the mass of the Earth, and *R* is the equatorial radius of the Earth. *P*_{nm}(sin ϕ) are the Legendre polynomials, and *C*_{nm} and *S*_{nm} are the Stokes geopotential coefficients of degree *n* and order *m* as function of the mass distribution within the Earth. and are hereafter referred to as the normalized Stokes coefficients. The unnormalized Stokes coefficients, *C*_{nm} and *S*_{nm}, are related to the normalized coefficients by the relation (, ) = (*C*_{nm}, *S*_{nm})/*N*_{nm}, where *N*_{nm} is the normalizing factor expressed as *N*_{nm} = [(2*n* + 1)(2 − *δ*_{0m})(*n* − *m*)!/(*n* + *m*)!]^{1/2}, where *δ*_{0m} = 1 for *m* > 0, and = 0 for *m* = 0. The second-degree Stokes coefficients have physical meaning and relate to Earth's inertia tensor. In particular, the degree 2 and order 1 geopotential coefficients can be expressed as *C*_{21} = −*I*_{xz}/*MR*^{2} and *S*_{21} = −*I*_{yz}/*MR*^{2}. Thus, the Earth's principal figure axis can be determined by the degree 2 and order 1 geopotential coefficient, and , where and .

[3] In the absence of external torques, the principal figure axis of the Earth would be very close to the rotational axis [*Wahr*, 1987]. However, as a consequence of excitations due to tidal and nontidal mass redistribution within the Earth system (dominated by seasonal variations), the two axes are not completely aligned, and the rotational axis precesses around the Earth's principal figure axis (causing the ∼435 day Chandler “wobble”). Although the frame defined by the principal axes of the Earth is convenient for theoretical discussion of the Earth's rotation, measuring the figure axis and its variations can improve the understanding of the relation of the figure axis oscillation to changes in the Earth's mass redistribution and provide an important global constraint on the properties of the core and core-mantle boundary as shown by *Wahr* [1987, 1990].

[4] Recent improvement in the accuracy of space geodetic measurements and related force modeling suggest that it is possible to directly measure the position of the figure axis and its variations rather than relying on calculations based on current solid Earth and ocean tide models. At present, the gravity field to degree and order, 60, along with its seasonal variations, is well determined by the Gravity Recovery and Climate Experiment (GRACE) mission [*Tapley et al.*, 2004], and a low-degree model can also be accurately determined from analysis of satellite laser ranging (SLR) data. The long-term SLR analyses from multiple geodetic satellites and results from GRACE measurements spanning more than 7 years are examined here to study the mean Earth's figure axis and its variations through the determination of spherical harmonic coefficients, and . Theoretically, the excitation of polar motion is caused by changes in the products of the inertia due to the relative motion and the surface mass redistribution, which is proportional to the changes in / [*Munk and MacDonald*, 1960; *Lambeck*, 1980]. More accurate Earth orientation parameters (EOP) determined from space geodetic techniques, such as GPS, SLR, LLR and VLBI, and the surface mass loading and motion (i.e., the atmospheric winds and ocean currents) data are also available for exploring the variations of /.

[5] As a means of validating the GRACE results, *Chen and Wilson* [2008] compared the seasonal variation in / determined by GRACE with results from SLR data and polar motion excitation function. The monthly SLR estimates available at that time for / were part of the routine estimation of the second-degree zonal harmonic coefficient () for GRACE science applications. A 5 year time series, spanning April 2002 to May 2007, was based on the GRACE RL01 standards where (1) the IERS1996 solid Earth pole tide model [*McCarthy*, 1996] was used and (2) the ocean pole tide model was not a part of the GRACE model standards at that time. The most important influence of the ocean pole tide is on /, while the effect on is small. *Chen and Wilson* [2008] found a poor agreement between the two results, with the SLR-derived annual variations differing from the GRACE results in amplitude (∼51% smaller) for and in phase (∼100°) for . Since there are significant rotational variations at the Chandler and annual frequencies, it is not unexpected that there would be poorer agreement with no ocean pole tide included and a less accurate solid Earth pole tide model used for the SLR data analysis. In addition, the time series of 5 years was relatively short and, over this period, the Chandler variation is the dominant signal. Separation of the Chandler and annual frequencies requires a time span longer than ∼6.4 years, so estimating the annual variation in / from a 5 year data set would be difficult. Understanding the contribution of the pole tides to the variations in / is necessary to correctly interpret the estimates of the seasonal variations in these harmonics.

[6] This paper presents an updated analysis of estimates of / determined from SLR tracking data of multiple satellites using models consistent with the current GRACE data processing (RL04) [*Bettadpur*, 2007], including the ocean pole tide and Atmosphere-Ocean De-aliasing (AOD) model [*Flechtner*, 2007]. Section 2 briefly reviews the model for the pole tides (rotational deformation), and the variations induced in the Earth's figure axis. Constraints on the “fluid Love number” and core rotation will be inferred based on the mean pole and the mean figure axis determined from SLR and GRACE data. Section 3 introduces the GRACE estimate of / and the AOD model used in the SLR and GRACE data processing. Analysis of SLR data is given in section 4. Section 5 analyzes the spectrum of variations in / based on the time series of weekly estimates of / from SLR data spanning approximately 17 years. In addition, the annual variation for a period of 8 years is determined based on the monthly estimates from SLR, GRACE, and Polar motion excitation function. The effects on / from the AOD background model are also compared. Section 6 discusses the secular change in / inferred from the motion of the mean pole, SLR, and GRACE, and section 7 summarizes the results.