Zonal Coefficients Research Articles

The shape, internal structure and gravity of the fast spinner β Pictoris b

Abstract A young extrasolar gas giant planet, β Pictoris b, recently discovered in the β Pictoris system, spins substantially faster than the giant gas planets Jupiter and Saturn. Based on the newly measured parameters – the rotation period of the planet, its mass and radius – together with an assumption that the gas planet β Pictoris b is in hydrostatic equilibrium and made of a fully compressible barotropic gas with a polytropic index of unity, we are able to compute, via a hybrid inverse method, its non-spherical shape, internal density/pressure distribution and gravitational zonal coefficients up to degree 8. Since the mass Mβ for the planet β Pictoris b is highly uncertain, various models with different values of Mβ are studied in this Letter, providing the upper and lower bounds for its shape parameter as well as its gravitational zonal coefficients. If Mβ is assumed to be 6MJ with MJ being Jupiter's mass, we show that the shape of the planet β Pictoris b is approximately described by an oblate spheroid whose eccentricity at the one-bar surface is $\mathcal {E}_{\beta }=0.369\,28$ with the gravitational coefficient (J2)β = +15 375.972 × 10−6. It follows that our results open the possibility of constraining or inferring the mass Mβ of the planet β Pictoris b if its shape can be measured or constrained. By assuming that the planet β Pictoris b will shrink to the size of Jupiter in the process of cooling down and, hence, rotate much faster, we also calculate the future shape and internal structure of the planet β Pictoris b.

ON THE VARIATION OF ZONAL GRAVITY COEFFICIENTS OF A GIANT PLANET CAUSED BY ITS DEEP ZONAL FLOWS

Rapidly rotating giant planets are usually marked by the existence of strong zonal flows at the cloud level. If the zonal flow is sufficiently deep and strong, it can produce hydrostatic-related gravitational anomalies through distortion of the planet’s shape. This paper determines the zonal gravity coefficients, J2n ,n = 1,2,3 ,..., via an analytical method taking into account rotation-induced shape changes by assuming that a planet has an effective uniform density and that the zonal flows arise from deep convection and extend along cylinders parallel to the rotation axis. Two different but related hydrostatic models are considered. When a giant planet is in rigid-body rotation, the exact solution of the problem using oblate spheroidal coordinates is derived, allowing us to compute the value of its zonal gravity coefficients ¯ J2n ,n = 1,2,3 ,... , without making any approximation. When the deep zonal flow is sufficiently strong, we develop a general perturbation theory for estimating the variation of the zonal gravity coefficients, ΔJ2n = J2n − ¯ J2n ,n = 1,2,3 ,... , caused by the effect of the deep zonalflows for an arbitrarily rapidly rotating planet. Applying the general theory to Jupiter, we find that the deep zonal flow could contribute up to 0.3% of the J2 coefficient and 0.7% of J4. It is also found that the shape-driven harmonics at the 10th zonal gravity coefficient become dominant, i.e., ΔJ2n ¯ J2n for n5.

Evaluation of the coefficients of horizontal exchange in the Black Sea according to the data of drifter experiments

We generalize the method used for the evaluation of the coefficients of horizontal diffusion of Brownian particles to the nonstationary case, as applied to the drifter experiments. The limits of applicability of the proposed procedure are determined. The coefficients of horizontal exchange are computed with an aim to compare them with the estimates obtained earlier by using the generalized Taylor theory. The mutual agreement of these estimates is demonstrated. It is shown that the difference between the values of the zonal (0.19⋅104 m2/sec) and meridional (0.11⋅104 m2/sec) coefficients of exchange in the Black Sea cannot be explained by the effect of increase in the longitudinal component of pulsations of the velocity as a result of its transverse shear relative to the mean current. A conclusion is made that the processes of horizontal exchange in the Black Sea are geographically anisotropic.

A simulation of Martian gravity field recovery by using a near equatorial orbiter

An improvement to the Martian gravity field may be achieved by means of future orbiting spacecraft with small eccentricity and low altitude exemplified through a newly proposed mission design that may be tested in upcoming reconnaissance of Mars. Here, the near equatorial orbital character (with an inclination approximating 10°, eccentricity as 0.01 and semi-major axis as 4000km) is considered, and its contribution to Martian gravity field solution is analyzed by comparing it with a hypothetical polar circular orbiter. The solution models are evaluated in terms of the following viewpoints: power spectra of gravity field coefficients, correlations of low degree zonal coefficients, precise orbit determination, and error distribution of both Mars free air gravity anomaly and areoid. At the same time, the contributions of the near equatorial orbiters in low degree zonal coefficients time variations are also considered. The present results show that the near equatorial orbiter allows us to improve the accuracy of the Martian gravity field solution, decrease correlation of low degree zonal coefficients, retrieve much better time variable information of low degree zonal coefficients, improve precise orbit determination, and provide more accurate Mars free air gravity anomaly and areoid around the equatorial region.

Low-degree gravity change from GPS data of COSMIC and GRACE satellite missions

This paper demonstrates estimation of time-varying gravity harmonic coefficients from GPS data of COSMIC and GRACE satellite missions. The kinematic orbits of COSMIC and GRACE are determined to the cm-level accuracy. The NASA Goddard's GEODYN II software is used to model the orbit dynamics of COSMIC and GRACE, including the effect of a static gravity field. The surface forces are estimated per one orbital period. Residual orbits generated from kinematic and reference orbits serve as observables to determine the harmonic coefficients in the weighted-constraint least-squares. The monthly COSMIC and GRACE GPS data from September 2006 to December 2007 (16 months) are processed to estimate harmonic coefficients to degree 5. The geoid variations from the GPS and CSR RL04 (GRACE) solutions show consistent patterns over space and time, especially in regions of active hydrological changes. The monthly GPS-derived second zonal coefficient closely resembles the SLR-derived and CSR RL04 values, and third and fourth zonal coefficients resemble the CSR RL04 values.

Numerical study on the mixed model in the GOCE polar gap problem

Gravity gradients acquired by the Gravity field and steady-state Ocean Circulation Explorer (GOCE) do not cover the entire earth because of its sun-synchronous orbit leaving data gaps with a radius of about 6.5° in the polar regions. Previous studies showed that the loss of data in the polar regions deteriorates the accuracy of the low order (or near zonal) coefficients of the earth gravity model, which is the so-called polar gap problem in geodesy. In order to find a stable solution for the earth gravity model from the GOCE gravity gradients, three models, i.e. the Gauss-Markov model, light constraint model and the mixed model, are compared and evaluated numerically with the gravity gradient simulated with the EGM2008. The comparison shows that the Best Linear Uniformly Unbiased Estimation (BLUUE) estimator of the mixed model can solve the polar gap problem as effectively as the light constraint model; furthermore, the mixed model is more rigorous in dealing with the supplementary information and leads to a better accuracy in determining the global geoid.

Improved GRACE science results after adjustment of geometric biases in the Level-1B K-band ranging data

The GRACE (Gravity Recovery and Climate Experiment) satellite mission relies on the inter-satellite K-band microwave ranging (KBR) observations. We investigate systematic errors that are present in the Level-1B KBR data, namely in the geometric correction. This correction converts the original ranging observation (between the two KBR antennas phase centers) into an observation between the two satellites’ centers of mass. It is computed from data on the precise alignment between both satellites, that is, between the lines joining the center of mass and the antenna phase center of either satellite. The Level-1B data used to determine this alignment exhibit constant biases as large as 1–2 mrad in terms of pitch and yaw alignment angles. These biases induce non-constant errors in the Level-1B geometric correction. While the precise origin of the biases remains to be identified, we are able to estimate and reduce them in a re-calibration approach. This significantly improves time-variable gravity field solutions based on the CNES/GRGS processing strategy. Empirical assessments indicate that the systematic KBR data errors have previously induced gravity field errors on the level of 6–11 times the so-called GRACE baseline error level. The zonal coefficients (from degree 14) are particularly affected. The re-calibration reduces their rms errors by about 50%. As examples for geophysical inferences, the improvement enhances agreement between mass variations observed by GRACE and in-situ ocean bottom pressure observations. The improvement also importantly affects estimates of inter-annual mass variations of the Antarctic ice sheet.

AIUB-CHAMP02S: The influence of GNSS model changes on gravity field recovery using spaceborne GPS

The gravity field model AIUB-CHAMP02S, which is based on six years of CHAMP GPS data, is presented here. The gravity field parameters were derived using a two step procedure: In a first step a kinematic trajectory of a low Earth orbiting (LEO) satellite is computed using the GPS data from the on-board receiver. In this step the orbits and clock corrections of the GPS satellites as well as the Earth rotation parameters (ERPs) are introduced as known. In the second step this kinematic orbit is represented by a gravitational force model and orbit parameters. In order to ensure full model consistency the GPS satellite orbits and clock corrections, which have been used for the generation of the kinematic LEO trajectories, were taken from the Center for Orbit Determination in Europe (CODE), located at AIUB ( Dach et al., 2009). In recent years many changes have taken place in the processing chain of global navigation satellite system (GNSS) data, e.g., the implementation of absolute antenna phase center modeling. Therefore a reprocessing of the GPS data to obtain state-of-the-art GPS satellite orbits and clock corrections was performed. From these updated GPS products new kinematic orbits of the CHAMP satellite were derived for the years 2002–2007. From the updated CHAMP trajectories spherical harmonic (SH) coefficients of the Earth’s gravity field were determined in exactly the same way as from the original LEO orbit. This allowed us to study the impact of the improved LEO orbits on the derived gravity field parameters and the generation of the multi-year gravity field model AIUB-CHAMP02S. The change of the IGS standards creates an inconsistency to existing global gravity field models, which mainly affects the zonal coefficients of low even degrees. The inconsistency is caused by the change to the absolute antenna phase center model and can be reduced by estimating the phase center variation of the CHAMP GPS antenna.

Martian gravity field model and its time variations from MGS and Odyssey data

We present results of several years of research and data processing aimed at modelling the Mars gravity field and its longest wavelength time variations. The new solution includes tracking data from Mars Global Surveyor (MGS) from 1998 to 2006 (end of mission) and from Mars Odyssey from 2002 to the spring of 2008; this is the longest analyzed data set from these two orbiter missions as compared to previous works. The new model has been obtained by a team working in Europe, independently from the works of groups at NASA Jet Propulsion Laboratory (JPL) and Goddard Space Flight Center (GSFC), also with totally independent software. Observations consist in two and three-way Doppler measurements (also one way for MGS), and range tracking data collected by the Deep Space Network and have been processed in 4 day arcs, taking into account all disturbing forces of gravitational and non-gravitational origins; for each arc the state vector, drag and solar pressure model multiplying factors, and angular momentum dump parameters are adjusted. The static field (MGGM08A) is represented in spherical harmonics up to degree and order 95 and is very close to previously published models (in terms of spectral components and also over specific features); correlations with the global Mars topography are established and apparent depths of compensation by degree are derived. Lumped zonal harmonics of degree two and three are solved for every 10 days, exhibiting variations in line with previous results (including authors’ ones); the work also shows the difficulty of finding clean signatures (annual and semi-annual) for the zonal coefficient of second degree. The k 2 Love number is also derived from the ensemble of data, as well as from subsets of them; values between 0.110 and 0.130 are found, which are consistent with the existence of a Martian fluid core of significant radius.

Postseismic signature of the 2004 Sumatra earthquake on low‐degree gravity harmonics

We perform an extensive analysis of the low‐degree gravity field harmonics measured by the GRACE mission, in order to find a signature of the postseismic relaxation following the 2004 Sumatra earthquake. We find a statistically significant perturbation in the secular trend of low‐degree zonal coefficients (Jl) in correspondence of the 2004 Sumatra earthquake and a similar perturbation, but with weak associated statistical significance, also in the nonzonal coefficients. Technical features and results of such analysis are discussed. The time‐dependent postseismic evolution of harmonic coefficients is modeled for various asthenosphere viscosity values, using a theoretical model of global postseismic deformation. The observed change in secular trend is found to be consistent with our modeling results but it cannot be used to discriminate between viscosities. A forward modeling of the perturbations to time‐dependent zonal variation rates following the Sumatra earthquake for various asthenospheric viscosities is provided. As a result, an evident signature of the Sumatra earthquake on l time series is expected for asthenospheric viscosity values below 1018 Pa s. Therefore, long term l time histories from satellite laser ranging will be able to put constraints on the asthenosphere viscosity, if such a signature is evidenced from data or, at least, put lower limits if no significant perturbation will be observed.

The effect of the internal structure of Mars on its seasonal loading deformations

Mars is continuously subjected to surface loading induced by seasonal mass changes in the atmosphere and ice caps due to the CO 2 sublimation and condensation process. It results in surface deformations and in time variations of gravity. Large wavelength annual and semi-annual variations of gravity (particularly zonal coefficients Δ J n ) have been determined using present day geodetic satellite measurements. However loading deformations have been poorly studied for a planet like Mars. In this paper, we compute these deformations and their effect on spacecraft orbiting around Mars. Loading deformations of terrestrial planet are typically investigated assuming a spherical planet, radially symmetric. The mean radial structure of Mars is not well known. In particular the radius of the liquid or solid core remains not precisely determined. One may then wonder what is the effect of these uncertainties on loading deformations. Moreover, Mars presents a strong topography and probably large lateral variations of crustal thickness (relative to the Earth). The paper answer the questions of what is the effect of such lateral heterogeneities on surface deformations, and is the classical way to calculate loading deformation well adapted for a planet like Mars. In order to answer these questions we have investigated theoretically loading deformations of Mars-like planets. We first investigated classical load Love numbers. We show that for degrees inferior to 10, the load Love numbers mainly depend on the radius of the core and on its state, and that for degree greater than 10, they depend on the mean radius of mantle–crust interface. Using a General Circulation Model (GCM) of atmosphere and ice caps dynamics we show that loading vertical displacements have a 4–5 cm magnitude and present a North–South pattern with periodic transitions. Finally we investigated the effect of lateral variations of the crustal thickness on these loading deformations. We show that thickness heterogeneities perturb the deformations and the time variation of gravity at about 0.5%. However this perturbation on Δ J n is only about 1‰ due to main direct attraction of surface fluid layers. We conclude that lateral variations of crustal thickness are today negligible. However, observation of load Love numbers would bring information on the radial internal structure of the planet, particularly on the core radius. Δ J n study would permit to infer the load Love number k n ′ , particularly for degree 2 and 3, knowing surface fluid layer dynamics. However k n ′ load Love numbers are quite small (about 0.05), and despite the present good agreement between GCM and Δ J n observations, will only be estimated in the near future when a slightly better precision in observation and modeling will make it possible to infer these numbers. The investigation of load Love number h n ′ , which are larger than k n ′ numbers, would be particularly interesting. It would permit to study degree 1 contribution of atmosphere and ice caps dynamics, which is the most important component of surface fluid dynamics on Mars. Surface displacement measurements would be necessary on a few places near the pole regions, which may be possible in the future, with a project involving precise positioning of a lander on the surface of Mars.

Basis functions for the consistent and accurate representation of surface mass loading

SUMMARY Inversion of geodetic site displacement data to infer surface mass loads has previously been demonstrated using a spherical harmonic representation of the load. This method suffers from the continent-rich, ocean-poor distribution of geodetic data, coupled with the predominance of the continental load (water storage and atmospheric pressure) compared with the ocean bottom pressure (including the inverse barometer response). Finer-scale inversion becomes unstable due to the rapidly increasing number of parameters which are poorly constrained by the data geometry. Several approaches have previously been tried to mitigate this, including the adoption of constraints over the oceanic domain derived from ocean circulation models, the use of smoothness constraints for the oceanic load, and the incorporation of GRACE gravity field data. However, these methods do not provide appropriate treatment of mass conservation and of the ocean’s equilibrium-tide response to the total gravitational field. Instead, we propose a modified set of basis functions as an alternative to standard spherical harmonics. Our basis functions allow variability of the load over continental regions, but impose global mass conservation and equilibrium tidal behaviour of the oceans. We test our basis functions first for the efficiency of fitting to realistic modelled surface loads, and then for accuracy of the estimates of the inferred load compared with the known model load, using synthetic geodetic displacements with real GPS network geometry. Compared to standard spherical harmonics, our basis functions yield a better fit to the model loads over the period 1997‐2005, for an equivalent number of parameters, and provide a more accurate and stable fit using the synthetic geodetic displacements. In particular, recovery of the low-degree coefficients is greatly improved. Using a nine-parameter fit we are able to model 58 per cent of the variance in the synthetic degree-1 zonal coefficient time-series, 38‐41 per cent of the degree-1 non-zonal coefficients, and 80 per cent of the degree-2 zonal coefficient. An equivalent spherical harmonic estimate truncated at degree 2 is able to model the degree-1 zonal coefficient similarly (56 per cent of variance), but only models 59 per cent of the degree-2 zonal coefficient variance and is unable to model the degree-1 non-zonal coefficients.

DASの最適軌道制御に関する一研究

DAS (Dive and Ascent Satellite) is a kind of satellite that can approach very close to the earth, which was designed by NAL (National Aerospace Laboratory of Japan) in later 1970’s. Now we paid attention to the function of the satellite that can descend and ascend, and proposed to use it for getting detailed information of the area which is suffered from a natural disaster. In this paper orbital control of DAS with minimum fuel consumption is discussed, where the earth is treated as an ellipsoid of revolution, the atmosphere drag and the second zonal coefficient is considered. The optimal orbital control problems are treated as nonlinear two point boundary problems, and solved by the steepest ascent method. The derivation of equations and results are shown in detail.

Long-Period Variations of the Eccentricity Vector Valid also for Near Circular Orbits around a Non-Spherical Body

This paper studies the long period variations of the eccentricity vector of the orbit of an artificial satellite, under the influence of the gravity field of a central body. We use modified orbital elements which are non-singular at zero eccentricity. We expand the long periodic part of the corresponding Lagrange equations as power series of the eccentricity. The coefficients characterizing the differential system depend on the zonal coefficients of the geopotential, and on initial semi-major axis, inclination, and eccentricity. The differential equations for the components of the eccentricity vector are then integrated analytically, with a definition of the period of the perigee based on the notion of “free eccentricity”, and which is also valid for circular orbits. The analytical solution is compared to a numerical integration. This study is a generalization of (Cook, Planet. Space Sci., 14, 1966): first, the coefficients involved in the differential equations depend on all zonal coefficients (and not only on the very first ones); second, our method applies to nearly circular orbits as well as to not too eccentric orbits. Except for the critical inclination, our solution is valid for all kinds of long period motions of the perigee, i.e., circulations or librations around an equilibrium point.

Ionospheric and induced field leakage in geomagnetic field models, and derivation of candidate models for DGRF 1995 and DGRF 2000

Abstract As part of the 9th generation of the IGRF defined by IAGA, we proposed a candidate model for DGRF 1995 and two candidate models for DGRF 2000. These candidate models, the derivation of which is described in the present note, are based on the “Comprehensive Model, Version 4 (CM4)”, and on the “Ørsted Main and Secular Variation Model (OSVM)”; two parent models that have been published elsewhere (Olsen, 2002; Sabaka et al., 2004; Lowes and Olsen, 2004). However, the main field part of OSVM is contaminated by “leakage” of the ionospheric field and its induced counterpart, which affects mainly the zonal coefficients g 1 0 , g 3 0 , …, by 1–2 nT. We describe the reason for this contamination, and present a method to correct for it. Since not only OSVM, but probably all main field models that are derived primarily from data around local midnight suffer from this effect, the presented scheme can also be applied to approximately correct these models.

On the physical meaning of the zonal components of the geopotential

Abstract The MacCullagh equation is of key importance in the study of the Earth. It connects the geometrical and the physical properties of the Earth through the geodynamical shape factor J 2 . This second zonal geopotential coefficient is closely related to the flattening and to the angular spin velocity of the Earth as well as to its equatorial ( A ) and polar ( C ) moments of inertia. Through these moments of inertia the gravitational potential V is connected to the mass density distribution within the Earth. The main target of the present study is to obtain a generalized form of the MacCullagh equation for even orders n ≥ 2 by including the higher order zonal coefficients J n connected with the higher ( n ≥ 2) degree moments of inertia C n and A n . The higher the degree n , the higher is the weight of the near-surface (i.e. shallow) mass density distribution in J n . The second part of this contribution deals with the temporal variations of J n , d J n /d t for n ≥ 2.

Precession, nutation, and space geodetic determination of the Earth's variable gravity field

Precession and nutation of the Earth depend on the Earth's dynamical flattening, H, which is closely related to the second degree zonal coefficient, J2 of the geopotential. A small secular decrease as well as seasonal variations of this coefficient have been detected by precise measurements of artificial satellites (Nerem et al. [CITE]; Cazenave et al. [CITE]) which have to be taken into account for modelling precession and nutation at a microarcsecond accuracy in order to be in agreement with the accuracy of current VLBI determinations of the Earth orientation parameters. However, the large uncertainties in the theoretical models for these J2 variations (for example a recent change in the observed secular trend) is one of the most important causes of why the accuracy of the precession-nutation models is limited (Williams [CITE]; Capitaine et al. [CITE]). We have investigated in this paper how the use of the variations of J2 observed by space geodetic techniques can influence the theoretical expressions for precession and nutation. We have used time series of J2 obtained by the “Groupe de Recherches en Géodésie spatiale” (GRGS) from the precise orbit determination of several artificial satellites from 1985 to 2002 to evaluate the effect of the corresponding constant, secular and periodic parts of H and we have discussed the best way of taking the observed variations into account. We have concluded that, although a realistic estimation of the J2 rate must rely not only on space geodetic observations over a limited period but also on other kinds of observations, the monitoring of periodic variations in J2 could be used for predicting the effects on the periodic part of the precession-nutation motion.

Estimation of the Earth's tensor of inertia from recent global gravity field solutions

The dynamic figure of the Earth, characterized by the principal axes and principal moments of inertia, is estimated from satellite-derived gravitational harmonic coefficients of second degree in recent global Earth gravity models and from the dynamic ellipticity resulting from the precession constant observed through very-long-baseline interferometry (VLBI). Closed, exact formulae for the determination of these parameters of the Earth's tensor of inertia are developed based on the exact solution of the eigenvalue–eigenvector problem, including a rigorous error propagation. These formulae are applied to determine (a) static components and accuracy of the Earth's tensor of inertia at epoch and (b) the variation with time of the Earth's tensor of inertia and its accuracy. The best-fitting principal moments of inertia and second-degree harmonic coefficients in the principal-axes system are found from an adjustment involving four global gravity field models and six different values for the dynamic ellipticity. The evolution with time of the dynamic figure of the Earth is determined from the mean pole path and the observed secular rate of change in the second-degree zonal coefficient. It is found that differences in the principal moments of inertia change significantly over the time interval from 1962 to 2000, whereas changes in the absolute values cannot be reliably resolved due to the uncertainty in the dynamic ellipticity.

The SLR secular gravity variations and their impact on the inference of mantle rheology and lithospheric thickness

A long history of SLR (Satellite Laser Ranging) observations of the geodetic satellites LAGEOS‐I, LAGEOS‐II, Starlette and Stella have been analyzed in order to estimate the time series of the low degree zonal coefficients in the Earth gravity field, up to degree six, and derive their secular drifts. The paper will point out the critical aspects of the analysis process and will compare the estimated zonal rates with other published results. Comparison of these zonal rates with the results of global, viscoelastic Earth models forced by Pleistocenic deglaciation, shows that the SLR retrieved zonals and the lumped odd zonals can be used to infer the upper mantle viscosity and lithospheric thickness. Discrepancies in the viscosity profiles, required to reproduce the different zonals, seem to indicate ongoing mass redistribution over the Earth.

Seasonal variations in low degree zonal harmonics of the Earth's gravity field from satellite laser ranging observations

Seasonal variations in the Earth's gravity field were determined using satellite laser ranging (SLR) observations from multisatellite. The time series for the variations of the even zonal harmonics, Jl(l = 2, 4, 6, and 8), were determined using the SLR data from the geodetic satellites, including Starlette, Ajisai, Stella, LAGEOS I and LAGEOS II, during the period from October 1993 to December 1996. Owing to uncertainties in the eccentricity excitation for LAGEOS I and II, the variations of J3 and J5 were determined using only the SLR data from Starlette, Ajisai, and Stella. The seasonal variations of J2, J4, J6 and J8 become separable using the existing multisatellite SLR data sets collected in 15‐day time intervals. The amplitude (normalized and in units of 10−10) and phase (in a cosine conversion and in units of degrees) for the annual variation in Jl (l= 2, 3, 4, 5, 6, and 8) are estimated to be (1.25 ± 0.1, 140 ± 10), (2.16 ± 0.21, 341 ± 19), (1.07 ± 0.1, 338 ± 15), (1.12 ± 0.16, 152 ± 16), (0.26 ± 0.17, 337 ± 9), and (1.03 ± .016, 209 ± 10), respectively. The observed annual variations of J3 and J5 are essentially opposite in phase. This phenomenon results in a different lumped sum effect for various satellites. For example, the lumped sum of J3 and J5 annual variation from LAGEOS I is twice as large as that from Starlette. The excitation due to the mass redistribution in the atmosphere and ocean and the changes in continental water storage were considered in this study using the available global geophysical data, which included the European Centre for Medium‐Rang Weather Forecasts atmospheric surface pressure, the TOPEX/Poseidon altimetry derived sea surface anomalies, and the World Monthly Surface Station Climatic Data. Overall, the variation in the even zonal coefficients due to the atmospheric mass redistribution is responsible for 30% to 60% of the observed annual variations in the node residual for Starlette and LAGEOS I. Comparison indicates that the oceanic mass movement, in particular, the continental water change produce comparable contributions to the seasonal zonal variations. The amplitude of the observed annual variation of J2 is found to fall between the values predicted from the models of the surface water, ocean, and atmosphere with and without the inverted barometer (IB) oceanic response, but the phase is in good agreement with the IB models. The nontidal mass redistributions in atmosphere, ocean, and continental water change can only account for ∼13% of the semiannual variation in J2 but are the primary excitation sources for semiannual variations in the higher‐degree zonal terms.