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The Earth's mechanical and geometrical parameters were estimated from the simultaneous adjustment of the 2nd-degree harmonic coefficients of six gravity field models and seven values of the dynamical ellipticity H D all reduced to the common MHB2000 precession constant at epoch J2000. The transformation of 2nd-degree harmonic coefficients in the case of a finite commutative rotation was developed to avoid uncertainty in the deviatoric part of inertia tensor. This transformation and the exact solution of eigenvalueeigenvector problem are applied to determine (a) the static components and accuracy of the Earth's tensor of inertia at epoch 2000 and (b) the time-dependent constituents of the inertia tensor. Special attention was given to the computation of temporally varying components of the Earth's inertia tensor, which are based on the time series of 2nd-degree coefficients through GRACE observations. A remarkable stability in time of the position of the axis A of inertia as the parameter of the Earth's triaxiality is discussed.

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<p class="Default">The GOCE satellite mission is one of the main achievements of the satellite geodesy for the Earth’s gravitational field recovery. Three different approaches have been developed for the estimation of harmonic coefficients from gradiometry data measured on board of GOCE-satellite. In this paper a special version of the space-wise method based on the second method of Neumann for fast determination of the harmonic coefficients <em>C_nm, S_nm</em> of the Earth’s gravitational potential is given based on the radial gravity gradients of the EGG_TRF_2 product, except of two polar gaps filled by radial gradients from the EGM2008 gravity model. In the pre-processing stage GOCE-based second degree radial derivatives were averaged to the regular grid through Kalman static filter with additional Gaussean smoothing of residual radial derivatives. All computations are made by iterations. As the first step the determination of the preliminary NULP-01S<strong> </strong>model up to degree/order 220 derived from the Gaussean grid of the GOCE radial derivatives with respect to the WGS-84 reference field was developed based only one of the radial gradients EGG_TRF_2 in the EFRF-frame. In the second iteration the same algorithm is applied to build the NULP-02S gravity field model up to degree/order 250 using the same Gaussean grid with respect to the NULP-01S reference field. The NULP-02S model was verified by means of applying various approaches for the construction of the gridded gravity anomalies and geoid heights in the Black sea area using processing of datasets from six altimetry satellite missions. Comparison of different models with GNSS-levelling data in the USA area demonstrates the independent verification of achieved accuracy of the constructed NULP-02S Earth’s gravity field model.</p>

The Earth's global density model given by the restricted solution of the 3D Cartesian moments problem inside the ellipsoid of revolution was adopted to preserve in this way the external gravitational potential up to second degree/order, the dynamical ellipticity, the geometrical flattening, and six basic radial jumps of density as sampled for the PREM model. Comparison of lateral density anomalies with estimated accuracy of density leads to the same order values in uncertainties and density heterogeneities. Hence, four radial density models were chosen for the computation of the Earth's gravitational potential energy E: LegendreLaplace, Roche, Bullard, and Gauss models. The estimation of E according to these continuous density models leads to the inequality with two limits. The upper limit E H agrees with the homogeneous distribution. The minimum amount E Gauss corresponds to the Gauss' radial density. All E-estimates give a perfect agreement between E Gauss , the value E derived from the piecewise Roche's density with 7 basic shells, and the values E based on the four simplest models separated additionally into core and mantle only.

The estimation of the Earth's gravitational potential energy E was obtained for different density distributions and rests on the expression E =  (W min + W) derived from the conventional relationship for E. The first component W min expresses minimum amount of the work W and the second component W represents a deviation from W min interpreted in terms of Dirichlet's integral applied on the internal potential. Relationships between the internal potential and E were developed for continuous and piecewise continuous density distributions. The global 3D density model inside an ellipsoid of revolution was chosen as a combined solution of the 3D continuous distribution and the reference PREM radial piecewise continuous profile. All the estimates of E were obtained for the spherical Earth since the estimated (from error propagation rule) accuracy  E of the energy E is at least two orders greater than the ellipsoidal reduction and the contribution of lateral density inhomogeneities of the 3D global density model. The energy E contained in the 2nd degree Stokes coefficients was determined. A good agreement between E = E Gauss derived from Gaussian distribution and other E, in particular for E = E PREM based on the PREM piecewise continuous density model and E-estimates derived from simplest Legendre-Laplace, Roche, Bullard and Gauss models separated into core and mantle only, suggests the Gaussian distribution as a basic radial model when information about density jumps is absent or incomplete.