# Estimation of the potential gravitational energy of the Earth based on reference density models

2008;
: 5-24
1
Department of Geodesy, Institute of Geodesy, Lviv Polytechnic National University
2
Engineering geodesy department of Lviv Polytechnic National University

The estimation of the Earth’s gravitational potential energy E based on the given density distribution is considered. The global density model was selected as combined solution of the 3D continuous distribution and reference radial piecewise profile with basic density jumps as sampled for the PREM density. This model preserves the external gravitational potential from zero to second degree/order, the dynamical ellipticity, the planet’s flattening, and basic radial density-jumps. The rigorous error propagation of adopted density parameterization was derived to restrict a possible solution domain. Comparison of lateral density anomalies with estimated accuracy of density leads to values of the same order in uncertainties and density heterogeneities. As a result, radial-only density models were chosen for the computation of the potential energy E. E-estimates were based on the expression E = -(Wmin +ΔW) derived from the conventional relationship for E through the Green’s identity. The first component Wmin expresses some minimum amount of the work W and the second component Δrepresents a deviation from Wmin treated via Dirichlet’s integral on the internal potential. Relationships for the internal potential and E, including error propagation were developed for continuous and piecewise densities. Determination of E provides the inequality with two limits for E-values corresponding to different density models. The upper limit EH agrees with the homogeneous distribution. The minimum amount EGauss corresponds to Gauss’ continuous radial density. All E-estimates were obtained for the spherical Earth since the ellipsoidal reduction gives two orders smaller quantity than the accuracy σE= ±0.0025×1039 ergs of the energy E. Thus, we get a perfect agreement between EGauss = −2.5073×1039 ergs, E = −2.4910×1039 ergs derived from the piecewise Roche’s density, EPREM = −2.4884×1039 ergs based on the PREM model, and E-values from simplest models separated into core and mantle only. Distributions of the internal potential and its first and second derivatives were derived for piecewise and continuous density models. Influence of the secular variation in the zonal coefficient C20 on global density changes is discussed using the adopted 3D continuous density model as restricted solution of the three-dimensional Cartesian moments problem inside the ellipsoid of revolution.

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