Estimation of the gravitational potential energy of the earth based on different density models

Abstract

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.

On approach to determine the internal potential and gravitational energy of ellipsoid

“…The above formulas (22)-(24), which coincide in content with the results of the work [17], allow us to write down the relation for determining the gravitational energy in each layer as well for a one-dimensional model (model PREM [18]) as for its modification an ellipsoid.…”

Gravity potential and its component of centrifugal force inside the ellipsoidal planet

“…Виділення невирішених раніше частин загальної проблеми. Потенціал всередині еліпсоїдальної планети визначено для постійного розподілу мас, причому на поверхні рівневого еліпсоїда подається замкнутою формулою [11]. Логічно постає завдання узагальнення цих виразів для потенціалу з постійною щільністю на випадок кускового неперервного радіального розподілу мас надр планети, фігура якої -еліпсоїд.…”

Geodynamics

“…In view of geophysics the energy E may also be applied as an estimate of a lower limit to the mantle viscosity obtained from the rate of timechanges of E [Rubincam, 1979]. The continuous Legendre-Laplace law, Roche's law, Bullard's model, Gaussian distribution of the Earth and the piecewise continuous Roche profile were applied in Marchenko (2009) for several estimations of E that led to the inequality with minimum limit corresponding to Gauss' continuous profile and very close to the piecewise PREM model [Marchenko, Zayats, 2011]. Therefore, the comparison of all the E-estimates gives GaussEarthHmin ££<-EEEW .…”