A generalized theory of the figures of the Earth’s interior to a third-order precision of ellipticity is proposed in accompanying paper in which all the odd degree and nonzero order spherical harmonic terms are included. As both the direct and indirect contributions of the asymmetric crust are included, this theory makes a significant improvement for calculating the asymmetric equilibrium figures of the real Earth comparing with the traditional theories which can only deal with the ideal symmetric Earth. The principal moments of inertia (PMOI: A, B, C) and global dynamical flattening (H) are important quantities in studying the rotating Earth. Precession and gravity observations give observation value of H (
 $$H_{\mathrm{obs}} \approx 1/305.4559$$
 ) with very high precision, while its theoretical calculated value (
 $$H_{\mathrm{theory}} \approx 1/308.5$$
 ) from traditional theories and a starting symmetric Earth model (like PREM model) is about $$1\%$$
 less than $$H_{\mathrm{obs}}$$
 . Using the new theory in accompanying paper and replacing the homogeneous outermost crust and oceanic layers in PREM with CRUST1.0 model, we recalculate the equilibrium figures of the Earth’s interior and finally get new values of PMOI and $$H_{\mathrm{theory}}$$
 (
 $${\approx } \,1/304.7167$$
 ) whose consistency with $$H_{\mathrm{obs}}$$
 are significantly improved to 0.24%. Furthermore, the asymmetric figures of some interesting boundaries, like inner core boundary, core-mantle boundary, are also given as by-products of this work as these boundaries’ figures are key input for studies of their topographic effect on global rotation and geodynamics, like nutation, normal modes, especially like free core nutation.

A generalized theory of the figure of the Earth: on the global dynamical flattening

The observed data of the Earth's gravitational field indicate that the figure of external equipotial surface of the Earth differs from the value given by the solution of hydrostatic equilibrium equation. Besides, the dynamical flattening of the Earth, as observed by geodetic techniques, is different by about 1 per cent from the value associated with the PREM density profile with hydrostatic equilibrium. Based on the theory of figure of the planets, the internal equipotential surface of a self gravitating, rotating mass in hydrostatic equilibrium is coincident with the equal density surface everywhere within the celestial body. Once the density distribution ρ(r) is known, one can obtain the profile of the flattening e(r) in the planetary interior. In this paper using the density distribution in the PREM, the author discusses two factors affecting the dynamical flattening of the Earth. From the calculated results the author comes to the following conclusions: (1)The density redistribution of the lower mantle, located under Earth's surface from 671 km to 2891 km, makes the value of dynamical flattening easily coincide with its observed one. (2)If the dynamical shape of the actual Earth represents the hydrostatic figure from a past geological age, it is the angular velocity from 9. 8×10 6 years ago that matches the Earth's dynamical flattening. This inference is difficult to confirm observationally, therefore nobody can confirm or deny it. However, it appears that the nonhydrostatic components of the Earth's shape not only relates to the present physical state within the Earth, but also reflects the effects of some factors during the whole history of its evolution. This result calls our attention to making careful investigation on the tide evolution of the Earth moon system, and checking whether or not some long term variation exists in Earth's physical parameters. (3) The results of the investigation with respect to the Earth's dynamical flattening can be used for reference in the domain of figure theory for other terrestrial planets. In particular, the influences of density redistribution within various interior zones on the Earth's flattening are different from each other. This conclusion might be used to study in the future the interior structure of Mars.

Zhangzhi Cheng

Papers

Remarks on the dynamical flattening of the earth

Remarks on the dynamical flattening of the Earth.