Temporal changes in the Earth’s tensor of inertia and the 3d density model based on the UT/CSR data

This study aims to derive the Earth’s temporally varying Earth’s tensor of inertia based on the dynamical ellipticity HD(t), the coefficients C2m(t), S2m(t), from UT/CSR data. They allow to find the time-varying Earth’s mechanical and geometrical parameters during the following periods: (a) from 1976 to 2020 based on monthly and weekly solutions of the coefficient C20 ; (b) from 1992 to 2020 based on monthly and weekly solutions of the non zero coefficients A20(t), A22(t)  related to the principal axes of inertia, allowing to build models their long-term variations. Differences between C20  and A20 , given in various systems, represent the average value  ≈2·10-15  , which is smaller than time variations of C20 or A20 , characterizing a high quality of UT/CSR solutions. Two models for the time-dependent dynamical ellipticity HD(t)  were constructed using long-term variations for the zonal coefficient  A20(t) during the past 44 and 27.5 years. The approximate formulas for the time-dependent dynamical ellipticity  HD(t) were provided by the additional estimation of each parameter of the Taylor series, fixing  HD=3.27379448x10-3 at epoch  t0=J2000 according to the IAU2000/2006 precession-nutation theory. The potential of the time-dependent gravitational quadrupole V2 according to Maxwell theory was used to derive the new exact formulas for the orientation of the principal axes A, B, C   via location of the two quadrupole axes. Hence, the Earth’s time-dependent mechanical and geometrical parameters, including the gravitational quadrupole, the principal axes and the principal moments of inertia were computed at each moment during the past 27.5 years from 1992 to 2020. However, their linear change in all the considered parameters is rather unclear because of their various behavior on different time-intervals including variations of a sign of the considered effects due to a jump in the time-series  during the time-period 1998 – 2002. The Earth’s 3D and 1D density models were constructed based on the restricted solution of the 3D Cartesian moments inside the ellipsoid of the revolution. They were derived with conditions to conserve the time-dependent gravitational potential from zero to second degree, the dynamical ellipticity, the polar flattening, basic radial jumps of density as sampled for the PREM model, and the long-term variations in space-time mass density distribution. It is important to note that in solving the inverse problem, the time dependence in the Earth's inertia tensor arises due to changes in the Earth's density, but does not depend on changes in its shape, which is confirmed by the corresponding equations where flattening is canceled.