Space Gravity Spectroscopy - determination of the Earth’s gravitational field by means of Newton interpolated LEO ephemeris Case studies on dynamic (CHAMP Rapid Science Orbit) and kinematic orbits

Abstract

Abstract. An algorithm for the (kinematic) orbit analysis of a Low Earth Orbiting (LEO) GPS tracked satellite to determine the spherical harmonic coefficients of the terrestrial gravitational field is presented. A contribution to existing long wavelength gravity field models is expected since the kinematic orbit of a LEO satellite can nowadays be determined with very high accuracy in the range of a few centimeters. To demonstrate the applicability of the proposed method, first results from the analysis of real CHAMP Rapid Science (dynamic) Orbits (RSO) and kinematic orbits are illustrated. In particular, we take advantage of Newton's Law of Motion which balances the acceleration vector and the gradient of the gravitational potential with respect to an Inertial Frame of Reference (IRF). The satellite's acceleration vector is determined by means of the second order functional of Newton's Interpolation Formula from relative satellite ephemeris (baselines) with respect to the IRF. Therefore the satellite ephemeris, which are normally given in a Body fixed Frame of Reference (BRF) have to be transformed into the IRF. Subsequently the Newton interpolated accelerations have to be reduced for disturbing gravitational and non-gravitational accelerations in order to obtain the accelerations caused by the Earth's gravitational field. For a first insight in real data processing these reductions have been neglected. The gradient of the gravitational potential, conventionally expressed in vector-valued spherical harmonics and given in a Body Fixed Frame of Reference, must be transformed from BRF to IRF by means of the polar motion matrix, the precession-nutation matrices and the Greenwich Siderial Time Angle (GAST). The resulting linear system of equations is solved by means of a least squares adjustment in terms of a Gauss-Markov model in order to estimate the spherical harmonics coefficients of the Earth's gravitational field. Key words. space gravity spectroscopy, spherical harmonics series expansion, GPS tracked LEO satellites, kinematicCorrespondence to: T. Reubelt (reubelt@gis.uni-stuttgart.de) orbit analysis, Newton interpolation Reference framesFirst, the beforehand mentioned transformation between the Inertial Reference Frame and the Body Fixed Reference Frame is considered for this contribution more an issue of the operational point of view (software development) since the underlying theoretical aspects are well known, eg. McCarthy (1996). The resulting transformation matrix R(t) contains the parameters of nutation, precession, polar motion and Greenwich siderial time. Corrections for the nutation model and the parameters for polar motion and Greenwich siderial time are delivered by the Bulletins of the International Earth Rotation Service (IERS). Representation of the Earth's gravitational field -the spherical harmonics series expansionFor the description of the gravitational field of the Earth we use a spherical harmonics series expansion. Equation (1) defines this series expansion of the gravitational potential U...