# Local Orbital Derivatives of the Earth Potential Expressed in Terms of the Satellite Cartesian Coordinates and Velocity

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## Local Orbital Derivatives of the Earth Potential Expressed in Terms of the Satellite Cartesian Coordinates and Velocity

In a satellite gradiometry mission the observables will be the second order derivatives of the Earth potential in the local orbital reference frame. The conventional expansions for these derivatives contain singular factors. They depend on the functions of the orbit inclination I and their first and second order derivatives. If the orbit eccentricity is taken into account then the functions of the eccentricity also involve these expressions. In the present paper more simple alternative expansions for the orbital derivatives are constructed, depending on the spherical coordinates and cos I. They have only two sums and do not have singular factors. These expansions depend on the Legendre functions of the latitude but do not depend on their derivatives. As compared to the earlier expressions of the authors the present ones have the form which is more convenient for computations. Besides, these expressions can be applied not only for the case when the satellite orbit is circular and π / 2 ≤ I ≤ π but for the arbitrary eccentricity (0 ≤ e < 1) and inclination (0 ≤ I ≤ π). After additional transformations the final expansions for the orbital derivatives represent, for the first time, simple functions of the cartesian coordinates of the satellite and the components of its velocity. These expressions may be convenient for inverting a huge amount of the GOCE data in the geopotential coefficients.

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## The Journal of Space Research Centre of Polish Academy of Sciences

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