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2 Artificial Satellites : Journal of Planetary Geodesy

2 2007

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Local orbital derivatives of the Earth potential expressed in terms of the satellite cartesian coordinates and velocity

Petrovskaya M. S., Vershkov A. N.

Artificial Satellites : Journal of Planetary Geodesy

2007

Vol. 42, No. 1

17--39

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.

Explicit construction of the modified spherical harmonic series for the gravity gradients and analysis of their characteristics

Petrovskaya M. S., Vershkov A. N.

Artificial Satellites : Journal of Planetary Geodesy

2007

Vol. 42, No. 4

185--214

The present paper complements the research carried out in PV2008 (Petrovskaya and Vershkov, 2008), concerning the expansion of the gravity gradients in the local northoriented reference frame in orthogonal series of modified spherical harmonics. In PV2008 procedures are developed for recovering the orthogonal bases of these series. Then an idea is briefly described how the spectral relations can be obtained between the gravity gradients and the geopotential. However no explicit procedures are demonstrated for their derivation. In the present paper successive transformations are described for each derivative which convert the initial non-orthogonal expansion into the orthogonal series. The resulting spectral relations are applied for evaluating the harmonic coefficients of these series at different altitudes, on the basis of the geopotential model EGM2008. The corresponding degree variances are estimated. The new simple expressions for the gravity gradients are convenient for various applications. In the present paper they are implemented for constructing digital colored maps for Fennoscandia region which attracts much attention of geophysicists. These maps visually demonstrate an anomalous behavior of the gravity gradients in this area.