The temporal variation of the spherical and Cartesian multipoles of the gravity field: the generalized MacCullagh representation |
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| Authors: | E Grafarend J Engels P Varga |
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| Institution: | (1) Department of Geodesy and Geoinformatics, Stuttgart University, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany e-mail: grafarend@gis.uni-stuttgart.de; Tel.: +49 711 1213390; Fax: +49 711 1213285, DE;(2) Geodetic and Geophysical Research Institute, Seismological Observatory, Meredek 18, H1112 Budapest, Hungary e-mail: varga@seismology.hu; Tel.: +36 1 3193382; Fax: +36 1 3193385, HU |
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| Abstract: | The Cartesian moments of the mass density of a gravitating body and the spherical harmonic coefficients of its gravitational
field are related in a peculiar way. In particular, the products of inertia can be expressed by the spherical harmonic coefficients
of the gravitational potential as was derived by MacCullagh for a rigid body. Here the MacCullagh formulae are extended to
a deformable body which is restricted to radial symmetry in order to apply the Love–Shida hypothesis. The mass conservation
law allows a representation of the incremental mass density by the respective excitation function. A representation of an
arbitrary Cartesian monome is always possible by sums of solid spherical harmonics multiplied by powers of the radius. Introducing
these representations into the definition of the Cartesian moments, an extension of the MacCullagh formulae is obtained. In
particular, for excitation functions with a vanishing harmonic coefficient of degree zero, the (diagonal) incremental moments
of inertia also can be represented by the excitation coefficients. Four types of excitation functions are considered, namely:
(1) tidal excitation; (2) loading potential; (3) centrifugal potential; and (4) transverse surface stress. One application
of the results could be model computation of the length-of-day variations and polar motion, which depend on the moments of
inertia.
Received: 27 July 1999 / Accepted: 24 May 2000 |
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| Keywords: | : Moments of inertia – Spherical harmonics – Potential field – Earth's rotation – Length of day |
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