Satellite Gravity and the Geosphere Contributions to the Study of the Solid Earth and Its Fluid Envelopes (1997) / Chapter Skim
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APPENDIX A
APPENDIX A
Pages 91-98
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From page 91... ...
is solved in sphencal coordinates. Gravitational potentials satisfy Laplace's equation in regions of free space, that is, where densities are zero; thus the spherical harmonic expansion of the Earls gravity field is a useful device for describing the field outside the Earth, such as is felt by orbiting satellites.
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From page 92... ...
The set is complete under the inner product norm, so that any function f(0,¢) whose square is integrable over the surface of He unit sphere can be approximated by an expansion In spherical harmonics in such a way Hat the mean squared error of He infinite series expansion varnishes.
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From page 93... ...
A.2 DEGREE VARIANCES From the above it can be shown that the spherical harmonics H~,m~n(~¢53 have m zeros on meridians in ~ radians of longitude and ~-m zeros on parallels in ~ radians of colatitude, thus the degree ~ > 0 counts He total number of zero crossings In any hemisphere bounded by a meridian plane, while the order O < m < ~ counts how many ofthose zero crossings are along meridians. The (0,¢3 coordinates of a point on the sphere depend on what point has been chosen as He pole Hi= 0)
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From page 94... ...
This is the potential of the anomalous gravitational attractions, which are those parts of the gravity field not accounted for by the ellipsoidal shape of the Earth or its rotation. Some authors call T the "disturbing potential," rather Han the "anomalous potential." We prefer the latter term, because in celestial mechanics the former term is used to denote all terms in With the exception of GA1'r; these terms cause satellite orbits to depart from Keplenan ellipses.
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From page 95... ...
The static field amplitudes are based on the degree variances of the anomalous potential fez, and are shown as the expected dimensionless amplitude, cat, the expected geoid height anomaly amplitude, are, and the expected gravity-anomaly amplitude, GM/-l. We also show error spectra that are analogous quantities derived from the estimated degree variances of the uncertainties In the anomalous potential coefficients expected for venous missions or gravity field models.
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From page 96... ...
This resolution length is the side of a square having the same area as a sphencal cap of radius ~/2 In order to form Gaussian-weighted averages from degree variances, we need the Legendre expansion of the weight function. Define ,0~ by ,~ = 1 ~2aexpf X1 x)
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From page 97... ...
We found ~at, for Me range of resolution lengths considered here, ~ = 1900 was sufficient.
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