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Harmonic Analysis of Orbital Data

A great deal of study has been done regarding the effect of the earth's gravitational attraction on satellite motion. The fact that there are a number of perturbing factors has already been mentioned – the uneven distribution of the earth's mass, the oblateness of the earth, atmospheric drag, the effects of the planets, sun and moon, and electromagnetic effects. The perturbations are measured by observing the position of the satellite in orbit around the earth. As observational data accumulates, orbital parameters become more precisely defined and reliable earth-centered positioning becomes available. An analysis of orbital data can also be used to develop an expression of the earth's external gravity field for a better interpretation of the shape of the geoidal surface through spherical harmonics. (The Appendix contains a discussion of spherical harmonics.) Although a complete analysis of orbital data requires consideration of all perturbing effects, the earth itself is the only perturbing body of major consequence in the study of near-earth satellite motion. The effects of the sun, moon, and atmosphere are removed so that only the effects of earth's shape and uneven mass distribution remain.

The uneven distribution of the earth's mass causes the force of gravity to vary from point to point on the surface and in external space. While force of gravity is measured at points on the surface with highly sensitive instruments, mathematical procedures are required to analyze orbital perturbations and to express the gravitational potential. The gravitational potential may be explained in terms of potential surfaces – surface to which the force of gravity is always perpendicular. If the earth were a perfect non-rotating sphere with homogeneous mass distribution, the potential surface would be spherical in shape. The fact that the earth is shaped more like an ellipsoid than a sphere causes the potential surface to be shaped more like an ellipsoid. Actually, the earth is neither spherical nor ellipsoidal. The potential surface bulges where there is excessive mass and it is depressed in areas of mass deficiency. The undulating surface described earlier as the geoid is a potential surface of the real earth. The diagram in Figure 30 illustrates the three surfaces just discussed.

The most convenient way to express the gravitational potential is in terms of a series of spherical harmonics mentioned above. The coefficients of the various harmonic terms are functions of the various orbital perturbations. A few are directly related to the shape of the earth and the remainder to the uneven distribution.

While it is possible to derive harmonic coefficients from observed gravity, the method is limited due to the lack of high quality worldwide gravity coverage. The computation of coefficients from satellite data also has its limitations. There are many coefficients that are not well defined from tracking data due to the small magnitude of the orbital perturbations at geodetic satellite altitudes. In addition, satellites orbiting at different inclinations are needed to reduce the correlation between the computed coefficients. For best results, the current practice is to combine tracking data with available surface-gravity data when solving for the spherical harmonic coefficients of the earth's gravitational field.