Figure 39. The geoid in the case of a homogeneous, plastic, rotating earth. In this case, also an idealized one, the geoid is a perfect oblate ellipsoid of revolution. of Rotation 山 Polar Flattening Equatorial thus forming a bulge. In a similar vein, near a mass deficiency, the level surface would show a depression. For a rotating, nonhomogeneous earth, the same reasoning applies. Mass concentrations in the crust produce bulges in the geoid while mass deficiencies create depressions (fig. 41). Thus, the actual geoid, which for the real earth is defined as the level surface that over the oceans coincides with mean sea level, furnishes a good means of visualizing variations in the structure of the earth. Since mass and gravity go together, these struc Figure 40. The geoid in the case of a nonhomogeneous, nonrotating earth. Variations in the earth's density generate bulges and depressions in the geoid. Solid line is Broken line is geoid with distortions exaggerated. Figure 41. The actual geoid. The real earth's geoid has numerous bulges and depressions, revealing by their existence appreciable variations in the material of the planet. + Indicates mass excess tural features are revealed in their influence on the earth's gravitational field, intensifying the field near mass excesses and weakening it near mass deficiencies. These perturbations in the earth's field produce corresponding perturbations in the orbits of satellites which revolve under the influence of gravity. A precise analysis of these orbital perturbations can yield the features of the field. From these one finally gets back to the geoid and the earth's structure. Mathematically the earth's gravitational field can be derived by calculus from what is called the geopotential function V. Physically the surfaces over which is constant are the level surfaces discussed earlier in defining the geoid. Thus the makeup of the geoid and that of the earth's surface geopotential are identical. The geopotential function can be expressed as the sum of an infinite number of terms (in general). Because these terms can be expressed in sines and cosines of latitude and longitude, they are referred to as spherical harmonics, by analogy with the harmonic analysis of a vibrating string where sines and cosines of the various multiples of the fundamental frequency of the string are used.* The amount of a specific harmonic in the expansion of the geopotential is given by a coefficient J. The most general harmonics correspond to distortions of the geoid in both latitude and longitude. Some, called sectorial harmonics, reveal major distortions in longitudefor example, an ellipticity in the earth's equator. Of special importance are the zonal harmonics, which correspond to coefficients Jnm for which m = 0 and which depend only on latitude. The second zonal harmonic corresponds to the earth's equatorial bulge caused by the earth's rotation. The third zonal harmonic, if present in the expansion of the geopotential, More completely, like the sines and cosines, the basic functions out of which spherical harmonics are constructed form what mathematicians call an orthogonal set. would add a pear-shaped component to the earth's figure, elevating the geoid at one pole and depressing it at the other. It was in regard to the reference ellipsoid and the coefficients in the spherical harmonic expansion of the earth's potential that satellite measurements could aid the geodesist. The most straightforward contribution was to provide a sighting point in the sky that could be used to make direct connections between remotely separated points of the earth, supplementing the method of triangulation along the earth's surface. For this purpose simultaneous sightings of a satellite from two widely separated points were most useful. But once the orbit of a satellite had been accurately determined, simultaneous sightings were not required. One could relate separate sightings by computing the time and distance along the orbit from one sighting to the other, and again proceed to compute the distance between the two observing stations on the earth. By this latter method continental and transoceanic distances could be spanned, clearly a powerful aid in tying together different geodetic nets of the world. The second major contribution that satellites could make was to help determine the different harmonic components of the earth's gravitational field, or alternately of the earth's gravitational potential. The orbit of a satellite is determined completely by its initial position and velocity and the forces operating on it. These forces include the gravitational influences of the earth, sun, and moon; atmospheric drag; solar radiation pressures; and self-generated disturbances such as those caused by gases escaping from the interior of the satellite. For a satellite near the earth yet well out of the appreciable atmosphere, the earth's gravity controls the orbit, the other effects amounting to corrections that have to be taken into account. As for the earth's field, Newton's inverse square law term constrains the satellite to an essentially elliptical orbit. But higher order terms also have their effects. The second order zonal harmonic or equatorial bulge causes the plane of the satellite's orbit and the line joining apogee and perigee to rotate in space. Still higher harmonics produce slight undulations in the satellite's orbit, which can be measured and analyzed to determine which harmonics, and how much of each, are producing the observed effects. The application was simple in principle, but mathematically very complicated. Satellite orbits and their perturbations were directly related to the geoid, while the positions of the tracking stations and geodetic nets were tied to the reference ellipsoid, and a major objective was to improve the quantitative definitions of both geoid and ellipsoid. Because of the complexities, the modern computer was required to take advantage of the satellite opportunities. But with the computer the complexities and important results were quickly sorted out. Some of the earliest came from the first Sputnik and Vanguard satellites. Using Sputnik 2, E. Buchar of Czechoslovakia was able to make an estimate of the earth's flattening. From the measured rate of precession of the satellite's orbit, which could be related mathematically to the flattening, Buchar obtained the value f = 1/(297.90±0.18) (12) which is to be compared to the previously accepted value of 1/297. From a more extensive analysis of the Vanguard 1 satellite, workers at the U.S. Army Map Service obtained f = 1/(298.38±0.07) (13) while a U.S. Naval Research Laboratory group got virtually the same answer. 34 While the measured flattening of the earth was smaller than that which had been in use, it was significantly greater than that which would exist in a plastic earth rotating at the present angular velocity of the earth, namely 1/300.35 The implication was that the earth's mantle was not perfectly plastic. For a perfectly plastic earth, the flattening indicated by the satellite measurements would correspond to an earlier, faster angular velocity of earth. Instead, changes in the earth's equatorial bulge lag by a substantial period-tens of millions of years—behind the changes in the centrifugal forces producing the bulge.36 Vanguard 1 data also showed that the eccentricity of the satellite's orbit varied by 0.00042 ± 0.00003 with a period of 80 days. John O'Keefe and his colleagues concluded that this variation had to be caused mostly by the third harmonic in the earth's gravitational field. The distortion corresponding to this harmonic was very slight, amounting to only about 20 meters elevation of the geoid at the north pole and an equivalent depression at the south pole-widely described in the newspapers as the earth's "pear-shaped" component-but was significant in that it might imply a considerable strength in the earth's interior. O'Keefe and his colleagues estimated that a crustal load of 2 X 10 dynes/cm2 (2 × 106 n/m2) was implied by their findings, producing stresses which they said "must be supported by a mechanical strength larger than that usually assumed for the interior of the earth or by large-scale convection currents in the mantle."37 It was possible by a detailed analysis of the orbital perturbations to derive a chart of the departures of the geoid above and below the reference ellipsoid, a chart which could suggest a great deal about the distribution of mass within the earth's crust. Using observations from five different satellites, William Kaula of Goddard Space Flight Center produced the chart shown in figure 42.38 The positive numbers give elevations of the geoid in meters above the reference ellipsoid, which was taken to have the flattening of 1/298.24 indicated by satellite measurements. The negative numbers give depressions of the geoid below the reference ellipsoid. As an example of what one can read from such a chart, the elevations and depressions of the geoid shown in the equatorial belt strongly suggest that the earth's equator is not a circle, but an ellipse. This is consistent with an analysis by |