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5.3.2 Satellite in Geosynchronous Orbit and Above

An important special case of the satellite motion as seen from the Earth's surface occurs for geostationary satellites, which hover approximately over one location on the Earth's equator. This will occur at an altitude of 35,786 km, for which the satellite period is 1,436 min, equaling the Earth's sidereal rotation period relative to the fixed stars. Chapter 6 describes the long-tram drift of geostationary satellites. We describe here the apparent daily motion of these satellites as seen by an observer on the Earth.

For convenience, we assume the observer is at the center of the Earth and compute the apparent motion from there. The detailed motion seen from a location on the Earth's surface will be much more complex because the observer is displaced relative to the Earth's center. (See Wertz [2001].) But the general results will be the same, and the variations can be computed for any particular location.

Zenith

Zenith

Horizon

A. Geometry on the Globe B. Geometry on the

Ground-Station-Centered Celestial Sphere

Fig. 5-18. Motion of a Satellite at 1,000 km as Seen on the Earth and by an Observer on the Surface of the Earth. See text for formulas.

Horizon

A. Geometry on the Globe B. Geometry on the

Ground-Station-Centered Celestial Sphere

Fig. 5-18. Motion of a Satellite at 1,000 km as Seen on the Earth and by an Observer on the Surface of the Earth. See text for formulas.

Orbit inclination and eccentricity are the principal causes of the apparent daily motion of a geosynchronous satellite. These two effects yield different-shaped apparent orbits, which can cause confusion if the source of the apparent motion is not clearly identified. As Fig. 5-19A shows, the inclination of the orbit produces a figure eight centered on the equator, as seen by an observer at the Earth's center. The half-height, hfo, and half-width, w^ of the figure eight due to an inclination, i, are given by hinc=±i (5-53)

tan Wfc = (Vsec 1 - Vcos 1) = tan2 (i / 2) i5"54)

where the approximation in the second formula applies to small i. The source of this figure eight or analemma is the motion of the satellite along its inclined orbit, which will alternately fall behind and then catch up to the uniform rotation of the Earth on its axis.

. The second factor which causes a nonuniform apparent motion is a nonzero eccentricity of the satellite orbit An eccentricity, e; causes an East-West oscillation, wecc, of magnitude

In general, the inclination and eccentricity motions are superimposed, resulting in two possible shapes for the motion of the geosynchronous satellite as seen from the Earth. If the nonzero inclination effect dominates, then the satellite appears to move in a figure eight If the eccentricity effect is larger than the inclination effect then the apparent motion is a single open oval, as shown in Fig. 5-19B.

For satellites above geosynchronous orbit, the rotation of the Earth on its axis dominates the apparent motion of the satellite. Consequently, it is most convenient in this case to plot the motion of the satellite relative to the background of the fixed stars.

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