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The Geostationary Orbit
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to compensate for the movement of the satellite about the nominal geostationary position. With the types of antennas used for home reception, the antenna beamwidth is quite broad, and no tracking is necessary. This allows the antenna to be fixed in position, as evidenced by the small antennas used for reception of satellite TV that can be seen fixed to the sides of homes. The three pieces of information that are needed to determine the look angles for the geostationary orbit are 1. The earth-station latitude, denoted here by lE 2. The earth-station longitude, denoted here by fE 3. The longitude of the subsatellite point, denoted here by fSS (often this is just referred to as the satellite longitude) As in Chap. 2, latitudes north will be taken as positive angles, and latitudes south, as negative angles. Longitudes east of the Greenwich meridian will be taken as positive angles, and longitudes west, as negative angles. For example, if a latitude of 40 S is specified, this will be taken as 40 , and if a longitude of 35 W is specified, this will be taken as 35 . In Chap. 2, when calculating the look angles for low-earth-orbit (LEO) satellites, it was necessary to take into account the variation in earth s radius. With the geostationary orbit, this variation has negligible effect on the look angles, and the average radius of the earth will be used. Denoting this by R: R 6371 km (3.5)
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The geometry involving these quantities is shown in Fig. 3.1. Here, ES denotes the position of the earth station, SS the subsatellite point, S the satellite, and d is the range from the earth station to the satellite. The angle is an angle to be determined. There are two types of triangles involved in the geometry of Fig. 3.1, the spherical triangle shown in heavy outline in Fig. 3.2a and the plane triangle of Fig. 3.2b. Considering first the spherical triangle, the sides are all arcs of great circles, and these sides are defined by the angles subtended by them at the center of the earth. Side a is the angle between the radius to the north pole and the radius to the subsatellite point, and it is seen that a 90 . A spherical triangle in which one side is 90 is called a quadrantal triangle. Angle b is the angle between the radius to the earth station and the radius to the subsatellite point. Angle c is the angle between the radius to the earth station and the radius to the north pole. From Fig. 3.2a it is seen that c 90 lE.
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aE aE SS h s d
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The geometry used in determining the look angles for a geostationary satellite.
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There are six angles in all defining the spherical triangle. The three angles A, B, and C are the angles between the planes. Angle A is the angle between the plane containing c and the plane containing b. Angle B is the angle between the plane containing c and the plane containing a. From
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c A B a b C
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= 90 + E
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Figure 3.2 (a) The spherical geometry related to Fig. 3.1. (b) The plane triangle s obtained from Fig. 3.1.
aGSO = aE + h
The Geostationary Orbit
Fig. 3.2a, B E SS. It will be shown shortly that the maximum value of B is 81.3 . Angle C is the angle between the plane containing b and the plane containing a. To summarize to this point, the information known about the spherical triangle is a c B 90 90
(3.6) lE
(3.7) (3.8)
Note that when the earth station is west of the subsatellite point, B is negative, and when east, B is positive. When the earth-station latitude is north, c is less than 90 , and when south, c is greater than 90 . Special rules, known as Napier s rules, are used to solve the spherical triangle (see Wertz, 1984), and these have been modified here to take into account the signed angles B and lE. Only the result will be stated here. Napier s rules gives angle b as b 5 arccoss cos B cos lEd and angle A as A 5 arcsina sin Z B Z b sin b (3.10) (3.9)
Two values will satisfy Eq. (3.10), A and 180 A, and these must be determined by inspection. These are shown in Fig. 3.3. In Fig. 3.3a, angle A is acute (less than 90 ), and the azimuth angle is Az A. In Fig. 3.3b, angle A is acute, and the azimuth is, by inspection, Az 360 A. In Fig. 3.3c, angle Ac is obtuse and is given by Ac 180 A, where A is the acute value obtained from Eq. (3.10). Again, by inspection, Az Ac 180 A. In Fig. 3.3d, angle Ad is obtuse and is given by 180 A, where A is the acute value obtained from Eq. (3.10). By inspection, Az 360 Ad 180 A. In all cases, A is the acute angle returned by Eq. (3.10). These conditions are summarized in Table 3.1.
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