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(a) Launch site A, showing launch azimuth Az; (b) enlarged version of the spherical triangle shown in (a). l is the latitude of the launch site.
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slight difference between geodetic and geocentric latitudes may be ignored here). The dotted line shows the satellite earth track, the satellite having been launched at some azimuth angle Az. Angle i is the resulting inclination. The spherical triangle of interest is shown in more detail in Fig. 3.13b. This is a right spherical triangle, and Napier s rule for this gives cos i cos l sin Az (3.23)
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For a prograde orbit (see Fig. 2.4 and Sec. 2.5), 0 i 90 , and hence cos i is positive. Also, 90 l 90 , and hence cos is also positive. It follows therefore from Eq. (3.23) that 0 Az 180 , or the launch azimuth must be easterly in order to obtain a prograde orbit, confirming what was already known. For a fixed l, Eq. (3.23) also shows that to minimize the inclination i, cos i should be a maximum, which requires sin Az to be maximum, or Az 90 . Equation (3.23) shows that under these conditions cos imin or imin l (3.25) cos l (3.24)
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Thus the lowest inclination possible on initial launch is equal to the latitude of the launch site. This result confirms the converse statement made in Sec. 2.5 under inclination that the greatest latitude north or south is equal to the inclination. From Cape Kennedy the smallest initial inclination which can be achieved for easterly launches is approximately 28 .
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The Geostationary Orbit
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3.1. Explain what is meant by the geostationary orbit. How do the geostationary orbit and a geosynchronous orbit differ 3.2. (a) Explain why there is only one geostationary orbit. (b) Show that the range d from an earth station to a geostationary satellite is given by
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d 2sR sin Eld2 hs2R hd R sin El,
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where R is the earth s radius (assumed spherical), h is the height of the geostationary orbit above the equator, and El is the elevation angle of the earth station antenna. 3.3. Determine the latitude and longitude of the farthest north earth station which can link with any given geostationary satellite. The longitude should be given relative to the satellite longitude, and a minimum elevation angle of 5 should be assumed for the earth station antenna. A spherical earth of mean radius 6371 km may be assumed. 3.4. An earth station at latitude 30 S is in communication with an earth station on the same longitude at 30 N, through a geostationary satellite. The satellite longitude is 20 east of the earth stations. Calculate the antenna-look angles for each earth station and the round-trip time, assuming this consists of propagation delay only. 3.5. Determine the maximum possible longitudinal separation which can exist between a geostationary satellite and an earth station while maintaining lineof-sight communications, assuming the minimum angle of elevation of the earth station antenna is 5 . State also the latitude of the earth station. 3.6. An earth station is located at latitude 35 N and longitude 100 W. Calculate the antenna-look angles for a satellite at 67 W. 3.7. An earth station is located at latitude 12 S and longitude 52 W. Calculate the antenna-look angles for a satellite at 70 W. 3.8. An earth station is located at latitude 35 N and longitude 65 E. Calculate the antenna-look angles for a satellite at 19 E. 3.9. An earth station is located at latitude 30 S and longitude 130 E. Calculate the antenna-look angles for a satellite at 156 E. 3.10. Calculate for your home location the look angles required to receive from the satellite (a) immediately east and (b) immediately west of your longitude. 3.11. CONUS is the acronym used for the 48 contiguous states. Allowing for a 5 elevation angle at earth stations, verify that the geostationary arc required to cover CONUS is 55 to 136 W.
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