The Geostationary Orbit 95

High earth orbit

Low earthorbit

1

Hohmann transferorbit

Figure 3.10 Hohmann transfer orbit.

in the same plane. As shown in Fig. 3.11, it takes 1 to 2 months for thesatellite to be fully operational (although not shown in Fig. 3.12, thesame conditions apply). Throughout the launch and acquisition phases,a network of ground stations, spread across the earth, is required to per-form the tracking, telemetry, and command (TT&C) functions.

Velocity changes in the same plane change the geometry of the orbitbut not its inclination. In order to change the inclination, a velocitychange is required normal to the orbital plane. Changes in inclinationcan be made at either one of the nodes, without affecting the otherorbital parameters. Since energy must be expended to make any orbitalchanges, a geostationary satellite should be launched initially with asIowan orbital inclination as possible. It will be shown shortly that thesmallest inclination obtainable at initial launch is equal to the latitudeof the launch site. Thus the farther away from the equator a launchsite is, the less useful it is, since the satellite has to carry extra fuel toeffect a change in inclination. Russia does not have launch sites southof 45°N, which makes the launching of geostationary satellites a muchmore expensive operation for Russia than for other countries which havelaunch sites closer to the equator.

Prograde (direct) orbits (Fig. 2.4) have an easterly component ofveloc-ity, so prograde launches gain from the earth's rotational velocity. Fora given launcher size, a significantly larger payload can be launched inan easterly direction than is possible with a retrograde (westerly)launch. In particular, easterly launches are used for the initial launchinto the geostationary orbit.

The relationship between inclination, latitude, and azimuth may beseen as follows [this analysis is based on that given in Bate et al. (1971)].Figure 3.13a shows the geometry at the launch site A at latitude A.(the

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Figure3.11 From launch to station of INTELSAT V (by Atlas-Centaur). (From Satellite Communications Technology, editedby K. Miya, 1981; courtesy of KDD Engineering & Consulting, Inc., Tokyo.)

Abbreviations:

AMF - apogee motor firingAKM- apogee kick motor

RF - radio frequencyPKS - perigee kick stage

RCS - reaction control system

STS - space transportation system

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98 Chapter Three

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Figure 3.13 (a) Launch site A, showing launch azimuth Az; (b) enlarged version of thespherical triangle shown in (a). Ais the latitude of the launch site.

slight difference between geodetic and geocentric latitudes may beignored here). The dotted line shows the satellite earth track, the satel-lite having been launched at some azimuth angle Az. Angle i is theresulting 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 AsinAz (3.23)

For a prograde orbit (see Fig. 2.4 and Sec. 2.5), 0 :s i :s 90°, and hencecos i is positive. Also, -90° :s A:S 90°, and hence cos A.is also positive.It follows therefore from Eq. (3.23) that 0 :S Az :S 180°, or the launchazimuth must be easterly in order to obtain a prograde orbit, confirm-ing what was already known.

For a fixed A, Eq. (3.23) also shows that to minimize the inclination i,cos i should be a maximum, which requires sin Az to be maximum, orAz = 90°. Equation (3.23) shows that under these conditions

cos imin = COS A (3.24)

or

~min= A (3.25)

Thus the lowest inclination possible on initial launch is equal to thelatitude of the launch site. This result confirms the converse statementmade in Sec. 2.5 under inclination that the greatest latitude north or southis equal to the inclination. From Cape Kennedy the smallest initial incli-nation which can be achieved for easterly launches is approximately 28°.

The Geostationary Orbit 99

~ - Explain what is meant by the geostationary orbit. How do the~rionary orbit and a geosynchronous orbit differ?

1.:. ~a)Explain why there is only one geostati~nary orbit. (b) Show that the~ d from an earth station to a geostationary satellite is given by

d = Y(RsinEl)2 + h(2R + h) - RsinEl,

r.zere R is the earth's radius (assumed spherical), h is the height of the~tionary orbit above the equator, and El is the elevation angle of the earth~n antenna.

~ Determine the latitude and longitude of the farthest north earth station-jch can link with any given geostationary satellite. The longitude should be&en relative to the satellite longitude, and a minimum elevation angle of 5°~d be assumed for the earth station antenna. A spherical earth of mean~us 6371 km may be assumed.

3.4. An earth station at latitude 300S is in communication with an earth~tion on the same longitude at 300N, through a geostationary satellite. The;;irellite longitude is 20° east of the earth stations. Calculate the antenna-look2!lgles for each earth station and the round-trip time, assuming this consists of~pagation delay only.

3.,5. Determine the maximum possible longitudinal separation which can exist!:!etween a geostationary satellite and an earth station while maintaining line-of-sight communications, assuming the minimum angle of elevation of the earth5t8.tion antenna is 5°. State also the latitude of the earth station.

3.6. An earth station is located at latitude 35°N and longitude 1000W Calculatethe antenna-look angles for a satellite at 67°W.

3.7. An earth station is located at latitude 12°S and longitude 52°W Calculatethe antenna-look angles for a satellite at 700W

3.8. An earth station is located at latitude 35°N and longitude 65°E. Calculatethe antenna-look angles for a satellite at 19°E.

3.9. An earth station is located at latitude 300S and longitude 1300E. Calculatethe antenna-look angles for a satellite at 156°E.

3.10. Calculate for your home location the look angles required to receive fromthe satellite (a) immediately east and (b) immediately west of your longitude.

3.11. CONUS is the acronym used for the 48 contiguous states. Allowing fora 5° elevation angle at earth stations, verify that the geostationary arc requiredto cover CONUS is 55° to 136°W

100 Chapter Three

--,

3.12. Referring to Prob. 3.11, verify that the geostationary arc required forCONUS plus Hawaii is 85° to 136° Wand for CONUS plus Alaska is 115° to136°W.

j 3.13. By taking the Mississippi River as the dividing line between east andwest, verify that the western region of the United States would be covered bysatellites in the geostationary arc from 136° to 163°W and the eastern regionby 25° to 55°W. Assume a 5° angle of elevation.

3.14. (a) An earth station is located at latitude 35°N. Assuming a polar mountantenna is used, calculate the angle of tilt. (b) Would the result apply to polarmounts used at the earth stations specified in Probs. 3.6 and 3.8?

3.15. Repeat Prob. 3.14 (a) for an earth station located at latitude 12°S.Would the result apply to a polar mount used at the earth station specified inProb. 3.7?

3.16. Repeat Prob. 3.14 (a) for an earth station located at latitude 300S.Would the result apply to a polar mount used at the earth station specified inProb. 3.9?

3.17. Calculate the angle of tilt required for a polar mount antenna used at yourhome location.

3.18. The borders of a certain country can be roughly represented by atriangle with coordinates 39°E, 33.5°N; 43.5°E, 37.5°N; 48.5°E, 300N. If ageostationary satellite has to be visible from any point in the country,determine the limits of visibility (i.e., the limiting longitudinal positions fora satellite on the geostationary arc). Assume a minimum angle of elevation forthe earth station antenna of 5°, and show which geographic location fixeswhich limit.

3.19. Explain what is meant by the earth eclipse of an earth-orbiting satellite.Why is it preferable to operate with a satellite positioned west, rather than east,of earth station longitude?

3.20. Explain briefly what is meant by sun transit outage.

3.21. Using the data given in Fig. 3.7, calculate the longitude for INTELSAT904.

3.22. Calculate the semimajor axis for INTELSAT 901.

3.23. Calculate the apogee and perigee heights for INTEL SAT 906.

3.24. Calculate the rate of regression of the nodes and the rate of rotation ofthe line of apsides for INTELSAT 907.

"The Geostationary Orbit 101

~~~. D. Mueller, and J. E. White. 1971.Fundamentals ofAstrodynamics. Dover,':~£'I'i" York.

~~ at http://celestrak.comlNORAD/elements/intelsat.txt.~ J., and J. Wild. 1984. "Commercial Launch ¥ehicles and Upper Stages." Space

~un. Broadcast., Vol. 2, pp. 339-362. -l~ G., and M. Bousquet. 1998. Satellite Communications Systems. Wiley, New York.~J. J. 1977. Digital Communications by Satellite. Prentice-Hall, Englewood Cliffs, NJ.Te;;L J. R (ed.). 1984. Spacecraft Attitude Determination and Control. D. Reidel, Holland.

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