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Orbits and Launching Methods
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2.42. Determine the GMT for the following local times and locations: (a) 7 A.M. Los Angeles, USA; (b) 1 P.M. Toronto, Canada; (c) 12 noon Baghdad, Iraq; (d) 3 P.M. Tehran, Iran.
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ADC USAF. 1980. Model for Propagation of NORAD Element Sets. Spacetrack Report No. 3. Aerospace Defense Command, U.S. Air Force, December. ADC USAF. 1983. An Analysis of the Use of Empirical Atmospheric Density Models in Orbital Mechanics. Spacetrack Report No. 3. Aerospace Defense Command, U.S. Air Force, February. Arons, A. B. 1965. Development of Concepts of Physics. Addison-Wesley, Reading, MA. Bate, R. R., D. D. Mueller, and J. E. White. 1971. Fundamentals of Astrodynamics. Dover, New York. Celestrak, at http://celestrak.com/NORAD/elements/noaa.txt Duffett-Smith, P. 1986. Practical Astronomy with Your Calculator. Cambridge University Press, New York. Schwalb, A. 1982a. The TIROS-N/NOAA-G Satellite Series. NOAA Technical Memorandum NESS 95, Washington, DC. Schwalb, A. 1982b. Modified Version of the TIROS-N/NOAA A-G Satellite Series (NOAA E-J): Advanced TIROS N (ATN). NOAA Technical Memorandum NESS 116, Washington, DC. Thompson, Morris M. (editor-in-chief). 1966. Manual of Photogrammetry, 3d ed., Vol. 1. American Society of Photogrammetry, New York. Wertz, J. R. (ed.). 1984. Spacecraft Attitude Determination and Control. D. Reidel, Holland.
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The Geostationary Orbit
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3.1 Introduction A satellite in a geostationary orbit appears to be stationary with respect to the earth, hence the name geostationary. Three conditions are required for an orbit to be geostationary: 1. The satellite must travel eastward at the same rotational speed as the earth. 2. The orbit must be circular. 3. The inclination of the orbit must be zero. The first condition is obvious. If the satellite is to appear stationary, it must rotate at the same speed as the earth, which is constant. The second condition follows from this and from Kepler s second law (Sec. 2.3). Constant speed means that equal areas must be swept out in equal times, and this can only occur with a circular orbit (see Fig. 2.2). The third condition, that the inclination must be zero, follows from the fact that any inclination would have the satellite moving north and south, (see Sec. 2.5 and Fig. 2.3), and hence it would not be geostationary. Movement north and south can be avoided only with zero inclination, which means that the orbit lies in the earth s equatorial plane. Kepler s third law may be used to find the radius of the orbit (for a circular orbit, the semimajor axis is equal to the radius). Denoting the radius by aGSO, then from Eqs. (2.2) and (2.4), a P 4 b 2
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(3.1)
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The period P for the geostationary is 23 h, 56 min, 4 s mean solar time (ordinary clock time). This is the time taken for the earth to complete one revolution about its N S axis, measured relative to the fixed stars (see Sec. 2.9.4). Substituting this value along with the value for given by Eq. (2.3) results in aGSO 42164 km (3.2)
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The equatorial radius of the earth, to the nearest kilometer, is aE 6378 km (3.3)
and hence the geostationary height is hGSO aGSO aE 42164 6378 35786 km
(3.4)
This value is often rounded up to 36,000 km for approximate calculations. In practice, a precise geostationary orbit cannot be attained because of disturbance forces in space and the effects of the earth s equatorial bulge. The gravitational fields of the sun and the moon produce a shift of about 0.85 /year in inclination. Also, the earth s equatorial ellipticity causes the satellite to drift eastward along the orbit. In practice, stationkeeping maneuvers have to be performed periodically to correct for these shifts, as described in Sec. 7.4. An important point to grasp is that there is only one geostationary orbit because there is only one value of a that satisfies Eq. (2.3) for a periodic time of 23 h, 56 min, 4 s. Communications authorities throughout the world regard the geostationary orbit as a natural resource, and its use is carefully regulated through national and international agreements.
3.2 Antenna Look Angles The look angles for the ground station antenna are the azimuth and elevation angles required at the antenna so that it points directly at the satellite. In Sec. 2.9.8 the look angles were determined in the general case of an elliptical orbit, and there the angles had to change in order to track the satellite. With the geostationary orbit, the situation is much simpler because the satellite is stationary with respect to the earth. Although in general no tracking should be necessary, with the large earth stations used for commercial communications, the antenna beamwidth is very narrow (see Chap. 6), and a tracking mechanism is required
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