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The polar mount antenna.
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The Geostationary Orbit
In the calculations leading to d, a spherical earth of mean radius 6371 km may be assumed and earth-station elevation may be ignored, as was done in the previous section. The value obtained for will be sufficiently accurate for initial alignment and fine adjustments can be made, if necessary. Calculation of the angle of tilt is illustrated in Example 3.3.
Example 3.3 Determine the angle of tilt required for a polar mount used with an
earth station at latitude 49 north. Assume a spherical earth of mean radius 6371 km, and ignore earth-station altitude.
Solution
Given data: aGSO 42164 km; R 6371 km; b lE = 49 .
49 ;
Equation (3.11) gives: d 26371
cos 49
> 38287 km From Eq. (3.12): El arccosa > 33.8 90 > 7 33.8 49 42164 sin 49 b 38287
3.4 Limits of Visibility There will be east and west limits on the geostationary arc visible from any given earth station. The limits will be set by the geographic coordinates of the earth station and the antenna elevation. The lowest elevation in theory is zero, when the antenna is pointing along the horizontal. A quick estimate of the longitudinal limits can be made by considering an earth station at the equator, with the antenna pointing either west or east along the horizontal, as shown in Fig. 3.6. The limiting angle is given by aE arccos a arccos 81.3
6378 42164
Three
aGSO
aE
aGSO
Illustrating the limits of visibility.
Thus, for this situation, an earth station could see satellites over a geostationary arc bounded by 81.3 about the earth-station longitude. In practice, to avoid reception of excessive noise from the earth, some finite minimum value of elevation is used, which will be denoted here by Elmin. A typical value is 5 . The limits of visibility will also depend on the earth-station latitude. As in Fig. 3.2b, let S represent the angle subtended at the satellite when the angle min 90 Elmin. Applying the sine rule gives S R arcsina a sin
GSO min b
(3.17)
A sufficiently accurate estimate is obtained by assuming a spherical earth of mean radius 6371 km as was done previously. Once angle S is known, angle b is found from b From Eq. (3.9): B arccosa cos b b cos lE (3.19) 180
(3.18)
Once angle B is found, the satellite longitude can be determined from Eq. (3.8). This is illustrated in Example 3.4.
The Geostationary Orbit
Example 3.4 Determine the limits of visibility for an earth station situated at mean sea level, at latitude 48.42 north, and longitude 89.26 degrees west. Assume a minimum angle of elevation of 5 .
Solution
Given data: 48.42 ; E 89.26 ; Elmin
5 ; aGSO 90
42164 km; R
6371 km
Elmin
Equation (3.17) gives: S arcsina 8.66 Equation (3.18) gives: b 180 76.34 Equation (3.19) gives: B arccosa 69.15 The satellite limit east of the earth station is at
6371 sin 95 b 42164
8.66
cos 76.34 b cos 48.42
20 approx.
and west of the earth station at
158 approx.
3.5 Near Geostationary Orbits As mentioned in Sec. 2.8, there are a number of perturbing forces that cause an orbit to depart from the ideal keplerian orbit. For the geostationary case, the most important of these are the gravitational fields of the moon and the sun, and the nonspherical shape of the earth. Other significant forces are solar radiation pressure and reaction of the satellite itself to motor movement within the satellite. As a result, station-keeping maneuvers must be carried out to maintain the satellite within set limits of its nominal geostationary position. Station keeping is discussed in Sec. 7.4. An exact geostationary orbit therefore is not attainable in practice, and the orbital parameters vary with time. The two-line orbital elements
Three
are published at regular intervals, Fig. 3.7 showing typical values. The period for a geostationary satellite is 23 h, 56 min, 4 s, or 86,164 s. The reciprocal of this is 1.00273896 rev/day, which is about the value tabulated for most of the satellites in Fig. 3.7. Thus these satellites are geosynchronous, in that they rotate in synchronism with the rotation of the earth. However, they are not geostationary. The term geosynchronous satellite is used in many cases instead of geostationary to describe these near-geostationary satellites. It should be noted, however, that in general a geosynchronous satellite does not have to be near-geostationary, and there are a number of geosynchronous satellites that are in highly elliptical orbits with comparatively large inclinations (e.g., the Tundra satellites). Although in principle the two-line elements could be used as described in Chap. 2 to determine orbital motion, the small inclination makes it difficult to locate the position of the ascending node, and the small eccentricity makes it difficult to locate the position of the perigee. However, because of the small inclination, the angles w and can be assumed to be in the same plane. Referring to Fig. 2.9 it will be seen that with this assumption the subsatellite point will be east of the line of Aries. The longitude of the subsatellite point (the satellite longitude) is the easterly
INTELSAT 1 26824U 2 26824 INTELSAT 1 26900U 2 26900 INTELSAT 1 27403U 2 27403 INTELSAT 1 27380U 2 27380 INTELSAT 1 27438U 2 27438 INTELSAT 1 27513U 2 27513 INTELSAT 1 27683U 2 27683 INTELSAT 1 28358U 2 28358
901 01024A 0.0158 902 01039A 0.0156 903 02016A 0.0362 904 02007A 0.0202 905 02027A 0.0205 906 02041A 0.0111 907 03007A 0.0206 1002 04022A 0.0079
05122.92515626 -.00000151 00000-0 10000-3 0 7388 338.7780 0004091 67.7508 129.4375 1.00270746 14318 05126.99385197 .00000031 00000-0 10000-3 0 6260 300.5697 0002640 112.8823 231.2391 1.00271845 13528 05125.03556931 .00000000 00000-0 10000-3 0 6249 171.6123 0002986 232.5077 157.1571 1.00265355 11412 05125.62541657 .00000043 00000-0 00000+0 0 5361 0.0174 0003259 40.3723 108.3316 1.00272510 11761 05125.03693822 .00000000 00000-0 10000-3 0 5812 164.2424 0002820 218.0675 189.4691 1.00265924 10746 05126.63564565 .00000012 00000-0 00000+0 0 324.7901 0003200 99.2828 93.4848 1.00272600 05124.32309516 .00000000 00000-0 10000-3 0 13.5522 0009594 61.6856 235.7624 1.00266570 05124.94126775 -.00000018 00000-0 00000+0 0 311.0487 0000613 59.4312 190.2817 1.00271159 4817 9803 3108 8131 1527 3289
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