Page 101 - Satellite Communications, Fourth Edition

The Geostationary Orbit 81

Fig. 3.2a, B . It will be shown shortly that the maximum value
SS
E
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 spher-
ical triangle is

a 90° (3.6)
(3.7)
c 90° l E
B SS (3.8)
E
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 lati-
tude 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 l . Only the result will be stated
E
here. Napier’s rules gives angle b as
b 5 arccoss cos B cos l d (3.9)
E

and angle A as
sin Z B Z
A 5 arcsina b (3.10)
sin b

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 A A.In Fig. 3.3b,
z
angle A is acute, and the azimuth is, by inspection, A 360° A.In
z
Fig. 3.3c,angle A is obtuse and is given by A 180° A, where A
c
c
is the acute value obtained from Eq. (3.10). Again, by inspection,
A z A c 180° A.In Fig. 3.3d, angle A d is obtuse and is given by
180° A, where A is the acute value obtained from Eq. (3.10). By
inspection, A 360° A 180° A. In all cases, A is the acute
d
z
angle returned by Eq. (3.10). These conditions are summarized in
Table 3.1.
Example 3.1 A geostationary satellite is located at 90°W. Calculate the azimuth
angle for an earth-station antenna at latitude 35°N and longitude 100°W.
Solution The given quantities are
SS 90° E 100° l E 35°

B E SS
10°