Determination of the Azimuth and Elevation angles (Look Angles)
Figure-1 shows spherical triangle bounded by points N, ES, and SS.
Here ES denotes the earth station. The point s denotes the satellite in geostationary orbit, and point SS the sub satellite point. To solve this triangle we have to use Napier's rules. By solving we get following results.
Let lE represents the latitude of the earth station, fE represents the longitude and of the earth station, fS the longitude of sub satellite point, and observing the sign convention stated previously, the angle B is given by
B can not exceed a theoretical limit of 81.3° , set by horizon. Napier's rules can then be used to show
Once angle A is determined, the azimuth angle AZ can be found. Four situations must be considered, the results for which can be summarized as follows:
(a) lE<0;B<0:AZ= A
(a) lE<0;B>0:AZ= 360°-A
(a) lE>0;B<0:AZ= 180°+A
(a) lE>0;B>0:AZ= 180°-A
These do not take into account the case when the earth station is on the equator. Obviously, when the sub-satellite, the elevation is 90° and Azimuth is irrelevant. When the west (B >0 ), the azimuth is 270°.
To find the range and elevation, it is necessary first to find side c of the quadrantal spherical triangle, and then use this in the plane triangle shown in fig. side c is obtained using the rule.
The equatorial radius RE= 6378.14 km, and geostationary height h = 35,786 km. Because of the flattening of the earth at the poles, the radius R varies with latitude. An equation that gives R to a close approximation is
Here RE is the earth's equatorial radius and R is the radius at the earth station.
The plane triangle shown in fig-2 can be solved using the plane trigonometry. Applying the cosine rule gives the distance d as.
The elevation angle is denoted by EI deg in fig . Application of the sine rule to the plane triangle gives