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Antenna look angles 

Determination of the Azimuth and Elevation angles (Look Angles)

	S			C     R		RE +h       S
b=90 deg- lE		a=90 deg
ES		SS			90deg-El

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 l_E represents the latitude of the earth station, f_E represents the longitude and  of the earth station, f_S the longitude of sub satellite point, and observing the sign convention stated previously, the angle B is given by

                                       B = f_E - f_S

B can not exceed a theoretical limit of 81.3^O , set by horizon. Napier’s rules can then be used to sho
                             Tan A  = - tan |B|/ sin l_E
Once angle A is determined, the azimuth angle A_Z can be found. Four situations must be considered, the results for which can be summarized as follows:

        (a)            l_E <0;B<0:A_Z = A
(b)            l_E <0;B>0:A_Z = 360^o - A
(c)             l_E >0;B<0:A_Z = 180^o + A
(d)            l_E >0;B>0:A_Z = 180^o – 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^o and Azimuth is irrelevant. When the west (B >0 ), the azimuth is 270^o. 

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. 

                                     Cos c = cos l_E cos B 

The equatorial radius R_E = 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             
                  R = R_E ( 1- sin²l_E/298.257)

Here R_E 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. 

               d= square root of( R² + (R_E + H)² –2R(R_E  +h) cos C) 

   The elevation angle is denoted by EI deg in fig . Application of the sine rule to the plane triangle gives

                                      Cos EI = (R_E⁺ h/designs C )
                                        EI =Cos^-1 (R_E⁺ h/designs C)

 Example Problem

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