Satellite TV  -  Scientific Analysis

3  A Fixed Dish

To recap, three settings (parameters) together completely describe the alignment of a fixed dish:

  1. Azimuth, the horizontal angle in degrees measured clockwise from True North;
  2. Elevation, the vertical angle in degrees measured upwards from the horizontal;
  3. Skew, the polarisation angle of the incoming signal, clockwise positive viewed from behind the dish.

All these values are completely determined by:

  1. Longitude difference, DLong, including sign, between the target satellite and the dish site
  2. Latitude, Lat, of the dish site

Because it depends on whether longitudes are west (-ve) or east (+ve), and latitudes south (-ve) or north (+ve), azimuth is rather unwieldy for the purposes of this discussion.  Consequently, a more convenient intermediate result in its calculation is used here, the apparent difference in longitude (DLongA), defined as the angle east or west of the dish site's meridian at which the target satellite appears.  The actual relationship between the two is:

Azimuth =DLong -veDLong ZeroDLong +ve
Lat -veDLongA0360 - DLongA
Lat Zero900270
Lat +ve180 - DLongA180180 + DLongA

All these angles are marked on the following diagram:

Satellite TV - Scientific Analysis - Fixed Dish Calculation Diagram

The settings can be derived as follows ...

Consider the spherical triangle OC'S':

  Sin(DLongA)
 Sin(DLong) 
= Sin(OS'C')
 Sin(Lat)  
Spherical Sine Law (18)
Sin(DLongA) = Sin(DLong).Sin(OS'C') / Sin(Lat)  
& Cos(DLongA) = -Cos(OC'S').Cos(OS'C')+Sin(OC'S').Sin(OS'C').Cos(DLong) Spherical Cosine Law (19)
Cos(DLongA) = Sin(OS'C').Cos(DLong) OC'S' = 90°
Sin(DLongA)
Cos(DLongA)
= [Sin(DLong).Sin(OS'C')]
[(Sin(Lat).Sin(OS'C').Cos(DLong)]
Divide first result by the second
Tan(DLongA) = Tan(DLong)/Sin(Lat) Tan(A) = Sin(A)/Cos(A)
DLongA = ArcTan[ Tan(DLong) / Sin(Lat) ]  

Consider again the spherical triangle OC'S':

  Cos(OES') = Cos(DLong).Cos(Lat) + Sin(DLong).Sin(Lat).Cos(OC'S') Spherical Cosine Law (11)
Cos(OES') = Cos(DLong).Cos(Lat) OC'S' = 90°

Consider the triangle EOS:

  SO² = Rc² + Re² - 2.Rc.Re.Cos(OES) Cosine Law
SO = √[Rc² + Re² - 2.Rc.Re.Cos(OES)]  
& Sin(EOS) / Rc = Sin(OES) / SO Sine Law
Sin(EOS) = Rc.Sin(OES) / SO  
Sin(EOS) = Rc.√[1 - Cos²(OES)] / SO Cos²(A) + Sin²(A) = 1 (5)
Sin(EOS) = Rc.√[1 - Cos²(OES)] / √[Rc² + Re² - 2.Rc.Re.Cos(OES)] Substitute SO
Sin(EOS) = Rc.√{ [1 - Cos²(OES)] / [Rc² + Re² - 2.Rc.Re.Cos(OES)] }  
Sin(EOS) = √{ [1 - Cos²(OES)] / [1 + (Re/Rc)² - 2.(Re/Rc).Cos(OES)] } Divide through by Rc
Sin(EOS) =
[1-Cos²(DLong).Cos²(Lat)]
[1+(Re/Rc)²-2.(Re/Rc).Cos(DLong).Cos(Lat)]
Substitute Cos(OES) = Cos(OES')
& EOS = 90 + Elevation By definition
Sin(EOS) = Sin(90 + Elevation)  
Sin(EOS) = Sin(90).Cos(Elevation) + Cos(90).Sin(Elevation) Compound Angles (1)
Sin(EOS) = Cos(Elevation)  
Cos(Elevation) =
[1-Cos²(DLong).Cos²(Lat)]
[1+(Re/Rc)²-2.(Re/Rc).Cos(DLong).Cos(Lat)]
 
Elevation =
ArcCos 
(
[1-Cos²(DLong).Cos²(Lat)]
[1+(Re/Rc)²-2.(Re/Rc).Cos(DLong).Cos(Lat)]
)
 

Consider the triangle EQR:

  QR = ER.Sin(DLong)  
QR = Re.Cos(Lat).Sin(DLong) Substitute ER
& Tan(Skew) = QR / OR Definition of Tan
Tan(Skew) = Re.Cos(Lat).Sin(DLong) / Re.Sin(Lat) Substitute QR & OR
Tan(Skew) = Sin(DLong) / Tan(Lat) Tan(A) = Sin(A)/Cos(A)
Skew = ArcTan[ Sin(DLong) / Tan(Lat) ]  

The following tables show examples of DLongA, Elevation, and Skew:

Apparent Difference In Longitude (DLongA)
Latitude
(Lat)
Actual Difference In Longitude (DLong)
0102030405060708090
00.0090.0090.0090.0090.0090.0090.0090.0090.0090.00
100.0045.4464.4973.2678.3181.7184.2786.3888.2590.00
200.0027.2746.7859.3667.8273.9978.8382.9086.5590.00
300.0019.4336.0549.1159.2167.2473.9079.6984.9690.00
400.0015.3429.5241.9352.5561.6669.6476.8383.5390.00
500.0012.9625.4137.0047.6157.2766.1474.4282.3190.00
600.0011.5122.8033.6944.1053.9963.4372.5081.3290.00
700.0010.6321.1731.5741.7651.7461.5271.1280.5990.00
800.0010.1520.2830.3840.4350.4360.3870.2880.1590.00
900.0010.0020.0030.0040.0050.0060.0070.0080.0090.00
Notes
  • DLong and DLongA only agree at 0° and 90° and at the poles;
  • On the equator, any satellite that is not directly overhead must lie either east or west;
  • At intermediate latitudes DLongA always lies between these limits.
Elevation
Latitude
(Lat)
Actual Difference In Longitude (DLong)
01020304050607080
090.0078.2366.5555.0343.7232.6921.9311.471.30
1078.2373.4263.9253.3442.5831.8921.4011.151.15
2066.5563.9257.3248.7439.3229.6019.8410.190.69
3055.0353.3448.7442.1534.3926.0117.368.630.05
4043.7242.5839.3234.3928.2821.4014.086.55
5032.6931.8929.6026.0121.4016.0510.194.02
6021.9321.4019.8417.3614.0810.195.821.15
7011.4711.1510.198.636.554.021.15
801.301.150.690.05
  • Interchangeable symmetry between Lat and DLong in the formula results in diagonal symmetry in the table.
Skew
Latitude
(Lat)
Actual Difference In Longitude (DLong)
0102030405060708090
00.0090.0090.0090.0090.0090.0090.0090.0090.0090.00
100.0044.5662.7370.5774.6677.0478.4979.3779.8580.00
200.0025.5143.2253.9560.4864.5967.2068.8369.7270.00
300.0016.7430.6440.8948.0753.0056.3158.4359.6260.00
400.0011.6922.1830.7937.4542.3945.9048.2449.5750.00
500.008.2916.0122.7628.3432.7336.0138.2639.5740.00
600.005.7311.1716.1020.3623.8626.5728.4829.6230.00
700.003.627.1010.3113.1715.5817.5018.8819.7220.00
800.001.753.455.046.477.698.689.419.8510.00
900.000.000.000.000.000.000.000.000.000.00
  • Skew is only zero at the poles and on the meridian
  • On the equator, any satellite that is not directly overhead will have a skew of ±90°
  • At intermediate latitudes Skew always lies between these limits

Note:  Sometimes where it is useful to illustrate a point, tables in this and the following page may include data for when a satellite would actually be invisible below the horizon.