ECE 5233 Satellite Communications
Prepared by:
Dr. Ivica Kostanic
Lecture 4: Look angle determination
(Section 2.2)
Spring 2014
Outline
Sub-satellite point
Motion of sub-satellite point
Calculation of elevation and azimuth
Look angle calculation spreadsheet
Look angles to geo-synchronous satellites
Examples
Important note: Slides present summary of the results. Detailed
derivations are given in notes.
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Sub-satellite point
 Point at which a line between the satellite and the center of
the Earth intersects the Earth’s surface
 Location of the point expressed in terms of latitude and
longitude
 If one is in the US it is common to use
o Latitude – degrees north from equator
o Longitude – degrees west of the Greenwich meridian
 Location of the sub satellite point may be calculated from
coordinates of the rotating system as:

zr
Ls   cos 
 x2  y2  z 2
2
r
r
 r

 Case 1

Case 2
ls  
Case 3
Case 4

1
xr
xr
xr
xr
 0, yr
 0, yr
 0, yr
 0, yr
0
0
0
0




 tan 1  yr / xr 
  tan 1  yr / xr 
  / 2  tan 1  yr / xr
 tan 1  yr / xr 


Q1
Q2
Q3
Q4
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Examples of sub-satellite point trajectories
 sub-satellite point used for 2D
map display of satellite path
 For most satellites the
trajectory is part of sinusoidal
 For geo-stationary satellites
the trajectory is a point
Sirius radio – two geo
stationary and three
highly inclined orbit
satellites
Note: maps are
generated using STK by
Analytic Graphics, Inc.
International space
station – LEO orbit
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Page 4
Look angles – elevation (El) and azimuth (Az)
 Az – angular distance of the satellite
from the north
o Az is between 0 and 360 degrees
 El – angular distance of the satellite
from the local horizontal plane
o El is between 0 and 90 degrees
 Az and El are required for proper pointing of
the Earth station antenna
 If the satellite is geo-stationary the antenna
is pointed once
Definition of Az and El
 If the satellite is on non stationary orbit, the
ground system needs to track the Az and El
in time
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Calculation of elevation
Given:
Example: Calculate El for the following data
Le – latitude of Earth Station
ES:
latitude: 28.06280 N (+0.4898 rad)
longitude: 80.62311 W (+1.4071 rad)
le – longitude of Earth station
SSP: latitude: 49.5432 N (+0.8647 rad)
Ls – latitude of sub-satellite point
longitude: 48.2967W (+0.8429 rad)
ls – longitude of sub-satellite point
radius, rs = 38000km
rs – distance to satellite
Step 1:
Step 1:
cos   0.8418006
  0.5702  32.6693
cos   cosLe  cosLs  cosls  le   sin Le sin Ls 
  cos 1 cos  
Step 2:
cosEl  
Step 2:
cosEl   0.9821
sin 
  r 2  r 

e
e
1     2  cos  
  rs 

 rs 
1/ 2
El  0.1896  10.8628
Where re is the radius of the Earth (6370km)
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Calculation of azimuth - cases
 Eight cases to consider
 Northern hemisphere – 4 cases
o At least one of the two points
(Earth station, sub-satellite point) is
in the northern hemisphere
 Southern hemisphere – 4 cases
o Both points (Earth station and subsatellite point) are in the southern
hemisphere
Given:
Le – latitude of Earth Station
le – longitude of Earth station
Ls – latitude of sub-satellite point
ls – longitude of sub-satellite point
rs – distance to satellite
Note: presented algorithm accommodates general case
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Page 7
Calculation of azimuth – northern hemisphere
tan 0.5Y  X  
sin 0.5LB  LA 
tan 0.5C  cos0.5LB  LA 
tan 0.5Y  X  
cos0.5LB  LA 
tan 0.5C sin 0.5LB  LA 
where
B west of A
A west of B
 l A  lB
C
360 - l A  l B
Note: B chosen to be north of A
Case
SSP
ES
Relations
Az (degrees)
1
A
B
A west of B
360-Y
2
B
A
A west of B
X
3
A
B
B west of A
Y
4
B
A
B west of A
360-X
if l A  l B  180
if l A  l B  180
1. Solve tan equations for X and Y
2. Identify the case and use table to determine
the AZ
SSP- sub-satellite point
ES – Earth station
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Calculation of azimuth – southern hemisphere
tan 0.5Y  X  
tan 0.5Y  X  
sin 0.5 LB  LA 
tan 0.5C  cos0.5 LB  LA 
cos0.5 LB  LA 
tan 0.5C sin 0.5 LB  LA 
where
B west of A
A west of B
Note: B chosen to be south of A
Case
SSP
ES
Relations
Az (degrees)
1
A
B
A west of B
180+Y
2
B
A
A west of B
180-X
3
A
B
B west of A
180-Y
4
B
A
B west of A
180+X
 l A  lB
C
360 - l A  l B
if l A  l B  180
if l A  l B  180
1. Solve tan equations for X and Y
2. Identify the case and use table to determine
the AZ
SSP- sub-satellite point
ES – Earth station
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Azimuth calculation - example
Example: Calculate Az for the following data
ES:
latitude: 28.06280 N (+0.4898 rad)
longitude: 80.62311 W (-1.4071 rad)
SSP: latitude: 49.5432 N (+0.8647 rad)
longitude: 48.2967W (-0.8429 rad)
radius, rs = 6738km
This is Case 2 of Northern hemisphere calculation:
C = |80.62311-48.2967|=32.326410.5642 rad
X=0.6982 rad
LB=49.54320.8647 rad
Y=2.0778 rad
LA=28.06280.4898 rad
For Case 2 of northern hemisphere:
Az = X = 0.6982 rad 40.0016
tan[0.5(Y-X)]=0.82510.5(Y-X)=0.6898 rad
tan[0.5(Y+X)]=5.41000.5(Y-X)=1.3880 rad
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Look angle worksheet
Earth station
Latitude
Longitude
Radius of Earth
Sub-satellite point
Latitude
Longitude
Radius to satellite
-24.870278
-113.703611
6378.4
degrees
degrees
km
-0.4340682
-1.9845024
rad
rad
Instructions
1. Black fields are inputs, blue fields are calculated
2. Make sure that proper case is identified. This implies chosing one of eight answers provide by the spreadsheet.
3. Possible answers are in bold
0
-110
42134
degrees
degrees
km
0
-1.9198622
rad
rad
cos(gamma) 0.9053675
gamma
0.4385519
rad
25.127171
Elevation calculation
Step 1
Step 2
cos(El)
El
0.4907114
1.0578903
rad
60.612647
3.703611
0
-24.870278
6.8202288
-140.26621
1.425211
-1.5636671
degrees
degrees
degrees
0.0646402
0
-0.4340682
rad
rad
rad
rad
rad
X
Y
-2.9888782
-0.1384561
degrees
Azimuth calculation (northern hemisphere)
C
LB
LA
tan(0.5(Y-X))
tan(0.5(Y+X))
0.5(Y-X)
0.5(Y+X)
rad
rad
360-X
360-Y
-171.25011
-7.9329506
531.25011
367.93295
degrees
degrees
degrees
degrees
SS point
A
B
A
B
Note:
ES
B
A
B
A
Relation
A west of B
A west of B
B west of A
B west of A
Aximuth (degrees)
360-Y
X
Y
360-X
B more north than A
Azimuth calculation (southern hemisphere)
C
LB
LA
tan(0.5(Y-X))
tan(0.5(X+Y))
0.5(Y-X)
0.5(Y+X)
3.703611
-24.870278
0
6.8202288
140.26621
1.425211
1.5636671
degrees
degrees
degrees
0.0646402
-0.4340682
0
rad
rad
rad
rad
rad
X
Y
0.1384561
2.9888782
SS point
A
B
A
B
rad
rad
180+Y
180-X
180-Y
180+X
7.9329506
171.25011
351.25011
172.06705
8.7498944
187.93295
degrees
degrees
degrees
degrees
degrees
degrees
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Note:
ES
B
A
B
A
Relation
A west of B
A west of B
B west of A
B west of A
Aximuth (degrees)
180+Y
180-X
180-Y
180+X
B more south than A
Page 11
Look angles to geo-stationary satellites
 Geo stationary satellites
o Occupy non-inclined geo-synchronous
orbit
o Always above same equatorial point
o Location specified using longitude of the
sub-satellite point and distance to the
satellite
 The El/Az calculation spreadsheet “works” for
geo-stationary satellites
 There are also many websites that calculate
El/Az pairs
o Example site:
http://www.sadoun.com/Sat/Installation/Satellite
-Heading-Calculator.htm
 VSAT broadcast terminals are usually
operating with Geo-stationary satellites
Example of “dish pointer”
website
Note: compare pointing results between class spreadsheet and dish-pointing
websites
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Page 12