ECE 5233 Satellite Communications Prepared by:

Dr. Ivica Kostanic

Lecture 4: Look angle determination(Section 2.2)

Spring 2014

Florida Institute of technologies

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Sub-satellite pointMotion of sub-satellite pointCalculation of elevation and azimuth Look angle calculation spreadsheetLook angles to geo-synchronous satellitesExamples

Outline

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 equatoro Longitude – degrees west of the Greenwich meridian

Location of the sub satellite point may be calculated from coordinates of the rotating system as:

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222

1cos2

rrr

rs

zyx

zL

41

31

21

11

/tan0,04 Case/tan2/0,03 Case

/tan0,02 Case/tan0,01 Case

QxyyxQxyyxQxyyxQxyyx

l

rrrr

rrrr

rrrr

rrrr

s

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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

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Sirius radio – two geo stationary and three highly inclined orbit

satellites

International space station – LEO orbit

Note: maps are generated using STK by Analytic Graphics, Inc.

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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 If the satellite is on non stationary orbit, the

ground system needs to track the Az and El in time

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Definition of Az and El

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Calculation of elevation

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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 Step 1:

coscos

sinsincoscoscoscos1

seesse LLllLL

Step 2:

2/12

cos21

sincos

s

e

s

e

l

rr

rr

E

Where re is the radius of the Earth (6370km)

Example: Calculate El 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: 48.2967W (+0.8429 rad)

radius, rs = 38000km

6693.325702.08418006.0cos

Step 2:

8628.101896.0

9821.0cos

l

l

EE

Step 1:

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Calculation of azimuth - cases

Eight cases to considerNorthern hemisphere – 4 cases

o At least one of the two points (Earth station, sub-satellite point) is in the northern hemisphere

Southern hemisphere – 4 caseso Both points (Earth station and sub-

satellite point) are in the southern hemisphere

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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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Calculation of azimuth – northern hemisphere

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A west of B B west of A

Note: B chosen to be north of A

AB

AB

LLCLLXY

5.0cos5.0tan

5.0sin5.0tan

AB

AB

LLCLLXY

5.0sin5.0tan

5.0cos5.0tan

where

180 if-360

180 if

BABA

BABA

llllllll

C

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

SSP- sub-satellite point

ES – Earth station

1. Solve tan equations for X and Y

2. Identify the case and use table to determine the AZ

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Calculation of azimuth – southern hemisphere

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A west of B B west of A

Note: B chosen to be south of A

AB

AB

LLCLL

XY

5.0cos5.0tan5.0sin

5.0tan

AB

AB

LLCLL

XY

5.0sin5.0tan5.0cos

5.0tan

where

180 if-360

180 if

BABA

BABA

llllllll

C

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

SSP- sub-satellite point

ES – Earth station

1. Solve tan equations for X and Y

2. Identify the case and use table to determine the AZ

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Azimuth calculation - example

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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: 48.2967W (-0.8429 rad)

radius, rs = 6738km

This is Case 2 of Northern hemisphere calculation:

C = |80.62311-48.2967|=32.326410.5642 rad

LB=49.54320.8647 rad

LA=28.06280.4898 rad

tan[0.5(Y-X)]=0.82510.5(Y-X)=0.6898 rad

tan[0.5(Y+X)]=5.41000.5(Y-X)=1.3880 rad

X=0.6982 rad

Y=2.0778 rad

For Case 2 of northern hemisphere:

Az = X = 0.6982 rad 40.0016

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Look angle worksheet

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Earth stationLatitude -24.870278 degrees -0.4340682 rad Instructions

Longitude -113.703611 degrees -1.9845024 rad 1. Black fields are inputs, blue fields are calculatedRadius of Earth 6378.4 km 2. Make sure that proper case is identified. This implies chosing one of eight answers provide by the spreadsheet.

Sub-satellite point 3. Possible answers are in boldLatitude 0 degrees 0 rad

Longitude -110 degrees -1.9198622 radRadius to satellite 42134 km

Elevation calculationStep 1

cos(gamma) 0.9053675gamma 0.4385519 rad 25.127171

Step 2cos(El) 0.4907114

El 1.0578903 rad 60.612647 degrees

Azimuth calculation (northern hemisphere)

C 3.703611 degrees 0.0646402 radLB 0 degrees 0 rad SS point ES Relation Aximuth (degrees)LA -24.870278 degrees -0.4340682 rad A B A west of B 360-Y

tan(0.5(Y-X)) 6.8202288 B A A west of B Xtan(0.5(Y+X)) -140.26621 A B B west of A Y

0.5(Y-X) 1.425211 rad X -2.9888782 rad -171.25011 degrees B A B west of A 360-X0.5(Y+X) -1.5636671 rad Y -0.1384561 rad -7.9329506 degrees

360-X 531.25011 degrees Note: B more north than A360-Y 367.93295 degrees

Azimuth calculation (southern hemisphere)

C 3.703611 degrees 0.0646402 rad SS point ES Relation Aximuth (degrees)LB -24.870278 degrees -0.4340682 rad A B A west of B 180+YLA 0 degrees 0 rad B A A west of B 180-X

tan(0.5(Y-X)) 6.8202288 A B B west of A 180-Ytan(0.5(X+Y)) 140.26621 B A B west of A 180+X

0.5(Y-X) 1.425211 rad X 0.1384561 rad 7.9329506 degrees0.5(Y+X) 1.5636671 rad Y 2.9888782 rad 171.25011 degrees Note: B more south than A

180+Y 351.25011 degrees180-X 172.06705 degrees180-Y 8.7498944 degrees180+X 187.93295 degrees

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Look angles to geo-stationary satellites

Geo stationary satelliteso Occupy non-inclined geo-synchronous

orbito Always above same equatorial pointo 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

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Example of “dish pointer” website

Note: compare pointing results between class spreadsheet and dish-pointing websites