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Satellite Look Angle Problem

The WESTAR III satellite is a geosynchronous satellite located at 91 degrees W longitude. Determine the look angles to this WESTAR III satellite from an earth station near the city of Los Angeles California at 120 degrees W, 34 degrees N. Also, determine the latitude and longitude of the northern most and the westerly most point at which the WESTAR III satellite is visible at a 10 degree elevation angle.


Solution:

a) In determining the look-angles to the geosynchronous satellite describe above:

Le = 34 degrees N
le = 120 degrees W
ls = 91 degrees W

Elevation (El):

El = cos (γ) = cos (Le) • cos (ls - le)

cos (γ) = cos (34) • cos (91 - 120)

γ = 43.52 degrees

Therefore,

cos (El) = sin(γ) / [ (1.02274 - 0.301596) cos(γ) ]1/2

El = 39.79 degrees


Azimuth (Az):

a = | ls - le | = 29 degrees

c = | Le - Ls | = 34 degrees

s = 0.5 ( a + c + γ ) = 53.26 degrees

α = 2 tan-1[ (sin (s - γ) sin (s - |Le|)) / (sin (s) sin (s - |le - ls|)) ]1/2

α = 2 tan-1[ 0.169]1/2

α = 44.75 degrees

Sense sub-satellite position is southeast of the earth station,

Az = 180 degrees - α = 135.25 degrees


b) In finding the lat and long of the northern most point and westerly most point, set the elevation equal to 10 degrees and solve for gamma (γ).

cos (El) = sin (γ) / [ (1.023 - 0.302) cos (γ) ]1/2

Let X = cos (γ)
[ 1 - X2 ]1/2 = sin (γ)
A = 10.23
B = 0.302
C = cos (El) = 0.985

We get the following expression: 1 - X2 = C2 (A - BX)

Which as written as a quadratic equation we get: X2 - BC2X + (AC2 - 1) = 0

Rewriting the equation into the quadratic formula we get: X = BC2 ± [ B2C4 - 4 (AC2 - 1) ]1/2 / 2

X = (0.293 + 0.340) / 2

γ = cos-1X = 71.54 degrees


Therefore, the northern most point is where le is equal to ls and Le is equal to γ

Latitude = 71.54 degrees N
Longitude = 91.00 degrees W




Now, the westerly most point must lie on the equator, Le = 0 degrees

cos (γ) = cos Le cos (ls - le)

cos (γ) = 1 cos (ls - le)

γ = | ls - le |

le = ls ± γ

le = 91.00 degrees ± 71.54 degrees

So: le = 162.54 degrees or 19.46 degrees

This equates to the westerly most point of,

Latitude = 0 degrees N
Longitude = 162.54 degrees W


Satellite key words: earth station look angle, visible, line of sight, site, geosynchronous, elevation angle, westar, coordinates, lat, long, satcom, absolute value, mathematical orbital computation, calculated, apogee, perigee, spacecraft, velocity, eccentricity, angular limits, engineering dynamics, scientific notation.


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