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I am trying to check my work on calculating the angle between two sats and a ground station observing those two sats. I have the az and elevation angles to each sat from the ground station. What would be the equation to get the angle Sat1-GS-Sat2?

I know that it uses the law of Cosines, but not sure if I was using the correct equation. Here is what I am using:

Angle = acos((abs(sin("V_Sat_1_EL")*sin("V_Sat_2_EL")))+(cos("V_Sat_1_EL")*cos("V_Sat_2_EL")*cos(abs("V_Sat_2_AZ"-"V_Sat_1_AZ"))))

Thanks

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2 Answers 2

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This figure from Wikipedia:Spherical law of cosines applies if the ground station is at the center of the sphere, u is the ground station's zenith, and v and w are the apparent positions of the satellites as seen from the ground station.

spherical triangle diagram

Then a and b are the satellites' zenith angles, i.e. 90° minus their elevation angles. C is the difference between the satellites' azimuths, and c is the angle you want.

Using the cosine rule for sides $$\cos c = \cos a \cos b + \sin a \sin b \cos C$$ and the rules for sines and cosines of complementary angles, the expression in the question looks correct if the absolute value operations are omitted.

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If you only need angles, you need interior angle triangle forumula, all angles in interior sum is equal to 180° . only you need Angle Sat 1 + Angle Sat 2 + Angle Sat 1-2 = 180°, if you need distances A, B , C you need use polar coordenates. lat, long and elevation.

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  • $\begingroup$ I don't know the angle between the two satellites, only the AZ and EL to each from the GS. Think of the GS as the center of a sphere, and the two satellites are points on that sphere and I have two angles from the center of the sphere that give the position of where each Satellite is on the surface of the sphere. I need the angle that is formed by that GS looking at each Satellite. $\endgroup$
    – Shaps
    Nov 4, 2019 at 16:54
  • $\begingroup$ AZ is Azimuth , and EL is Elevation ? because Azimuth is a angular meassure $\endgroup$
    – Adrian R
    Nov 4, 2019 at 17:43
  • $\begingroup$ I should clarify. I know the AZ (angle) and Elevation Angle to each. So imagine my Ground Station is sitting at the center. My AZ to one satellite is the angle I rotate myself to look at it and then my elevation is how high I need to look at it. I have both of those for each Satellite. I need to figure out the angle that is formed between the first satellite, the Ground Station, and then the second satellite. I hope that this makes sense. $\endgroup$
    – Shaps
    Nov 4, 2019 at 21:44
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    $\begingroup$ This answer is for Euclidian planar geometry but the question is about spherical trigonometry; triangles drawn on the surface of a unit sphere. $\endgroup$
    – uhoh
    Nov 5, 2019 at 1:08

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