Angle of Satellite Visibility

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Equation / Last modified by KurtHeckman on 2019/05/01 00:23
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eng.Angle of Satellite Visibility
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The Angle of Satellite Visibility calculator computes the angular extent of visibility when an observer can see a satellite based on the altitude (h) of the satellite.

INSTRUCTIONS: Choose units and enter the following:

  • (h) This is the altitude of the satellite

Angle of Visibility (η): The calculator returns the angle in degrees.  However this can be automatically converted to compatible units via the pull-down menu.

The Math / Science

The Earth's shape is approximated as a sphere in this calculation.  USING THE CALCULATOR, `h` is the height of the viewing location.  

`delta`is the minimum elevation limit of your line-of-sight to the satellite.  the minimum elevation angle could represent obstruction in the line-of-sight. `delta` defaults to 0 degrees.

If you are looking at the horizon from the top of a mountain, the height of the viewing location should be the height of the mountain plus the height of your eye or viewing instrument.

This equation uses the radius of the Earth (`R_E`) and the altitude of the location from which the horizon is observed (`h`).

Since the site line is tangent to the circle, the triangle including angle `alpha` is a right triangle. The angle, `alpha`, can be computed with line through the tangent point having a length equal to the Earth's radius.  The hypotenuse of the triangle is then the sum of the radius of the Earth and the altitude of the viewing location.

satellite visibility angle.png

`R_E` is the radius of the Earth, which is the vCalc constant: Earth - mean spherical radius (Km)

`h` is the height above the Earth's mean radius

cosine (`alpha`) = `R_E / (R_E + h) -> alpha` = acos (`R_E / (R_E + h)`) = η

angle computation.pngOnce we know the angle `alpha`, we use the fact that both triangles including `alpha` and the triangles including `beta` are all right triangles. So, we can compute `c_1` and `c_2` using the Pythagorean theorem:

`c_1 = tan(alpha) / R_E  = `

`c_2 = tan(beta) / R_E `

`c = c_1 + c_2 = tan(alpha) * R_E +tan(beta) * R_E`

`c/R_E = tan(alpha) +tan(beta)`

EXAMPLE

If we were to view the horizon from the top of a mountain that is 3 kilometers at its summit, this Central Angle of Horizon equation would tell us the central angle of the tangent point at the horizon is approximately 3.51591974 degrees from the viewing location.

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