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Geometry, Trigonometry, and Amateur Satellites

Back to Part III. I'm still playing around with building a satellite tracking application. I'm helping with a special event station to help raise awareness for the Friends of Science East's effort to restore Tesla's last lab, Wardenclyffe, in Shoreham, NY.

First, the good news, the satellite pass finder and viewer is up and running at!

Just drag the yellow thumbtack to your location and click the 'Passes' button and all the radio visible satellite passes for your location will be listed. By clicking on the map checkbox for any pass, you can display it on the globe. You can see how visible a pass will be from a city building by positioning yourself near the building and looking up for the pass. The app is still very beta, so please let me know if you see anything that could be better.

One of the big steps in getting the application to work was determining if a satellite would be visible over the horizon. To do this took some geometry and a little bit of trigonometry. The very same rules that they tell you to memorize in high school, but you can't see where you would ever use them. This is where!

For the mathematically curious, here's what I had to do. It follows from the proof drawn up by the folks at the Northern Virginia Community College. If you think about where you can see a satellite from on earth, it's where the earth, (the horizon), isn't getting in the way. If you draw a line from the satellite that's tangent with the earth, then, where that line meets the earth, anyone to the left of it in the picture below won't be able to see the satellite, and anyone to the right will.

The question is, given the height of the satellite the radius of the Earth, and the location on the Earth directly below the satellite, how do you figure out where the line meets the earth? The queston framed another way is how do you draw a line from a point so that it is tangent to a circle? First, draw a line from the point, (the satelite), to the center of the circle. Then, draw a second circle that has the new line as its diameter.

Where the second circle meets the first circle is where the tangent line from the satellite will just meet the Earth. How do we know that? This is where a few geometry rules come in handy. First, we should know that to find the tangent line of a circle, we draw a radius from the center of the circle to the point where we want to find the tangent. The tangent will be a line that forms a right angle with the radius line. Second, any triangle drawn inside a circle so that one of it's sides is the diameter of the circle and all three of its points touch the circle is a right triangle.

To prove that we actually found the tangent location, first draw a line from the intersection of the two circles back to the center of the smaller circle, (the Earth). Since it touches the edge of the smaller circle, and its center, this is a radius line and the tangent will be perpendicular to it. Next, draw a line from the intersection of the two circles back to the far side of the diameter of the bigger circle, (the location of the satellite). This newest line, the radius line and the diameter form a triangle. The angle that is opposite the diameter is a right angle. Remember, the tangent line makes a right angle with the radius of the small circle. That same angle is the right angle between the radius of the circle and our line back to the satellite, that's our tangent line!

If we can find out the distance from the location directly below the satellite, (remember, that's one of our givens), to the horizon, we can compare that to the distance between our location and the location directly below the satellite. If we're closer to the satellite ground location than the horizon, then we can see the satellite. If we're not, we can't, (ignoring diffraction effects we're not going to talk about yet).

To figure out the distance to the horizon, we'll go back to trigonometry. If we know the angle from the location on the ground directly below the satellite, then we can figure out the distance. The distance along a circle is the angle swept out by the circumference of the circle times the radius of the circle. We're in luck because we still have our tangent right triangle and it will let us solve the problem.

The short side of the triangle is the radius of the earth which we know. The hypotenuse is the radius of the earth plus the altitude of the satellite which we also know. We can plug those two numbers into the formula for the cosine of an angle and solve. The cosine of the angle is equal to the short side divided by the hypotenuse. The value for the angle is the arccosine of the same quotient, the Earth's radius divided by the hypotenuse, (the Earth's radius plus the satellite's altitude). Finally, by multiplying the angle by the radius of the Earth, we get the distance from the point directly below the satellite to the horizon.


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