Satellite Footprint Formula

[KJ6KQE]

Does anyone know a formula I could use to determine the maximum possible footprint of a satellite at a given height?

It's just for kicks mostly, but I'd like to find out how far one could get with a repeater connected to a balloon.

So what is the formula to find the length of arc AB given radius r and elevation e?

 

[GnG8d]

If you don't get an answer, I'll dig up my machinists handbook when I get home. Fun stuff

 

[Captain Kilowatt]

Try this to determine the distance to the radio horizon which is what you really want.

Distance to the Horizon Calculator

 

[KJ6KQE]

OK, I found the answer. It took a while, and CK's post was the "lightbulb" that helped me work through it.
Consider this situation:

A satellite at point B is at a height of c2 above the earth, with a radius of c1. (OK, OK, the earth is not a perfect sphere, but let's assume for the moment it is)
in this case, c1 = b, and c=c1+c2 (and read a^2 as a-squared)
The question is, given c2 and a known value c1, what is the length of arc CE.

The answer is arc CE = 2*arc CD.
arc CD = b*angle BAC (in radians)
angle BAC (in radians) = arc-cosine (a^2+c^2-b^2)/2ac
a^2=c^2-b^2

So, assuming the earth's radius of 7,917.5 miles, a balloon at 96,000 feet, has a theoretical foot print of 757 miles.
AO-27, at 494 miles, has a footprint of 3,764 miles, which is a mere three miles short of what my gpredict software says. I guess the earth is not perfectly round after all!

 

[W1MSG]

Have you taken into effect the antenna radiation pattern ?? Wouldn't that have a certain amount of effect on the actual radios foot print ?

Just curious.

Craig

 

[KJ6KQE]

No, I did not.
This is a theoretical maximum. In fact if you look at a satellite like DirecTV, the radiation pattern of an antenna like a dish can choke the footprint dramatically.
Great point, Craig!