Designing a GPS constellation - a question for math nerds.

[DerekL1963]

I'm designing a GPS constellation for KerbalGPS Revived - thanks to USN I understand the geometry of CPA, and that leads me to realize there's a problem:

If the transmitter has a range of 2,500km and I put it in a 2,500km orbit - it's only in range of a given receiver when it's exactly overhead.  The maximum orbit height (yielding horizon-to-horizon coverage) is actually defined by a tangent 2500km long, originating at the receiver.   So my question would be, what is the orbital altitude defined by the endpoint of the tangent?

[Gaarst]

Use Pythagoras.

You have a triangle CSR (Kerbin Centre, Satellite and Receiver) which has a right angle at R (because tangent).

So: CS^2 = RS^2 + CR^2. SC (distance of sat to Kerbin centre) is x + 600, SR (distance of sat to receiver) is 2500 and RC (distance of receiver to Kerbin centre) is 600.

(x + 600)^2 = 2500^2 + 600^2. In the end the altitude x of the sat is 1970 km.

Edited October 3, 2018 by Gaarst

[DerekL1963]

Well, that turned out to be simpler than I imagined...     Thanks!