Originally Posted by
Shadoko
Hope I didn't make errors there and this could help!
Looks good to me. You have the radius of the geosynchronous orbit off by about 100 km, but it does not affect the results in any significant way. Intermediate calculations can be reduced to a single formula through "law of cosines".
And, just to be clear, yes, it's entirely possible that the satellite has a clock that could measure signal travel delay down to a fraction of your 620 microseconds. But that assumes the signal being reflected back in a passive way or by a "dumb" device. You can bounce a laser beam off a reflector on the surface of the Moon and use it measure the distance with sub-meter accuracy. That's because the reflector is passive. If the reflector is a computer system that needs to "think" about the answer, that's a source of uncertainty which may or may not be longer than 620 microseconds. Or maybe there's no reflector at all, we just have a transmitter on the aircraft with its own clock that is used to timestamp the transmission, and we're making deductions based on the timestamp and the reception time. Then we also have to wonder how accurate the clock in the transmitter is.
So, the "geometry" of the process, so to speak, is pretty clear, but there are other things which are not.