Calculate the orbital radius of a synchronous satellite

[james111]

I'v been trying to work out this question for ages, but nothings quite working for me, here goes..

q. a) calculate the orbital radius of a synchronous satellite (one period of 24hr, so appears stationary above anyone point). Approximately how many radii of Earth is this orbital radius? Why does the satellite have to be in the plane of the Equator?

b) draw a scale diagram to estimate the angle above the horizontal that a receiving aerial in latitude 45 deg must point in order to receive signals from the satellite.

For a) I figured that I could use the eqn a = (v^2)/r for circular motion substituting v = 2rPi / (24x60x60) ... but this gives r as 1853078.528km I think... which would be wrong. I don't know wot to do.

thanks for any help

 

[berkeman]

IIRC, geosynchronous satellites are something like 24,000 miles up (dunno if that's above the Earth's surface or measured from the center of the Earth, though). Because the acceleration due to gravity falls off in space, you need to figure out where the central acceleration due to that lower gravitational acceleration is correct for the orbital velocity. Do you know the equation that gives the gravitational acceleration in terms of two masses m and M and the gravitational constant G and the radius r?

 

[berkeman]

Hey, I just checked wikipedia, and I was pretty close with my guess. James, *after* you work out this problem and get an answer close to my guess, check out the page at wikipedia.org about "geosynchronous".

 

[james111]

GMm/R ?? I still can't figure how we can apply this. Sorry, this is quite a new area for me, and I still haven't fully figured it out.

 

[berkeman]

GMm/R ?? I still can't figure how we can apply this. Sorry, this is quite a new area for me, and I still haven't fully figured it out.

Um, no. Close, but not correct. Do you have a textbook for this class? It should definitely be giving you this information before asking you this question.

I went back to the wikipedia page about geosynchronous orbits, and followed a link at the bottom of that page to a web page about orbital mechanics. It's a nice write-up, and it has the correct equation (similar to yours but with one term changed) part-way down the page. Try reading through this link to see if it helps this question make more sense.

http://www.braeunig.us/space/orbmech.htm