question on lunar colonies?
When we eventually build colonies on the moon, we will probably need communications satellites. Communications satellites are used to bounce radio waves from the earth’s surface to send messages around the curvature of the earth. In order to be available all the time, they must remain above the same point on the earth’s surface and must move in a circle above the equator.
Using the fact that the moon spins on its axis once every 27.3days, calculate the height above the lunar surface at which these satellites should be placed in orbit.
I got 1.725 x 10^7, i don't understand why is it wrong
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What are the units of you answer? Miles? feet? meters?
In reality, a synchronous orbit around the moon would not work, because the gravity of the Earth would interfere. A communications satellite, in order to be synchronous with the moon, would need to be at one of the Lagrange points. L1, L2, and L3 are not stable, and L4 and L5 are too far away.
If the moon was all by itself, the sync orbit radius would be 2.104 times Earth's sync orbit radius. The formula is:
R^3 = [ GM / (4pi^2) ] T^2
The moon is 1/80 as massive as the Earth.
2.104^3 = [1/80] 27.3^2
2.104 * 42,164 km = 88,720 km.
Minus the moon's radius gives the height above the surface:
88,720 - 1,738 = 86,982 km
Such an orbit would carry the satellite 1/4 of the way to the Earth. But in order to orbit around the moon, a satellite must stay within 60,000 km (see Hill sphere).
Without talking about the answer...
Note that you've discussed synchronous communications satellites. There are also Low Earth Orbit communications satellites, used, for example by Iridium and other satellite phone services. Basically the group of these works like a bunch of cell towers that receive calls and route them from one satellite to another until they are at a satellite which can relay them to the ground network. A way different idea than the synchronous type you talk about.
For the orbital distance, consider that earth's synchronous satellites are about 22,000 miles up and make one orbit per day, while you'd have to be much higher on earth to make one orbit per 27 days.
Since the lunar gravity is much less, so would the altitude, but you knew that.
Good luck with the calculations.. Kepler would have been proud.