Communications satellites to the far side of the Mun

[SnappingTurtle]

I'm using the RemoteTech mod and setting up a communications network, only as far as the Mun for now. I made a 3 point geostationary network based on Scott Manley's tutorial but I have a kethane mining operation on the far side of the Mun that needs communications.

I've thought of two ideas for this. I could send up two satellites, one to lead the Mun and one to trail it, on the same orbit as the Mun but outside its sphere of influence. Hopefully they would reach far enough around to the Mun base. I tried this but as one of my satellites was coming in it suddenly got an encounter that wasn't predicted by the node system which threw me off.

Otherwise I could try and do a 3-point system around the Mun. Not geostationary but hopefully they'd have enough coverage. But I don't know the mathematics of what orbit to put my deployment vessel into to space the satellites evenly. If I choose an orbit of 2000 km for my satellites around the moon, what periapsis would I need for an orbit to have the same apoapsis (2000 km) but 2/3 or 5/6 the period?

Thoughts?

Edited July 28, 2013 by SnappingTurtle

[metaphor]

Orbital period is proportional to a^(3/2), where a is the semi-major axis. A circular orbit at 2000 km would have a semi-major axis of 2200 km (since the Mun's radius is 200 km), so to get a period of 2/3 the circular period, you want the semi-major axis to be 2200 * (2/3)^(2/3), or 1679 km. If you want to keep a 2000 km apoapsis, that means your periapsis will be 2*1679 - 2*200 - 2000 = 958 km.

To get a period of 5/6 the circular period, you want the semi-major axis to be 2200 * (5/6)^(2/3), or 1948 km. If you want to keep a 2000 km apoapsis, that means your periapsis will be 2*1948 - 2*200 - 2000 = 1496 km.

So to get a period of x times the circular period, you want the semi-major axis to be x^(2/3) times as big. (Semi-major axis = periapsis + apoapsis + body's diameter)

[SnappingTurtle]

Thanks, that worked great.