Moon-Earth is 384,400 km, and Moon-L2 is 61,347 km. The moon has a radius of 1,737 km. From the L2 point, a beam shearing the moon's surface will pass earth at a distance of 12,642 km from its center - or 6,272 km above the surface. That is well within the orbit of geostationary and GNSS satellites. So yes, you do indeed need a relay, but it can piggyback off of a MEO constellation like O3b, or make use of one of the many TV broadcast sats. Use two 90-degree out-of-phase GEO sats and your comms beam passes a comfortable 2,363 km above the moon's surface.
Usually sats like those (there have been several dozen) aren't at the L2 point, but are in a Lissajous orbit around it. This avoids the worst stability issues, although they are still dynamically unstable so they'll require minor station keeping adjustments from time to time.
Comments
Moon-Earth is 384,400 km, and Moon-L2 is 61,347 km. The moon has a radius of 1,737 km. From the L2 point, a beam shearing the moon's surface will pass earth at a distance of 12,642 km from its center - or 6,272 km above the surface. That is well within the orbit of geostationary and GNSS satellites. So yes, you do indeed need a relay, but it can piggyback off of a MEO constellation like O3b, or make use of one of the many TV broadcast sats. Use two 90-degree out-of-phase GEO sats and your comms beam passes a comfortable 2,363 km above the moon's surface.
Usually sats like those (there have been several dozen) aren't at the L2 point, but are in a Lissajous orbit around it. This avoids the worst stability issues, although they are still dynamically unstable so they'll require minor station keeping adjustments from time to time.