Chang'e 4, a Lunar orbiter and lander on the far side of the Moon, will put a relay satellite in a halo orbit around EML2 that also will do some low frequency radio astronomy. Since the Chang'e 4 satellite will also be a relay satellite, it obviously will have Earth in lone of sight at least part time. More generally, could any potential telescope around EML2 use a halo orbit that takes advantage of the Moon shadowing radio disturbances from Earth, as if it were on the surface on the far side of the Moon?
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The Earth-Moon L2 is located about 1.16 times as far from the Earth as the centre of the Moon (wikipedia, so the Earth is about 7 times as from L2 as the Moon is. Since the Earth is less than 7 times the Moon's diameter the Moon would protect a telescope at L2 (or in any small halo orbit around it) from all radio noise directly from the Earth and from LEO. Satellites in GEO would still be visible, which could be useful for data relay, but might also be a source of noise.
It's a reasonably easy calculation how far from L2 you can go and still be in radio shadow. Roughly, if you move x km from L2 (in a non-radial direction) you can see Earth's limb if $7*(1700-x) > 6000$ which gives something like $x < 800$. This is unfortunately not far enough to reach any of the stable Halo orbits (which are probably not that stable for the Moon anyway, due to perturbation from the Sun), so you would need to use a small amount of fuel actively station-keeping. On the other hand, a radio astronomy satellite probably has a rather accurate idea of its position, as a pre-requisite for doing its job, so it could stay very precisely on L2, using very little fuel.
answered Apr 18, 2018 at 12:41
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$\begingroup$ Nice answer! The OP doesn't constrain frequency, but if it were the same range of low frequencies described in the OP's link for Chang'e 4 radio observations, (roughly 200 MHz and (much) lower) there may not be much noise at all from the GEO satellites. It's hard to find a good reference for the attenuation of these frequencies in (mostly dielectric) rock, but thousands of km should safely do it. There's a script that could be modified for the Moon in this answer if you are interested but the Moon's orbit is so elliptical it's probably not worth $\endgroup$– uhohApr 18, 2018 at 13:40
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1$\begingroup$ At very low frequencies, diffusion around the Moon would become a concern. Also possibly noise from the ionosphere. $\endgroup$ Apr 18, 2018 at 13:49
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$\begingroup$ Are there highly eccentric halo orbits around L2 such that a communication satellite there could communicate with Earth at apoapsis of the orbit with Earth in line of sight, and do radio astronomy at periapsis when it is in the Earth's radio shadow of the Moon? $\endgroup$ Apr 18, 2018 at 17:34