Giant variation of the proposed eLISA mission using reflectors on Earth and the Moon possible?

I was thinking about the proposed ESA mission eLISA, which is essentially a space version of the LIGO experiment, and it occurred to me that trailing three satellites behind the Earth's orbit seemed excessively complicated in comparison to using a reflector on the ground and another on the moon (if I am not mistaken, there is even already a mirror on the moon, from the Lunar Laser Ranging experiment) and a single satellite. Is there not a geosynchronous orbit with a line of sight to the moon at all times? It seems to me the added range would provide better sensitivity. Provided you know the exact distance to the reflector, you can calculate individual frequencies for the Earth and Moon lasers, downconvert the faster frequency through a PLL of sorts, and still use the resulting data in an interferometry analysis. Is there something major I am missing here that would make this arrangement more difficult than the proposed eLISA mission? It seems to shift a lot of the physical risk and cost of the mission to computational costs, which by comparison are much, much cheaper.

This would not be a giant version of eLISA, but a small (and complicated) one.

The distance between the LISA satellites will be 1 million kilometers, while the moon is on average 380.000km from Earth. So that arm would be less than half as long. Geosynchronous orbit is at 36.000km, not even 10% of the distance to the moon. The difference in length between your proposed detector arms is much shorter than the coherence length of any laser, so your interferometer would not work.

Further problems:

• Positioning of mirror on Earth: Where do you want to put the mirror such that the moon is always visible in the sky?

• Positioning of satellite: I doubt (without checking) that you can see the moon at all times from geosynchronous orbit. Maybe you could work with one of the Earh-Moon Lagrange points.

• Constant mirror adjustment: Earth is rotating, and the Moon revolves around the Earth. So your interferometer arms would change angles constantly. (And you'd need to dig lots of tunnels straight through Earth, see above.)

• Vibration: The mirrors have to be as free from vibration as possible. That is much easier to achieve with a free floating satellite than on earth or the moon. (Did you know there are moon quakes?)

• Ellipticity: The distance to the moon varies between 360.000km and just over 400.000km during its orbit. So you'd be constantly adjusting the interferometer. A constant drift is not necessarily problematic, but the huge difference in distance might be.

Lastly, it is not clear to me how you want to replace precision measurement by calculation. What is PLL? How do you get meaningful interference patterns with lasers of different wavelengths? Also note that the "exact distance", if you want to count waves, must be better than half the wavelength of your laser, so a few hundred nanometers. At the distance of the moon, that is better than $1 : 10^{15}$ (hope that calculation is correct).

• I didn't realize the distance between satellites was 1 million kilometers! Yes that makes a lot of sense then, there is no gain. As for a PLL, it is a phase-locked loop, it can be used in an analog filter to downconvert a signal band from a higher frequency to a lower one, it is a basic RF circuit, used in every radio telescope, computer, and HF transceiver on earth. Thank you for the precision! Aug 14 '16 at 10:21
• +1 also atmosphere. The light would be passing through the atmosphere, which refracts and slows light. Aug 14 '16 at 10:43
• True, forgot about the atmosphere. Thanks for the explanation of PLL! The 1million km are actually in your eLISA link ;-)
– Alex
Aug 14 '16 at 14:11