Having attended an outreach talk on gravitational waves, we had a couple of questions:
1) The moon is a massive orbiting object near to the Earth. Does it produce measurable gravitational waves? If so, do these waves affect the measurements taken by LIGO?
2) The sun is changing mass due to solar wind, flares and nuclear reactions. Is it possible to measure this effect with current (or near future) gravitational wave detectors? We appreciate that it won't be an oscillation but it will be changing the curvature of space-time as the Earth orbits it over time.
Glad that you enjoyed the lecture. In answer to your questions:
However, more directly, the Moon does affect LIGO though its gravitational pull. This is a direct force that can pull at the mirrors at the end of the arms and move them slightly. Fortunately, since we know where the Moon is, we can take this effect into account. In fact, the detectors are so sensitive, that not only must we take into account the Moon, but also the tides. The tides are the movement of water in response the the gravitational pull of the Moon and the Sun. At different points in the cycle there is a different distribution of water across the Earth, and the pull from having extra water in regions close to detectors can be measured.
In conclusion, the Moon orbitting will produce gravitational waves, but these are not of the right frequency to be detected. However, the Moon's gravity still needs to be taken into account.
Regarding gravitational wave measurement, you've hit a very important point. There shouldn't be any (significant) oscillation. To excite gravitational waves you must have something like a binary (technically where the mass quadrupole is changing) and not something spherically symmetric. The Sun's mass loss is very nearly perfectly spherical, and so there shouldn't be any real gravitational wave emission.