This is another one of my off-the-wall questions, one for which I don't have any definitive answer but, given my track record, may still get me into trouble with some expert or another. That said though, while I have vague inkling as to where the answer might lie, I don't pretend to know it’s precise nature. And I may be completely wrong as to where the answer lies as well.
Gyrocompasses use gyroscopic precession to identify true north. Basically as the earth rotates around its axis, gyroscopic torque is induced on a gyroscope that is somehow limited in its motion, precessing it such that it points to true north, or to the north celestial pole. If weights are used to force the gyroscope to point to the centre of the Earth in one axis, the gyroscope's other two axes will end up parallel with the Earth's surface and it will point true north. If a viscous fluid (or equivalent) is used, the relevant axis of the gyroscope will become parallel with the Earth's axis of rotation, pointing north. The viscous fluid is used, as far as I can tell, to dampen overshoot and allow the gyroscope to settle on north rather than dithering on either side of it.
Now, this precession is due to the rotation not so much of the Earth around its axis as the rotation of the gyroscope around the Earth's axis of rotation. In effect, if not in actuality, it is the centripetal force on the gyroscope that leads to the precession. However, the gyroscope in question is not only rotating around the Earth's axis of rotation, it is also rotating around the Earth's axes of orbit (of the centre of mass of the Earth and the Moon and the centre of mass of the Earth and the Sun). There's also the orbit around the centre of the galaxy, but this is a pretty slow orbit, so I think we can disregard it.
My question is this: would a highly precise, dampened gyroscope still have a detectable (even if extremely minor) wobble related to the combination of the Earth's orbit(s) and the Earth's rotation – or would it stay fixed on a celestial north pole defined only by the Earth's rotation? If there is no wobble, why isn't there one? In other words, why is the rotation around the Sun not a factor (if it isn't)?
The title, by the way, is related to apparent fact that a gyroscope on the Moon might be forced to point to the Earth's axis of rotation, since Moon is tidally locked to the Earth, and shows only one face to us (with a slight wobble, so that over time we see slightly more than 50% of it). In effect, a gyroscope on the Moon would be describing same sort of path as one on the Earth, just at a different speed. Since the Earth and the Moon both rotate in the same direction, and the Moon is tidally locked, they share the same celestial north pole anyway. If there is a wobble related to the orbit around the Sun, then this might be more noticeable on the Moon, because the looping would be more extreme.
I am aware that the Earth's axis of rotation is not the same as the axis of orbit around the sun. If it were, then we might still have summers and winters (due to the elliptical orbit), but they'd be at the same time for both the north and the south hemispheres. As it is, the southern hemisphere is tilted towards the Sun during summer (the definition of summer) and the Earth is also closest to the Sun in January (which could be a definition of summer if we had no tilt to our axis, if we didn't call it summer, there would still be a term that means "when temperatures are generally higher than the annual average").