PeterDonis
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DrStupid said:In the rest frame of Earth
Which is not inertial over the relevant time scale.
DrStupid said:there is no reason to prefer the rest frame of Sun
Yes, there is: that it is inertial over the relevant time scale (a month), while the rest frame of the Earth is not.
More precisely, the inertial frame in which the barycenter of the solar system is at rest is the frame in which the orbit of the Moon has the property I stated, and it is inertial over the relevant time scale. Strictly speaking, the rest frame of the Sun is not, because of the Sun's movement relative to the barycenter, but it is still much closer to being inertial over a time scale of a month than the Earth's rest frame, because the Sun is so much closer to the barycenter than the Earth is (so the non-inertial terms are much, much smaller).
DrStupid said:The Moon orbits Earth - even according to your own definition.
No, it doesn't, because my definition requires an inertial frame.