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I realize the Earth and the Moon both orbit around their shared center of mass, and that in the case of the Earth and Moon this center is "inside" the Earth. However, I'm looking for an approach to determine the distance the Earth travels in one "orbit around the moon" (as small as it may be).

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  • $\begingroup$ The title of your question assumes that your frame of reference is fixed to Moon. Then, the radius of Earth's orbit around Moon is the same as that of Moon's orbit around Earth. Following from this, the circumference of Earth's orbit around Moon is also the same as that of Moon's orbit around Earth. If you mean that your frame of reference is the centre of mass of Earth+Moon system, then [Steve Linton's answer][astronomy.stackexchange.com/a/25378] is adequate. $\endgroup$ – garyF Mar 5 '18 at 11:45
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    $\begingroup$ @garyF - Re-read the question. $\endgroup$ – Hot Licks Mar 5 '18 at 13:35
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    $\begingroup$ If anyone knows of an animation that shows the Earth and the moon both moving correctly, I'd love to see that. Seems like the Earth is always stationary. $\endgroup$ – JPhi1618 Mar 5 '18 at 15:54
  • $\begingroup$ Found this video which shows the motion pretty well. Also the Wiki article on barycenter has some basic animations. $\endgroup$ – JPhi1618 Mar 5 '18 at 16:37
  • $\begingroup$ @JPhi1618 You'd need a pretty high resolution frame to show it clearly, as the scale of the Earth's motion is quite small compared to the separation. $\endgroup$ – dmckee Mar 5 '18 at 18:23
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The mass of the Earth is about 81 times that of the Moon and the distance between their centres of mass is typically 384 400 km, so the centre of mass will be about 1/82 of this distance from the centre of mass of the Earth, which is about 4700km.

The orbits are not too far from circular, so the Earth travels about $2 \pi$ times that each month, so a little under 30000km.

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  • $\begingroup$ Why is it 1/82, and not 1/81? $\endgroup$ – Strawberry Mar 5 '18 at 13:11
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    $\begingroup$ @Strawberry because both the moon and Earth are off-center from the COM, but the Earth-Moon distance measurement is simply between bodies. So if you take the Earth's off-center distance as the baseline ("one unit") and the Moon's off-center distance is 81 times that, and the total of all 82 "units" is the Earth-Moon distance (center-to-center of course). $\endgroup$ – Asher Mar 5 '18 at 13:26
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    $\begingroup$ It feels like everything is spinning. My head hurts. I need to go lie down for a bit... ;-) $\endgroup$ – Strawberry Mar 5 '18 at 14:21
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    $\begingroup$ @Strawberry, the fact that Earth wobbles that much is a little off-putting. I feel like every "moon rotating around the Earth" model or animation I've ever seen is a lie. $\endgroup$ – JPhi1618 Mar 5 '18 at 15:50
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    $\begingroup$ There's a second lie in most of the figures you've seen that somewhat counteracts the failure to show the wobble of the earth. If you have seen an old LP record, the hole in the middle is roughly the size of the earth if the moon's orbit is the rim of the record. On that scale, the wobble would certainly be noticeable, but small compared to the whole scene. $\endgroup$ – Mark Foskey Mar 5 '18 at 17:21

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