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The masses of the Earth and Moon are in a ratio of about 81:1, so they can be thought of as orbiting around their common barycenter which is inside the Earth, but about 2/3 of the way to the surface.

When two bodies have somewhat comparable masses, like the Earth-Moon system, and a semimajor axis is reported to many significant figures (e.g. 384,399 km), is this more like the distance between the centers of the two bodies, or is it more like the Moon's distance from the Earth-Moon barycenter about which it orbits?

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The semi-major axis is the average distance between the barycenters of the two bodies, in this case the Moon and the Earth.

We can double check that:

The Earth-Moon distance is known to submeter accuracy. If you have access to the DE or EPM or INPOP ephemerides, you can do all the calculations.
For example, let’s do the calculations with the SPICE library and the de430.bsp data file.

Consider 1 orbit of the Moon that starts at 2018-07-27 00:00:00.000 UTC and ends at 2018-08-23 06:14:14.849 UTC (27.25989 days).

The semi-major axis is 383906.157 km (the average distance from the Earth’s barycenter to the Moon’s barycenter; I calculate it by numerical integration). While if I calculate the distance between the Moon’s barycenter and the EMB, I get 377687.223 km.

If I randomly sample 10000 orbits in the current year (10000 random starting points for the orbit), I get that the semi-major axis ranges from 383684.426 km to 385612.081 km and its average is 384396.002 km which is quite close to the 384,399 stated in Wikipedia.

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    $\begingroup$ Please feel free to edit further, I just wanted to give an idea of what I meant. This seems conclusively the answer, thank you very much! $\endgroup$ – uhoh Sep 7 '18 at 11:00
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    $\begingroup$ I added that the sma is the average distance. $\endgroup$ – Cristiano Sep 7 '18 at 11:03

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