Maybe you remember that viral internet graphic from a few years ago showing how the other 7 major planets in the Solar System could fit neatly between the Earth and the Moon. I’m not sure if I’m supposed to be surprised by that, but it’s a mildly interesting coincidence that the distances are so close.
But under what circumstances do they fit? The Earth-Moon distance varies considerably, and some planets’ equatorial diameters are considerably larger than their polar diameters.
The planetary diameters are well known, and easy to look up. You can quibble about whether we should include the atmospheres of the rocky planets, but that doesn’t make enough difference to matter.
For a non-professional-astronomer like me, though, figuring out the Earth-Moon distance is harder. Lots of sources will tell you a range of numbers for the “apogee” and “perigee” of the Moon, but they’re not so clear about what exactly those numbers represent. Are they measured from the surface of the Moon to the surface of the Earth? Or the center of the Moon to the center of the Earth? Or the center of the Moon to the barycenter of the Earth-Moon system? Those are all different things.
After some research, I think it’s supposed to be the Moon center to Earth center. But I did a few calculations as a sanity check, and now I’m more confused than ever.
I would have expected that the average of the perigee and apogee (for some given orbit of the moon), minus the distance from the Earth’s center to the barycenter (the perigee and apogee overlap in this segment, and we only want to count it once), should equal the moon’s semi-major axis.
How do I get the perigee and apogee for some orbit of the moon? I don’t really know. And maybe I’m on the wrong track anyway. My guess is/was that, roughly speaking, the minimum perigee would occur in the same orbit as the maximum apogee.
I’ll use 356400km as the perigee, and 406700km as the apogee. The barycenter is on average about 4671km from the Earth’s center (it would be a little larger than average in this case, but not significantly). The Moon’s semi-major axis is 384399km. So we should have:
(356400 + 406700)/2 − 4671 ≈ 384399 376879 ≈ 384399
That’s off by more than 7500km. Something is seriously wrong.
I give up. For now. I won’t try to figure out what I did wrong. I’ll assume the usual numbers for perigee and apogee do accurately represent the distance from the center of the moon to the center of the Earth.
I plugged the data into a spreadsheet, and came up with this:
By my calculations, if we use average values for everything, they probably do not quite fit, but it’s so close that I can’t be sure.
- At the minimum Earth-Moon distance, the planets never fit.
- At the average Earth-Moon distance, the planets only fit if we turn some of them sideways.
- At the maximum Earth-Moon distance, the planets always fit.