Wikipedia:Reference desk/Archives/Mathematics/2021 November 8

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November 8

When will the moon touch Mars's orbit??

The moon is moving away from the earth at a speed of 1.5 inches a year. This means it will take 42,240 years for it to reach a mile. How long will it be before it touches Mars's orbit?? Georgia guy (talk) 19:43, 8 November 2021 (UTC)

The question is covered here. The upshot is that the sun will become a red giant and swallow up both the Earth and the Moon long before the Moon leaves Earth's orbit. --RDBury (talk) 22:06, 8 November 2021 (UTC)

It says 6 billion years; that would make a distance of 142,045 miles assuming the current speed. Georgia guy (talk) 22:34, 8 November 2021 (UTC)

Actually we are not really sure about that. See Future of the Earth: the Earth may survive the red giant Sun. But solar atmospheric drag would be so much that the Moon's orbit would decay and it would crash back into the Earth.

And even if that did not happen, tidal evolution in the Earth-Moon system alone would stop when the system reached mutual tidal locking, i.e. the current state of the Pluto-Charon system. The Moon wouldn't reach Mars' distance. Thereafter, tides from the remnant Sun would slowly take away angular momentum from the system until, once again, the Moon crashed into the Earth. So the ultimate fate of the Moon, assuming that the Sun doesn't swallow the Earth whole, seems to be uniting with the Earth in a sad bookend to its origin. Double sharp (talk) 20:12, 9 November 2021 (UTC)

This is really a science question, not a math question. The mechanism driving the Moon away from the Earth is the fact that the Earth completes its daily rotation in less time than the Moon takes to complete its monthly orbit around Earth. Considering the effect of tidal forces and friction, the result is a transfer of angular momentum from the Earth to the Moon. Without interference from the expanding Sun, this process would end when the Earth's rotation had slowed enough to match the Moon's orbital period. See Orbit of the Moon#Tidal evolution. This would not remove the Moon from orbiting around the Earth, it would just make the orbit somewhat larger and slower. It would never reach the orbit of Mars. --184.145.50.17 (talk) 23:04, 8 November 2021 (UTC)

This sort of thing always reminds me of Mark Twain's remarks on extrapolation:

In the space of one hundred and seventy-six years the Lower Mississippi has shortened itself two hundred and forty-two miles. That is an average of a trifle over one mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oolitic Silurian Period,' just a million years ago next November, the Lower Mississippi River was upwards of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.

--Trovatore (talk) 02:02, 9 November 2021 (UTC)