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My question is simple:

Given the gravitational relationship between Earth and the moon, how wide is Earth's roche limit? How close does the moon have to orbit before getting disintegrated by Earth's gravitational pull, and how far does it have to orbit before being free from Earth's orbit entirely?

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    $\begingroup$ The Roche limit and formulae for its calculation are given at en.m.wikipedia.org/wiki/Roche_limit $\endgroup$ – Rob Jeffries Feb 3 '17 at 8:41
  • $\begingroup$ I'm more concerned about the Seveneves possibility :-) $\endgroup$ – Carl Witthoft Feb 3 '17 at 12:33
  • $\begingroup$ @RobJeffries It wasn't straightforward. $\endgroup$ – JohnWDailey Feb 3 '17 at 15:07
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    $\begingroup$ In what way? $R_{\rm roche} = r_m (2M/m)^{1/3} = 9500$ km. There are uncertainties depending on exactly how the Moon's mass is distributed and its rigidity. Is that what you are asking about? How accurately do you need the answer? $\endgroup$ – Rob Jeffries Feb 3 '17 at 15:18
  • $\begingroup$ How close does the moon have to be to Earth before getting disintegrated, and how far does it have to be to Earth before drifting away to freedom? $\endgroup$ – JohnWDailey Feb 3 '17 at 15:21
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A little reading and effort is recommended and the Wikipedia page is straight forward enough, if you're not completely math-phobic.

But if you want an answer, Rob Jeffries 9,500 km is right, however, that's center point to center point. Surface to surface, that would only be less than 2,000 km, which would be crazy close. That close the Moon would take up about (guessing) close to 50% of the sky, maybe 70 degrees or so, when directly overhead. But that 2000 km number doesn't take into account tidal stretching or rotation, as the Moon would almost certainly be tidally locked and rotating in sync with it's orbit like it does now and the Earth would be similarly tidally locked and have a tidal and gravitational bulge in the direction of the Moon.

Taking into account rotation, that pushes it out to the cube root of 3, not 2 or about 14% further, but that's center to center. The surface to surface distance would be increased to about 3,000 KM. Still insanely close. Any orbital eccentricity would increase instability though, so for a safe distance, Earth to Moon where the Moon wouldn't break apart at all, but it would still stretch measurably, I wouldn't want it any closer than about 5,000 or 6,000 km from the Earth's surface. That's a ballpark guess with tidal forces, tidal locking, solar perturbations and orbital eccentricity all in mind and even there, some loose moon dust and rocks might regularly blow off and shower onto the Earth. 6,000 KM (surface to surface) is less than twice the Moon's diameter, so it would still be enormous in the sky, but I think that's about the distance where the Moon would hold together in orbit. The math says even closer, but that doesn't take into effect various issues like eccentricity, rigidity, etc. It gets rather hard to calculate with rock solid accuracy taking into account all the details. But, point is, the Moon would need to be stupid/gonzo/cray close before it begins to break apart. Some 50-60 times closer (surface to surface) than it is now.

Now as far as escape, the math of the Hill Sphere or Sphere of Influence is pretty straight forward, but the long-term stable zone is about 1/2 to 1/3rd of the Hill Sphere calculation, some 500,000 to 750,000 km. Somewhere in there or perhaps just beyond that range the Moon probably becomes unstable and escapes the Earth's orbit becoming a very large and potentially very dangerous near earth object. That doesn't mean that the Moon would just fly away past 750,000 km. It would remain in orbit for multiple orbits, hundreds, maybe thousands even past that distance, but past the stable region it would probably destabilize over time and escape eventually.

The Moon is perhaps as much as 70% of the way towards where it could eventually escape the Earth, but that final 30% is a LOT. The moon isn't going anywhere anytime soon. It's still comfortably inside the long-term stable region and should remain orbiting the Earth for billions of years.

Hope that helps.

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