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So my friend and I had a debate. He stated that we don't have to worry about the sun consuming the Earth ─ we'll already be broken apart by then. He states that as the sun expands, its Roche limit will also expand. The Earth will have been broken apart long before the sun touches its orbit.

I'm arguing that the sun isn't changing mass (ignoring Mercury, Venus, and solar winds) and its gravitational force won't change, ergo eventually the boundary of the sun will be further than the Roche limit. The Earth will stay intact as it gets cooked.

Which, if either, of us is correct?

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You are correct, and your friend is not.

So long as the Sun remains spherically symmetric, its gravitational field can be replaced with the field of a point mass at its centre of mass, which is what determines the Roche limit (so, in particular, it fixes both the gravitational field and the gravitational field gradient at any position you care to name). The radial extent of the mass distribution is irrelevant so long as the spherical symmetry is retained.

I should also say I find it dubious that the Earth would break apart even if it did skim the Roche limit of some other body, because the Roche limit sets the break-apart point for systems that are bound gravitationally, i.e. for loose piles of rubble held together by their own gravity. This is not the case for the Earth, which is held together by strong chemical bonds in the rock of the mantle and core. Given a strong enough gravitational field gradient you can imagine the Earth getting torn apart, but this will happen a good bit further in than the Roche limit, I should think.

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    $\begingroup$ And due to the conservation of angular momentum the sun's rotation will slow as it expands do it will become more spherically symmetric in the red giant phase. $\endgroup$ – dmckee Sep 18 '17 at 15:27
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    $\begingroup$ If the Earth were to spiral into the towards the Roche limit, first the tides would increase in size, alternately flooding and then draining land. We would have devastation orders of magnitude higher than the worst tsunami happening twice a day. The concept of a "continent" would cease to be relevant, as a continental shelf would simply be an area that spends slightly more time not covered in water. Once the Earth went past the Roche limits, the oceans would eventually be ripped away Earth. After that, it would probably take a while before the tidal forces were able to destroy the Earth. $\endgroup$ – Acccumulation Sep 18 '17 at 22:26
  • $\begingroup$ based on the projected time when we believe this would happen, would the Earth still have water and an atmosphere present? $\endgroup$ – kevchadders Sep 19 '17 at 11:01
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    $\begingroup$ "...the Earth, which is held together by strong chemical bonds in the rock of the mantle and core..." is wrong! The earth would have to be one big and cold diamond to have a comparable chemical bonding energy as her gravitational binding energy. $\endgroup$ – klanomath Sep 19 '17 at 15:15
  • $\begingroup$ @klanomath Either way, the chemical binding energy is nonzero, which means that the Roche limit does not apply. But maybe you can provide a quantitative estimation of the chemical binding energy and show that the change is negligible? Absent that, I don't see the point of your comment. $\endgroup$ – Emilio Pisanty Sep 19 '17 at 15:18

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