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Recently some exoplanets were discovered which are thought to be the leftover cores of giant planets that had survived a star becoming a Red Giant and then shrinking back to a subdwarf.

If giant planet cores can last the lifetime of a Red Giant, then maybe smaller planets can survive at least for a while.

When the Sun becomes Red Giant, if the Sun engulfs the Earth and Moon before they have evaporated, what will happen to the Earth-Moon orbit ? Will they be crashed into each other, or will they drift apart or will the orbit remain stable for a while before they are melted and evaporated away?

Edit: Newly found planets are 'roasted remains' BBC News, 21 December 2011

Edit: More planetary systems around white dwarfs (WD 0421+162 and WD 0431+126) that shows signs of planets being consumed by the stars. http://news.discovery.com/space/alien-life-exoplanets/hubble-discovers-planet-graveyard-130509.htm

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The problem with answering your question is that red giant stars don't have a well defined edge. It's a bit like Earth's atmosphere. The density gets gradually lower as you move away from the centre but it's hard to draw a line and say "this the edge". So when you say the Sun engulfs the Earth, it depends on how deep into the Sun the Earth gets and how long it stays there.

I'd guess you're talking about the planet orbiting V391 Pegasi, and this does seem to have originally orbited at about the Earth's orbit. However it's much much heavier than the Earth so there would have been more of it to evaporate and it would survive longer than the Earth. We don't know exactly what the planet's original orbit was, or how big V391 Pegasi got during it's red dwarf phase, and we don't know how much of the planet was evaporated.

I think the best that we can say is it neither proves the Earth-Moon system will be destroyed nor proves that it will definitely survive. It does suggest that the possibility exists.

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I've added a link to news story about discovery of planet remains. –  vtt Sep 18 '12 at 15:33
    
There's more detail at nature.com/nature/journal/v480/n7378/full/nature10631.html but this doesn't change the overall conclusion, which is that we don't know whether the Earth-Moon system would survive. –  John Rennie Sep 18 '12 at 16:31
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