Red giant Habitable zone

I was looking a bit at the evolution of Stars and one thing I sometimes see floated is this idea that in the far flung future, when the sun has ballooned into a red giant and the earth burned to a crisp, the solar system's habitable zone has moved far out enough that Saturn's moon Titan has become a pretty nice place to live.

There are some books that work with this idea, having some sort of weird alien species develop on titan long after all life has been extinguished from Earth. Stephen Baxter's book on Titan is a good example.

What I want to know is whether or not a Red giant could be counted upon to provide a suitable habitable zone. Now I understand that Red giants are not long lived, so we'd probably only be looking at around 1 billion-1.5 billion years max that it would maintain itself before giving up the ghost. Can it sustain a stable Habitable zone over that time? I understand that there are stages where it will inflate and deflate in size and ones where it will blast off huge amounts of matter which I assume would send things screwy and would probably sterilize a world before anything interesting happened, but could there be a sweet spot of about a billion years of relative stability when looking at a Red giant formed from a star about sun size or smaller?

I'm imagining some sort of world in the outer solar system that may develop life up to maybe a Eukaryotic stage (being generous here) in a subsurface Ocean before its star puffs out and a habitable zone expands to where it is, allowing liquid water and atmosphere to exist on the surface. At this point might it be conceivable for photosynthesis to emerge, oxygenation of the atmosphere, Multicellular life of increasing complexity, maybe culminating in an intelligent race before the clock runs out? Is this conceivable with a red Giant, would life emerging billions of years before mean that they would be ahead of the curve of some the developments on earth, and would a red Giant possibly be able to offer stable enough conditions over a long enough period for complex life to evolve?

Thanks!

posted over 9 years ago

Khwarezm‭

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