I have read that our Sun will increase in size to a red giant. But after it goes through the helium flash it will contract somewhat. Then after that grow again to be even larger before finally losing its outer layer as a planetary nebula and becoming a white dwarf. I've seen estimates that the red giant phase will consume Mercury, Venus, and probably Earth. (Even with Earth's expanded orbit due to the Sun losing mass to the solar wind.) My question is how large will that first stage be prior to the helium flash? Which planets, if any, will it engulf?
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Low mass stars like the Sun do become very large prior to He ignition in the core. The exact value depends a bit on models for mass loss from the extended atmosphere (e.g. Guo et al. 2016), but estimates of 250 times its current size are possible (Schroeder & Smith 2008; Spiegel & Madhusudhan 2012).
At this radius both Mercury and Venus are engulfed. The mass loss from the Sun means Earth would orbit a bit further out than now. Nevertheless, it too would be engulfed in Schroeder and Smith's model at that point, because tidal drag dissipates energy from the orbit and it "falls" into the Sun very quickly.
Interestingly, the Sun achieves its maximum size at the tip of the RGB, rather than in the AGB phase.
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$\begingroup$ So the red giant phase is where the Sun reaches it's maximum size, not the asymptotic red supergiant phase? That's surprising, I assumed the description supergiant would have meant bigger! I had wondered if the Earth would survive the red giant (not Mercury or Venus though) and the horizontal branch, but be engulfed during the asymptomatic phase. The sci fi thought experiment I always have is an advanced species using en.wikipedia.org/wiki/Star_lifting to prevent Earth being engulfed and through shading of the sun protect life there. Thank you! $\endgroup$ Nov 10, 2020 at 14:50
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2$\begingroup$ @BrooksNelson I suspect the detail of which gets bigger - the RGB or AGB depends exactly on how much mass is lost during the RGB. Different models may yield different results. The whole calculation is as well mass-dependent, so whilst this may be true for the Sun it may not be true for stars of different mass or composition. $\endgroup$– ProfRobNov 10, 2020 at 14:57
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$\begingroup$ @Brooks The Earth will become uninhabitable long before the Sun leaves the main sequence. It will be rather unpleasant here in a billion years or so, and the planet will be pretty well baked by around a billion years before the Sun starts its red giant phases. See en.wikipedia.org/wiki/… for details. $\endgroup$– PM 2RingNov 10, 2020 at 15:51
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$\begingroup$ @PM 2Ring Yes, which is why I mentioned shading the Earth from the Sun. But if we use star lifting techniques to reduce the Sun's mass now, even at a low rate. The Sun would stay on the main sequence longer, develop a smaller red giant, and Earth's orbit would move further out. All of which would extend life's time on Earth by many millions of years and prevent Earth from being engulfed by the Sun. $\endgroup$ Nov 12, 2020 at 2:07