Solar expansion and habitability
Is there a feasible setup where the Sun has expanded enough to make the Earth uninhabitable, but some of the other rocky bodies in our solar system (past Earth) become more reasonable colonisation prospects?
For the purposes of this hypothetical, let's hand-wave the why and focus on different solar radii and what they do for habitability through the system. I suspect most of the useful range sits between what we have now and substantially less than red giant, but let's work that through!
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A decent proxy for habitability and long-term colonizability is the effective temperature of the planet - essentially the surface temperature. A planet's effective temperature scales as $T\propto (L/r^2)^{1/4}$, where $L$ is the luminosity of the star and $r$ is the planet's orbital radius. We want our planet, moon, asteroid, etc. to have an effective temperature of roughly Earth's. Knowing the distance between a body and the star makes it easy to calculate how bright the Sun needs to be to make that body habitable.
Let's take an example: Mars. Mars lies 1.52 AU from the Sun. If we plug that in, we see that it could reach Earth-like temperatures (assuming identical albedo and greenhouse effect - more on that later) when the Sun reaches $L\sim2.3L_{\odot}$. That's going to happen at the end of the Sun's subgiant phase - a portion of time after it's left the main sequence but before it becomes a red giant. To take another example, consider Europa, an oft-discussed place for life to arise in the future. Europa requires the Sun to reach a luminosity of $L\sim27L_{\odot}$, in the early stages of the red giant branch. The same holds for any of the moons of Jupiter, and Saturn wouldn't be too far behind.
Towards the far end of its life, when the Sun ends the red giant phase and enters a brief portion of its life we refer to as the asymptotic giant branch, it will reach peak a peak luminosity of $L\sim5000L_{\odot}$. That's . . . well, large. At this point, a body about 70 AU from the Sun would receive roughly the same flux as Earth does right now, and therefore have a similar surface temperature.
The point is, if you wait long enough, virtually any body in the Solar System you'd want to colonize will reach Earth-like temperatures.
Up to this point, we've ignored two things: the albedo of the planet (how well it reflects and absorbs light) and the greenhouse effect. Thinner atmospheres mean less of a greenhouse effect, so for many bodies, our estimate is a little bit off. Still, it's a decent enough approximation, for our purposes - give or take a factor of a few.
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