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One billion years into the future and the Sun has swollen in size and it is now not possible to live on Earth due to the heat. Mankind has relocated to Mars where the temperature is more favorable now than on Earth. My question is how much time has this gotten us before we can't live on Mars any longer because the Sun is dying ? The next step I guess will be to relocate to another Solar System ?

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  • $\begingroup$ It would be easier to shield Earth from radiation IMO. $\endgroup$
    – Mithoron
    Feb 10 '15 at 20:59
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As it gets older, the core of the Sun starts to fill with Helium ash. This increases the average mass per particle and hence the core temperature must increase to maintain the pressure. This increases the nuclear reaction rate and the Sun becomes more luminous, at almost a constant surface temperature.

The habitable zone is controlled not only by the luminosity of the star, but also by the atmosphere of the planet. It's doubtful Mars would ever become "habitable" in that sense (without our intervention), but what I will assume is that you want the equilibrium temperature to be warmer than 263K, but say cooler than 303K (i.e. between -10 and 30 Celsius).

The details of finding the blackbody equilibrium temperature can be found here. The formula we need is $$ T = \left(\frac {L_{\odot} (1-a)}{16\pi \sigma D^2}\right)^{1/4},$$ where $L_{\odot}$ is the luminosity of the Sun at any time, $D$ is the distance to the planet, $a$ is the albedo and $\sigma$ is the Stefan-Boltzmann constant. $T$ is in Kelvin.

I will assume that the average albedo of Mars is 0.25 (though it varies considerably with wavelength, depends on icecap coverage etc.) and that $D=2.27\times10^{11} m$. We can then rearrange the formula above to give the luminosity of the Sun for a given equilibrium temperature. $$L_{\odot} =21.3\ \pi \sigma D^2 T^4$$ This means that for $T>263 K$, then $L_{\odot}>9.35\times 10^{26}\ W$, but for $T<303\ K$, we require $L_{\odot}<1.65\times10^{27}$\ W$. That is the Sun's luminosity should be somewhere between 2.44 and 4.30 times its current luminosity.

The next step is to look at a stellar evolutionary model for a star like the Sun.You can generate one here. I find that the Sun will have a luminosity in this range from ages 8.9 billion (at the cool end) to 10.0 billion years (at the hot end).

Obviously you can play with the numbers (upper and lower temperature bound, albedo) to get different answers (the big assumption was to just use the equilibrium temperature, but an atmosphere could warm things up a bit), but you should be able to follow this prescription using whatever numbers you wish.

Incidentally, at the "cool end" of the Mars calculation - the Earth's equilibrium temperature would be 315K (42 Celsius).

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The Sun is not dying when the Earth supposedly becomes uninhabitable, it is just getting hotter as it evolves on the main sequence (core hydrogen burning, the Sun has something like 5+ billion years before reaching the red giant stage of its evolution)). It will continue to get more luminous and at a guess if Mars could be made habitable it would remain so for millions, possibly another billion years. But then do you really think humanity (or rather our descendant civilization/s) if it survives for another billion years will need a planetary surface to survive?

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  • $\begingroup$ Yes, I am aware of this also ,that it would be simpler for us just to move into an artificial habitat in space but don't you think that it would be nicer to live on a Planet ? $\endgroup$
    – Peter U
    Feb 10 '15 at 20:49
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One billion years into the future and the Sun has swollen in size and it is now not possible to live on Earth due to the heat.

The problem isn't so much the Sun swelling in size. The key problems are that the Sun's luminosity increases over time and that the Earth is covered with oceans. The Earth will become uninhabitable long before the Sun turns into a red giant. Ever increasing luminosity will result in an increased surface temperature, which in turn will result in increased atmospheric water vapor. Right now this isn't a problem because increased atmospheric water vapor means more clouds, which increases albedo, which counters the effect of increased luminosity.

There is another effect to consider, however. Water vapor is an extremely powerful greenhouse gas. At some point, increased surface temperature will increase atmospheric water vapor to the point where the increased greenhouse effect increases the surface temperature, which increases vaporization even more, and so on. A loop! And a loop of the very worst kind, a positive feedback loop.

When this will happen is subject to debate, perhaps as short as 500 million years (Goldblatt 2013), perhaps a billion years (Kasting 1988), or perhaps as long as 1.5 to 2 billion years (Leconte 2013) in the future.

Mankind has relocated to Mars where the temperature is more favorable now than on Earth.

Speculating what humanity will do or become 500 million years from now is a bit much. It's not science; nobody will live to see whether there speculations turn out to be true. But it is a bit fun. Why would we relocate to Mars? Why not relocate Earth instead (Korycansky 2001)? We might eventually have to worry about Mars getting in the way. In that case, simply move Mars, too!


References

Goldblatt, Colin, et al (2013), "Low simulated radiation limit for runaway greenhouse climates," Nature Geoscience, 6.8:661-667.

Kasting, James F. (1988), "Runaway and moist greenhouse atmospheres and the evolution of Earth and Venus," Icarus, 74.3:472-494.

Korycansky, D. G., Gregory Laughlin, and Fred C. Adams (2001), "Astronomical engineering: a strategy for modifying planetary orbits," Astrophysics and Space Science, 275.4:349-366.

Leconte, Jérémy, et al. (2013), "Increased insolation threshold for runaway greenhouse processes on Earth-like planets," Nature, 504.7479:268-271.

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  • $\begingroup$ Yes, this would be so much better to just knock Mars out of the way and move the Earth into its orbit farther from the Sun. How would this be done though ? $\endgroup$
    – Peter U
    Feb 10 '15 at 20:51
  • $\begingroup$ @PeterU - Read the referenced article. The answer to your question is there. $\endgroup$ Feb 10 '15 at 21:26
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If you postulate some things that began as humans are still around in 1,000,000 years, I would also guess that way off topic of astronomy (but not for exploration) that "people" will be safely basically immortal in virtual worlds and have sought out the safest place to do their thinking and playing. Perhaps headed outside the galaxy or the plane of the galaxy to wherever the least junk is flying about.

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  • $\begingroup$ +1 I doubt the outside of the galaxy though. Two things to consider here. One is the strong radiation from the center of the galaxy, which is shielded by the starts in the plane of rotation, so the plane is the safest place. The other is entropy. Self evolving systems can exist only by dissipating a concentrated energy from a low-entropy source. This is the reason life evolved on the Earth as the right balance between the Sun and space. The same principle on a larger scale would keep the super-intelligent creatures of the future somewhere inside the galaxy(s) in the best entropy balance. $\endgroup$ Feb 23 '18 at 0:44

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