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If some advanced race were to make a solid opaque shell around our Sun, say around Mercury's orbit, something like a Dyson Sphere, how long would it take for Earth to freeze over and end most life?

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Here, it is useful to treat the planet as a black body. The Stefan-Boltzmann law states that a black body will have a luminosity $L$ directly related to its temperature $T$ and radius $R$: $$L=4 \pi R^2 \sigma T^4$$ where $\sigma$ is the Stefan-Boltzmann constant. We have to be careful picking $R$ and $T$, though, because the atmosphere and climate have to be considered. I would suggest using the mean radius of Earth and the mean temperature of Earth - not just the surface temperature.

This shows us that luminosity is a function of temperature. Luminosity is expressed in units of watts per second, meaning that it is a energy over time: $$L=\frac{U}{t}$$ The $U$ here is the thermal energy of the Earth. Now you can write time as a function of temperature: $$t=\frac{U}{4 \pi R^2 \sigma T^4}$$ and pick the temperature you want.

There are other corrections you have to make, though - for example, taking the atmosphere into account, as well as the fact that the Dyson Sphere, too, will radiate.

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    $\begingroup$ +1 for pointing out that the Dyson Sphere radiates. There are several really clever extensions to the dyson sphere which take advantage of that detail to harness more heat energy. $\endgroup$
    – Cort Ammon
    Jan 11, 2015 at 20:15
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HDE always has great mathematical answers. I suspect that most life here would be be Popsicle's before the Dyson Sphere is completed. We can change the climate on earth by putting more CO2 in the air to raise temps, and we can cool it off by filling it with particulates, I believe at least one summer was missed because of a volcanic eruption.

So Going whole days, weeks or months without direct sunlight would be catastrophic for the environment. Plants use oxygen and emit CO2 when they are not actually producing energy from sunlight. This can happen even during the day if the cloud cover is poor enough. This of course would help hold in the heat, but a couple weeks are going to be very bad for all flora and our plants will start to die off.

Of course the biggest issue would be any civilization that could create a Dyson sphere, ring or swarm would use up all the raw material already present orbiting the star in question, this would include the earth, no reason for this little planet to go to waste!

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