How long to freeze a planet?
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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