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We can make a rough estimate by looking at the moon. The coldest parts of the moon are always in the shade (near the poles inside craters), and are about -240 degrees Centigrade, or -400 Fahrenheit. If the earth had no sunlight, it might approach this temperature, but there are some other factors that would make the earth's temperature in darkness higher than the moon's.

The molten core of the earth releases its own heat energy, and the atmosphere would help to trap that energy and keep the surface warmer than the -240C on the moon. Of course given enough time this heat would eventually escape, the core would freeze, and the earths temperature would drop to close to -240 C.

The atmosphere may also become liquid at that temperature, but only near the surface of the earth at the correct pressures.

Source:http://www.newscientist.com/article/dn17810-moon-is-coldest-known-place-in-the-solar-system.html

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