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Does anyone know the amount of thermal energy that the Earth's mantle and core possess? I don't mean the maximum limit of electrical power we could generate with geothermal plants, but rather: if you took the Earth and magically cooled it down so that its temperature became homogeneously 0K, what would the change in the Earth's internal energy be? (ignoring the Sun)

If we don't have that data, does anyone have an idea how to perform an order of magnitude estimate?

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As a start, you could look up specific heats for the substances that make up the mantle, outer core, and inner core along with their temperatures and attempt to calculate the heat that would be lost if they were to be cooled to absolute zero. This neglects phase transitions and probably other subtleties...but it's a start. –  joshphysics Jan 23 '13 at 1:14
you might also start with this article scientificamerican.com/article.cfm?id=why-is-the-earths-core-so –  anna v Jan 23 '13 at 4:39
@joshphysics I thought about that, but I don't know how to calculate the mass of the mantle, outer core and inner core. I could use a state equation (we do know the volume, temperature and pressure) but I have zero idea what state equation to use for molten stuff under huge pressures and temperatures. –  Sandra Jan 23 '13 at 16:10

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The amount of thermal energy in the Earth's core and mantle is determined primarily by the temperature. The three-dimensional temperature field inside the earth is imprecisely known. One dimensional "Onion skin" models of the earth's interior are based upon empirical evidence from seismology, geodesy, and mineral physics, but there are almost certainly lateral variations in temperature which are important to understand.

The article mentioned in the comments can provide you with estimated temperatures at important discontinuities inside the earth. Connect these with constant gradients, or find an alternative model temperature profile.

You should be able to find some specific heat capacity measurements for periodite (mantle), perovskite, and liquid iron. To an order of magnitude these should be like $10^2$ - $10^3$ $\text{J} \text{ kg}^{-1} \text{K}$.

You'll also need the density profile, and can get that from a one dimensional model of the density of the earth. You might try Preliminary Reference Earth Model (PREM) or The Reference Earth Model Website. The latter reference models include three dimensional models.

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