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Because we can calculate the core's pressure, the temperature question is answered by determining the melting point of metallic iron at the pressure of the inner-outer core boundary. To do this, we must worry about the purity of the iron as impurities lower the melting temperature. We know that there is a little nickel--iron meteorites, which come from the cores of fragmented planets in the asteroid belt have about 10%. We suspect that the other impurities in the outer core are mostly sulfur and oxygen.
How can we determine the melting temperatures of these metals at core pressures? One of the most useful devices for doing this is a diamond anvil cell. In it, we can squeeze a small sample between two diamonds up to pressures as high as the earth's core and then heat the sample by shining a powerful laser beam right through one of the diamonds to see if the sample melts. Doing it isn't so hard. What is hard is knowing the temperature accurately and knowing for sure that what you see is really melting. It has taken years to perfect this technique but now it is able to give us some pretty good melting temperatures of the materials that we believe make up the inner core, outer core, and mantle, the rocky layer over the core that makes up most of the earth. When we combine all of this information, we come up with a temperature of about 3550C (6400F) at the core-mantle boundary and 4500C (7200F) at the inner-outer core boundary. This is not the final answer - geologists are still working on the problem - but it seems to be the best answer we have right now.
The heat from the earth's interior is escaping through the surface of the earth. It is small compared to the heat from the sun's rays but it can be measured and it is consistent with other measurements like those described above.
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