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Assuming that the earth is spherical, that its temperature is continuous, and that some other more or less realistic conditions hold, we might think that the Earth's core temperature should be about the average of its surface temperatures.

This is not the case, as the core is hotter than all but a few spots on the surface. Can someone explain where the assumptions break down? Is it that the temperature is not static?

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The question says: "the core is hotter than all but a few spots on the surface." Are there spots on the surface of the Earth that are as hot as the core? –  becko May 28 '13 at 2:22
    
Yes, if we count the almost negligible artificial places in labs where these temperatures have been generated. If I left these out I felt someone would surely object. –  daniel May 28 '13 at 2:24
    
@daniel But those spots are not on the planet's surface; surely they are elevated on some lab benchtop. –  Kaz May 28 '13 at 6:34
    
@Kaz: Good point. I would tend to include anything within a few miles of the surface as being on it. After all, Mt. Everest is 39,000 feet above sea level, and the earth's diameter is about 7900 miles. It's an approximation. –  daniel May 28 '13 at 10:05
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up vote 11 down vote accepted

In the limit of very, very long time you can expect that situation to obtain, but

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Equilibration is an exponential-decay process, so it doesn't really matter much how high the initial temperature was. What matters much more is the rate of exponential decay. The reason the rate of equilibration is so slow is that the earth has a low surface to volume ratio. Bodies like Mars and the moon no longer have molten cores, because they have lower surface to volume ratios. –  Ben Crowell May 28 '13 at 2:23
    
I would also add that it is not only decays of long lived radioactive materials that are adding to the heat but possibly also fission of these en.wikipedia.org/wiki/Natural_nuclear_fission_reactor . –  anna v May 28 '13 at 3:35
    
Also that the heat content of Mars is under investigation en.wikipedia.org/wiki/InSight . also the moon onlinelibrary.wiley.com/doi/10.1029/JZ072i012p03301/abstract –  anna v May 28 '13 at 3:38
    
@annav The KamLAND and Borexino results convincingly rule out a steady central reactor of significant size contributing to the current geothermal heat. That said, I've seen proposals for non-central or intermittent reactors that can't be ruled out by the existing data. –  dmckee May 28 '13 at 3:40
    
I am just pointing out that they would increase the internal heat or even the errors, if measurements depend on locality. –  anna v May 28 '13 at 3:42
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Your assumption would be accurate in the extreme far future. The surface of a sphere after a long time has a roughly uniform temperature where the heat it absorbs balances with the heat it loses and the temperature would be uniform throughout.

With the Earth though, a huge amount of heat was generated when the planetary disk coalesced. Because the only (essentially only anyways) way the Earth can lose heat energy is through black-body radiation it takes a very long time to shed the excess heat from formation. The Earth is a very large sphere so it has a small ratio of surface area to volume ratio which makes the surface somewhat of a heat loss bottleneck.

Also, there are a few factors that dramatically slow the heat loss. The Earth's atmosphere traps a lot of heat captured by the Sun's radiation. The Sun is adding heat to the Earth so in order for the Earth to cool it must shed all of the heat it's absorbing as well as the extra heat from the core as it slowly makes its way to the surface.

Also, there are many radioactive isotopes in the Earth's core generating large amounts of heat through fission.

You should check out the Wikipedia article on the geothermal gradient.

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Thank you I am up-voting both answers as responsive and helpful and accepting the other only on the basis of time priority. –  daniel May 28 '13 at 2:21
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