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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?

Brandon Enright
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daniel
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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? – a06e May 28 '13 at 02:22
  • 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 02:24
  • @daniel But those spots are not on the planet's surface; surely they are elevated on some lab benchtop. – Kaz May 28 '13 at 06:34
  • @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

2 Answers2

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

In the time since I first wrote this, David Hammen provided a much more complete discussion of the geothermal heat budget. What you see here leaves off the latent heat of fusions at the inner core boundary and the conversion of gravitational potential to heat as the Earth settles out into density segregated layers.

Community
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dmckee --- ex-moderator kitten
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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. –  May 28 '13 at 02:23
  • 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 http://en.wikipedia.org/wiki/Natural_nuclear_fission_reactor . – anna v May 28 '13 at 03:35
  • Also that the heat content of Mars is under investigation http://en.wikipedia.org/wiki/InSight . also the moon http://onlinelibrary.wiley.com/doi/10.1029/JZ072i012p03301/abstract – anna v May 28 '13 at 03:38
  • @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 --- ex-moderator kitten May 28 '13 at 03:40
  • 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 03:42
  • I just read the abstract, but I would think that neutrinos from uranium decay would not exclude some uranium to fission? Are there further arguments in the publication? – anna v May 28 '13 at 03:49
  • It can't rule out fission on any kind of absolute basis, but it can put strict limits on the fission power (on-going chain reaction fission generates a higher energy neutrino spectrum than un-triggered decay). The claim is that there is not a large amount of fission going one in the inner core. Like I said, since those results people have been floating models that move the fission out or have it turning on and off. It would take a much bigger, much cleaner detector to be really definitive about those kinds of models. – dmckee --- ex-moderator kitten May 28 '13 at 03:52
  • @BenCrowell: FYI, Mars still has a molten core. – Mark Adler May 28 '13 at 05:55
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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.

Brandon Enright
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