A science question

The other mechanism that's listed in most discussions is fractionation; heavier elements are still sinking ot the core and that si another source of heat.

The Earth is so hot on the inside because heat is still being generated there, and because the mechanisms for transporting heat away from the Earth take time. I have no idea when, if ever, the core will be cool to the touch; we may be engulfed by the Sun turning into a red giant (in five billion or so years) before the core cools.There's a long and interesting history of the answer to your questions.Isaac Newton speculated on how long it would take a globe of red-hot iron, the size of the Earth, to cool. He pretty much guessed, and came up with 50,000 years.Georges-Louis Leclerc, Comte de Buffon, made an attempt in the late 1700's. He had a foundry build him iron spheres of different diameters (1/2 inch to 5 inches). He heated them to white heat and measured the cooling time. He found a roughly linear relationship between diameter and cooling rate. Assuming the Earth was like an iron sphere and extrapolating to the Earth's size, he came up with 96,670 years (the precision is obviously not justified). Later he repeated the experiment with rock spheres and including the ffect of the Sun's heat, and modified his result to 74,832 years. This wildly wrong result was actually very important in the development of modern science and geology. From "The Age of the Earth", G. Brent Dalrymple, Stanford University Press, 1991, page 31:

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Buffon's major contribution to science, however, was much more important and lasting than his several ages for the Earth. Buffon believed and demonstrated that nature was rational and could be understood in terms of ordinary physical processes force, motion, chemical reaction, heat, and other forms of energy operating over geologic time. Invoking unique, supernatural, or extraordinary causes to explain natural history was to Buffon both unnecessary and unproductive. Thus, Buffon was the first to construct a history of the Earth based on the application of observable processes to explain the effects produced by like processes in the past. He also was the first to apply laboratory experimentation to the problem of the age of the Earth, and in doing so became one of the founders of geophysics. Finally, Buffon clearly separated the appropriate roles of theology and science in explaining nature science was perfectly capable of answering the questions of how and when; the question of why was reserved for theology. Although Buffon's age for the Earth and much of his detailed history of the planet as set forth in Epochs are now known to be incorrect, the techniques he developed for deciphering Earth history through application of the law of cause and effect are a keystone of the modern scientific method.

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The most famous early result about the Earth's cooling was from William Thomson (later Lord Kelvin) in 1862. He used measured properties of rocks, measurements of temperature from wells and mines, and the relatively new science of thermodynamics to calculate a cooling time of 98 million years. He realized that his data was inadequate, and estimated that the cooling time was no less than 20 million years and no more than 400 million years.This was a tremendous problem for geologists and "evolutionists". Lord Kelvin was the Einstein of his day, and his results and pronouncements were held in high repute. But the results of geologists studying the Earth and "evolutionists" studying the fossil record strongly suggested the Earth was much older. There were lively debates.Later papers by others refined Kelvin's calculations, and probably the "best" estimate based on the assumptions and knowledge of the day was 24 million years, by Clarence King in 1893.So, we can see that if the Earth is as old as we say it is today, then there must be some source of heat other than the influx of solar energy and the initial heat content when the Earth formed. And that is ... radioactivity. Well, mostly. Dalrymple puts it well on pages 46-47:

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But the most serious flaw in Kelvin's method is that its basic assumptions are invalid. The thermal budget of the Earth is far more complex than Kelvin, King, Chamberlain, or any nineteenth-century scientist could possibly have imagined. One complication is that there are far more sources of heat within the Earth than were known in Kelvin's time. One important source, recognized by Kelvin, is primordial, i.e. heat left over from the formation of the Earth. Radioactivity, gravitational energy from compaction, chemical energy from the segregation of the iron—nickel core, and mechanical energy from meteoritic impacts during the period that the Earth was still sweeping up large quantities of material from its orbital path all contributed to this primordial heat. These sources probably generated enough heat to raise the temperature of the outer part of the Earth to near the melting point within 100 to 200 Ma of its formation. Much of this primordial heat has not yet escaped from the Earth.

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A second and probably the most important source of continuing heat production is radioactivity. This heat is generated by the radioactive decay of primarily uranium, thorium, and potassium contained in the rocks of the Earth. Although the exact distribution of these radioactive elements within the Earth is not well known, there is no problem in constructing reasonable models that attribute most or even all of the heat now flowing outward from the Earth to radioactive decay.

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In addition to primordial heat and heat from radioactivity, contraction of the Earth due to cooling and release of gravitational energy as the core grows may also be important contributors to the Earth's thermal budget. Thus, even though the relative importance of the various sources of heat is poorly known, there is little doubt that they are sufficient to permit an Earth many billions of years old.

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Another factor that invalidates Kelvin's approach is that convection is a more important mechanism for the loss of heat from within the Earth than conduction. About three-fourths of the heat lost from the Earth leaves through the ocean basins. Virtually all of this heat comes from the mantle and is brought close to the ocean floors by convection. Conduction brings the heat to the surface, where it is lost into space. Approximately two-thirds of the heat lost from the continents is generated within the continental crust by radioactivity. The remaining third is brought to the base of the crust by convection, whereupon it is conducted upward through the crust to the surface. But even though the balance of mechanisms by which heat is transferred from the interior of the Earth outward is known in a semi-quantitative way, current knowledge is insufficient to permit an exact description of heat loss from the Earth.

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Finally, there are other poorly known factors, including the exact composition and structure of the Earth as well as the relevant physical properties of the rocks at various depths, such as conductivity, specific heat, and viscosity. Thus, the thermal history and budget of the Earth are very complicated, and our knowledge of the relevant details remains far too inadequate to permit any valid estimates of the age of the Earth from thermal calculations.

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We know somewhat more now that we did in 1991, but Dalrymples remarks are still pretty much on the money.

Rutherford spoke before the Royal Insitution in 1904, and recalled:

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I came into the room, which was half dark, and presently spotted Lord Kelvin in the audience and realized that I was in for trouble at the last part of my speech dealing with the age of the earth, where my views conflicted with his. To my relief, Kelvin fell fast asleep, but as I came to the important point, I saw the old bird sit up, open an eye and cock a balefule glance at me! Then a sudden inspiration came, and I said Lord Kelvin had limited the age of the earth, provided no new source (of energy) was discovered. That prophetic utterance refers to what we are now considering tonight, radium! Behold! the old boy beamed upon me.

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Copied from The ‘demise’ of Kelvin.

Vacuum essentially prevents the transfer of heat by conduction, which requires molecules that physically touch each other (at least occasionally). Nothing prevents transfer of heat by radiation (not the radiation produced by radioactivity, the radiation produced by all things that are above absolute zero). Note that we receive quite a bit of heat, via radiation, from the Sun ... through the vacuum of space. That warm feeling on your skin at the beach in the summer? Radiation.

Not to date, but stay tuned.They tried with the Mohole, but that was a flop.I'm pretty sure that the deepest we've gone is 5.6 miles, in the KTB Borehole. In Germany, where that hole was drilled, the crust is about 25 miles thick (The Earth's Crust).

I replied: "Not to date, but stay tuned". I was just assuming that someday we'll do it. Well, someday seems to be coming. From A Sea Change in Ocean Drilling:

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IODP may even take another shot at penetrating the Mohorovicic discontinuity. With the lubricating drilling mud circulating through its riser, Chikyu should be able to reach 6 kilometers and into the upper mantle. Such a hole would help refine knowledge of the structure, composition, and physical properties of the oceanic crust.

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See also The Deep-Sea Drilling Vessel Chikyu report.

It's been done, except it was done before the discovery of radioactivity. The equations were both more and less sophisticated than yours; yours don't include the rates at which energy is conducted/convected from the interior of the Earth to the surface, but the "E" term for the enregy generated in the Earth was left out. Both turn out to be very important in the calculation. The variation of thermal conductivity with depth is also important, and that's tough to measure. You might want to look back at Message 11 in this thread.It turns out that there's a scanned copy of the chapter from which I took message 11 on the Web, minus some tables: CHAPTER TWO, Age of the Earth by G. Brent Dalrymple. It includes a lot more than I did. The Inner Workings of the Earth may also be interesting.I doubt that anybody has the resoures to get the values needed for a highly accurate calculation.

<*cough*>convection<*cough*>

Note the link in Message 40.

"L" is not a constant, it's a function of the temperature of the Earth's surface and atmosphere, which is itself a function of the rate of heat transfer from the interior (and other things). The rato of heat transfer from the interior depends on lots of complex stuff. One of the several inadequacies of early models is that they didn't realize that the the mantle, although solid, is plastic enough to transfer a lot of heat by convection, which is much more efficient than conduction. The amount of heat lost through the Earthss surface is very different for the continents than it is for the ocean, mostly because oceanic crust is much thinner than continental crusst.

Yup .. but you're going to have to wait a few hundred thousand years or more to get an accurate set of measurments for predicting what happens next, and maybe not even by then. We know it's non-constant and at least approximately cyclical. There's a whole group of periodic changes that together are called the Milankovitch cycles,and the period of the overall sum of those cycles is way up there. The Earth's orbit changes over time; the largest frequency component is at 100,000,000 years with strong components at 43,000 years and 413,000,000 years. See Milankovitch cycles and ice ages, especially figure 2.2. The precession of the Earth's axis and the tilt of the Earth's axis from the orbital plane also vary cyclically.AFAIK climatologists think that Milankovitch cycles are far from the whole story, but they are definitely important. There's feedback mechanisms that amplify changes.

I was making the implicit assumption that we were talking about suficient precision to predict average global temperatures within a few degrees. Sure sounds wrong to me. The Earth is generating a significant amount of its own heat by radioactive decay. If it weren't, we'd be a lot colder, and the core of the Earth would no longer be warm. Lord Kelvin showed that in the late 1800's. I know I've posted stuff about this before, but here's a good link again: No webpage found at provided URL: http://thermo.gg.utk.edu/courses/Ge475/Dalrymple.htmlCHAPTER TWO, Age of the Earth by G. Brent Dalrymple, the "Cooling of the Earth and Sun" heading. I question the precision of that number, I doubt that we know the Earth's energy balance to one part in 100,000. If you want to assume that number is correct and extrapolate that it will stay the same for millions of years (or that it has stayed the same for millions of years) yes, you could do a calculation ... but it would take some fancy dancin' to convince me, for one, that such extrapolation is justified!