Radioactivity as a heat source for planets

[clint]

I recently read this wikipedia article about plate tectonics.
What caught my attention was the part about radioactivity as a heat source and its implications for the age of the Earth:

The discovery of radioactivity and its associated heating properties in 1895 prompted a re-examination of the apparent age of the Earth,[4] since this had previously been estimated by its cooling rate and assumption the Earth's surface radiated like a black body.[5] Those calculations had implied that, even if it started at red heat, the Earth would have dropped to its present temperature in a few tens of millions of years. Armed with the knowledge of a new heat source, scientists realized that the Earth would be much older, and that its core was still sufficiently hot to be liquid.

The difference in estimations for the age of the Earth before and after the discovery of radioactivity as a heat source are huge (tens of millions vs. billions of years).
Is radioactivity really such a big heat source for a planet?

[geonuc]

Is radioactivity really such a big heat source for a planet?

Yes, it is. At least for this one.

[ngc3314]

And geophysical neutrinos have provided a direct measure of the current decay rates of ²³⁸U and ²³²Th from the KamLAND experiment. Some of my colleagues working on neutrino experiments give me funny looks when I point out that they have now detected neutrinos from three astrophysical sources, instead of just the two they always talk about.

[Ken G]

The difference in estimations for the age of the Earth before and after the discovery of radioactivity as a heat source are huge (tens of millions vs. billions of years).
Is radioactivity really such a big heat source for a planet?

No, it really isn't, your intuition is right on the mark here-- the importance of radioactivity is highly oversold. It would indeed alter the estimates and allow the Earth to be older, but not nearly enough to resolve Kelvin's problem. By far the largest error in the infamous Lord Kelvin estimate had to do with the rate that energy leaves the Earth's core. This is a subtle story, it was originally told by one of Lord Kelvin's assistants, John Perry. I saw it in an article in GSA Today by England, Molnar, and Richter, in January 2007.

First of all, you have to realize that Kelvin's beef with the age of the Earth had nothing to do with the interior temperature, which he had no way of knowing. It had to do with the temperature near the surface-- it was awfully warm. Not right at the surface, that's controlled by the Sun, but down in the mines-- it gets hot in the mines, and that is what Kelvin's age estimate was all about. Given what the core temperature of the Earth must have started out being (the temperature where rock molecules can stick together), if the Earth were billions of years old, the surface layers should have cooled much more than they have-- assuming that all the heat transport in the interior works the same way it does at the surface. That was Kelvin's key false assumption-- the absence of radioactive heating sources was a relatively minor issue and including it does not fix the problem.

What really happens is, the Earth's interior carries heat by convection, which is much more efficient than the conduction that Kelvin could see at the surface. Convection implies a much shallower temperature gradient from surface to core. So when Kelvin neglected convection, he basically got a model where after 5 billion years, the core stays extra hot, but the surface gets extra cool. This latter part is what was not observed, and forces the shorter age estimate. But what really happens is, convection makes the core cooler than Kelvin thought, providing a huge heat source for keeping the surface layers as warm as is observed after 4.7 billion years. We thus have a cooler core, and a warmer surface, than Kelvin's calculation imagined, so the observed warm surface is consistent with the older age. If you bottle up the core heat, you are forced by surface observations to conclude the Earth is younger than it is, but if you let the core heat out (via convection), you allow an old Earth to still have warm surface mines.

On the other hand, radioactivity in the Sun is of course hugely important, and coincidentally, its neglect gave Kelvin a similar (wrong) age for the Sun as for the (wrong) age of the Earth. That unfortunate coincidence only served to strengthen his infamous resolve on this issue. But radioactivity in the Earth plays nowhere near the importance that it has in the Sun, so is pretty much a complete red herring in the whole "age of the Earth" saga.

[mugaliens]

Given what the core temperature of the Earth must have started out being (the temperature where rock molecules can stick together), if the Earth were billions of years old, the surface layers should have cooled much more than they have-- assuming that all the heat transport in the interior works the same way it does at the surface. [I]

Unless we were slammed by something that melted us to the core all over again. You know, like the Moon...

What really happens is, the Earth's interior carries heat by convection, which is much more efficient than the conduction that Kelvin could see at the surface.

...or like convection, yes. The mantle is reasonably efficient at that...

[Ken G]

Unless we were slammed by something that melted us to the core all over again. You know, like the Moon...

That actually wouldn't matter much, as the age of the Moon is thought to be very close to the age of the Earth.

[clint]

Great answer, Ken, thank you!
Wikipedia is a great source for a first approach, but as this shows again, it's important to always double-check the details...

On the other hand, radioactivity in the Sun is of course hugely important, and coincidentally, its neglect gave Kelvin a similar (wrong) age for the Sun as for the (wrong) age of the Earth. That unfortunate coincidence only served to strengthen his infamous resolve on this issue.

Interesting how in this case two unrelated errors ended up as two pieces of evidence for the same wrong conclusion.

[Ken G]

Interesting how in this case two unrelated errors ended up as two pieces of evidence for the same wrong conclusion.

Yes, it's a cautionary tale of how hard it is to ever really be certain of something in science, especially if a different scientific branch reaches a different conclusion. And kudos to your horse sense for recognizing that radioactivity in the Earth could not explain a factor of 100 error-- that story is essentially the academic equivalent of an urban legend (an "academic legend?").

[mugaliens]

That actually wouldn't matter much, as the age of the Moon is thought to be very close to the age of the Earth.

So two 1,000 def F blobs slamming into one another at 25,000 mph wouldn't heat them up to, say, 2,000 deg F? The numbers are hypothetical, Ken G, but the concept remains valid.

Regardless, we know from the production of high-temperature igneous rocks such as komatites that the Earth's internal temperature was significantly higher than it is today (the Earth is too cool to product komatites, today). Based on this and other well-known factors, including ratios of major heat-producing isotopes and their half-lives, we conclude that 20% of Earth's current internal heat is residual from planetary accretion, while the remaining 80% is due to the heat produced from radioactive decay. We also know that three billion years ago the heat produced by decay was about twice what it is today.

That 80% factor is a handy fact to keep in mind when talking with anti-nuclear folks.

[Ken G]

So two 1,000 def F blobs slamming into one another at 25,000 mph wouldn't heat them up to, say, 2,000 deg F? The numbers are hypothetical, Ken G, but the concept remains valid.

Which do you think released more energy, the formation of the Earth itself, or the formation of the Moon that followed shortly thereafter? Then add the fact that radioactive sources represent an even larger energy than both of those, but still not nearly enough to resolve Lord Kelvin's problem if his assumptions about the heat transport had been accurate.

Regardless, we know from the production of high-temperature igneous rocks such as komatites that the Earth's internal temperature was significantly higher than it is today (the Earth is too cool to product komatites, today). Based on this and other well-known factors, including ratios of major heat-producing isotopes and their half-lives, we conclude that 20% of Earth's current internal heat is residual from planetary accretion, while the remaining 80% is due to the heat produced from radioactive decay.

And this was basically Clint's point in asking the OP-- a factor of 4 more radioactive heat could not possibly explain a factor of 100 error in the age of the Earth. And it doesn't, that was not the primary problem in Lord Kelvin's estimate.

[George]

And geophysical neutrinos have provided a direct measure of the current decay rates of ²³⁸U and ²³²Th from the KamLAND experiment. Some of my colleagues working on neutrino experiments give me funny looks when I point out that they have now detected neutrinos from three astrophysical sources, instead of just the two they always talk about.

"Down to Earth" astrophysics. What in Earth is not astrophysics?

We thus have a cooler core, and a warmer surface, than Kelvin's calculation imagined, so the observed warm surface is consistent with the older age.

Thanks. This clears up some mental mud due to the erroneous views given in general readership science books, which use radioactivity to explain the age discrepancy. It isn't too surprising they assume this to be true given that it was Rutherford, apparently, who initiaed such an older view for the Earth due to radiation. [This also came in a speech he gave with Kelvin in attendance; new clout emerging among old clout.]

[mugaliens]

Which do you think released more energy, the formation of the Earth itself, or the formation of the Moon that followed shortly thereafter?

The formation of the substantially larger mass of the Earth, of course.

Then add the fact that radioactive sources represent an even larger energy than both of those, but still not nearly enough to resolve Lord Kelvin's problem if his assumptions about the heat transport had been accurate.

You're skirting the 83% I mentioned previously, aren't you?

And this was basically Clint's point in asking the OP-- a factor of 4 more radioactive heat could not possibly explain a factor of 100 error in the age of the Earth. And it doesn't, that was not the primary problem in Lord Kelvin's estimate.

Yes. So please detail the problem in Lord Kelvin's estimate.

[Ken G]

The formation of the substantially larger mass of the Earth, of course.

Correct, so already we see the Moon is irrelevant.

You're skirting the 83% I mentioned previously, aren't you?

No, that's where the factor of 4 I mentioned came from (before you said 80%, if it's 83%, then it's a factor of 5).

Yes. So please detail the problem in Lord Kelvin's estimate.

See post #4. The bottom line is, Kelvin's low estimate of how fast heat could leave the interior worked to bottle up heat in the core, causing the near-surface mines to be much cooler than they actually are, forcing him to imagine a much younger Earth.

[Ken G]

It isn't too surprising they assume this to be true given that it was Rutherford, apparently, who initiaed such an older view for the Earth due to radiation. [This also came in a speech he gave with Kelvin in attendance; new clout emerging among old clout.]

Yes, that's probably what happened-- new physics trumping the old physics, but all the while the geology record was clear enough on the point.

[neilzero]

It seems a bit strange to me that we assume only two sources of interior heat for Earth, even though the others may be less than 1% each. Some that come to mind are gravitational compression, internal tidal heating by the Moon and Sun, the rare neutrinos that are absorbed instead of passing completely though Earth and other as yet not discovered phenomenon. Surely it is not rigorous that 17% to 20% of the internal heat budget is radioactive disintegration of nucleus?
Other than neutrinos, and Earth's surface temperature, why is the sun a large factor in calculating the heat loss from the core of Earth? Would the core be hundreds of degrees k cooler, if the surface temperature of Earth was 10 degrees k? Neil

[Ken G]

There were a few questions there, but most of them sound like very minor considerations. The one you ended with is interesting-- you might think that the way the Sun warms the surface of the Earth would affect the heat flow from the interior, but it does not. That's because the interior of the Earth heats up well above what the Sun warms it to, by the time you get down into a deep mine. That's not very far down, so basically the heat transport from the interior is completely ambivalent to anything the Sun is doing to the Earth's surface.

[George]

Yes, that's probably what happened-- new physics trumping the old physics, but all the while the geology record was clear enough on the point.

Another thought I had on the misdirection favoring a radioactive explanation was a likely underestimation of the core temperature. I don't know what value Kelvin started with only that he assumed a molten beginning.

Since the energy content increases with the 4th power of temperature, a relatively small error in underestimating the core temperature would make a very big error in heat content. Although the transfer rate would be greater with greater core temperatures, an assumption of conductivity for the transfer would also greatly affect the time because conductivity is a linear transfer. As you said [per the reference], he did not consider convective flow, which would indeed make a significant difference. It certainly does for the Sun. [Edit: This is way wrong.]

[The Earth's outer core temperature is about that of the Solar limb temperature (photosphere): 5000K. The inner core is about 7000K.]

Another curious thought is how important sizzle is to a viewpoint. Radioactivity has sizzle.

[EricFD]

And geophysical neutrinos have provided a direct measure of the current decay rates of ²³⁸U and ²³²Th from the KamLAND experiment. Some of my colleagues working on neutrino experiments give me funny looks when I point out that they have now detected neutrinos from three astrophysical sources, instead of just the two they always talk about.

LOL Bill! You know, I never thought of that before. But that certainly would affect the neutrino count! hehe

Eric

[Ken G]

Another thought I had on the misdirection favoring a radioactive explanation was a likely underestimation of the core temperature. I don't know what value Kelvin started with only that he assumed a molten beginning.

That could be a contributor, but I'll bet he had it right to at least a factor of 2, so that's even less of an issue than the radioactivity.

Since the energy content increases with the 4th power of temperature, a relatively small error in underestimating the core temperature would make a very big error in heat content.

The 4th power of T only appears when the energy content is primarily radiative. That is not the case in stars like the Sun, and it's also not the case in the Earth's core, so the energy content only scales like the 1st power of T. All the Earth temperature estimates will scale in proportion to the initial core T.

Another curious thought is how important sizzle is to a viewpoint. Radioactivity has sizzle.

Yes, and it's easier to explain in ten words or less, so those are all reasons why the story got mangled.

[George]

That could be a contributor, but I'll bet he had it right to at least a factor of 2, so that's even less of an issue than the radioactivity.

Considering he was eventually off by a factor of 200 on the Earth's age, this seems low.

The 4th power of T only appears when the energy content is primarily radiative. That is not the case in stars like the Sun, and it's also not the case in the Earth's core, so the energy content only scales like the 1st power of T. All the Earth temperature estimates will scale in proportion to the initial core T.

Yikes, what an error! Obviously, I've been out in the Sun much too much! [At least it's a light error. (bad pun, too.)]

Yes, and it's easier to explain in ten words or less, so those are all reasons why the story got mangled.

Yes, brevity adds sparkle to sizzle. Can't beat that.

[Ken G]

Considering he was eventually off by a factor of 200 on the Earth's age, this seems low.

But the problem was never really about the core temperature, it was about the contrast between the core and the deep mines. Kelvin used too little coupling between them, so got too much contrast-- forcing him to blow the age of the Earth because the mines we so warm. Correct coupling raises the temperature in the mines so requires an old Earth to avoid having mines that humans couldn't even live in, whereas Kelvin wondered why the mines were warm at all. It's not that Kelvin thought the core should have cooled off by now, that's only part of the mangling of the story.

[George]

But the problem was never really about the core temperature, it was about the contrast between the core and the deep mines.

Yes, I did understand your prior explanation.

It is a bit confusing to be off by a factor of 200 for the age by only being off in temperature by perhaps a factor of only 2. Yet your estimate is logical since he did understand temperature increases caused by gravitational expenditures. [I think he did similar calculations for the Sun.]

Again, as you say, the key is the convective heat transfer difference keeping the mantle much warmer than simple conductivity would produce.

[Ken G]

It is a bit confusing to be off by a factor of 200 for the age by only being off in temperature by perhaps a factor of only 2.

It is indeed a subtle issue, it has to do with the fact that the constraint being applied deals with the temperature in the deep mines, which are at a distance only perhaps a ten millionth of the Earth's radius. So we are taking data that is only one ten millionth as deep as we'd like, and extrapolating the rest-- that opens up a huge potential for false extrapolation if a key parameter isn't right (here the thermal conductivity). That made Kelvin reason that if the Earth was billions of years old, the core T would have to be orders of magnitude larger than is reasonable, not just a factor of 2.

[StupendousMan]

It is indeed a subtle issue, it has to do with the fact that the constraint being applied deals with the temperature in the deep mines, which are at a distance only perhaps a ten millionth of the Earth's radius.

Hmmm. Radius of Earth is about 6.37 x 10^(6) meters = 6370 km.

One millionth of that is ... about 6.4 meters.

Ten millionths of that is ... about 64 meters.

I suspect that even in Kelvin's time, the deepest mines were closer to 640 meters deep than 64 meters deep, so the proper factor of ten to choose would be "about a hundred millionth".

Four orders of magnitude is still quite enough to cause serious errors when extrapolating, of course.

[Ken G]

I suspect that even in Kelvin's time, the deepest mines were closer to 640 meters deep than 64 meters deep, so the proper factor of ten to choose would be "about a hundred millionth".

Yeah, I wasn't doing the calculation very carefully. It was just a lot of extrapolation, that's my point. Whenever one sees such a small part of the "elephant", one needs to beware.

[George]

I want to say that the mines were surprisingly deep in Kelvin's day, so his temp. gradient was somewhat accurate for such a small portion of the planet. I tried to find a reference but found, instead, this article that shows that Perry on or before 1895 realized that the convective transfer would be the more logical choice -- a floating mantle was possible based on numerous lines of evidence known prior to this time -- and would allow an age of over 2 billion years. He published his work in Nature (is this the paper you mentioned Ken?), but he never could convince Kelvin, and he, apparently, did not push it too hard out of respect for Kelvin's views.

[Ken G]

He published his work in Nature (is this the paper you mentioned Ken?

Yes, although I saw it in a more recent article that updated the earlier work into more modern language.

[George]

I was surprised to see it contemporary to Kelvin. The real story is often more amazing than an assumed one. Perry was ATM but was right. [The negative views towards Darwin's theory contributed much to this problem, no doubt.]

[Ken G]

Perry was ATM but was right. [The negative views towards Darwin's theory contributed much to this problem, no doubt.]

Even that depends on which "M" you consider-- the geologists were almost unanimously inclined to take Perry's explanation as the right one, it was only the astrophysicists, for whom Kelvin was a superstar, who would see Perry's position as ATM.

[publiusr]

So, for instance, could a small world still have have tectonic activity if nothing but radioactive substances were left behind--pulsar planet style?