The age a cooling Earth gives

The markers that move because atoms are unequal followed a diffusion experiment measured in years inside a bar of brass. The summer that reaches the cellar in December used the same equation on the scale of a season, finding the depth at which the temperature cycle lags by six months. Between them they suggest a much older application, one of the most consequential and controversial in the history of physics. Heat escaping from the surface of the Earth is measured in every mine, and in the 1860s it was already clear that the temperature rises as miners go deeper, by something like one degree per 30 metres. In a solid object cooling from the outside, that gradient has a precise relation to how long the object has been cooling. A geothermal gradient is a clock.

William Thomson read that clock in 1862, in a paper with the plain title “On the secular cooling of the Earth”. His method was mathematically straightforward, coming directly from Fourier’s 1822 treatment of heat, and it produced an age of about 100 million years with a wide range. Over the following decades he defended it and then shrank it, and he used it to attack the assumption of the geologists and Charles Darwin that the Earth had been around for immensely longer. It has become a famous error. Few people, though, are told what the error actually was, since the usual story, about radioactivity, is only half of the answer.

A cooling planet, read from the top #

Take a half-space of rock at a uniform starting temperature $T_0$, with its surface kept suddenly at zero. A half-space is an excellent approximation for depths much less than the Earth’s radius, and it has an exact solution for all time:

$$T(z, t) = T_0\,\mathrm{erf}\left(\frac{z}{2\sqrt{\kappa t}}\right),$$

where $\kappa$ is the rock’s thermal diffusivity. The profile has only one length in it, $\sqrt{\kappa t}$, so at different times it is the same curve at a different scale.

Cooling reaches downward as the square root of time. After a million years it has reached about 60 km. After a hundred million years it has reached several hundred kilometres. Below that, the rock has not yet noticed that the surface exists, and it has stayed at its original temperature. Conduction is so slow on this scale that even now most of the rock in a conductive Earth would never have felt the surface. The equation that only runs forwards found this behaviour in the same equation with a different meaning: a random walker’s distance goes as the square root of time, which is why a thing that cools by conduction takes four times as long to go twice as far.

Differentiating the error function at the surface gives the gradient there,

$$\left.\frac{\partial T}{\partial z}\right|_{z=0} = \frac{T_0}{\sqrt{\pi\kappa t}}.$$

Everything Kelvin needed to know is in this one line. The gradient falls as the inverse root of time. Solve for $t$:

$$t = \frac{T_0^2}{\pi\kappa g^2}.$$

Three numbers go in, and an age comes out.

The number Kelvin got, and how firm it was #

Kelvin took one degree Fahrenheit per fifty feet for the gradient, about 36 K/km, from a survey of Scottish mines and from others. He took for the rock’s diffusivity about $1.2 \times 10^{-6}\ \text{m}^2/\text{s}$, from his own measurements on Edinburgh rock with thermometers buried in the ground. The starting temperature had to be supposed. He picked 7,000 °F (3,870 °C), near the melting point of rock as then estimated for high pressures, on the sensible grounds that the Earth had once been molten and solidified with its interior near its melting point. The formula gives about 97 million years. Allowing for what he knew about the uncertainties, he stated a range of 20 to 400 million years. Later, as data on rock melting improved, he revised the figure downward, and in 1897 he settled on 20 to 40 million years.

The formula makes the age depend very strongly on its inputs. It goes as the square of the gradient and the square of the temperature. A factor of 2 in gradient gives a factor of 4 in age, as the plot shows on its logarithmic scale. It was a real concern, and Kelvin allowed for it, since the geothermal gradient varies from place to place by about a factor of three. Neither could the geologists do anything about this. Their own estimates for how long it would take rivers to carve valleys and sediments to accumulate ran to hundreds of millions of years at the least, and Darwin in the first edition of On the Origin of Species estimated about 300 million years for the erosion of the Weald, in a passage he subsequently withdrew under pressure. They had qualitative arguments and no way to attack a differential equation.

At the time, Kelvin’s position looked like hard physics against soft estimation, and in a sense that is exactly what it was. His calculation of the age of the Sun, by the same reasoning about how long gravitational contraction could fuel it, reached a similar answer. The ball of gas that heats up as it cools deals with that Kelvin–Helmholtz timescale, which remains a correct physical quantity, though its application to the Sun’s age was wrong for a clearly stated reason. For the Earth the two independent estimates seemed to reinforce each other, which is why this reasoning had such force for forty years.

What a moving interior does to the answer #

The textbook version says that the discovery of radioactivity overturned this. In truth, an objection had been raised eight years before, and it needed no new physics at all. John Perry, an engineer who had been Kelvin’s own assistant, wrote to Nature in 1895. Suppose the interior of the Earth conducts heat much better than its surface. In that case the deep rock can supply heat upward much faster than conduction in a solid. The top part of a body is what controls the surface gradient, and the heat being fed into its base would slow down the fall in that gradient very greatly. Perry’s mechanism could be either higher conductivity in hot rock or, as he noted, “fluidity”: convection.

The numerical integration in the figure does what he did with a pencil, in a slightly more modern form. It solves the heat equation on a stretched grid for a lid of 65 km, with Kelvin’s own diffusivity, above an interior whose diffusivity is raised by a constant factor, and it was first checked against the uniform case, where it must return Kelvin’s number, and does within a few per cent. With an interior only ten times better at carrying heat, the age rises to 379 million years. At a hundred times, it is about 3 thousand million years. A factor of a thousand gets it to more than 5 thousand million, near what happens when a sphere’s whole heat capacity begins to be exhausted. All of those figures are from a lid with the same surface gradient as a mine shows. Note how the curve flattens: the gradient hardly changes for a very long time after the initial period, because the lid is in a steady state with a nearly constant supply of heat from below.

A fluid interior was physically possible, though it was hard to demonstrate, since seismic waves showed the mantle to be rigid on short time scales. This is the same paradox as the liquid that remembers discussed, the idea that a material can be solid in response to fast forces and fluid in response to slow ones, with mantle rock flowing at centimetres a year. Philip England, Peter Molnar and Frank Richter examined the history in 2007 in GSA Today, in a paper with a pointed title, “John Perry’s neglected critique of Kelvin’s age for the Earth”. They redid the calculations, showing that Perry’s reasoning alone was sufficient to reach an age of 2–3 thousand million years. Kelvin himself was polite and replied that he could not disprove it. Only much later, when plate tectonics was accepted in the 1960s, did the required convection become a measured part of the Earth.

The right mathematics, in the right place #

The story has a surprising final section. Kelvin’s mathematics turned out to be perfectly right for something he never knew existed: an oceanic plate.

At a mid-ocean ridge, hot mantle rock rises, at a temperature of about 1,300 °C, and spreads sideways as a plate. Each piece of the plate begins with a uniform temperature, near the surface, and thereafter cools from above, carried off at a few centimetres a year. It is a half-space cooling from the surface. Replace Kelvin’s Earth with a moving slab, and his equations apply, with the plate’s age on the ocean floor taking the role of time. The heat flow is

$$q = \frac{kT_m}{\sqrt{\pi\kappa t}},$$

about 480 mW/m² divided by the square root of the age in millions of years. Measurements of heat flow on the ocean floor are consistent with this out to about 80 million years, past which there is a floor at about 48 mW/m², mostly because convection underneath has begun to resupply heat to the plate. There is a second consequence. A cooler plate is denser, by thermal contraction, and so it sinks into the mantle. That predicts an ocean depth that increases with the square root of age, and that is what is observed, with a ridge at about 2,500 m and a depth increasing as about 350 m times the square root of the age in millions of years. Barry Parsons and John Sclater showed this in 1977, and it is often quoted as one of the clearest confirmations of plate tectonics. Its derivation could be done on the back of Kelvin’s 1862 paper.

So the method was correct and was applied to the wrong object. The Earth as a whole does not cool like a solid sphere. A part of its surface does, and it has been doing so repeatedly at the ridges, in the form of plates, whose sinking back into the mantle carries the cold down and sustains the convection that destroys the original calculation. A plate is about 100 km thick at its thickest, about what the integrations give for 80 million years of conduction. Its base is at the depth at which Kelvin’s calculation stops being applicable.

Most of the Earth’s heat loss happens exactly there, through young ocean floors with their high gradient, not through continents. The continental gradients Kelvin used are set in part by radioactive heating within the crust, and this is a further reason why Scottish mines were a dubious place from which to deduce an age. It was also a fact about the upper crust not known to anyone in his time.

What radioactivity did, and did not, settle #

Pierre Curie and Albert Laborde reported in 1903 that radium gives off a steady heat, and within a year Ernest Rutherford was speaking at the Royal Institution, in front of an elderly Kelvin, and suggesting that this removed the limit. In his later recollection he tactfully noted that Kelvin had limited the age of the Earth only on condition that no new source of heat was discovered, and that a new source was the evening’s subject. That is a much-repeated story, with some uncertainty as to its details. It is also true that radiometric dating, based on the same decays, then measured the age directly, and with the chain that runs at its slowest member’s rate in place of a thermometer, it settled on a figure of 4.5 thousand million years by the 1950s. The half-life that chemistry can change describes one of the subtleties in those clocks.

Radioactivity’s contribution to the heat budget, however, is only about half. In 2011 the KamLAND experiment in Japan measured electron antineutrinos coming from the Earth, products of uranium and thorium decay in the rock, and it found about 20 TW from those. With roughly 4 TW more estimated from potassium, not directly measured, about half of the 47 TW leaving the Earth remains. That remainder is still primordial heat, being lost as the planet cools. So in part, Kelvin’s calculation was correct: the Earth really is a cooling object, and about half of its present heat flow really does come from cooling, as he supposed. A heating source alone does not remove the cooling problem, for the cooling continues, and a planet supplied with half its present flow would still have lost its gradient long ago unless something other than conduction were carrying the heat. Radioactivity gave a means of dating and a useful source of heat, while the answer to the actual physical issue came from the stirring.

This has something to teach about how science corrects itself. Kelvin’s position fell for reasons that were better on the second inspection than the story told at the time, and they were not those that people gave. A wrong assumption that looks like a detail can hide inside a very clean calculation, and the clean calculation can carry the wrong assumption for a very long time. Precision about one part of an argument does not transfer to the rest.

The mathematics that was already old #

Fourier’s work was the source of the error function solution. His 1807 memoir on heat, published as a book in 1822, introduced the method of separating a function into sinusoids that decay independently, which is how the equation was first solved. Kelvin as a teenager had published work defending Fourier against a criticism, and his thesis on the distribution of heat in solids was among his first papers. How far a molecule gets reaches the same square-root law from a molecular direction, which is the modern way of seeing it. So does the walk that comes home, in terms of a random walker’s return. Kelvin thought of the equation as a macroscopic fact about solids, and used it as such. Its molecular meaning was developed only near the end of his life.

Another point of interest concerns the planet’s structure. The melting temperature of rock rises with pressure, so a cooling molten Earth would solidify from the bottom up, or at some intermediate depth, and not from the surface. Kelvin knew this, and that was why he supposed the interior to be at its melting point and then to have solidified in a way that left no molten interior. Seismologists have since established that the outer core is liquid iron, and the shadow that shows the core is liquid gives the evidence. His guess of a solid interior was thus partly wrong for that reason as well, though the mantle above is the part that matters for the surface.

What the integration assumes #

The half-space solution assumes uniform rock properties, a uniform start, a fixed surface temperature and no internal heat sources, and Kelvin knew all of those were approximations. The two-layer integration replaces all the complexity of mantle convection with a single higher diffusivity below 65 km, which is Perry’s device. It is a crude model of what is really transport by flowing rock, and it gives only an order-of-magnitude guide to how a convecting interior affects a lid. Actual mantle convection has a temperature-dependent viscosity, phase changes, plates and subducting slabs. Radioactive sources are not included in the integration either, so the ages drawn are for a purely cooling body. A proper calculation includes those and the changes in convective vigour over time, and it would give a more complicated answer.

The integration is also stopped when the gradient matches the present value, which has no physical significance beyond being Kelvin’s own criterion. The ocean-floor figure uses typical values for conductivity and diffusivity, whose true values vary with temperature, and the measured heat flow near young ridges is lower than the curve, since hydrothermal circulation carries away some of it in water, and that is not recorded by a probe in the sediment. The heat budget has uncertainties of several terawatts, and the mantle’s share of potassium remains estimated.

Still open: how fast the mantle is cooling now #

The present split between radiogenic heating and secular cooling has large consequences for the mantle’s past. If heat production has been falling, as it must with decaying isotopes, the mantle must have been hotter and more vigorous in the Archaean, and the plate-driven heat loss rate ought to have been much greater. Simple extrapolation backward with the current heat flow and a standard scaling law for convection yields a catastrophic “thermal runaway” at about a billion years ago, where the mantle becomes implausibly hot. That is contrary to the evidence from ancient rocks. To resolve this, some have proposed that plate tectonics ran more slowly or in a different form in the past. Whether plate tectonics was operating for the first 2 thousand million years at all is under active debate.

Better measurement of geoneutrinos should narrow the radiogenic share. New detectors such as JUNO in China, and those in Canada and Italy, may be able to separate crustal contributions from the mantle’s, and that would fix the planet’s overall composition. It would be a remarkable piece of closure if a detector could measure directly how much of what Kelvin was trying to measure from the bottom of a mine is in fact a remnant of the Earth’s original heat.

A half-space cooling from a uniform start has a surface gradient $T_0/\sqrt{\pi\kappa t}$, and Kelvin’s three numbers give 97 million years. The calculation was correct, and it assumed a still interior. An interior a hundred times better at carrying heat gives about 3 thousand million years with the same surface, as Perry pointed out in 1895. The same curve fits the ocean floor out to 80 million years, because a plate really is what Kelvin thought the Earth was. Radioactivity gives about half the heat and has settled the dates; the stirring settled the rest.