Right Clock, Wrong Body

In 1862 Kelvin read the age of the Earth off the heat leaking out of it and got 98 million years. The Earth is about 4.5 billion. But his equation is the one geophysicists now use to date the ocean floor, so we handed it 17,940 real seafloor heat-flow measurements. It reads floor 40 to 60 million years old at 0.93 of its true age. On older floor it falls further and further behind, and on land it reads tens of millions of years almost everywhere. The clock was sound. It was timing the cold lid, not the planet.

The argument was the most respected number in Victorian science, and it was simple enough to do on the back of a letter. Go down a mine and it gets warmer, about one degree Fahrenheit for every fifty feet. So heat is flowing up and out. If the Earth began molten and has been cooling ever since, the rate at which it leaks now tells you how long it has been leaking: a body that has cooled for longer has a gentler gradient at its skin. Kelvin put the numbers into a solution Fourier had already written down, and the geologists, who wanted hundreds of millions of years for their rivers to cut and their sediments to pile up, spent forty years arguing against arithmetic they could not fault.

Everyone now knows he was wrong. What is less well known is how. The popular story says radioactivity was the missing heat source; that story is not right (section 5). And the equation itself was never wrong at all: a century later it turned up again as the standard description of the sea floor. This page puts the equation in your hands, then hands it a planet's worth of thermometers.

1. The clock

Here is Kelvin's recipe with his own three inputs. The starting temperature is his estimate for melting rock; the gradient is the "rough mean" from mines and wells; the diffusivity (how quickly heat spreads through rock, which he wrote as κ) he had worked out from Principal Forbes's thermometers buried in three places near Edinburgh. Change any of them.

With his values it says 97.5 million years, and he printed 98:

“If we suppose the temperature of melting rock to be about 10,000° Fahr. (an extremely high estimate), the consolidation may have taken place 200,000,000 years ago. Or, if we suppose the temperature of melting rock to be 7000° Fahr. (which is more nearly what it is generally assumed to be), we may suppose the consolidation to have taken place 98,000,000 years ago.”W. Thomson, “On the Secular Cooling of the Earth” (read 1862), §10, as reprinted in Mathematical and Physical Papers III (1890), p. 300

and he bracketed it, from the gentlest and steepest gradients anyone had measured:

“the consolidation cannot have taken place less than 20,000,000 years ago, or we should have more underground heat than we actually have, nor more than 400,000,000 years ago, or we should not have so much as the least observed underground increment of temperature.”ibid., §11

The dashed line is the whole trick. Carry today's surface gradient straight down and it reaches the starting temperature at a depth D; the age is just D² ÷ πκ, the time heat takes to diffuse across D. The clock does not know how old the Earth is. It knows how deep the cold has got.

2. Hand it a seafloor

New ocean floor is made hot, at the mid-ocean ridges, and then carried away from them, cooling from the top as it goes. That is Kelvin's problem exactly, with one difference: the floor has a birthday, and we know it, because the rock froze in the direction of the Earth's magnetic field and the field keeps flipping. Those stripes, dated, are the colours on the map below (Seton and colleagues' 2020 age grid). The white dots are every seafloor heat-flow measurement in the International Heat Flow Commission's 2024 database that lands on oceanic crust: 17,940 probes pushed into the mud by seventy years of research ships (the papers behind them run from 1954 to 2023).

Kelvin's clock needs a starting temperature, a conductivity and a diffusivity. We fixed them before looking, at round textbook values for the mantle: 1,300 degrees, 3.3 W per metre per kelvin, 1 mm² per second. Then the clock turns a heat flow q into an age: 431 ÷ q, squared, in millions of years. Click anywhere on the ocean.

3. What 17,940 probes say

Each faint dot below is one probe: across, the age of the floor from its stripes; up, the age Kelvin's clock reads from its heat flow. A clock that was simply right would put every dot on the dashed diagonal. The white circles are the medians in nine age bands, and the band edges, the parameters and four predictions were written down and committed before the two datasets were joined (the pre-registration).

Three things happen, and all three were predicted.

On young floor the clock runs fast. Near the ridges the crust is cracked and open, and cold seawater circulates through it and carries heat out sideways, through vents, where a probe in the mud never sees it. The probe reads too little heat, so the clock reads too old. This is well known: about a third of the heat the oceanic crust loses is thought to leave this way, and the effect fades once sediment seals the crust, at around 65 million years (Stein 1995). How large the over-read looks depends on how you count. Surveys over vent fields put hundreds of probes into one small patch of sea floor (the busiest one-degree square, on the Juan de Fuca Ridge off Vancouver Island, holds 1,574), and letting each one-degree square vote only once turns the median over-read on floor under 5 million years from 1.7 into 7.4.

On middle-aged floor it reads true. For floor 40 to 60 million years old the median reading is 0.93 of the stripe age, with a bootstrap interval of 0.89 to 0.97, and 0.98 when each square votes once. Where exactly the clock reads true depends on the constants: with the constant Stein and Stein fitted in 1992 (510 instead of 431), the best band moves to 60 to 80 million years (0.96). That is not the point. The point is that a formula written in 1862 to date the planet, fed probes of real mud, reads seafloor ages of tens of millions of years to within a few percent somewhere in the middle, and on floor older than 20 million years never misses by much more than a factor of two in the median, whichever constant and however you count.

On old floor it falls behind, and then nearly stops. Floor 120 million years old (median) reads 63; floor 161 million years old reads 77. The oldest open-ocean floor, in the western Pacific east of the Mariana Islands, is about 190 million years old, and the 263 probes on floor older than 180 read, in the median, 0.47 of it. (The grid's oldest crust of all is a remnant in the eastern Mediterranean, dated there at up to 339 million years; its 23 probes fall outside the pre-registered bands.) Section 4 is why.

Predictions, written before looking: P1, the median reading ÷ age exceeds 1 on floor under 20 Myr: . P2, past 100 Myr it is below 1 and the readings sit between 50 and 150 Myr: . P3, some band from 20 to 80 Myr reads within a factor of 1.5: . P4, readings rise with age (Spearman ρ = 0.67): . P2 promised the readings would be "roughly flat" past 100 Myr; they rise from 63 to 77, which is flatter than the floor ages (121 to 161) but not flat, and the page says so rather than claiming it.

4. Why it stops: a lid on a stirred interior

Kelvin's Earth is solid all the way down, so the cold can keep creeping deeper forever and the surface gradient keeps getting gentler. The real sea floor is a lid, about a hundred kilometres thick, riding on mantle that is hot and slowly stirring. Once the cold reaches the bottom of the lid, the mantle underneath keeps it supplied with heat and the gradient stops changing. Then the clock stops too, at a ceiling that has nothing to do with time: L² ÷ πκ, the thickness of the lid expressed in years.

The ceiling runs backwards, too. A clock stuck at 77 million years is the ceiling of a lid 88 km thick (with κ = 1 mm² per second), which is the right size for the oceanic plate: Stein and Stein's 1992 model uses 95 km, Parsons and Sclater's 1977 one 125 km. The measured floor is less tidy than the ideal lid, which has all but stopped by 150 million years while the probes are still creeping up, and the curve in section 3 shows that too.

Seeing this did not need plate tectonics. It was argued in print in 1895, by John Perry, a former assistant of Kelvin's, in Nature. He did not doubt the sums (“I have usually said that it is hopeless to expect that Lord Kelvin should have made an error in calculation”). He doubted the solid interior:

“we have reason to believe in very much greater quasi-conductivity inside than of true conductivity in the surface rocks, and if there is even only ten times the conductivity inside, it would practically mean that Lord Kelvin's age of the earth must be multiplied by 56.”J. Perry, “On the Age of the Earth,” Nature 51 (3 January 1895), pp. 224 to 227

A stirred interior is Perry's “quasi-conductivity” taken to its limit, and with a thin enough lid on a well-enough stirred interior, the same surface gradient is compatible with an Earth billions of years old. Kelvin was not persuaded. Two years later his estimate had come down, not up: “more than 20 and less than 40 million years ago; and probably much nearer 20 than 40” (“The Age of the Earth as an Abode Fitted for Life,” 1897, printed in Science in 1899). And Perry's argument, as England, Molnar and Richter put it in 2007, was “neglected or forgotten”.

5. What radioactivity did, and did not do

The story most people have heard is that radioactivity, discovered in 1896, supplied the missing heat. Its source is partly Rutherford himself, telling it years later about a lecture at the Royal Institution in 1904:

“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 baleful glance at me! Then a sudden inspiration came, and I said Lord Kelvin had limited the age of the earth, provided no new source was discovered. That prophetic utterance refers to what we are now considering tonight, radium! Behold! the old boy beamed upon me.”as told by Rutherford, in A. S. Eve, Rutherford (1939), p. 107

Radioactivity did two real things: it gave geology a clock that works (section 7), and it is indeed a large part of the heat coming out of the ground today. What it did not do is rescue Kelvin's calculation. England, Molnar and Richter (2007) showed that putting the known amount of radioactive heating into Kelvin's solid Earth does not change his answer much, and the arithmetic is short. Spread heat production A through the half-space and the surface drains what is made within about √(κt) of it, adding 2A√(κt/π) to the heat flow. At the mantle's average of about 0.02 microwatts per cubic metre (their figure), that is 1.3 mW m⁻² after 100 million years, and after 4.5 billion years of conduction the total surface heat flow would still be only about 15 mW m⁻², a quarter of what the median land probe measures. A solid, conducting Earth cannot be 4.5 billion years old and this warm, radioactive or not. A stirred one can, which was Perry's point, made before radioactivity was known. (For the Sun the popular story is the right one: there the missing source was real, and it was nuclear. Section 7 comes back to the Sun.)

6. On land, the clock reads tens of millions almost everywhere

Kelvin's gradient was not the problem either. The same database holds 39,939 heat-flow sites on the continents. Read with the seafloor constants, their median is 47 million years; nine in ten read under 129 million, and only 269 of them (0.7%) read over a billion. And the 29,536 land gradients the database reports have a median of 30 K per km (a quarter of them under 21.7, a quarter over 50), a little gentler than Kelvin's 1 °F per 50 feet, which is 36.5 K per km. Put that median into his own recipe, with his 7,000 °F and his Edinburgh rock, and it reads 144 million years. The Earth's surface, probed anywhere, says tens of millions of years, because that is how long it takes the cold to cross a lid, and the continents are lids too. Kelvin read the instrument correctly. It was measuring something else.

7. The other clocks of the argument

Kelvin's cooling Earth was not the only clock in the nineteenth-century argument, and every one of the others that came out at tens of millions of years was also timing something real.

The Sun. In the same year Kelvin estimated how long the Sun could have shone on the energy of its own contraction, and concluded “that the sun has not illuminated the earth for 100,000,000 years, and almost certain that he has not done so for 500,000,000 years” (“On the Age of the Sun's Heat,” Macmillan's Magazine, March 1862). The modern form of that clock, the gravitational energy GM²/R divided by the luminosity, is the Kelvin–Helmholtz time:

That is not the age of the Sun (4.6 billion years), but it is not nothing: it is about how long a young star can shine on shrinking before its core is hot enough to burn hydrogen. In Baraffe and colleagues' 2015 stellar models a star of one solar mass contracts from about three times the Sun's radius at half a million years to its settled size, 0.894 of it, at about 45 million years. Kelvin's sun clock timed the Sun's childhood.

The salt. In 1899 John Joly divided the sodium in the oceans by the sodium the rivers bring in each year and got “between 80 and 90 millions of years”. That quotient is exactly what oceanographers now call a residence time: the sea loses sodium about as fast as it gains it, so the division says how long an atom of sodium stays, not how long the sea has existed. The Water Leaves, the Salt Stays recomputes it for every ion.

The one that worked. In 1956 Clair Patterson measured lead isotopes in meteorites and got “4.55 ± 0.07 × 10⁹ yr”. It worked for the reason the others did not: the lead in a meteorite has been accumulating from the decay of uranium in a sealed reservoir that nothing has stirred, refilled or reset since it formed. Two Clocks in One Crystal works that arithmetic. Every clock measures the time since its reservoir last forgot. Heat in a lid forgets in tens of millions of years, sodium in the sea in tens of millions, gravity in a star in tens of millions. Lead in a meteorite has not forgotten yet.

This observation about the equation is not new, and it should be credited: England, Molnar and Richter wrote in 2007 that the seafloor problem “is mathematically identical to Kelvin's problem, though the age involved is that of the ocean floor, rather than of the Earth.” What this page adds is the measurement: the identity, held to 17,940 probes, with its three regimes read off real mud and its predictions written down first.