A Moon measured in banded iron

The Moon is receding from the Earth at 3.83 centimetres a year, measured by laser pulses bounced off the reflectors left on its surface. Run backwards at today’s rate, or at today’s tidal efficiency scaled to the closer distances of the past, that recession gives a history that cannot be right: the efficiency scaled with distance brings the Moon down to the Earth about 1.5 billion years ago, when the rocks show nothing of the kind. The tides must have worked less hard for most of the Earth’s history than they do today, and the question is how much less, and when.

The essay on the rhythm in the rock followed one answer. The rate at which the Earth’s axis precesses depends on the Moon’s distance, because the Moon’s pull on the Earth’s equatorial bulge is the larger part of the torque; climate responds to the precession through the timing of the seasons relative to perihelion; and sediments that record the climate’s rhythms record the precession’s period, and therefore the Moon’s distance. Sediments 1.4 billion years old from northern China put the Moon at 341,000 kilometres. That essay ended on the earliest Moon, and on whether older rocks could be read the same way.

In 2022 they were. Layers in a banded iron formation 2.46 billion years old, in the Hamersley Basin of Western Australia, carry a precession signal that puts the Moon at about 321,800 kilometres, with an uncertainty of about six thousand. This essay works out what that one point, added to the others, says about the tides — and why the answer is more interesting for what it does not fit than for what it does.

A rhythm read in iron #

Banded iron formations are an archive of a particular epoch. They were laid down in the oceans of the Archaean and early Proterozoic, when the deep water held dissolved iron that precipitated as oxides wherever oxygen, made by photosynthesis near the surface, reached it. Their characteristic banding — alternating iron-rich and silica-rich layers, from microscopic laminae to beds a metre thick — records changes in the supply of iron and silica, and some of the larger-scale alternations repeat with a regularity that is hard to explain except as a climatic rhythm.

The Joffre Member of the Brockman Iron Formation contains two such rhythms in a fixed ratio. One, with a period of about four hundred thousand years, matches the longest and most stable of the Earth’s orbital eccentricity cycles, set by the interaction of Jupiter and Venus and essentially unchanged over billions of years; it calibrates the sediment’s accumulation rate and turns thicknesses into time. The other, about thirty times shorter, matches the climatic precession: about eleven thousand years, against the nineteen to twenty-three thousand years of today’s. A shorter precession period means a faster precession of the axis, which means a stronger lunar torque, which means a nearer Moon. The conversion uses the same physics as the 1.4-billion-year measurement, run at a distance twenty thousand kilometres closer.

The precession period also depends on the Earth’s rotation rate and on the flattening of its equatorial bulge, both of which were different when the day was shorter. Those are not independent unknowns: the conservation of the angular momentum shared between the Earth’s spin and the Moon’s orbit ties the day’s length to the Moon’s distance, and the bulge’s flattening follows from the spin. One measured period gives one distance.

The oldest point on the curve #

Put the new point beside the older ones and the two backward extrapolations.

The three points lie well above the curve that today’s tidal efficiency draws, and they lie close together. Between 1.4 and 2.46 billion years ago the Moon receded by only nineteen thousand kilometres — an average of about 1.8 centimetres a year over a billion years, half of today’s rate, at distances where a constant efficiency would have made it faster than today’s, not slower.

The solid curve takes the efficiency that fits those two points and holds it fixed further back. The tidal torque grows as the inverse sixth power of the distance, so at a fixed efficiency the recession speeds up enormously as the Moon gets closer, and the curve plunges towards the Earth. It reaches it 4.78 billion years ago. The Moon is about 4.5 billion years old, so the curve is wrong — but it is wrong by a few hundred million years in a history of four and a half billion, after an extrapolation over two billion years beyond the oldest point. That is close. It says the efficiency of the middle Proterozoic is roughly the efficiency the earliest Earth needed, if a little low.

When the old tides would have had to start #

The oldest point carries an uncertainty, and the extrapolation is sensitive to it.

The fitted efficiency depends on the difference between the sixth-and-a-half powers of the two distances, and a few thousand kilometres at 2.46 billion years changes that difference by tens of per cent. If the Moon was nearer than the central value, the recession between the two points was larger, the efficiency higher, and the extrapolated start later — 4.1 billion years ago at the lower end of the error. If it was further, the efficiency was lower and the start recedes to six billion years, before the Solar System existed. The Moon’s actual age falls inside the range the point allows.

So the measurement does not contradict a simple history with a single efficiency from the Moon’s formation to 1.4 billion years ago. It does not require one either. What it rules out, firmly, is today’s efficiency held throughout: no value within its error lets the present tides run for more than a fraction of the Earth’s history.

How hard the tides have worked #

The points can be read another way, as average efficiencies over the intervals between them.

The last 620 million years averaged about half of today’s efficiency; the interval before it about two thirds; and the billion years between 1.4 and 2.46 billion years ago about a fifth. Every interval is below today’s. That is consistent with the explanation now generally given: today’s ocean basins are close to a resonance with the twice-daily tide. The North Atlantic in particular has a natural period of oscillation near twelve hours, so the tide sloshes in it with a large amplitude and dissipates its energy on the shelves round it, far more efficiently than an ocean shaped differently would. As the continents drifted, basins moved into and out of resonance, and for most of the past the tides were further from it than now.

The billion years from 2.46 to 1.4 billion years ago stand out as particularly idle. A fifth of today’s efficiency, sustained for a billion years, is what an Earth with no basins near resonance, or with a single large ocean whose natural period was far from twelve hours, would give. The Proterozoic supercontinent cycles shuffled the basins, but a low average over a billion years suggests that resonant basins were rare in that interval — or that the oceans’ stratification, which changes how the tide is converted into internal waves and dissipated in the deep ocean, was different.

Why the present is unusual #

The comparison carries a lesson about inference from the present. Today’s tidal dissipation of 3.7 terawatts is measured with great precision by three independent methods, and for decades it was tempting to treat it as the Earth’s normal state. The points show it is not. The present rate is about twice the long-term average, and two and a half to five times the rate of the Proterozoic.

That matters well beyond the Moon. The rate at which the day lengthens, the rate at which the Earth’s rotational energy is turned into heat in the oceans, and the energy available to mix the deep ocean are all set by the same efficiency. If the present is a resonant interval, all of them are near a high point now, and models of the past that scale from present values overestimate the tide’s role by a factor of several.

Theory had anticipated the shape. Models that compute the ocean’s tidal response in reconstructed basins, as the continents moved, find efficiencies that wander by factors of several over hundreds of millions of years, with resonant episodes like today’s as peaks against a lower background. What the geological points add is the background’s level, measured rather than modelled.

A seventeen-hour day #

The Moon’s distance and the day’s length are two ends of one exchange.

The tide raised on the Earth by the Moon is carried slightly ahead of the line between them by the Earth’s rotation, and the Moon’s pull on that displaced bulge brakes the Earth’s spin while accelerating the Moon along its orbit. Angular momentum leaves the spin and enters the orbit, and the Moon’s distance and the day’s length are tied by its conservation. At 321,800 kilometres the day was about seventeen hours long and the year held about five hundred and twenty of them.

The figure makes the history’s shape plain. The day lengthened by less than two hours in the billion years between the two oldest points, and by more than five in the 1.4 billion since. The lengthening has accelerated towards the present, not because the Moon has come closer — it has receded — but because the efficiency has risen. A day that stood near eighteen hours for much of the Proterozoic is consistent with sediment-based estimates of the day’s length from other formations of that era.

A faster wobble and a shorter month #

The same numbers fix the rest of the Earth–Moon clockwork. Today the Earth’s axis takes about twenty-six thousand years to sweep once round its cone, which is why the pole star has a shelf life. With the Moon at 321,800 kilometres and the Earth spinning faster, the torque on the bulge was larger and the bulge itself flatter, and the axis precessed in about twelve thousand years — twice as fast. The climatic precession that the sediments record, which is the axis’s precession combined with the slow turning of the Earth’s orbit, was correspondingly shorter: about eleven thousand years against today’s nineteen to twenty-three thousand. A pole star in the age of banded iron kept its title for half as long.

The month was shorter too. A Moon at 84 per cent of its present distance went round the Earth in 20.9 of today’s days, by Kepler’s third law, and the interval from new Moon to new Moon was about twenty-two of today’s days. Counted in the seventeen-hour days of the time, that is about thirty-one days: the number of days in a lunar month, which is a number calendars have had to tabulate because it is not a whole one, was then a little larger than now, because the day had shrunk faster than the month. The year, fixed by the Earth’s orbit round the Sun and unaffected by the tides, held about sixteen or seventeen of those months.

A retreat that was fastest at the start #

The steep dependence of the torque on distance has a consequence for the earliest history.

The Moon formed close to the Earth, at a few Earth radii, from the debris of a giant impact. At that distance the tide raised on the Earth was thousands of times larger than now and the torque correspondingly enormous, so the Moon retreated quickly: half its present distance within tens of millions of years at any plausible efficiency. Most of its retreat happened while the Earth was still cooling from its formation. On the way out it passed through stages that left their own marks: the point at about fourteen Earth radii where the Moon’s orbital plane had to jump from the Earth’s equator towards the ecliptic, and the resonances that may explain why the Moon’s orbit is tilted by five degrees when the simplest history leaves it in the ecliptic. The long, slow part of the history — from about three hundred thousand kilometres outward — is the part sediments can record.

That is why the extrapolation of the middle-aged efficiency backwards is not as reckless as it looks. The early fast retreat is insensitive to the efficiency’s exact value, because any efficiency gets the Moon most of the way out within a hundred million years; what the efficiency controls is the slow drift afterwards. A history that starts at the right time needs only that the average efficiency between the end of the fast phase and the Proterozoic be close to what the two oldest points measure.

The same history with a nearer point #

The sensitivity to the oldest distance is easiest to see by moving it.

At the lower end of its uncertainty the point implies a larger recession between 2.46 and 1.4 billion years ago, a higher efficiency, and an extrapolated start about 4.1 billion years ago — too late, now, by a few hundred million years in the other direction. Between the two ends lies a value at which a single efficiency from the Moon’s formation to 1.4 billion years ago would fit everything. The data do not yet say whether nature chose it, or whether the earliest efficiency differed from the Proterozoic’s in either direction.

What the points assume #

The precession signal is the precession. The identification of an eleven-thousand-year rhythm in the iron formation as climatic precession rests on its ratio to the four-hundred-thousand-year eccentricity cycle and on statistical tests that it is not noise or a different cycle. That identification has been tested on younger sediments where the answer is known, and the method’s practitioners consider it robust; it is still an inference from layers of rock.

The conversion from period to distance needs the Earth’s shape. The precession rate depends on the equatorial bulge’s flattening, which depends on the spin rate and on the Earth’s internal structure. An Earth whose interior was hotter and differently stratified 2.46 billion years ago would have had a slightly different flattening at the same spin, and the inferred distance would shift by a fraction of its quoted error.

The metronome is assumed steady. The four-hundred-thousand-year eccentricity cycle calibrates the sediment’s accumulation rate, and its period is taken as today’s. It comes from the slow interaction of Venus’s and Jupiter’s orbits and is the most stable term in the solar system’s chaotic evolution, but chaos still blurs it over billions of years by a per cent or so. A metronome a per cent off shifts the inferred precession period, and so the distance, by a similar fraction — well inside the point’s quoted uncertainty, and the reason the uncertainty is not smaller.

One efficiency per interval. The figures assume a constant efficiency between points. The real history wandered within each interval; the points measure the average and nothing finer.

The Sun’s tide is folded in. The Sun also raises a tide on the Earth and brakes its spin without moving the Moon, at about a fifth of the Moon’s effect. The angular-momentum bookkeeping that ties the day to the Moon’s distance treats it approximately.

Still open: the Moon before the iron #

The banded iron formation pushes the measured history of the Earth–Moon distance back to 2.46 billion years, more than halfway to the Moon’s formation. What it shows is a Proterozoic in which the tides worked at about a fifth of today’s efficiency and the Moon receded at half today’s rate, and a backward extrapolation that misses the Moon’s formation by a few hundred million years — within the point’s own uncertainty, but not comfortably.

Older cyclic sediments exist. Banded iron formations and other rhythmically layered rocks of three billion years and more have been studied for orbital signals, and the eccentricity cycle’s stability makes them, in principle, as readable as the Joffre Member. A single well-measured point at three or three and a half billion years would test whether the efficiency in the Archaean was the Proterozoic’s, or higher, as the extrapolation’s near miss suggests — and with it whether the early oceans, on an Earth with little continental crust, were for a while as well tuned to the Moon’s tide as today’s.