A rhythm in the rock that measures the Moon

The Earth’s axis turns round the pole of the ecliptic once in about 25,800 years. The pole star has a shelf life because of it, and the stars’ coordinates drift by fifty arcseconds a year. The cause is a torque: the Sun and the Moon pull on the Earth’s equatorial bulge, trying to bring it into the plane of their orbits, and the spinning Earth responds the way a spinning top does, by precessing instead of tipping.

Two-thirds of that torque is the Moon’s. And the Moon has not always been where it is. The tides it raises have been carrying it outward for billions of years, at a few centimetres a year now, and slowing the Earth’s spin by the same exchange. A Moon that was nearer pulled harder on a bulge that was larger, because a faster-spinning Earth is flatter. The axis turned faster in the past.

That faster precession left a record. The Earth’s precession, combined with the slow turning of its orbit, sets the season at which the Earth is closest to the Sun, and that drives a climate cycle of about twenty thousand years — one of the rhythms ice sheets and sediments follow. Some rocks record it in their layers. Those layers, laid down more than a billion years ago, can be read for how fast the axis was turning, and from that for where the Moon was.

The rate, and what it depends on #

The precession constant — the rate at which the axis turns — is the torque divided by the spin angular momentum it acts on. The torque from a body of mass $m$ at distance $a$ on a bulge of dynamical flattening $H$ goes as $H\,Gm/a^3$; the spin angular momentum goes as the spin rate $\omega$. So

$$k = \frac{3}{2}\,\frac{H\cos\varepsilon}{\omega}\left(n_\odot^2 + \frac{Gm_{\rm M}}{a_{\rm M}^3}\right),$$

with $\varepsilon$ the obliquity and $n_\odot$ the Earth’s orbital frequency. Today it comes to about fifty arcseconds a year: fifty and a quarter, measured.

Going into the past, three things in that expression change, and they are tied together. The Moon’s distance was smaller, and its term grows as the inverse cube. The spin was faster, by conservation of the angular momentum the tides have been moving from the spin to the orbit. And the flattening was larger, because a planet in hydrostatic equilibrium is flattened by its rotation in proportion to the square of the spin rate — so $H/\omega$ grows as $\omega$.

The conservation law does nearly all the work here. The angular momentum of the Earth–Moon system is fixed, apart from the small part the Sun’s tide removes from the spin, and the Moon’s orbital angular momentum grows as the square root of its distance. So every distance has its own day, without any model of how fast the tides acted, and every distance also has its own precession constant.

The dashed curve deserves a moment, because it shows where the sensitivity lies. If the Earth’s flattening stayed at today’s value while it spun faster, a nearer Moon’s stronger pull would be almost exactly cancelled by the larger spin angular momentum it had to turn, and the precession constant would barely move. It is the flattening following the spin — the faster Earth being fatter at the equator — that makes the constant change by two-thirds. The Earth’s mantle relaxes to the shape its rotation demands in tens of thousands of years, which is fast enough that over the millions of years between samples it is always in equilibrium; but the conversion from a measured constant to a distance depends on that, and on how the Earth’s internal density structure set the size of the bulge for a given spin.

Faster precession, shorter cycles #

The precession constant is not what a rock records directly. What a rock records is climate, and the climatic precession cycle is the precession of the axis combined with the precession of the Earth’s orbit.

The orbit’s perihelion turns slowly too, driven by the planets, and its motion is a sum of terms at frequencies set by the planets’ own interactions — the secular frequencies $g_i$, a few to twenty arcseconds a year. The season at which the Earth passes perihelion then cycles with frequencies $k + g_i$, one for each term. Today the main ones give periods of about 23.8, 22.4 and 19 thousand years. The obliquity oscillates with the precession constant plus a nodal frequency $s_3$ that is negative, at about forty-one thousand years.

The obliquity cycle’s sensitivity is worth noticing. Today the precession constant is fifty and the node term is minus nineteen, so the obliquity frequency is thirty-one arcseconds a year. An increase of thirty-four in the constant raises it to sixty-five and halves the period. The precession cycles, whose frequencies are sums, change by less.

A record with no clock in it #

Here is the difficulty. A sedimentary rock records its rhythms as variations in layer thickness or composition against depth, not against time. The rate at which the sediment accumulated is unknown, and almost never steady, so a measured cycle is a cycle in metres, and its period in years is exactly what is wanted and exactly what the rock does not supply.

What the rock does supply is ratios. If several cycles are recorded in the same stretch of sediment, their ratio of wavelengths is a ratio of periods, independent of the accumulation rate. And one of the cycles available is a metronome that does not depend on the Moon at all.

The Earth’s orbital eccentricity varies as the perihelion terms beat against each other, with frequencies $g_i - g_j$ — differences of planetary frequencies alone, with no precession constant in them. The strongest, $g_2 - g_5$, is the beat between the perihelion motions driven by Venus and by Jupiter, and it has a period of 405,000 years that has been stable for hundreds of millions of years because it involves the two most massive and most regular contributions. An eccentricity cycle is visible in sediments as a modulation of the amplitude of the precession cycle — the precession cycle matters more for climate when the orbit is more eccentric — and it supplies the scale.

So a sedimentary sequence that records the precession rhythm and its eccentricity modulation gives the ratio of the precession periods to the eccentricity periods, and with the eccentricity periods known from the planets, the precession periods in years follow, and the precession constant from them. The layered deposits on Mars are read the same way, with the same problem — a rhythm without dates — and the same solution in principle.

How a climate cycle becomes a layer #

The chain from the axis to the rock has a link in the middle that is not astronomy, and it is worth stating how it works, because the method depends on it.

The precession cycle does not change the total sunlight the Earth receives in a year. It changes how that sunlight is shared between the seasons and the hemispheres: when perihelion falls in northern summer, northern summers are more intense and shorter, and the average over the year is untouched. A climate that responds to seasonal contrast — the strength of a monsoon, the rate of weathering on land, the runoff of rivers into a basin, the productivity of plankton in the surface water — therefore oscillates with the precession period, and so does whatever falls to the sea floor.

In the Xiamaling shales the signal is chemical: the proportions of elements delivered by rivers and by wind rise and fall with depth, as the supply of land-derived sediment rose and fell with the monsoon. The deposit was laid down in deep, quiet water under an ocean with little oxygen, which helps, because nothing burrowed through the mud and mixed the layers after they settled. A record of that quality is rare. Most sediments of that age have been churned, eroded, heated or folded, and the rhythm in them, if any, has been lost.

The cycles then have to be shown to be astronomical rather than some other rhythm of the basin, and the eccentricity modulation is what does it: a precession signal whose amplitude swells and fades on the scale of the 405,000-year cycle, with the ratio of periods the planets predict, is hard to produce any other way.

Checking the method where the answer is known #

A method that turns a ratio of rhythms into a precession constant can be tested on rocks young enough for the answer to be known. For sediments of the last few tens of millions of years, the orbital solutions computed by integrating the solar system backwards give the precession constant and the planetary frequencies directly, and the same statistical analysis, run blind on those records, recovers a constant close to today’s fifty arcseconds a year within its uncertainty.

That calibration matters because the analysis has free parameters that could absorb a wrong answer: the sedimentation rate, allowed to vary, and the planetary frequencies, allowed to drift within bounds. A method that could fit any constant to any rhythm would be useless. One that returns fifty for young rocks and eighty-six for old ones is measuring something, and the something agrees with what angular-momentum conservation and the tidal history say it should be.

The bulge the torque acts on #

The torque in the precession constant acts on the Earth’s dynamical flattening, the fractional difference between the moments of inertia about the polar and equatorial axes. Today it is known to six figures, from the precession itself and from the nutation that rides on it, and from it comes the free wobble period of the Earth, the Chandler wobble, which depends on the same ratio.

Today’s flattening is not quite the hydrostatic value for the present spin — part of the Earth’s shape is not explained by its spin at all, but by the flow and density contrasts of the mantle. In the deep-time calculation that excess is an unknown. If it was larger in the past, the precession was faster than the hydrostatic model says for a given distance, and the Moon inferred from a measured constant would be slightly further away than the figures put it. The effect is at the level of a per cent in the flattening, smaller than the measurement’s uncertainty now; it would matter for a better record.

The measurement at 1.4 billion years #

The Xiamaling Formation in northern China is a sequence of finely laminated marine shales about 1.4 billion years old, dated by radiometric ages from ash layers within it. Its composition varies cyclically with depth, and a statistical analysis published in 2018 fitted the variations with a model in which the precession constant and the sedimentation rate were free and the eccentricity frequencies were constrained by the planetary dynamics. The fit gave a precession constant of 85.8 ± 2.7 arcseconds a year.

The precision is striking for a measurement made from layers of mud. The distance is fixed to about one per cent and the day to a quarter of an hour, because the precession constant changes steeply with the distance: a three per cent uncertainty in the constant is less than a one per cent uncertainty in the distance.

That steepness is the reason the method works at all. A looser measurement still says something useful.

What the point says about the tides #

A single distance at a known date tests the history of the Moon’s recession, and the test is not trivial.

The Moon recedes today at 3.83 centimetres a year, measured by laser ranging to the reflectors left on its surface. If that rate had held for 1.4 billion years, the Moon would have been at about 331,000 kilometres then. If instead the tides had always dissipated as efficiently as they do now, scaled to the closer distance — the tidal torque grows as the inverse sixth power of the distance — the Moon would have been nearer still, and further back would have been at the Earth’s surface 1.5 billion years ago.

That the Moon was not at the Earth 1.5 billion years ago is obvious; the Moon is 4.5 billion years old. The problem was noticed in the 1960s and is usually resolved by noting that today’s tidal dissipation is anomalously high — the oceans’ basins happen to sit near a resonance with the semi-diurnal tide, and in most past configurations of the continents they did not. What the deep-time precession adds is a measurement of how much lower the dissipation was, at a date long before the oldest tidal laminae. An average recession of about three centimetres a year over 1.4 billion years, against the 3.83 now, says the tides have been running at about four-fifths of today’s rate on average — and since they must have been stronger when the Moon was nearer, at fixed efficiency, the efficiency itself was lower for much of the interval.

The same calculation, applied to the cycles a calendar keeps #

The ice ages of the last few million years are paced by the same cycles: precession at nineteen to twenty-four thousand years, obliquity at forty-one, eccentricity at a hundred thousand and four hundred thousand. How ice sheets respond to them is a question about the climate system, and the periods themselves are usually taken as fixed.

They are not quite fixed even over that interval. The Moon recedes about forty kilometres in a million years, which changes the precession constant by a few hundredths of an arcsecond a year in every million years, and the orbital solutions used to date the last few tens of millions of years of sediments include this tidal term along with the Earth’s changing flattening as ice sheets grow and melt. Over fifty million years the drift in the precession periods is a few per cent, comparable to the precision with which the cycles in the best records can be measured, so the tidal history is already a parameter in the astronomical dating of young rocks as well as a result from old ones.

Where the model stops #

An idealised torque. The precession constant here uses circular orbits and an ecliptic Moon, scaled to today’s measured value; the real expression carries the Moon’s eccentricity and inclination and a small planetary term, which is why the inversion gives 339,000 kilometres where the published analysis, with a fuller model, gives 341,000.

Angular momentum conserved between two bodies. The Sun’s tide on the Earth slows the spin without moving the Moon, and removes a few per cent of the spin angular momentum over the Earth’s history; the day at a given distance is therefore slightly longer than drawn.

A hydrostatic Earth. The flattening is taken to follow the square of the spin. The Earth’s real flattening exceeds the hydrostatic value by about one per cent today, a mantle-convection effect that need not have been the same in the past, and the moment of inertia it is measured against depends on the density structure, which was different when the core was smaller.

Planetary frequencies held fixed. The perihelion and node frequencies of the inner planets drift chaotically and cannot be computed beyond about fifty million years. The analysis of the 1.4-billion-year record allows them to vary within bounds, relying on the 405,000-year term, which involves only Venus and Jupiter and is stable; the figures here use today’s values.

The surprising connection #

The thickness of mud layers on the floor of a sea 1.4 billion years ago depended on the distance to the Moon. Not through the tides — through the climate, which was paced by the season of perihelion, which was set by how fast the axis turned, which was set by how hard the Moon pulled on an equatorial bulge whose size was set by how fast the Earth spun, which was set by how far the Moon had already carried the Earth’s angular momentum away. Every link in that chain is a piece of classical mechanics, and the end of it is a measurement of the Moon’s distance from a rock with no clock in it, accurate to one per cent.

Still open: the earliest Moon #

The Xiamaling measurement and the 620-million-year tidal laminae are the two firm points in the Moon’s history between its formation and the present, and they show a recession slower on average than today’s. What happened in the first three billion years is constrained by almost nothing: the Moon formed close to the Earth, perhaps at a few Earth radii, and must have receded rapidly while the oceans, if any, and the continents were unlike anything since. Older cyclic sediments exist, and some have been analysed for precession signals; whether the rhythms they show are astronomical, and whether the eccentricity metronome can be identified in them reliably enough to calibrate the precession, is what decides whether the Moon’s distance at two or three billion years can be measured the same way.