A day five hours long

A tide is a difference of gravity rather than gravity itself, and the first rung of this anchor got two bulges out of that subtraction while the second got a distance inside which a moon cannot hold itself together. Both are statements about a single instant. This is the rung where the tide is allowed to run for a while.

Friction drags the bulge a few degrees ahead of the Earth–Moon line, and a bulge that leads exerts a couple. The couple hands the Earth’s spin angular momentum to the Moon’s orbit at a rate now measured to a fraction of a millimetre a year, and the natural thing to do with a measured rate is integrate it backwards.

The result is one line and two readings of it, and they have completely different standing.

Two quantities, and only one of them is refuted #

The solid curve is calibrated to a measurement and ends in an absurdity. Extrapolated back at a constant tidal quality factor, the Moon leaves the Earth’s surface 1.54 Gyr ago, crossing the Roche limit at 2.88 Earth radii four years before that. The Moon is three times older than the answer, so the extrapolation is wrong, and what is wrong in it is the constancy.

That failure is easy to over-read. It does not show that the recession is mismeasured, and it does not show that tidal friction is the wrong mechanism. It shows only that the present rate is anomalously high, which is a claim about the current arrangement of the Earth’s oceans rather than about the physics of tides.

Now read the same figure’s other curve. The day length is not obtained by integrating a friction law at all. It is obtained by insisting that what the Earth’s spin loses, the Moon’s orbit gains — and that identity fixes the day as a function of the separation with no rate in it anywhere. Whatever the dissipation was doing at any epoch, a Moon at half its present distance means a day of 9.84 hours, and a Moon at the Roche limit means 4.97.

The title is that number, and it is the robust half of the story. The refuted half is the horizontal axis.

The couple, read from either end #

The mechanism is a cancellation that nearly succeeds. Each bulge pulls the Moon along its own line; the near bulge is closer and pulls harder, so a component survives, and that surviving component is the whole of the recession.

The two-sided reading matters. A torque on the Moon and a torque on the Earth are not two effects to be added but one internal couple seen from its two ends, and the figure’s own check is that they sum to zero. That is what licenses the conservation argument below: the pair exchanges angular momentum with nothing outside itself.

It is also where the sign of the whole story comes from. The Earth turns faster than the Moon goes round, so the bulge runs ahead and the couple is prograde. Reverse that inequality and everything reverses with it.

Five hours is a conservation law #

Write the total angular momentum as the Earth’s spin plus the orbit’s, using the second law’s conserved quantity for the second term:

$$L = C\,\omega_E + \frac{mM}{m+M}\sqrt{G(M+m)\,a},$$

with $C = 0.3307\,MR^2$ the Earth’s measured polar moment of inertia — not the $0.4MR^2$ of a uniform sphere, because the mass is concentrated inward, and the coefficient comes from satellite geodesy and from the rate of precession of the equinoxes.

The orbital term carries the reduced mass and the total mass because both bodies move, which is the correction that turns a one-body problem into a genuine two-body one. With the measured masses and the present separation, 83.0 per cent of the system’s angular momentum is in the orbit and 17.0 per cent in the Earth’s spin. Solving the same equation for the spin gives the day at any separation,

$$\omega_E(a) = \frac{L - \dfrac{mM}{m+M}\sqrt{G(M+m)\,a}}{C},$$

and nothing on the right is a rate, a friction or a time. It is an identity between two numbers a diagram can carry: where the Moon is, and how fast the Earth turns.

That is why five hours is a stronger result than 1.54 Gyr. The young Earth’s day can be known without knowing anything whatever about the mechanism that lengthened it — a history without a model of the process, which is rarer than it sounds.

The timeline is a friction, and the friction is a fitted number #

The horizontal axis needs a rate law, and the standard one holds the lag angle fixed, which is what a constant quality factor $Q$ means. Then

$$\frac{da}{dt} = 3\frac{k_2}{Q}\frac{m}{M}\left(\frac{R}{a}\right)^{5} n a,$$

and since the mean motion $n$ goes as $a^{-3/2}$ by the third law, the right-hand side goes as $a^{-11/2}$. Integrating makes $a^{13/2}$ exactly linear in time, so the whole history is a straight line in a variable nobody plots, and the hero’s generator checks that straightness to one part in $10^{10}$ before drawing anything.

Calibrating that law to the measured recession rather than to a modelled dissipation gives $k_2/Q = 0.0257$, or $Q = 11.6$ for the Earth’s measured tidal Love number of 0.299 — a startlingly dissipative planet, where most solar-system bodies are quoted at hundreds. A mean $Q$ of 34, drawn dashed in the hero, pushes the crossing back to 4.51 Gyr, which is the sort of value the Moon’s age requires.

Why the present rate is high is geography rather than physics. Dissipation happens overwhelmingly in shallow shelf seas, and the present continents bound basins whose natural periods sit near the semidiurnal forcing. One number standing for that arrangement cannot be constant across an interval in which the continents rearranged themselves repeatedly.

What a corner cube and a shadow’s track measure #

Two numbers here come from instruments, and neither of them is a day length.

The recession is a range. Apollo 11, 14 and 15 left arrays of corner-cube retroreflectors on the Moon and the Lunokhod rovers left two more; a laser pulse fired at one returns after about 2.5 seconds, and timing that round trip gives the distance to a few millimetres. Fifty years of such ranges, fitted with a model carrying the Moon’s libration, the solid Earth’s own tide, the station’s motion and the relativistic corrections, yield a secular drift of 3.83 cm per year. What is measured is a light travel time; the recession is one coefficient in a large fit to it.

The lengthening of the day is measured a different way and much further back. An ancient record of a total eclipse fixes where the umbra fell, and the umbra is a small patch on a turning Earth. Compute such an eclipse from modern gravitational theory with a uniformly rotating Earth and the track lands in the wrong place. That discrepancy, expressed as a clock error, is $\Delta T$, and its growth with age is the observed lengthening. Babylonian, Chinese, Arab and medieval European records of eclipses and of lunar occultations carry the measurement back some 2,700 years — and the recurrence that repeats an eclipse a third of a world away is what makes such records identifiable at all.

Three rates that ought to be one #

The recession requires the day to lengthen by 2.10 milliseconds per century. The tidal total is about 2.3, because the Sun raises a tide too and that one brakes the Earth without giving anything to the Moon. And the ancient records show 1.75.

Each gap is separate physics, and the second one is the surprise. If the tides remove 2.3 ms per century and the record shows 1.75, something is speeding the Earth up by 0.55 — and the something is the last ice age. Unloading kilometres of ice from Fennoscandia and northern Canada let the mantle beneath begin flowing back, and it is still flowing; mass moving from the equatorial bulge toward the poles lowers the Earth’s moment of inertia, and a body that contracts spins faster.

So the Pleistocene is in the length of the day. It also means $C$ above is not a constant — which is precisely the assumption the discrepancy convicts.

A count in a rock, and a date to check it against #

The eclipse record reaches thousands of years; the hero’s curve claims billions. Between them sits one measurement with a date on it.

Tidal rhythmites are laminated estuarine sediments in which each lamina is one tidal cycle and the thicknesses are modulated by the fortnightly spring–neap beat and by the year. Counting laminae between successive neap minima counts tidal cycles per lunar month; counting the months in an annual bundle counts months per year. Neither is a rate. Both are integers read off a rock dated independently, which makes a rhythmite a direct measurement of an ancient day.

The Elatina and Reynella formations of South Australia, at about 620 Myr, give a day of 21.9 hours and a Moon at 96.5 per cent of its present distance. The constant-rate history reads 20.1 hours and 92.4 per cent — too fast and too close, in the same direction and for the same reason as the zero crossing, at a date somebody has dated. That is the better refutation, because a wrong answer 1.5 Gyr ago can be blamed on an unknown early Earth and 620 Myr cannot.

Where the dissipation actually happens #

The claim that the present rate is anomalously high because of geography deserves more than an assertion, because it is the whole reason the timeline fails.

The open ocean is a poor dissipator. A tidal wave in deep water propagates with very little loss; the energy is removed where the wave runs onto a continental shelf and breaks down into smaller-scale motion, and where a shallow basin’s own natural period is close enough to twelve hours for the response to be resonant. Those places are few and they are small: the Bay of Fundy, the Patagonian shelf, the European shelf seas, the Yellow Sea, the northwest Australian shelf. A substantial fraction of the Earth’s entire tidal dissipation happens in a handful of continental margins occupying a per cent or two of the ocean’s area.

A resonance is exactly the kind of thing that cannot be assumed constant. A basin’s natural period depends on its length and its depth; move the continents and the period moves with them, and a basin that is near resonance now was not near resonance in the Cretaceous and will not be in fifty million years. The measured $Q$ of about twelve is therefore the value at a moment when several large shelf systems happen to be tuned, and there is no reason for the average over four and a half billion years to resemble it.

There is a second dissipation channel that behaves quite differently, and it is the one that makes the whole picture harder rather than easier. Part of the surface tide is converted into internal waves at abrupt topography — ridges and seamounts — and those propagate into the deep ocean and dissipate far from where they were generated. That channel does not depend on shelf geometry in the same way, so it provides a floor that the shelf term sits on top of. How large the floor is decides whether the ancient Earth’s $Q$ was a hundred or a thousand, and it is estimated from satellite altimetry over the last three decades and extrapolated across the Proterozoic.

It also explains why the solid Earth contributes so little. The rocky body has its own tide of about thirty centimetres and its own lag, and its dissipation is roughly a twentieth of the ocean’s — so a planet with no ocean would recede its moon twenty times more slowly, and the Earth’s tidal history is a history of its water rather than of its rock.

The number that dates the Moon’s recession is therefore an average over a variable nobody can reconstruct, which is why the rhythmite measurements matter so much: they are the only direct readings of the integral.

What the picture cannot show #

The lag angle is drawn and not measured. Nothing observes the bulge running three degrees ahead; the angle is a repackaging of the dissipation, which is what the timeline turns on. Drawn at twice the value the arithmetic barely notices, and that insensitivity is the point.

A single $Q$ stands for a planet whose geography changed. Constant-$Q$ friction is not a model of an ocean but one number chosen so that a closed form exists, and the closed form’s elegance is a property of the assumption rather than of the Earth.

The bulge is drawn 3.5·10⁶ times its real height. The equilibrium ocean tide is 0.36 m on a radius of 6,371 km, or 5.7·10⁻⁸ of it — thinner than the stroke the sphere is drawn with. Every tidal figure ever published has this problem, and the honest response is the number.

The day-length curve carries today’s moment of inertia at every epoch. It has to, because nothing else is known, and the section above shows that assumption to be measurably false now. Five hours is robust against the friction and not against the Earth’s internal structure, and those are different robustnesses.

Nothing here says the Moon formed at the Roche limit. The curve stops at 2.88 radii because it stops meaning anything there, not because that is a beginning. What happened before is a question about a giant impact and the disc it left, and no part of this figure addresses it.

Where the exchange runs the other way #

Everything above depends on one inequality: the Earth turns faster than the Moon orbits. Break it and the couple changes sign.

Phobos goes round Mars in 7.65 hours, well inside Mars’s 24.6-hour rotation, so the bulge it raises trails behind it instead of leading. The couple is retrograde, orbital angular momentum flows into the planet’s spin, and Phobos is spiralling inward toward the distance at which a body held together by its own gravity comes apart. Same tide, same algebra, opposite sign, and the outcome is a ring rather than a receding moon. At the other end the exchange stops. Once a body’s rotation matches its orbital period the bulge no longer sweeps and the couple vanishes. The Moon reached that state long ago, which is why its face is very nearly fixed — and the “very nearly” is the interesting part. The same secular exchange runs wherever a tide is raised on something that rotates: it circularises the orbits of planets that arrived where one cannot form, it locks the inner moons of every giant planet, and it is why a close-in exoplanet’s rotation cannot be assumed. It also belongs to a category this collection returns to — a secular rate too small to see in one pass, read out of a long enough series of arrival times, which is exactly how a binary pulsar’s orbit is measured to be shrinking.

One more lag angle brackets the range the tidal torque is fitted over.

Where the ladder goes next #

Later rungs on this anchor: the tidal potential expanded in Legendre polynomials, which is where $k_2$ and $Q$ acquire definitions rather than values. The spin–orbit resonances other than 1:1, and why Mercury sits in a 3:2. Locking timescales and their sixth power of distance. And the day as a quantity with several definitions, since a solar day and a sidereal day already differ by four minutes before any of this is applied.

Edmond Halley noticed in 1695 that ancient eclipses could not be reconciled with a uniform lunar motion, and read the discrepancy as an acceleration of the Moon; it is now read as a deceleration of the Earth, the same observation with the clock and the calendar exchanged. George Darwin worked the tidal history out in the 1870s and got a timescale of order 50 Myr — shorter even than the 1.54 Gyr above, and wrong for the same reason. What has changed in a century and a half is not the error but the number it fails against: the Moon’s age is now measured too.