A migration timed by the planet and not the moon

Every tidal rate in astronomy is proportional to one combination of two unknowns — the Love number $k_2$, which says how far a body deforms, and the quality factor $Q$, which says how late the deformation lags behind the force. The ratio $k_2/Q$ sets how fast a moon recedes, how fast a planet’s spin slows, how much heat a moon’s interior generates. For decades the working assumption was that each planet has a value, roughly constant, and that every moon around it sees the same one.

For Saturn that assumption makes a specific, checkable prediction about its moons: the distant ones should hardly move. The prediction has now been checked, and it fails by a factor of twenty for the moon where it is easiest to test.

Why a distant moon should barely move #

A moon raises a tide on its planet. If the planet’s response lags — if the bulge is carried ahead of the moon by the planet’s faster rotation — the bulge pulls the moon forward along its orbit, and the moon gains energy and moves outward. The rate of gain depends on how large the bulge is, how far ahead it is carried, and how strongly it pulls on the moon:

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

Every factor of distance works against a distant moon. The tide it raises falls as the cube of its distance; the bulge’s pull on it falls again; and the angular momentum it must gain to move outward grows. The migration time $a/\dot a$ grows as the distance to the thirteen-halves power, and inversely with the moon’s mass.

The thirteen-halves power is what makes the test sharp. Titan is two and a half times farther than Rhea and five times farther than Enceladus, and it is two hundred times heavier than Rhea. On the constant-Q lines, the mass helps it — a heavy moon raises a larger tide — but the distance overwhelms the help. Its migration time should be of order two hundred billion years, fifteen times the age of the universe. In the four and a half billion years of the solar system’s existence it should have moved outward by about two per cent of its orbit.

Eleven centimetres a year #

Measuring a moon’s migration means measuring a slow change in its orbit against the background of every other perturbation — the other moons, the Sun, the planet’s oblateness. The change accumulates quadratically in the moon’s longitude: a moon whose orbit is slowly expanding lags further and further behind where a fixed orbit would put it. Over decades, the lag becomes measurable in astrometry — the precise positions of the moons on photographic plates and CCD images going back more than a century — and in the radio tracking of spacecraft that flew past them.

For Titan, two independent analyses — one from a century of astrometric positions, one from the Cassini spacecraft’s radio tracking during its close flybys — gave the same answer: Titan is moving outward at about eleven centimetres a year, with an uncertainty of about two. That is a migration time of about eleven billion years, and an effective $k_2/Q$ for Titan’s tide about twenty times larger than for the inner moons’. Saturn’s measured Love number, about 0.38, makes Titan’s $Q$ about a hundred; the inner moons see a $Q$ of a couple of thousand.

Seen from the Earth, the effect is small. A moon whose orbit expands slowly falls behind a fixed-orbit prediction by an angle growing as the square of time — three-quarters of the fractional migration rate times the mean motion times the elapsed time squared. For Titan over a century that is about a ten-thousandth of a radian along its orbit, a hundred and twenty kilometres, which at Saturn’s distance is under two hundredths of an arcsecond. No single photographic plate of the last century can measure that. Thousands of positions, reduced against a modern star catalogue with accurate proper motions, can — and the radio tracking of Cassini, which measured Titan’s position directly during flybys a few thousand kilometres away, confirmed it independently.

So Saturn has no single Q. The inner moons see one value and Titan another, twenty times smaller. A planet whose interior were a uniform viscous fluid could not do that, because a uniform fluid’s dissipation depends smoothly on the forcing frequency, and the frequencies at which Saturn is tidally forced by its moons differ by less than a factor of two — they are all set by the planet’s ten-and-a-half-hour rotation, which is much faster than any moon’s orbit.

A mode that follows the moon #

The explanation now favoured, put forward by Jim Fuller, Jing Luan and Eliot Quataert in 2016 before Titan’s rate was measured, is that the dissipation is not smooth at all. Saturn’s interior has normal modes of oscillation — internal waves trapped in its stably stratified regions — and a mode whose frequency happens to match a moon’s tidal forcing frequency responds far more strongly than the smooth background. Near the match, the tidal response grows by orders of magnitude, and so does the dissipation.

A resonance of that kind would be fleeting if nothing changed. But two things change. As the moon migrates outward, its orbital frequency falls and the forcing frequency shifts. As Saturn cools and contracts over billions of years, its internal structure changes and the mode frequencies shift too. If the two shifts happen to keep pace, the moon stays locked to the mode: it migrates exactly fast enough to keep its forcing frequency on the mode as the mode drifts.

Why a smooth dissipation cannot single out Titan is worth spelling out, because it is the crux. A moon’s tide is felt by the planet at twice the difference between the planet’s spin rate and the moon’s orbital rate. Saturn spins in ten and a half hours; Mimas orbits in twenty-two and a half, Titan in sixteen days. The tidal forcing periods are therefore about ten hours for Mimas and five and a half for Titan — less than a factor of two apart. Any dissipation that varies smoothly with frequency, as a viscous fluid’s does, can change by at most a modest factor over so narrow a range. A twentyfold difference between two moons whose forcing frequencies differ by eighty per cent requires a response with structure on that scale — a resonance.

That is resonance locking, and its central prediction is that the migration rate is set not by the moon’s mass or distance but by the rate at which the planet’s mode frequencies change. A moon migrating faster would leave the resonance and lose the enhanced dissipation; a moon migrating slower would fall deeper into resonance and speed up. The lock is stable, and the migration time is a few times the time over which the planet’s structure evolves — the same for every locked moon.

The figure inverts the usual question. Instead of asking what $Q$ the planet has, it asks what $Q$ each moon must see if they all migrate together, and the answer is a smooth sequence falling with distance — exactly what a constant-$Q$ picture forbids and resonance locking requires. The measured values follow the sequence’s trend, within the scatter of the measurements. The planet does not have a quality factor; each moon finds the planet as dissipative as the mode it is locked to.

A time scale that belongs to the planet #

If a moon’s migration time is set by how fast the planet’s mode frequencies drift, then the measured migration time is a measurement of the planet’s evolution. Eleven billion years for Titan means, in the simplest locked model, that Saturn’s relevant mode frequencies change by order unity over a few billion years — comparable to the planet’s age, which is what a planet that is still slowly cooling and contracting should show.

That consistency is not trivial. A migration time of eleven million years would have required a planet changing a thousand times faster than any model of Saturn allows; one of eleven trillion would have required a planet effectively frozen. The measured value lands in the narrow range where a lock is plausible, and it does so for a reason that has nothing to do with the moon: Saturn has been cooling since it formed, its interior structure has been changing at a rate set by how fast it radiates its internal heat, and the moons inherit that rate.

This turns the usual logic around. In the constant-Q picture a moon’s migration is a property of the moon’s mass and distance, and the planet contributes a single number. In the locked picture the migration is a property of the planet’s history, and the moon contributes only the choice of which mode it is listening to. A measured migration time becomes a clock reading for the planet’s interior — a cooling rate, read off a moon’s position on a century of photographic plates.

The surprising part: the same physics as the rings #

The modes that would lock to the moons are not hypothetical. Saturn’s rings have measured some of them.

The planet’s internal oscillations perturb its external gravity field, and a perturbation rotating at the right rate excites waves in the rings at resonant radii, exactly as a moon does. Waves in Saturn’s C ring were found in Cassini data at locations no moon could explain, with patterns that identified them as driven by oscillations of the planet itself. The frequencies of those oscillations, compared with models of Saturn’s interior, require a stably stratified region deep inside the planet — a dilute core, where heavy elements are mixed gradually into the hydrogen envelope rather than sharply separated.

A stably stratified region is precisely what supports internal gravity waves, and internal gravity waves are the modes whose frequencies evolve as the planet cools. The ring that weighs Saturn’s interior by the wave crossing it is therefore measuring the same structure that would lock the moons. The ring seismology and the tidal migration are two readings of one stratified core — one through its free oscillations, the other through its forced ones.

Enceladus is paid for by the same tide #

There is a second, independent piece of evidence that Saturn’s dissipation is much stronger than the old constant-$Q$ values assumed, and it comes from the opposite end of the system.

Enceladus is a small moon, five hundred kilometres across, venting water from fractures near its south pole and radiating ten to sixteen gigawatts of heat from that region — far more than radioactive decay in a body that small could supply. The heat is tidal, generated by the flexing of Enceladus by Saturn as its slightly eccentric orbit carries it nearer and farther. The eccentricity would damp away in a geologically short time if nothing maintained it; a moon radiating tidal heat is spending something it cannot have saved. Enceladus’s eccentricity is maintained by its 2:1 resonance with Dione, which pumps eccentricity as Saturn’s tides push the pair outward.

In equilibrium, the heat Enceladus radiates is the orbital energy the resonance diverts from the pair’s outward migration, and that energy is proportional to Saturn’s $k_2/Q$. With the old value of $Q$, near eighteen thousand, the equilibrium heat would be about a gigawatt, a tenth of what is measured. With the value the inner moons’ astrometry gives, it is about ten — consistent with the measurement. Enceladus’s warmth is evidence of Saturn’s dissipation, measured from the opposite end of the system and by an entirely different instrument.

Where Titan was #

If the migration has run at the locked rate for the age of the solar system, Titan has moved a long way.

The locked history is an exponential: a constant $a/\dot a$ means the distance grows by a fixed fraction per unit time. Under the constant-$Q$ history, by contrast, the migration rate falls so steeply with distance that a moon that is now far out has always been nearly where it is. The difference between the two curves is the difference between a moon system that formed roughly as it is and one whose present arrangement is the latest frame of a slow expansion.

Whether the rate has held is the obvious question, and resonance locking has an answer: a lock persists as long as the planet’s modes keep evolving at a similar rate, which for a planet cooling over billions of years is a reasonable expectation, but locks can also be broken — when a moon passes through a resonance with another moon, or when the mode it is locked to changes character. The history drawn here is the simplest case, a single lock held throughout.

The Earth’s tide has the same problem #

The Moon recedes from the Earth by 3.8 centimetres a year, measured to a few hundredths of a millimetre by laser ranging to the reflectors left on its surface. Run that rate backward with a constant quality factor and the Moon would have been at the Earth’s surface about one and a half billion years ago — impossible, since the Moon is far older. The present rate cannot have held.

The resolution is that the Earth’s tidal dissipation is dominated by its oceans, and the oceans’ response is resonant. The Atlantic and other basins have natural sloshing periods close to the twelve-and-a-half-hour lunar tide, so they respond far more strongly than a basin out of resonance would, and the tide at any harbour is a resonance as much as an astronomical forcing. The present dissipation is anomalously high because the present ocean basins happen to be near resonance; in the past, with different continental arrangements and a shorter day shifting the forcing frequency, they were further from it, and the Moon receded more slowly.

The Earth and Saturn are thus two cases of one phenomenon: a tidal dissipation set not by the bulk material’s viscosity but by a resonance in a fluid layer, which makes the present rate both much larger than the smooth estimate and dependent on the system’s history. The difference is in what moves the resonance. For the Earth, plate tectonics rearranges the basins and the lengthening day moves the forcing; there is no reason for the two to stay matched, and the dissipation has risen and fallen over geological time. For Saturn, the planet’s cooling moves the modes and the moon’s migration moves the forcing, and the two are coupled: the migration is driven by the very dissipation that the match produces, which is what keeps them matched.

A tilt that a moon could give a planet #

The consequences reach beyond the moons. Saturn’s spin axis is tilted by 26.7 degrees, and the tilt has no satisfactory explanation from the planet’s formation. One proposal ties it to Titan’s migration.

Saturn’s axis precesses, mainly because of the Sun’s torque on its equatorial bulge, which Titan enhances by its own pull. The precession rate depends on how far Titan is from the planet. Neptune’s orbit also precesses, slowly, and its precession tugs on Saturn’s orbital plane. If the two precession rates came to match, the axis would be captured into a spin–orbit resonance and dragged to larger tilt as Titan’s migration continued to change the axial precession. Titan migrating outward at the measured rate would have brought Saturn into that resonance within the last billion years or so, and the resonance would then have tilted it to its present value — and would continue to tilt it further.

The idea is not established, and competing accounts propose a different history. But it shows how the measured migration rate, a number about one moon, becomes a statement about the planet’s orientation in space. A constant-$Q$ Titan, barely moving, could not do it.

What the figures assume #

Saturn’s constants and the moons’ masses are as measured, and the constant-$Q$ curves use a single $k_2/Q$ fitted to the inner moons. The measured inner-moon values scatter about that fit by factors of a few; some analyses find Rhea, in particular, migrating faster than its neighbours, which a single lock time would also allow.

The lock time is taken from Titan alone and applied to every moon in the comparison. Resonance locking predicts similar, not identical, times for different locked moons, since each is locked to a different mode that may evolve at a different rate. The figures show the time scale the measurements share, not a prediction of each moon’s rate.

The Enceladus heating is the equilibrium value, the heat the resonance supplies when the eccentricity is steady. Enceladus need not be in equilibrium: its heat output may oscillate over tens of millions of years as the eccentricity and the interior’s response feed back on each other, and the measured output could sit above or below the mean.

The histories assume the present rate held throughout. A planet’s modes evolve fastest when it is young, so a lock early in Saturn’s history would have driven faster migration than now; and moons can pass through mutual resonances that break locks and capture others.

Still open: whether every moon is locked, and to what #

Titan’s rate is measured; the inner moons’ rates are measured with larger uncertainties; and the theory predicts that each locked moon migrates on a time scale set by its own mode. What has not yet been done is to identify, for any moon, the mode it is locked to — a frequency in Saturn’s spectrum of oscillations that matches the moon’s tidal forcing to the precision a lock requires.

Ring seismology has measured a few of Saturn’s modes; interior models predict thousands more. Matching one to a moon would turn resonance locking from an explanation of the migration rates into a measurement of the planet’s interior through its moons — and would test the prediction that the same planet’s tides can be weak for one moon and twenty times stronger for another, because the planet is listening to each of them through a different note.