A ring weighed by the wave crossing it

Almost every property of Saturn’s rings resists measurement in the same way. They are made of pieces from a centimetre to a few metres across, which no telescope and no camera has ever resolved. They are vertically thin — tens of metres, against a radial extent of sixty thousand kilometres — so no spacecraft has flown through them, and none ever will deliberately. And their total mass is small enough that it does not measurably perturb anything: Saturn is a hundred million times heavier, and the moons that orbit outside the rings feel a pull from them that is far below the accuracy of any orbit determination.

So the mass of the rings ought to be unknowable. It is known, to a few per cent in places, and the instrument is a wave.

This is a familiar shape in this collection — a quantity recovered not from the object but from something the object does to a signal that was passing anyway, as a shadow’s edge gives a shape or a line width gives a distance. What is unusual here is how little is assumed. The measurement below rests on Newtonian gravity, on the moons’ orbits, and on the geometry of a wave, and on nothing about what ring particles are made of or how large they are.

Why a ring has waves at all #

A ring particle on a circular orbit does not know that a moon exists in any way that matters. It feels a periodic tug, once per synodic period, and the tug averages to nothing.

Unless the periods are commensurate. If the particle completes exactly five orbits while the moon completes three, the tug arrives at the same phase of the particle’s orbit every time, and the small kicks accumulate rather than cancelling — the mechanism that clears a gap in one place and locks a moon in another. What accumulates is the eccentricity, and an eccentric orbit in a disc of otherwise circular ones is a place where the disc has been compressed. The distinction between accumulating and averaging away is the whole of resonance, and it is the same distinction that decides whether an element of an orbit drifts or merely oscillates. That much would produce a disturbance at one radius. What makes it a wave is the ring’s own gravity. A compressed region attracts material into itself from either side, which compresses the neighbouring region, which does the same — so the disturbance propagates outwards, and the restoring force for that propagation is the self-gravity of the ring rather than any pressure. Rings have essentially no pressure: their particles are on ballistic orbits and collide rarely.

The dispersion relation, and the one unknown in it #

For a tightly wound wave in a nearly Keplerian disc the relation between frequency and wavenumber is

$$(\omega - m\Omega)^2 \;=\; \kappa^2 - 2\pi G\sigma|k|,$$

where $\sigma$ is the mass per unit area of the disc. Near an inner Lindblad resonance the left-hand side vanishes, and it vanishes linearly in the distance from resonance, so

$$k(r) \;=\; \frac{3(m-1)\,\Omega_L^2\,(r-r_L)}{2\pi G\sigma\,r_L}.$$

The wavenumber grows in proportion to the distance travelled. The wavelength therefore shrinks as the reciprocal of that distance, and the phase — which is the integral of the wavenumber — accumulates as the square.

The appearance of $\sigma$ there is the interesting part, and it is worth being clear about where it comes from. It is not a pressure term and it is not a correction: it is the ring pulling on itself, the same self-gravity that in a much denser disc would be the restoring force behind a spiral arm. A ring of test particles with no mass would have no dispersion relation and no wave; the disturbance would sit at the resonance and never travel.

Every quantity in that expression except $\sigma$ is known to several figures. The resonance radius comes from the moon’s semi-major axis, which comes from tracking. The orbital frequency comes from the resonance radius and Saturn’s mass. The integer $m$ comes from which resonance it is. So the rate at which the crests crowd together is a measurement of the ring’s mass per unit area, and of nothing else.

Thirty-five kilograms per square metre is about the weight of a paperback resting on a square metre of table. Integrated over the A ring it is something like a third of the mass of Mimas, and the total mass of the main rings is now known well enough to be an argument about their age rather than a hole in the accounting.

How the crests are counted #

The waves are not photographed. At their outer ends the wavelength is a kilometre or less, which no camera resolves from Saturn orbit, and in any case what is wanted is not an image but a profile.

The technique has a floor, and it is not the instrument.

The rest of the chain is arithmetic on the light curve. What the detector records is a count rate against time; converting that to optical depth against radius needs the star’s position, the spacecraft’s, and the ring plane’s orientation, and the conversion is where the precision is won or lost. A tenth of a per cent error in the geometry is a tenth of a per cent error in every radius, which for a wave whose crests are a kilometre apart at fifty thousand kilometres is fatal. The reconstruction is done against ring features of known radius — the sharp edges — which turns the whole ring system into its own ruler.

Radio occultations, in which the spacecraft’s own transmitter is the source and the Earth the receiver, do better in one respect and worse in another: the longer wavelength makes the Fresnel zone larger, but the signal is coherent, so the diffraction can be undone by processing rather than merely suffered. That is how the finest ring profiles were obtained.

The resonances that make edges rather than waves #

A wave is what a resonance produces where the ring can carry one. Where it cannot — or where the resonance is strong enough — the result is an edge.

The mechanism is angular momentum. A resonance is a channel through which the ring and the moon exchange it, and the exchange has a sign: material inside a resonance loses angular momentum to the moon and material outside gains it. A ring edge held at a resonance is therefore being actively confined — the moon’s torque balances the ring’s own tendency to spread — and the moon is being pushed outwards in return. That is the same accounting as the tidal transfer that is lengthening the day and lifting the Moon, with a resonance in place of a bulge.

The same arithmetic, elsewhere #

Two other systems in this collection are the same calculation with different numbers in it.

Where the ring stops #

The rings have an outer edge for the reason above and an inner extent for a different one. The limit is a good deal softer than the phrase suggests. Real ring particles are held together partly by material strength and partly by gravity, and the small moons just outside the limit — Pan, Atlas, Prometheus — are unreasonably low in density and unreasonably elongated, which is what a body assembled marginally outside a Roche limit ought to look like.

Both halves of the measurement — the dispersion relation that turns a wavelength into a surface density, and the wave whose crests are counted — are worth reading at a second density, because the whole method is a claim that the two agree.

The moons on the other end of the torque #

A resonance exchanges angular momentum, so a ring edge held by a moon is a moon being pushed by a ring. That side of the transaction is measurable too, and measuring it has produced a result nobody expected.

The moons’ orbits are tracked over decades by imaging and by radio ranging, and their semi-major axes are known well enough that a secular drift of centimetres per year is detectable. What the tracking found is that Saturn’s inner moons are migrating outwards considerably faster than the standard tidal calculation allows — by factors of several to ten, depending on the moon.

The tidal calculation’s weak point is a single parameter: the efficiency with which Saturn dissipates the tide its moons raise on it, conventionally a quality factor. Assuming it is a constant, and calibrating it on one moon, predicts the migration rate of every other. The observed rates do not follow that pattern.

The explanation now favoured is that the dissipation is not a constant at all but a resonance in the planet. Saturn’s interior has normal modes, and a mode whose frequency happens to match a moon’s tidal forcing dissipates far more strongly than the smooth background does. As the moon migrates, the forcing frequency changes; as the planet’s structure evolves, the mode frequency changes; and if the two track each other the moon stays locked to the resonance and migrates at whatever rate keeps it there.

That is resonance locking, and it predicts a migration rate set by the planet’s evolutionary timescale rather than by its dissipation efficiency — which is why the observed rates are similar for moons of very different masses, a pattern the constant-quality-factor picture cannot produce.

The rings and the moons are two ends of one accounting, and the surprise came from the end that could be tracked rather than the end that had to be inferred.

What the measurement then said #

The mass that the density waves returned is the reason the rings are now thought to be young.

A ring spreads. Collisions between particles transfer angular momentum outwards, exactly as viscosity does in any disc, and the spreading time depends on the surface density: a more massive ring spreads faster. Meanwhile the innermost material rains onto Saturn, and the outer edge is held by moons that are themselves being pushed outwards by the torque. Put the measured mass into that accounting and the main rings cannot have looked like this for four and a half billion years. There is a second and independent argument to the same conclusion, and it is about cleanliness: micrometeoroids fall onto the rings continuously, carrying silicates and organics, and a ring exposed for the age of the solar system should by now be visibly dirty. The main rings are more than ninety-five per cent water ice. Either the pollution rate is far lower than measured, or the rings have not been out there very long. The numbers point to something between ten million and a few hundred million.

Which converts a static object into an event. If the rings are young, something happened — a moon disrupted, a comet captured and torn apart — recently enough that the solar system was already essentially as it is now. The alternative is that the accounting is missing a term, and both possibilities are live.

It is worth noticing how the argument runs, because the shape of it recurs. A quantity that could not be measured directly was extracted from a wave; the quantity turned out to set a rate; the rate turned out to be fast compared with the age of everything around it; and a structure that had been treated as permanent scenery became a recent event with a cause to be found. Nothing about the rings changed. What changed was that a number had been put on them.

The mass measured the other way #

Everything above weighs the rings locally, wave by wave, and then integrates. There is a second route that weighs them all at once, and its agreement with the first is what makes the age argument stand.

A spacecraft passing between a planet and its rings feels the gravity of both. Saturn’s field is enormous and the rings’ is tiny — the ratio is of order a hundred million — but the two have different geometries: the planet’s pull is very nearly radial, while the rings’ is a thin sheet whose attraction is directed towards the ring plane and falls off differently with distance. A trajectory threading the gap therefore carries a small signature that no adjustment of the planet’s own harmonics can imitate.

Extracting it is a Doppler measurement of the kind the navigation of any deep-space mission rests on: the spacecraft’s radio carrier is returned coherently, the range rate is measured to a fraction of a millimetre a second, and the whole trajectory is fitted with the ring mass as one free parameter among many.

The difficulty is that the ring mass is strongly correlated with the planet’s own gravity field, particularly with the zonal harmonics that describe its oblateness, and with the deep winds that produce some of those harmonics. Separating them required many passes at different geometries, and the published uncertainty is dominated by that correlation rather than by the Doppler noise.

The answer that came out is close to the total obtained by summing the wave measurements — around four tenths of Mimas’s mass for the whole main ring system. The two methods share nothing: one is a local dispersion relation read off an occultation profile, the other a global perturbation read off a radio link. Where two measurements of one quantity have no systematic in common and agree, the quantity is measured, and it is that agreement rather than either result alone that licenses the argument about the rings’ age.

It is worth being precise about what “close” means here, because the agreement is not a coincidence of central values. The wave method’s uncertainty is dominated by the B ring, whose opacity defeats the occultation profiling over much of its extent, so its integrated total carries an error bar of tens of per cent. The gravitational method’s uncertainty is dominated by the correlation with Saturn’s own zonal harmonics, and is a few per cent. The two overlap comfortably, and the gravitational result is the tighter — so what the comparison establishes is not that the wave method is right in detail but that the local measurements can be integrated without a missing component.

That is the specific worry the second method was needed to remove. A ring could in principle contain a substantial mass in a form the waves do not see: material so opaque that no occultation profiles it, or so finely divided that it contributes surface density without contributing structure. A global gravitational measurement is blind to how the mass is arranged and sensitive only to how much there is, which is exactly the complement required.

Where the picture stops #

The particles are not smoothly distributed, and the wave equation assumes they are. At the densities of the A and B rings the ring’s own self-gravity organises the particles into transient elongated clumps — self-gravity wakes — a few tens of metres long, forming and shearing apart on an orbital timescale. They make the ring’s opacity depend on the direction it is viewed from, which is measured, and they change the effective viscosity, which is not.

The B ring is opaque and its mass is the hardest to get. Density waves need the ring to be transparent enough for an occultation to profile it and massive enough to carry a wave. The B ring is too opaque over much of its extent, and its surface density is the largest single uncertainty in the ring system’s mass.

And the wave is linear only near the resonance. A few wavelengths out the amplitude is no longer small and the wave steepens into something closer to a shock, at which point the clean relation between crest spacing and surface density acquires corrections. The fits use the first several crests for exactly this reason.

And the same construction at a different resonance entirely, since the method’s independence of which moon drives it is part of what makes it a measurement.

One more configuration shows the other way a ring’s surface density becomes measurable.

Where this ladder goes next #

Later rungs on this anchor: bending waves, which are the vertical rather than the radial version of the same dispersion relation and which measure the same surface density through a different geometry; the shepherding of a narrow ring by a pair of moons, and why the arrangement is more fragile than it looks; self-gravity wakes and the direction-dependent opacity they produce; the propellers, which are the wakes of moonlets too small to open a gap and which have been watched migrating; and the rings of the other three giant planets, each of which is a different answer to the same question about what material inside a Roche limit does.