An edge is a balance, not a boundary

Saturn’s A ring ends. Not gradually, over thousands of kilometres, but over a few tens — a distance smaller than the ring’s own thickness is large. That is a strange thing for a fluid to do — rings sit inside the distance at which a body cannot hold itself together, so there is nothing cohesive about the material at all — and the previous rung of this anchor, which weighed a ring by counting the crests of a wave crossing it, established the property that makes it strange: a ring is viscous, and a viscous disc spreads.

For Daphnis and the Keeler gap the balance lands twenty-one kilometres out against a measured half-width of twenty-one. That agreement is the essay.

Why a ring spreads #

Ring particles collide. Each collision is inelastic and low-velocity — random speeds in Saturn’s rings are millimetres per second, which is why the rings are ten metres thick and a hundred thousand kilometres across — but collisions exchange momentum between particles on neighbouring orbits, and neighbouring orbits have different specific angular momenta.

The result is a viscous stress. Momentum flows from the faster inner material to the slower outer material, which is to say angular momentum flows outward, which is to say the inner material sinks and the outer material moves out. It is the same requirement that an accretion disc has to satisfy before anything can fall in, on a ring that has nowhere to fall to. The ring spreads in both directions, and the rate is set by a kinematic viscosity.

The spreading time is short. A ring of Saturn’s surface density and viscosity doubles its width in something like ten million years, which against the age of the solar system means the rings should long since have become a diffuse sheet from the planet’s surface out past every moon. They have not.

What pushes back #

A satellite orbiting just outside a ring exerts a torque on it. The mechanism is resonant: at each radius where a ring particle’s orbital period is a simple ratio of the satellite’s, the satellite’s perturbation is coherent from one encounter to the next and builds up. That is the mechanism that clears a gap in one place and locks a moon in another, applied where the resonances are packed too closely to be told apart. Summing the first-order resonances over the region between the ring edge and the satellite gives a torque per unit mass proportional to the square of the satellite’s mass and to the inverse cube of the gap. The steepness is what makes an edge possible: a torque that fell as slowly as the viscous one does would produce a gradient rather than a boundary.

The balance, and what it measures #

Setting the two torques equal gives a gap half-width proportional to the satellite’s mass to the two-thirds and inversely to the viscosity to the one-third. Everything in that expression is measurable except the viscosity, which is why the balance is useful.

The Keeler gap is the cleanest case. It is 42 kilometres wide, Daphnis sits in the middle of it, and Daphnis’s mass is known from the amplitude of the waves it raises on the gap’s edges — a measurement made from images rather than from any dynamical fit. Putting those in and solving for the viscosity gives about twelve square centimetres a second.

That number cannot be measured any other way. It is the collision rate among metre-sized particles a billion kilometres from here, and the only handles on it are this balance, the damping length of density waves, and the vertical thickness of the ring inferred from occultations.

The width, worked through #

The scaling is worth doing by hand once, because it explains why some rings have visible shepherds and most do not.

Set the satellite’s torque per unit ring mass equal to the viscous one. The satellite’s term carries the square of its mass ratio to the planet and the inverse cube of the gap in units of the orbital radius; the viscous term carries the dimensionless viscosity, which is the kinematic viscosity divided by the square of the radius times the orbital rate. Solving,

$$\frac{\Delta}{a} \;\sim\; \left(\frac{q^2}{\tilde\nu}\right)^{1/3},$$

with $q$ the satellite-to-planet mass ratio. The two-thirds power on the mass is the important part. A shepherd ten times lighter opens a gap only four and a half times narrower, so gaps do not scale away quickly — but the same shepherd is a thousand times fainter, so it disappears from the images long before its gap does.

That asymmetry is the whole of the missing-shepherd problem. Uranus’s epsilon ring has a width consistent with a pair of satellites a few tens of kilometres across, and those were found. The other rings are consistent with satellites a few kilometres across, which Voyager could not have seen and no subsequent mission has looked for.

Momentum has to go somewhere, and it goes to the moon #

The confinement is not free. Every unit of angular momentum the satellite removes from the ring’s outer edge it keeps, so the satellite migrates outward while the ring is held in.

The rate is small and it is not negligible over the age of the solar system. A shepherd of Daphnis’s mass, exchanging angular momentum with the A ring at the rate the torque balance implies, moves outward by an appreciable fraction of its own orbital radius in a few hundred million years — the same secular outward drift that tidal friction gives the Moon, driven by a ring rather than by a bulge — which is short compared with four and a half billion.

That leads directly to the age problem the last section flags. Either the rings are much younger than the planet, or the exchange is slower than the balance says, or something returns angular momentum to the rings. The first is currently the favoured answer, and its evidence is independent: the rings are far too clean to have been collecting interplanetary dust for four billion years.

The other narrow rings, and the moons that are missing #

Shepherding was proposed before it was seen. Goldreich and Tremaine argued in 1979 that Uranus’s narrow rings — some of them only a few kilometres wide, with edges sharper than any diffusive process allows — required confining satellites, and predicted their approximate masses and positions.

Voyager found two of them, Cordelia and Ophelia, flanking the epsilon ring exactly as required. It did not find shepherds for the other nine.

That absence is the honest state of the subject. Some narrow rings have identified shepherds; most do not, and the candidates are below the detection limit of every mission that has looked. The alternatives proposed — self-gravity of the ring holding it together, or confinement by a resonance with a distant satellite rather than a nearby one — work for particular cases and not as a general answer.

The other way to hold an edge #

Nine of Uranus’s eleven narrow rings have no known shepherd, and the alternative that has held up best is that a ring can confine itself.

The epsilon ring is the test case, and its awkward property is not its narrowness but its eccentricity. It is a slightly elliptical ring whose width varies with position — about twenty kilometres at periapsis and ninety-six at apoapsis — and whose apsidal line precesses as a single rigid unit. A ring is not rigid, and the planet’s oblateness makes an inner orbit precess faster than an outer one, so a set of nested elliptical orbits should shear apart in a few hundred years.

The proposed answer is the ring’s own gravity. The inner edge is pulled outward by the material beyond it and the outer edge inward by the material within, and those forces contribute to the precession with the opposite sign to the planet’s oblateness. For one particular surface density the two effects cancel differentially, and the whole ring precesses at a single rate.

That makes the ring’s mass a prediction rather than a parameter, and it is the reason the model is taken seriously: the required surface density can be compared with the one inferred from occultation optical depths. The comparison is not clean — the required mass came out several times larger than the observed material appeared to supply — but it is a genuine test that the model could have failed outright and did not.

A self-confining ring needs no undiscovered moon and does need an unusually precise surface density, and which of those is the more comfortable assumption is a question the observations have not yet settled.

There is a general point in the comparison, and it is about what counts as an explanation. Shepherding explains an edge by naming an object; self-gravity explains it by naming a condition on the ring itself. The first is falsified by looking and failing to find, which is a weak test when the object is a few kilometres across and a billion kilometres away. The second is falsified by measuring a surface density, which is hard but is not a matter of detection limits — and that difference is why the self-gravity model has been argued about productively for forty years while the missing shepherds have mostly been argued about by assertion.

What a shepherd does to the material it does not confine #

Daphnis holds the Keeler gap open, and it also does something more visible and less important, which is worth separating out because the two effects are routinely conflated.

Every ring particle near the gap edge passes the moon at the synodic period, and each passage gives it a small kick. The kicks are coherent along the edge, so the edge is left with a wavy pattern whose wavelength is the distance the edge material travels relative to the moon in one synodic period. Because the moon’s orbit is slightly inclined, part of that kick is vertical, and the edge is corrugated out of the ring plane as well as within it.

Those are the waves whose shadows were photographed at Saturn’s equinox, and they are what the moon’s mass was measured from. The amplitude is set by the moon’s mass and the distance to the edge, both of which are known, so the comparison is a check rather than a fit.

The waves also decay. Collisions among ring particles damp the pattern as the edge material moves downstream, and the damping length is another route to the viscosity — the same quantity the gap width gives, measured over metres of amplitude rather than kilometres of width.

What the waves are not is the confinement. The confining torque is a resonant, secular exchange summed over many passages; the waves are the immediate kinematic response to one. A picture showing a spectacular scalloped edge is showing the second, and the first is invisible.

What the balance assumes #

Three things, and each is known to be violated somewhere.

The first is that the viscosity is a constant. It is not: the transport in a dense ring is dominated by self-gravity wakes — transient clumps sheared out by the differential rotation — and their contribution depends on the surface density, so the viscosity varies across a ring and is not even a single-valued function of the local state.

The second is that the torque can be summed over resonances treated independently. Close to the satellite the resonances overlap, and overlapping resonances are the standard route to chaos rather than to a clean sum. The third is that the system is in steady state. A ring whose edge is being confined is losing angular momentum to the satellite, and the satellite is gaining it and moving outward. The Keeler gap’s edge is therefore moving, and the balance drawn in the hero is an instantaneous one rather than a permanent arrangement. There is a third possibility that has gained ground as the ring systems have been mapped in more detail, and it is that the question is badly posed. Not every sharp edge need have the same cause. Saturn’s A ring outer edge sits at a strong resonance with a known moon; the Keeler gap has a shepherd sitting in it; the Uranian rings have neither and have a plausible self-gravity model; and the Encke gap has a moon that is too small to account for the gap’s full width on the balance above. A mechanism that explains one of those need not be wrong because it fails on another, and treating “what confines a narrow ring” as a single question with a single answer is how the missing shepherds came to look like a crisis rather than a list of separate cases.

The methodological point generalises past rings. A phenomenon that appears in several systems invites a common explanation, and the invitation is worth resisting until the systems have been shown to be alike in the respects the explanation depends on. Here they are not: the ratio of shepherd mass to ring mass, the strength of the nearest resonance, and the ring’s own optical depth all vary by orders of magnitude across the four cases, and each of those quantities appears in the balance.

What was actually measured #

The chain from images to a viscosity is worth writing out, because it is unusually short for this subject and every step is geometric.

Daphnis’s mass comes from the amplitude of the vertical waves it raises on the gap edges. Those waves were photographed at Saturn’s equinox, when the Sun was in the ring plane and vertical structure cast shadows tens of kilometres long across an otherwise flat sheet — the one geometry in twenty-nine years that makes a hundred-metre feature visible from a spacecraft. The amplitude scales with the satellite’s mass and the shadow length gives the amplitude.

The gap width comes from stellar occultations: a star’s light is monitored as the rings pass in front of it, and the sharpness of the transition at the edge is limited by the projected size of the star rather than by the instrument. Widths are known to hundreds of metres, which is far finer than any image resolves — the edge of a shadow is a diffraction pattern rather than a line, and reading it as one is how these measurements reach below the resolution limit.

The orbital elements come from imaging over years.

The viscosity is the only inferred quantity, and it is inferred from a balance that assumes the ring is in equilibrium. If the Keeler gap is still opening or still closing, the number is wrong by however far from equilibrium it is — and nothing in the observations says whether it is.

It is worth noticing that the balance is a stability argument as well as an equilibrium one. Displace the edge inward, toward the shepherd, and the satellite’s torque rises as the inverse cube while the viscous torque does not move — so the material is pushed back out. Displace it outward and the reverse happens, and viscosity carries it in. The crossing is therefore an attractor rather than a coincidence, which is the reason a real edge stays sharp under all the perturbations a ring system supplies rather than being smeared by the first passing moon. That stability is also what makes the width usable as a measurement. A quantity that sits at an unstable equilibrium tells an observer only where it happened to be put; one that sits at a stable equilibrium tells them about the balance that holds it there. The Keeler gap’s width is a measurement of ring viscosity for exactly that reason, and the same argument fails for features held open by a single resonance rather than by a nearby body.

One more shepherd mass shows how the balance at the edge responds.

An edge that is a balance rather than a boundary can be moved, and the fact that these edges have not moved measurably in four decades of observation is itself a constraint on how fast the two terms in the balance can change.

Where the ladder goes #

The next rung is the ring’s own self-gravity, which the balance above treats as part of the viscosity and which is a separate mechanism: the wakes that dominate the transport are gravitational instabilities, and their properties depend on the surface density in a way that makes the ring’s response to a satellite nonlinear.

The other direction is the age problem. Every process discussed here removes angular momentum from the rings and gives it to the moons, so the rings are being ground down and the moons pushed out. Running that backwards makes the rings far younger than the solar system, which is either a strong result about their origin or a sign that something in the accounting is missing.