A ring that is thinner from one side

Saturn’s A ring has been seen to be brighter in two opposite quarters of its circle than in the other two since the 1950s, when observers measuring it through telescopes on the ground found the asymmetry and could find no reason for it. A ring of particles in circular orbits is the same all the way round. Its brightness can depend on the angle between the Sun and the observer, and on the tilt of the ring to both, but not on longitude around the planet, because nothing in an axisymmetric ring knows which way it is facing.

The explanation, proposed in 1976 and confirmed by every spacecraft since, is that the ring is not axisymmetric on small scales. It is made of slabs — elongated clumps a few dozen metres across, held together by the particles’ mutual gravity, tilted to the direction of orbital motion by the shear of the differential rotation, and all tilted the same way. A ring made of identically tilted slabs is not the same in every direction, and looking along the slabs shows more gap than looking across them.

This essay is about those slabs: why they exist, why they have the size they have, why they come and go in an orbit rather than growing into moons, and what they do to every measurement of the ring that treats it as a uniform sheet. The ring weighed by the wave crossing it took the ring’s surface density from the wavelength of a density wave launched at a moon’s resonance. The same ring’s own gravity, acting on itself with no moon involved at all, produces a structure a hundred times smaller and far more consequential for how the ring looks.

The measurement that changes with direction #

The clearest evidence is not the old brightness asymmetry but the occultations, because an occultation measures something simpler — the same measurement that reads an asteroid’s outline from the edge of its shadow, turned on a ring. A star passes behind the ring, and the fraction of its light that comes through is recorded every few milliseconds. That fraction, the transmission, is converted to an optical depth by assuming the ring is a uniform layer: a slant path through a layer of normal optical depth $\tau$ at elevation $B$ transmits $e^{-\tau/\sin B}$, and the measured transmission gives $\tau$ directly.

When the Cassini spacecraft observed dozens of stellar occultations through the same rings, from many directions, the optical depth of the same radius in the A ring came out different for different stars — by factors of two, in a pattern that depended on the direction of the sight line in the ring plane and not on anything else. A uniform layer cannot do that, since its slant path length depends only on the elevation.

The model behind the figure is the one fitted to the occultations, and it is crude on purpose. The ring is replaced by infinitely long rectangular bars of width $W$ and height $H$, repeated at a spacing $\lambda$, nearly opaque, with a thin uniform layer of particles in the gaps between them. A ray at elevation $B$ whose direction in the ring plane makes an angle $\psi$ with the bars’ long axis is blocked wherever it meets a bar, and the shadow a bar casts across the gap has width $H\cot B\,|\sin\psi|$. The open fraction of each period is therefore

$$f_{\text{open}} = \max\!\left(0,\; 1 - \frac{W + H\cot B\,|\sin\psi|}{\lambda}\right),$$

and the transmission is that fraction times the transmission of the gap layer.

Every feature of the figure is in that expression. Along the bars $\sin\psi = 0$, the shadows vanish, and the open fraction is simply the gap fraction. Across them the shadows grow, and at a low enough elevation they cover the gaps completely. The pattern repeats every half-turn because a bar looks the same from either end.

Why there are slabs at all #

The slabs are the ring’s response to its own gravity, and the length on which they form was worked out for galaxies before anyone applied it to rings.

A thin rotating disc of self-gravitating material is pulled towards clumping by its gravity and resisted by two things. One is the random motion of its particles, which acts like a pressure and disperses any clump smaller than a certain size. The other is the shear: in a disc where the inner parts orbit faster than the outer, a clump that is too wide in radius is pulled apart by the difference in orbital speed across it. Gravity wins only between those two limits, and the largest scale on which it can win against the rotation is Toomre’s critical wavelength,

$$\lambda_{\text{cr}} = \frac{4\pi^2 G\Sigma}{\kappa^2},$$

where $\Sigma$ is the surface density and $\kappa$ the epicyclic frequency, which in a ring orbiting a planet is equal to the orbital frequency. It was derived for the stellar discs of galaxies, where it comes to kiloparsecs; in a planetary ring the surface density is modest but the orbital frequency is enormous, because the ring sits deep in the planet’s gravity, and the wavelength comes out tiny.

Sixty metres is a few dozen of the ring’s larger particles side by side. The ring is fourteen and a half thousand kilometres wide and some tens of metres thick, so on the scale of the whole A ring the instability is microscopic — invisible to every camera that has imaged the rings, including those on the spacecraft that flew within a few thousand kilometres of them. It is seen only statistically, through what it does to light, which is exactly the situation of the ring weighed by its wave: a structure nobody can photograph, measured by a property it imposes on a signal that passes through.

The tilt comes from the shear. A clump that forms in a differentially rotating ring is stretched in the direction of motion, because its inner edge orbits faster than its outer edge, and its own gravity holds it together only along the shorter axis. The result is an elongated structure whose outer end lags behind its inner end — a trailing tilt, of about twenty degrees from the orbital direction in simulations and in the fits to observations. The angle is set by a balance between the shear, which keeps rotating the clump towards the orbital direction, and the clump’s self-gravity, which keeps reforming it on a new tilt, and since both are set by the orbital frequency, every slab in a given part of the ring has roughly the same angle.

A ring that is deep in the unstable region #

Whether a disc forms wakes at all depends on how much random motion its particles have, and the measure is Toomre’s parameter,

$$Q = \frac{c\,\kappa}{3.36\,G\Sigma},$$

for a disc of particles with velocity dispersion $c$. A disc with $Q$ well above one is stable against clumping; below about two, particle simulations show the wakes appearing.

The point the figure makes is that the A ring is not marginally unstable. In a gas, the velocity dispersion is the sound speed and can be anything; in a ring, it is set by collisions between inelastic ice particles, and those are almost dead — a collision at a few millimetres a second returns only a fraction of the energy, so random velocities decay until they are comparable to the speed at which the shear brings neighbours together, which is the particle size times the orbital frequency. For metre-sized particles in the A ring that is a fraction of a millimetre per second.

The ring’s own self-gravity then makes it colder still in one sense and hotter in another. The wakes stir the particles: a particle falling into a forming slab and then flung out when it is sheared apart gains random velocity from the encounter, and this heating by the wakes is what holds the dispersion up against the collisional cooling. The ring settles into a state in which the wakes heat it just enough to keep $Q$ near one or two. It is a self-regulating instability — a balance between a driving and a damping of the kind that holds a shepherded ring’s edge in place, set here inside the ring rather than at its edge — and the same one the stellar discs of galaxies run on, where spiral arms heat the disc that makes them until the disc is too hot to make them.

So the slabs are not a disturbance of the ring but its normal condition. Every patch of the A ring at any moment is a field of slabs forming and dissolving, and that is the reason the azimuthal asymmetry is permanent and the same every time it is measured: the individual slabs live for an orbit, but the statistics of their size and tilt are fixed by the ring’s surface density and its distance from the planet.

Why the slabs never become moons #

A clump that forms by gravitational instability in a disc can grow into a permanent body — that is how planets are thought to begin in some discs and how stars begin in galactic ones — and in Saturn’s rings it does not. The slabs dissolve. Why they dissolve is the question of where a body is close enough to its planet to be pulled apart, asked of two touching particles rather than of a moon.

The right quantity is the Hill parameter, built from the sphere inside which a body’s own gravity dominates the planet’s: the radius of the joint Hill sphere of two touching particles, divided by the distance between their centres. If it exceeds one, the pair’s mutual gravity reaches beyond their surfaces and they can stay together against the tide. If it is less than one, the tide pulls them apart as soon as anything disturbs them.

That the Hill parameter has no size in it is worth dwelling on. The joint Hill sphere of two particles grows with their mass, which grows as the cube of the radius, so its radius grows linearly with the particles’ radius — exactly as fast as the separation between their centres. Two pebbles and two boulders at the same distance from Saturn are equally able or unable to hold each other, and what decides it is how dense the material is and how deep in the planet’s gravity it sits. This is the Roche limit seen from the inside, and the same absence of size that fixes the fastest spin a rubble pile can keep: the radius inside which a body of a given density cannot hold itself together is the same radius at which the particles it would be assembled from cannot hold each other.

The particles of the A ring are porous ice, with densities thought to be well below that of solid ice, so the Hill parameter in most of the A ring is below one but not far below. That is the regime in which gravity is strong enough to gather particles into slabs every orbit but too weak to keep them, and the slabs shear out and reform continually rather than collapsing. Bodies that do survive in the ring are larger and denser than the particles around them, and announce themselves the way the small moons that clear gaps at resonances do — by what they do to the material nearby. Further out, beyond the A ring’s outer edge, the parameter passes one for any plausible density, and the F ring — narrow, clumpy, with small moonlets embedded in it — sits just past it. Further in, in the C ring, the parameter is well below one and the surface density is lower too, and there is no evidence of wakes there at all.

The A ring is therefore a narrow window in the ring system: far enough out for self-gravity to matter, close enough in for the tide to undo what it builds. The structure exists because the ring sits on the boundary that separates a ring from a moon.

What the slabs look like #

The field of wakes is not drawn in any photograph, but its statistics are fitted from the occultations, and it can be drawn as the model describes it.

The fits that produced numbers like these are worth describing, because they are the only way the slabs have been measured. Each occultation gives one transmission at each radius, along one direction at one elevation. A single occultation cannot separate a uniform ring of moderate optical depth from a slab-filled ring with dense slabs and clear gaps, because both can produce the same transmission. Dozens of occultations through the same radius from different directions can, because the uniform ring predicts the same normal optical depth from every direction and the slab-filled ring predicts the pattern in the first figure. Fitting the pattern gives the slabs’ tilt, the ratio of their height to their width and the optical depth of the gaps.

The answers are consistent with the theory and fix its free parameters. The tilt comes out near twenty degrees, trailing, as the simulations say. The gaps have optical depths of a tenth or two — they are not empty, but they hold far fewer particles than the slabs. And the slabs are flat: wider than they are tall by a factor of several, so that a slab is less a wall than a low ridge lying across the orbit.

The particles a uniform model loses #

Here is where the slabs change the measurement of the ring’s mass.

A uniform sheet with a given transmission has a given optical depth, and an optical depth is the total cross-sectional area of the particles per unit area of ring. Convert it to mass with a size distribution and a density, and the uniform reading weighs the ring. But in a ring made of nearly opaque slabs and nearly empty gaps, the light that gets through has passed through the gaps and has never sampled the slabs at all. The transmission measures the gaps, and the slabs, which hold most of the particles, contribute only by taking up space.

The consequence is that optical depth is a poor guide to mass in the parts of the rings where the wakes are strongest. It has never been the main route to the ring’s mass — the density waves weigh the ring through its gravity, which the slabs cannot hide — but it is the only route in the regions where no wave propagates, and in the densest part of the B ring, where the optical depth is so high that occultations see almost nothing, the surface density was for decades a matter of argument between models that differed by factors of several. The total mass of the rings was eventually measured by the gravity they exert on a spacecraft flying between them and the planet, at about forty per cent of the mass of Mimas — at the lower end of the range the optical-depth arguments allowed, and a measurement that needed no model of how the particles are arranged.

The same saturation affects the age of the rings. Rings exposed to the steady rain of interplanetary dust are darkened by it, and the purity of the ice says how long they have been exposed — but the conversion from purity to age goes through the mass of ice that the dust has been mixed into. A ring that holds more mass than its optical depth suggests is a ring that has diluted more dust, and looks younger than its exposure for the same reason a large glass of water looks clearer than a small one with the same drop of ink in it.

The old asymmetry, explained #

The brightness asymmetry that started this is the same geometry seen in reflected sunlight rather than transmitted starlight.

An observer looking at the lit face of the ring sees a mixture of slab tops, slab sides and gaps, in proportions that depend on the direction of view relative to the slabs. Where the sight line runs along the slabs it looks down the gaps and sees more of the dark sky beyond and of the sparse particles there, and the ring appears fainter; across the slabs the view is filled by their sides and the ring appears brighter. Since the slabs are tilted about twenty degrees from the orbital direction everywhere, the two directions are reached at the same two longitudes for every observer at a given geometry — two minima and two maxima round the ring, offset from the points where the ring is seen edge-on by about the tilt. That is the quadrant asymmetry measured from the ground.

The asymmetry is strongest in the middle of the A ring and weaker towards its edges, much weaker in the B ring and absent in the C ring. Both facts fit the instability: the A ring has the right combination of surface density and distance from Saturn, and the B ring, though denser, is so optically thick that slabs and gaps alike are opaque and nothing is seen through either. The weakening at the A ring’s outer edge is thought to be where density waves from the moons stir the ring and raise its random velocities, pushing $Q$ up and smoothing the wakes.

It is a satisfying sequence: a puzzle from ground-based photometry, an explanation from the theory of galactic discs, and confirmation decades later from occultations and from simulations with hundreds of thousands of particles. Each step came from a different instrument, and all three describe the same tilted slabs.

What depends on the elevation #

The figures show that the direction of the sight line matters only at moderate and low elevations. At sixty degrees above the ring the inferred optical depth varies by about twenty per cent round the circle; at twelve degrees it varies by more than a factor of ten, because the shadow of each slab lengthens as $\cot B$ and eventually spans the whole gap.

This has a practical consequence for anyone comparing observations of the rings made at different times. The ring’s opening angle as seen from the Earth changes through Saturn’s twenty-nine-year orbit, from edge-on to twenty-seven degrees and back. Measurements of the A ring’s brightness made at different opening angles are measurements of different mixtures of slab and gap, and a change in the ring’s apparent properties between two epochs need not be a change in the ring.

Where the model stops #

The slabs are not rectangular bars. The model replaces a field of irregular, transient, clumped structures with an idealised periodic one. The fitted parameters are effective values for that idealisation — the real distribution of slab widths, heights and spacings is broad, and the occultations constrain its average shape rather than its individual members. Simulations are the main check that the effective values are meaningful, and they agree at the level of the tilt and the gap optical depth; they disagree in detail about the slabs’ vertical structure.

The dimensions drawn here are representative, not fitted. The slab width, height, spacing and gap optical depth used in the figures are of the order of those the occultation fits return in the middle of the A ring, and they vary with radius across it. The figures show what the geometry does; the measured values come from the fits.

The surface density is taken from elsewhere. The critical wavelength is proportional to the surface density, and the surface density used here is the one density waves measure. Where no wave propagates, the wavelength is a model inference, and the circularity noted above — the slabs hide the mass that sets their own size — is real.

Collisions are summarised by one number. The coefficient of restitution of porous ice at millimetre-per-second speeds, the friction between rotating particles and the particles’ size distribution all enter the velocity dispersion, and none is known well. The argument that the ring sits deep in the unstable region is robust to them because the margin is large; anything quantitative about the slabs’ shape is not.

The surprising connection #

The theory that explains the A ring’s quadrant asymmetry is the theory of spiral structure in galaxies, applied to a system ten trillion times smaller.

The quantities are the same and their roles are the same. A stellar disc’s surface density and epicyclic frequency set a critical wavelength of a few kiloparsecs; a ring’s set one of sixty metres. A stellar disc’s random velocities are raised by the spiral structure until the disc is barely stable; a ring’s are raised by its wakes until it is barely unstable. In both, the instability is self-regulating, the structure transient, and the pattern statistically steady while every individual feature lives for about a rotation. The equations were written for galaxies in the 1960s and borrowed for rings in the 1970s with no change but the numbers.

The difference is that in a galaxy the structure is the thing being explained, visible in any photograph, while in a ring it is a hidden variable — invisible directly, inferred from how the light that passes through it varies with direction. The same theory is a description of spiral arms in one case and a correction to an optical depth in the other.

Still open: how deep the wakes go into the B ring #

The B ring is the densest part of the system and the one whose mass mattered most to the argument about the rings’ age. It is too opaque for the azimuthal pattern to be measured in most of it — every direction sees an opaque ring — and the few semi-transparent regions show a variation with direction that is weaker than in the A ring and harder to interpret. Whether the B ring is a field of wakes so dense that gaps and slabs are both opaque, or whether its particles behave differently enough that a different kind of structure takes over — viscous overstability, which produces axisymmetric waves a hundred metres long rather than tilted slabs, is the leading alternative and has been seen in parts of the A and B rings — is not settled. The total ring mass is now measured by gravity and does not depend on the answer; how that mass is distributed across the B ring, and so how much dust each part of it has diluted, does.