A narrow ring that holds its own ellipse together

Saturn’s main rings are weighed by the waves crossing them: moons excite spiral density waves at resonances, the wavelength of a wave depends on the ring’s surface density, and counting the crests gives the mass. The rings of Uranus offer no such waves. They are narrow — most less than ten kilometres across, the widest under a hundred — and they were found not by imaging but by a star that blinked before it should have, when Uranus passed in front of it in 1977 and nine brief dimmings on either side revealed a system of thin rings nobody had suspected.

The occultations showed something else about those rings that is unusual in a disc. Several of them are eccentric, and the largest, the ε ring, is eccentric in a particular way: it is narrowest where it passes closest to Uranus, about twenty kilometres wide, and widest at the opposite side, nearly a hundred. Its inner and outer edges are two ellipses with their long axes aligned and their eccentricities slightly different. That arrangement is not what the planet’s gravity would leave alone.

This essay follows the argument Goldreich and Tremaine made in 1979: the planet’s flattening tries to pull the ring’s edges out of alignment, the alignment persists, and the only force that can hold it is the ring’s own gravity. How much gravity it takes is a mass, and the mass is a measurement of a ring that nothing else weighs.

A planet that turns every ellipse #

A particle orbiting a perfectly spherical planet follows a fixed ellipse. Uranus is not spherical: its rapid spin has flattened it, and the extra mass near its equator pulls on a nearby orbit in a way that makes the ellipse’s long axis turn slowly forward. The rate depends on the planet’s flattening coefficient $J_2$ and on how close the orbit is:

$$\dot\omega = \frac{3}{2}\, n\, J_2 \left(\frac{R}{a}\right)^2 .$$

For the ε ring, at 51,000 kilometres, the ellipse turns at about 1.35 degrees a day, a full revolution in under nine months. The rate was measured from successive occultations, and in fact was used to measure Uranus’s $J_2$ before any spacecraft arrived. Because the rate falls as the distance to the power three and a half, it is not the same across the ring’s own width. The inner edge, a few dozen kilometres closer to Uranus, is turned faster than the outer edge — by about five thousandths of a degree a day.

How fast the edges would part #

Five thousandths of a degree a day sounds negligible. It accumulates.

If nothing but Uranus’s flattening acted, the inner edge’s ellipse would run ahead of the outer edge’s, and within half a century the two would be at right angles; in a century their pericentres would be opposite, so that the ring’s width, instead of varying smoothly from twenty to ninety-six kilometres round its circumference, would pinch to almost nothing at two points and swell elsewhere. Every particle between the edges would be on its own slightly different ellipse turning at its own rate, and the ring would spread into a band whose edges had no particular shape at all.

The ε ring has been observed since 1977, through dozens of stellar occultations and by Voyager 2 in 1986 and the space telescopes since, and its edges have stayed aligned throughout. Rings of Uranus are thought to be at least millions of years old, which is tens of thousands of the times it would take their edges to separate. Something is holding the ellipses together.

The signature of an aligned ring #

The width variation is itself the evidence of alignment.

Two aligned ellipses with the same centre and different eccentricities lie closest together on the side where both are closest to the planet, if the outer one is the more eccentric. The width then varies as the cosine of the longitude from pericentre, which is what the occultation profiles show to within their errors. The outer edge’s eccentricity is larger than the inner’s by $7.5\times10^{-4}$ on a mean eccentricity of $7.9\times10^{-3}$: the whole ring is a nested family of ellipses whose eccentricity grows by about a tenth from inside to outside, all pointing the same way.

That gradient is the part that matters for the dynamics. It is what gives the ring’s gravity something to act on.

The ring’s gravity as a lever #

The ring’s own gravity is tiny compared with the planet’s, and it does not need to be large; it needs only to supply the small difference in precession rate between the edges. The mechanism is the same one that couples the orbits of the planets in the secular theory of the solar system: two nearby orbits with aligned ellipses exert slow torques on each other that change each other’s precession rates, and the effect depends on the difference of their eccentricities and on how close they are.

Treat the ring as two thin wires, its inner and outer edges, each holding half its mass. The outer wire, more eccentric, pulls on the inner one in a way that slows its precession; the inner, less eccentric, pulls on the outer one in a way that speeds it up. Both effects work against the planet’s differential precession. In the linear theory, which assumes the eccentricity gradient is small, the difference they make is

$$\Delta\dot\omega_{\rm self} \approx \frac{n}{2\pi}\,\frac{m}{M}\left(\frac{a}{\delta a}\right)^2 \frac{\delta e}{e},$$

and setting it equal to the planet’s differential precession gives the ring mass needed:

$$m = 2\pi M\,\frac{\Delta\dot\omega_{J_2}}{n}\left(\frac{\delta a}{a}\right)^2 \frac{e}{\delta e} .$$

The factor $(a/\delta a)^2$ is the lever: close wires interact strongly, so a narrow ring needs little mass. The factor $\delta e/e$ is the grip: if the edges had the same eccentricity, their mutual pull could not change their precession rates differently, and no amount of mass would align them. For the ε ring, with an eccentricity gradient $q = a\,\delta e/\delta a$ of about two-thirds, the linear estimate is about two hundred grams per square centimetre — a ring mass of a few times $10^{16}$ kilograms, equivalent to a moon of icy material some forty kilometres across.

Why the answer is an upper estimate #

The linear theory assumes the gradient is small, and the ε ring’s is not. With $q$ near two-thirds, the edges nearly approach each other at pericentre — the ring is four times narrower there than at apocentre — and the mutual pull of the streamlines is concentrated where they are close, which the linear expansion underweights. Nonlinear treatments of the same problem find that the self-gravity is more effective at large $q$, so less mass does the same work.

There is a second effect. A ring of colliding particles has pressure, and at a sharp edge the pressure gradient is large. Chiang and Goldreich showed in 2000 that the collisions near the edges add a contribution to the differential precession that the gravitational calculation alone omits, concentrated in the outermost few hundred metres of each edge where the particles are packed most densely. Including it lowers the mass the ring needs by a substantial factor below the simple linear value, and makes the answer depend on how the particles behave at the edges as well as on how much there is of them.

So the figures here are upper estimates with a definite direction of error. What they establish robustly is that self-gravity of a plausible ring can hold the alignment, and that the surface density needed is of the order of what a ring of icy particles a metre or so across, packed a few deep, would have. A ring a hundred times less massive could not do it; one a hundred times more would be a moon.

How the requirement scales #

The formula shows how the alignment depends on the ring’s geometry and its planet.

The differential precession grows in proportion to the width, and the lever of self-gravity weakens as the width squared, so a ring twice as wide needs four times the mass, twice the surface density. Narrow rings are cheap to align and wide ones expensive, which is consistent with the fact that the eccentric rings of Uranus and the eccentric ringlets in the gaps of Saturn’s rings are all narrow. A ring that spread would lose its alignment, and an aligned ring cannot be wide.

The planet’s flattening sets the strength of the differential precession and so the requirement in direct proportion. Saturn is five times flatter than Uranus, and an eccentric ring there needs five times the mass per area to stay aligned at the same relative distance — which is part of why Saturn’s eccentric ringlets, such as those in the Maxwell and Huygens gaps, are only tens of kilometres wide.

How a ring is measured that nobody has photographed #

Everything in the argument rests on measurements made by watching stars disappear. When Uranus passes in front of a star, each ring blocks the starlight for a few seconds, and the timing of the dimming at a given site gives the ring’s distance from the planet’s centre along one line across the system. Many sites, and many occultations over years, give many chords, and each ring’s shape and orientation are fitted to them. The ε ring’s eccentricity, its pericentre’s longitude, the rate at which that longitude turns and the width at each point round the ring all come from such fits, made from the ground, from aircraft and from orbit.

The precision is remarkable for an object never imaged in detail: ring radii to a few hundred metres, precession rates to parts in a hundred thousand. That precision is what turned the ε ring into a probe of Uranus’s interior — the precession rates of several eccentric rings at different distances measure $J_2$ and $J_4$, and the rotation period of the planet’s interior, before Voyager arrived — and it is what makes the alignment so striking. A misalignment of a degree between the edges would have been seen. None has been.

The edges themselves are sharp on scales of tens of metres, sharp enough to diffract starlight, which is another argument that something is actively confining them. A free ring of colliding particles spreads into soft, gradual edges in a few thousand years. Sharp edges on an eccentric ring are two confinements at once, radial and orientational, and the second is the one this essay is about.

Where narrow eccentric rings come from #

The rings of Uranus are generally thought to be young, or at least continually renewed, and the most likely source of their material is the disruption of small moons that strayed inside the distance at which a moon stops holding together. A moon broken up by tides leaves its debris on nearly its own orbit, with the moon’s eccentricity, and the debris collides, spreads and settles into a ring. If the moon’s orbit was eccentric, so is the young ring.

The eccentricity would not survive by itself. Collisions between particles on slightly different ellipses damp the relative motions that keep them apart, and a free ring’s eccentricity would decay; but a ring held between shepherds that are themselves on eccentric orbits, or near a resonance with one, can have its eccentricity excited and maintained. The ε ring’s eccentricity may be a remnant of its origin, maintained by its shepherds, or excited by them from a circular start; in either case the alignment has to be maintained by the ring’s self-gravity once the eccentricity is there.

Saturn’s broad rings show the same physics at a different scale. Self-gravity wakes — tilted, transient clumps of particles pulled together by their own gravity and sheared apart within an orbit — are the local expression of the same mutual attraction that, integrated across a narrow ring, aligns its edges. In the broad rings it organises the particles into structures tens of metres long; in a narrow ring it holds an ellipse fifty thousand kilometres across together.

Rings that keep their shape and rings that keep their edges #

The alignment argument is about the shape of a ring, not about where its edges are. That is a separate problem with a separate answer. A narrow ring left to itself spreads, because collisions transfer angular momentum outward the way viscosity does in any disc; its edges are a balance between that spreading and the torques of small shepherd moons on either side. Goldreich and Tremaine predicted those moons for the ε ring, and Voyager 2 found two, Cordelia and Ophelia, nine years later — about as clean a confirmation as the subject has.

The two arguments together make the ε ring a self-regulating object: the shepherds hold its edges in place against spreading, and its own gravity holds the edges’ ellipses in line against the planet’s flattening. The first needs moons of a definite size; the second needs a ring of a definite mass. Both quantities came from requiring a narrow structure to persist, before either had been directly measured.

There is an echo here of the large-scale solar system. The planets’ orbits are aligned or anti-aligned in combinations that secular theory calls modes, and the ε ring’s streamlines are locked in the simplest such mode, all pointing the same way. A ring is a continuum of orbits doing what a handful of planets do, with self-gravity in place of the planets’ mutual pull and the planet’s flattening in place of the Sun’s — and with the timescales shortened from millions of years to decades.

What the estimate leaves out #

The ring is reduced to two wires, each carrying half its mass, which is the simplest model that captures the mechanism. A real ring has a distribution of mass across its width, and the edges, where the particles are densest, carry a disproportionate share; the continuous calculation changes the coefficient but not the scaling. The precession uses $J_2$ alone; Uranus’s next harmonic $J_4$ changes the rate by about a per cent and the differential rate by a similar fraction. The linear secular theory is used at an eccentricity gradient where it is known to be inaccurate, as discussed, and the collisional term is not included in the figures.

The ring is also assumed to be in a steady state, with the self-gravitational and planetary precession exactly balanced. Small departures from balance would make the ring’s shape oscillate, its edges librating about alignment, and the amplitude of such librations — if they could be measured over decades of occultations — would give the ring’s mass independently of the steady-state formula.

Still open: a mass measured some other way #

The ε ring’s surface density has never been measured directly. Voyager 2’s radio occultation through the ring, a single pass, measured its optical depth at two wavelengths and constrained the size of its particles; turning that into a surface density requires assuming how the particles are packed. The alignment argument gives a mass only through a model of the ring’s dynamics, whose answer moves by a factor of several depending on which effects are included.

The two together would test each other. A ring whose particles are a few centimetres to a metre across, packed a few deep, has a surface density set by its optical depth and its particles’ size distribution; the alignment requires a particular mass; agreement would confirm both the dynamics and the particle model. A future spacecraft at Uranus could measure the ring’s mass directly, from the perturbation it exerts on the spacecraft’s own trajectory or on the shepherd moons, or from density waves too small for Voyager’s single pass to have seen. The same reasoning applies, with different numbers, to every narrow eccentric ring now known — round Saturn, round Uranus, and round two small bodies of the outer solar system, Chariklo and Haumea, whose rings were found by the same method of stellar occultation. A narrow ring round a body with a large flattening and no visible shepherds is a puzzle of exactly this kind: either its edges are not aligned, or something holds them, and the something has a mass. Each such ring is a small experiment in whether self-gravity, collisions and unseen moons are enough.

Until then the ε ring’s mass is known in the way the masses of the outer planets were known before any spacecraft visited them: from what has to be true for the motions to be what they are.