Saturn’s dynamical rings

The Cassini spacecraft has been in orbit around Saturn for a little more than three years now. During that time, it has returned data revealing details of several dynamical processes that determine the structure and evolution of the planet’s rings. Those phenomena not only give clues about the physical nature of the ring material but also provide a severe test of planetary physicists’ understanding of the gravitational and collisional physics involved. In that sense, Saturn’s rings act as a dynamical laboratory in which researchers can observe processes and apply their knowledge to other disk systems under more extreme conditions.

Self-gravity

All the giant planets in our solar system have nearly circular ring systems. In addition, all have small moons orbiting among or near the rings and larger moons beyond. The rings could be material that failed to accrete into moons or even the debris from fragmentation due to tidal forces of moons or passing objects such as comets.

Any orbiting object can be broken up if it moves so close to the planet that tidal forces overcome its self-gravity; for large objects, the distance of closest safe approach is called the Roche limit. Thus, material within the Roche limit cannot accrete into larger bodies; the same idea also applies to material in a disk orbiting a star.

James Clerk Maxwell proved that Saturn’s rings could not be solid because shear forces would tear them apart. Instead they had to be composed of individual particles orbiting the planet. The shear arises because, as a result of Kepler’s third law, material at the inner edge of the rings orbits the planet at a greater speed than material at the outer edge. Although the orbital velocities are large, typically about 20 km/s, collisional velocities are less than a few centimeters per second.

The particles themselves comprise mostly water ice and range in size from millimeters to meters. The gravitational effect of the ring particles on each other can be important, especially in regions such as Saturn’s A and B rings, where the local surface density of particles is higher. (These outermost of Saturn’s principal rings are evident in the cover image.) Cassini‘s UV occultation experiments provide good evidence for the existence of so-called self-gravity wakes, in which loose accumulations of material continually gather and disperse. Sustained growth is almost possible; under different conditions those structures could well stabilize as they accrete and retain more material. But the dominating effects of shear prevent such growth from occurring in Saturn’s rings.

Any collisionally evolved system of masses, such as Saturn’s rings, should have a characteristic spectrum of sizes. In particular, the rings should contain a relatively small population of particles as large as 100 m or so. Numerical models of the gravitational influence of such objects on nearby ring material show that as each object perturbs particles passing upstream and downstream of its location, it imprints a characteristic propeller-like feature in the rings. Therefore, although the 100-m objects may be difficult to resolve, their presence can be inferred. In fact, four propeller features were seen in the highest-resolution images taken by Cassini as it arched above the rings shortly after it entered orbit on 1 July 2004. On the basis of the tiny fractional area imaged, a population of some 10⁷ objects tens of meters across could exist in Saturn’s A ring.

Perhaps the most important aspect of the observation is its relevance to the formation of planetary systems. Simulations of evolving protoplanets accreting material from a surrounding primordial disk yield analogous propeller structures even though the physical parameters are drastically different—not a surprising result given that the basic dynamics is the same. Still, Cassini has provided the first clear evidence of the phenomenon in any astrophysical disk.

Embedded satellites

What about the dynamical effect of even larger objects in the rings? Planetary scientists know of two small satellites that orbit in gaps in Saturn’s A ring: Pan, whose diameter is about 20 km, orbits in the Encke Gap, and Daphnis, with a diameter of about 7 km, is in the narrower Keeler Gap. For near-circular rings, the gravitational effect of an embedded satellite on a passing particle depends on just two parameters: the mass of the satellite and its radial separation from the particle.

As illustrated in the figure, particles moving inside the orbit of the satellite have a larger velocity than the satellite and so catch up to it. If the particle has a sufficiently large radial separation, it will feel the satellite’s perturbation almost as an impulse and its orbit will acquire a small eccentricity; the smaller the separation, the larger the impulse and the larger the eccentricity. The next particle’s orbit will get the same eccentricity but slightly later, leading to a phase difference, and so on.

The resulting path composed of all the perturbed particles takes the form of a simple “edge” wave moving ahead of the satellite’s location; its wavelength λ = 3πs depends only on the radial separation s. Particles outside the orbit of the satellite move slower than the satellite but otherwise the dynamics of the encounter are identical to those of the interior case. A similar wave results, but it travels in the opposite direction to the interior edge wave. Increasing s yields a wave with longer wavelength; the amplitude, which also depends on satellite mass, decreases with increasing s. The net result is to produce wakes associated with each disturbance. Those wakes are distinct from the earlier mentioned self-gravity wakes detected in the rings, although both are the result of gravitational interactions between ring materials. Edge waves such as those shown in the figure had been observed by Voyager in the Encke and Keeler gaps. Indeed, it was those earlier observations that suggested the presence of embedded satellites.

In addition to producing edge waves, an embedded satellite scatters and thereby removes particles from its immediate radial vicinity—the result is a cleared gap in the rings. However, in one radial location, ring material can remain almost unperturbed, at least in the short term. In a narrow region centered on the orbit of the satellite, particles move in orbits in which, relative to the satellite, they execute a horseshoe-shaped path. Pan, for example, maintains such a ring of material in its orbit. Those co-orbital rings are also convenient refuges for material ejected from a satellite’s surface by impacting interplanetary material, which probably explains why faint rings are associated with some of Saturn’s other small moons.

Although most of the dynamics of embedded satellites can be understood in the context of the classic gravitational three-body problem—Saturn, the satellite, and a ring particle being the three interacting bodies—self-gravity and collisional dynamics also play key roles. The edge waves in the Encke and Keeler gaps dampen due to interparticle collisions, and self-gravity can also be an important determinant of edge-wave structure. An additional factor is the influence of the surrounding ring material on the orbit of the embedded satellite. Understanding those processes will give insight into the physical properties of rings and, as with propellers, help planetary scientists to understand other disks.

It is not just the embedded satellites that influence the rings. The numerous small moons that orbit just outside the main rings also exert a noticeable effect. At locations in the rings where the orbital period of a particle is a simple fraction of the orbital period of the satellite, resonances occur that alter the local gravitational potential. The outer edge of the A ring, for example, is associated with a 7:6 resonance with Janus, but other nearby moons such as Atlas, Prometheus, and Pandora also give rise to a series of density waves that determine the fine structure of the A ring. Cassini observations of such waves provide important clues about local physical environments in the rings. Density waves associated with resonances are not confined to ring systems. They also play a key role in transporting angular momentum in the early stages of planetary formation—another example of how context may differ but the essential dynamics is the same.

The online version of this Quick Study provides links to further resources, including Cassini images of Saturn’s rings.