DIRECT EVIDENCE FOR GRAVITATIONAL INSTABILITY AND MOONLET FORMATION IN SATURN's RINGS

Saturn's ring system is the only local example of an astrophysical disk in which the competing effects of collisions and self-gravity can be studied in situ by spacecraft (Porco et al. 2005). In the main rings, the near-circular orbits of the ring particles ensure that internal collisions are less disruptive because the relative velocities are small (∼mm s⁻¹). However, self-gravity is important and is believed to be responsible for the wake structures in the A ring indirectly detected using Cassini data (Colwell et al. 2006; Hedman et al. 2007; Porco et al. 2008). In addition, unusual structures at the edge of the A ring and the Encke gap have been attributed to enhanced kinematical and self-gravitational effects associated with the squeezing of streamlines (Porco et al. 2005; Lewis & Stewart 2005). What has often been assumed, but never verified by observation, is the process of accretion in a planetary ring.

Located between the orbits of its two shepherding satellites, Prometheus and Pandora, the F ring lies 3500 km beyond the edge of Saturn's main ring system. Initial observations from the Cassini spacecraft (Porco et al. 2005) revealed a multi-stranded ring with a bright core and periodic azimuthal structure associated with perturbations from the inner shepherding satellite, Prometheus. Every 14.7 hr the satellite reaches its apoapse and imposes a characteristic "streamer-channel" pattern on the ring, its "strands", and surrounding material. The subsequent Keplerian shearing of this perturbed material downstream from the encounter can easily be detected in images of the ring, and every 67.6 days the same region experiences another Prometheus encounter (Murray et al. 2005, 2008); more distant and less massive Pandora produces its own, less obvious pattern.

Several lines of evidence suggest the existence of additional objects in the F ring system: (1) the strands are now known to be spiral in nature and the result of sheared jets of debris from collisions between several small (radius ∼5–10 km) objects orbiting in the region and the core of the F ring (Murray et al. 2008; Charnoz et al. 2005; Charnoz 2009); (2) additional analysis of high-resolution Cassini images of the F ring (Figure 3 of Murray et al. 2008) has revealed the presence of "fans" (sequences of sheared channels with a common point of intersection covering ∼05 in longitude) indicative of the perturbing effect of embedded masses (radii ∼5 km assuming a density comparable to that of Prometheus) on orbits close to that of the F ring core but with eccentricities differing from it by ∼10⁻⁴ (Williams 2009)²; and (3) numerous (∼10²) bright features in the F ring have been attributed to the existence of ∼1 km diameter objects associated with the F ring core (Murray et al. 2008). At least some of these objects are likely related to occultation features detected close to the F ring (Esposito et al. 2008).

Many of these lines of evidence can be seen in Figure 1 which shows the same 40° section of the F ring captured in five successive sets of observations between 2007 February 27 and 2007 May 5; the radial extent of each reprojected mosaic is 400 km. A shearing jet is seen above the core in Figures 1(b)–(d) and two fans have been detected in the images. Numerous features are detectable in the core and their spacing of ∼3° is consistent with Prometheus perturbations. Furthermore, many of these features can be seen moving in either direction with respect to the core. These drifts correspond to a typical displacement from the semimajor axis of the core of ∼5 km. In Figure 1(a) Prometheus has a corotating longitude of 168°, ∼33° to the right of the displayed segment. During successive sequences, it traverses the F ring before encountering the displayed ring segment again between 2007 April 18 and May 5 (the dates of Figures 1(d) and (e)). Note that most of the features in the core survived another passage of Prometheus, although clearly the core in Figure 1(e) appears as disrupted and distorted as that in Figure 1(a), which has also just experienced passage by Prometheus. Therefore there is good evidence that Prometheus produces features in the F ring's core and that these bright objects are long-lived, perhaps surviving multiple passages of the satellite.

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The existence of a moonlet belt at this location was originally proposed to explain the charged particle environment measured by Pioneer 11 (Cuzzi & Burns 1988). Although such a system could result from a collision between larger objects, its location at 2.32 Saturn radii means that the F ring is in the Roche zone (Murray & Dermott 1999) where accretion processes due to self-gravity continually compete with disruption due to collisions and tidal forces (Barbara & Esposito 2002; Canup & Esposito 1995). While it is tempting to consider some or all of the numerous objects in the F ring region as accretion products, until now there has been no direct observational evidence of such a process in any planetary ring.

Figure 2 shows three separate mosaics of 28° sections of the F ring constructed from reprojected Cassini Imaging Science Subsystem (Porco et al. 2004) images taken on three occasions between 2008 July and 2009 April. The bright, central core is always visible within ∼50 km of zero relative radius. The strands below the core are spiral in nature and are the sheared jets resulting from a collision in the F ring that we believe occurred in 2007. The more diffuse strands above the core are likely to be from older collisions. Each mosaic shows the channels that are a characteristic of the streamer-channel phenomenon produced by a recent passage of Prometheus (moving from left to right in this frame); Prometheus itself is in Figures 2(a) and (c). In each instance, the channel is ∼1° in azimuthal width (∼2450 km) and separated from the next channel by the characteristic wavelength of 327. All mosaics show at least one "fan" indicating the presence (Murray et al. 2008) of an object embedded in the core perturbing nearby material (up to ∼200 km beyond the core). Furthermore, the base of each fan is located at the left-hand edge of a channel. The fans appear in material exterior to the ring's core implying a perturber with a semimajor axis near the inner edge of the core. Other processes including observational selection effects and proximity to Prometheus may account for the bias. Figures 2(b) and (c) show shadows associated with each fan, as well as bright objects in the inner strand, and, in Figure 2(c), all bright objects in the ring segment. The optical depth, τ, for the F ring core is ∼0.1 (Bosh et al. 2002). Theoretically, the maximum width of each fan channel is known to be ∼10 times the Hill's radius of the embedded object (Murray et al. 2008; Williams 2009) implying a clump of mass (0.5–1.0) × 10¹⁶ kg produces the fan in Figure 2(b). FWHM measurements of the shadows in the raw images suggest a width of 73 ± 25 km. Measurements of shadow lengths in Figures 2(b) and (c) for the two solar elevation angles imply that they are caused by structures extending ∼5–10 km from the F ring plane. Together these observations are consistent with an azimuthally extended clump with mass comparable to that of an ∼10–20 km radius moonlet. The clear association of these objects with channel edges strongly suggests that Prometheus is responsible for their creation. Furthermore, the clumps persist until at least the next encounter with Prometheus (see Figure 1).

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To understand this process, we carry out numerical integrations of test particles filling an ∼10° section of a 40 km-wide model F ring after it has undergone passage by Prometheus. The state of the ring segment at four separate times is shown in Figure 3. The times have been deliberately chosen to be at the point within a cycle when the channels are most open. As previously demonstrated (Murray et al. 2005), the streamer-channel structure changes dramatically over one orbital period of the F ring particles (14.8 hr). Several effects are obvious in Figure 3: (1) there is a concentration of test particles at the edges of the channels at these points in the cycle, except for Figure 3(d) where particles are starting to leak into the channels; (2) the tilt of the channels continues to decline due to Keplerian shear; and (3) the channels get wider and consequently the ring sections between channels get narrower with time. All of these effects can be understood by undertaking a quantitative examination of the consequences of an encounter between F ring particles and Prometheus.

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Figure 4 shows the changes in the semimajor axis, a, that result as Prometheus encounters a 3° section of the F ring in a numerical simulation for five different relative orientations of the two orbits. (Differential precession results in the periapses being anti-aligned every ∼19 years.) The method we used is similar to that of Showalter & Burns (1982). Our results show that the acquired changes in orbital elements depend on the separation at encounter and the relative orientation of the orbits. (The maximum changes in the semimajor axis, eccentricity, and longitude of periapse for different relative orientations were quoted in Murray et al. 2008.) The induced eccentricity change is comparable to that required to form fans. Furthermore, the sign of each change in the semimajor axis differs on either side of the zero relative longitude. In particular, if we consider the behavior of a within 05 of the zero position close to the anti-alignment configuration, then a clear gradient of changes in a is imposed on either side with a maximum and minimum falling within ∼01 of the center. Most importantly, Δa > 0 for negative relative longitude and Δa < 0 for positive relative longitude. (This is consistent with both directions of relative drift shown in Figure 1.) In our chosen frame, this results in material from the right edge moving to the right as material from the left edge moves to the left. This explains why the channels widen and the inter-channel regions narrow as the simulation progresses (Figures 3(a)–(c)); there is also evidence for this in Figure 2 where widening occurs as the separation from Prometheus (i.e., time since encounter) increases; see also Showalter & Burns (1982). Eventually the test particles from one channel edge meet the retreating edge from the next channel and start to fill the channel again (Figure 3(d)). While our simulations neglect the influence of the ring's self-gravity and collisions, this kinematic model suggests that the streamer-channel pattern created by Prometheus periodically triggers local increases in the surface density. This effect is particularly important at channel edges and may be sufficient to allow self-gravity to become effective and growth to occur in these regions. Nevertheless, future work explicitly including self-gravity and collisions would be useful in exploring how these processes influence the development and persistence of these condensations.

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Cassini observations of both fans as well as bright regions capable of casting shadows at regions associated with perturbations by Prometheus all suggest circumstances in which gravitational condensations could occur. In Figure 5, we examine in detail how it is possible for the conditions for collapse to arise. The increase in the maximum number density following the passage of Prometheus across the ring segment is clearly seen in Figure 5(a). Examination of the variation in maximum number density and the minimum values of the dispersion in the radial velocity over one cycle (Figures 5(b) and (c)) shows that near-zero dispersion in the radial velocity coincides with the times when the number density is at a maximum. It has been verified that this occurs for the same cells. The observed minimum values of the radial velocity dispersion (⩽0.02 m s⁻¹) are at least 2 orders of magnitude less than the escape velocity of an ∼10 km icy satellite implying that collisions would quickly result in a local condensation. Further, if Prometheus' perturbations create regions with bulk densities greater than the local Roche density (=3ρ(R/a)³ ≈ 0.15 g cm⁻³ at the F ring, where ρ is the density of the planet, R is its radius) and low dispersion velocities, gravitational instability and collapse under self-gravity may occur. Note that this is equivalent to the conditions specified by the Toomre parameter (Toomre 1964; Karjalainen & Salo 2004; Lewis & Stewart 2009) frequently used in the context of stability in disks.³

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A picture of Prometheus' cumulative effect on the F ring now emerges. At the closest approach during each orbital period, the gravitational effect of the satellite significantly perturbs an ∼1° segment of the ring and the streamer-channel pattern is repeated every 327. Each passage triggers the formation of objects at the channel edges and these now evolve with respect to the core of the F ring. The synodic period of this process is 67.6 days after which Prometheus returns to repeat the process, perturbing a different section of the ring, parts of which may already contain objects that have accreted material since their formation at previous encounters. Furthermore, we have detected bright objects and accompanying shadows around the ring implying that these objects can persist over several synodic periods. Particles suffer mutual collisions on a timescale of t_col ∼ 1/τΩ, where Ω is the orbital frequency. So, optically thick regions, susceptible to gravitational instability, can dissipate relative velocities via inelastic collisions and become more tightly bound in a few orbital periods. The observational evidence suggests that, at least initially, these objects are formed by Prometheus and can be considered as coherent, enhanced density clumps which are still an integral part of the perturbed core, rather than fully formed moonlets. It is important to note that perturbations from Prometheus can also excite velocities between particles (note the variation over one cycle in Figure 5(c)). Depending on the phase, subsequent Prometheus encounters could either drive the disruption of such objects or enhance their growth; the balance of these processes ultimately gives rise to a characteristic distribution of different sizes observed in the present F ring (Murray et al. 2008).

The process by which Prometheus triggers local condensations in Saturn's F ring has direct applicability to the more general problem of accretion in systems such as a protoplanetary disk being perturbed by a nascent protoplanet, where the embedded gas giant perturbs a marginally stable gas disk (Armitage & Hansen 1999). Gravitational instability provides a mechanism for the rapid formation of large bodies via the rapid collapse of an over-dense region of an astrophysical disk. Various modes of instability have been suggested to account for the formation of 10–100 km planetesimals (Goldreich & Ward 1973; Cuzzi et al. 2008; Johansen et al. 2007) and the formation of gas giant planets (Boss 2008). In this light, the clumps in the F ring may provide an observed celestial example of such an instability, at least in the case where gravitational growth is limited by the internal density and physical size of the particles. The sustained creation and destruction of these collapsing objects reveals the F ring as a natural laboratory for studying the processes that facilitate and frustrate such rapid growth.

This Letter is dedicated to the memory of K. Beurle. This work was funded by the UK Science and Technology Facilities Council and by NASA/JPL. We thank M. Showalter and the Cassini VIMS team for their invaluable help in obtaining observations. We also thank J. Burns and R. Nelson for useful discussions and suggestions.

Facilities: Cassini - Cassini-Huygens