Plato's cube and the natural geometry of fragmentation

Plato envisioned Earth's building blocks as cubes, a shape rarely found in nature. The solar system is littered, however, with distorted polyhedra-shards of rock and ice produced by ubiquitous fragmentation. We apply the theory of convex mosaics to show that the average geometry of natural two-dimensional (2D) fragments, from mud cracks to Earth's tectonic plates, has two attractors: "Platonic" quadrangles and "Voronoi" hexagons. In three dimensions (3D), the Platonic attractor is dominant: Remarkably, the average shape of natural rock fragments is cuboid. When viewed through the lens of convex mosaics, natural fragments are indeed geometric shadows of Plato's forms. Simulations show that generic binary breakup drives all mosaics toward the Platonic attractor, explaining the ubiquity of cuboid averages. Deviations from binary fracture produce more exotic patterns that are genetically linked to the formative stress field. We compute the universal pattern generator establishing this link, for 2D and 3D fragmentation.

Cassini-Huygens' exploration of the Saturn system: 13 years of discovery

The Cassini-Huygens mission to Saturn provided a close-up study of the gas giant planet, as well as its rings, moons, and magnetosphere. The Cassini spacecraft arrived at Saturn in 2004, dropped the Huygens probe to study the atmosphere and surface of Saturn's planet-sized moon Titan, and orbited Saturn for the next 13 years. In 2017, when it was running low on fuel, Cassini was intentionally vaporized in Saturn's atmosphere to protect the ocean moons, Enceladus and Titan, where it had discovered habitats potentially suitable for life. Mission findings include Enceladus' south polar geysers, the source of Saturn's E ring; Titan's methane cycle, including rain that creates hydrocarbon lakes; dynamic rings containing ice, silicates, and organics; and Saturn's differential rotation. This Review discusses highlights of Cassini's investigations, including the mission's final year.

Close-range remote sensing of Saturn's rings during Cassini's ring-grazing orbits and Grand Finale

Saturn's rings are an accessible exemplar of an astrophysical disk, tracing the Saturn system's dynamical processes and history. We present close-range remote-sensing observations of the main rings from the Cassini spacecraft. We find detailed sculpting of the rings by embedded masses, and banded texture belts throughout the rings. Saturn-orbiting streams of material impact the F ring. There are fine-scaled correlations among optical depth, spectral properties, and temperature in the B ring, but anticorrelations within strong density waves in the A ring. There is no spectral distinction between plateaux and the rest of the C ring, whereas the region outward of the Keeler gap is spectrally distinct from nearby regions. These results likely indicate that radial stratification of particle physical properties, rather than compositional differences, is responsible for producing these ring structures.

In situ collection of dust grains falling from Saturn's rings into its atmosphere

Saturn's main rings are composed of >95% water ice, and the nature of the remaining few percent has remained unclear. The Cassini spacecraft's traversals between Saturn and its innermost D ring allowed its cosmic dust analyzer (CDA) to collect material released from the main rings and to characterize the ring material infall into Saturn. We report the direct in situ detection of material from Saturn's dense rings by the CDA impact mass spectrometer. Most detected grains are a few tens of nanometers in size and dynamically associated with the previously inferred "ring rain." Silicate and water-ice grains were identified, in proportions that vary with latitude. Silicate grains constitute up to 30% of infalling grains, a higher percentage than the bulk silicate content of the rings.

Steady oscillations in aggregation-fragmentation processes

We report surprising steady oscillations in aggregation-fragmentation processes. Oscillating solutions are observed for the class of aggregation kernels K_{i,j}=i^{ν}j^{μ}+j^{ν}i^{μ} homogeneous in masses i and j of merging clusters and fragmentation kernels, F_{ij}=λK_{ij}, with parameter λ quantifying the intensity of the disruptive impacts. We assume a complete decomposition (shattering) of colliding partners into monomers. We show that an assumption of a steady-state distribution of cluster sizes, compatible with governing equations, yields a power law with an exponential cutoff. This prediction agrees with simulation results when θ≡ν-μ<1. For θ=ν-μ>1, however, the densities exhibit an oscillatory behavior. While these oscillations decay for not very small λ, they become steady if θ is close to 2 and λ is very small. Simulation results lead to a conjecture that for θ<1 the system has a stable fixed point, corresponding to the steady-state density distribution, while for any θ>1 there exists a critical value λ_{c}, such that for λ<λ_{c}, the system has an attracting limit cycle. This is rather striking for a closed system of Smoluchowski-like equations, lacking any sinks and sources of mass.

Stationary mass distribution and nonlocality in models of coalescence and shattering

We study the asymptotic properties of the steady state mass distribution for a class of collision kernels in an aggregation-shattering model in the limit of small shattering probabilities. It is shown that the exponents characterizing the large and small mass asymptotic behavior of the mass distribution depend on whether the collision kernel is local (the aggregation mass flux is essentially generated by collisions between particles of similar masses) or nonlocal (collision between particles of widely different masses give the main contribution to the mass flux). We show that the nonlocal regime is further divided into two subregimes corresponding to weak and strong nonlocality. We also observe that at the boundaries between the local and nonlocal regimes, the mass distribution acquires logarithmic corrections to scaling and calculate these corrections. Exact solutions for special kernels and numerical simulations are used to validate some nonrigorous steps used in the analysis. Our results show that for local kernels, the scaling solutions carry a constant flux of mass due to aggregation, whereas for the nonlocal case there is a correction to the constant flux exponent. Our results suggest that for general scale-invariant kernels, the universality classes of mass distributions are labeled by two parameters: the homogeneity degree of the kernel and one further number measuring the degree of the nonlocality of the kernel.

Oscillations in Aggregation-Shattering Processes

We observe never-ending oscillations in systems undergoing collision-controlled aggregation and shattering. Specifically, we investigate aggregation-shattering processes with aggregation kernels K_{i,j}=(i/j)^{a}+(j/i)^{a} and shattering kernels F_{i,j}=λK_{i,j}, where i and j are cluster sizes, and parameter λ quantifies the strength of shattering. When 0≤a<1/2, there are no oscillations, and the system monotonically approaches a steady state for all values of λ; in this region, we obtain an analytical solution for the stationary cluster size distribution. Numerical solutions of the rate equations show that oscillations emerge in the 1/2<a≤1 range. When λ is sufficiently large, oscillations decay and eventually disappear, while for λ<λ_{c}(a), oscillations apparently persist forever. Thus, never-ending oscillations can arise in closed aggregation-shattering processes without sinks and sources of particles.

Size distribution of particles in Saturn's rings from aggregation and fragmentation

Saturn's rings consist of a huge number of water ice particles, with a tiny addition of rocky material. They form a flat disk, as the result of an interplay of angular momentum conservation and the steady loss of energy in dissipative interparticle collisions. For particles in the size range from a few centimeters to a few meters, a power-law distribution of radii, ~r(-q) with q ≈ 3, has been inferred; for larger sizes, the distribution has a steep cutoff. It has been suggested that this size distribution may arise from a balance between aggregation and fragmentation of ring particles, yet neither the power-law dependence nor the upper size cutoff have been established on theoretical grounds. Here we propose a model for the particle size distribution that quantitatively explains the observations. In accordance with data, our model predicts the exponent q to be constrained to the interval 2.75 ≤ q ≤ 3.5. Also an exponential cutoff for larger particle sizes establishes naturally with the cutoff radius being set by the relative frequency of aggregating and disruptive collisions. This cutoff is much smaller than the typical scale of microstructures seen in Saturn's rings.

Gravitational vortices and clump formation in Saturn's F ring during an encounter with Prometheus

Saturn rings are most beautiful and dynamic places in the solar system, consisting of ice particles in a constant battle between the gravitational forces of Saturn and its many moons. Fan, spiral, propellers, moonlets and streamer-channels observed by CASSINI in the F-ring have been attributed to encounters by Prometheus on the F ring, with investigations of optical thickness revealing large populations of transient moonlets. Taking into account gravitational interaction between particles and a multi-stranded F-ring structure we show that Prometheus' encounters create rotational flows, like atmospheric vortices and the self-gravity enhances the accelerated growth and size of moonlets. Vortex patches form caustics, which is a primary cause of the transient particle density clumps of 20 km width and 100 km length, and they are elongated to cover an area of 1600 km by 150 km, which may eventually combine into a vortex sheet.

Twisted Disks

Ripple patterns in Saturn's and Jupiter's rings result from collisions with comets.

Origin of Saturn's rings and inner moons by mass removal from a lost Titan-sized satellite

The origin of Saturn's rings has not been adequately explained. The current rings are more than 90 to 95 per cent water ice, which implies that initially they were almost pure ice because they are continually polluted by rocky meteoroids. In contrast, a half-rock, half-ice mixture (similar to the composition of many of the satellites in the outer Solar System) would generally be expected. Previous ring origin theories invoke the collisional disruption of a small moon, or the tidal disruption of a comet during a close passage by Saturn. These models are improbable and/or struggle to account for basic properties of the rings, including their icy composition. Saturn has only one large satellite, Titan, whereas Jupiter has four large satellites; additional large satellites probably existed originally but were lost as they spiralled into Saturn. Here I report numerical simulations of the tidal removal of mass from a differentiated, Titan-sized satellite as it migrates inward towards Saturn. Planetary tidal forces preferentially strip material from the satellite's outer icy layers, while its rocky core remains intact and is lost to collision with the planet. The result is a pure ice ring much more massive than Saturn's current rings. As the ring evolves, its mass decreases and icy moons are spawned from its outer edge with estimated masses consistent with Saturn's ice-rich moons interior to and including Tethys.

The four hundred years of planetary science since Galileo and Kepler

For 350 years after Galileo's discoveries, ground-based telescopes and theoretical modelling furnished everything we knew about the Sun's planetary retinue. Over the past five decades, however, spacecraft visits to many targets transformed these early notions, revealing the diversity of Solar System bodies and displaying active planetary processes at work. Violent events have punctuated the histories of many planets and satellites, changing them substantially since their birth. Contemporary knowledge has finally allowed testable models of the Solar System's origin to be developed and potential abodes for extraterrestrial life to be explored. Future planetary research should involve focused studies of selected targets, including exoplanets.