It is often heard that planetary rings (PR) are local dynamical laboratories, important for understanding protoplanetary disks (PPD). While they are very different in profound ways, PR and PPD indeed do inform, illustrate, or complement each other, theoretically and observationally.
1) Con o senza gas? The most profound difference between PR and most PPD is that gas plays a critical role in most PPD and a very minor one in most PR. There are however important PPDs that lack gas (debris disks) in which there are PR connections in the areas of collisional cascades and small particle size distributions.
2) Fluid dynamics is fundamental to PR as it is to PPD; in PR the "fluid" pressure and viscosity are generated by very gentle collisions between cm-m size particles, acting like molecules in a dense gas. In PPD, the gas provides pressure and viscosity, but the viscosity is strongly dependent on the intensity of PPD gas turbulence - which is poorly understood and hard to observe.
3) Radial transport of mass and angular momentum is similar in PR and PPD: in PRs, collision-scale viscosity is analogous to molecular kinematic viscosity and there is not really any equivalent to turbulence. However, angular momentum in both PR and PPD can also be powerfully transported over longer ranges by gravitational effects, including spiral waves that obey the same physics.
4) In spite of various attempts to ignore it, the messy physics of collisional particle sticking and bouncing is important in both environments. The outcome of growth by sticking in PPD - that determines the path to planetesimal formation - depends critically on the intensity of turbulence, whereas in PR growth is limited by tidal forces.
5) Large scale radial structure (strong surface density fluctuations and empty gaps) is generated by gravitational torques associated with local or nonlocal large objects. In both PR and PPD, local gravitational instabilities may be important. There is abundant smaller-scale structure revealed by Cassini in Saturn's rings that we do not understand, that may lead to further insights.
6) PRs and PPDs are not closed systems - infall of matter from beyond the system, and loss of matter to the central object, are very important to the compositional and radial evolution of both systems. Indeed, infall may be our best constraint on the age of Saturn's rings, and also limit their lifetime.
7) Short timescales are surprisingly important. We see features in PR evolving before our eyes. Saturn's rings may be both geologically young AND short-lived. In PPD, the most youthful stages - still the least well observed or understood - may be the key to many of the puzzles of the meteorite record, planetesimal formation, and the path to planets.
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