@article{BrilliantovKrapivskyBodrovaetal.2015,
  author    = {Brilliantov, Nikolai V. and Krapivsky, P. L. and Bodrova, Anna and Spahn, Frank and Hayakawa, Hisao and Stadnichuk, Vladimir and Schmidt, Jurgen},
  title     = {Size distribution of particles in Saturn's rings from aggregation and fragmentation},
  series = {Proceedings of the National Academy of Sciences of the United States of America},
  volume    = {112},
  journal   = {Proceedings of the National Academy of Sciences of the United States of America},
  number    = {31},
  publisher = {National Acad. of Sciences},
  address   = {Washington},
  issn      = {0027-8424},
  doi       = {10.1073/pnas.1503957112},
  pages     = {9536 -- 9541},
  year      = {2015},
  abstract  = {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, similar to r(-q) with q approximate to 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.},
  language  = {en}
}
@article{SeissSpahn2011,
  author    = {Seiss, Martin and Spahn, Frank},
  title     = {Hydrodynamics of saturn's dense rings},
  series = {Mathematical modelling of natural phenomena},
  volume    = {6},
  journal   = {Mathematical modelling of natural phenomena},
  number    = {4},
  publisher = {EDP Sciences},
  address   = {Les Ulis},
  issn      = {0973-5348},
  doi       = {10.1051/mmnp/20116409},
  pages     = {191 -- 218},
  year      = {2011},
  abstract  = {The space missions Voyager and Cassini together with earthbound observations revealed a wealth of structures in Saturn's rings. There are, for example, waves being excited at ring positions which are in orbital resonance with Saturn's moons. Other structures can be assigned to embedded moons like empty gaps, moon induced wakes or S-shaped propeller features. Furthermore, irregular radial structures are observed in the range from 10 meters until kilometers. Here some of these structures will be discussed in the frame of hydrodynamical modeling of Saturn's dense rings. For this purpose we will characterize the physical properties of the ring particle ensemble by mean field quantities and point to the special behavior of the transport coefficients. We show that unperturbed rings can become unstable and how diffusion acts in the rings. Additionally, the alternative streamline formalism is introduced to describe perturbed regions of dense rings with applications to the wake damping and the dispersion relation of the density waves.},
  language  = {en}
}