@article{MakuchBrilliantovSremcevicetal.2006,
  author    = {Makuch, Martin and Brilliantov, Nikolai V. and Sremcevic, Miodrag and Spahn, Frank and Krivov, Alexander V.},
  title     = {Stochastic circumplanetary dynamics of rotating non-spherical dust particles},
  series = {Planetary and space science},
  volume    = {54},
  journal   = {Planetary and space science},
  number    = {9-10},
  publisher = {Elsevier},
  address   = {Oxford},
  issn      = {0032-0633},
  doi       = {10.1016/j.pss.2006.05.006},
  pages     = {855 -- 870},
  year      = {2006},
  abstract  = {We develop a model of stochastic radiation pressure for rotating non-spherical particles and apply the model to circumplanetary dynamics of dust grains. The stochastic properties of the radiation pressure are related to the ensemble-averaged characteristics of the rotating particles, which are given in terms of the rotational time-correlation function of a grain. We investigate the model analytically and show that an ensemble of particle trajectories demonstrates a diffusion-like behaviour. The analytical results are compared with numerical simulations, performed for the motion of the dusty ejecta from Deimos in orbit around Mars. We find that the theoretical predictions are in a good agreement with the simulation results. The agreement however deteriorates at later time, when the impact of non-linear terms, neglected in the analytic approach, becomes significant. Our results indicate that the stochastic modulation of the radiation pressure can play an important role in the circumplanetary dynamics of dust and may in case of some dusty systems noticeably alter an optical depth. (c) 2006 Elsevier Ltd. All rights reserved.},
  language  = {en}
}
@book{SpahnHertzschBrilliantov1995,
  author    = {Spahn, Frank and Hertzsch, Jan-Martin and Brilliantov, Nikolai V.},
  title     = {The role of particle collisions for the dynamics in planetary rings},
  series = {Preprint NLD},
  volume    = {14},
  journal   = {Preprint NLD},
  publisher = {Univ.},
  address   = {Potsdam},
  pages     = {23 S.},
  year      = {1995},
  language  = {en}
}
@article{BodrovaSchmidtSpahnetal.2012,
  author    = {Bodrova, Anna and Schmidt, J{\"u}rgen and Spahn, Frank and Brilliantov, Nikolai V.},
  title     = {Adhesion and collisional release of particles in dense planetary rings},
  series = {Icarus : international journal of solar system studies},
  volume    = {218},
  journal   = {Icarus : international journal of solar system studies},
  number    = {1},
  publisher = {Elsevier},
  address   = {San Diego},
  issn      = {0019-1035},
  doi       = {10.1016/j.icarus.2011.11.011},
  pages     = {60 -- 68},
  year      = {2012},
  abstract  = {We propose a simple theoretical model for aggregative and fragmentative collisions in Saturn's dense rings. In this model the ring matter consists of a bimodal size distribution: large (meter sized) boulders and a population of smaller particles (tens of centimeters down to dust). The small particles can adhesively stick to the boulders and can be released as debris in binary collisions of their carriers. To quantify the adhesion force we use the JKR theory (Johnson, K., Kendall, K., Roberts, A. [1971]. Proc. R. Soc. Lond. A 324, 301-313). The rates of release and adsorption of particles are calculated, depending on material parameters, sizes, and plausible velocity dispersions of carriers and debris particles. In steady state we obtain an expression for the amount of free debris relative to the fraction still attached to the carriers. In terms of this conceptually simple model a paucity of subcentimeter particles in Saturn's rings (French, R.G., Nicholson, P.D. [2000]. Icarus 145, 502-523; Marouf, E. et al. [2008]. Abstracts for "Saturn after Cassini-Huygens" Symposium, Imperial College London, UK, July 28 to August 1, p. 113) can be understood as a consequence of the increasing strength of adhesion (relative to inertial forces) for decreasing particle size. In this case particles smaller than a certain critical radius remain tightly attached to the surfaces of larger boulders, even when the boulders collide at their typical speed. Furthermore, we find that already a mildly increased velocity dispersion of the carrier-particles may significantly enhance the fraction of free debris particles, in this way increasing the optical depth of the system.},
  language  = {en}
}
@article{SpahnVieiraNetoGuimaraesetal.2014,
  author    = {Spahn, Frank and Vieira Neto, E. and Guimaraes, A. H. F. and Gorban, A. N. and Brilliantov, Nikolai V.},
  title     = {A statistical model of aggregate fragmentation},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {16},
  journal   = {New journal of physics : the open-access journal for physics},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/16/1/013031},
  pages     = {11},
  year      = {2014},
  abstract  = {A statistical model of fragmentation of aggregates is proposed, based on the stochastic propagation of cracks through the body. The propagation rules are formulated on a lattice and mimic two important features of the process-a crack moves against the stress gradient while dissipating energy during its growth. We perform numerical simulations of the model for two-dimensional lattice and reveal that the mass distribution for small-and intermediate-size fragments obeys a power law, F(m) proportional to m(-3/2), in agreement with experimental observations. We develop an analytical theory which explains the detected power law and demonstrate that the overall fragment mass distribution in our model agrees qualitatively with that one observed in experiments.},
  language  = {en}
}
@article{GuimaraesAlbersSpahnetal.2012,
  author    = {Guimaraes, Ana H. F. and Albers, Nicole and Spahn, Frank and Seiss, Martin and Vieira-Neto, Ernesto and Brilliantov, Nikolai V.},
  title     = {Aggregates in the strength and gravity regime Particles sizes in Saturn's rings},
  series = {Icarus : international journal of solar system studies},
  volume    = {220},
  journal   = {Icarus : international journal of solar system studies},
  number    = {2},
  publisher = {Elsevier},
  address   = {San Diego},
  issn      = {0019-1035},
  doi       = {10.1016/j.icarus.2012.06.005},
  pages     = {660 -- 678},
  year      = {2012},
  abstract  = {Particles in Saturn's main rings range in size from dust to kilometer-sized objects. Their size distribution is thought to be a result of competing accretion and fragmentation processes. While growth is naturally limited in tidal environments, frequent collisions among these objects may contribute to both accretion and fragmentation. As ring particles are primarily made of water ice attractive surface forces like adhesion could significantly influence these processes, finally determining the resulting size distribution. Here, we derive analytic expressions for the specific self-energy Q and related specific break-up energy Q(star) of aggregates. These expressions can be used for any aggregate type composed of monomeric constituents. We compare these expressions to numerical experiments where we create aggregates of various types including: regular packings like the face-centered cubic (fcc), Ballistic Particle Cluster Aggregates (BPCA), and modified BPCAs including e.g. different constituent size distributions. We show that accounting for attractive surface forces such as adhesion a simple approach is able to: (a) generally account for the size dependence of the specific break-up energy for fragmentation to occur reported in the literature, namely the division into "strength" and "gravity" regimes and (b) estimate the maximum aggregate size in a collisional ensemble to be on the order of a few tens of meters, consistent with the maximum particle size observed in Saturn's rings of about 10 m.},
  language  = {en}
}
@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}
}