Size Distribution of Particles in Saturn's Rings from Aggregation And

Size distribution of particles in Saturn’s rings from aggregation and fragmentation Nikolai Brilliantov ∗, P. L. Krapivsky †, Anna Bodrova ‡§, Frank Spahn § , Hisao Hayakawa ¶, Vladimir Stadnichuk ‡ , and Jurgen¨ Schmidt k § ∗University of Leicester, Leicester, United Kingdom,†Department of Physics, Boston University, Boston, USA,‡Moscow State University, Moscow, Russia,§University of Potsdam, Potsdam, Germany,¶Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan, and kAstronomy and Space Physics, University of Oulu, Oulu, Finland Submitted to Proceedings of the National Academy of Sciences of the United States of America Saturn’s rings consist of a huge number of water ice particles, as observed in Saturn’s rings, is universal for systems where a with a tiny addition of rocky material. They form a flat disk, as balance between aggregation and disruptive collisions is steadily the result of an interplay of angular momentum conservation and sustained. Hence, the same size distribution is expected for any the steady loss of energy in dissipative inter-particle collisions. ring system where collisions play a role, like the Uranian rings, the For particles in the size range from a few centimeters to a few q recently discovered rings of Chariklo and Chiron, and possibly meters, a power-law distribution of radii, r− with q 3, has been inferred; for larger sizes, the distribution∼ has a steep≈ cutoff. rings around extrasolar objects. 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 Results and Discussion have been established on theoretical grounds. Here we propose a Model. Ring particles may be treated as aggregates1 built up of pri- model for the particle size distribution that quantitatively explains 2 the observations. In accordance with data, our model predicts the mary grains [9] of a certain size r1 and mass m1. Denote by exponent q to be constrained to the interval 2.75 q 3.5. Also mk = km1 the mass of ring particles of "size" k containing k pri- ≤ ≤ an exponential cutoff for larger particle sizes establishes naturally mary grains, and by nk their density. For the purpose of a kinetic with the cutoff-radius being set by the relative frequency of aggre- description we assume that all particles are spheres; then the radius3 1/3 gating and disruptive collisions. This cutoff is much smaller than of an agglomerate containing k monomers is rk = r1k . Systems the typical scale of micro-structures seen in Saturn’s rings. composed of spherical particles may be described in the framework of the Enskog-Boltzmann theory [25, 26, 27]. In this case the rate of Saturn’s rings planetary rings fragmentation | | binary collisions depends on particle dimension and relative velocity. The cross-section for the collision of particles of size i and j can be 2 2 2 1/3 1/3 2 ombardment of Saturn’s rings by interplanetary meteoroids written as σij = (ri + rj ) = r1(i + j ) . The relative speed B[1, 2, 3] and the observation of rapid processes in the ring system (on the order of 0.01 0.1 cm/s [16]) is determined by the velocity 2 − 2 [4] indicate that the shape of the particle size distribution is likely not dispersions vi and vj for particles of size i and j. The velocity primordial or a direct result from the ring creating event. Rather, ring dispersion quantifies the root mean square deviation of particle veloc- particles are believed to be involved in an active accretion-destruction ities from the orbital speed ( 20 km/s). These deviations follow a dynamics [5, 6, 7, 8, 9, 10, 11, 12, 13] and their sizes vary over a certain distribution, implying∼ a range of inter-particle impact speeds, few orders of magnitude as a power-law [14, 15, 16, 17], with a sharp and thus, different collisional outcomes. The detailed analysis of cutoff for large sizes [18, 19, 20, 21]. Moreover, tidal forces fail an impact shows that for collisions at small relative velocities, when to explain the abrupt decay of the size distribution for house-sized the relative kinetic energy is smaller than a certain threshold energy, particles [22]. One would like to understand: (i) can the interplay Eagg, particles stick together to form a joint aggregate [28, 11, 29]. between aggregation and fragmentation lead to the observed size dis- This occurs because adhesive forces acting between ice particles’ sur- tribution, and (ii) is this distribution peculiar for Saturn’s rings, or is faces are strong enough to keep them together. For larger velocities, it universal for planetary rings in general? To answer these questions particles rebound with a partial loss of their kinetic energy. For still quantitatively, one needs to elaborate a detailed model of the kinetic larger impact speeds, the relative kinetic energy exceeds the threshold processes in which the rings particles are involved. Here we develop a energy for fragmentation, Efrag, and particles break into pieces [29]. theory that quantitatively explains the observed properties of the par- Using kinetic theory of granular gases one can find the collision ticle size distribution and show that these properties are generic for a frequency for all kinds of collisions and the respective rate coeffi- arXiv:1302.4097v5 [astro-ph.EP] 3 Aug 2015 steady-state, when a balance between aggregation and fragmentation holds. Our model is based on the hypothesis that collisions are binary and that they may be classified as aggregative, restitutive or disruptive Reserved for Publication Footnotes (including collisions with erosion); which type of collision is realized depends on the relative speed of colliding particles and their masses. We apply the kinetic theory of granular gases [23, 24] for the multi- component system of ring particles to quantify the collision rate and the type of collision. Significance Although it is well accepted that the particle size distribution in Saturn’s rings is not primordial, it remains unclear whether the 1The concept of aggregates as Dynamic Ephemeral Bodies (DEB) in rings has been proposed observed distribution is unique or universal. That is, whether it is in [7]. determined by the history of the rings and details of the particle 2Observations indicate that particles below a certain radius are absent in dense rings [16]. 3In principle, aggregates can be fractal objects, so that r k1/D , where D 3 is the k ∼ ≤ interaction, or if the distribution is generic for all planetary rings. fractal dimension of aggregates. For dense planetary rings it is reasonable to assume that We show that a power-law size distribution with large-size cutoff, aggregates are compact, so D = 3. www.pnas.org/cgi/doi/10.1073/pnas.0709640104 PNAS Issue Date Volume Issue Number 1–8 cients: Cij for collisions leading to merging and Aij for disruptive the emergence of the steady state. The constant λ here characterizes collisions. The coefficients Cij give the number of aggregates of the relative frequency of disruptive and aggregative collisions. With- size (i + j) forming per unit time in a unit volume as a result of out loss of generality we set C0 = 1. Solving the governing equations aggregative collisions involving particles of size i and j. Similarly, for mono-disperse initial conditions, nk(t = 0) = δk,1, one finds Aij quantify disruptive collisions, when particles of size i and j col- λ /λ 1 1 λt 1 − 2 1 lide and break into smaller pieces. These rate coefficients depend on n1(t) = λ1 1 + λ− λ2− e λ− /2 , [5] masses of particles, velocity dispersions and threshold energies, Eagg − and Efrag: where λ1 = λ/(1+λ) and λ2 = 2λ/(1+2λ). Utilizing the recursive nature of Eqs. (3), one can determine nk(t) for k > 1. The system Cij = νij (1 (1 + Bij Eagg) exp ( Bij Eagg)) demonstrates a relaxation behavior: After a relaxation time that scales − − 1 Aij = νij exp ( Bij Efrag) as λ− , the system approaches to a steady state with n1 = λ1, the − other concentrations satisfying 2 2 2 νij = 4σ π v + v /6 [1] ij h i i j 1 q1 1 0 = ninj (1 + λ)nkN. [6] mi− + mj− 2 − B = 3 . i+j=k ij 2 2 X vi + vj h i h i Here N = λ2 is the steady-state value of the total number density These results follow from the Boltzmann equation which describes of aggregates, N = k 1 nk. We solve (6) using the generation ≥ evolution of a system in terms of the joint size-velocity distribution function technique to yield P function (see the section below and SI Appendix). The governing rate k 1 N 2n1 Γ(k 2 ) equations for the concentrations of particles of size k read: nk = (1 + λ) − . [7] √ (1 + λ)N Γ(k + 1) 4π dnk 1 ∞ Now we assume that disruptive collisions in rings are considerably = Cij ninj Ckinink [2] dt 2 − less frequent than aggregative ones, so that λ 1 (this assump- i+j=k i=1 X X tion leads to results that are consistent with observations); moreover, k k 1 for most of the ring particles. Using the steady-state values, ∞ ∞ Akinink (1 δk1) + ni Aij nj xk (j) n1 = λ1 and N = λ2, one can rewrite Eq. (7) for k 1 as − − i=1 i=1 j=k+1 X X X λ λ2k 3/2 1 nk = e− k− . [8] + A n n [x (i) + x (j)] . √π 2 ij i j k k i,j k+1 2 X≥ Thus for k < λ− , the mass distribution exhibits power-law behavior, 3/2 nk k− , with an exponential cutoff for larger k.

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