Size Distribution of Particles in Saturnts Rings from Aggregation

Size distribution of particles in Saturn’s rings from aggregation and fragmentation Nikolai Brilliantova, P. L. Krapivskyb, Anna Bodrovac,d, Frank Spahnd, Hisao Hayakawae, Vladimir Stadnichukc, and Jürgen Schmidtf,1 aDepartment of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom; bDepartment of Physics, Boston University, Boston, MA 02215; cFaculty of Physics, Moscow State University, Moscow, 119991, Russia; dInstitute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany; eYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan; and fAstronomy and Space Physics, University of Oulu, PL 3000, 90014, Oulu, Finland Edited by Neta A. Bahcall, Princeton University, Princeton, NJ, and approved June 12, 2015 (received for review February 26, 2015) Saturn’s rings consist of a huge number of water ice particles, with a spheres; then the radius of an agglomerate containing k monomers 1=3 tiny addition of rocky material. They form a flat disk, as the result of is rk = r1k . (In principle, aggregates can be fractal objects, so that 1=D an interplay of angular momentum conservation and the steady loss rk ∼ k ,whereD ≤ 3 is the fractal dimension of aggregates. For of energy in dissipative interparticle collisions. For particles in the dense planetary rings it is reasonable to assume that aggregates are size range from a few centimeters to a few meters, a power-law compact, so D = 3.) Systems composed of spherical particles may −q – distribution of radii, ∼ r with q ≈ 3, has been inferred; for larger be described in the framework of the Enskog Boltzmann theory – sizes, the distribution has a steep cutoff. It has been suggested that (25 27). In this case the rate of binary collisions depends on particle dimension and relative velocity. The cross-section for the this size distribution may arise from a balance between aggregation i j and fragmentation of ring particles, yet neither the power-law de- collision of particles of size and can be written as σ2 = r + r 2 = r2 i1=3 + j1=3 2 pendence nor the upper size cutoff have been established on theo- ij ð i jÞ 1ð Þ . The relative speed [on the order of 0.01 − 0.1 cm=s (16)] is determined by the velocity dispersions retical grounds. Here we propose a model for the particle size v2 v2 i j distribution that quantitatively explains the observations. In accor- h i i and h j i for particles of size and . The velocity dispersion quantifies the root-mean-square deviation of particle velocities dance with data, our model predicts the exponent q to be con- ∼ = ≤ q ≤ from the orbital speed ( 20 km s). These deviations follow a strained to the interval 2.75 3.5. Also an exponential cutoff certain distribution, implying a range of interparticle impact speeds for larger particle sizes establishes naturally with the cutoff radius and, thus, different collisional outcomes. The detailed analysis of being set by the relative frequency of aggregating and disruptive an impact shows that for collisions at small relative velocities, when collisions. This cutoff is much smaller than the typical scale of mi- the relative kinetic energy is smaller than a certain threshold en- crostructures seen in Saturn’srings. ergy, Eagg, particles stick together to form a joint aggregate (11, 28, 29). This occurs because adhesive forces acting between ice parti- planetary rings | kinetic theory | coagulation–fragmentation cles’ surfaces are strong enough to keep them together. For larger velocities, particles rebound with a partial loss of their kinetic en- ombardment of Saturn’s rings by interplanetary meteoroids ergy. For still larger impact speeds, the relative kinetic energy ex- E B(1–3) and the observation of rapid processes in the ring system ceeds the threshold energy for fragmentation, frag, and particles (4) indicate that the shape of the particle size distribution is likely break into pieces (29). not primordial or a direct result of the ring-creating event. Rather, Using kinetic theory of granular gases one can find the colli- ring particles are believed to be involved in active accretion–de- sion frequency for all kinds of collisions and the respective rate – coefficients: Cij for collisions leading to merging and Aij for struction dynamics (5 13) and their sizes vary over a few orders of C – disruptive collisions. The coefficients ij give the number of magnitude as a power law (14 17), with a sharp cutoff for large i + j – aggregates of size ð Þ forming per unit time in a unit volume as sizes (18 21). Moreover, tidal forces fail to explain the abrupt a result of aggregative collisions involving particles of size i and j. decay of the size distribution for house-sized particles (22). One Similarly, Aij quantify disruptive collisions, when particles of size wishes to understand the following: (i) Can the interplay between aggregation and fragmentation lead to the observed size distri- Significance bution? And (ii) is this distribution peculiar for Saturn’srings,oris it universal for planetary rings in general? To answer these questions quantitatively, one needs to elaborate a detailed model Although it is well accepted that the particle size distribution in Saturn’s rings is not primordial, it remains unclear whether the of the kinetic processes in which the ring particles are involved. observed distribution is unique or universal, that is, whether it is Here we develop a theory that quantitatively explains the observed determined by the history of the rings and details of the particle properties of the particle size distribution and show that these interaction or whether the distribution is generic for all plane- properties are generic for a steady state, when a balance between tary rings. We show that a power-law size distribution with aggregation and fragmentation holds. Our model is based on the large-size cutoff, as observed in Saturn’s rings, is universal for hypothesis that collisions are binary and that they may be classified systems where a balance between aggregation and disruptive as aggregative, restitutive, or disruptive (including collisions with collisions is steadily sustained. Hence, the same size distribution erosion); which type of collision is realized depends on the relative is expected for any ring system where collisions play a role, like speed of colliding particles and their masses. We apply the kinetic the Uranian rings, the recently discovered rings of Chariklo and theory of granular gases (23, 24) for the multicomponent system of Chiron, and possibly rings around extrasolar objects. ring particles to quantify the collision rate and the type of collision. Author contributions: N.B. designed research; N.B., P.L.K., A.B., H.H., V.S., and J.S. performed Results and Discussion research; N.B., P.L.K., A.B., H.H., V.S., and J.S. contributed new reagents/analytic tools; Model. Ring particles may be treated as aggregates built up of N.B., P.L.K., A.B., V.S., and J.S. analyzed data; and N.B., P.L.K., F.S., and J.S. wrote the paper. primary grains (9) of a certain size r1 and mass m1. [Observations The authors declare no conflict of interest. indicate that particles below a certain radius are absent in dense This article is a PNAS Direct Submission. rings (16).] Denote by mk = km1 the mass of ring particles of “size” 1To whom correspondence should be addressed. Email: [email protected]. k k n containing primary grains and by k their number density. For This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. the purpose of a kinetic description we assume that all particles are 1073/pnas.1503957112/-/DCSupplemental. 9536–9541 | PNAS | August 4, 2015 | vol. 112 | no. 31 www.pnas.org/cgi/doi/10.1073/pnas.1503957112 Downloaded by guest on September 24, 2021 i and j collide and break into smaller pieces. These rate co- loss of generality we set C0 = 1. Solving the governing equations for efficients depend on masses of particles, velocity dispersions, and monodisperse initial conditions, nkðt = 0Þ = δk,1,onefinds E E threshold energies, agg and frag: " # −λ =λ À À Á À ÁÁ λ−1 2 1 n t = λ + λ−1 λ−1eλt − [5] Cij = νij 1 − 1 + BijEagg exp −BijEagg 1ð Þ 1 1 2 , À Á 2 Aij = νij exp −BijEfrag vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where λ = λ=ð1 + λÞ and λ = 2λ=ð1 + 2λÞ. Using the recursive na- u 1 2 u 2 2 ture of Eq. 3, one can determine nkðtÞ for k > 1. The system tπ vi + vj ν = σ2 demonstrates a relaxation behavior: After a relaxation time that ij 4 ij −1 6 scales as λ , the system approaches a steady state with n1 = λ1, the other concentrations satisfying −1 −1 mi + mj Bij = 3 . [1] X v2 + v2 1 i j 0 = ninj − ð1 + λÞnkN. [6] 2 i+j=k These results follow from the Boltzmann equation, which describes evolution of a system in terms of the joint size–velocity Here N = λ2 is the steady-stateP value of the total number density distribution function (section below and SI Text). The governing N = n 6 of aggregates, k≥1 k. We solve [ ] using the generation rate equations for the concentrations of particles of size k read function technique to yield X X∞ dnk 1 k = Cijninj − Ckinink N 2n1 Γðk − 1=2Þ dt 2 nk = pffiffiffiffiffi ð1 + λÞ . [7] i+j=k i=1 4π ð1 + λÞN Γðk + 1Þ X∞ Xk X∞ − Akininkð1 − δk1Þ + ni AijnjxkðjÞ [2] Now we assume that disruptive collisions in rings are consider- i=1 i=1 j=k+1 ably less frequent than aggregative ones, so that λ 1 (this as- X 1 sumption leads to results that are consistent with observations); + Aijninj½xkðiÞ + xkðjÞ.

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