Predator-Prey Model for Saturn’s A Ring Haloes LW Esposito, ET Bradley, JE Colwell, M Sremcevic, P. Madhusudhanan UVIS Team Meeting 5 June 2013

Cassini Observed ‘Haloes’ in Saturn’s A Ring Annuli of increased brightness were seen by VIMS and UVIS at Saturn Orbit Insertion Found at strongest density waves, but not at Mimas 5:3 bending wave

A Ring Brightness from Cassini UVIS Saturn Insertion

UVIS SOI (150 km resolution elements)

Close-up of UVIS SOI reflectance at Janus 5:4 density wave 300 km, Peak I/F = 0.0090 300 km, Peak I/F = 0.0082 150 km, I/F = 0.0077

VIMS effective grain size at Janus 5:4 resonance From Hedman etal Icarus 2013

‘Straw’ in images

Modified Predator-Prey Equations for Ring Clumping M= ∫ n(m) m2 dm / <M>; Vrel2= ∫ n(m) Vrel2 dm / N dM/dt= M/Tacc – Vrel2/vth2 M/Tcoll [accretion] [fragmentation/erosion] dVrel2/dt= -(1-ε2)Vrel2/Tcoll + (M/M0)2 Vesc2/Tstir [dissipation] [gravitational stirring] - A0 cos(ωt) [forcing by streamline crowding]

In the Predator-Prey Model Periodic forcing from the moon causes streamline crowding This damps the relative velocity, and allows aggregates to grow About a quarter phase later, the aggregates stir the system to higher relative velocity The limit cycle repeats each orbit, with relative velocity ranging from nearly zero to a multiple of the orbit average: 2-10x is possible

Phase plane trajectory V2 M

Upgrades to Predator-Prey Model – Collisions among Ring Particles Add stochastic forcing to simulate aggregate collisions: Random outcome doubles or halves aggregate mass. Previously, no collisions. Add threshold for gravity-bound aggregates: above this it is harder to disrupt aggregates. Previously, erosion of aggregates from Blum (2007) This allows us to find the fixed points, their stability, basins of attraction, and asymptotic behavior, not easy for N-body codes

Log plot of updated system trajectories

The original equations (see above) are: Re-writing in dimensionless form, with threshold depending on mass: With: Gives fixed points: And Jacobian:

Stability for Gravity-Bound Aggregates Stable for ε < Cycles at forcing frequency, with a phase lag Otherwise, unstable These represent ‘outbreak’ fixed points, resembling ‘absorbing’ states of Esposito 2011 Next, try ε(v), and search for stable fixed points

Stable fixed point for ε = 0.5

Unstable for ε = 0.9

Effects on Ring Particle Regolith In the perturbed region, collisions erode the regolith, removing smaller particles The released regolith material settles in the less perturbed neighboring regions Diffusion spreads these ring particles with smaller regolith into a ‘halo’ This process resembles thermal diffusion in a granular system

Brownian motion model Model the thermal diffusion with a random walk on the line (Feller 1971) At regular intervals regolith particles receive a kick from interparticle collisions that throws them a jump distance, with a spread about the mean jump The jump and spread vary, based on the relative velocity expected from the Predator-Prey model: assume a triangular distribution peaked XD outside the resonance

Regolith depth after 91 years for triangular forcing

Steady-State Regolith Depth compared to 1/Seff

VIMS grain size Seff compared to early and steady state

Thermal diffusion is insufficient The steady state regolith depth does not resemble VIMS as much as some of the early intermediate states The haloes are broader than the likely throw distances from 1 m/sec collisions UVIS is not sensitive to regolith depth and grain size: this model does not explain UVIS photometry, although it is consistent with no UVIS spectral differences seen in the haloes

Markov Chain transport model shows halo build-up, similar to inverted VIMS. Differences from the stationary distribution may show an intermediate stage matches the halo, or we need to include production

Solution: Add a production term An additional effect of the regolith removal from aggregates in the strongest density waves is that their surfaces will be exposed directly to meteoritic bombardment This will eject bright new grains These grains will be thrown further, gradually mixed vertically in the regolith, and diffuse radially

Summary ISS, VIMS, UVIS spectroscopy and occultations show haloes around the strongest density waves. Based on a predator-prey model for ring dynamics, we offer the following explanation: Cyclic velocity changes cause perturbed regions to reach higher collision speeds at some orbital phases, which preferentially removes small regolith particles This forms a halo around the ILR, if the forcing is strong Surrounding particles diffuse back too slowly to erase the effect Predicts no UVIS spectroscopic change longward of H2O absorption edge, only photometric brightening of 10-50%, consistent with UVIS SOI observations Predicts larger effective size at ring edges maintained by resonances, too. Maybe the equinox objects are the largest aggregates?

Ring dynamics and history implications Moon-triggered clumping occurs at perturbed regions in Saturn’s rings Cyclic system trajectories forced around the stable point create both high velocity dispersion and large aggregates at these distances We observe their effects: both small and large particles are found at the perturbed locations This confirms the triple architecture of ring particles: a broad size distribution of particles; aggregate into temporary rubble piles; coated by a regolith of dust Aggregates can explain the dynamic nature of the rings and can renew rings by shielding and recycling