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
150 km, I/F = 0.0077
300 km,
Peak I/F
= 0.0090
300 km,
Peak I/F
= 0.0082
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]
- A0 cos(ωt)
[gravitational stirring]
[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 < n
• 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 PredatorPrey 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