A Predator-Prey Model for
Clumping in Saturn’s Rings
Larry W. Esposito
LASP
14 November 2012
Key Cassini Observations
• High resolution images of propellers,
straw, embedded moons, F ring objects.
These show aggregation in the rings!
• Occ’s confirm structure, self-gravity
wakes, B ring propellers, ghosts. More
evidence for aggregates.
• Equinox images: embedded objects
• Key question: Are Saturn’s rings young
or old?
Motivation
• Processes observed in in the planetary ring
systems parallel those that occurred at the
time of origin of the planets.
• Many processes occuring in the rings
resemble those in the solar nebula, like the
interaction between the disk and the
protoplanets.
• Rings are our best local laboratory for
studying phenomena in flattened disks:
– Detailed ring observations help develop good
models for the processes in the rings, which may
be useful in understanding these processes
Sub-km structure seen in wavelet
analysis varies with longitude
• Wavelet analysis from multiple UVIS
occultations is co-added to give a
significance estimate
• For the B ring edge, the significance of
features with sizes 200-2000m shows
maxima at 90 and 270 degrees ahead of
Mimas. Same distance where objects are
seen in the images!
Edges also show structure
• Some of this can be explained by
multiple modes
• Other sharp features appear stochastic,
likely caused by local aggregates
From
Albers etal
2012
F ring Observations
• 27 significant features in F ring: ‘Kittens’
from 22m to 3.7km, likely they are
elongated and transient
• They have weak correlation to
Prometheus, may evolve into moonlets
New Features
I Gatti di Roma: temporary features in an ancient structure
We identify our ‘kittens’ as transient clumps
Prometheus excites F ring structures
‘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 Modified 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
Predator-Prey model of Moon-triggered Accretion?
Phase plane trajectory
V2
M
Amplitude proportional to forcing
B ring phase plane trajectories
Wavelet power seen is proportional to resonance torques
Predicted Phase Lag
• Moon flyby or density wave passage
excites forced eccentricity; streamlines
crowd; relative velocity is damped by
successive passes through crests
• This drives the collective aggregation/
dis-aggregation system at a frequency
below its natural limit-cycle frequency
• Model: Impulse, crowding, damping,
aggregation, stirring, disaggregation
• Aggregate M(t) lags moon by roughly π
F ring profiles show streamline crowding (Lewis & Stewart)
F ring phase lag
Rare accretion can renew rings
• Solid aggregates are persistent, like the
absorbing states in a Markov chain
• Even low transition probabilities can
populate the states: e.g., 10-9 per
collision to an absorbing state
• These aggregates can renew the rings:
– shield their interiors from meteoritic dust
pollution
– release pristine material when disrupted by
an external impact
Predator-prey model explains bright haloes of density waves in Saturn’s Rings
VIMS sees brighter and fresher water ice spread
in a halo on both sides of strong density waves.
The Cassini image at left shows that km-size clumps form by compaction in
the crests of the strong density waves. The clumps stir the collision speeds of
nearby particles far more than on average, eroding the particles. The clumps
can then grow again. This cycling is described by the “predator-prey” model.
Lynx and hare populations fluctuate in time
“Haloes” of relatively pure water ice are seen by UVIS and VIMS on both sides of strong density waves. It is believed these
are created by unusually violent collisions excited by huge, but transient, clumps that form in densely compacted crests of
density waves. The grains then spread radially, forming a halo around the density wave. The stirring creates a cycle of high
speed collisions and erosion that resembles the economic ‘boom-bust’ cycle of predator and prey in an ecological system.
Esposito etal Predator-Prey Model for Saturn’s A Ring Haloes 2012 DPS; Hedman etal Connection between spectra & structure in Saturn's main rings based on Cassini VIMSata2012 Icarus,in
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,
threshold for erosion of aggregates from
Blum (2007)
Predator-Prey Equations
M= ∫ n(m) m2 dm / <M>;
Vrel2= ∫ n(m) Vrel2 dm / N
dM/dt=
– Vrel2/vth2 M/Tcoll
M/Tacc
[accretion]
[fragmentation]
[collisional disruption & accretion]
M(stochastic)
dVrel2/dt= -(1-ε2)Vrel2/Tcoll + (M/M0)2 Vesc2/Tstir

[dissipation]
- A0 cos(ωt)
[gravitational stirring]
[forcing by streamline crowding]
Updated Model Results
Analogy: Coast Redwoods
1 in 104 seeds
grows to a tree!
Like Beijing, rings contain
both new and
ancient structures!
Conclusions
• Although rings show youthful features,
this may imply renewal instead
• Changes since Voyager and abundant
embedded objects indicate accretion
• A predator-prey model shows how
moons can trigger accretion today
• Are rings young or old? Yes, maybe
both!
• Key measurement: Ring mass in 2017.
Conclusions: Haloes
• Cyclic velocity changes cause perturbed
regions to reach higher collision speeds
• Which preferentially removes small
regolith particles
• This forms a bright halo around the ILR
• Surrounding particles diffuse back too
slowly to erase the effect