Boom and Bust Cycles in
Saturn’s rings?
Larry W. Esposito, Bonnie
Meinke, Nicole Albers, Miodrag
Sremcevic
LASP/University of Colorado
DPS Pasadena
5 October 2010
Cassini UVIS occultations
• UVIS has observed over 100 star
occultations by Saturn’s rings
• Time resolution of 1-2 msec allows
diffraction- limited spatial resolution of
tens of meters in the ring plane
• Multiple occultations provide a 3D ‘CAT
scan’ of the ring structure
• Spectral analysis gives characteristics
of ring structures and their dimensions
F ring Kittens
• UVIS occultations initially found 13
statistically significant features
• Interpreted as temporary clumps and a
possible moonlet, ‘Mittens’
• Meinke etal (2010) now catalog 25
features from the first 102 stellar
occultations
• For every feature, we have a location,
width, maximum optical depth (opacity),
nickname
New Features
I Gatti di Roma
We identify our ‘kittens’ as temporary clumps
Features Lag Prometheus
• 12 of 25
features have
=180º ±
20º
• The maximum
optical depth
is at =161º
• Sinusoidal fit
gives Δλ=191º
but r2 only 0.1
Sub-km structure seen in wavelet
analysis varies with longitude
• Wavelet analysis from multiple 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
• For density waves, significance correlated to
resonance torque from the perturbing moon
Observational Findings
• F ring kittens more opaque trailing Prom by π
• Sub-km structure, which is seen by wavelet
analysis at strongest density waves and at B
ring edge, is correlated with torque (for
density waves) and longitude (B ring edge)
• Structure leads Mimas by π/2, equivalent to
π in the m=2 forcing frame
• The largest structures could be visible to ISS:
we thought they might be the equinox objects
Do Saturn’s rings resemble a
system of foxes and hares?
• In absence of interaction between size
and velocity, prey (mean aggregate
mass) grows; predator (velocity) decays
• When they interact, a stable equilibrium
exists with an equilibrium for the size
distribution and a thermal equilibrium
Model Approach
• We model accretion/fragmentation
balance as a predator-prey model
• Prey: Mean aggregate mass
• Predator: Mean random velocity (it
‘feeds’ off the mean mass)
• Calculate the system dynamics
• Compare to UVIS HSP data: wavelet
analysis (B-ring), kittens (F-ring)
• Relate to Equinox aggregate images
Predator-Prey Model
• Simplify accretion/fragmentation
balance equations, similar to approach
used for plasma instabilities
• Include accretional aggregate growth,
collisional disruption, dissipation,
viscous stirring
• Different from Showalter & Burns (1982)
moons perturb the system, not just the
orbits
Phase plane trajectory
V2
M
Amplitude proportional to forcing
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
What Happens at Higher
Amplitudes?
• The moonlet perturbations may be
strong enough to force the system into
chaotic behavior or into a different basin
of attraction around another fixed point
(see Wisdom for driven pendulum); or
• Individual aggregates in the Roche zone
may suffer random events that cause
them to accrete: then the solid body
would orbit at the Kepler rate
Conclusions
• UVIS sees aggregation/disaggregation
• In a predator-prey model, moon
perturbations excite cyclic aggregation
at the B ring edge and in the F ring; this
explains phase lag
• Stochastic events in this agitated
system can lead to accreted bodies that
orbit at Kepler rate: equinox objects
(ISS), ring renewal (Charnoz)? Not
shown yet…
Backup Slides
Lotka-Volterra equations
describe a predator-prey system
• This system is neutrally stable around
the non-trivial fixed point
• Near the fixed point, the level curves
are ellipses, same as for pendulum
• The size and shape of the level curves
depend on size of the initial impulse
• The system limit cycles with fixed period
• Predators lag prey by π/2
Lotka-Volterra Equations
M= ∫ n(m)
m2
dm ; V 2= ∫ n(m) V 2 dm
rel
rel
m
dM/dt= M/Tacc
– Vrel2/vth2 * M/Tcoll
dVrel2/dt= (1-ε2)Vrel2/Tcoll + M2/M02 *Vrel2/Tstir

M: mean aggregate mass;
Vrel2: velocity dispersion;
Vth: fragmentation threshold; ε: restitution
coeff; M0: reference mass (10m);
Tacc: accretion; Tcoll: collision; Tstir: viscous
stirring timescales
Better Model
dM/dt= M/Tacc
– Vrel2/vth2 * M/Tcoll
dVrel2/dt= (1-ε2)Vrel2/Tcoll + M2/M02 *Vesc2/Tstir
In the second equation, we replaced Vrel2 by
Vesc2: This is equivalent to viscous stirring
by aggregates of mass M.
This is no longer a pure predator-prey
model, but it better mimics the ring
dynamics.