1. Boom and Bust Cycles in Saturn’s rings?
Larry W. Esposito
LASP / University of Colorado
Cassini PSG Munich
9 June 2010

2. Accretion in Rings
Accretion is possible in the Roche zone, where rings are found, but limited by tides and collisions
F ring shows clumps and moonlets; A ring has propellers; Embedded moonlets are elongated
Self-gravity wakes in A,B rings indicate temporary aggregations
Does the size of aggregates represent an equilibrium between accretion and fragmentation? Is this equilibrium steady?

3. 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

4. Features in F ring
UVIS occultations initially identified 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

5. F Ring Search Method Search was tuned for 1 VIMS-confirmed event:
Optimal data-bin size
 threshold
VIMS
UVIS
Pywacket
-15 km km

6. Fluffy
Butterball

7. New Features

8. We identify our ‘kittens’ as temporary clumps

10. Features Lag Prometheus
12 of 25 features in the range =180º ± 20º
The maximum optical depth is at =161º

11. Phase Lag of Kittens
Most features cluster at Prometheus’ antipode

12. Feature optical depth relative to Prometheus
fit = 0.42 cos(+191°)
tau_fit = *sin(x )
Curvefit
No weights
4.9=280deg
All 25 features
Simple icicle
Multi-icicle

13. Sub-km structure seen in wavelet analysis varies with time, 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 m increases since 2004; and shows maxima at 90 and 270 degrees ahead of Mimas
For density waves, significance correlated to resonance torque from the perturbing moon

16. We identify this structure as temporary clumps

17. 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

20. Possible explanations
The clumping behaviour arises from the moon forcing the ring particles to jam together, like ‘traffic jams’ seen in the computer simulations (Lewis and Stewart 2005)
Attraction among the ring particles creates temporary clumps that further perturb the rings around them

22. Summary
Cassini occultations of strongly forced locations show accretion and then disaggregation: scales of hours to weeks
Moons may trigger accretion by streamline crowding (Lewis & Stewart); which enhances collisions, leading to accretion; increasing random velocities; leading to more collisions and more accretion.
Disaggregation may follow from disruptive collisions or tidal shedding

23. 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

26. 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) we drive the system, not the orbits

27. Rings resemble a system of foxes and hares
In absence of interaction between size and velocity, prey (mean mass) grows; predator (velocity) decays
When they interact, a stable equilibrium exists with an equilibrium for the size distribution and a thermal equilibrium

28. Lotka-Volterra Equations
M= ∫ n(m) m dm; Vrel2= ∫ n(m) Vrel2 dm
dM/dt= M/Tacc – Vrel2/vth2 * M/Tcoll
dVrel2/dt= (1-ε2)Vrel2/Tcoll + M2/M02 *Vrel2/Tstir
M: mean mass; Vrel2: velocity dispersion; Vth: fragmentation threshold; ε: restitution coeff; M0: reference mass (10m); Tacc: accretion; Tcoll: collision; Tstir: viscous stirring timescales

29. 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

31. F ring Predator-Prey model, forced by Prometheus

32. Better Ring Model
A more realistic ring model is modified from the Lotka-Volterra equations
Now the fixed point is a spiral stable point; Stability analysis is the same as for a forced damped pendulum at origin: The system spirals to the stable point
Driving this system at low amplitude gives a steady state solution at the forcing frequency in the synodic frame with mass max trailing velocity min by roughly π/2

33. 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.

34. Better F ring model,
forced by Prometheus

36. 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
Mass aggregation M(t) lags by π for pure sinusoidal forcing at synodic period

37. 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 may suffer events that cause them to accrete: then the solid body orbits at the Kepler rate

38. Summary UVIS occultations show aggregation/disaggregation
A predator-prey model idicates moon perturbations can excite cyclic aggregation at the B ring edge and in the F ring, with a natural phase lag
Model: Impulse, crowding, damping, aggregation, stirring, disaggregation
Stochastic events in this agitated system can lead to accreted bodies that orbit at Kepler rate: equinox objects?

39. Backup slides

40. Listener Beware!
My explanation is speculative and not in agreement with some other proposals…
Beurle etal consider fluid instability criterion
Spitale and Porco emphasize normal modes (driven and free)
I prefer a kinetic theory picture, emphasizing individual random events

41. We conclude that the agitation by the moons at both these locations in the F ring and at the B ring outer edge drives aggregation and disaggregation in the forcing frame.
This agitation of the ring material allows fortuitous formation of solid objects from the temporary clumps, via stochastic processes like compaction, adhesion, sintering or reorganization that drives the denser parts of the aggregate to the center or ejects the lighter elements.
Such processes can create the equinox objects seen at the B ring edge and in the F ring, explain the ragged nature of those ring regions and allow for rare events to aggregate ring particles into solid particles, recycling the ring material and extending the ring lifetime.  

42. F ring Predator-Prey model with periodic forcing

46. B ring
Impulsive forcing

47. MASS (t)

48. Vrel2(t)

51. Structure increasing near B ring edge

52. Are Saturn’s Rings Young or Old?
Voyager found active processes and inferred short lifetimes: we concluded the rings were created recently
It is highly unlikely a comet or moon as big as Mimas was shattered recently to produce Saturn’s rings; Are we very fortunate?
New Cassini observations show a range of ages, some even shorter… and even more massive rings!

53. Ring Age Implications
Saturn’s rings appear young… but don’t confuse ‘age’ with most recent renewal!
Like the global economy, rings may not have a stable accretion equilibrium
Instead, boom/bust cycling triggered by moon forcing may allow stochastic events to create big particles
Recycling of ring material means the rings can be as old as the Solar System