Boom and Bust Cycles in
Saturn’s rings?
Larry W. Esposito
LASP / University of Colorado
7 May 2010
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 show temporary
aggregations
• Does the size of aggregates represent an
equilibrium between accretion and
fragmentation? Is this equilibrium steady?
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
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
F Ring Search Method
• Search was
tuned for 1
VIMSconfirmed
event:
– Optimal
data-bin
size
–  threshold
VIMS
UVIS
Pywacket
-15 km
0
15 km
Butterball
Fluffy
New Features
We identify our ‘kittens’ as temporary clumps
Features Lag Prometheus
• 12 of 25
features in the
range
=180º ±
20º
• The maximum
optical depth
is at =161º
Features in each
quadrant
15
10
60
24
0:
3
40
12
0:
2
12
0
5
0
0:
• Most features
cluster at
Prometheus’
antipode
Number of
significant features
Phase Lag of Kittens
Distance from
Prometheus (deg)
Feature optical depth relative to
Prometheus
fit = 0.42 cos(+191°) + 1.47
Simple icicle
Multi-icicle
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 200-2000m 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
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
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
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
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
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
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
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
• 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
F ring Predator-Prey model, forced by Prometheus
Summary
• UVIS occultations show
aggregation/disaggregation
• A predator-prey model shows how
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?
Backup slides
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
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
π/2 to π
F ring Predator-Prey model with periodic forcing
B ring
Impulsive forcing
MASS (t)
Vrel2(t)
Structure increasing near B
ring edge
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!
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
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