Boom and Bust Cycles in
Saturn’s rings?
Larry W. Esposito
LASP / University of Colorado
Cassini Science Highlight
PSG Munich, 10 June 2010
Revised 30 June 2010.
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 have
=180º ±
20º
• The maximum
optical depth
is at =161º
• Sinusoidal fit
gives Δλ=191º
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
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
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
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, 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
F ring Predator-Prey model, forced by Prometheus
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
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.
F ring model,
forced by Prometheus
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
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
Living at the Edge of Saturn’s Rings
F ring
The F ring is a dusty ringlet located just beyond the main ring system at a distance of 140,200km from the center of Saturn. At this dynamically peculiar
location, the Roche zone, the tendency for ring particles to merge due to gravity is balanced by the tidal effects of the planet acting to tear them apart. The F
ring has a complicated, continually changing structure dominated by a bright, ~50km-wide core; the adjacent strands have a spiral structure and are thought
to the icy debris from small objects which have impacted the ring. The gravity of the small moon Prometheus (86km wide) produces “channels” across the F
ring when it passes and these shear over time. However, new results from the Cassini spacecraft have shown that Prometheus does more than just create
pretty patterns and that it has an interesting dynamical history. The F ring itself may be the “signature” of a process by which new moons are formed regularly
at the outer edge of Saturn’s ring system.
Cassini scientists have shown that Prometheus triggers the gravitational
collapse of ring particles in the F ring core. The resulting objects have
Prometheus
sufficient mass to perturb the surrounding material and create “fan”
“channels”
structures which reveal their presence. Similar processes on a much
larger scale are thought to have operated in the early history of the solar
system as planets formed out of clouds of dust and gas.
Additional research has shown that there are checks and balances that
determine the lifetimes of the objects formed in the F ring by Prometheus.
F ring core
As more objects form there is an increase in the number and speed of
collisions. Over time this ultimately acts to decrease the number of
objects, but then they start to grow again. The whole process can be
studied using “predator-prey” models from ecology systems theory.
Prometheus is just one of many small, icy moons with elongated shapes on
the outskirts of Saturn’s rings. Recent work has shown that such moons
must have accreted in the outer part of Saturn’s main rings (perhaps
agitated by predator-prey cycles) and evolved outwards due to their
sheared “channels”
gravitational interaction with the rings within the last 10 million years,
creating the F ring on their path outwards. In many ways Saturn’s rings
behave like a miniature protoplanetary disk.
Janus
“fan”
F ring core
Two images of a 40,000km-long section of Saturn’s F ring taken by
Cassini’s cameras on June 1st, 2010. The small moon Prometheus can
be seen in the upper image along with several channel structures it has
created in the ring. In the lower image, taken 50 minutes later,
Prometheus has moved out of the field of view but a “fan” structure in the
rings indicates the presence of an object in the ring’s core.
Atlas Prometheus Pandora
Epimetheus
20km
A selection of the small, icy, irregularly-shaped moons orbiting just outside
the main ring system. Their odd shapes are due to the strong tides of Saturn.
Conclusions
• UVIS occultations show
aggregation/disaggregation
• A predator-prey model indicates 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
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
Living at the edge of Saturn’s rings
Cassini finds clues to the way moons form and evolve
F
ring
Saturn’s F ring is at a special location in the rings where there is a balance between self-gravity causing ring
particles to accrete into moons, and tidal forces trying to tear them apart.
•
As the moon Prometheus (86km wide) passes the F
Prometheus
ring its gravity creates a signature “channel” pattern
(see upper left), similar to the wake made by a boat
“channels”
moving across a lake.
•
Prometheus triggers the collapse of the icy ring
material to form mountain-sized snowballs at the core
F ring core
of the F ring. These leave tell-tale “fan” structures in
the ring (see lower left).
•
Over time the number of objects can rise and fall in
cycles, similar to the behavior of ‘predator-prey’
models of ecological systems.
sheared “channels”
•
Many of Saturn’s small moons (see below) are now
thought to have formed at the outer edge of the rings
and evolved outwards over the last 10 million years.
F ring core
“fan”
Two Cassini images of Saturn’s F ring taken 50 minutes apart
on June 1, 2010
Saturn’s rings behave like a miniature version of
the disk out of which the planets formed – the
same physical processes operate even though
the scales are vastly different.
Janus
Atlas
Prometheus
Pandora
Epimetheus
20km
Cassini images of the small moons of Saturn that lie just outside
the main rings. The images are to the same scale.
• 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.
Structure increasing near B
ring edge
F ring Predator-Prey model with periodic forcing
B ring
Impulsive forcing
MASS (t)
Vrel2(t)