AC10 · Ice in differential rotation

Planetary Ring Waves

How can a small moon leave a wave in a ring of moving particles? A moon nudges orbiting particles, and faster inner material shears past slower outer material. Repeated orbital forcing can organize those nudges into a resonant wave around a real ring. A small local patch shows the initial crowding; the same concentration removes more starlight during an occultation.

Drag a moon through a collisional ring patch. Watch differential rotation stretch its disturbance, then read an occultation made from the same particles.

Saved tracers $1024$

Accumulated impacts $6674$

Mean transmission $0.495$

Lowest estimated transmission $0.338$

Disturb the moving ice

Moon gravity

0.35

GM_{\rm m}/(\Omega^2L^3)

Normal restitution

0.35

\epsilon

Starlight elevation

35

B\;[{}^\circ]

Drag the amber moon through the icy patch. Warmed-up particles, a projected star track and saved photon counts respond together.

Position and model limits

Moon radial offset

0.7

x_{\rm m}/L

Moon azimuthal offset

-1.8

y_{\rm m}/L

A moon-centered rotating Hill sheet uses exact free epicycles and symmetric softened-moon kicks. Equal-mass tracer collisions lose normal relative energy according to restitution. Tracers are finite numerical clouds, not observed ring particles; there is no mutual self-gravity, global resonance solver, spin or inferred viscosity. Azimuth is periodic, radial motion is open.

Read the ring-wave course and real Cassini replay

Model assumptions

This is a small planar moon-centered Hill sheet with softened prescribed moon gravity, exact free epicycles and frictionless equal-mass tracer collisions. The particles are numerical clouds, not measured individual ring particles. Azimuth is periodic; radial motion is open. A Gaussian kernel maps the same saved positions to a normal optical column; oblique attenuation then produces exact Poisson photon counts. Opacity weights and collision radii are teaching parameters. There is no particle self-gravity, global resonant density-wave propagation, calibrated viscosity or ring-age measurement. The separate tutorial replays actual Cassini UVIS v3.0 binned photon signals, background/star models, geometry and flags. Scene length is100m per coordinate unit, time is accelerated, and the draggable moon marker is enlarged.

White ice points are saved simulated tracer positions; short trails use their actual velocities. Amber flashes mark recent collisions. Mint starlight brightness uses the saved photon count at the moving projected chord location. All lengths are schematic multiples of one hundred metres; the moon marker is enlarged.

Saved synthetic photons

Optical column and its noisy estimate

Gray is the ideal Gaussian-kernel column from the same snapshot. Mint points are noisy Poisson measurements with counting-only error bars. Each sample has four hundred unocculted photons plus four known background photons. Negative net transmission is retained; its optical-depth logarithm is omitted.

What this small patch can tell us

The saved particle positions generate both curves. Collisional tracers visualize local moon-forced crowding and shear; no mutual particle gravity is present. Global resonant density waves, self-gravity wakes and ring-age inference require additional physics discussed in the course. The normal-depth weight is independent of the numerical collision radius, so neither one should be read as a measured particle size or ring mass.

Start with the essentials

Focus question: Does a darker occultation require more mass, a lower viewing angle, or both?
One-sentence intuition: Trajectories set the optical column; geometry sets the path through it.

Core mathematical model

Kepler shear

\dot y=-\frac32\Omega x

In a frame rotating at the reference orbit, material outside that orbit drifts backward and material inside moves ahead.

Hill motion

\ddot x-2\Omega\dot y-3\Omega^2x=F_x,\qquad\ddot y+2\Omega\dot x=F_y

The local tide and Coriolis acceleration accompany the moon force. Turning off that force recovers the free shear exactly.

An oblique occultation

T=\exp\!\left(-\frac{\tau_n}{|\sin B|}\right)

At a lower opening angle the same normal optical column attenuates more light. This slab relation does not turn optical depth into mass without an opacity model.

Common difficulties

The stripes are not one family

Typical misconceptionAny striped pattern proves that particle self-gravity is present.

Better mental modelA prescribed moon can organize local kinematic crowding without particle mutual gravity. Genuine self-gravity wakes need that mutual force, and global density waves also require a ring-wide propagation model.

Run the experiment

01
Turn off the moon
Compare Moon-forced crowding with Shear without a moon. Watch the actual velocity trails on both sides.
What to observe: Shear remains when the moon force vanishes; moon-centered deflection disappears.
02
Move the perturber
Drag the moon toward the outer edge and across the projected star track.
What to observe: The changed trajectories alter the sampled column and photon counts together.
03
Change the line of sight
Pause particles and lower the starlight elevation. Compare normal depth with transmitted fraction.
What to observe: The same positions give the same ideal normal column, while a longer slant path transmits less light.

Planetary Ring Waves

A disturbance travels through moving ice

Core mathematical model

Kepler shear

Hill motion

An oblique occultation

Common difficulties

The stripes are not one family

Run the experiment

Turn off the moon

Move the perturber

Change the line of sight

Related experiments and articles

From ring particles to starlight