---
title: "Derive the motion of a small patch of a planetary ring"
description: "Derive Kepler shear and Hill motion, solve the free epicycle, and follow how moon kicks and inelastic collisions change a local ring patch."
canonical_url: "https://sandboxphysics.com/stories/planetary-rings/shearing-orbits/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/planetary-rings/shearing-orbits/"
last_modified: "2026-10-28"
---

[Saturn’s ring waves: from orbits to starlight](https://sandboxphysics.com/stories/planetary-rings/)

AC10 · L2

# Derive the motion of a small patch of a planetary ring

Sandbox Physics · 2026-10-28

Start with two circular orbits. Their relative speed gives the shear. Then let a particle move away from its circular path, keeping the Coriolis acceleration that appears in a rotating frame. This produces a solvable local model into which a moon and collisions can be added.

## Before you begin

Newtonian gravity, circular motion, derivatives and simple harmonic motion. Use an outward radial coordinate and a prograde azimuthal coordinate, in a frame corotating with the moon’s reference orbit.

## Subtract the orbit before expanding it

At the reference radius, circular balance gives the angular frequency below. Write a nearby radius as the reference radius plus a small outward displacement. Expand angular frequency to first order in that displacement. In the rotating frame, multiply the angular-speed difference by the reference radius to obtain local azimuthal speed; the discarded corrections are second order in patch size divided by orbital radius.

The minus sign matters: a positive radial displacement drifts backward in the prograde coordinate. The origin can follow a different moon orbit when you drag the moon relative to the sampled ring strip. The model’s radial coordinate is always measured from that moon’s circular reference orbit, so a drawn offset is not a stationary object held against orbital shear.

$$
\Omega(r)=\sqrt{\frac{GM_S}{r^3}},\qquad \frac{d\Omega}{dr}=-\frac{3\Omega}{2r}
$$

Differentiate the power of radius before making the local approximation.

$$
\dot y\simeq r_0[\Omega(r_0+x)-\Omega(r_0)]=-\frac32\Omega_0x
$$

A circular orbit becomes a straight shear trajectory in the local coordinates.

Reading: Rein & Tremaine, Symplectic integrators in the shearing sheet, arXiv v2 · NASA NSSDCA Saturn Fact Sheet

## Keep the rotating-frame accelerations

A displaced particle also has radial and azimuthal velocity. The Coriolis acceleration couples those velocities. Expanding central gravity and the centrifugal acceleration about the reference circular orbit leaves a radial tide. Combining these terms gives Hill’s equations below; the right-hand side denotes the additional acceleration from the moon or other local forces, not the already included planetary gravity.

The browser keeps the planar terms. It omits vertical motion, oblateness corrections, particle mutual gravity, spin and friction. Those omissions make the mechanism readable but exclude a quantitative simulation of self-gravity wakes or a fitted Saturn-ring viscosity. Length is 100 m per scene unit and time is one inverse reference angular frequency.

$$
\ddot x=2\Omega_0\dot y+3\Omega_0^2x+F_x,\qquad \ddot y=-2\Omega_0\dot x+F_y
$$

The radial tide and Coriolis terms are required together; ordinary velocity-independent leapfrog splitting would handle this system poorly.

$$
r_0=1.30\times10^8\,\mathrm m,\quad GM_S=3.7931208\times10^{16}\,\mathrm{m^3\,s^{-2}},\quad\Omega_0\simeq1.31\times10^{-4}\,\mathrm{s^{-1}}
$$

One simulation time unit represents about 2.11 hours at these adopted reference values; screen playback is accelerated.

Reading: Rein & Tremaine, Symplectic integrators in the shearing sheet, arXiv v2 · NASA NSSDCA Saturn Fact Sheet

## Solve the free motion before adding a force

Set the additional forces to zero. The azimuthal equation says that the combination below is constant. Substitute the resulting azimuthal velocity into the radial equation. The resulting harmonic oscillator moves about a guiding radius.

The solution contains an oscillation and a steady guiding-center drift. A circular initial condition has zero radial velocity and the shear azimuthal velocity. Its oscillation amplitude vanishes exactly. An impulsive moon encounter generally changes that constant combination and the radial velocity, so the subsequent path oscillates and drifts.

Our integrator advances this unforced solution exactly, then symmetrically places half a softened-moon kick on either side. An independent small-step Runge–Kutta solution checks the forced trajectory, while the zero-moon shear and free epicycle energy provide analytic tests. Passing those tests checks this local approximation, not the missing global physics.

$$
C=\dot y+2\Omega_0x,\qquad\ddot x+\Omega_0^2x=2\Omega_0C,\qquad x_g=\frac{2C}{\Omega_0}
$$

Integrate the azimuthal equation once, then eliminate azimuthal velocity from the radial equation.

$$
x(t)=x_g+(x_0-x_g)\cos(\Omega_0t)+\frac{\dot x_0}{\Omega_0}\sin(\Omega_0t)
$$

The oscillation frequency equals the circular frequency in the Kepler limit.

$$
y(t)=y_0-3Ct-2(x_0-x_g)\sin(\Omega_0t)-\frac{2\dot x_0}{\Omega_0}[1-\cos(\Omega_0t)]
$$

Integrate the azimuthal velocity using the radial solution; keep the secular guiding-center drift.

Reading: Rein & Tremaine, Symplectic integrators in the shearing sheet, arXiv v2

## Work an orbit and a collision through the numbers

Place a particle one hundred metres outside the reference orbit. Its local azimuthal speed is about minus 0.0197 m/s. Over one hour it drifts about seventy-one metres backward. With no moon force and no collision, its radial position stays fixed. The long time represented by a short screen motion is why the Lab labels playback as accelerated.

For a frictionless equal-mass collision, only the relative velocity along the contact normal changes. Momentum conservation fixes the common center-of-mass motion; restitution sets the outgoing normal relative speed. The lost kinetic energy follows by subtracting the before and after relative energies.

As a numerical example, take two 1 kg bodies with normal relative approach speed 0.02 m/s and restitution 0.35. The collision loses about 87.8 microjoules. Tangential relative velocity is unchanged. This is an example of the collision law, not an adopted mass or size for each displayed tracer.

$$
u_n'=-\epsilon u_n,\qquad\Delta K=\frac{m}{4}(1-\epsilon^2)u_n^2
$$

The reduced mass of two equal bodies is half of either body’s mass. Perfect restitution loses no normal relative energy.

$$
\Delta K=\frac{1\,\mathrm{kg}}4(1-0.35^2)(0.02\,\mathrm{m\,s^{-1}})^2=8.78\times10^{-5}\,\mathrm J
$$

Evaluate the energy loss after fixing which velocity is relative and normal.

Reading: Rein & Tremaine, Symplectic integrators in the shearing sheet, arXiv v2 · Salo & Mondino-Llermanos, Photometric modelling of self-gravity wakes and overstable oscillations

## Know which pattern this calculation can produce

The local model gives a prescribed moon, periodically repeated azimuthal encounters and finite collision tracers. Radial motion is open. It can produce deflection, epicyclic crowding and collisional rearrangement. A Gaussian density kernel then measures those same positions. The numerical collision radius and optical weight are independent teaching choices, so counting the drawn disks does not measure a physical particle size.

Global resonant density waves require a collective ring response and a pattern speed tied to an orbital resonance. Self-gravity wakes require mutual particle attraction. Neither is generated by silently adding a decorative stripe. The next chapter connects the local optical column to a detector and then introduces the separate global dispersion relation.

Reading: Rein & Tremaine, Symplectic integrators in the shearing sheet, arXiv v2 · Goldreich & Tremaine, The excitation and evolution of density waves · Salo & Mondino-Llermanos, Photometric modelling of self-gravity wakes and overstable oscillations

## Exercises and explanations

1. Move the same circular particle 100m inward. After one hour, which way has it drifted?

   Read the explanation

   The radial displacement changes sign, so the shear speed becomes positive 0.0197 m/s. It moves about 71m ahead in the prograde coordinate, retaining its radius in the force-free limit.

2. Set the additional force to zero but give a radial velocity kick. Should the particle travel on a straight radial line?

   Read the explanation

   No. The kick excites the epicycle, and the Coriolis acceleration changes azimuthal velocity as the radial position changes. The radial coordinate oscillates around the guiding center.

3. A collision conserves pair momentum but loses kinetic energy. Has the implementation violated a conservation law?

   Read the explanation

   Not for an inelastic collision. Center-of-mass momentum is conserved while normal relative kinetic energy becomes unmodeled internal energy. Check the restitution energy formula; a perfectly elastic case should lose none.

## Try the prediction

### [Separate shear, forcing and dissipation](https://sandboxphysics.com/labs/planetary-ring-waves/)

1. Compare the moon-free and moon-forced presets, then predict how a displaced tracer should drift.
2. Compare elastic and inelastic collisions. Use the local model’s limits when explaining any resulting concentration.

[Previous: How a small moon writes a stripe in Saturn’s rings](https://sandboxphysics.com/stories/planetary-rings/moving-stripes/) [Next: From a ring’s moving particles to its missing starlight](https://sandboxphysics.com/stories/planetary-rings/columns-and-waves/)

## Sources and calculation scope

1. [Rein & Tremaine, Symplectic integrators in the shearing sheet, arXiv v2](https://arxiv.org/pdf/1103.1376v2)

   19April2011 revision; Sections2–3, Hill equations and exact epicycle splitting; Section4 and Figures1–4, free/perturbed trajectory tests. We implement a planar exact-free-motion/symmetric-kick method with an explicitly softened prescribed moon, not their full many-body calculation.

2. [NASA NSSDCA Saturn Fact Sheet](https://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html)

   Updated18March2025, consulted28October2026. Bulk GM and radius establish the physical scale; the Lab adopts the fixed gravitational parameter and reference radius displayed in the course as teaching inputs, not an ephemeris fit or updated gravity solution. The source lists a rounded GM.

3. [Goldreich & Tremaine, The excitation and evolution of density waves](https://authors.library.caltech.edu/records/ef5pg-y1e07)

   Published ApJ222,850–858,15June1978; full author-archive PDF read, SectionII equations4–17 (especially9,11–15): thin self-gravitating sheet, tightly wound dispersion, resonances and wave winding. Course gives its own Kepler-limit expansion; finite pressure, oblateness and nonlinear waves remain separate limits.

4. [Salo & Mondino-Llermanos, Photometric modelling of self-gravity wakes and overstable oscillations](https://doi.org/10.1051/0004-6361/202452328)

   Published3March2025, A&A695,A37. Selected original indexed sections4.2/Figure5 (weak-gravity asymmetry and correlation checks), Figure4 (photometric regimes),8.3/AppendixB (transport contributions), and hybrid-model limitations. Primary full-text HTML was available through indexed excerpts; the whole29-page numerical study was not reproduced.