---
title: "Differential rotation and orbital resonances"
description: "Derive angular and epicycle frequencies for a flat rotation curve, then locate corotation and the circular Lindblad reference radii."
canonical_url: "https://sandboxphysics.com/stories/galactic-bar-orbits/differential-rotation-and-orbital-resonances/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/galactic-bar-orbits/differential-rotation-and-orbital-resonances/"
last_modified: "2026-11-05"
---

[Galactic bars, resonances and stellar migration](https://sandboxphysics.com/stories/galactic-bar-orbits/)

Galactic bar orbits · L1

# Differential rotation and orbital resonances

[Open the rotating-bar Lab](https://sandboxphysics.com/labs/galactic-bar-orbits/)

Sandbox Physics · 2026-11-05

A star's speed along its orbit and a pattern's angular speed answer different questions. Comparing their angles tells us how often the star encounters a bar or spiral. Comparing that encounter rate with a small radial oscillation identifies the simplest orbital resonances.

## Before you begin

Begin with the concept chapter. We use circular motion, a derivative as a local slope, and a second derivative as the curvature of a potential.

## Fix the plane and the clock

Take a right-handed plane with coordinates $x,y$. Positive azimuth $\phi$ runs counterclockwise. Radius $R$ measures distance from the centre. A star's tangential speed is $v_\phi$; angular speed is $\Omega=d\phi/dt=v_\phi/R$. The latter has inverse-time units.

Choose a length $R_0$ and speed $v_0$. Time is measured in $R_0/v_0$. Our calculations set both scales to one. A dimensionless pattern speed of one therefore means $v_0/R_0$, not one revolution per unit time. A revolution contains $2\pi$ radians.

$$
\Omega(R)=\frac{v_0}{R},\qquad T_{\rm orb}=\frac{2\pi R}{v_0}
$$

These relations use a flat circular-speed curve. They describe the axisymmetric circular reference, before a bar perturbs an orbit.

Reading: Dehnen (1999v1/2000), The Effect of the Outer Lindblad Resonance of the Galactic Bar on the Local Stellar Velocity Distribution · Sandbox Physics independent galactic-bar scientific reader v1

## A material line winds up

Let two circular stars begin at the same angle, at radii $R_0$ and $2R_0$. After $2\pi R_0/v_0$, the inner star has made one revolution and the outer star half a revolution. The original line now spans an angular difference of $\pi$.

A rigidly rotating pattern has a different rule: every part of its shape advances through the same angle in the same time. A collective gravitational disturbance can have such a pattern speed even when its members orbit at different rates. Temporary arms can also wind and change their pattern speed. We will use one constant pattern as a controlled first calculation.

Reading: Sellwood & Binney (2002), Radial mixing in galactic discs, astro-ph/0203510v2

## Count encounters in the rotating frame

A frame rotating at pattern speed $\Omega_p$ assigns the star angle $\phi'=\phi-\Omega_pt$. For a nearly circular orbit, that angle changes at $\Omega-\Omega_p$. If the pattern has $m$ equivalent arms, the star encounters an equivalent arm after a relative angle $2\pi/m$.

For a two-fold pattern with $\Omega_p=1$, a circular star at $R=0.8$ has $\Omega=1.25$. Its relative angular speed is $0.25$ and its encounter time is $4\pi\simeq12.57$. At $R=1.2$ the relative rate is $-1/6$; the pattern overtakes the star and the encounter time becomes $6\pi\simeq18.85$.

The circular encounter time diverges at corotation. A real perturbed star there can librate: its angle relative to the pattern swings between turning points. That requires solving the orbit; a long encounter time alone does not prove capture.

$$
T_{\rm encounter}=\frac{2\pi}{m|\Omega-\Omega_p|},\qquad \Omega(R_{\rm CR})=\Omega_p
$$

The absolute value gives a positive interval; the sign of the relative angular speed tells us who overtakes whom.

Reading: Dehnen (1999v1/2000), The Effect of the Outer Lindblad Resonance of the Galactic Bar on the Local Stellar Velocity Distribution

## A small radial kick has its own frequency

Remove the bar for this step. The flat circular-speed potential is $\Phi_0=v_0^2\ln(R/R_0)$. All energies here are per unit mass. Axisymmetry preserves the specific angular momentum $L_z$, so radial motion sees an effective potential $U=\Phi_0+L_z^2/(2R^2)$.

At the reference circular radius $R_c$, the first derivative of $U$ vanishes and $L_z=R_cv_0$. Displace the star by a small $\delta R$. The first nonzero restoring force is the negative curvature of $U$ times that displacement. Differentiating twice gives $U''(R_c)=2v_0^2/R_c^2$.

The resulting small oscillation is called an epicycle. Its frequency $\kappa$ is larger than the circular angular frequency by $\sqrt2$. The independent reader gives a radial kick of $10^{-4}v_0$ at $R_c=1.3R_0$ and measures $\kappa\simeq1.08785658\,v_0/R_0$ from successive radial maxima (apocentres).

The general epicycle formula follows from the same argument. The circular-force relation is $\Phi'_0=R\Omega^2$. Differentiating once gives $\Phi''_0=\Omega^2+R\,d\Omega^2/dR$. The angular-momentum part of the effective potential contributes another $3\Omega^2$ at the circular orbit. Adding them gives the expression below.

This is an angular frequency: one small radial oscillation takes $2\pi/\kappa$, not $1/\kappa$. A positive squared frequency gives a restoring oscillation in this local radial approximation. A finite bar can couple that oscillation to the azimuthal motion, which is why the reference frequencies are useful starting points for a resonance calculation.

$$
\frac{d^2\delta R}{dt^2}+\kappa^2\delta R=0,\qquad \kappa^2=R\frac{d\Omega^2}{dR}+4\Omega^2=2\frac{v_0^2}{R^2}
$$

The final equality applies to the flat-speed potential. The oscillation equation keeps only first-order displacement terms.

Reading: Sellwood & Binney (2002), Radial mixing in galactic discs, astro-ph/0203510v2 · Sandbox Physics independent galactic-bar scientific reader v1

## Locate the circular resonance references

A two-fold perturbation repeats at frequency $2|\Omega-\Omega_p|$ along a circular orbit. If that equals $\kappa$, radial kicks can recur at the same epicycle phase. The inner and outer Lindblad reference radii solve the equations below.

With $R_0=v_0=\Omega_p=1$, the radii are $R_{\rm ILR}=1-1/\sqrt2\simeq0.2929$, $R_{\rm CR}=1$ and $R_{\rm OLR}=1+1/\sqrt2\simeq1.7071$. The experiment admits whole trajectories only outside $R=0.5$, so its inner Lindblad circle lies outside that admitted branch.

These are linear, circular-orbit references. A finite bar changes orbital frequencies and creates finite resonant regions. The circles are useful landmarks, not exact boundaries for nonlinear capture.

$$
\Omega_p=\Omega-\frac{\kappa}{2}\quad({\rm ILR}),\qquad \Omega_p=\Omega\quad({\rm CR}),\qquad \Omega_p=\Omega+\frac{\kappa}{2}\quad({\rm OLR})
$$

ILR and OLR mean inner and outer Lindblad resonance. Each equation refers to the unperturbed near-circular frequencies.

Reading: Dehnen (1999v1/2000), The Effect of the Outer Lindblad Resonance of the Galactic Bar on the Local Stellar Velocity Distribution · Sellwood & Binney (2002), Radial mixing in galactic discs, astro-ph/0203510v2

## Exercises and explanations

1. At fixed flat speed, double $\Omega_p$. How do all three circular reference radii change?

   Read the explanation

   Each radius is proportional to the inverse pattern speed. They all halve. Their ratios remain the same for this particular flat-speed reference.

2. A star crosses two arms at the same frequency as its radial oscillation. Does it have a closed orbit?

   Read the explanation

   The frequency agreement identifies a linear resonance condition. Closure also requires the complete position and velocity to return in a specified frame. Finite-amplitude motion and additional frequencies must be checked by integration.

## Try the prediction

### [Predict a moving corotation circle](https://sandboxphysics.com/labs/galactic-bar-orbits/)

1. Before changing the pattern speed, predict the inverse change in the corotation radius.
2. Drag the corotation handle and compare the accepted radius with your prediction.
3. Select a star near a reference circle. Follow its actual radial motion and compare it with the reference circle.

[Previous: Why stars pass through galactic spiral arms](https://sandboxphysics.com/stories/galactic-bar-orbits/why-stars-cross-spiral-arms/) [Next: Torque and energy in a rotating bar](https://sandboxphysics.com/stories/galactic-bar-orbits/torque-energy-and-a-rotating-bar/)

## Sources and calculation scope

1. [Dehnen (1999v1/2000), The Effect of the Outer Lindblad Resonance of the Galactic Bar on the Local Stellar Velocity Distribution](https://arxiv.org/abs/astro-ph/9911161v1)

   Fixed 14-page astro-ph/9911161v1, all text read in parameter preflight; equation 3, section 3 and original orbital figures. The own model uses a fixed bar radius 0.5 and counterclockwise convention. The paper's historical Milky Way inference is not treated as a current measurement.

2. [Sellwood & Binney (2002), Radial mixing in galactic discs, astro-ph/0203510v2](https://arxiv.org/abs/astro-ph/0203510v2)

   Fixed 13-page v2, complete original read; section 2 specific energy/action exchange and sections 3–4 transient self-gravitating waves. The paper's live-disk simulations are distinct from our prescribed external-bar release.

3. [Sandbox Physics independent galactic-bar scientific reader v1](https://sandboxphysics.com/stories/galactic-bar-orbits/materials/galactic-bar-orbits-reader-science-v1.zip)

   Own zero-network Python integration and saved-record analysis: one full 256-tracer population, ten independent DOP853 controls, exact circle/epicycle, prescribed-growth work, six fresh periodic roots and three own synthetic figures. No App, completed orbit input, original paper or observed data.