---
title: "Galactic Bar Orbits and Corotation Interactive Lab | Sandbox Physics"
description: "Why can a rotating galactic bar change a star’s orbit? A bar pulls partly along a star’s orbit and changes its angular momentum."
canonical_url: "https://sandboxphysics.com/labs/galactic-bar-orbits/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/galactic-bar-orbits/"
last_modified: "2026-11-05"
---

Stars passing a rotating gravitational pattern

# Galactic Bar Orbits and Corotation

**Why can a rotating galactic bar change a star’s orbit?** A bar pulls partly along a star’s orbit and changes its angular momentum. Relative rotation changes how that pull accumulates. Some paths close in the bar’s rotating frame; a fixed-field model lets us follow the exchange between a prescribed pattern and massless stars.

Drag corotation or change the rotating field. Follow the stars, select one and compare its angular momentum with the torque acting along its path.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The stars are massless tracers released in a prescribed planar logarithmic field and a rotating quadrupole. They do not generate the field, and the release is not a fitted stellar distribution or a self-consistent galaxy. Positions and velocities use reference units; energy and angular momentum are per unit mass. The cold release contains coincident pairs; the warm preset gives paired opposite radial velocities. Within each history, field strength and pattern speed stay fixed. Each physical change recomputes the full finite history from the same release. Two actual integration resolutions are compared. Trajectories entering the excluded field-radius region are refused, and the preceding complete result stays visible. The warm release with the fastest, strongest field is one such refusal. Rings mark circular-orbit corotation and outer Lindblad resonance of the background field. Eccentric orbits need not have these exact boundaries. Fixed-field periodic references are separately shot and integrated through a full cycle. Only the two exact baseline parents have offline linear-elliptic evidence; neighboring parents remain unclassified and nonlinear stability is unproved. The two finite sequences do not trace a whole bifurcation or carry named literature-family labels. Particle brightness and arrow lengths are display conventions, not luminosity or a quantitative speed scale.

Physics tutorial

## A rotating pull changes angular momentum

**Background** A circular background field pulls straight toward the centre. Add a rotating bar-shaped contribution and some of the pull runs along a star’s path. Watch a selected star gain or lose angular momentum as it passes the pattern.

**Why it matters** A star can keep moving forward while drifting relative to the bar. That relative motion determines how the changing torque adds up.

Start with the essentials

- **Focus question**: How does the pull along the path add up to an angular-momentum change?
- **One-sentence intuition**: The torque acts along the whole trajectory. A steady rotating field conserves one combination of energy and angular momentum, even as each changes.

### Core mathematical model

#### The prescribed rotating field

$$
\begin{gathered}\Phi=v_0^2\ln(R/R_0)+\Phi_{\rm b}\\\Phi_{\rm b}=A f(R)\cos[2(\phi-\Omega_{\rm p}t)]\\ f(R)=-\left(\frac{R_{\rm b}}R\right)^3,\ R>R_{\rm b}\end{gathered}
$$

The first term gives a flat circular-speed curve. The second has two opposite directions of stronger pull and rotates rigidly. The displayed trajectories stay outside the fixed field radius.

#### Tangential pull changes angular momentum

$$
\frac{dL_z}{dt}=-\frac{\partial\Phi}{\partial\phi},\qquad\Delta L_z=\int\tau_z\,dt
$$

Energy and angular momentum here are per unit mass. A radial pull has no torque about the centre. The bar contribution can act tangentially, with its sign changing along the path. The time plot compares the actual angular-momentum change with the accumulated torque.

#### Corotation is a relative-speed condition

$$
\begin{gathered}\Omega(R)=\frac{v_0}{R},\quad\kappa(R)=\frac{\sqrt2v_0}{R}\\ R_{\rm CR}=\frac{v_0}{\Omega_{\rm p}}\\ R_{\rm OLR}=\left(1+\frac1{\sqrt2}\right)R_{\rm CR}\end{gathered}
$$

Corotation means a circular star keeps pace with the pattern. The outer Lindblad reference combines radial oscillation with relative rotation. These radii describe small departures from circular background orbits; an eccentric orbit requires its actual history.

#### Energy and angular momentum change together

$$
E_J=E-\Omega_{\rm p}L_z,\qquad\Delta E=\Omega_{\rm p}\Delta L_z
$$

This Jacobi combination is constant for one rigidly rotating, steady-amplitude field. For example, in reference units a pattern speed of one and an angular-momentum gain of 0.05 imply an energy gain of 0.05. Changing the field starts a different calculation.

#### Measure an angular response

$$
Q_2=\left|\sum_jw_j e^{2i(\phi_j-\Omega_{\rm p}t)}\right|,\qquad w_j=\frac1{256}
$$

Equal angular spacing makes this response vanish initially. A two-direction preference can raise it. Every tracer has the same weight; the number describes their angles, not a measured bar mass or luminosity fraction.

### Common difficulties

#### A path that looks closed

**Typical misconception** A curve returning near its starting point in the picture is a periodic orbit.

**Better mental model** A periodic parent is a closed reference for nearby motion. It must return in position and velocity in the bar frame. The reference presets solve that condition and then integrate a whole cycle. The inner and outer groups are separate local samples.

#### Changing the pattern during a run

**Typical misconception** The Jacobi value should remain fixed after a pattern-speed drag.

**Better mental model** That conservation law belongs to one steady field. Changing a physical control starts a complete new trajectory from the original release conditions.

### Run the experiment

1. 01

   #### Remove the tangential pull

   Pause the replay, set Potential amplitude to zero, then replay the saved times. Predict what happens to the selected angular-momentum change.

   **What to observe:** The circular cold release keeps nearly fixed angular momentum. Restoring the bar gives the selected star a varying torque; its sign and net result depend on the whole path.

2. 02

   #### Move the place that keeps pace

   Drag the cyan corotation handle outward. Compare Corotation moves outward with A faster inner pattern.

   **What to observe:** A slower pattern moves circular corotation outward. For example, a pattern speed of 0.7 puts it near 1.43 reference radii and the outer Lindblad reference near 2.44. The stars and angular-response curve change with the recomputed motion.

3. 03

   #### Compare two actual periodic samples

   Choose Perpendicular closed parents, then Parallel closed parents. Read the radial-departure plot below the stage.

   **What to observe:** At this same fixed field the inner sample extends slightly farther perpendicular to the bar; the outer sample extends slightly farther along it. The separate plot resolves the small distortion. Only the two exact baselines have offline linear-elliptic evidence; neighboring stability remains unknown.

4. 04

   #### Read one star and its torque

   Select a visible star, pause on a saved time and export. Compare its angular-momentum change with the torque integral.

   **What to observe:** The particle, plots and export follow the same trajectory. Their close torque agreement also reflects the discrete kick-drift-kick update. Independent reference integration checks the trajectory’s physical accuracy.

Physical conclusion

A rotating bar changes angular momentum through a tangential pull. Follow the whole trajectory, and distinguish a released star from a genuinely solved periodic parent.

## Related experiments and articles

Galactic bars, resonances and stellar migration · Reading path

### Galactic bars, resonances and stellar migration

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why stars pass through galactic spiral arms**](https://sandboxphysics.com/stories/galactic-bar-orbits/why-stars-cross-spiral-arms/)
2. [Article **Differential rotation and orbital resonances**](https://sandboxphysics.com/stories/galactic-bar-orbits/differential-rotation-and-orbital-resonances/)
3. [Article **Torque and energy in a rotating bar**](https://sandboxphysics.com/stories/galactic-bar-orbits/torque-energy-and-a-rotating-bar/)
4. [Article **Periodic orbits and radial migration**](https://sandboxphysics.com/stories/galactic-bar-orbits/periodic-orbits-and-radial-migration/)
5. [Article **Recalculate a bar orbit from saved states**](https://sandboxphysics.com/stories/galactic-bar-orbits/replay-a-bar-orbit-from-saved-states/)
6. [Article **Reading the 2026 bar-resonance tree rings**](https://sandboxphysics.com/stories/galactic-bar-orbits/reading-bar-resonance-tree-rings-2026/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)