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.

Selected angular-momentum change $\Delta L_z/(R_0v_0)=-0.115$

Selected radial velocity $v_R/v_0=0.0105$

Circular corotation radius $R_{\rm CR}/R_0=1$

Weighted angular response $Q_2=0.0855$

A rotating gravitational field

Pattern speed

1.00

\Omega_{\rm p}R_0/v_0

Potential amplitude

0.0400

A/v_0^2

Replay time

12.6

t/(R_0/v_0)

Drag the cyan corotation handle or the gold strength handle; select a star to follow its torque.

Computing the complete default trajectories.

Frame, units and model limits

The display rotates coordinates only. Arrows follow velocity in that display frame; their length is compressed, while color always means physical radial velocity.

$R/R_0,\quad v/v_0,\quad t/(R_0/v_0)$

$R_{\rm b}=0.5R_0,\qquad t_{\rm end}=8\pi R_0/v_0$

All tracers have equal weight. The warm release gives opposite radial velocities, not a Gaussian distribution. Brightness is a display choice.

Circles are near-circular background references. The inner Lindblad radius lies inside the excluded field region throughout these controls. The field axis marks an imposed force pattern, not a measured mass shape.

The strength-handle length is an indicator; the field radius stays fixed. A new physical input rebuilds the history from the same release. Some warm, rapid-pattern combinations enter the excluded radius and are refused.

The refinement estimate is a local numerical check, not a uniform error theorem over every continuous input. The original same-sign wide shooting bracket failed and remains documented; fixed-field references use separately registered narrow brackets.

Dehnen 1999 v1: prescribed field and orbit families

Sellwood & Binney 2002 v2: transient-wave migration

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.

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 misconceptionA curve returning near its starting point in the picture is a periodic orbit.

Better mental modelA 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 misconceptionThe Jacobi value should remain fixed after a pattern-speed drag.

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

Run the experiment

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.
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.
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.
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.

Galactic Bar Orbits and Corotation

A rotating pull changes angular momentum

Core mathematical model

The prescribed rotating field

Tangential pull changes angular momentum

Corotation is a relative-speed condition

Energy and angular momentum change together

Measure an angular response

Common difficulties

A path that looks closed

Changing the pattern during a run

Run the experiment

Remove the tangential pull

Move the place that keeps pace

Compare two actual periodic samples

Read one star and its torque

Related experiments and articles

Galactic bars, resonances and stellar migration