Differential rotation and orbital resonances

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.

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.

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.

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.

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