Torque and energy in a rotating bar

Specify the gravitational field

Work in the dimensionless units $R_0=v_0=1$. The background potential is $\ln R$. Add a two-fold bar with dimensionless amplitude $A$, fixed radius $R_b=0.5$ and phase $\theta_b$. A positive pattern speed rotates counterclockwise, with $\dot\theta_b=\Omega_p$.

The outer radial shape falls as the inverse cube. The inner expression joins it with a continuous value and first derivative at $R_b$; its second derivative jumps. The interactive trajectories use only $R>R_b$. This is a specified force on massless tracers; their positions do not rebuild the gravitational field.

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

The angular derivative gives the torque

Specific angular momentum is $L_z=xv_y-yv_x=R^2\dot\phi$. A radial force has no moment about the centre. The tangential force is $F_\phi=-(1/R)\partial\Phi/\partial\phi$; multiplying by its lever arm $R$ gives the specific torque $\tau_z$.

Differentiating the cosine produces a sine. Ahead of and behind a bar axis, its sign changes. A star therefore gains angular momentum during part of an encounter and loses it during another part. The net change requires the force along the complete trajectory.

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

A changing potential does work

The star's specific energy is $E=|\boldsymbol v|^2/2+\Phi$. Differentiating along its motion produces a kinetic term $\boldsymbol v\cdot\dot{\boldsymbol v}$ and a spatial-potential term $\boldsymbol v\cdot\nabla\Phi$. Newton's equation makes those cancel. The remaining rate is the explicit time derivative of the potential.

When only the pattern rotates, that derivative is $\Omega_p\tau_z$. Consequently a gain $\Delta L_z=0.05$ at fixed $\Omega_p=1.2$ requires $\Delta E=0.06$. The star's energy itself is not constant.

The conserved combination is the Jacobi integral. It is an excellent diagnostic only after its physical conditions are established.

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

Write the conserved quantity in bar coordinates

A rotating observer measures a velocity different from the inertial velocity. At a fixed point the two differ by $\boldsymbol\Omega_p\times\boldsymbol r$. Substitute that velocity relation into the inertial energy and subtract $\Omega_pL_z$. The cross term cancels, leaving rotating kinetic energy plus an effective potential.

The extra negative centrifugal term comes from changing frames; it is not a new gravitational mass. At fixed Jacobi value, a point where that value is below the effective potential has no real rotating speed available. This gives a useful allowed-region test, although it does not determine the trajectory inside an allowed region.

The Coriolis force also appears in the rotating equation of motion. It turns velocity and does no work because it is perpendicular to that velocity. Keeping this distinction prevents a rotating orbit plot from being mistaken for an inertial force diagram.

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

Growth and slowdown add explicit terms

If the amplitude grows, $\partial\Phi/\partial t$ contains $\dot A f\cos[2(\phi-\theta_b)]$. If the pattern speed also changes, differentiating $E-\Omega_pL_z$ contributes $-\dot\Omega_p L_z$. Both terms have to be included.

The reader's controlled growth example raises $A$ smoothly from zero to $0.04$ over eight time units, at constant $\Omega_p=1$. Its largest Jacobi change among ten test states is about $0.006897$. The independent time integral of the amplitude work agrees with that change to about $2.6\times10^{-12}$. Demanding constant Jacobi during this growth would reject a correct solution.

The interaction changes physical parameters by preparing a new history from the same release. Dragging its pattern-speed control is not a simulation of a bar slowing down. A slowdown calculation must specify the entire time-dependent phase and include the corresponding work.

Reading: Sandbox Physics independent galactic-bar scientific reader v1 · Chiba, Fujii, Baba, Dubinski & Schönrich (2026), Tree-ring structure of Galactic bar resonance in N-body simulations, arXiv:2604.01294v1

Angular momentum change can move the guiding radius

In an axisymmetric flat-speed reference, a circular orbit has $L_z=R_gv_0$, so $R_g=L_z/v_0$ defines a guiding radius. This is a reference label, not the current distance to the centre.

For a near-circular orbit, write energy as the circular-orbit energy plus radial-oscillation energy. The slope of circular energy against angular momentum is $dE_c/dL_z=\Omega$. Combining it with the steady-pattern exchange gives $\Delta(E-E_c)\simeq(\Omega_p-\Omega)\Delta L_z$ to first order.

At corotation that first-order radial-heating cost vanishes. A star can change its guiding radius without a comparably large increase in radial motion. Away from corotation, heating generally accompanies the exchange. Temporary waves can leave a lasting change when they fade; a permanently present wave can allow a later return.

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