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Use the position second-moment tensor $I_{ij}=\int \rho x_i{x}_j{d}^{3}x$, not the engineering moment-of-inertia convention. If a density pattern rotates steadily about its third principal axis, $I(t)=R(\Omega t)I(0)R(\Omega t{)}^T$. Differentiating twice at a principal-axis instant gives the displayed tensor. The tensor virial theorem then determines the pattern speed from the difference of its first two diagonal components. This density-pattern speed is distinct from stellar mean streaming; in the axisymmetric limit $I_{11}=I_{22}$, this difference cannot determine it.

For the radial epicyclic frequency, introduce the effective potential at fixed specific angular momentum $l$: ${\Phi }_{\mathrm{eff}}=\Phi +l^2/(2r^2)$. A circular orbit at $r_c$ satisfies ${\Phi }^{\prime }(r_c)=l^2/r_c^3=r_c{\Omega }^2(r_c)$. A small radial displacement $\xi$ obeys $\ddot{\xi }=-{\Phi }_{\mathrm{eff}}^{\prime \prime }(r_c)\xi$. Differentiate at fixed $l$ before inserting the circular-orbit relation:For a flat rotation curve, $\Omega \propto r^{-1}$, so $\kappa =\sqrt{2}\Omega$.

In a perturbation proportional to $\exp [i(\int kdr+n\varphi -\omega t)]$, $n$ is the integer azimuthal mode number, giving $|n|$ arms for a nonaxisymmetric pattern. Its angular pattern speed is ${\Omega }_p=\omega /n$ for a real frequency and $n\ne 0$; $a_0$ is the unperturbed gas sound speed, or the specified effective fluid pressure-response speed. For growing patterns the angular phase speed uses $\mathrm{Re}\omega /n$. Axisymmetric modes have $n=0$: their pattern speed is undefined, and one must use $(\omega -n\Omega {)}^2\to {\omega }^2$, not set the pattern-speed expression to zero.

Define the Toomre parameter and Toomre wavenumber byThen $q=|k|/k_T$ givesFor $a_0>0$, the minimum is negative precisely when $Q<1$. Thus this condition is necessary and sufficient for an exponentially growing axisymmetric mode within the given local razor-thin fluid model; $Q=1$ is marginal. It is not a guarantee against global or nonaxisymmetric instabilities, and the collisionless stellar-disk criterion has a different pressure-response normalization.

The Toomre unstable band and fastest growth follow by minimizing ${\omega }^2$:Here the growing branch has $\omega =i{\gamma }_{\ast }$ under the stated Fourier convention. Both signs of the radial wave number have the same growth rate. The unstable band is $q_{-}<q<q_{+}$, where $q_{\pm }=2(1\pm \sqrt{1-Q^2})/Q^2$.

For the minimum Toomre parameter in an exponential flat-rotation disk with constant $a_0$, put $x=r/r_0$:The logarithmic derivative is $d\log Q/dx=1-1/x$, so the unique minimum occurs at $r=r_0$. Both $Q\to \infty$ as $r\downarrow 0$ and $Q\to \infty$ as $r\to \infty$. Hence inner and outer regions are stable within this idealized rotation law, even when an intermediate annulus is unstable. The whole disk is strictly stable in this local axisymmetric test ifEquality makes the ring at $r_0$ marginal. The central conclusion uses the assumed flat rotation curve all the way inward; real central rotation laws need not retain that limit.

differential rotation winds a material arm because $d\Omega /dr\ne 0$. A spiral density wave instead has a pattern speed different from the material speed: stars and gas can pass through its crests. Collective self-gravity, orbital resonances and shear can sustain or repeatedly amplify non-axisymmetric patterns; a visible spiral need not be a single permanent rigid material structure.

For a razor-thin gas disk, local tightly wound disturbances satisfyRotation stabilizes long wavelengths and pressure stabilizes short ones. Minimizing the axisymmetric right-hand side over $|k|$ gives $|k{|}_{\ast }=\pi G\Sigma /c_s^2$ andfor local axisymmetric stability. A collisionless stellar disk instead has $Q_{\ast }={\sigma }_R\kappa /(3.36G{\Sigma }_{\ast })$; the radial random velocity dispersion is not literally a gas sound speed.^[ref]

Local axisymmetric stability does not guarantee global disk stability. Gas and stars gravitate together, thickness modifies the short-scale force, and a disk with $Q>1$ may still have spiral or bar responses. Dissipation in gas can generate shock waves and concentrate star formation near a spiral crest; stellar random motions tend to reduce the collective response.

A spiral density wave is an organized density and gravitational-potential perturbation through which stars and gas can move; it is not generally a permanently labelled set of stars. Differential rotation winds a material arm rapidly. In a wave description, a perturbation has phase $m\varphi +\int kdr-\omega t$, with pattern speed ${\Omega }_p=\omega /m$ distinct from the local stellar angular speed.

For a thin fluid disk the local tightly wound dispersion relation isRotation and pressure resist self-gravity. A collisionless stellar disk requires its stellar response function instead of simply replacing $c_s$ by a dispersion; the familiar local axisymmetric stability parameter is $Q_{\star }={\sigma }_R\kappa /(3.36G\Sigma )$. This is distinct from the gas $Q_g=c_s\kappa /(\pi G\Sigma )$.

Waves exchange angular momentum at Lindblad resonances and near corotation radius. Swing amplification can strengthen a leading disturbance as shear turns it into a trailing one. Bars or companions can force a coherent spiral; recurrent self-excited disturbances also occur. Thus a single indefinitely steady wave is one model, not a description established for every spiral galaxy. Gas compression and shocks along arms can organize star formation, while gravitational torques redistribute angular momentum. Arm brightness, recent star formation and stellar mass density need not trace precisely the same pattern.^[ref]

The restriction $k_{\phi }=\pm m$ describes an $m$-fold rotating perturbation of a galactic disk, for example a transient spiral density wave or, for $m=2$, a galactic bar. Its pattern speed is ${\Omega }_{\mathrm{p}}$, its amplitude peaks at $t_{\mathrm{peak}}$, and its Gaussian function envelope has duration of order $\tau$. The coefficients $u_{\mathbf{k}}(\mathbf{J})$ describe its strength along different stellar orbits. A real perturbing potential requires $u_{-\mathbf{k}}=u_{\mathbf{k}}^{\ast }$.

Write $g(t)=\exp [-(t-t_{\mathrm{peak}}{)}^2/(2{\tau }^2)]$. The product entering c isThe phase therefore measures detuning from the rotating pattern:Substitution givesThe strongest long-lived response occurs near an orbital resonance, ${\omega }_{\mathbf{k}}=0$, where successive force contributions do not rapidly cancel in phase.

For $k_R=0$, the detuning is ${\omega }_{m0}=m({\Omega }_{\phi }-{\Omega }_{\mathrm{p}})$. The Gaussian resonance broadening factor selects the corotation resonance, where the star's azimuthal frequency matches the pattern speed. The derivative driving the flux is just $m\partial f_0/\partial J_{\phi }$. In fact this contribution alone has the formA single Fourier series harmonic changes the action variables in the direction $(m,k_R)$, because $\dot{\mathbf{J}}=-i\mathbf{k}\delta {\varphi }_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{\theta }}$ plus its conjugate. For this harmonic, $\Delta J_R=0$: stars change angular momentum and shift their epicyclic guiding center without a leading-order increase of random radial motion. This is stellar radial migration driven by the corotation resonance. The resonance condition explains the absence of leading heating: $\Delta E\simeq {\Omega }_{\phi }\Delta J_{\phi }={\Omega }_{\mathrm{p}}\Delta J_{\phi }$ at the corotation resonance.

For tightly wound perturbations in a locally uniform, razor-thin gas disk, continuity equation and rotating-fluid momentum equations couple the surface-density perturbation to radial and azimuthal velocities. The sheet potential is $\delta \Phi =-2\pi G\delta \Sigma /|k|$. Eliminating these perturbations gives the displayed dispersion relation. radial epicyclic frequency supplies the rotation term, the sound-speed term resists short waves, and self-gravity supplies the negative term. For an axisymmetric mode $n=0$, use ${\omega }^2$ directly; the pattern speed $\omega /n$ is undefined.

For a bar rotating with constant pattern speed ${\Omega }_b$, use ${\psi }_{\mathrm{eff}}=\psi +{\Omega }_b^2{R}^2/2$ in the force convention $+\nabla \psi$. The centrifugal acceleration is its added gradient. This is the negative of the usual physical effective potential ${\Phi }_{\mathrm{eff}}$; Coriolis acceleration is separate.