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Gravitational instability of an astrophysical disk occurs when a density enhancement attracts enough additional matter for self-gravity to overcome the disk's restoring motions. It can generate spiral density waves or, under suitable thermal conditions, lead to cooling-limited disk fragmentation. Gas pressure and orbital motion oppose collapse on different scales.

For local axisymmetric disturbances of an inviscid razor-thin disk with surface density of a disk $\Sigma$, barotropic sound speed $c_s$ and radial epicyclic frequency $\kappa$, the compressive dispersion relation isThe razor-thin disk Poisson kernel supplies the attractive $|k|$ term. Pressure dominates at large wavenumber, while epicyclic motion stabilizes sufficiently small wavenumbers. Minimizing over $|k|$ gives $|k{|}_{\ast }=\pi G\Sigma /c_s^2$ and ${\omega }_{\min }^2={\kappa }^2[1-Q^{-2}]$, where the gas Toomre parameter is $Q=c_s\kappa /(\pi G\Sigma )$. Thus $Q<1$ admits exponential local axisymmetric growth; $Q=1$ is marginal. Collisionless stellar disks require a different random-motion treatment and numerical normalization.^[ref]

Linear growth does not guarantee bound fragments. Compression and shocks heat the gas; continued collapse depends on radiative cooling, transport and the tidal environment. Cooling-limited disk fragmentation therefore asks a further nonlinear question beyond whether an infinitesimal disturbance grows.^[ref]

Jerry A. Sellwood studies the dynamics of galactic disks, including the origin and evolution of spiral density waves. His review with Karen L. Masters compares observations, dynamical theory and simulations of recurring spiral patterns.

The review emphasizes a physical feedback: transient spiral activity increases random stellar motion, which can make further spiral activity harder to sustain. A dissipative gas component can alter that evolution. This connects the visible shape of a spiral galaxy to the changing dynamical state of its stars and gas, rather than treating the arms as a permanently fixed decoration.^[ref]

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.