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At fluid Toomre parameter $Q=1$, the local spiral-wave dispersion relation becomes $n^2({\Omega }_P-\Omega {)}^2=(a_0|k|-\kappa {)}^2$. The minus-sign, longer-wavelength root is nonnegative only between the two Lindblad resonance conditions. The plus-sign, shorter-wavelength root remains positive outside that interval. Thus this dispersion relation alone does not prove that every density wave is confined between Lindblad resonances; an additional physical restriction is needed.

Type I migration is orbital angular-momentum exchange between an embedded low-mass planet and its disk without a deep gap. The planet excites spiral density waves near Lindblad resonances; the disk's reaction torques act back on the planet. The outer wake usually supplies a negative torque and the inner wake a positive one, with their small imbalance often driving inward migration. Gas on horseshoe orbits in a planetary disk near corotation adds a corotation torque that depends on entropy and vortensity gradients.

For a thin disk, define $q=M_p/M_{\star }$, $h=H/a$ andThe coefficient $C$ is signed and depends on disk gradients, thermodynamics and diffusion; it is not a universal constant. For a circular orbit with fixed planet mass, $L_p=M_p{a}^2\Omega \propto a^{1/2}$. The Type I migration timescale from disk torque follows from ${\dot{L}}_p=\Gamma$:Thus a more massive embedded core generally migrates faster within this regime, unlike a picture in which migration starts only after giant-planet formation.

In a simple locally isothermal smooth disk the net torque is commonly negative, potentially removing cores before they acquire gas. In a non-isothermal disk an unsaturated entropy-related corotation torque can balance or exceed the negative wave torque, giving a planet trap or outward migration. Viscosity and thermal diffusion help maintain the background gradients across the horseshoe region; without adequate replenishment corotation-torque saturation can suppress that contribution. The linear low-mass scaling requires a small disturbance, roughly $q\ll h^3$; larger planets, gap formation and strong disk feedback require a different treatment. Type I migration is therefore neither universally inward nor literally viscous advection of the planet.^[ref]

Radial epicyclic motion. At fixed specific angular momentum, use ${\Phi }_{\mathrm{eff}}=\Phi +{\ell }^2/(2r^2)$ and $\ddot{r}=-{\Phi }_{\mathrm{eff}}^{\prime }$. A circular orbit at $r_c$ obeys ${\Phi }^{\prime }(r_c)={\ell }^2/r_c^3=r_c{\Omega }^2$. Expanding $r=r_c+\xi$ gives $\ddot{\xi }+{\kappa }^2\xi =0$, whereFor ${\kappa }^2>0$ the radial epicyclic frequency describes simple harmonic motion. The angular correction follows from conservation of angular momentum, $\dot{\varphi }\simeq \Omega (1-2\xi /r_c)$, and describes an epicyclic motion around the guiding circular orbit.

Stabilizing terms. In the given local fluid dispersion relation, ${\kappa }^2$ is rotational support, important at long wavelength; $a_0^2{k}^2$ is random-motion or pressure support, important at short wavelength. The term $-2\pi G\Sigma |k|$ is destabilizing self-gravity. Although the wording refers to stellar dispersion, this algebra uses the supplied fluid approximation; a collisionless stellar treatment has a reduction factor and a different numerical normalization.^[ref]

Axisymmetric threshold. For $n=0$, set $q=|k|\ge 0$. Completing the square givesThus $Q>1$ gives local axisymmetric oscillatory stability; $Q<1$ gives a growing band, withAt $Q=1$, the minimizing mode has zero frequency and is marginal, not a strictly restoring oscillation. Local axisymmetric stability does not establish stability to every global nonaxisymmetric mode.

long-wave and short-wave branches at marginal Toomre stability. For $Q=1$ and $n\ge 1$,The long-wavelength branch, using the minus sign, requiresThese endpoints are the inner and outer Lindblad resonance conditions. The stated dispersion relation does not restrict every wave solution to that interval. Its plus-sign short-wavelength branch has $q>0$ even outside it: for example, ${\Omega }_P=\Omega +2\kappa /n$ gives $q=3\kappa /a_0$ and satisfies the equation. Additional branch selection or resonant absorption physics would be needed for an absolute confinement claim. No such condition is supplied.

Toomre stability radius of a flat-rotation disk. With $\Omega =v_0/r$, $\kappa =\sqrt{2}{v}_0/r$. Using $G=4.3009\times 10^{-3}\mathrm{pc}(\mathrm{km}{\mathrm{s}}^{-1}{)}^2{M}_{\odot }^{-1}$,The supplied local model therefore predicts stable radii below about $4.19\mathrm{kpc}$, marginality at that radius, and axisymmetric instability farther out while its assumed constant surface density and thin-disk approximation remain applicable. An infinite constant-density, flat-rotation disk is an idealization, not a global self-consistent galaxy model.

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]