Bar Instabilities and Disk Dynamics

A galactic disk is one of nature's most finely balanced structures: a nearly two-dimensional sheet of stars circling a common center, held out against gravity almost entirely by rotation and only slightly by random stellar motions. That near-perfect rotational order is exactly what makes it dangerous. A disk cold enough to look like a razor-thin platter of ordered orbits is also cold enough for self-gravity to organize those orbits into growing collective patterns — trailing spirals and, most dramatically, a rotating central bar. Roughly two-thirds of nearby disk galaxies carry such a bar, so barring is not an exotic accident but the ordinary fate of a settled disk. This article treats the two levels at which the physics must be read. Locally, one linearizes the collisionless Boltzmann equation about an axisymmetric equilibrium and asks which tightly wound density waves grow; the answer is Toomre's number $Q$ and, for shearing wavelets, swing amplification. Globally, the whole disk rings like a struck bell: an open $m=2$ disturbance feeds back on itself and locks into a standing mode tumbling at a single pattern speed $\Omega_p$ . The empirical Ostriker–Peebles ratio $t\equiv T_\mathrm{rot}/|W|$ tells us which disks cross into that instability, and once a bar exists it becomes an engine — torquing gas inward, exporting angular momentum to the halo, reshaping the galaxy over many rotations. Bars are the cleanest laboratory we have for collisionless self-gravitating dynamics.

Key takeaways

A rotationally cold, self-gravitating stellar disk is prone to collective gravitational instabilities; the most violent is the bar instability, a growing $m=2$ mode that reshapes the inner disk into a rotating triaxial bar.
Local axisymmetric stability is set by Toomre's parameter $Q=\sigma_R\kappa/(3.36\,G\Sigma)$ : $Q<1$ lets rings and clumps collapse, while $Q\sim1$ disks swing-amplify shearing waves into spirals and bars.
Global bar stability is captured by the Ostriker–Peebles energy ratio $t=T_\mathrm{rot}/|W|$ : disks with $t\gtrsim0.14$ bar quickly, whereas a hot bulge or dominant halo pushes $t$ down and stabilizes.
A bar is a self-consistent family of trapped $x_1$ orbits living between the inner Lindblad resonance and corotation, not a passing ripple; it tumbles rigidly at pattern speed $\Omega_p$ .
Through gravitational torques a bar moves angular momentum outward — funneling gas to the center to build pseudobulges, heating the outer disk, and being braked by dark-halo dynamical friction.
The whole story is collisionless: no two-body relaxation is involved, only the coherent self-gravitating response of the distribution function through the CBE and Poisson's equation.
Key idea: a rotationally cold, self-gravitating disk is globally unstable to a growing $m=2$ mode — when $T_\mathrm{rot}/|W|\gtrsim0.14$ (equivalently when Toomre $Q$ is low over an extended region), collective self-gravity overwhelms the stabilizing effects of dispersion and shear and produces a rotating bar that then redistributes angular momentum secularly.

The core insight

A rotationally cold, self-gravitating disk is globally unstable to a growing $m=2$ mode: when $T_\mathrm{rot}/|W|\gtrsim0.14$ (equivalently when Toomre $Q$ is low over an extended region), collective self-gravity overwhelms the stabilizing effects of dispersion and shear, producing a rotating bar that then redistributes angular momentum secularly.

Where this lives today

Bars are the most accessible drivers of internal galaxy evolution, and bar dynamics remains a live quantitative frontier. Large near-infrared surveys put the local bar fraction near two-thirds, and deep imaging with Hubble and now JWST finds substantial bar fractions already at $z\sim1$ – $2$ , showing that disks settled and became bar-unstable earlier than once thought. Integral-field spectroscopy (MaNGA, ATLAS3D) measures bar pattern speeds directly via the Tremaine–Weinberg method, and the persistent finding that most bars are fast ( $\mathcal{R}\lesssim1.4$ ) is in genuine tension with cosmological simulations, where friction against dense cold-dark-matter halos tends to slow bars — making bar speeds a clean test of halo densities. High-resolution N-body and hydrodynamic simulations reproduce spontaneous bar growth from disk noise, vertical buckling into boxy/peanut bulges, and bar-driven gas inflow feeding central starbursts. Closer to home, the Milky Way's own bar — mapped by stellar streaming in Gaia data and the peanut-shaped bulge in infrared counts — is the nearest full-scale laboratory for all of this.

The physical picture

Picture a spinning platter of stars, where three influences fight over every small overdensity. Self-gravity tries to pull it together. Rotation resists: displace a star inward and conservation of angular momentum flings it back out, so near-circular orbits behave like springs oscillating at the epicyclic frequency $\kappa$ — a rotational stiffness. And the stars' random velocities, the disk's "pressure," smear out anything smaller than an epicyclic drift. Toomre's $Q$ referees this three-way contest at each wavelength: when $Q<1$ a band of intermediate wavelengths escapes both rotation (which only stiffens long waves) and pressure (which only stiffens short waves), and that band collapses. Non-axisymmetric disturbances add a fourth actor: shear. Because the disk rotates differentially, a leading wavelet is continuously sheared through the radial orientation into a trailing one. During that brief radial passage the wave's self-gravity is maximally coherent and, if the disk is responsive enough ( $Q$ near unity), it is transiently but enormously amplified — swing amplification. Feed the amplified trailing wave back through the center and you have a closed loop: an amplifier plus feedback is an oscillator, and the standing $m=2$ oscillation of a cold disk is a bar. Once formed, a bar is a resonant trap. Over a wide range of radii the elongated $x_1$ orbits precess at nearly the same rate, so many stars can be held on orbits aligned with one axis tumbling at the pattern speed $\Omega_p$ ; the aligned mass reinforces the very potential that traps it — a self-sustaining standing wave, not a ripple that winds up and fades. Because that mass distribution is triaxial and tumbling, it exerts torques: the wakes it raises in the surrounding gas and halo are not symmetric, so material inside corotation loses angular momentum and sinks while material outside gains it. Angular momentum, not mass, is the currency it redistributes. A cold disk is thus a stack of coins on edge — ordered but eager to topple into a bar; a massive bulge or dense halo raises $\kappa$ , lowers $t=T_\mathrm{rot}/|W|$ , and props the coins back up.

Figure

Dispersion relation for axisymmetric disk waves ω²/κ² = 1 − (2/Q)x + x² with x = kσ_R/κ: for Q = 1.3 the curve stays positive so every wavelength is stable, while for Q = 0.7 it dips below zero between x = 0.41 and x = 2.45 and that band of wavelengths grows; the minimum value equals 1 − 1/Q² and changes sign exactly at Q = 1

Figure

Lindblad resonance diagram for a flat rotation curve v_c = 220 km/s so that κ = √2 Ω: the three curves are Ω, Ω − κ/2 and Ω + κ/2 versus radius R, and a bar pattern speed Ω_p = 40 km/s per kpc cuts them at the inner Lindblad resonance R ≈ 1.6 kpc, corotation R ≈ 5.5 kpc, and the outer Lindblad resonance R ≈ 9.4 kpc — close to the Milky Way's bar

The core equations

Local stability of an axisymmetric stellar disk to tightly wound density waves is decided by one dimensionless number, weighing the stabilizing dispersion $\sigma_R$ and rotational stiffness $\kappa$ against the destabilizing surface self-gravity $G\Sigma$ . The coefficient $3.36$ is the stellar-kinetic value; the fluid analogue carries $\pi$ .

Q=\dfrac{\sigma_R\,\kappa}{3.36\,G\,\Sigma} \qquad (1)

where: $\sigma_R$ — radial velocity dispersion of the stars; $\kappa$ — epicyclic frequency; $G$ — gravitational constant; $\Sigma$ — disk surface mass density; $Q$ — dimensionless stability parameter. Reading it: $Q\ge1$ means the disk is locally stable to axisymmetric perturbations at every wavelength; $Q<1$ opens a band of unstable wavelengths that collapse into rings or clumps. The rotational stiffness in $Q$ is the epicyclic frequency, the rate at which a slightly non-circular orbit oscillates in radius. It is fixed entirely by the shape of the rotation curve, and its ratio to the circular frequency $\Omega$ is among the most useful facts about a disk.

\kappa^2=R\,\frac{\mathrm{d}\Omega^2}{\mathrm{d}R}+4\Omega^2=\frac{2\Omega}{R}\frac{\mathrm{d}(R^2\Omega)}{\mathrm{d}R} \qquad (2)

where: $\Omega(R)$ — circular angular frequency at radius $R$ ; $\kappa$ — radial (epicyclic) oscillation frequency of a near-circular orbit; $R$ — galactocentric radius. Reading it: a flat curve gives $\kappa=\sqrt2\,\Omega$ , solid-body $\kappa=2\Omega$ , Keplerian $\kappa=\Omega$ , so $\kappa$ always lies between $\Omega$ and $2\Omega$ . Local stability says nothing about whether the disk will bar. Global bar stability is captured empirically by an energy ratio — how much of the disk's binding energy is invested in ordered rotation — which Ostriker and Peebles distilled from N-body experiments into a single threshold.

t\equiv\dfrac{T_\mathrm{rot}}{|W|}\lesssim0.14\quad\Rightarrow\quad\text{bar-stable} \qquad (3)

where: $T_\mathrm{rot}$ — ordered rotational kinetic energy of the disk; $W$ — gravitational potential (binding) energy, negative; $t$ — the rotational-to-potential energy ratio. Reading it: a disk with $t\gtrsim0.14$ is rotationally cold enough that self-gravity organizes a global $m=2$ mode; a hot bulge or dominant halo lowers $t$ and suppresses barring. A bar, once present, is a rigidly rotating pattern with a single angular speed $\Omega_p$ , and its dynamical skeleton is the set of radii where a star's epicyclic motion resonates with the pattern's forcing. These are the Lindblad resonances and corotation.

m\big(\Omega_p-\Omega(R)\big)=\pm\,\kappa(R),\qquad \Omega_p=\Omega(R_\mathrm{CR}) \qquad (4)

where: $\Omega_p$ — pattern (bar) angular speed; $m$ — azimuthal wavenumber ( $m=2$ for a bar); the $-$ sign gives the inner Lindblad resonance (ILR) and $+$ the outer Lindblad resonance (OLR); $R_\mathrm{CR}$ — corotation radius. Reading it: the resonances always order as $R_\mathrm{ILR}<R_\mathrm{CR}<R_\mathrm{OLR}$ , and the bar-supporting orbits live between the ILR and corotation. The most unstable axisymmetric wavelength — the size of the clumps a $Q<1$ disk breaks into — follows directly from the dispersion relation, the natural fragmentation scale of a rotating self-gravitating sheet.

\lambda_\mathrm{crit}=\frac{4\pi^2 G\Sigma}{\kappa^2}=\frac{2\pi}{k_\mathrm{crit}},\qquad k_\mathrm{crit}=\frac{\kappa^2}{2\pi G\Sigma} \qquad (5)

where: $\lambda_\mathrm{crit}$ — wavelength of the marginally/most unstable axisymmetric mode; $k_\mathrm{crit}$ — corresponding wavenumber. Reading it: this is the largest scale that rotation can no longer stabilize; longer waves are held up by $\kappa$ , shorter waves by pressure. For non-axisymmetric shearing waves the control parameter is the swing-amplification parameter, the ratio of the critical wavelength to the azimuthal wavelength — a measure of how strongly shear can pump a wave.

X\equiv\frac{k_\mathrm{crit}R}{m}=\frac{\kappa^2 R}{2\pi G\Sigma\,m} \qquad (6)

where: $X$ — swing-amplification parameter; $m$ — number of arms / azimuthal wavenumber. Reading it: amplification peaks for $X\approx1$ – $2$ ; too large $X$ (heavy shear, low density) or too small $X$ (too many arms) both quench growth, which is why disks favour two-armed patterns. Once a bar exists it does work on its surroundings through the torque its non-axisymmetric potential exerts on gas and halo — the master equation of secular evolution.

\frac{\mathrm{d}L_z}{\mathrm{d}t}=\tau=-\int \Sigma\,\frac{\partial\Phi_\mathrm{bar}}{\partial\phi}\,\mathrm{d}^2x \qquad (7)

where: $L_z$ — angular momentum of the responding component (gas or halo); $\tau$ — torque from the bar; $\Phi_\mathrm{bar}$ — non-axisymmetric part of the potential; $\phi$ — azimuthal angle. Reading it: a trailing phase offset between the bar and the wake it raises makes $\tau\ne0$ , draining angular momentum from material inside corotation and depositing it beyond. Finally, observers classify bars by how far corotation sits beyond the bar's end — a dimensionless rotation-rate parameter that, through dynamical friction, constrains the halo.

\mathcal{R}\equiv\frac{R_\mathrm{CR}}{a_\mathrm{bar}},\qquad 1.0\le\mathcal{R}\le1.4\ \text{(fast bars)} \qquad (8)

where: $R_\mathrm{CR}$ — corotation radius; $a_\mathrm{bar}$ — bar semi-major axis; $\mathcal{R}$ — dimensionless bar speed. Reading it: $\mathcal{R}\lesssim1.4$ ("fast") means the bar ends near corotation and has not been braked; $\mathcal{R}>1.4$ ("slow") signals a bar slowed by a dense halo.

Derivation AToomre Q from the tightly wound dispersion relation

The goal is to derive the local axisymmetric stability threshold of a razor-thin rotating disk from first principles: to find the dispersion relation $\omega^2(k)$ for small tightly wound density waves, identify when it admits a growing mode ( $\omega^2<0$ ), and reduce the condition for stability at every wavelength to the single dimensionless number $Q$ . We work with the fluid disk first, then quote the stellar-kinetic correction that turns the fluid coefficient $\pi$ into $3.36$ ; the three agents — rotation, self-gravity, and pressure — appear as three separate terms whose competition is the whole story.

Step 1. Linearize the continuity, Euler, and Poisson equations for a razor-thin fluid disk of surface density $\Sigma$ , sound speed $c_s$ , and epicyclic frequency $\kappa$ , taking perturbations $\propto\exp[i(kR-\omega t)]$ with $|kR|\gg1$ (the tight-winding, WKB limit). The result is the dispersion relation $\omega^2=\kappa^2-2\pi G\Sigma|k|+k^2 c_s^2$ . Why: each term is one actor — rotation gives the positive $\kappa^2$ , the self-gravity of a sheet the destabilizing $-2\pi G\Sigma|k|$ (linear in $k$ ), and pressure the positive $k^2c_s^2$ .

Step 2. A mode grows when $\omega^2<0$ , i.e. when $2\pi G\Sigma|k|>\kappa^2+k^2c_s^2$ for some $k$ . Why: instability requires self-gravity at some wavelength to exceed the combined rotational and pressure support; long waves are stabilized by $\kappa^2$ and short waves by $k^2c_s^2$ , so any instability must live at intermediate $k$ .

Step 3. Find the most dangerous wavenumber by minimizing $\omega^2$ over $k$ : $\mathrm{d}\omega^2/\mathrm{d}k=-2\pi G\Sigma+2kc_s^2=0$ gives $k_*=\pi G\Sigma/c_s^2$ , and substituting back gives $\omega^2_{\min}=\kappa^2-(\pi G\Sigma)^2/c_s^2$ . Why: if even the most unstable wavelength has $\omega^2\ge0$ , no wavelength can grow, so the whole stability question collapses onto the value of $\omega^2$ at this single $k_*$ .

Step 4. Demand $\omega^2_{\min}\ge0$ : this is $\kappa^2c_s^2\ge(\pi G\Sigma)^2$ , i.e. $Q_\mathrm{gas}\equiv c_s\kappa/(\pi G\Sigma)\ge1$ . A full stellar treatment replaces the fluid pressure by the reduction factor of a velocity-dispersion distribution, softening the coefficient from $\pi$ to $3.36$ and $c_s$ to $\sigma_R$ : $Q_\mathrm{star}=\sigma_R\kappa/(3.36\,G\Sigma)$ . Why: a collisionless stellar disk exerts no isotropic pressure, so it is slightly easier to destabilize than a gas disk of the same "temperature" — the marginal $Q$ is still $1$ but the coefficient softens. *The key results check by direct substitution: $k_*=\pi G\Sigma/c_s^2$ , $\omega^2_\mathrm{min}=\kappa^2-(\pi G\Sigma)^2/c_s^2$ , and the marginal condition $Q_\mathrm{gas}=1$ all follow consistently from the dispersion relation derived above.*

Go deeper · Derivation B · 1 minEpicyclic frequency and the flat rotation curve value

Derivation BEpicyclic frequency and the flat rotation curve value

The goal is to obtain the epicyclic frequency $\kappa$ that appears in every stability criterion, starting from the radial equation of motion of a star on a nearly circular orbit and reducing it to the curvature of an effective potential, then to evaluate $\kappa$ for the three canonical rotation curves — flat, solid-body, and Keplerian — so that the ubiquitous relation $\kappa=\sqrt2\,\Omega$ for a flat curve is grounded rather than asserted.

Step 1. For planar motion with conserved specific angular momentum $L_z=R^2\dot\phi$ , the radial dynamics obey $\ddot R=-\mathrm{d}\Phi_\mathrm{eff}/\mathrm{d}R$ with the effective potential $\Phi_\mathrm{eff}(R)=\Phi(R)+L_z^2/2R^2$ . The circular orbit sits at the guiding radius $R_g$ where $\mathrm{d}\Phi_\mathrm{eff}/\mathrm{d}R=0$ , i.e. $\Omega^2R=\mathrm{d}\Phi/\mathrm{d}R$ . Why: folding the centrifugal term into an effective potential turns a two-dimensional orbit problem into a one-dimensional oscillation about a potential minimum, where the frequency is set by the local curvature.

Step 2. Expand about $R_g$ with $x=R-R_g$ : $\ddot x=-(\mathrm{d}^2\Phi_\mathrm{eff}/\mathrm{d}R^2)_{R_g}\,x\equiv-\kappa^2 x$ , so $\kappa^2=(\mathrm{d}^2\Phi/\mathrm{d}R^2+3L_z^2/R^4)_{R_g}$ . Using $L_z^2=R^4\Omega^2$ this collapses to $\kappa^2=R\,\mathrm{d}\Omega^2/\mathrm{d}R+4\Omega^2$ . Why: the curvature of $\Phi_\mathrm{eff}$ is the spring constant of the radial oscillation, and rewriting it in $\Omega(R)$ makes $\kappa$ computable straight from an observed rotation curve.

Step 3. For a flat curve, $\Omega=v_c/R$ : then $R\,\mathrm{d}\Omega^2/\mathrm{d}R=-2\Omega^2$ and $\kappa^2=4\Omega^2-2\Omega^2=2\Omega^2$ , giving $\kappa=\sqrt2\,\Omega$ . Why: this governs most of a real galaxy, whose curve is flat over the optical disk, and fixes the resonance geometry used throughout.

Step 4. Check the limits: solid-body ( $\Omega=\mathrm{const}$ ) gives $\kappa=2\Omega$ , and Keplerian ( $\Omega\propto R^{-3/2}$ ) gives $\kappa=\Omega$ . Why: the closed Kepler orbit ( $\kappa=\Omega$ , no precession) and rigid inner rotation ( $\kappa=2\Omega$ ) are the anchoring cases that confirm the formula behaves correctly. *All three canonical results follow directly from $\kappa^2=R\,\mathrm{d}\Omega^2/\mathrm{d}R+4\Omega^2$ : flat curve $\kappa=\sqrt2\,\Omega\approx1.41\Omega$ , solid body $\kappa=2\Omega$ , and Kepler $\kappa=\Omega$ .*

Go deeper · Derivation C · 1 minLindblad resonances for an m equals two bar pattern

Derivation CLindblad resonances for an m equals two bar pattern

The goal is to locate the dynamically special radii of a rigidly rotating $m=2$ bar — the inner and outer Lindblad resonances and corotation — by transforming to the frame rotating with the bar and asking where a star's epicyclic motion is resonantly forced by the pattern. This is what fixes the physical size of the bar (it must live inside corotation) and where gas shocks and angular-momentum deposition occur; the argument matches the frequency at which a star meets successive crests to its natural radial frequency $\kappa$ .

Step 1. In the frame rotating at $\Omega_p$ the bar potential is static, and a star at radius $R$ meets the pattern's $m$ crests at the forcing frequency $\omega_\mathrm{force}=m[\Omega(R)-\Omega_p]$ . Why: it is the relative angular speed between star and pattern, times the number of crests $m$ , that sets how often the star feels a kick from the non-axisymmetric potential.

Step 2. A resonance occurs when this forcing matches the natural radial frequency, $m[\Omega(R)-\Omega_p]=\pm\kappa(R)$ , with the case $\Omega(R)=\Omega_p$ defining corotation. Why: as in any driven oscillator, energy and angular momentum are exchanged resonantly when the drive matches the natural frequency, so these radii most strongly reshape orbits.

Step 3. Set $m=2$ to get $\Omega_p=\Omega\mp\kappa/2$ : the minus sign is the ILR and the plus the OLR, so with $\kappa=\sqrt2\,\Omega$ the ILR sits at $\Omega_p=\Omega(1-1/\sqrt2)\approx0.293\,\Omega$ and the OLR at $\Omega_p=\Omega(1+1/\sqrt2)\approx1.707\,\Omega$ . Why: since $\Omega$ decreases outward, $\Omega_p=0.293\,\Omega$ places the ILR at small $R$ and $1.707\,\Omega$ places the OLR at large $R$ , bracketing corotation.

Step 4. The ordering is thus $R_\mathrm{ILR}<R_\mathrm{CR}<R_\mathrm{OLR}$ , and the bar-supporting $x_1$ orbits live between the ILR and corotation; a strong central mass can create two, one, or zero ILRs. Why: whether an ILR exists decides if the growing wave is absorbed at small radius or reflected to close the feedback loop — so it controls whether a global bar mode survives at all. *The flat-curve factors $1\mp1/\sqrt2=0.293,\,1.707$ and the ordering $R_\mathrm{ILR}<R_\mathrm{CR}<R_\mathrm{OLR}$ follow numerically from the resonance conditions derived above.*

Limiting cases

Different regimes of the stability parameters produce qualitatively different disk behaviour.

Case	Result	Meaning
Jeans / collapse limit	$Q<1$	A band of intermediate axisymmetric wavelengths grows: the disk fragments into rings or clumps as self-gravity beats both rotational and pressure support.
Marginal / swing-amplified	$Q\sim1$ – $2$	Axisymmetrically stable, but non-axisymmetric shearing waves are strongly amplified; transient spirals form and, if feedback closes, a global bar grows.
Hot / stable disk	$Q\gg1$	Large dispersion or a small disk-to-halo mass fraction quenches both local and global instabilities; the disk stays smooth and featureless.
Ostriker–Peebles threshold	$t=T_\mathrm{rot}/\|W\|\gtrsim0.14$	Global $m=2$ bar instability sets in for rotationally cold, self-gravitating disks; a dominant hot halo or bulge lowers $t$ and stabilizes.
Fast vs slow bar	$\mathcal{R}=R_\mathrm{CR}/a_\mathrm{bar}$	$\mathcal{R}\lesssim1.4$ (fast) means a bar unbraked by the halo; $\mathcal{R}>1.4$ (slow) signals strong dynamical friction against a dense halo.

Worked examples

1 · Is the solar neighbourhood locally stable? (fully worked)

Estimate Toomre $Q$ for the stellar disk near the Sun, using the local surface density $\Sigma\approx50\,M_\odot\,\mathrm{pc}^{-2}$ , radial dispersion $\sigma_R\approx38\ \mathrm{km\,s^{-1}}$ , and the Oort constants $A=15.3$ , $B=-11.9\ \mathrm{km\,s^{-1}\,kpc^{-1}}$ .

Step 1. Get $\kappa$ from the Oort constants, using $\kappa^2=-4B(A-B)$ : $\kappa^2=-4(-11.9)(15.3+11.9)=47.6\times27.2=1295\ (\mathrm{km\,s^{-1}\,kpc^{-1}})^2$ , so $\kappa=\mathbf{36.0\ km\,s^{-1}\,kpc^{-1}}=0.0360\ \mathrm{km\,s^{-1}\,pc^{-1}}$ . This is the local realization of $\kappa=\sqrt2\,\Omega$ with $\Omega\approx26\ \mathrm{km\,s^{-1}\,kpc^{-1}}$ .

Step 2. Use $G=4.30\times10^{-3}\ \mathrm{pc}\,M_\odot^{-1}\,(\mathrm{km\,s^{-1}})^2$ so every quantity is in the same units. Numerator: $\sigma_R\kappa=38\times0.0360=1.368\ (\mathrm{km\,s^{-1}})^2\,\mathrm{pc}^{-1}$ . Denominator: $3.36\,G\Sigma=3.36\times4.30\times10^{-3}\times50=0.722\ (\mathrm{km\,s^{-1}})^2\,\mathrm{pc}^{-1}$ .

Step 3. Divide: $Q=1.368/0.722=\mathbf{1.9}$ . The local disk is comfortably above the marginal value, so it is stable to axisymmetric collapse — consistent with the smooth, non-fragmenting disk near the Sun. Q ≈ 1.9, locally stable. Per the warning box, this does not prove the inner Galaxy is bar-stable — and indeed the Milky Way hosts a strong central bar.

2 · Resonances of the Milky Way bar (fully worked)

The Galactic bar tumbles at a pattern speed $\Omega_p\approx40\ \mathrm{km\,s^{-1}\,kpc^{-1}}$ in a disk with a flat rotation curve $v_c\approx220\ \mathrm{km\,s^{-1}}$ . Locate corotation and the two Lindblad resonances, and classify the bar given a semi-major axis $a_\mathrm{bar}\approx5\ \mathrm{kpc}$ .

Step 1. Corotation is where $\Omega(R)=\Omega_p$ . With $\Omega=v_c/R$ , $R_\mathrm{CR}=v_c/\Omega_p=220/40=\mathbf{5.5\ kpc}$ . This is the outer boundary of the bar-supporting orbits.

Step 2. For the flat curve $\kappa=\sqrt2\,\Omega$ , so the ILR satisfies $\Omega_p=\Omega(1-1/\sqrt2)=0.293\,\Omega$ . Then $\Omega(R_\mathrm{ILR})=40/0.293=136.6\ \mathrm{km\,s^{-1}\,kpc^{-1}}$ and $R_\mathrm{ILR}=220/136.6=\mathbf{1.6\ kpc}$ . Gas shocks and the nuclear ring form near here.

Step 3. The OLR satisfies $\Omega_p=\Omega(1+1/\sqrt2)=1.707\,\Omega$ , so $\Omega(R_\mathrm{OLR})=40/1.707=23.4\ \mathrm{km\,s^{-1}\,kpc^{-1}}$ and $R_\mathrm{OLR}=220/23.4=\mathbf{9.4\ kpc}$ . The full ordering is $1.6<5.5<9.4\ \mathrm{kpc}$ , exactly as required.

Step 4. Classify with $\mathcal{R}=R_\mathrm{CR}/a_\mathrm{bar}=5.5/5=\mathbf{1.1}$ . Since $\mathcal{R}\lesssim1.4$ , the Milky Way bar is a fast bar — corotation just beyond the bar's end, so it has not been strongly braked by friction, arguing for a sub-dominant (not maximally dense) inner dark-matter halo.

Physics of the portrait

Bar dynamics has many parents — Lindblad supplied resonances, Lin and Shu the density-wave picture, Ostriker and Peebles the energy criterion — but one figure sits at its analytical centre, having written both the local stability criterion that bears his name and the theory of swing amplification.

Origins · 1 minAlar Toomre and how this was discovered

Alar Toomre

born 1937, Estonian-American applied mathematician · the stability and self-organization of rotating stellar disks

Alar Toomre was born in Estonia in 1937 and trained as an applied mathematician at Manchester and MIT, where he spent his career reshaping galactic dynamics from the outside. In 1964, independently of the density-wave programme of C. C. Lin and Frank Shu, he derived the local criterion for the axisymmetric stability of a rotating disk and introduced the dimensionless number $Q=\sigma_R\kappa/(3.36\,G\Sigma)$ that every later paper on disk stability would use. He did not stop there: in 1981 he articulated swing amplification, the transient but powerful boosting of a shearing wavelet as differential rotation carries it from leading to trailing through the radial orientation, explaining why cold disks grow open spirals and bars even when $Q>1$ . With his brother Juri he also produced, in 1972, the celebrated N-body models showing that tidal encounters throw out the bridges and tails of interacting galaxies. Toomre's hallmark was physical economy — distilling the behaviour of $10^{11}$ stars into a few dimensionless numbers and clean mechanisms — and the modern picture of bars as amplified, feedback-sustained global modes rests directly on the local criterion and the swing mechanism he supplied.

History

1927

— Bertil Lindblad explains spiral structure through orbital resonances in a rotating disk

1964

— Toomre and, independently, Lin and Shu derive the local axisymmetric stability parameter Q

1971

— Frank Hohl's early N-body disks spontaneously buckle into a strong central bar

1973

— Ostriker and Peebles propose the rotational-to-potential energy bar criterion near 0.14

1981

— Toomre formulates swing amplification of shearing density waves

Going further

Texts. Binney and Tremaine, Galactic Dynamics, 2nd ed., Ch. 5–6 (disk stability, spiral structure, bars) · Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 25 (galactic dynamics and structure) · Mo, van den Bosch and White, Galaxy Formation and Evolution, Ch. 11 (disk dynamics and secular evolution) · Dodelson, Modern Cosmology (for the collisionless Boltzmann framework the disk analysis specializes).

Papers. Toomre (1964), ApJ 139, 1217 — local axisymmetric stability and the $Q$ criterion · Lin and Shu (1964), ApJ 140, 646 — density-wave theory of spiral structure · Ostriker and Peebles (1973), ApJ 186, 467 — the bar-instability energy criterion · Toomre (1981), in The Structure and Evolution of Normal Galaxies — swing amplification · Kormendy and Kennicutt (2004), ARA&A 42, 603 — secular evolution and pseudobulges. All equations and thresholds are standard. The Toomre parameter uses fluid coefficient $\pi$ and stellar coefficient $3.36$ ; minimizing the dispersion relation over $k$ recovers $Q_\mathrm{gas}=c_s\kappa/(\pi G\Sigma)$ . The worked solar-neighbourhood value $Q\approx1.9$ follows from $(\Sigma,\sigma_R,\kappa)=(50\,M_\odot\,\mathrm{pc}^{-2},38,36)$ in mixed disk units. The resonances $R_\mathrm{ILR}\approx1.6$ , $R_\mathrm{CR}\approx5.5$ , $R_\mathrm{OLR}\approx9.4\ \mathrm{kpc}$ correspond to $\Omega_p=40\ \mathrm{km\,s^{-1}\,kpc^{-1}}$ and $v_c=220\ \mathrm{km\,s^{-1}}$ . The bar fraction ( $\sim2/3$ ) and fast-bar range ( $\mathcal{R}\lesssim1.4$ ) are standard observational values.

Prerequisites. Collisionless Boltzmann Equation · Distribution Function and Phase Space · Jeans Equations · Orbits in Axisymmetric and Triaxial Potentials · Action-Angle Variables · Virial Theorem in Stellar Systems. Leads to. Two-Body Relaxation Time · Gravitational N-Body Problem · Galactic Potential Models.

Ask the tutor: Ask me to walk through why a disk with $Q>1$ everywhere can still topple into a bar, or to trace a parcel of gas from the bar's dust-lane shock down to the nuclear ring.

What is the difference between the Toomre Q criterion and the Ostriker–Peebles criterion?

They answer different questions. Toomre $Q$ is local and axisymmetric: it tells you whether short, tightly wound density waves at a given radius collapse, and $Q\ge1$ guarantees only that. The Ostriker–Peebles ratio $t=T_\mathrm{rot}/|W|$ is global: it weighs the whole disk's ordered rotation against its binding energy and predicts whether an open $m=2$ bar mode grows. A disk can pass the local test everywhere ( $Q>1$ ) and still fail the global one ( $t>0.14$ ), which is why a smooth-looking disk can spontaneously bar. Use $Q$ for fragmentation and spiral seeding, $t$ for the bar as a whole.

Why does the epicyclic frequency come out as √2 times Ω for a flat rotation curve?

The epicyclic frequency is the rate a slightly non-circular orbit oscillates in radius, set by the curvature of the effective potential: $\kappa^2=R\,\mathrm{d}\Omega^2/\mathrm{d}R+4\Omega^2$ . The $4\Omega^2$ term is the centrifugal stiffness of pure rotation; the first term corrects for how fast $\Omega$ changes with radius. For a flat curve $\Omega\propto1/R$ , so $R\,\mathrm{d}\Omega^2/\mathrm{d}R=-2\Omega^2$ and the two combine to $2\Omega^2$ , giving $\kappa=\sqrt2\,\Omega\approx1.41\,\Omega$ . This sits between rigid rotation ( $\kappa=2\Omega$ ) and Kepler ( $\kappa=\Omega$ , closed orbits), which is why disk orbits are slowly precessing rosettes.

What are the x1 orbits, and why do they matter?

The $x_1$ orbits are the closed, elongated orbits aligned along the bar's major axis in the bar's rotating frame. They live between the inner Lindblad resonance and corotation and form the bar's structural skeleton: a bar is essentially the self-gravitating superposition of stars trapped on $x_1$ orbits, whose aligned mass reinforces the potential that traps them. If a strong central mass concentration builds an inner Lindblad resonance that disrupts the $x_1$ family — replacing them with orbits elongated perpendicular to the bar (the $x_2$ family) — the bar loses its backbone and can dissolve. The $x_1$ orbits thus make a bar both self-consistent and destructible.

How does a bar drive gas toward the galactic center?

Through gravitational torque and shocks. In the rotating non-axisymmetric potential gas orbits are not closed: streaming along the bar, gas runs into the steep gradient and forms two offset shocks (the straight dust lanes seen in barred galaxies). At the shock it loses angular momentum, which the bar's torque carries outward, so the gas sinks toward the inner Lindblad resonance. There it piles into a nuclear ring, fuels a circumnuclear starburst, and over gigayears builds a rotationally supported "pseudobulge" — the essence of secular evolution: the galaxy rearranges itself internally, without any merger, by having the bar shuffle angular momentum around.

Do bars last forever, or can they be destroyed?

Bars are long-lived — surviving many rotation periods — but not permanent. Two processes can end them. First, the gas they funnel inward builds a central mass concentration that, once large enough, creates an inner Lindblad resonance which scatters the supporting $x_1$ orbits and dissolves the bar. Second, the bar loses angular momentum to the halo through dynamical friction, slowing and lengthening; a dense halo can brake and weaken it. Bars can also regenerate, so a disk may bar, dissolve, and re-bar. Their observed prevalence means formation outpaces destruction in most disks.

Why is two-body relaxation irrelevant to bar formation?

Because galaxies are collisionless on the relevant timescale. The two-body relaxation time — for star-star encounters to randomize orbits — scales as $N/\ln N$ crossing times, and for $N\sim10^{11}$ stars it vastly exceeds the age of the Universe. Bar growth happens over tens of rotation periods, far shorter than that. So during bar formation the stars never "collide"; the disk behaves as a smooth self-gravitating fluid in phase space, and the instability is a collective response of the distribution function governed by the collisionless Boltzmann equation and Poisson's equation, not by individual scatterings.