Gravitational Stability of Rotating Disks and the Toomre Criterion

Astrophysical disks are ubiquitous structures spanning an enormous range of sizes and compositions. Protoplanetary disks around young stars, the gaseous and stellar disks of spiral galaxies, and even the accretion disks that feed supermassive black holes can all be approximated as thin, rotating sheets of matter. Despite their diversity, these systems share a common susceptibility to self-gravity. Under the right conditions, a small perturbation in surface density may grow, fragmenting the disk and giving rise to bound objects such as planets, stars, or giant molecular clouds. Understanding when such fragmentation occurs is a fundamental problem, and one of the most powerful tools for addressing it is the Toomre $Q$ parameter. Originally derived by Alar Toomre in 1964 in the context of stellar disks, the criterion has since been generalized to gaseous disks and remains a cornerstone of disk stability analysis.

The Toomre parameter encapsulates the competition between stabilizing forces—thermal pressure and rotational shear—and the destabilizing influence of self-gravity. In a razor-thin disk with surface density $\Sigma$, sound speed $c_s$, and epicyclic frequency $\kappa$ (which generalizes the orbital angular frequency for non-Keplerian potentials), the parameter is defined as

$Q=\frac{c_s}{\kappa }\frac{1}{\pi }$

When $Q$ is less than unity, axisymmetric perturbations can grow, indicating that the disk is gravitationally unstable. Conversely, $Q>1$ implies stability against local axisymmetric collapse. This simple inequality hides a wealth of physics. The numerator of $Q$ measures the ability of pressure and rotation to resist compression: higher sound speed or faster differential rotation stabilize the disk. The denominator packages the disk’s self-gravity through the surface density $\Sigma$; more massive disks are more prone to fragmentation. The gravitational constant $G$ serves as the proportionality factor linking mass to gravitational strength.

The derivation of the Toomre criterion starts by considering small perturbations of the form ${\delta }_{\Sigma }(r,\varphi )$ superimposed on a background disk. Linearizing the equations of motion and continuity, and assuming axisymmetry for simplicity, leads to a dispersion relation for perturbations with radial wavenumber $k$:

${\omega }^2={\kappa }^2-2\pi G\Sigma |k|+{c_s}^2{k}^2$

Instability occurs when the squared frequency ${\omega }^2$ becomes negative for some wavenumber. The first term on the right-hand side represents rotational support, the second term encodes self-gravity (which promotes collapse), and the third term describes pressure support. By minimizing this expression with respect to $k$, one obtains the condition for the onset of instability, yielding the Toomre parameter. The most unstable wavelength, where the growth rate of perturbations is maximal, follows from the same analysis:

$\lambda \_\mathrm{crit}=\frac{2}{{c_s}^2}$

This critical scale indicates the characteristic size of fragments that form when $Q<1$. In protoplanetary disks, $\lambda \_\mathrm{crit}$ might correspond to the birth size of gas giant planets or brown dwarfs. In galactic disks, it can mark the scale of giant molecular clouds, the star-forming factories of the Milky Way. The associated fragment mass can be approximated by taking a circular patch of radius $\lambda \_\mathrm{crit}/2$ and multiplying by the surface density, giving $M\_\mathrm{frag}=\pi \frac{{\lambda }^{\_}}{2}\Sigma$. Our calculator evaluates both $\lambda \_\mathrm{crit}$ and $M\_\mathrm{frag}$ alongside $Q$.

Although the basic Toomre criterion is derived for infinitesimally thin, axisymmetric disks, real astrophysical environments require additional considerations. Finite thickness tends to stabilize the disk by diluting self-gravity, effectively increasing $Q$. Non-axisymmetric modes, magnetic fields, and radiative cooling can also modify the stability threshold. For example, spiral density waves in galaxies are inherently non-axisymmetric and can grow even when $Q$ slightly exceeds unity. In protoplanetary disks, rapid cooling may allow fragmentation at $Q$ values modestly above one. Consequently, the oft-quoted criterion $Q<1$ should be interpreted as a guideline rather than a strict boundary. Nonetheless, it provides a valuable first-order diagnostic of disk behavior.

Users of this calculator can input any combination of surface density, sound speed, and epicyclic frequency to evaluate $Q$ and related quantities. The epicyclic frequency reduces to the angular rotation rate in a purely Keplerian disk, where $\kappa =\Omega$, but it differs in the presence of a non-Keplerian potential or significant radial pressure gradients. In galactic dynamics, $\kappa$ can be computed from the rotation curve via ${\kappa }^2=R\frac{d{\Omega }^2}{dR}+4{\Omega }^2$. The surface density $\Sigma$ may represent either gas or stars, or a combination of both. Many studies compute separate $Q$ values for gas and stars and then combine them into an effective multi-component stability parameter.

To illustrate the sensitivity of $Q$ to its inputs, the following table lists values for a hypothetical galactic disk patch with varying surface densities and sound speeds, assuming a fixed epicyclic frequency of 1×10⁻¹⁵ s⁻¹. The classification column indicates whether the disk is expected to be stable according to the ideal Toomre criterion.

Σ (kg/m²)	cs (km/s)	Q	Status
500	0.5	0.48	Unstable
1000	1	0.95	Marginal
500	2	1.9	Stable

These examples emphasize how doubling the sound speed can double $Q$, potentially stabilizing the disk even if the surface density is high. Conversely, increasing the surface density by a factor of two halves $Q$, making instability more likely. This interplay between thermal support, rotation, and self-gravity underlies the diverse morphologies observed in astrophysical disks.

The physical interpretation of the critical wavelength and fragment mass is also instructive. Taking the second row of the table as an example (Σ = 1000 kg/m², c_s = 1 km/s), we find , which corresponds to roughly a few astronomical units. The associated fragment mass is on the order of Jupiter’s mass, suggesting that such a disk could, in principle, produce giant planets via gravitational instability. In contrast, galactic disks with lower sound speeds and higher surface densities yield critical scales of tens to hundreds of parsecs, aligning with the size of giant molecular clouds.

Beyond its role in predicting fragmentation, the Toomre parameter is widely used to interpret observations. Measurements of gas surface density and velocity dispersion in nearby galaxies allow astronomers to map $Q$ across disks, revealing where star formation is likely to occur. Regions with $Q\lesssim 2$ often correlate with intense star-forming activity, while higher values correspond to quiescent zones. In the Milky Way, the spiral arms are thought to trace loci of lower $Q$, indicating ongoing gravitational instabilities that gather gas into dense clouds. Similarly, the outskirts of galaxies, where the surface density falls off and the rotation curve flattens, generally exhibit $Q$ well above unity, explaining their relative tranquility.

Our calculator distills these concepts into a simple interface. After entering the desired parameters, the script computes $Q$, the critical wavelength $\lambda \_\mathrm{crit}$, and the corresponding fragment mass. The output also includes a brief classification indicating whether the disk is likely to be stable. Because the entire calculation occurs client-side in JavaScript, the tool can be used offline and customized for educational demonstrations or rapid exploratory analyses. While no calculator can capture all the complexities of real astrophysical disks, the Toomre Q parameter provides a remarkably powerful first estimate of gravitational stability, making it an essential concept for students and researchers alike.