Gravitational Instability in Protoplanetary Disks

• Gravitational instability in protoplanetary disks is characterized by self-gravity overcoming thermal pressure and rotational shear to form large-scale spiral arms, clumps, and rings.
• Key parameters such as the Toomre Q (<1–1.7), disk-to-star mass ratio (q ≳ 0.1), and cooling timescale (β ≲ 1) determine the onset and strength of GI-induced features.
• The process drives efficient angular momentum transport, episodic accretion, and creates conditions favorable for the direct formation of gas giants, brown dwarfs, and dust-enhanced planetesimal rings.

Gravitational instability (GI) in protoplanetary disks refers to the process whereby the disk’s self-gravity, under suitable conditions of mass, temperature, and cooling, overcomes the stabilizing influences of thermal pressure and rotational shear, leading to the growth of large-scale non-axisymmetric density structures. These instabilities regulate angular momentum transport, mass redistribution, and potentially the direct formation of gas giant planets and brown dwarfs. The GI phenomenon is mediated by the interplay between basic disk parameters such as the disk-to-star mass ratio, thermal properties (including cooling timescales), irradiation, and infall from the surrounding envelope. GI manifests observationally as spiral arms, clumps, and density rings, and is a key driver of disk evolution in the earliest stages of star and planet formation.

1. Stability Criteria, Key Quantities, and Physical Regimes

The onset and character of GI are governed primarily by the Toomre Q parameter: $Q = \frac{c_s \Omega}{\pi G \Sigma}$ where $c_s$ is the local sound speed, $\Omega$ is the Keplerian angular velocity, $G$ the gravitational constant, and $\Sigma$ the surface density (Harsono et al., 2010, Kimura et al., 2012, Steiman-Cameron et al., 2023). Disks become unstable when $Q \lesssim 1$–$1.7$, although nonlinear effects can cause instabilities at slightly higher Q (Steiman-Cameron et al., 2023, Su et al., 18 Mar 2025).

A second key criterion involves the spatial scale of the unstable modes: $\lambda \gg H \qquad (H = c_s/\Omega)$ with $H$ the disk scale height, so that global instabilities require modes with wavelengths much greater than the local vertical scale-height. These “global” modes (dominated by low azimuthal wavenumbers $m=1$–$4$) transfer angular momentum across wide radial ranges, contrasting with local, high-$m$ modes in lighter disks (Harsono et al., 2010, Dong et al., 2015).

A disk’s proclivity for GI is set largely by its mass relative to the central star. The threshold lies at $q = M_\mathrm{disk}/M_\star \gtrsim 0.1$, with stronger, more global instabilities, and visible grand-design spirals requiring $q \gtrsim 0.25$ (Harsono et al., 2010, Dong et al., 2015, Speedie et al., 3 Sep 2024, Yoshida et al., 24 Sep 2025).

Another stabilizing factor is the cooling timescale, generally parameterized as: $t_\mathrm{cool} = \beta \Omega_K^{-1}$ with $\beta$ the dimensionless cooling time and $\Omega_K$ the Keplerian angular velocity. Short cooling times promote instability and fragmentation, while longer cooling times stabilize the disk against fragmentation but allow quasi-steady gravitoturbulence (Su et al., 18 Mar 2025).

2. Nonlinear Outcomes: Spiral Arms, Clumps, and Rings

Once GI is triggered, it can lead to the rapid development of spiral density waves (spiral arms), with their morphology and kinematics set by the disk mass, temperature, and angular momentum profile (Dong et al., 2015, Dong et al., 2016, Xu et al., 26 Apr 2025). Observationally, these arms are most prominent for $q \gtrsim 0.25$, display pitch angles of $\sim$10°–15°, and can have contrasts in surface density or NIR scattered light of a factor of $2$–$3$ (Dong et al., 2015).

If the disk is cold enough (i.e., $t_\mathrm{cool} \Omega_K \lesssim 1$), strong spiral shocks can become nonlinear and fragment into gravitationally bound clumps (fragments), with masses typically above a few Jupiter masses (Zhu et al., 2011, Galvagni et al., 2013, Su et al., 18 Mar 2025). The fate of these clumps is determined by the competition between:

• Migration (often initially at a type-I rate, $\sim 10^3-10^4$ yr at $100$ AU, slowing as the clump mass grows) (Zhu et al., 2011, Galvagni et al., 2013)
• Accretion (with rates of $\dot{M}_c \sim 10^{-5}$–$10^{-4}$ $M_\odot/\text{yr}$, rapidly growing the clump)
• Tidal disruption (when the clump’s radius exceeds a fraction of the Hill radius during inward migration)
• Possible stalling if the clump opens a gap (more likely in low-viscosity disks) (Galvagni et al., 2013)

Most clumps in simulations either become very massive, open gaps and stall (becoming brown dwarfs or binary companions), are tidally disrupted, or migrate in and are lost to the central star. In situ giant planet formation by GI is therefore challenged except under a narrow set of conditions (Zhu et al., 2011, Galvagni et al., 2013).

In addition to spiral arms and clumps, GI drives the formation of axisymmetric and non-axisymmetric rings. These structures emerge particularly through the nonlinear growth of secular GI (mediated by dust–gas friction), and through the two-component viscous GI (TVGI) where both turbulent viscosity and friction are critical (Tominaga et al., 2017, Tominaga et al., 2019, Tominaga et al., 2020). The radial location, width, and longevity of these rings depend sensibly on local cooling conditions, dust dynamics, and radial drift.

3. Angular Momentum Transport and Gravitoturbulence

GI is widely recognized as a major agent for angular momentum transport, especially in early, massive disks (Harsono et al., 2010, Steiman-Cameron et al., 2023, Longarini et al., 10 Jun 2024). Angular momentum is transferred outward nonlocally by the torques of low-$m$ spiral arms, allowing accretion rates onto the star to exceed what would be expected from local transport alone; the effective Shakura–Sunyaev viscosity parameter ($\alpha_\text{GI}$) ranges from $0.01$ up to $0.1$ in active regions (Harsono et al., 2010, Steiman-Cameron et al., 2023, Longarini et al., 10 Jun 2024).

The character of this transport is complex:

• Infall from an envelope creates strong vertical shear (due to sub-Keplerian rotation above/below the disk midplane), exciting global, low-$m$ spiral modes that are dramatically more efficient at transporting angular momentum than in isolated disks (Harsono et al., 2010).
• The resulting mass-accretion rate can approach the infall supply rate $\dot{M}_\mathrm{infall} \sim c_s^3/G$, far exceeding the local steady-state GI limit of $\dot{M}_\mathrm{acc,max} \sim 0.1\,c_s^3/G$ (Harsono et al., 2010).
• The spiral modes erupt stochastically (via swing amplification), leading to a highly variable, gravitoturbulent state with time-variable torque strengths (Steiman-Cameron et al., 2023, Xu et al., 26 Apr 2025).

Empirically, the amplitude of density/velocity perturbations (as traced by “GI wiggles” in line kinematics) provides a direct diagnostic of $\alpha_\text{GI}$ and enables a comparison with the observed stellar accretion rate in systems such as Elias 2-27 and AB Aurigae (Longarini et al., 10 Jun 2024, Speedie et al., 3 Sep 2024). Agreement between kinematically inferred $\alpha$ and accretion rates supports GI as the dominant mass transport process in those disks.

4. Cooling, Irradiation, and the Regulation of GI

Thermal physics—especially the efficiency and character of cooling—regulates the onset and nature of GI. The key parameters are the cooling time $t_\mathrm{cool}$ (or its dimensionless analog $\beta$) and the irradiation floor provided by the central star (Zhu et al., 2011, Su et al., 18 Mar 2025).

• For $\beta \lesssim 1$ (i.e., $t_\mathrm{cool} \sim$ orbital period or less), GI grows rapidly and fragmentation is possible. Shorter cooling timescales favor lower azimuthal wavenumber spirals ($m$), so global modes prevail (Su et al., 18 Mar 2025).
• Slow cooling ($\beta \gg 1$) suppresses fragmentation, leading to a quasi-steady state of gravitoturbulence marked by continually excited, but generally nonfragmenting, spiral structure (Xu et al., 26 Apr 2025).
• In the presence of stellar irradiation, the outer disk cools less efficiently, raising the critical radius beyond which GI can fragment the disk, sometimes to hundreds of AU depending on accretion/infall rates (Zhu et al., 2011, Kimura et al., 2012).
• Transitional GI modes, dominant in the outer disk for $\overline Q \approx 1$ and $\beta \approx 1$, are especially effective at driving substructure and fragmentation in planet/brown-dwarf forming regions (Su et al., 18 Mar 2025).

The interplay between local viscous heating, irradiation, and radiative cooling sets the disk's equilibrium structure and azimuthal stability profile, reflected in a discontinuous jump in the critical surface density for GI near the “snow line” ($r_c \sim 20$–$24$ AU where ices form) (Kimura et al., 2012).

5. GI-Driven Substructure: Observational Diagnostics

GI produces a wealth of observable substructures whose signatures vary with scale, evolutionary stage, and wavelength:

• Spiral Arms: Prominent in NIR scattered light and mm/sub-mm continuum, these arms appear bright due to raised photospheres, and are often aligned with the local Keplerian motion—supporting a GI as opposed to planet-induced scenario (Dong et al., 2015, Dong et al., 2016, Yoshida et al., 24 Sep 2025).
• Clumps/Fragments: If formed, these manifest as local depressions in NIR scattered light (from reduced photosphere height) but appear bright in mm emission due to column density enhancement (Dong et al., 2016).
• Rings: Multiple narrow dust rings can arise from secular GI and TVGI, especially in systems with significant dust-to-gas ratio and turbulence; the ring width corresponds well to most unstable GI wavelengths, often $\sim 10$ AU far from the star (Tominaga et al., 2019, Tominaga et al., 2020).
• Kinematic Signatures (GI Wiggles): Deviations from Keplerian velocity, particularly sinusoidal fluctuations along disk minor axes, provide a robust diagnostic of GI and allow indirect mass inference (Longarini et al., 10 Jun 2024, Speedie et al., 3 Sep 2024).
• Gravitoturbulence and Clumpiness: GI-saturated disks often show a clumpy, non-grand-design appearance, with localized overdensities that may be misidentified as embedded companions or stochastic brightness variations (Xu et al., 26 Apr 2025).

Observed systems such as Elias 2-27, AB Aurigae, and IM Lup display kinematic and morphological features best explained by active GI (Longarini et al., 10 Jun 2024, Speedie et al., 3 Sep 2024, Yoshida et al., 24 Sep 2025).

6. Dust-Gas Interactions, Secular GI, and Planetesimal Formation

The evolutionary impact of GI extends beyond gas dynamics:

• Dust Concentration: Turbulent GI-generated circulation and inertial waves significantly increase the vertical scale height of solids ($H_\mathrm{dust} \sim 0.6 H_\mathrm{gas}$ for cm-size grains), impeding efficient dust settling and planetesimal formation unless secondary instabilities or local enhancements occur (Riols et al., 2020).
• Secular GI and Ring Formation: The coupling between gas and dust (frictional drag) enables the secular GI, a slow-acting process gathering dust into narrow rings even when conventional GI is inactive. These rings can reach line masses close to the critical filament value $2 c_d^2/G$, creating locally dust-dominated environments ripe for gravitational collapse into planetesimals (Tominaga et al., 2017, Tominaga et al., 2019, Tominaga et al., 2020).
• Nonlinear Outcomes: Whereas global GI is gas-dominated, secular and TVGI act preferentially on the solid component, often without driving comparable gas substructure. The presence of pronounced dust rings, without corresponding gas gaps, points to these mechanisms as key contributors to observed ring architectures (Tominaga et al., 2020).

Given the fragility and slow migration of these dust rings, secular GI may alleviate the radial drift barrier and enable efficient planetesimal assembly (Tominaga et al., 2017).

7. Evolutionary and Theoretical Implications

GI provides a paradigm for disk evolution in which self-gravity regulates mass distribution, drives episodic accretion, sculpts observable substructures, and sets the conditions for direct planet/brown dwarf formation (Harsono et al., 2010, Bae et al., 2014, Steiman-Cameron et al., 2023, Su et al., 18 Mar 2025). Major theoretical and practical consequences include:

• Explaining rapid mass transport and accretion bursts in embedded phases, reconciling high observed protostellar accretion rates with disk physics (Bae et al., 2014, Steiman-Cameron et al., 2023).
• Setting the outer disk regions as preferred sites for GI-induced fragmentation (given the drop in critical surface density due to opacity changes at the snow line) (Kimura et al., 2012, Su et al., 18 Mar 2025).
• Providing a pathway for both rapid and late-stage planet (and possibly brown dwarf) formation, even in regions where core accretion is inefficient due to long dynamical times.
• Demonstrating, through direct kinematic measurements and morphological mapping, that GI operates in observed disks and is a principal agent of substructure and planet formation in young systems (Longarini et al., 10 Jun 2024, Speedie et al., 3 Sep 2024, Yoshida et al., 24 Sep 2025).

Ongoing improvements in global radiative 3D simulations, observational resolution, and analytical diagnostics continue to refine the quantitative correspondence between theory and observation, establishing GI as a central actor in the early evolution of protoplanetary disks.