Gravitational instability in planet-forming discs

Abstract

During the earliest phases of star and planet formation, a young star is surrounded by a disc that can become gravitationally unstable. Indeed, if the protoplanetary disc is sufficiently massive, its self-gravity triggers the formation of large scale spiral arms, transporting angular momentum, trapping solid particles and, potentially, collapsing to form sub-stellar companions. In this paper, we review the theoretical and observational advances in the study of gravitational instability in protoplanetary discs over the last 10 years, since the advent of ALMA. These developments have transformed our understanding of gravitational instability, moving beyond its historical role as a theoretical mechanism for giant planet formation toward a physically rich framework that can be directly tested against high-resolution observations.

1 Introduction

Protoplanetary discs are the site of planet formation, providing the physical link between protostars and the emergence of planetary systems. Constraining their dynamical properties is therefore essential to understand how planets form, how efficiently material is converted into planets, and what types of planetary systems are expected to emerge.

During the last decade, the observational landscape has been revolutionised by the Atacama Large Millimetre Array (ALMA), which has enabled spatially and spectrally resolved observations of gas and dust in protostellar environments (Andrews et al., 2018), uncovering their morphology and the physical processes taking place within them. In particular, high-resolution continuum and molecular line observations have revealed a rich variety of substructures in protoplanetary discs (Andrews, 2020), including rings, gaps (ALMA Partnership et al., 2015; Huang et al., 2018a; Clarke et al., 2018; Guzmán et al., 2018; Long et al., 2018), spirals (Pérez et al., 2016; Huang et al., 2018b), and asymmetries (Casassus et al., 2013; Isella et al., 2013; Marino et al., 2015; Curone et al., 2025), opening new windows onto the dynamics of planet-forming systems.

In this paper, we focus on one of the most fundamental aspects of disc physics: the role of the disc self-gravity. Protoplanetary disc mass sets the total reservoir available for planet formation, regulates the aerodynamical coupling between gas and dust, and determines whether key physical processes, such as hydrodynamical instabilities, can operate. In particular, when the disc becomes sufficiently massive, its self-gravity (SG) can no longer be neglected. In this regime, the gravitational potential of the disc becomes comparable to that of the central star, fundamentally modifying its dynamics. One of the most important consequences is the onset of gravitational instability (GI), a mechanism that can drive angular momentum transport, generate large-scale spiral structures, and potentially lead to fragmentation under the right physical conditions.

Gravitational instability is expected to be especially relevant in the earliest stages of disc evolution. Young discs are likely more massive, as protoplanetary discs are accretion discs whose mass decreases over time due to disc dispersal processes, such as viscous accretion or disc winds (Pringle, 1981; Blandford and Payne, 1982). Hydrodynamical simulations of star formation, that follow the collapse of molecular clouds (Bate, 2018; Lebreuilly et al., 2024) typically result in the formation of star + disc systems, where the disc mass is a substantial fraction of the total mass. This trend is supported observationally by comparisons of disc mass functions¹ across star-forming regions of different ages: younger regions host systematically more massive discs, while more evolved regions are dominated by lower-mass systems (Tobin et al., 2020), as shown in Figure 1. This makes SG and GI not only a theoretical possibility, but a key ingredient in understanding these early stages.

FIGURE 1

The very early stages of protoplanetary disc lifetime are fundamental to understand the process of planet formation. Direct imaging of protostellar discs, in scattered light and especially at sub-mm wavelengths, has opened debate regarding the timescale for planet formation. Many of the large and bright discs observed with ALMA and SPHERE show substructures (Andrews et al., 2018; Andrews, 2020) that is consistent with, and often interpreted as, the theoretical signpost of planet-disc interaction (Dong et al., 2015b,a; Lodato et al., 2019; Paardekooper et al., 2023). A subset of these discs, including HL Tau (ALMA Partnership et al., 2015), IRS 63 (Segura-Cox et al., 2020) and GY91 (Sheehan and Eisner, 2018) are young, with ages estimated to be less than 1 Myr. Assuming a planet origin for the substructure, a substantial part of the planet formation process must overlap with the time when protostellar discs are still young, and likely self-gravitating.

In this paper, we focus on the physics of gravitational instability in young protostellar discs, combining theoretical developments and observational advances over the past decade. The paper is structured as follows. In Section 2, we provide a brief overview of gravitational instability in protostellar discs. In Section Equation 3, we present recent theoretical progress, focusing on numerical developments, the connection between GI and dust dynamics, planet formation, and thermodynamics. In Section 4, we discuss observational advances, highlighting three sources as representative examples of GI in discs, and review methods to infer disc masses in bright systems. Finally, in Section 5, we summarise our conclusions. Gravitational instability in protoplanetary discs has been the subject of extensive study, and we refer the reader to the comprehensive review by (Kratter and Lodato, 2016) for a thorough overview. The goal of this work is to present the developments and advances that have emerged since 2016.

2 Summary of self gravity and gravitational instability in protostellar discs

Self-gravity is fundamental in astrophysics, from cosmic structures to stars and planetary rings. In the context of protoplanetary discs, self-gravity is particularly important in the early stages of the disc lifetime, when the disc-to-star mass ratio is high. A crucial effect of the disc self-gravity is the development of gravitational instability. This mechanism is characterised by the formation of large scale spiral structures, transporting angular momentum throughout the disc. The foundations of our understanding of self-gravity and gravitational instability in discs comes from the field of galactic dynamics. Indeed, during the second half of the last century, the so-called “density wave theory” was developed to explain the origin of spiral structure in galaxies (Lin and Shu, 1964; Toomre, 1964). In this context, massive protoplanetary discs can be treated, to first approximation, in analogy with galactic discs, provided that the appropriate rescaling of physical quantities and gravitational potentials is taken into account.

In this section, we provide a concise overview of the main properties of gravitational instability in protostellar discs. We introduce the instability criterion and the key physical quantities that characterise the GI regime. This is not intended to be a comprehensive review of the subject, but rather a minimal framework to support the discussion in the following sections.

2.1 Linear theory of gravitational instability

To introduce the linear behaviour of gravitational instability, it is worth discussing the dispersion relation, i.e., how the perturbation frequency depends on the radial wavenumber . This relationship was first obtained in the context of galactic dynamics by Safronov (1960), and then formalised by Lin and Shu (1964). In the razor thin disc limit, in the WKB approximation, the dispersion relation readswhere is the number of spiral arms, the sound speed of the fluid, the disc surface density and the epicyclic frequency, that for a Keplerian disc . The dispersion relation is a quadratic in , and each term has a physical interpretation. The left side term is the Doppler shifted perturbation frequency, and can be re-written by underlying the spiral pattern frequency , where is the spiral pattern frequency. The first term on the right side of Equation 1 is the stabilizing pressure contribution. It is quadratic in and hence it is significant for large wavenumbers - short wavelengths. The second term is the disc self-gravity, that is linear with the wavenumber. Its contribution is more significant at intermediate wavelengths, and its role is destabilizing, contributing with a negative sign. Finally, the third term is related to the disc rotation and represents epicyclic oscillations. It does not depend on and hence its stabilizing effect is dominant at large wavelengths. The competition of these three effects determines the stability of the system. This is particularly clear when studying the instability threshold for axisymmetric perturbations, namely, . The perturbation is unstable ifwhere is the so-called Toomre parameter (Toomre, 1964). The instability threshold is hence . In a protostellar disc, the sound speed and the surface density can be described as power laws with radius. We usually assume that , and : this implies that . If , the Toomre parameter decreases with the radius, meaning that the outer part of the disc is more prone to be unstable. Conversely, if , the disc will be more likely gravitationally unstable in the inner regions. If we consider a standard power-law exponent for temperature , it is likely that the condition for instability will be met in the outer (inner) regions of the disc when . Observations of discs have typically yielded power-law exponents within the range of [0, 2] (Andrews and Williams, 2007), making it more likely for the condition to be met in the outer regions of a disc.

For the case of an unstable disc, the most unstable wavenumber (i.e., where the right-hand side of Equation 1 is at its minimum) occurs atWhen the disc is marginally unstable , only modes close to are excited. If we evaluate Equation 1 for and , we get the condition , which means that all excited modes are expected to be close to co-rotation. It is worth noting that the most unstable wavenumber is exactly the inverse of the disc thickness in the self-gravitating limit . Equivalently, it is possible to define the most unstable wavelength . and are respectively the most unstable wavenumber and wavelength, often called Jeans wavenumber and wavelength.

2.2 Non-linear evolution of gravitational instability

Now, we ask what is the non-linear evolution of GI, and whether it can lead to transport of angular momentum. To tackle this question, it becomes imperative to employ numerical simulations of gravitationally unstable discs. Over the past 2 decades, this topic has gained considerable attention, employing various numerical techniques, ranging from global 3D SPH simulations (Rice et al., 2003, 2004; Lodato and Clarke, 2011), global grid based simulations (Kratter and Matzner, 2006; Kratter et al., 2008; Meru and Bate, 2010; 2012), local grid simulations (Booth and Clarke, 2019; Leedham et al., 2025; Rice et al., 2025).

A useful starting point is the Toomre stability parameter, , which scales proportionally with the sound speed (and hence temperature) and inversely with the disc surface density. As a consequence, cold and massive discs are more prone to gravitational instability. This naturally identifies two primary pathways to trigger GI: a reduction in temperature (cooling-driven GI) or an increase in surface density (infall- or mass-loading-driven GI). A morphological comparison between these two regimes is shown in Figure 2.

FIGURE 2

The non-linear outcome of GI broadly falls into two regimes. In the first, the disc settles into a self-regulated state, in which spiral density waves reach a quasi-steady amplitude and transport angular momentum outward. In this regime, for relatively light discs, GI can be described in terms of an effective turbulent viscosity (Lodato and Rice, 2004), often parameterised through an prescription. In the second regime, the instability grows exponentially, leading to fragmentation: spiral perturbations collapse into gravitationally bound clumps (Gammie, 2001; Rice et al., 2003), potentially giving rise to the formation of companions (Boss, 1997).

In the following, we first discuss the self-regulated regime, highlighting the differences between cooling-driven and infall-driven GI, and then examine the conditions under which fragmentation occurs.

2.2.1 Cooling-driven

Self-regulation can be achieved through cooling. Let us consider an initially hot and gravitationally stable disc that progressively cools. As the temperature decreases, the value of correspondingly drops, eventually approaching the marginal stability threshold . At this point, gravitational instability turns on: the disc develops spiral density structures which, through compression and shocks, dissipate energy and heat up the gas.

This establishes a feedback loop in which cooling drives the disc toward instability, while GI-induced heating counteracts further cooling (Bertin, 2000). As a result, the disc evolves toward a quasi-steady state in which heating and cooling balance each other, maintaining close to unity. In this sense, gravitational instability acts as a thermostat, activating efficient angular momentum transport and dissipation only when the disc becomes sufficiently cold.

We can model the cooling through the parameter, that is the cooling timescale in unit of the dynamical one, defined as . Cossins et al. (2009) showed that in the self-regulated state, the amplitude of the spiral perturbation defined as is solely determined by the cooling, according toFaster cooling (i.e., lower values of the cooling parameter ) leads to larger-amplitude spiral perturbations. In order to maintain thermal balance, the disc must generate stronger gravitational stresses to compensate for the increased cooling rate. This results in more prominent spiral structures and enhanced angular momentum transport.

The self-regulated state described above can be quantified by relating the strength of angular momentum transport to the cooling rate. In particular, if no additional heating sources are present, thermal equilibrium requires that the heating provided by gravitational stresses balances radiative cooling. Under this assumption, the viscous heating timescale must match the cooling timescale, i.e., . This condition leads to a direct relation between the effective viscosity and the cooling time. The resulting parameter associated with gravitational instability can be written as (Gammie, 2001; Lodato and Rice, 2004)This expression makes explicit that faster cooling (lower ) requires stronger gravitational stresses, and therefore a higher effective , to maintain thermal balance. This provides a quantitative framework for the self-regulated state described above.

2.2.2 Infall-driven

Self-regulation can also be achieved in discs that are continuously fed by mass infall from the surrounding envelope. In this case, instead of reducing the temperature, gravitational instability is triggered by an increase in the disc surface density. As mass is injected into the disc, the Toomre parameter decreases, eventually approaching the marginal stability threshold , at which point GI develops.

Similarly to the cooling-driven case, the onset of GI leads to the formation of spiral density structures that transport angular momentum and redistribute mass within the disc. This transport acts to process the incoming material, preventing an indefinite build-up of mass and driving the system toward a quasi-steady state. In this sense, infall-driven GI can also establish a self-regulated configuration, in which the disc adjusts its structure to accommodate the imposed mass accretion rate.

A useful framework to characterise this regime was introduced by Kratter et al. (2010), who defined two dimensionless parameters describing the strength of accretion relative to the disc dynamical properties:where is the infall rate, is the sound speed, is the total system mass (disc plus central object), and is the local orbital frequency. The parameter measures the infall rate in units of the isothermal collapse rate, while compares the mass-loading timescale to the dynamical timescale of the system.

Using three-dimensional simulations of discs undergoing self-similar accretion, Kratter et al. (2010) showed that the system can either settle into a quasi-steady, self-regulated state or undergo fragmentation, depending on the values of these parameters.

More recently, the nature of self-regulation in infall-driven discs has been revisited in Longarini et al. (2025b), where it is shown that continuous mass loading can maintain gravitational instability over extended timescales, establishing a tight coupling between disc mass, stellar mass, and accretion rate. We defer a detailed discussion of this regime to Section 3.5.

2.2.3 GI as a companion formation scenario - gas fragmentation

If the disc cools too efficiently, or receives too much mass from the surrounding envelope, gravitational instability cannot reach a self-regulated state, resulting in the exponential growth of spiral perturbations. In this regime, the spiral arms fragment into gravitationally bound objects (Durisen et al., 2007).

In the cooling driven scenario, the critical value of below which fragmentation occurs has been the subject of extensive debate over the past 2 decades. Firstly, Gammie (2001) performed shearing box simulations of gravitational instability using the cooling prescription. He found that if the cooling timescale is faster than approximately 3 times than the dynamical one, fragmentation occurs. Afterwards, Rice et al. (2005) tackled this problem through three-dimensional global SPH simulations. They found that, depending on the value of the coefficient, the critical value of is between 6 and 10. Meru and Bate (2010) and Meru and Bate (2011) performed high resolution SPH simulations, and they did not find convergence in their results when using a different number of gas particles. Lodato and Clarke (2011) proposed that this behaviour could be driven by resolution effects. One year later, Meru and Bate (2012) obtained evidence that convergence with increasing resolution occurs with SPH, as long as the effects of numerical viscosity are taken into account. In particular, the authors found that reducing the dissipation from the numerical viscosity leads to larger values of the critical cooling time at a given resolution. This happens because the numerical dissipation in SPH depends on resolution. Additionally, they also found convergence for grid based codes, however the results were in contrast with the particle based ones. From this analysis, they found that is larger than previously thought. In particular, for , their critical was higher than 20, implying that young discs are very likely to fragment. Finally, Deng et al. (2017) obtained convergence between SPH and grid based codes, and the value of below which fragmentation occurs isas firstly proposed by Gammie (2001). A minimum value of implies a maximum value of , i.e., the maximum amount of angular momentum that can be transported through GI. For , the maximum value of isfor .

In the infall-driven regime, Kratter et al. (2010) showed that the dimensionless parameters and are strong predictors of fragmentation. In particular, fragmentation is found to occur when

In the ’90s, the discovery of the first extrasolar giant planet (Mayor and Queloz, 1995) presented a significant challenge to theorists, as the widely accepted core accretion model could not explain the formation of such a massive planet, because of the too long timescale. Boss (1997) proposed gravitational instability as an alternative paradigm for giant planet formation. For disc-to-star mass ratios of the order of (i.e., comparable to the disc aspect ratio ), the outer part of the disc is likely to be marginally gravitationally unstable, according to the -criterion, for a power-law surface density profile. For a more realistic surface density profile, e.g., a self-similar one, a somewhat higher disc-to-star mass ratio, , is typically required to reach in the outer disc, for the same aspect ratio. If the thermal saturation is not reached, because of a too short cooling time, gas fragmentation can happen. Boss (1997) proposed gas fragmentation as a way to rapidly form planets in the outer disc. One natural question to ask is: what is the typical mass of a GI fragment? We expect that the mass of the fragment is of the order of the Jeans mass. According to the quadratic dispersion relationship, the Jeans mass isthis means that for a solar mass star, the typical Jeans mass is of the order of . It is important to note that this value is just an initial mass. Indeed, the GI fragment will start accreting material in the disc, that is dense because the disc is massive. This process will lead to runaway accretion, and to the formation of a stellar companion (Kratter and Matzner, 2006) rather than a planet. Indeed, the theoretical minimum mass a star must have to undergo fusion at the core, is estimated to be about , and it can be rapidly reached in those conditions. It is worth noting that the historical motivation for gravitational instability, namely, the difficulty of core accretion in forming massive cores within the disc lifetime, has been substantially reconsidered in light of more recent developments in the core accretion paradigm (e.g., Venturini et al., 2020; Drkazkowska et al., 2023. While this is true in the inner disc, the standard core accretion scenario still struggles in explaining planet formation in the outer discs, in cases such WISPIT2 (van Capelleveen et al., 2025) and HR 8799 (Marois et al., 2008). Although it circumvents the timescale problem, the standard gravitational instability scenario is not likely to explain the formation of planets, without invoking ad hoc mechanisms, such as tidal downsizing Nayakshin (2010).

3 Theoretical advances on GI

3.1 GI wiggle and kinematics

During the last 10 years, ALMA and SPHERE data showed a huge number of protoplanetary discs with spiral structures (Huang et al., 2018b). The origin of such structures remains a subject of debate. While planet-disc interactions, stellar flybys, and gravitational instability can all produce similar morphologies, gas kinematics offers a powerful diagnostic to break this degeneracy. Specifically, the kinematic signatures of GI are expected to be distinct and clearly identifiable.

Using hydrodynamics simulations coupled with radiative transfer, Hall et al. (2020) first demonstrated that a disc undergoing GI exhibits clear kinematic signatures in molecular line observations. This feature, referred to as the “GI Wiggle”, manifests as a distinctive zig-zag pattern in the emission lines (isovelocity contours), most commonly seen in the CO roto-vibrational lines in the mm band. A key property that distinguishes the GI wiggle from planet-driven perturbations (e.g., Bollati et al., 2021) is its global nature. While a planet induces a localised perturbation that decays with distance from the source, GI-induced spirals perturb the disc across its entire radial and azimuthal extent, resulting in a coherent, large-scale kinematic signal.

Longarini et al. (2021) provided an analytical description of the GI wiggles in the linear regime of GI, under the assumption of self-regulation. As a wave-like perturbation, the wiggle is characterised by both its wavelength and its amplitude. Since the kinematic perturbation follows the density enhancement of the spiral arms, the number of arms and their pitch angle directly determine the spatial frequency and the position of the wiggles in the velocity field. These parameters are not arbitrary; they scale with the disc-to-star mass ratio, with more massive discs typically displaying fewer and more open spiral arms (i.e., larger pitch angles). The amplitude of the GI wiggle is a particularly critical quantity since it allows to constrain fundamental disc properties. Longarini et al. (2021) showed that this quantity is determined by the disc-to-star mass ratio and the spiral density contrast, that is related to the angular momentum transported by the spirals, according towhere measures the amount of angular momentum transported by gravitational instability, and is the Keplerian azimuthal velocity.

In a self-regulated state, more massive discs are typically hotter², leading to a higher sound speed. This results in an increased perturbation amplitude, as the kinematic response is proportional to the local sound speed. Furthermore, the velocity perturbation scales with the normalised density fluctuation, . This quantity reflects the specific mechanism triggering the instability (as shown in Equation 4), but it is most fundamentally related to the gravitational stress exerted by the spirals. When the disc reaches a self-regulated state where angular momentum transport can be described via a pseudo-viscous -prescription, the following relation holdswhere is the effective viscosity produced by gravitational instability. This connection is profound: it implies that by measuring the amplitude of the kinematic “wiggles” in molecular line observations, we can potentially “weigh” the disc and simultaneously measure the efficiency of its internal transport mechanisms.

However, the applicability of this analytical framework to real observations depends on whether the disc operates within the linear regime of the instability. While the linear theory provides a clear link between kinematics and disc properties, its validity must be carefully verified against global, non-linear models. Figure 3 illustrates this by comparing the analytical predictions of the wiggle with synthetic observations derived from 3D hydrodynamical simulations (Speedie et al., 2024). Although GI is an inherently non-linear process characterised by shocks and strong density gradients, the large-scale kinematic response retains the features predicted by the linear theory. A further test of this analytical framework is provided in Terry et al. (2022), where the expected scaling of the wiggle amplitude with disc mass is remarkably well recovered in numerical simulations. By measuring the velocity deviations in synthetic observations of discs with varying masses, they confirmed that the kinematic response remains sensitive to the underlying gravitational potential even as the instability enters the non-linear regime.

FIGURE 3

It is important to underline that identifying these kinematic signatures from observations, and disentangling them from other kinds of signatures, is not straightforward. Indeed, high spatial and spectral resolution ALMA observations are required, and the geometry of the disc—such as its extent and its inclination with respect to the line of sight—plays an important role. For these reasons, the number of sources for which this method is currently applicable remains very small.

Altogether, the GI wiggle stands out as a unique kinematic signature, encoding in a single observable the disc mass, the spiral morphology, and the efficiency of gravitational stress. This makes kinematics an indispensable tool in the interpretation of self-gravitating discs.

3.2 GI and dust, planetary cores formation

Planet formation through gravitational instability is often synonymous with gas fragmentation, however this process typically results in the formation of stellar-mass companions or brown dwarfs rather than planetary-mass objects (Kratter and Lodato, 2016). Nevertheless, the role of GI in planet formation extends far beyond the direct collapse of the gaseous component. In this section, we explore the interplay between dust and gravitational instability, highlighting how GI-induced structures can alter the evolution of the solid component and provide a rapid pathway for the formation of planetary cores.

Rice et al. (2004) first proposed and demonstrated that gas spiral arms act as efficient dust traps. As pressure maxima, these structures naturally collect dust particles, with the trapping process being particularly effective for grains with a Stokes number (, where is the stopping time) close to unity. Contrary to planetary-driven spirals, GI spirals efficiently trap dust particles because their pattern speed is close to corotation (Cossins et al., 2009), making the relative velocity between the pattern and the dust particles very small, whereas planetary spirals propagate at a fixed pattern speed. Beyond simple concentration, the authors suggested that the dust density within these arms could reach a critical threshold, leading to the direct gravitational collapse of the solid component into planetary cores. This hypothesis was further explored in Rice et al. (2006), confirming the viability of the mechanism (left panel of Figure 4). While those early numerical simulations were limited by resolution and could not fully resolve the dust Jeans length, they opened a new avenue for planet formation in young, massive discs: the collapse of dust within spiral structures. This hybrid scenario combines the advantages of both traditional theories, offering a mechanism that is as fast and efficient as gravitational instability, while being fundamentally dust-based like core accretion.

FIGURE 4

To understand the physics behind this process, it is necessary to consider the gravitational stability of a two-fluid disc. This problem has been thoroughly explored in the context of galactic dynamics, where the two components are typically stars and gas (Jog and Solomon, 1984; Bertin and Romeo, 1988). These works investigated the gravitational stability of a two-phase system where the components interact primarily through their shared gravitational potential. The two fluids can be classified according to their relative abundance and “dynamical temperature.” In galactic discs, stars represent the dominant and “hot” component, characterised by high velocity dispersion, while the gas is less abundant and “cold”. Conversely, in protoplanetary discs, the roles are reversed: the gas is the dominant, hot component, while the dust is the less abundant, cold component. While dust is often treated as a collisionless fluid in these environments, for the purpose of gravitational stability it can be assigned an effective velocity dispersion which operates analogously to a sound speed. Bertin and Romeo (1988) demonstrated that gravitational instability can be driven by the cold component provided it is sufficiently abundant and cold. Specifically, the onset of this regime requires that

Longarini et al. (2023b) revisited this scenario by incorporating the role of the drag force, parameterised by the Stokes number . They found that for small Stokes numbers (i.e., strongly coupled particles), the destabilising effect of the solid component is negligible. However, as the Stokes number increases and the coupling weakens, the dust component contribution to the instability becomes increasingly significant. While , , and are treated as independent variables for the purpose of stability analysis, they are interdependent in a physical disc: the dust concentration typically peaks near , and the velocity dispersion is itself a function of the gas–dust coupling. Furthermore, the authors derived the Jeans mass for these dust-driven perturbations. Due to the significantly lower dynamical temperature of the dust compared to the gas, the resulting Jeans mass is considerably smaller than in the gas-only fragmentation scenario, falling within the range of 1–, making this mechanism a viable way to create solid cores in young discs. These numerical models of dust dynamics in GI discs include dust backreaction. The dust particles are collisionless and inviscid, meaning that any diffusive mechanism arises solely from aerodynamic coupling with the gas.

These theoretical models highlight the critical role of the dust particles’ kinematics. From a physical perspective, the interaction between gas spiral arms and the solid component induces relative motions among dust grains. These motions can be categorised into two distinct quantities: velocity dispersion and collisional velocity. The velocity dispersion (characterised by the standard deviation of particle relative velocities at a given location) represents a macroscopic pseudo-thermal property of the dust layer. This former quantity is what defines the effective sound speed , which provides the pressure-like support against gravitational collapse discussed in the two-fluid stability analysis. Conversely, the collision velocity is defined on a pair-wise basis as the relative speed between two individual grains; it is the primary factor governing whether particles will grow through coagulation or undergo fragmentation upon impact.

The magnitude of is regulated by the level of gravito-turbulence, set by the cooling timescale (where faster cooling leads to higher excitation Booth and Clarke, 2016), and the Stokes number. Typically, mirrors the gas turbulence for . It then has a minimum for before increasing for decoupled particles due to stochastic gravitational “kicks” from the spiral arms (Walmswell et al., 2013; Booth and Clarke, 2016; Riols et al., 2020). This velocity dispersion is the primary factor governing the stability of the solid component. Specifically, the criterion (Equation 13) defines the stability of the dust layer through the interplay between the relative dispersion and the local dust-to-gas ratio. There is a growing consensus within the community, supported by diverse numerical approaches, that dust grains with Stokes numbers in the range are the most effective candidates for rapid planetary cores formation. Indeed, particles in that range undergo the strongest concentration while maintaining a low velocity dispersion. This unique combination allows the solid component to satisfy the dust-driven GI criterion, overcoming the “dynamical heating” of the disc and leading to the direct gravitational collapse into planetary cores (Longarini et al., 2023a; Rowther et al., 2024, right panel of Figure 4).

To properly assess the conditions to activate dust collapse, recent works have employed local shearing box simulations, which allow for the high resolution necessary to capture the small-scale dynamics of the dust layer. Baehr et al. (2022) demonstrated that, in the presence of gravito-turbulence, dust grains can reach densities high enough to trigger collapse even when the gas itself is stable, confirming the analytical picture. Specifically, Baehr et al. (2022) and Baehr et al. (2025) identified a lower limit for the Stokes number required for collapse, finding that while particles are the most prone to fragmentation into clumps, the mechanism remains viable down to . Below this threshold, the aerodynamic coupling is too strong, and the “dynamical heating” from the gas turbulence prevents the dust from settling into a sufficiently thin and dense layer to satisfy the condition in Equation 13. Furthermore, Rice et al. (2025) demonstrated that the success of the hybrid scenario depends on the dust size distribution. Direct core formation only occurs if a significant mass fraction is concentrated in the range . Conversely, a high abundance of small grains can inhibit collapse by increasing the effective velocity dispersion and preventing the dust from reaching critical densities. This underlines the importance of the dust growth process, to allow gravitational collapse to happen.

The viability of this hybrid scenario fundamentally depends on whether dust grains can grow to the required sizes within the short timescales of the GI-active phase. For a typical disc mass of , a Stokes number of 0.1 corresponds to a physical grain size of roughly 0.5 mm, a significant growth milestone for such a young system. We underline that the grain size estimate strongly depends on the radial distance from the central star. Standard dust growth models in smooth discs struggle to reach these sizes so rapidly (Birnstiel, 2024), as they are further hindered by the radial drift barrier, especially in the outer disc. However, gas spiral arms act as pressure traps that not only halt radial drift but also create high-density environments where grain growth could be significantly accelerated. To determine if this acceleration leads to successful growth or to a fragmentation bottleneck, it is necessary to investigate the collisional velocity between grains.

The collisional velocity, , defined on a pair-wise basis as the relative speed between two individual grains, is the primary factor governing whether particles grow through coagulation or undergo fragmentation upon impact. In gravito-turbulent discs, the dynamics of collisions are significantly more violent than in laminar flow, creating a potential bottleneck for the hybrid planet formation scenario.

Booth and Clarke (2016) provided the first systematic characterisation of the collisional velocity in the context of GI, demonstrating that it scales as . This scaling is physically rooted in the nature of the gas dynamics; as shown by Booth and Clarke (2019) through spectral analysis, GI-driven turbulence follows a Kolmogorov-like cascade. Since dust grains are accelerated by the turbulent gas velocity field, their relative motions reflect the properties of the turbulent cascade, leading to the square-root dependence on the Stokes number. However, a crucial distinction must be made between mono-disperse and multi-disperse dust populations. In a mono-disperse distribution, where all grains share the same Stokes number, relative velocities are the smallest, because all the particles react identically to gas fluctuations. Conversely, in a multi-disperse environment, the difference in aerodynamic coupling between grains of different sizes significantly increases their relative motion. This makes the fragmentation barrier a big obstacle for grain growth. For silicate-dominated grains, which fragment at velocities of approximately (Gundlach and Blum, 2015; Musiolik and Wurm, 2019), GI-induced motions can easily exceed the threshold at Stokes numbers as low as . This suggests that reaching the critical regime required for dust-driven GI might be extremely challenging for rocky material. On the contrary, for icy aggregates with higher fragmentation thresholds (, Wada et al., 2013; Musiolik, 2021), growth remains a viable pathway, particularly within the high-density environment of spiral arms where the concentration of solids can significantly enhance the growth rate.

Even though self-gravity is not involved, it is worth comparing this hybrid planetary cores formation scenario with the streaming instability (SI), as both mechanisms ultimately lead to the direct gravitational collapse of concentrated solids into planetesimals or planetary cores. Yet, they differ substantially in their physical origin and in the disc conditions under which they operate. The SI is fundamentally an aerodynamic instability, driven by the mutual drag force between gas and dust: local overdensities of solids exert a backreaction on the gas that reduces its effective sub-Keplerian velocity, which in turn reduces the relative drift velocity of nearby particles, reinforcing the clumping of solid particles. Crucially, this mechanism does not rely on the self-gravity of the gas disc, and it is most efficient when the local dust-to-gas ratio approaches or exceeds unity, a condition often achieved through independent concentration processes. Dust-driven GI, by contrast, is intrinsically related to the large-scale self-gravity of a massive disc: the concentration of solids is a direct consequence of the spiral structure induced by the gas gravitational instability, and the subsequent collapse of the dust layer is governed by its own effective Jeans criterion. In this sense, the two mechanisms are in some respects complementary, rather than competing: dust-driven GI is expected to operate preferentially at very early times, when the disc is still massive enough to be self-gravitating, whereas the SI is likely to become relevant later in the disc lifetime, once the disc has evolved towards lower masses and dust has had time to settle and grow in a smoother disc.

Taken together, these results establish gravitational instability as a viable path to form planets, not through direct gas fragmentation, but by creating the physical conditions for the gravitational collapse of the solid component. This offers a rapid pathway to form planetary cores in the earliest stages of disc evolution.

3.3 GI and thermodynamics

The non-linear evolution of gravitational instability and its ability to lead to fragmentation are fundamentally governed by the thermodynamics of the disc. Historically, the most common approach to model it has been the cooling prescription (Gammie, 2001), where the cooling timescale is considered proportional to the dynamical timescale, with the proportionality factor. While numerically efficient and useful to identify the basic physics of gravitoturbulence, this is certainly not realistic. Indeed, this method neglects the complex physics of radiative cooling and shock heating, which in real discs depends strongly on local properties such as density, temperature, and, most critically, the optical depth , which varies significantly across the radial extent of the disc.

Firstly, Stamatellos et al. (2007) tackled this problem in 3D global SPH simulations, introducing more realistic thermodynamics through a local radiative cooling and heating approximation. By estimating the optical depth at every particle location in the simulation, according to the local disc properties, this method allowed for the first time to incorporate a physically motivated energy equation into SPH simulations without the prohibitive cost of full radiative transfer. Despite its success, this method relies on a simplified estimate of the optical depth based on the local column density, which may not fully capture the complex 3D geometry of radiation field in a highly perturbed disc.

As a direct evolution of the Stamatellos et al. (2007) approach, Young et al. (2024) introduced a refined thermodynamic scheme, building upon the work of Lombardi et al. (2015), to improve the accuracy of radiative cooling in global hydrodynamical simulations. Their approach is based on a polytropic cooling approximation, under the assumption that each gas parcel resides within a “spherical pseudo-cloud” to estimate the local optical depth. This hybrid framework overcomes the limitations of earlier models by providing higher accuracy in optically thick environments, where the original Stamatellos et al. method often faced challenges. In this framework, heating is treated as a prescribed mechanical response to the gas dynamics. Rather than being derived from a radiation field, the thermal energy input is calculated locally through artificial viscosity, which converts kinetic energy into heat at shock fronts, and adiabatic compression ( work). This approach ensures numerical efficiency by coupling the thermodynamics directly to the hydrodynamic variables, though it bypasses radiative feedback mechanisms.

A more fundamental approach is adopted by Xu et al. (2025a), Xu et al. (2025b), who transition from approximate prescriptions to solving the full radiation transport equation. These studies employ grid-based global simulations using the Athena++ code to resolve the interaction between gas and radiation along 32 discrete angles. In this high-fidelity framework, the thermodynamics is not prescribed but is instead an emergent property of the system. Rather than relying on local shock-heating recipes, the thermal state is self-consistently determined by solving the radiative transfer equation throughout the disc. This approach captures non-local feedback mechanisms, such as radiative “self-heating”, where energy released in one region can travel and warm distant areas, a process entirely neglected by previous methods. While this represents the current “Gold Standard” for capturing the internal physics of self-gravitating discs, it comes at a formidable computational cost. Solving the transport equation along dozens of angles requires orders of magnitude more resources than approximate methods, currently limiting these simulations to shorter evolution.

A significant novelty has been introduced by Rowther et al. (2024), who moved the focus from the disc internal energy balance to the influence of the stellar irradiation. While previous models concentrate on how the disc manages heat generated internally, this study employs the SPH code phantom coupled with the radiative transfer code mcfost to perform live Monte-Carlo Radiative Transfer (MCRT) simulations. This approach allows for the fundamental inclusion of stellar irradiation. In this scenario, the disc thermodynamics is no longer dictated solely by GI self-regulation, but is instead dominated by the radiative flux from the central star. This irradiation imposes a “thermal floor” that is often significantly higher than the heat generated locally by GI-induced shocks. This external heating breaks the feedback loop required for gravitational instability to self-regulate, suppressing or entirely preventing the formation of spiral arms in the disc. This approach comes at an incredibly high computational cost. A summary of the three aforementioned thermodynamic implementations is given in Table 1.

Before (Rowther et al., 2024), the role of stellar irradiation have been investigated extensively. Rice et al. (2011) were the first to systematically investigate the fragmentation boundary in irradiated self-gravitating discs, showing that irradiation weakens disc self-gravity and that the critical for fragmentation decreases as irradiation increases, meaning that strongly irradiated discs require more vigorous cooling to fragment. Building on this, Forgan and Rice (2011) extended the Jeans mass formalism to irradiated discs, demonstrating that the addition of stellar irradiation raises the equilibrium disc mass required for self-gravity to operate at a given accretion rate, and shifts the region of parameter space where fragmentation is expected to occur. Building upon those works, Haworth et al. (2020) and Cadman et al. (2020) extended the investigation of irradiation effects to explore how the susceptibility of a disc to gravitational instability and fragmentation depends on the mass of the host star. They found that discs become increasingly prone to the effects of fragmentation as the stellar mass increases. This happens because, at fixed Toomre and disc-to-star mass ratio, discs around more massive (and hence more luminous) stars are hotter, requiring higher heating and cooling rates to sustain energy balance. Since the effective scales directly with this cooling rate, this results in a higher or discs around high-mass stars. In the optically thin limit, stellar irradiation dramatically stabilises discs around low-mass stars. However, even in this strongly irradiated regime, discs around stars remain prone to fragmentation, preferentially producing wide-orbit giant planets and brown dwarfs. A complementary consequence of this result is that low-mass stars may support high disc-to-star mass ratios without triggering GI, providing large mass reservoirs for planet formation.

A complementary perspective on the role of stellar irradiation is offered by Leedham et al. (2025), who address a more fundamental question: how does the mathematical implementation of irradiation heating affect the outcome of GI simulations? Working within the framework of 2D local shearing-sheet simulations with -cooling, they compare two prescriptions for the irradiation heating term, a constant rate per unit mass versus per unit area of the disc. This distinction, rooted in whether the disc is optically thin or thick to the incident stellar flux, has profound consequences for the disc stability. In the former case, overdense regions are preferentially heated, possibly suppressing GI, in line with the earlier findings of Rice et al. (2011). In the latter case, gravitoturbulence persists even at high irradiation levels, with the disc reaching a self-regulated state at values of , well above the classical threshold. Crucially, fragmentation remains possible in this highly irradiated, optically thick regime, challenging the conventional expectation that strongly irradiated outer discs are stable against collapse.

A related but distinct aspect of irradiation heating concerns its effect on the linear stability of the disc. Lin and Kratter (2016) performed a stability analysis of discs subject to external heating, demonstrating the existence of growing modes at scales below the thermal Jeans length , a regime not captured by standard GI theory. Löhnert et al. (2020) confirmed these growth rates through 2D hydrodynamical simulations of externally irradiated discs, finding significant mode growth even when . Crucially, in the saturated state, this instability drives angular momentum transport corresponding to an effective , even for Toomre parameters as high as .

Taken together, these studies underscore the central role of thermodynamics in shaping the outcome of gravitational instability. While the cooling prescription remains a simplified approximation, it continues to provide an invaluable conceptual framework, isolating the fundamental physics of gravito-turbulence before the added complexity of realistic radiative transfer is introduced.

3.4 GI and planets: migration, ejection and interaction with spiral arms

Gravitational instability has the potential to create embedded companions, either through the direct collapse of the dust component (Rice et al., 2006; Longarini et al., 2023a), forming super-Earth mass objects, or through gas fragmentation (Boss, 1997), forming Jupiter-like planets and brown dwarfs. However, the fate of these bodies within a massive, strongly perturbed disc remains an open question.

Planet migration in self-gravitating discs is fundamentally different from its counterpart in standard, non-self-gravitating discs. In massive discs where GI is active, the gravitational stresses associated with spiral density waves generate strong, time-varying torques on any embedded companion. These stochastic torques, arising from the perturbed density field of the disc, can significantly alter the migration rate and direction compared to the smooth, laminar predictions of classical Type I and Type II migration theory (Paardekooper et al., 2023; Benitez-Llambay, 2026). The first systematic numerical studies of planet migration in self-gravitating discs were carried out by Baruteau et al. (2011) and Malik et al. (2015), using two- and three-dimensional simulations with a globally constant cooling parameter . Their results showed that planets of a wide range of masses migrate inward extremely rapidly, with migration rates well in excess of classical Type I predictions. More strikingly, gap opening, the standard mechanism by which massive planets transition to the slower Type II migration regime, was found to be severely inhibited, even for substellar companions as massive as .

A key limitation of these studies is the use of a globally constant . This prescription forces the disc to be gravitationally unstable everywhere, generating spiral structure throughout its entire radial extent. This is, however, not expected from physically motivated models of protostellar discs, where the cooling efficiency decreases strongly toward the inner regions due to the increasing optical depth. As a consequence, real self-gravitating discs are expected to be gravitationally unstable only in their outer parts, with the inner disc remaining in a stable, hotter state (Rafikov, 2005; Clarke, 2009; Kratter and Lodato, 2016). Rowther and Meru (2020) addressed this issue by implementing a radially varying cooling parameter which ensures that the cooling timescale in the inner disc is long enough to prevent gravitational instability, while the outer disc remains unstable and develops the spiral structures. In this setup, planets with different initial masses, ranging from to initially undergo strong Type I migration, consistent with Baruteau et al. (2011). However, when they reach the gravitationally stable inner region, migration significantly slows down, without requiring the planet to open a gap (Rowther and Meru, 2020). This happens because in the outer turbulent region, strong asymmetric density fluctuations produce a negative net torque on the planet, driving rapid inward migration. Conversely, in the inner, stable regions, the density field is much smoother and the corotation torque becomes more symmetric, reducing the net torque and allowing the planet to reach a quasi-stationary state. These results indicate that the thermodynamics of the disc plays a fundamental role in determining the migration rate of planets embedded in self-gravitating discs.

Nevertheless, the interaction between an embedded planet and a self-gravitating disc is not unidirectional. While the disc torques drive planetary migration, the planet itself can significantly alter the structure of the disc. Rowther et al. (2020) showed that an embedded giant planet can suppress the spiral structure driven by gravitational instability. As the planet interacts with the disc, the spiral density waves it generates propagate radially and create shocks, depositing energy and heating in the surrounding gas. This increase in sound speed raises the Toomre parameter above the instability threshold, causing the disc to become gravitationally stable even though its mass has not changed. The net result is a disc that, despite being massive enough to be self-gravitating in the absence of a planet, appears largely axisymmetric with a ring and gap structure in continuum observations. This has a profound implication for the interpretation of ALMA observations, since spiral arms are often considered the smoking gun of gravitational instability (Rowther et al., 2023). A massive disc hosting a giant planet may therefore appear gravitationally stable, making the two scenarios observationally degenerate, as shown in figure Figure 5.

FIGURE 5

Planet migration is not the only physical process occurring to bodies formed or embedded in self-gravitating discs. Indeed, the fate of the planetary objects is governed by a competition between several timescales: the gaseous collapse timescale, the grain growth and settling timescales, the migration timescale, and the mass loss timescale due to tidal stripping. This framework, known as tidal downsizing (Nayakshin, 2010), predicts a wide zoo of final objects depending on how these timescales are ordered. Forgan and Rice (2013), Forgan et al. (2018a) explored this scenario through a population synthesis model that incorporates fragment-fragment gravitational interactions during the disc phase and tidal downsizing (see also the recent DIPSY model by Schib et al., 2025). Their key finding is that dynamical scattering between fragments dominates over migration in shaping the final population: typical systems produced by GI consist of one or two massive objects at large separations, with a small fraction of terrestrial-type bodies and a significant population of ejected free-floating planetary mass objects (Nayakshin et al., 2026). In addition, the chemical composition of GI fragments is also expected to deviate significantly from the bulk disc material. Ilee et al. (2017) showed that fragments can undergo a “cold start” phase after formation, during which temperatures are not high enough to fully desorb volatile ices from grain surfaces. If dust sedimentation occurs on a comparable timescale, solid cores can form with a C/O ratio that differs from that of the local disc, meaning that the atmospheric composition of planets formed via GI does not necessarily reflect the bulk composition of the disc from which they formed.

3.5 GI and infall

Star and planet formation is the result of a complex, multi-scale process involving gravitational collapse and angular momentum transport. The transport of material from the large scales of molecular cloud cores down to the smaller scales of protostellar discs is a fundamental aspect of the earliest phases of disc assembly. In this context, the disc cannot be treated as an isolated system; rather, its evolution is strongly linked to the interaction with the parental envelope through mass transfer. Consequently, it is of great interest to understand the regime where gravitational instability is triggered and sustained by infall, rather than being purely regulated by cooling. This “infall-driven” GI is expected to be the dominant mechanism during the Class 0 and I phases, where the rapid mass infall rates can drive the disc toward instability long before secular cooling processes become significant. The connection between infall and gravitational instability was first explored in a series of numerical studies by Vorobyov and Basu (2005a), Vorobyov and Basu (2005b), Vorobyov and Basu (2007); Kratter et al. (2008), (2010), who modelled the self-consistent collapse of rotating molecular cloud cores into protostellar discs. These simulations showed that continuous mass loading from the envelope drives the disc into recurrent gravitational instability, leading to fragmentation and the inward migration of dense clumps, which generate episodic accretion bursts analogous to those observed in FU Ori-type objects.

Significant efforts have been made in the last few years to study the formation of protoplanetary discs in simulations of molecular cloud collapse. Kuffmeier et al. (2018) used large-scale “zoom-in” simulations to demonstrate that discs formed during the collapse of molecular cloud cores are naturally prone to gravitational instability. Their results show that the primordial infall of material from the envelope frequently drives the disc into an unstable regime, leading to episodic accretion bursts. These bursts are a fundamental mechanism for processing the rapidly accumulating mass, providing a possible physical explanation for the observed variability in young protostellar objects. This picture is further refined by the recent work of Lebreuilly et al. (2024), who investigated the early “build-up” phase of protoplanetary discs. Their models indicate that during the first years of disc assembly, these systems are exceptionally massive, often reaching a disc-to-star mass ratio . In this extreme regime, discs are inherently prone to gravitational instability and potentially to early fragmentation. However, the authors demonstrate that radiative feedback plays a crucial role in regulating this process. By accounting for the heat generated by accretion onto the protostar (namely, accretion luminosity), the disc temperature increases, acting as a powerful stabiliser that significantly reduces the occurrence of fragmentation compared to simpler barotropic models.

Building on this framework, Longarini et al. (2025b), following the works of Kratter et al. (2008), (2010), Zhu et al. (2012), formalised the concept of “infall-driven” gravitational instability, highlighting a fundamental self-regulation mechanism. Unlike isolated cooling-driven discs, where the stability is primarily governed by the cooling timescale , infall-driven discs are regulated by the rate of mass transferred from the envelope. In this scenario, the disc self-adjusts to a state of marginal stability by balancing the incoming mass with GI-induced angular momentum transport, redistributing it between the central star and the outer disc. The resulting effective viscosity and the amplitude of the spiral structures are thus dictated by the infall rate rather than local thermodynamics. We underline that there is an important underlying hypothesis, that is the locally isothermal equation of state. We expect that in a scenario where also temperature can evolve, the temperature regulation should also play a role in the infall-driven scenario. This self-regulation ensures that the disc can efficiently process the material injected from the large-scale environment, maintaining its stability throughout the high-mass-accretion phase of Class 0 and I sources.

Infall from the environment is known to produce misalignments and warps in protostellar discs (Kuffmeier et al., 2021). Rowther et al. (2022) studied the interplay between warps and gravitational instability, showing that as a warp propagates through a self-gravitating disc it heats the gas, raising the Toomre parameter above the instability threshold and suppressing the spiral structure entirely. The disc consequently appears axisymmetric in continuum observations, providing yet another mechanism, alongside planet-disc interactions, by which the signatures of GI can be hidden in young, massive discs.

These results signal a growing consensus that the environment surrounding a young disc cannot be ignored. Infall and gravitational instability are not independent processes, but act together in shaping the early disc structure and the condition for planet formation.

4 Observational advances on GI

4.1 Dynamical masses

Throughout the paper, we stressed how important is the disc mass to assess whether the disc is gravitationally unstable. However, Weighing protoplanetary discs is challenging (Miotello et al., 2023) because molecular hydrogen (), that is the main gaseous component, is invisible in such cold environments. Consequently, alternative tracers must be used to probe the disc mass. Common gas tracers include CO isotopologues (Miotello et al., 2017), which are chemically simple and among the most abundant molecules in discs. Another option is HD (e.g., Bergin et al., 2013), the closest analogue to , which provides a more direct probe of the total mass but the main transition is not currently covered by any operating facilities. Dust can also be used to infer the disc mass, under the assumption of optically thin emission and an assumed gas-to-dust ratio (e.g., 100 as in the ISM). However, all these methods rely on hypothesis regarding disc composition, optical depth, and tracer-to--conversion factors, which are often uncertain.

A powerful and more direct approach, free from these assumptions, has been developed in the last 5 years, and consists in estimating the disc mass dynamically. This method consists in thoroughly modelling the rotation curve of the disc (analogously to what done in galactic discs), including the self-gravitating contribution, in order to constrain its mass. Indeed, a gas particle at a distance from the central star will experience the Keplerian gravitational force of the star, and also the gravitational pull of the disc material within its orbit. In addition, the pressure gradient must be considered, slowing down the azimuthal gas rotation.

The first application of the model was done for the Elias 2–27 disc in Veronesi et al. (2021) and later refined in a series of papers. Here we present the latest model of the rotation curve of a protoplanetary disc, presented in Martire et al. (2024), as a generalisation of the vertically isothermal model presented in Lodato et al. (2023) and Veronesi et al. (2021). This method has been used to constrain dynamical disc masses of discs within MAPS and exoALMA sample (Martire et al., 2024; Longarini et al., 2025a), and benchmarked against numerical models (Veronesi et al., 2024; Andrews et al., 2024). According to the centrifugal balance equation, the rotation curve of a protoplanetary disc can be written aswhere the first contribution on the right hand side is the stellar one, namely, the Keplerian rotation at a given position in the disc, where is the stellar (Keplerian) gravitational potential. The second contribution corresponds to the self-gravitating contribution of the disc, where is the gravitational potential of the disc, proportional to the disc mass. Finally, the third contribution is the pressure gradient, where is the gas pressure.

An example of the fitting procedure is shown in figure Figure 6. This method has proven highly effective in providing dynamical mass estimates with relatively small uncertainties. Indeed, stellar masses are retrieved with a typical uncertainty of . It is important to emphasize that these stellar masses represent the most accurate estimates available to date, as all the different components contributing to the rotational profiles are taken into account. This marks a significant improvement compared to a simple Keplerian fit of the rotational data, which can result in discrepancies on the order of (Longarini et al., 2025a). As for disc masses, the typical uncertainty is of the order of (Veronesi et al., 2024; Andrews et al., 2024). It appears that the retrieval of the thermal structure is the dominant source of uncertainty for the disc mass, making it crucial to accurately characterise the 2D temperature structure. Pezzotta et al. (2025) employed seven molecular lines to retrieve the thermal structure of the HD 163296 disc, subsequently extracting the rotation curves for these same molecules to fit for the disc mass. Their findings demonstrate that increasing the number of molecular tracers leads to a more robust characterisation of the thermal structure, which in turn significantly reduces the uncertainty associated with the disc mass estimate.

FIGURE 6

Table 2 collects the 18 discs with available dynamical mass measurements. When comparing the masses of protoplanetary discs that can be probed with this dynamical method with standard continuum-based estimates, it is evident that the sample populates the very high end of both the disc and stellar mass distributions (Figure 7). Indeed, the dynamical mass estimates (orange dots) consistently correspond to the most massive systems within the broader Manara et al. (2023) PPVII sample (grey dots). Primarily, it is a physical limitation: the self-gravitating contribution becomes significantly detectable only in the most massive discs, where the disc-to-stellar mass ratio is high enough to induce measurable deviations from a purely Keplerian potential. Secondly, there is an observational limitation. The successful application of this method relies on high spatial and spectral resolutions. The target discs must be sufficiently extended and bright in molecular line emission to provide as many resolution elements as possible across the disc’s surface. These requirements naturally favour large, gas-rich systems around intermediate/high-mass stars, which are precisely the objects located at the very high end of the standard populations.

FIGURE 7

Dynamical masses and thermal structures of protoplanetary discs provide a direct means to assess the onset of gravitational instability. Computing the Toomre parameter requires knowledge of three key quantities: the midplane temperature profile, the surface density profile, and the stellar mass. It is worth noting that all three are model dependent. The thermal structure is inferred under the assumption that the functional form of the two-dimensional temperature profile is known (Galloway-Sprietsma et al., 2025), with the midplane temperature extrapolated from this function, since molecular line observations probe the ¹²CO and ¹³CO emitting layers rather than the midplane directly. The surface density profile is prescribed as a self-similar solution, meaning that the dynamical estimates of the disc and stellar mass depend on this choice. These assumptions are physically well motivated, but their cumulative effect means that the resulting Toomre parameter values should be interpreted with appropriate caution.

Table 2 collects the minimum Toomre parameter for the discs with available dynamical mass measurements. It is remarkable that the systems with the lowest values of , namely, Elias 2–27, GM Aur, IM Lup, and WaOph 6, are precisely those showing spiral structures, either in dust continuum emission (IM Lup, WaOph 6), molecular line emission (GM Aur), or both (Elias 2–27). The apparent outlier is HD 97048: pezzotta in prep note that this system hosts a known embedded protoplanet, as evidenced by kinematic signatures (Pinte et al., 2019). As discussed in Section 3.4, the interplay between an embedded planet and gravitational instability can suppress the spiral structure of the disc, hiding the signatures of the instability (Rowther et al., 2020).

4.2 The case of Elias 2–27

Elias 2–27 represents one of the most compelling targets for investigating gravitational instability (GI). Orbiting an M0-type star, its protoplanetary disc is one of the most massive observed to date. Initial estimates by Andrews et al. (2009), based on unresolved dust continuum emission, indicated a disc-to-star mass ratio of . Subsequently, Pérez et al. (2016) provided the first resolved millimeter observations of the system using ALMA (left panel Figure 8), revealing two large-scale spiral arms and confirming the high dust mass previously reported. The presence of these prominent spiral features, coupled with the exceptionally high disc-to-star mass ratio, suggests that gravitational instability is a primary physical mechanism operating within the system.

FIGURE 8

Meru et al. (2017) investigated the origin of the spiral structure through 3D SPH simulations, evaluating three distinct scenarios: an embedded Jovian planet (internal companion), a stellar-mass companion at a large separation (external companion), and gravitational instability (GI). Their results demonstrated that an internal companion alone cannot replicate the observed morphology, effectively ruling it out as the sole mechanism in Elias 2–27. Conversely, both the GI and external companion models successfully reproduced the spiral features. Given the high disc mass, the authors suggest that the disc might be gravitationally unstable or may have undergone fragmentation, with the resulting fragment acting as the external companion driving the spirals.

To further discriminate between these scenarios, Forgan et al. (2018b) mapped the positions of the spiral density maxima to evaluate the radial dependence of the pitch angle. While an external companion is expected to produce asymmetric spirals with a pitch angle that deviates from a constant value, the observed morphology of Elias 2–27 exhibits a high degree of symmetry. These symmetric features lend further support to the GI hypothesis. Furthermore, Hall et al. (2018) used synthetic ALMA imaging to demonstrate that GI-generated spirals are only observable within a narrow parameter space of low opacity and efficient cooling. They also noted that significant dust trapping is required to produce the observed emission contrast, implying that the visibility of these structures depends heavily on the coupling between gas and dust.

Recent ALMA observations from the DSHARP large program (Huang et al., 2018b) have provided unprecedented high-resolution images of the dust continuum in Elias 2–27. This work characterised the spiral morphology with exceptional clarity and confirmed the presence of a gap at , suggesting that a planetary companion may coexist with the mechanism driving the spiral arms. Similar symmetric spiral structures were also identified in two other DSHARP discs, IM Lup and WaOph 6. Kinematic analysis of molecular line emission by Pinte et al. (2020) revealed enhanced non-Keplerian signatures co-located with the Elias 2–27 gap; however, the system’s complex dynamics and the limited spectral resolution of the DSHARP data precluded definitive conclusions.

To address these limitations, Paneque-Carreño et al. (2021) presented high-sensitivity and high-spectral-resolution gas observations (right panel Figure 8). They identified global kinematic perturbations co-located with the dust spirals, which reminded of “GI wiggles” (Hall et al., 2020), providing further evidence for gravitational instability. Furthermore, Paneque-Carreño et al. (2021) reported a tentative detection of grain growth within the spiral arms, consistent with theoretical models of particle trapping (Dipierro et al., 2015). These kinematic data were later used by Veronesi et al. (2021) to perform the first rotation curve analysis aimed at constraining the dynamical mass of a protoplanetary disc. Their results yield a disc-to-star mass ratio of and a Toomre parameter of , which, while above the classical threshold, is considered sufficiently low to support GI as a plausible origin for the observed features.

Building upon these results, Longarini et al. (2024) applied the analytical framework for “GI wiggles” developed in Longarini et al. (2021) to constrain angular momentum transport in Elias 2–27. By adopting the dynamical disc mass determined by Veronesi et al. (2021), the authors were able to break the degeneracy between disc mass and viscosity, finding that an effective viscosity of accurately reproduces the amplitude of the observed kinematic perturbations. Furthermore, by assuming the self-similar surface density profile from Veronesi et al. (2021) and the temperature profile from Pérez et al. (2016), they calculated the predicted mass accretion rate onto the central star. The inferred value form the kinematic modelling is , and the observed value is . The remarkably good agreement between their model and the observed accretion rate suggests that the angular momentum transport driven by GI-induced spirals is sufficient to account for the stellar accretion in this system. Moreover, Longarini et al. (2024) point out that the disc-to-star mass ratio and the effective -viscosity are within the regime where dust particles can undergo direct gravitational collapse inside the gas spiral arms, provided the Stokes number is sufficiently large. The predicted mass of these dust fragments is on the order of , a range that is remarkably consistent with the mass estimate for a potential planetary companion inferred from the width of the observed dust gap.

Elias 2–27 is a significant example of how different signatures of gravitational instability line up in a self consistent and coherent way.

4.3 The case of AB Aur

AB Aurigae (AB Aur) is one of the most studied and debated systems in the context of gravitational instability. It is a Herbig Ae star of intermediate mass with an estimated age of 2.5–4.4 Myr (Garufi et al., 2024). Despite its relatively late evolutionary stage as a Class II young stellar object, its disc is exceptionally rich in gas and shows a complex morphology, making it one of the most compelling targets for studying the connection between disc self-gravity and planet formation.

The disc around AB Aur has been studied extensively across a wide range of wavelengths. Early near-infrared observations revealed prominent large-scale spiral arms in scattered light (Fukagawa et al., 2004), subsequently confirmed and characterised with ALMA (Boccaletti et al., 2020). The system also harbours a large central dust cavity as well as dust asymmetry (Tang et al., 2017). The origin of these substructures has been the subject of considerable debate, with proposed explanations including planetary companions, central binary, and gravitational instability. Currie et al. (2022) used the Subaru Telescope and the Hubble Space Telescope to find evidence for a Jovian protoplanet, AB Aur b, orbiting at a wide projected separation of au, possibly responsible for multiple planet-induced features in the disc. Cadman et al. (2021) assessed the formation history of this candidate planet, finding that core accretion timescales for such a massive object at wide separation typically exceed the system age, while gravitational instability, which operates on timescales of – yr, remains a viable formation channel. Using SPH simulations, they estimated a critical disc mass for fragmentation of .

The most direct kinematic evidence for gravitational instability in AB Aur was presented by Speedie et al. (2024), as shown in Figure 9. Using deep ALMA observations of ¹³CO and O line emission, they identified kinematic signatures strongly resembling the predictions of GI simulations and analytical models (Hall et al., 2020; Longarini et al., 2021). From quantitative comparisons, they inferred a disc mass of up to one third of the stellar mass enclosed within the observed region, placing AB Aur firmly in the regime where self-gravity cannot be neglected. The observed kinematic perturbations are consistent with the large-scale GI wiggle (Longarini et al., 2021), and are distinct from the localised signatures expected from planet-disc interactions alone. Independent constraints on the disc mass come from Stapper et al. (2024), who used CO isotopologue observations of a large sample of Herbig discs to estimate their gas masses via thermochemical modelling, finding the AB Aur disc mass to lie at the upper boundary of the model grid, consistent with a self-gravitating regime.

FIGURE 9

However, AB Aur is not an isolated system. Speedie et al. (2025) used deep ALMA ¹²CO and SO observations to map the large-scale kinematics. By separating disc-like from non-disc-like motions, they identified three large-scale streamers extending to au in radius, interpreted as late infall from the surrounding environment. Where these infalling structures merge with the disc, a brightness enhancement in SO emission, a known shock tracer, is detected at the azimuthal average, providing direct evidence of shocks at the streamer-disc interface. This picture, in which late infall replenishes the disc, triggers or sustains gravitational instability, and modifies the conditions for planet formation, connects AB Aur directly to the infall-driven GI framework discussed in Section 3.5.

4.4 The case of IM Lup

IM Lup is a young pre-main sequence star ( Myr) located in the Lupus star-forming region. It hosts an unusually large disc, extending out to au in the dust continuum and out to au in the gas (Cleeves et al., 2016), with a dynamical stellar mass of approximately a solar mass. The dust continuum emission shows clear evidence of a spiral morphology, which may be triggered by gravitational instability (Huang et al., 2018b). Cleeves et al. (2016) first estimated the disc mass from millimetre visibilities, finding a massive disc of . Verrios et al. (2022) proposed instead that the spiral structure could be generated by an embedded protoplanet. Through SPH simulations of planet-disc interactions, post-processed to reproduce CO, dust, and scattered light emission, they found that a high disc mass of is nonetheless required to match the scattered light image, so that sub-micron sized grains remain well coupled in the upper disc layers.

A more direct constraint on the disc mass was obtained by Lodato et al. (2023) and Martire et al. (2024), who applied rotation curve modelling to the molecular line kinematics of IM Lup. Their analysis yields a disc-to-star mass ratio of , and a minimum Toomre parameter of , close enough to unity to place IM Lup in a regime where gravitational instability is plausible, even if not guaranteed. Unlike Elias 2–27 and AB Aur, the kinematics of IM Lup do not show evident non-Keplerian perturbations, making it difficult to conclusively distinguish between gravitational instability and planet-driven spirals as the origin of the observed substructure.

This degeneracy could in principle be broken by studying the time evolution of the spiral pattern. Spiral arms induced by a companion are long-lived, coherent structures that rotate at the orbital angular velocity of the companion, maintaining a constant pattern speed over time. In contrast, spirals generated by gravitational instability are transient: their instantaneous pattern speed is set by the local Keplerian velocity of the disc material, meaning that, if maintained, they would rapidly wind up due to differential rotation (Cossins et al., 2009). Instead, GI spirals continuously form, evolve, and dissipate on dynamical timescales. Monitoring the temporal evolution of the spiral morphology in IM Lup, for example, through multi-epoch ALMA observations, could therefore provide a powerful discriminant between the two scenarios. Yoshida et al. (2025) identified a winding motion of the spiral arms in dust continuum emission (see Figure 10), that is consistent with the Keplerian rotation of the disc, in line with the predictions of gravitationally unstable discs.

FIGURE 10

5 Conclusion and future perspectives

Over the past decade, the study of gravitational instability in protoplanetary discs has undergone a profound transformation. The advent of ALMA has provided, for the first time, the angular resolution and sensitivity required to spatially resolve the kinematic and morphological signatures of self-gravity in individual discs. Simultaneously, advances in numerical algorithms and computational resources have enabled hydrodynamical simulations of unprecedented level of detail, incorporating increasingly sophisticated treatments of thermodynamics, dust dynamics, and the coupling between large-scale infall and disc evolution. Together, these developments have shifted the field from a largely theoretical exercise, where gravitational instability was invoked primarily as a formation mechanism for giant planets and brown dwarfs, to an observationally grounded research area where the physical conditions for GI can be directly measured, tested, and compared against theoretical predictions with quantitative precision.

On the theoretical side, significant progress has been made across several fronts. The long-standing debate over the critical cooling timescale for fragmentation has been resolved (Deng et al., 2017), recovering the original value of proposed by Gammie (2001) and establishing a firm upper limit on the angular momentum transport efficiency of gravito-turbulence. The role of thermodynamics has been greatly refined, moving beyond the idealised -cooling prescription toward simulations incorporating local radiative transfer approximations (Young et al., 2024), full radiation transport (Xu et al., 2025a), and live Monte Carlo radiative transfer (Rowther et al., 2024). These studies have clarified the central role of stellar irradiation in regulating, and in some cases suppressing, the self-regulated state of GI (Cadman et al., 2020; Leedham et al., 2025). Building on the pioneering work of Rice et al. (2004), (2006), the connection between gravitational instability and the solid component of the disc has been substantially advanced: GI-induced spiral arms act as efficient dust traps, concentrating solid material and triggering the direct gravitational collapse of the dust layer into planetary-mass cores (Longarini et al., 2023a; Baehr et al., 2022; Rice et al., 2025), establishing gravitational instability as a viable mechanism for the rapid formation of planetary cores in the earliest stages of disc evolution. Finally, the picture of GI as a process driven solely by internal cooling has given way to a more complete framework in which infall from the parental envelope plays a central role (Kuffmeier et al., 2018), sustaining gravitational instability over extended timescales and establishing a tight coupling between disc mass, stellar mass, and accretion rate (Longarini et al., 2025b).

On the observational side, it is worth noting that gravitationally unstable discs do not represent the majority of systems observed by ALMA. This is partly a selection effect: the bulk of high-resolution ALMA surveys (Andrews et al., 2018) have preferentially targeted relatively evolved, Class II discs, with typical ages exceeding 1 Myr, at which point the disc mass has already decreased substantially through accretion onto the central star and the role of gravitational instability is expected to be marginal (Tobin et al., 2020). The youngest, most massive discs, Class 0 and Class I sources, where GI should be at play, remain comparatively under-sampled at high angular resolution, and even when observed are often subject to severe dust optical depth effects, which can hide spiral substructure even in discs that are young and massive enough to be gravitationally unstable (Ohashi et al., 2023). At the same time, we acknowledge that this observational bias cannot fully account for the absence of spiral structure in some of the youngest, most deeply embedded and massive systems that have been imaged at sufficient resolution. In these cases, the lack of observed spirals may instead reflect genuine thermodynamic stabilisation of the disc.

Nevertheless, within this sample, the dynamical mass measurement technique has allowed us to identify a population of exceptionally massive discs whose kinematics deviate measurably from pure Keplerian rotation (Veronesi et al., 2021; Martire et al., 2024; Longarini et al., 2025a). For the most massive systems, spatially resolved kinematic perturbations consistent with GI-induced spiral structure have been detected, providing observational evidence that gravitational instability is an active physical process in these discs. Three systems, Elias 2–27, AB Aur, and IM Lup, stand out as particularly compelling cases, where multiple independent lines of evidence, including spiral morphology, non-Keplerian kinematics, and high disc-to-star mass ratios, converge toward a coherent GI picture.

5.1 Future perspectives

Despite the significant progress reviewed here, several fundamental questions remain open. In the following, we outline the most pressing theoretical and observational challenges that we expect will drive the field forward in the coming years.

5.1.1 Theoretical perspective

5.1.2 Observational perspective

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