Revisiting gravitational instability in protostellar discs with improved radiative cooling models

The paper Revisiting gravitational instability in protostellar discs with improved radiative cooling models by Alison K. Young, Ken Rice, Richard Booth, and Farzana Meru investigates the conditions under which the gravitational instability (GI) is active in young protostellar discs, using an improved method to approximate radiative cooling within hydrodynamics simulations.

The authors note that young discs are expected to be significantly more massive than those observed at > 1 Myr, and planet formation likely begins during this early stage. Such massive discs may be susceptible to GI, so determining the disc and stellar properties for which GI is active is important for understanding early disc evolution and planet formation. Prior work was limited by model assumptions and inaccuracies in the thermodynamics of protostellar discs.

The study uses a modified version of the SPH code phantom, combining a radiative cooling approximation with flux-limited diffusion (FLD) radiative transfer. The key improvement is the modified Lombardi method for estimating the pseudo-mean column density, which provides more accurate estimates of the optical depth and therefore the radiative cooling/heating rate compared to earlier methods (Stamatellos, Lombardi, and combined methods). The authors explore a wide parameter space, including stellar masses from 0.1 to 1.0 M⊙, disc-to-star mass ratios from 0.1 to 1.4, and disc outer radii of 50, 100, and 200 au, with stellar irradiation included.

The main results are:

1. Differences from earlier simulations: The parameters for which discs form spirals and fragment differ from those obtained with earlier methods. The Stamatellos method overestimates the optical depth, leading to cooler mid-planes and making discs more susceptible to GI. With the more accurate modified Lombardi method, discs remain stable for higher disc-to-star mass ratios.

2. Fragmentation in compact discs: The outer regions of discs with radii of 50 au may be susceptible to fragmentation, meaning GI-driven planet formation is not restricted to only the most extended discs. Fragmentation occurs for Md/M∗ ≳ 0.4 in 50 au discs, with fragments forming between 20-30 au (in the region of Uranus and Neptune's orbits in our solar system). More extended discs (100 and 200 au) fragmented only if Md/M∗ ≳ 0.5.

3. Stabilising effect of stellar irradiation: The additional thermal support due to stellar irradiation increases the disc mass that remains stable against GI. Discs may reach up to ≳ 0.4 M∗ without fragmenting, providing a considerable quantity of material for building planets. For stars of < 0.5 M⊙, 200 au discs can support discs of the same mass as the star without significant development of GI.

4. Spiral morphology: Large-scale spiral arms only developed for M∗ ≲ 0.3 M⊙, except in the most compact discs. The long-lived spiral structures that form tend to be flocculent and compact, indicating that large-scale spiral arms should not be considered a typical outcome of GI. Most spirals are faint, higher-order (m > 2) structures, which may be difficult to observe.

5. Self-regulation: Some discs show evidence of a self-regulated state, with the gravitational stress fluctuating in a zig-zag pattern indicative of the disc heating and cooling around Q ∼ 1. Long-lived spirals can persist for at least 10-20 outer rotation periods in weakly irradiated discs.

6. Accretion rates: Discs with faint and strong spirals have higher accretion rates than axisymmetric discs, with quasi-steady self-gravitating discs expected to have accretion rates of at least 10−7 M⊙ yr−1.

The authors conclude that because massive discs around the youngest protostars may be stable, there is a large reservoir of planetary building material with the potential for forming massive planets quickly. Since large disc masses and low stellar luminosities are required for GI to develop, GI is expected to play a key role in driving the evolution of the youngest discs (< 1 Myr old). The paper recommends evolving simulations for at least 10 dynamical timescales to capture the long-term state, and suggests that the modified Lombardi method is a suitable radiative cooling approximation for future work.

Improvements to AI Systems:

1. Enhanced Radiative Cooling Estimation in Simulation Codes

• Integrate the modified Lombardi method into AI-driven hydrodynamic simulation frameworks (e.g., SPH or grid-based codes) to replace outdated approximations (Stamatellos, Lombardi, combined).

• The improved AI system can automatically compute pseudo-mean column densities and optical depths with higher accuracy, leading to more realistic disc thermodynamics and stability predictions.

2. Predictive Parameter-Space Mapping for Gravitational Instability

• Train a machine learning model on the simulation results (stellar mass, disc-to-star mass ratio, outer radius, irradiation) to predict the onset of spiral formation and fragmentation.

• The improved AI can rapidly classify whether a given protostellar disc will be stable, self-regulated, or fragmenting—without running expensive full simulations—enabling large-scale parameter sweeps and observational target selection.

3. Real-Time Disc Evolution Forecasting

• Use the self-regulation and accretion rate findings (e.g., quasi-steady accretion ≥ 10−7 M⊙ yr−1) to build a surrogate model that forecasts disc mass evolution, spiral lifetime, and fragment survival over 10–20 outer rotation periods.

• The AI can then provide early warnings for fragment formation in compact discs (e.g., 50 au, Md/M∗ ≥ 0.4) and estimate the mass reservoir available for planet formation.

4. Observational Signature Simulator

• Incorporate the result that large-scale spirals are rare (only for M∗ ≤ 0.3 M⊙) and that most spirals are faint, higher-order (m > 2) structures into an AI-based image generator.

• The improved system can synthesize realistic ALMA or JWST observations of young discs, helping astronomers distinguish GI-driven spirals from other phenomena (e.g., planet–disc interactions) and test detection limits.

5. Optimized Simulation Time-Stepping

• Implement an adaptive AI controller that decides when to stop or extend simulations based on the self-regulated zig-zag pattern of gravitational stress (Q 1).

• The AI can automatically extend runs to at least 10 dynamical timescales, as recommended, and terminate early only when stability is conclusively reached, saving computational resources.

6. Stellar Irradiation Effect Calculator

• Build a neural network that quantifies the stabilizing effect of stellar irradiation on disc mass limits (e.g., up to 0.4 M∗ for 0.5 M⊙ stars, or equal-mass discs for 200 au around < 0.5 M⊙ stars).

• This AI can be used in population synthesis models to predict which young stars are likely to host massive, GI-active discs and thus have high planet-formation potential.

Young discs are expected to be significantly more massive than those observed at >1 Myr and it is at this earliest stage that planet formation likely begins. Such massive discs may be susceptible to the gravitational instability (GI), therefore we need to determine the disc and stellar properties for which the GI is active to understand its role in early disc evolution and planet formation. Prior work has been limited by model assumptions and inaccuracies due to the complex nature of the thermodynamics of protostellar discs so we now revisit this question using an improved method to approximate radiative cooling within hydrodynamics simulations. We have explored a wide parameter space, representative of young protostellar discs of 0.1 to 1 M and include irradiation from the host star. The parameters for which discs form spirals and fragment were found to differ to those obtained from earlier simulations. The outer regions of discs with radii of 50 au may be susceptible to fragmentation, meaning that GI-driven planet formation is not restricted to only the most extended discs. The additional thermal support due to stellar irradiation increases the disc mass that remains stable against GI: discs may reach up to 0.4 M* without fragmenting, providing a considerable quantity of material for building planets. Large scale spiral arms only developed for M* 0.3 M, except in the most compact discs. Furthermore, the long-lived spiral structures that form tend to be flocculent and compact, indicating that large-scale spiral arms should not be considered a typical outcome of GI.