The dispersal of compact protoplanetary discs

Technical Summary: The Dispersal of Compact Protoplanetary Discs

Problem Statement
Protoplanetary discs are the sites of planet formation, yet their dispersal mechanisms remain a subject of active investigation. Observational evidence, particularly from colour-colour evolution studies of young stellar populations, indicates that disc dispersal proceeds preferentially from the inside-out, creating a central cavity before the outer disc is eroded. However, standard theoretical models often assume discs with large outer radii (hundreds of astronomical units), well beyond the gravitational radius ($R_g$) where photoevaporation is most efficient. Recent surveys (e.g., ALMA) reveal that a substantial fraction of discs are "compact," with outer radii comparable to or smaller than $R_g$. Existing models, which typically assume unconstrained mass-loss profiles extending to large radii, fail to accurately reproduce the evolutionary tracks and dispersal timescales of these compact systems. Specifically, standard prescriptions often predict outside-in erosion for compact discs, contradicting observational diagnostics.

Methodology
The authors employed a multi-stage approach combining high-resolution radiation-hydrodynamic simulations with 1D viscous population synthesis models:

1. Radiation-Hydrodynamic Simulations: Using the PLUTO code coupled with the MOCASSIN radiative transfer framework, the authors simulated X-ray-driven photoevaporation for a $1 M_\odot$ star. They explored four distinct initial disc cut-off radii ($R_{cut}$): 10, 50, 100, and 200 au. The simulations tracked the thermal structure, density, and velocity fields until a quasi-steady state was reached. The gravitational radius was adopted as $R_g = 5$ au.
2. Derivation of Scaling Relations: From the hydrodynamic simulations, the authors analyzed the cumulative mass-loss rate distributions. They determined how the total wind mass-loss rate depends on the disc's outer cut-off radius.
3. Population Synthesis: These findings were incorporated into a 1D viscous evolution code (SPOCK). The authors performed a population synthesis of 4,096 synthetic discs using a Quasi-Monte Carlo sampling strategy (Sobol sequences) to explore a 9-dimensional parameter space (stellar mass, X-ray luminosity, initial disc mass, cut-off radius, viscosity, external FUV field, etc.).
4. Prescriptions Tested: The models compared two scenarios:
   • Standard Prescription: Internal photoevaporation mass-loss rates applied without regard to the disc's finite outer radius (effectively assuming the disc extends to infinity or beyond the mass-loss peak).
   • Cut-off Dependent Prescription: The mass-loss rate is truncated and normalized based on the specific cut-off radius of the disc, derived from the hydrodynamic scaling relations.
   • External Photoevaporation: Some models included mild external FUV photoevaporation (using the FRIED grid) to simulate nearby OB stars.

Key Contributions and Results

• Hydrodynamic Scaling: The simulations revealed that the surface mass-loss profiles are nearly independent of the disc size; the wind morphology and local mass-loss rates in the inner regions remain consistent regardless of the outer cut-off. However, the integrated total mass-loss rate is significantly reduced for compact discs because the wind is truncated at the cut-off radius. The cumulative mass-loss rate scales with radius, growing as $\dot{M}_w \propto R^3$ near the gravitational radius and flattening to $\dot{M}_w \propto R^{0.5}$ at larger radii.
• Evolutionary Tracks:
  • Standard Prescription: When applied to compact discs ($R_{cut} < 20-30$ au), the standard model predicts that the wind mass-loss rate exceeds the viscous spreading rate outside the cut-off radius. This leads to outside-in dispersal, where the outer edges are eroded first, contradicting observational evidence.
  • Cut-off Dependent Prescription: When the mass-loss rate is scaled to the cut-off radius, the total wind rate is reduced sufficiently to allow viscous spreading to dominate initially. This results in inside-out dispersal, where a gap opens near the gravitational radius and expands outward, consistent with observational diagnostics.
• Population Synthesis Outcomes:
  • The cut-off dependent prescription, combined with a reduced internal photoevaporation rate (scaled by a factor of 10 to align with recent chemical models), reproduces observed disc fractions and accretion rates as a function of cluster age without parameter tuning.
  • The median disc lifetime in the improved model is $2.70 \pm 0.13$ Myr, which aligns well with observational estimates ($2.24 \pm 0.35$ Myr) and is significantly longer than predictions from models without the rate reduction ($1.13 \pm 0.05$ Myr).
  • The inclusion of mild external photoevaporation does not prevent disc spreading when the cut-off radius prescription is used but results in a gentler increase in disc radii over time and a higher fraction of compact discs persisting to later ages.

Significance
The paper argues that the dispersal of compact protoplanetary discs cannot be accurately captured by standard, unscaled photoevaporation models. The authors propose a simple, physically motivated scaling relation that accounts for the finite outer radius of the disc. By incorporating this dependence, the models successfully reconcile theoretical predictions with the observed inside-out clearing of young stellar populations and the prevalence of compact discs. The study concludes that dispersal prescriptions must explicitly include the dependence on the disc cut-off radius to correctly predict the evolution and lifetime of compact discs, providing a necessary correction to current population synthesis models.