Critical Fragmentation in Protoplanetary Discs

• Critically fragmenting discs are gaseous structures where self-gravity, quick local cooling, and dust evolution combine to form bound clumps for potential substellar companion formation.
• Analytical models and 3D simulations reveal that disc fragmentation is highly sensitive to opacity, surface density profiles, and the efficiency of radiative cooling.
• Dust growth reduces opacity, enabling efficient cooling that lowers the critical mass and radius thresholds for planet formation in protoplanetary environments.

A critically fragmenting protoplanetary disc is a gaseous circumstellar structure in which the interplay of self-gravity, thermal physics, radiative cooling, and local mass distribution brings the system to the threshold of gravitational fragmentation. At this regime, the disc can rapidly break up into bound clumps capable of forming substellar companions (giant planets, brown dwarfs, or low-mass stars). The empirical and theoretical criteria for criticality, as well as the location, frequency, and evolution of this process, are highly sensitive to detailed disc properties (opacity, mass profile, turbulence, infall physics), the local thermodynamics, and the presence of dust growth or feedback processes.

1. Physical Criteria for Disc Fragmentation

Fragmentation in a protoplanetary disc is conventionally assessed via two principal requirements: global instability (typically measured by the Toomre Q parameter) and efficient local cooling.

• Gravitational instability: The Toomre parameter $Q$ (for a Keplerian disc),

$Q = \frac{c_s \kappa}{\pi G \Sigma}$

where $c_s$ is the sound speed, $\kappa$ the epicyclic frequency (equal to the local orbital frequency $\Omega$ in Keplerian discs), $G$ the gravitational constant, and $\Sigma$ the surface mass density, must fall below a critical threshold ($Q \lesssim 1$) for local patches to become self-gravitating.

• Cooling requirement: The local cooling time $$t_\mathrm{cool} = u / (\mathrm{d}u_\mathrm{cool}/\mathrm{dt})$$ (where $u$ is the specific internal energy) must be shorter than a few local orbital periods to permit contraction before centrifugal and pressure forces disrupt the instability. This is parameterized by $\beta = t_\mathrm{cool}\Omega$. Critical values of $\beta$ for fragmentation are not universal; rather, they scale empirically with the local disc-to-star mass ratio and surface density,

$$\beta < \eta \left( \frac{\Sigma R_f^2}{M_*} \right)^\delta$$

where $R_f$ is the fragment formation radius, $\delta \approx 1/2$, and $\eta \approx 47$ (Meru et al., 2010).

• Surface density profile impact: The radial profile $\Sigma \propto R^{-p}$ determines where instability is most likely:
  • For shallow profiles ($p < 2$), the fragmentation condition $H/R \lesssim (\pi \Sigma_0 R_0^p / M_*) R^{2-p}$ is more easily satisfied at large $R$, favoring fragmentation in the outer disc.
  • For steeper profiles ($p \gtrsim 2$), the instability migrates inward, possibly producing fragments at small radii if the disc can cool rapidly enough.
• Angular momentum considerations: Discs formed from clouds with rotational-to-gravitational energy ratios, $$\lambda = E_\mathrm{rot}/E_\mathrm{grav} \gtrsim 5 \times 10^{-3}$$, generally reach sufficient radial extent for critical fragmentation, essentially providing a lower angular momentum bound on fragmenting systems (Forgan et al., 2011).

2. Role of Dust Growth and Opacity in Critical Fragmentation

The opacity of disc material, governed predominantly by the dust grain population, controls the radiative cooling rate and thus the critical properties for fragmentation. Dust evolution alters the Rosseland mean opacity $\kappa(T, a_\mathrm{max})$:

• Dust growth effect: As grains grow (from ISM-like, $a_\mathrm{max} \lesssim 10 \ \mu$m, to $a_\mathrm{max} \sim$ cm), the total cross-sectional area, and thus opacity, decreases (for fixed dust mass, area $\propto a^{-1}$). The drop in $\kappa$ can be more than an order of magnitude, substantially enhancing disc cooling (Lee et al., 11 Sep 2025).

$$\kappa(a_\mathrm{max}, T) = \begin{cases} \kappa_0 (T/155\,\mathrm{K})^{p_l} & T \leq 155~\mathrm{K} \ \kappa_0 & 155~\mathrm{K} < T < 353~\mathrm{K} \ \kappa_0 (T/353\,\mathrm{K})^{p_h} & T \geq 353~\mathrm{K} \end{cases}$$

where $\kappa_0, p_l, p_h$ depend on $a_\mathrm{max}$.

• Disc conditions for fragmentation: When opacity is reduced, the disc can remain marginally gravitationally unstable ($Q \sim 1$) and become capable of cooling efficiently at lower surface densities and cooler temperatures, allowing fragmentation at smaller disc radii (down to $\sim$30 au with sufficient dust growth) and lower overall disc masses and accretion rates.
• Analytical scaling: The critical surface density and accretion rate for fragmentation shift lower:

$$\dot{M}_\mathrm{frag} \propto T_\mathrm{frag}\Sigma_\mathrm{frag}$$

and the initial fragment mass (using $$M_\mathrm{frag} = 57 \Sigma_\mathrm{frag} H_\mathrm{frag}^2$$) can drop into the gas giant planet mass regime, in contrast to higher masses found in ISM-dust models (Lee et al., 11 Sep 2025).

3. Numerical Simulation Results and Key Fragmentation Outcomes

Global, three-dimensional SPH and grid-based simulations confirm and extend these analytical criteria:

• Fragmentation loci: For shallow surface density profiles and large, massive discs, fragmentation initiates in the outer disc (typically $R \sim 20{-}100$ au for $p=1{-}1.5$). For steeper profiles, the onset moves inward, but rapid cooling is harder to achieve, constraining practical fragmentation to outer disc regions (Meru et al., 2010).
• Fragment mass spectrum: Simulations produce a broad range (from $\sim$1 $M_\mathrm{Jup}$ up to the brown-dwarf/low-mass stellar regime), with masses scaling with local disc temperature and surface density. Dust opacity modifications shift the minimum fragment mass downward (Lee et al., 11 Sep 2025).
• Disc evolution post-fragmentation: Discs rapidly lose mass and contract as a result of fragmentation (from $\sim$0.25 $M_\odot$ and $\sim$100–150 au down to $0.002{-}0.07~M_\odot$ and $20{-}70$ au within $10^4$ yr) (Stamatellos et al., 2010). These truncated discs resemble observed compact discs in young systems, many of which may be post-fragmentation remnants.
• Fragment fate and migration: Fragments interact gravitationally, leading to scattering, mergers, ejections, and inward migration. Tidal stripping during migration can convert massive clumps into giant protoplanets at tens of au, sometimes causing FU-Orionis type accretion outbursts when the envelope is accreted by the star (Vorobyov et al., 2018).

4. Implications for Planet and Companion Formation

The predicted fragment masses, locations, and survival probability bear directly on the planetary initial mass function and formation pathways:

• Mass–radius relationship: With opacity reduction via grain growth, fragmentation can explain giant planet formation at $\sim$30 au in discs with realistic (i.e. relatively low) masses and accretion rates, bridging the gap between observations and classical theory (Lee et al., 11 Sep 2025).
• Efficiency and occurrence rate: Population synthesis studies—incorporating disc infall rates, heating from stellar accretion, and realistic size distributions—find fragmentation fractions spanning $\sim$0.1–11% of discs, with the fraction dominated by early disc size (determined by infall location) and accretion heating efficiency (Schib et al., 2022).
• Observational constraints: The rarity of massive, extended discs may reflect transient fragmentation events or formation suppression (e.g. through magnetic braking) (Stamatellos et al., 2010). Many observed compact discs—after accounting for underestimation bias in mass measurements due to opacity, dust temperature, and inclination—are consistent with initial conditions supportive of fragmentation (Dunham et al., 2014).
• Limiting factors: Strong radiative feedback from forming objects, especially continuous feedback, can suppress subsequent fragmentation; episodic feedback scenarios allow bursts of cooling and intermittent fragmentation, reducing the average fragment number per system (Mercer et al., 2016).

5. Limitations, Model Dependencies, and Future Directions

Significant uncertainties persist regarding precise fragmentation conditions and outcomes:

• Analytical models: Current semi-analytic treatments often adopt radial grain size variations and smooth surface density profiles, neglecting vertical structure, turbulence-driven diffusion, and localized dust concentration in spiral arms. Non-uniformities in opacity and dynamical feedback from dust back-reaction could further modulate the fragmentation threshold (Lee et al., 11 Sep 2025).
• Numerical limitations: Simulation outcomes depend critically on cooling prescriptions, resolution, and initial angular momentum (the latter controlling disc size and mass distribution). Environmental factors, such as infall geometry (regulated by magnetic or hydrodynamic transport) and episodic vs. continuous accretion, significantly alter the fragment frequency and mass spectrum (Schib et al., 2022, Whitworth et al., 2016, Forgan et al., 2011).
• Fragment survival and evolution: The ultimate fate of fragments (growth via gas accretion, tidal stripping, ejection, or merger) requires coupled simulation of hydrodynamics, grain dynamics, radiative processes, and N-body gravitational interactions. The post-formation mass function and architecture of bound companions thus remain open to model assumptions.
• Key future directions: Advances will require:
  • Simulations including dust growth, mobilization, and self-consistent radiative transfer feedback (Lee et al., 11 Sep 2025).
  • Improved observational statistics on early disc sizes and masses, including high-resolution kinematic and interferometric surveys (Schib et al., 2022).
  • Deeper understanding of turbulence, magnetic fields, and their control over disc thermal and dynamical evolution.

6. Astrophysical and Planetary System Implications

The revised fragmentation paradigm provided by dust evolution and realistic cooling physics implies that:

• Architectures of planetary systems: Gas giant planets formed via disc fragmentation may be more common at moderate distances (20–50 au) than previously assumed, impacting models for the origin of wide-separation giant planets and brown dwarfs.
• Remnant disc properties: Systems observed today with compact, low-mass discs and low stellar multiplicity may be the relics of earlier fragmentation phases.
• Formation of rocky planets and satellites: Prior to (or during) fragmentation, chemical evolution such as the preservation of prebiotic molecules on grains can imprint signatures on planetesimals and forming cores (Quénard et al., 2018).
• Exoplanet population synthesis: The criticality of observed discs, their ability to form fragments at smaller radii due to dust growth, and the resulting mass/orbit distribution of substellar companions have direct implications for interpreting planet demographics from direct imaging and microlensing surveys.

In conclusion, the critical fragmentation of protoplanetary discs is a highly non-local phenomenon governed by the complex interplay of local thermodynamics, dust microphysics, disc mass and profile, and central stellar properties. The inclusion of evolving dust opacity fundamentally alters where and how discs can fragment, lowering the barrier to giant planet formation at modest radii, and providing a nuanced theoretical underpinning for interpreting both observations and the diversity of observed exoplanetary systems [(Lee et al., 11 Sep 2025); (Meru et al., 2010); (Stamatellos et al., 2010); (Schib et al., 2022)].