Global Planet Formation Model

• Global planet formation model is a physics-based framework that integrates gas disk evolution, solids growth, and N-body dynamics to simulate planetary system formation.
• It incorporates modular sub-models for disk physics, core and envelope accretion, and orbital migration, enabling calibrated predictions of exoplanet properties.
• Population synthesis via Monte Carlo sampling produces statistical outcomes for planetary masses, radii, and orbital periods that are directly comparable to observations.

A global planet formation model is a comprehensive, physics-based meta-framework designed to synthesize the evolution of planetary systems from first principles, coupling all relevant physical processes in a protoplanetary disk to simulate the statistical outcomes observed among exoplanets and solar system planets. Such models integrate detailed prescriptions for the evolution of gas disks, solid evolution (dust, pebbles, and planetesimal formation), planetary growth (core and envelope accretion), migration, and N-body gravitational dynamics, thereby enabling direct population-wide predictions of planetary masses, radii, orbits, luminosities, and compositions that are statistically comparable to exoplanet survey results (Mordasini et al., 2014).

1. Physical Components and Sub-Models

Global planet formation models are modular meta-models constructed from specialized sub-models, each describing a core physical process involved in planet formation and evolution:

• Protoplanetary Disk Evolution: The viscous, irradiated, and photoevaporating gas disk is described via the time-dependent surface density $\Sigma(r,t)$ derived from the viscous diffusion equation:

$$\frac{\partial \Sigma}{\partial t} = \frac{3}{r} \frac{\partial}{\partial r} \left[ \sqrt{r} \frac{\partial}{\partial r} (\nu\Sigma\sqrt{r}) \right] - \dot{\Sigma}_{\rm w}(r) - \dot{\Sigma}_{\rm planet}(r)$$

Here, $\nu = \alpha c_s H$ with $\alpha \sim 10^{-4}-10^{-2}$ encoding turbulent viscosity, $c_s$ the sound speed, and $H$ the disk scale-height. Photoevaporation is modeled as EUV/FUV-driven mass loss, removing gas at defined radii; $\dot{\Sigma}_{\rm planet}$ accounts for planetary accretion of disk gas.

• Solids Disk Evolution: The solids-to-gas ratio $f_{D/G}$ (set by initial [Fe/H], often $f_{D/G} \simeq 0.01$) evolves via growth, drift, condensation, and planetesimal formation. Outside the iceline, $\Sigma_{\rm solid}$ is enhanced by a factor of $2-4$. Planetesimal surface density evolves by differential accretion and ejection:

$$\frac{d\Sigma_{\rm solid}}{dt} = - \frac{1}{2\pi a \Delta a} \dot{M}_{\rm acc} - \dot{M}_{\rm ej}$$

• Solid-Core Growth: Core mass $M_{\rm core}$ grows via oligarchic/runnaway accretion of planetesimals/pebbles:

$$\dot{M}_{\rm core} = \pi R_{\rm acc}^2 \Sigma_{\rm solid} P_{\rm coll} \Omega$$

where $R_{\rm acc}$, $P_{\rm coll}$ account for aerodynamic/gravitational focusing, and $\Omega$ is the local Keplerian frequency. $R_{\rm acc}$ can be substantially enhanced by the proto-envelope.

• Gas Envelope Accretion: Once $M_{\rm core}$ approaches the critical value ($\sim10$--$15\,M_\oplus$), slow Kelvin-Helmholtz contraction transitions to runaway gas accretion:

$$\tau_{\rm KH} \approx 10^c \left(\frac{M_{\rm core}}{M_\oplus}\right)^{-d}$$

($c\sim8$–10, $d\sim2$–3). The gas accretion rate is capped by the disk supply rate:

$$\dot{M}_{\rm gas} = {\rm min}\big[\dot{M}_{\rm KH},\,\dot{M}_{\rm disk}\big]$$

• Orbital Migration: Low-mass (Type I) and gap-opening (Type II) migration are calculated using analytic torque formulae, e.g., Paardekooper et al. (2010):

$$\Gamma_I = \Gamma_0 \left[2.5 - 1.7\beta + 0.1\alpha \right] / \gamma$$

where $\Gamma_0 \propto (q/h)^2\Sigma a^4\Omega^2$ and $\alpha = -d\ln\Sigma/d\ln r$, $\beta = -d\ln T/d\ln r$. Planets exceeding $20–50\,M_\oplus$ open gaps, transitioning to Type II migration at the disk viscous velocity.

• N-body Interactions: The model incorporates gravitational interactions among multiple embryos/planets, including migration, eccentricity and inclination damping, resonance trapping, perfect merging or fragmentation in collisions, and scattering.

2. Statistical Synthesis and Population Modeling

A defining feature of the global approach is planetary population synthesis: Monte Carlo sampling of initial disk and embryo parameters, followed by evolution of tens of thousands of independent systems under the same physics, to predict and calibrate statistical outcomes. Key modeling steps:

1. Initial Conditions: Disks are drawn from observed distributions of gas mass, metallicity, and lifetime. Embryos are seeded at randomized orbital radii.
2. Evolution: Each disk+planet system is integrated through $10^7$ years (formation phase) plus $10^9$ years (evolution phase).
3. Synthetic Observations: Models include survey-specific detection filters (e.g., RV, transit, microlensing selection effects) for direct and unbiased comparison with exoplanet census data.
4. Statistical Analysis: Model populations are compared to observations via mass functions, radius distributions, period diagrams, and occurrence rates, leading to quantitative constraints on uncertain theoretical parameters (e.g., turbulence $\alpha$, grain opacity factors, migration efficiency).

3. Major Successes and Observational Predictions

Global planet formation models have yielded several robust statistical predictions directly comparable to current exoplanet survey data:

• Planetary Mass Function: A power-law rise is predicted toward low masses, with a "desert" around $20$–$50\,M_\oplus$ (the gap where rapid envelope accretion initiates), and a high-mass cutoff. This reproduces observed exoplanet mass functions when $\tau_{\rm KH}$ and $\dot{M}_{\rm gas,max}$ are set to allow critical cores of $\sim 10$–$15\,M_\oplus$ (Mordasini et al., 2014).
• Mass–Radius Relation: Synthetic mass–radius diagrams show the full observed spread, from rocky planets with $R\sim R_\oplus$ to puffy sub-Neptunes ($2$–$4\,R_\oplus$) and gas giants ($R\sim R_J$). Matching the observed locus for Jupiters requires grain opacity reductions $f_\kappa\sim 10^{-3}$, consistent with microphysical growth models.
• Period Distributions: The interplay of migration regimes yields the "hot Jupiter" pile-up at $0.03$–$0.05$ AU, an inner edge at $\sim 0.01$ AU (disk/stellar truncation), and stalling of low-mass planets at convergence zones or disk edges. This agrees with period and multiplicity statistics from Kepler.
• Additional Observables: Models that include coupled evolution predict direct imaging luminosities (e.g., for $\beta\,$Pic b), radial velocity amplitudes, and atmospheric compositions, enabling detailed comparison with direct imaging, microlensing, and transmission spectroscopy data.

4. Model Extensions: Pebble Accretion, Fragmentation, and Dust-Gas Coupling

Recent developments have expanded global models to include additional physics crucial for planetary diversity:

• Pebble Accretion: Incorporation of pebble accretion (efficient capture of mm–cm solids) accelerates core growth, enabling the rapid formation of giant-planet cores at larger distances and shorter timescales, as demonstrated by population-synthesis studies (Guilera, 2015). Pebble isolation mass (the mass at which a core halts inward pebble flux) sets a key limit; its radial dependence controls the efficiency and spatial range of gas giant formation.
• Planetesimal Fragmentation and Drift: Two-bin treatments of planetesimal-fragment evolution and radial drift under gas drag reveal that fragmentation can both enhance (by increasing accretion cross-sections) and inhibit (by rapid loss of small fragments) core growth, particularly beyond the ice line (Kaufmann et al., 2023). Outcome depends critically on fragment size distribution and coupling to the gas.
• Dust and Migration Torques: Inclusion of dust- and pebble-induced torques modifies migration, particularly by promoting outward migration for low/intermediate-mass cores in dust-rich inner disks (Guilera et al., 27 Jan 2025). These effects depend on local dust-to-gas ratios, pebble drift and sublimation, and dynamical gas-dust feedbacks.

5. N-body/Post-Oligarchic Evolution and Architectural Diversity

The final dynamical architecture of planetary systems is determined both during and after the disk phase:

• Post-Gas N-body Evolution: After gas dispersal, planetary systems undergo gravitational instabilities, giant impacts, and orbital reconfiguration. High-resolution N-body integrations and validated semi-analytical models reproduce the observed multiplicity, mass dispersion, mutual spacing, and eccentricity distributions of super-Earth and sub-Neptune systems (Kimura et al., 5 Jul 2025).
• Solar System Analogs: Population synthesis frameworks identify the narrow disk and migration parameters that favor Solar System–like architectures: massive disks, low planetesimal sizes, suppressed Type I migration, and extended disk lifetimes. Only a small percentage of parameter space yields a giant planet exterior to 1.5 AU and dry inner terrestrials, matching key Solar System features (Ronco et al., 2017).

6. Open Challenges, Uncertainties, and Future Directions

Despite these advances, several key uncertainties and modeling limitations remain:

• Planetesimal Formation Physics: The mechanisms (streaming instability vs. turbulent coagulation) and initial size distributions decisively affect core-accretion timescales. Self-consistent, multi-bin coagulation–fragmentation models are needed to replace simplified prescriptions (Mordasini et al., 2014, Kaufmann et al., 2023).
• Pebble Accretion and Isolation: Quantitative modeling of pebble flux regulation, the transition from pebbles to planetesimals, and improved estimates of pebble isolation mass across a range of disk conditions remain outstanding issues (Guilera, 2015, Voelkel et al., 2020).
• Migration in Non-Ideal Disks: Migration prescriptions for magnetized, layered, or non-isothermal disks, and their impact on torque saturation, outward migration zones, and disk substructure (e.g., pressure traps) require further 3D MHD hydrodynamic calibration.
• Envelope Opacity and Chemistry: Full Smoluchowski models for grain growth and fragmentation in proto-envelopes, and coupling to disk chemistry/mineralogy, are required to predict bulk/atmospheric C/O ratios and final compositions.
• N-body Effects and Envelope Loss: Realistic outcomes for compact multi-planet systems depend on efficient N-body integrators, stochastic impact modeling, and self-consistent modeling of atmospheric escape, tidal interaction, and bloating during and after the dynamical phase.
• Circumbinary and Binary Environments: Fully coupled global models are actively being extended to planet formation in stellar binaries and circumbinary disks, which introduce dynamical barriers to growth, non-axisymmetric disks, and strong external forcing (Silsbee et al., 2021, Coleman et al., 2023).

7. Meta-Model Synthesis and Theoretical Significance

The global planet formation model serves as a meta-model: its predictive and diagnostic power depends on continual improvement in the component sub-models for disk physics, solids evolution, accretion processes, dynamical interactions, and time evolution. By enabling population-level forward modeling, global frameworks provide a rigorous means for connecting uncertain microphysical prescriptions to concrete statistical outcomes, integrating all available constraints from exoplanet demographics and Solar System chronology. Current models successfully reproduce a broad swath of observed exoplanet properties, validate key theoretical expectations, and provide context for the interpretation of next-generation exoplanet surveys. However, because the models encode multiple layers of theoretical uncertainty, further progress will rely on advances in disk structure, planetesimal formation, accretion physics, and N-body dynamics (Mordasini et al., 2014).