Generation III Bern Planet Formation Model

• Generation III Bern model is a comprehensive, end-to-end framework for simulating multi-planet formation and evolution based on the core-accretion paradigm.
• It integrates viscous disc evolution, planetesimal dynamics, gas and solid accretion, migration, and N-body interactions to predict planetary masses, radii, and luminosities.
• Monte Carlo population synthesis using varying embryo counts reveals key demographics, metallicity trends, and architecture–composition links in planetary systems.

In planetary science, the Generation III Bern model is the “global end-to-end” Bern planetary formation and evolution framework developed in the NGPPS series to synthesize the formation, dynamical assembly, and long-term evolution of entire multi-planet systems within the core-accretion paradigm. It couples viscous protoplanetary disc evolution, planetesimal dynamics, solid and gas accretion, disc-driven migration, and full $N$-body interactions during formation, then follows the surviving planets individually to gigayear ages, predicting not only masses and orbits but also radii, luminosities, magnitudes, atmospheric escape, and related observables self-consistently from the formation history (Emsenhuber et al., 2020). In the nominal solar-mass-star application, the model is used to generate Monte Carlo populations with different initial numbers of Moon-mass embryos per disc, and its main demographic conclusions include converged giant-planet statistics for $\gtrsim 10$ embryos, the necessity of 100 embryos to robustly reach the terrestrial giant-impact regime, a characteristic $N \propto M^{-2}$ mass function between $5$ and $50\,M_\oplus$, and a strong metallicity dependence of giant-planet occurrence (Emsenhuber et al., 2020).

1. Lineage and conceptual scope

The Generation III Bern model is the newest version of the Bern planetary formation framework and was designed to reunify branches that had diverged after the original Generation I model. In the Bern lineage summarized in the NGPPS overview, Generation I focused on one planet per disc and formation only, Generation Ib added long-term thermal evolution and observables such as radii and luminosities, and Generation II added multiple embryos and $N$-body interactions but not the long-term evolution. Generation III recombines and extends those branches so that multi-planet formation and post-formation evolution are treated within one framework (Emsenhuber et al., 2020).

Its governing picture is global core accretion. Solids are represented by a planetesimal disc from which embryos accrete; once sufficiently massive, embryos accrete gas and may enter runaway accretion; meanwhile, migration and embryo-embryo interactions reshape the architecture; after gas-disc dispersal, the planets continue to cool, contract, lose atmosphere, and undergo tidal evolution. The model therefore addresses a large mass range, from about Mars mass to super-Jupiters and up to the deuterium-burning regime, and a broad orbital range from close-in orbits to wide orbits (Emsenhuber et al., 2020).

A central design goal is population synthesis rather than single-system reconstruction. The model is intended to provide a framework for comparing theoretical consequences of specific physical processes with observed exoplanet demographics, including radial-velocity, transit, and direct-imaging observables. This suggests why Generation III places unusual emphasis on radii, luminosities, and magnitudes in addition to masses and semi-major axes (Emsenhuber et al., 2020).

2. Coupled formation and long-term evolution

During the formation phase, lasting $\SI{20}{\mega\year}$, the model evolves the gas disc, planetesimals, multiple embryos, gas accretion, orbital migration, and $N$-body interactions simultaneously. The gas disc is a 1D, axisymmetric viscous $\alpha$-disc whose surface density obeys the viscous diffusion equation with sink terms for photoevaporation and planetary gas accretion,

$$\frac{\partial \Sigma}{\partial t} = \frac{1}{r}\frac{\partial}{\partial r} \left[ 3r^{1/2}\frac{\partial}{\partial r} \left(r^{1/2}\nu\Sigma\right) \right] -\dot{\Sigma}_{\rm pe}-\dot{\Sigma}_{\rm pl},$$

with viscosity parameterized as $\gtrsim 10$0. Generation III also computes the disc vertical structure, including viscous heating, optical-depth effects, and stellar irradiation, and uses stellar evolution tracks so that stellar luminosity, radius, and temperature evolve consistently with time (Emsenhuber et al., 2020).

Planetary growth follows the core-accretion paradigm. Solid accretion is computed in the oligarchic regime with planetesimal accretion plus embryo collisions, while gas accretion is obtained by solving the 1D internal structure equations for each planet. The model distinguishes attached, detached, and evolutionary phases of the envelope and includes reduced grain opacity in the protoplanetary envelope, with an opacity reduction factor fixed to $\gtrsim 10$1 in the NGPPS population setup (Emsenhuber et al., 2020).

Migration and gravitational dynamics are treated concurrently. The model uses Type I and Type II migration prescriptions, the Crida et al. gap-opening criterion, and a full $\gtrsim 10$2-body calculation with the Mercury hybrid integrator. Gas-driven migration and eccentricity/inclination damping enter the $\gtrsim 10$3-body equations as additional forces, so collisions, scattering, ejections, and resonance capture occur within the same evolving disc environment (Schlecker et al., 2020).

After $\gtrsim 10$4, each surviving planet is followed individually through a thermodynamic evolution phase to $\gtrsim 10$5, including cooling and contraction, atmospheric escape, bloating, Roche-lobe overflow, deuterium burning, and stellar tides. This long-term phase is what allows Generation III to predict radii, luminosities, magnitudes, and evaporation rates self-consistently, rather than treating formation outputs only as masses and orbits (Emsenhuber et al., 2020).

3. Initial conditions and Monte Carlo population synthesis

A defining use case of the Generation III Bern model is the NGPPS Monte Carlo experiment on solar-like stars, where the stellar mass is fixed at $\gtrsim 10$6 and five synthetic populations are generated with different initial numbers of Moon-mass embryos per disc: 1, 10, 20, 50, and 100. Each embryo begins with $\gtrsim 10$7, is placed with probability uniform in $\gtrsim 10$8 between the disc inner edge and $\gtrsim 10$9, and must be separated from neighboring embryos by at least 10 Hill radii. The 100-embryo population is the nominal case and contains 1000 systems; the single-embryo case contains 30000 systems so that the total number of embryos remains statistically useful (Emsenhuber et al., 2020).

The adopted disc model and population priors are fixed in this experiment and are central to the interpretation of the results.

Quantity	Adopted value or distribution	Notes
Stellar mass	$N \propto M^{-2}$0	Solar-mass hosts
Viscosity	$N \propto M^{-2}$1	Constant-$N \propto M^{-2}$2 disc
Gas surface-density slope	$N \propto M^{-2}$3	Power-law index
Solids slope	$N \propto M^{-2}$4	Steeper than gas
Solid-disc radius	Half gas-disc radius	More compact solids
Planetesimal radius	$N \propto M^{-2}$5	Fixed size
Disc mass range	$N \propto M^{-2}$6 to $N \propto M^{-2}$7	Upper end gravitationally stable
Metallicity range	$N \propto M^{-2}$8	Truncated normal
Mean stellar rotation period	4.7 days	Inner edge near $N \propto M^{-2}$9

The planetesimals have density $5$0 inside the ice line and $5$1 outside it. The gas disc inner edge is set from stellar rotation/corotation arguments and is truncated so that no disc edge lies inside $5$2. External photoevaporation rates are chosen so that disc lifetimes match the observed few-Myr distribution, with a log-normal centered around $5$3 and $5$4 (Emsenhuber et al., 2020).

This setup makes the initial number of embryos an explicit control parameter. The model’s statistical comparison across 1, 10, 20, 50, and 100 embryos is therefore not a secondary detail but one of the central structural experiments in the NGPPS program (Emsenhuber et al., 2020).

4. Demographic predictions and convergence properties

The most prominent structural result is that the 100-embryo population is the only one that robustly reaches the terrestrial giant-impact regime. With fewer embryos, the bodies are too widely spaced and evolve more like isolated cores; with 100 embryos, the inner disc becomes dynamically packed enough for repeated collisions, late accretion, and broad diversity in compositions and radii. For inner terrestrial planets, the 100-embryo population is therefore essential, whereas for giant planets the results are already close to converged once there are at least 10 embryos per disc (Emsenhuber et al., 2020).

The nominal 100-embryo population yields the following system fractions and multiplicities.

Planet class	Definition	100-embryo nominal result
Earth-like	$5$5–$5$6	90.1% of systems; multiplicity 5.17
Super-Earth	$5$7–$5$8	82.1% of systems; multiplicity 5.61
Neptunian	$5$9–$50\,M_\oplus$0	30.3% of systems; multiplicity 1.45
Sub-giant	$50\,M_\oplus$1–$50\,M_\oplus$2	8.5% of systems; multiplicity 1.25
Giant	$50\,M_\oplus$3	18.1% of systems; multiplicity 1.57

Inside $50\,M_\oplus$4, Earth-like planets occur in 57.2% of systems with multiplicity 2.83, while super-Earths occur in 66.1% with multiplicity 3.66. The average planetary system in the 100-embryo case contains 8.4 planets with mass $50\,M_\oplus$5, and the average number of planets per system in that category is 32.03 embryos remaining at $50\,M_\oplus$6, although most are low-growth outer embryos (Emsenhuber et al., 2020).

For giant planets, the model finds that 18.1% of systems host at least one giant planet, but only 1.6% host a giant beyond $50\,M_\oplus$7 and 0.8% beyond $50\,M_\oplus$8. The median giant semi-major axis shifts outward as embryos are added, increasing from 0.49 au in the 1-embryo population to 1.26 au in the 100-embryo population. Even so, the converged multi-embryo cases still place giants typically interior to the ice line because gas-driven migration remains effective (Emsenhuber et al., 2020).

The mass function is another key quantitative signature. Across the synthetic population,

$50\,M_\oplus$9

for planet masses between $N$0 and $N$1, while at lower and higher masses the slope is closer to $N$2. The paper also emphasizes a “planetary desert” in the $N$3–$N$4 range, whose depth depends strongly on the interplay between migration and gas accretion (Emsenhuber et al., 2020).

5. Migration physics, metallicity trends, and architecture–composition links

Multi-embryo dynamics alter migration in two coupled ways. First, mean-motion resonances between planets slow the inward migration of outer embryos by sharing torque across several bodies, reducing the rapid inward drift that dominates the single-embryo case. Second, competition for the same local reservoir of planetesimals lowers the solids accretion rate and therefore the planetesimal accretion luminosity, allowing runaway gas accretion to begin at a lower core mass. In the NGPPS interpretation, this is why the 100-embryo population forms giant planets farther out and with smaller cores than the one-embryo case (Emsenhuber et al., 2020).

The metallicity dependence is correspondingly structured. The occurrence of Neptunian, sub-giant, and giant planets increases monotonically with $N$5, and the mean host-star metallicity of the 100-embryo giant-planet sample is $N$6, rising to $N$7 for giants beyond $N$8 and $N$9 beyond $\SI{20}{\mega\year}$0. By contrast, the occurrence of smaller planets is non-monotonic: terrestrial planets peak at $\SI{20}{\mega\year}$1, and super-Earths peak near $\SI{20}{\mega\year}$2. At low metallicity, the limiting factor is a lack of solids; at high metallicity, the growth of more massive dynamically active planets suppresses the survival of small planets through accretion, scattering, and ejection. The surviving multiplicity of giant planets is anti-correlated with metallicity once giants form, because systems that form multiple giants more readily also destabilize them more strongly (Emsenhuber et al., 2020).

A major application of the Generation III Bern model is the study of warm super-Earths and cold Jupiters in 1000 multi-planet systems, after imposing an observational radial-velocity detection bias. In that synthetic population, there are 32,030 surviving bound planets after 5 Gyr, including 538 super-Earths in 291 systems and 182 cold Jupiters in 140 systems. The model reproduces broad occurrence rates, with $\SI{20}{\mega\year}$3 and $\SI{20}{\mega\year}$4, but yields only a positive but weak correlation between the two populations, with $\SI{20}{\mega\year}$5 and $\SI{20}{\mega\year}$6 (Schlecker et al., 2020).

The physical explanation offered is dynamical survival rather than mere formation efficiency. Warm and dynamically active giant planets frequently disrupt inner systems, especially in high-metallicity environments. Among systems with cold Jupiters but no surviving super-Earths, the missing super-Earths usually did form and were later removed: 29% were ejected, 11% were accreted by the star, and 60% were lost in collisions with other planets. The same study also reports a strong architecture–composition link: super-Earths in systems with cold Jupiters tend to be drier, denser, and more likely to have formed inside the water ice line, whereas super-Earths without giant companions are more often volatile-rich. This leads to the prediction that high-density super-Earths are more likely to host an outer giant companion (Schlecker et al., 2020).

6. Limits, interpretation, and nomenclature

The Generation III Bern model is explicitly a self-consistent semi-analytic and 1D-physics framework rather than a fully hydrodynamical simulation. Its planetesimals are represented statistically rather than as individual bodies; pebble accretion and the self-consistent formation of embryos from dust are not yet included; the planetary envelope is simplified to pure H/He without detailed compositional gradients or enrichment; and some close-in and high-eccentricity processes remain approximate. The formation phase is terminated at $\SI{20}{\mega\year}$7, so some distant, slow-growing systems will not have completed all dynamical evolution within the modeled formation window (Emsenhuber et al., 2020).

The NGPPS results also identify model-specific interpretive cautions. Giant-planet statistics are robust already for $\SI{20}{\mega\year}$8 embryos per disc, but terrestrial and super-Earth populations converge much more slowly and require the 100-embryo setup to reach the late giant-impact phase. Multi-embryo systems broaden the mass–radius relation, produce more planets with stripped or partially stripped envelopes, and make the evaporation valley blurrier. The paper notes that the model may overproduce envelope stripping in giant impacts, because the evaporation valley appears too populated compared with observations (Emsenhuber et al., 2020).

A separate nomenclature issue is that the phrase “Generation III Bern model” is not unique across arXiv. In exoplanet research it denotes the Bern global model of planet formation and evolution developed in the NGPPS series (Emsenhuber et al., 2020). In unrelated literatures, closely similar wording is used for a third-generation Physics-Informed Neural Network Gravity Model, PINN-GM-III, for gravitational potential learning (Martin et al., 2023), and for a semi-analytical Bermudan swaption pricing framework based on swap-rate distributions and correlations (Feldman, 13 Oct 2025). Within planetary population synthesis, however, the term refers specifically to the Bern core-accretion, multi-planet, long-term evolution framework described above.