GIB–Bern Model: Planetary Evolution Synthesis

• The GIB–Bern model is a comprehensive, global framework that simulates planetary system formation by coupling protoplanetary disc evolution, planetesimal dynamics, and N-body interactions.
• It integrates diverse physical processes including migration, gas and solid accretion, and atmospheric escape to capture the formation of both terrestrial and giant planets.
• The model enables statistical population synthesis with direct comparison to observational surveys, offering insights into the diversity of exoplanetary system architectures.

The GIB–Bern Model is a physically comprehensive, global end-to-end computational framework for the coupled formation and long-term evolution of planetary systems. Developed as the Generation III Bern model within the New Generation Planetary Population Synthesis (NGPPS) effort, it synthesizes multiple domains of planetary science, including protoplanetary disc physics, planetesimal dynamics, planetary internal structure, multi-body gravitational interactions, migration theory, and post-formation evolutionary processes. Its design enables quantitative prediction and statistical population synthesis of planetary system properties spanning from terrestrial to giant planets, under realistic astrophysical and microphysical constraints (Emsenhuber et al., 2020).

1. Integrated Model Architecture

The GIB–Bern model explicitly couples the following components throughout planetary system evolution:

• Protoplanetary disc: Evolving 1D viscous α-disc with gas dynamics, photoevaporation (internal EUV and external FUV), and direct stellar irradiation; gas surface density $\Sigma_g(r,t)$ is evolved with all key source/sink terms.
• Planetesimal disc: Evolving statistical ensemble (surface density $\Sigma_s(r,t)$, RMS eccentricity $e$, inclination $i$), dynamically excited/damped by gas drag, planet stirring, and self-stirring mechanisms.
• Planetary embryos: Simultaneous N-body integration of multiple embryos interacting gravitationally, accreting solids and gas, subject to disc-driven migration forces (type I/II), and mutual collisions/mergers.
• Planetary interior and atmospheric structure: 1D radially-symmetric stellar structure equations for envelope evolution, gas and planetesimal accretion (including attached, detached, and isolated evolutionary phases), and post-formation processes.
• Disc and planet feedback: Gas and solid accretion rates directly modify the protoplanetary disc structure, while disc characteristics modulate accretion and migration regimes.
• Long-term evolution: Thermal contraction, atmospheric escape (X-ray/EUV-limited, radiation-recombination limited), radius inflation in strongly irradiated giants, Roche-lobe overflow, and tidal orbital decay over gigayear timescales.

2. Governing Physical Equations

Central equations and prescriptions include:

• Gas disc evolution equation:

$$\frac{\partial \Sigma_g}{\partial t} = \frac{1}{r} \frac{\partial}{\partial r} \left[ 3 r^{1/2} \frac{\partial}{\partial r} (\nu \Sigma_g r^{1/2}) \right] - \dot\Sigma_w(r) - \dot\Sigma_p(r)$$

with $\nu = \alpha c_s H$ and detailed radiative equilibrium for midplane temperature.

• Planetesimal stirring/damping: Differential equations for $e$ and $i$ with analytic prescriptions for gas-drag, protoplanetary viscous stirring, and self-stirring.
• Accretion rates:
  • Planetesimal accretion in the oligarchic regime via

  $$\dot M_{\rm core} = \Omega\,\Sigma_s\,b\,R_H^2\,P_{\rm coll}(e,i)$$ - Gas accretion determined by Kelvin–Helmholtz contraction,

  $$\dot M_{\rm gas} = \frac{M_{\rm env}}{\tau_{\rm KH}}$$

  and disc-limited cap, with self-consistent feedback on the gas surface density.

• Orbital migration:

  • Type I and II torques implemented per Paardekooper et al. (2011), Coleman & Nelson (2014), with explicit handling of eccentric/inclined orbits, gap formation criteria, and evolution of migration rates.
• N-body integration and events:
  • Mercury symplectic hybrid integrator for multi-planet dynamics, with close encounters triggering Bulirsch–Stoer integration, energy-conserving collision/merger treatment, and incorporation of migration/damping forces as additional accelerations.
• Post-formation evolution:
  • Internal structure evolved with grey-Eddington atmospheres, surface boundary evolution, energy-limited and recombination-limited atmospheric escape, empirically and theoretically motivated radius inflation, and tidal decay (Emsenhuber et al., 2020).

3. Initial and Boundary Conditions, Numerical Protocols

System initialization encompasses:

• Protoplanetary disc: $$\Sigma_g(r) \propto r^{-\gamma} \exp\left[-(r/r_c)^{2-\gamma}\right] \times (1-\sqrt{r_{\rm in}/r})$$, typically with $r_{\rm in} \sim 0.1$ AU, characteristic disc mass and outer cutoff.
• Planetesimals: $\Sigma_s(r) \propto r^{-1.5} \exp(-(r/r_s)^2)$; monodisperse radius $\sim 300$ m; initial dynamical temperature (eccentricity, inclination).
• Embryos: $N=10$–$100$ per disc, mass $10^{-2}M_\oplus$, logarithmic spacing from $r_{\rm in}$ to $40$ AU, separated by at least $10$ mutual Hill radii.
• Stellar properties: Pre-computed evolutionary tracks for radius, temperature, and luminosity.
• Numerical: 3400-point logarithmic radial grid (disc), timestep adaptivity, symplectic integrator step size of order $1/20$ of the innermost planetary period.

The model tracks the coupled evolution from initial core emplacement through $20$ Myr (formation stage) and forward to $10$ Gyr (evolutionary stage).

4. Model Validation and Empirical Comparisons

Key validation outcomes:

• Terrestrial planet formation: For $N \geq 23$ embryos, giant impact growth after mutual stirring efficiently depletes planetesimals within $1$ AU to produce $5$–$10$ Earth-mass bodies, reproducing multiparameter trends (spacing, eccentricity) of late-stage N-body simulations.
• Giant planet formation and migration: Single-embryo systems require rapid core accretion ($\sim 10$–$20\,M_\oplus$) just before disc dispersal to avoid excessive type I migration; multiple embryos allow for migration-slowing resonant chains and embryo-embryo interaction, qualitatively capturing the diversity of Solar System formation scenarios.
• Population synthesis and diversity: Systems with solely terrestrial planets show ordered mass/spacing; giant planet–bearing systems demonstrate wide diversity including hot and cold Jupiters, ejected objects, and various merger histories. The framework enables direct comparison to radial velocity, transit, direct imaging, and microlensing surveys (Emsenhuber et al., 2020).

5. Notable Physical Processes and Their Treatment

Process	Governing Equation/Prescription	Effects Captured
Photoevaporation	EUV and FUV mass-loss, radiative transfer (Clarke 2001; Matsuyama 2003)	Disc dispersal, planet isolation
Migration	Lindblad/corotation torques; gap-opening criteria (Paardekooper et al.)	Type I/II drift, resonance trapping, pileups
Atmospheric escape	X-ray/EUV-limited, recombination-limited flow (Jin et al., Murray-Clay)	Envelope loss, super-Earth to mini-Neptune path
Radius inflation	Empirical energy addition (Thorngren & Fortney; Sarkis & Bercovici)	Hot Jupiter bloating
Tidal Evolution	Orbital decay via quality-factor scaling (Ferraz-Mello et al.)	Hot Jupiter fate, period shrinkage

The modular separation/closeness of these processes allows sensitivity studies and physics-motivated cross-comparison.

6. Principal Results and Scientific Significance

The GIB–Bern model demonstrates that physically consistent, end-to-end coupling of disc evolution, planetesimal and embryo dynamics, structure equations, and long-term processes is necessary to realistically reproduce both the architecture of Solar System analogs and the observed exoplanet population diversity. The model establishes the necessity for fine-tuned core formation/migration synchrony for giant planet survival and exposes the range of planetary system order vs. diversity arising from initial conditions and N-body interactions.

A central outcome is the ability to generate statistically representative synthetic populations of planetary systems with direct, multi-observable comparability to data, enabling robust constraints on planet formation and evolutionary scenarios (Emsenhuber et al., 2020). A plausible implication is that population-level empirical diversity, especially the dichotomy between ordered terrestrial-only and chaotic giant-planet–bearing systems, finds a natural explanation within the coupled stochastic and dynamical interplay encoded in GIB–Bern.

7. Context within Planetary Population Synthesis

The GIB–Bern model represents a significant advance over prior “lite” and semi-analytic population synthesis tools by directly solving the underlying system of coupled physical and dynamical equations without decoupling disc, migration, accretion, or evolutionary timescales. It thereby addresses requirements for capturing correlated observable planetary properties—including masses, orbits, radii, and volatile loss histories—across the spectrum of planetary masses and orbital configurations, furthering the statistical calibration of planet formation theory against the rapidly expanding exoplanet data landscape (Emsenhuber et al., 2020).