PPOLs Model: Pebble Accretion in Disks

• The PPOLs Model is an integrated framework that simulates planetary seed growth via pebble accretion, accounting for dust physics, filtering, and dynamic snow line evolution.
• It utilizes modular components like pebble predictors and accretion recipes to capture the diversity of planetary architectures across varying disk and stellar parameters.
• The model's predictions, including 'peas in a pod' systems and Solar System analogs, highlight its potential to unify our understanding of exoplanet formation mechanisms.

The PPOLs Model is an integrated computational framework designed to paper the simultaneous growth of protoplanetary seed masses by pebble accretion in evolving protoplanetary disks. Incorporating dust physics, accretion efficiencies, filtering effects, and a dynamic water ice line (“snow line”), the model provides predictive insight into the architectural diversity of planetary systems as they emerge from the initial disk phase. It spans a wide parameter space in stellar mass ($0.125$–$2.0\,M_{\odot}$) and disk mass fraction ($1$–$40\%$), capturing both the bulk composition and growth trajectories of planets as functions of disk conditions and formation environment (McCloat et al., 17 Sep 2025).

1. Model Architecture and Core Components

The PPOLs Model synthesizes established pebble formation and accretion schemes with disk evolution to follow the concurrent development of many planetary seeds distributed throughout the disk. The approach is modular, integrating:

• A pebble-predictor submodule that calculates local pebble properties (composition, size, drift rates) as a function of disk position and evolution, including dust coagulation and fragmentation.
• Pebble accretion recipes that determine the accretion efficiency ($\epsilon(t)$) for each seed, adapted from Ormel/Liu’s 3D formalism to distinguish between settling and drift regimes.
• Pebble flux tracking, wherein the local pebble flux $f_{\rm peb}(t)$ is updated sequentially: as pebbles drift inward, seeds accrete mass at a rate $dM/dt = f_{\rm peb}(t) \cdot \epsilon(t)$, and each outer seed filters and reduces the available flux for those further inward according to $$f_{\rm peb}(i, t+1) = f_{\rm peb}(i, t+1)\,(1 - \epsilon_o(t))$$.
• Pebble isolation mass: Once a seed achieves the critical mass $M_{\rm peb\, iso}$ (see Section 7), it carves a pressure bump in the gas disk that halts further inward drift of pebbles, thereby quenching the rapid growth of seeds both at and interior to its orbit.

This configuration allows for the concurrent simulation of $N$ seeds (e.g., 100), capturing the emergent system-level structure.

2. Pebble Accretion and Filtering Dynamics

Pebble accretion is predicated on the assembly and radial migration of pebbles, which are standardized into two chemical populations:

• Icy pebbles: Present in regions beyond the evolving snow line; these feature higher fragmentation velocities and can be accreted more efficiently due to their size.
• Rocky pebbles: Confined to the inner disk or produced by the sublimation of ices as the snow line advances inward.

The model computes the accretion efficiency for each seed, depending on its mass, disk parameters, and pebble composition, using three-dimensional dynamical formulas. Crucially, the sequential filtering by planetary seeds modifies the pebble flux, introducing competitive growth and highly nontrivial mass distributions across the system. This interplay is unique to models capable of tracking multiple seeds and directly influences the architectural emergence.

Pebble isolation occurs when a seed perturbs the disk sufficiently ($M_{\rm peb\, iso}$) to block further inward migration of pebbles, sharply contrasting with isolated, single-seed modeling efforts.

3. Evolution of the Water Ice Line (Snow Line)

The snow line’s position, $r_{\rm SL}$, is calculated self-consistently according to the disk’s evolving physical state:

$$r_{\rm SL} = 4.74\,\left(\frac{\alpha}{10^{-2}}\right)^{0.61} \left(\frac{\Sigma_{0,\rm gas}}{1000\,\mathrm{g\,cm^{-2}}}\right)^{0.704} \left(\frac{f_{DG}}{0.01}\right)^{0.37}\quad\mathrm{au}$$

where $\alpha$ is the Shakura-Sunyaev turbulent viscosity, $\Sigma_{0,\rm gas}$ the reference gas surface density, and $f_{DG}$ the dust-to-gas ratio.

As the disk evolves (with gas depletion and dust coagulation), the snow line migrates inward. This migration enables icy pebbles to reach inner seeds, instigating water delivery mechanisms (“pebble snow”). The model tracks composition swaps (from icy to rocky) at the snow line and initializes seed properties according to formation environment, e.g., water mass fractions up to $50\%$ for seeds originating outside the snow line.

4. Predicted System Architectures

The PPOLs Model robustly predicts three major classes of planetary system architectures as determined by initial disk mass fraction:

Architecture Type	Disk Mass Fraction	Resulting Planetary Features
Low-mass	Low ($\sim1$%)	Many Mars/Earth-mass cores, wide water variance
Middle-mass	Moderate ($\sim7$–$15$%)	Bimodal peak: inner Earth-mass, outer massive core
High-mass	High ($>20$–$40$%)	Water-rich proto-gas giants at snow line, starved inner disk

• Low-Mass Disk Architecture yields several short-period, Mars–Earth-mass embryos with a broad spread in water content and is the sole regime producing regular, compact “peas in a pod” systems akin to those seen by Kepler.
• High-Mass Disk Architecture leads to efficient outer seed growth, rapid attainment of pebble isolation, and proto-gas giant formation; inner disk regions remain depleted due to pebble quenching.
• Middle-Mass Disk Architecture produces a bimodal mass peak: inner, Earth-mass embryos alongside an outer, often snow-line-adjacent, core that grows up to an order of magnitude larger, closely paralleling the Solar System’s mass distribution.

The persistence of this sequence across all stellar masses highlights the dominant role of initial disk mass fraction in sculpting planetary system diversity.

5. Stellar Mass and Disk Mass Dependencies

Modeling is performed across a wide grid, with stellar mass ($M_{\rm star}$) ranging from $0.125$ to $2.0\,M_{\odot}$ and disk mass ($M_{\rm disk}$) directly scaled as $M_{\rm disk} = f_{\rm disk}\, M_{\rm star}$.

• For lower-mass stars (M dwarfs), lower solid content yields inner embryos of Mars-like mass; higher-mass stars and disks facilitate formation of Earth-mass to super-Earth-mass cores and more pronounced water delivery.
• Across all masses, the characteristic three-architecture progression is retained, with only middle-mass disks around F and G stars reliably reproducing Solar System-like configurations. A plausible implication is that multi-Earth systems (“peas in a pod”) should be most frequent around low-mass stars with low disk fractions, whereas giant outer cores should appear for high disk fractions independent of host star mass.

6. Comparison with Observed Planetary Systems

The architectural outcomes align with and help explain observed patterns:

• “Peas in a Pod” systems: Multi-planet systems with regularly spaced, near-equal-mass terrestrial planets and significant water content diversity, as seen in Kepler data, are naturally generated within low-mass disk architectures.
• Solar System analogs: The bimodal mass distribution from middle-mass disks reproduces inner rocky planets and a singular, massive outer core, providing initial conditions for further dynamical histories (e.g., the Grand Tack or collisional evolution) required for exact Solar System replication.

This suggests that pebble accretion physics—when implemented in a multi-seed framework with evolving ice chemistry—may underpin the universal mechanisms behind terrestrial planet and giant planet formation diversity.

7. Analytical Formulations and Computational Efficiency

Critical mathematical descriptions employed in the PPOLs Model include:

• Seed mass growth:

  $\frac{dM}{dt} = f_{\rm peb}(t) \cdot \epsilon(t)$

• Pebble filtering:

  $$f_{\rm peb}(i, t+1) = f_{\rm peb}(i, t+1) \, (1 - \epsilon_{o}(t))$$

• Pebble isolation mass:

  $$M_{\rm peb\, iso} = 20\,M_{\oplus}\,\left(\frac{M_{\rm star}}{M_{\odot}}\right)\left(\frac{H_{\rm gas}}{0.05}\right)^3$$

• Disk mass scaling:

  $M_{\rm disk} = f_{\rm disk}\, M_{\rm star}$

• Snow line evolution:

  $$r_{\rm SL} = 4.74\,\left(\frac{\alpha}{10^{-2}}\right)^{0.61} \left(\frac{\Sigma_{0,\rm gas}}{1000\,\mathrm{g\,cm^{-2}}}\right)^{0.704} \left(\frac{f_{DG}}{0.01}\right)^{0.37}\quad\mathrm{au}$$

By exploiting efficient grid-based tracking and modular recipes, the model achieves rapid computation even when following hundreds of seeds, enabling statistical sampling over diverse disk and stellar parameters.

8. Significance and Implications

The PPOLs Model advances the theoretical and computational paper of planetary system formation by providing a generalizable mechanism for mapping disk properties to planetary architectures. Its inclusion of pebble filtering, isolation, and snow line evolution captures the principal physical determinants affecting bulk composition and system structure. The predictions elucidate observed phenomena such as “peas in a pod” systems and Solar System analogs, suggesting unified formation pathways across different stellar environments. Additionally, the model’s analytical transparency enables comparisons with observations and motivates further investigation into late-stage dynamical and atmospheric evolution.

By bridging simulation with exoplanet demographics, the PPOLs Model constitutes a foundational tool for ongoing research into planetary formation via pebble accretion (McCloat et al., 17 Sep 2025).