A multi-fluid approach for polydisperse pebble accretion: From particles to fluids, establishing the multifluid framework

S.-J. Paardekooper; T. J. Konijn

REVIEW 3 major objections 1 minor 1 references

A multi-fluid approach for polydisperse pebble accretion: From particles to fluids, establishing the multifluid framework

T0 review · 3 major / 1 minor · reviewed 2026-07-01 · grok-4.3

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Pith's one-line read Multi-fluid model shows a protoplanet's gas perturbation lowers polydisperse pebble accretion rates for MRN distributions.

desk verdict The multi-fluid polydisperse setup in FARGO3D recovers prior static-disc results and shows gas perturbation effects on accretion, but the binning of the MRN distribution lacks any convergence checks. read the letter →

arxiv 2604.25742 v2 pith:M4WGFRHE submitted 2026-04-28 astro-ph.EP

T. J. Konijn , S.-J. Paardekooper This is my paper

classification astro-ph.EP

keywords pebbleaccretionpolydispersepebblesmulti-fluidhydrodynamicsplanetformationprotoplanetarydiscsefficiencyMRNdistribution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper develops and validates a multi-fluid hydrodynamic framework that treats a continuous pebble size distribution as several discrete fluid species, each with its own Stokes number, all coupled to the gas in 2D global disc simulations. When the gas disc remains static the resulting accretion efficiencies match those found in earlier single-size studies. Allowing the planet to perturb the gas flow reduces efficiency for larger pebbles while raising it for smaller ones, producing an overall lower accretion rate under an MRN size distribution. The framework is presented as a practical route to higher planet masses and to runs that include multiple pebble species simultaneously interacting with the gas.

What carries the argument

Multi-fluid treatment of polydisperse pebbles as separate fluid species with distinct Stokes numbers evolved inside modified FARGO3D global disc simulations.

What would settle it

A side-by-side comparison of the multi-fluid accretion rate against a high-resolution particle simulation using the identical continuous size distribution would reveal whether the chosen discretization produces a measurable error.

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Extended reading notes

Core claim

We constructed a multi-fluid model framework capable of accurately simulating polydisperse pebble accretion consistent with previous studies. This framework offers advantages for simulating higher planet masses and for modelling multiple pebble species coupled to the gas. We find that the protoplanet's perturbation of the gas-disc lowers the accretion rate when assuming an MRN-distribution of solids.

Load-bearing premise

Representing a continuous pebble size distribution by a modest number of discrete fluid species introduces negligible error in the total accretion rate and in the response of the gas disc.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The core advance is a working multi-fluid extension inside FARGO3D that tracks several discrete Stokes-number bins at once while letting them exchange momentum with the gas. For a static disc the efficiencies line up with earlier monodisperse calculations, which at least shows the implementation is not obviously broken. When the gas is allowed to respond, the planet’s perturbation reduces accretion for St ≳ 0.3 and increases it for smaller St, and the net polydisperse-to-monodisperse ratio ends up larger than previous estimates. That directional change with planet mass is the quantitative result that was not already in the literature.

The code handles multiple species coupled to the gas, which is useful for runs at higher planet masses where single-fluid or test-particle approximations start to fail. The finding that the largest St bins still dominate the total rate is consistent with what one would expect from the MRN slope.

The main weakness is that the paper never tests whether the modest number of bins (St from 0.01 to 1) is sufficient. Accretion is stated to be dominated by the high-St end, and the gas back-reaction grows with planet mass, so even modest shifts in bin placement or number could change the reported efficiencies. No resolution study, no sensitivity run with more bins, and no comparison to a moment-based or finer discretization appear in the abstract or the summary. Without those, the claim that the framework “accurately” represents a continuous distribution rests only on matching static-disc literature.

This is the sort of methods paper that a small group working on global pebble-accretion simulations would want to read for the implementation details. A population-synthesis modeller could use the directional trends as a rough guide but would need the convergence numbers before adopting the efficiencies. The work is coherent on its own terms and shows honest engagement with the existing literature, so it deserves referee time once the authors supply the missing numerical checks.

Referee Report

3 major / 1 minor

Summary. The paper develops a multi-fluid extension of FARGO3D to simulate polydisperse pebble accretion by representing a continuous MRN size distribution with a modest number of discrete fluid species at selected Stokes numbers (St ∈ [0.01,1]). It validates the approach by recovering accretion efficiencies consistent with prior monodisperse studies in static discs, then explores the effects of an evolving gas disc with solid-to-gas back-reaction, reporting lower efficiencies for St ≳ 0.3, higher efficiencies for St ≲ 0.3 (with the difference growing at higher planet masses), dominance of accretion by the largest St, and a higher polydisperse-to-monodisperse accretion-rate ratio than earlier estimates.

Significance. If the discretization into a small number of St bins introduces negligible error relative to a true continuous distribution, the framework would enable simulations at higher planet masses and with multiple coupled pebble species, clarifying how protoplanet-induced gas perturbations alter accretion under realistic polydisperse conditions.

major comments (3)

[Abstract] Abstract: the claim of consistency with earlier monodisperse studies is stated only qualitatively, with no reported quantitative metrics (relative errors, direct efficiency comparisons, or resolution details) to substantiate that the multi-fluid representation reproduces prior results within stated tolerances.
[Methods / Validation] Validation and methods sections: no bin-convergence tests, sensitivity studies on number of St bins, or bin-placement variations are presented for the range St ∈ [0.01,1]. Because accretion is stated to be dominated by the highest St and the MRN back-reaction effect strengthens with planet mass, this omission directly affects the reliability of both the static-disc validation and the reported changes in the evolving-disc case.
[Results] Results on evolving disc: the reported directional changes in efficiency (lower for St ≳ 0.3, higher for St ≲ 0.3) and the elevated polydisperse/monodisperse ratio rest on the assumption that the discrete-bin representation introduces negligible error in the gas response; without convergence data this assumption remains untested and load-bearing for the central claim.

minor comments (1)

[Methods] Notation for the Stokes-number bins and the precise mapping from MRN distribution to fluid species should be defined explicitly with an equation or table.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed report. We address each major comment below and commit to revisions that directly strengthen the validation and robustness of the multi-fluid framework.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of consistency with earlier monodisperse studies is stated only qualitatively, with no reported quantitative metrics (relative errors, direct efficiency comparisons, or resolution details) to substantiate that the multi-fluid representation reproduces prior results within stated tolerances.

Authors: We agree that quantitative support is needed. In the revised manuscript we will expand the abstract and add a dedicated validation paragraph (or table) reporting relative differences in accretion efficiency, direct numerical comparisons to the cited monodisperse benchmarks, and the exact number of St bins and grid resolution employed. revision: yes
Referee: [Methods / Validation] Validation and methods sections: no bin-convergence tests, sensitivity studies on number of St bins, or bin-placement variations are presented for the range St ∈ [0.01,1]. Because accretion is stated to be dominated by the highest St and the MRN back-reaction effect strengthens with planet mass, this omission directly affects the reliability of both the static-disc validation and the reported changes in the evolving-disc case.

Authors: This is a fair criticism. While the bin selection was motivated by the MRN distribution and prior single-fluid work, explicit convergence tests were not included. We will add a new subsection presenting results for 3, 5, and 7 bins, together with shifts in bin placement, to demonstrate that the key accretion efficiencies and back-reaction trends converge for the adopted parameter range. revision: yes
Referee: [Results] Results on evolving disc: the reported directional changes in efficiency (lower for St ≳ 0.3, higher for St ≲ 0.3) and the elevated polydisperse/monodisperse ratio rest on the assumption that the discrete-bin representation introduces negligible error in the gas response; without convergence data this assumption remains untested and load-bearing for the central claim.

Authors: We accept that the evolving-disc conclusions depend on the discretization error being small. The bin-convergence tests described above will directly test this assumption for both the static and perturbed cases. If any sensitivity is found, we will increase the number of bins and re-run the relevant simulations before final submission. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: numerical framework validated against independent prior studies

full rationale

The paper implements a multi-fluid extension in FARGO3D to represent a continuous MRN pebble distribution via discrete Stokes-number bins and performs 2D hydrodynamic simulations. Accretion efficiencies for the static-disc case are reported as consistent with earlier (external) monodisperse studies; the evolving-disc results, including the effect of gas perturbation and back-reaction, are obtained directly from the new runs rather than from any fitted parameter or self-referential equation. No derivation step reduces a claimed prediction to a quantity defined or fitted inside the present work, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The discretization assumption is stated explicitly but is not presented as a derived result.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The multi-fluid discretization itself is an implicit modeling choice whose accuracy is asserted but not derived from first principles in the provided text.

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Cite this review

Pith. "Pith review of A multi-fluid approach for polydisperse pebble accretion: From particles to fluids, establishing the multifluid framework." pith.science (2026). https://pith.science/paper/M4WGFRHE

@misc{pith2026260425742,
  author       = {Pith},
  title        = {Pith review of: A multi-fluid approach for polydisperse pebble accretion: From particles to fluids, establishing the multifluid framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M4WGFRHE}},
  note         = {Machine review of arXiv:2604.25742}
}

abstract

Pebble accretion offers an efficient pathway to form planets, driven by a constant supply of inward drifting mass and an accretion efficiency enhanced by gas drag. While most studies assume a single pebble size (monodisperse), real discs contain a range of sizes (polydisperse) that drift, interact, and accrete at different rates. We aim to model polydisperse pebble accretion with a fluid approach, validating the method and exploring how gas disc evolution, solid-to-gas back-reaction, and a polydisperse size distribution affect growth. We used FARGO3D, modified to allow pebble accretion, to run 2D hydrodynamic simulations in a global disc with multiple pebble species representing an underlying continuous pebble size distribution. With our multi-fluid approach, we find values for pebble accretion efficiency consistent with earlier studies for a static gas disc. This confirms that our approach gives an accurate representation of pebble accretion. Evolving the gas disc, we find lower efficiencies compared to an unperturbed gas disc for high Stokes numbers ($\gtrsim 0.3$) and higher efficiencies for smaller Stokes numbers ($\lesssim0.3$). This effect increases for higher planet masses. The accretion rate is mostly dominated by the highest Stokes numbers in our parameter study ($\mathrm{St}\in[10^{-2},10^0]$). The ratio we find between the polydisperse and monodisperse pebble accretion rates is higher than previous estimations. We constructed a multi-fluid model framework capable of accurately simulating polydisperse pebble accretion consistent with previous studies. This framework offers advantages for simulating higher planet masses and for modelling multiple pebble species coupled to the gas. We find that the protoplanet's perturbation of the gas-disc lowers the accretion rate when assuming an MRN-distribution of solids.