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Dust transport in envelopes of disk-embedded planets: I. Convectively stable envelopes

T0 review · 1 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The envelope of an embedded Earth-mass planet becomes strongly depleted in dust, by two to four orders of magnitude in the deep interior, because small grains are screened out by the recycling gas flow while large grains settle and…

desk verdict Solid, genuinely new simulations show strong inward dust depletion in convectively stable envelopes; the main caveat is that the result depends on the prescribed beta=1 thermal regime. read the letter →

arxiv 2608.05661 v1 pith:EG4NDBJR submitted 2026-08-06 astro-ph.EP

Ayumu Kuwahara , Michiel Lambrechts This is my paper · ORCID
classification astro-ph.EP
keywords protoplanetarydisksplanetformationdusttransportenveloperecyclingflowdust-to-gasratiogasaccretionmultifluidhydrodynamicsconvectivelystable
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 argues that the gaseous envelope around an embedded Earth-mass planet, when it is nearly isothermal and convectively stable, becomes strongly depleted in dust. In 2D and 3D multifluid simulations, the dust-to-gas ratio falls monotonically inward, reaching values two to four orders of magnitude below the envelope's outer edge inside 0.1 Bondi radius. The reason is size-selective delivery: small grains stay locked to the recycling gas flow and never enter the envelope, while large grains enter but settle onto the core too quickly to accumulate. This matters because dust is the main source of envelope opacity; a dust-depleted envelope cools faster and may trigger runaway gas accretion sooner, and it sets a stringent route for enriching the deep envelope with volatiles.

What carries the argument

The central object is the radial dust-to-gas density ratio profile inside the Bondi sphere, and the mechanism that sets it is a two-population filter. A buoyancy barrier (positive entropy gradient) near the Bondi radius, maintained by a prescribed short cooling time β=1, stops the polar recycling inflow from penetrating the inner envelope; this is the gate that excludes small grains. For grains that do enter, the key identity is the constant inwards dust mass flux F_d = 4π $r^{2}$ v_in(r) ρ_d(r) (or 2π r ... in 2D), which, together with an infall speed v_in = min(v_term, v_ff) and a collision-rate prescription P_col(St) for grains entering from the disk, yields the 1D analytic formula for the dust density. The Stokes number St = t_s Ω_0 is the dimensionless stopping time that separates the two behaviours: St≲$10^{-3}$ grains track the gas and are screened out, St≳$10^{-2}$ grains decouple and sediment.

What would settle it

Run the same standard runs with a shorter cooling time (β=$10^{-2}$) or with self-consistent radiative cooling: if small grains then penetrate the envelope and the dust-to-gas ratio at 0.1 R_B stays within a factor of ten of its value at the Bondi radius, the central depletion claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms: in a convectively stable (nearly isothermal) envelope around a low-mass planet, dust depletion is a robust outcome of the gas–dust dynamics, not a boundary artifact. The simulations resolve an outer recycling flow that is blocked from the inner envelope by a positive entropy (buoyancy) barrier near the Bondi radius, and an inner envelope that is therefore shielded from incoming small grains. Grains with Stokes number St≲$10^{-3}$ remain entrained in the recycling flow and do not cross into the envelope; grains with St≳$10^{-2}$ do enter but sediment rapidly onto the core along the midplane. The result is that the envelope is depleted of both small and large grains, with the dust-to-gas ratio reduced by more than two to four orders of magnitude at r<0.1 R_B compared with the value at the envelope edge. The depletion is robust across 2D and 3D simulations, across Stokes numbers from $10^{-3}$ to $10^{-1}$, and across fixed-size runs, and it is reproduced by a simple 1D model equating a radially constant dust mass flux with an infall velocity set by the smaller of the terminal and free-fall speeds.

Load-bearing premise

The load-bearing premise is that the envelope really is convectively stable with a sharp buoyancy barrier, which is imposed by hand through a short cooling time (β=1); if real opacity or accretion heating sets a different thermal structure, the recycling flow can penetrate deeper and the strong dust depletion can shrink or disappear.

Editorial extensions

If this is right

  • A convectively stable envelope is depleted of dust throughout most of its volume, so the interior dust opacity is far below the ISM-like value often assumed; envelope cooling is faster and the onset of runaway gas accretion can occur earlier, especially in the outer disk (≳10 au).
  • Because small grains are screened out, the deep envelope (<0.1 R_B) can only be enriched in dust or volatiles by large pebbles that fragment or sublimate once they have penetrated that deep.
  • The depletion profile is insensitive to the assumed Stokes number or grain size over the tested range, so the result is a generic expectation for radiative envelopes rather than a fine-tuned outcome.
  • The 1D analytic model (constant dust mass flux with terminal/free-fall infall speed and a collision rate P_col) reproduces the simulated profiles, giving a cheap tool for modelling dust distributions in envelope evolution calculations.
  • The dust-to-gas ratio is anisotropic in 3D, peaking at the midplane, which allows polar radiation escape and may further enhance cooling.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the depletion persists in self-consistent thermal models, published envelope-cooling timescales that assume a constant dust-to-gas ratio may be overestimates, and population syntheses should treat the dust-to-gas ratio as a radially decreasing function of the local envelope structure.
  • The same size-selective filtering should operate for other volatile carriers: only solids that arrive as large pebbles and disaggregate deep inside can enrich the atmosphere, which predicts a link between the pebble size distribution in the disk and the atmospheric metallicity of the planet.
  • The 1D collision-rate framework could be extended to predict depletion as a function of planet mass and disk location, offering a testable scaling before full 3D runs are done.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper presents two- and three-dimensional multifluid (gas and dust) hydrodynamical simulations of an Earth-mass planet embedded in a protoplanetary disk, with a prescribed dimensionless cooling time beta=1 that maintains a nearly isothermal, convectively stable envelope. The central result is that the dust-to-gas density ratio decreases radially inward within the envelope, by more than two to four orders of magnitude at radii below 0.1 Bondi radius relative to the envelope outer edge. The proposed mechanism is that small grains (St <= 1e-3) remain entrained in the outer recycling flow and are shielded from entering the envelope, while larger grains (St >= 1e-2) penetrate but settle rapidly onto the core. The authors also construct a one-dimensional semi-analytic model based on a constant dust mass flux and a collision-rate prescription, and they discuss implications for envelope opacity, the onset of runaway gas accretion, and atmospheric metallicity.

Significance. If the result holds for realistic radiative envelopes, it is significant: it directly links envelope-scale gas dynamics to the dust opacity that regulates envelope cooling, and it provides a plausible pathway to sub-stellar atmospheric metallicities through dust filtering in the outer envelope. The paper benefits from a consistent demonstration of the depletion trend in both 2D and 3D, resolution and inner-boundary sensitivity tests in Appendix A, additional tests of headwind, adiabatic index, and cooling time in Appendix D, and unusually transparent statements of its limitations. In particular, the paper explicitly identifies the restricted opacity range needed for the beta=1 regime and the non-convergence of the smallest-grain run. The central depletion result is a direct simulation outcome and does not depend on the analytic model; however, the quantitative headline is conditional on the prescribed beta=1 radiative end-member.

major comments (1)
  1. [Abstract; Sect. 5; Appendix D] The quantitative headline, namely a dust-to-gas reduction of more than two to four orders of magnitude at <0.1 R_B, is established only for the prescribed cooling time beta=1 (Sect. 2, Eq. 3). The paper's own analysis shows that this result is beta-dependent: Eq. (36) states that the recycling-radiative boundary scales as beta^0.22, and Appendix D (Fig. D.1c) states that the radius where the dust-to-gas ratio starts to decline moves inward as beta decreases. Because Sect. 2 further restricts the regime to a narrow opacity range (kappa ~ 1e-3 cm2/g at 10 au, from the companion paper KL26c), the abstract's unqualified statement that the depletion occurs in 'convectively stable envelopes' overreaches the presented support. I request either a quantitative statement of the depletion at 0.1 R_B as a function of beta (for example for beta=1, 0.1, and 0.01) or an explicit restriction of the abstract and conclusions to the beta=1 radiative end-member.
minor comments (5)
  1. [Appendix C, Eq. (C.10)] The value xi=0.08m is introduced as 'chosen to match our numerical results', and the same 3D simulations are then used to validate the 1D model in Sect. 4 and Conclusions. The agreement shown in Fig. 5b is therefore partly a calibration rather than an independent test; this should be stated explicitly at the point where the model is described as reproducing the numerics.
  2. [Table D.1] The beta column for the fiducial runs appears to read '100' rather than '1', which is inconsistent with Table 1 and with the text of Sect. 2; this should be corrected.
  3. [Sect. 2, after Eq. (3)] The term 'nearly isothermal' is used for the beta=1 envelope, but with gamma=1.43 and a finite cooling time the gas is not strictly isothermal. Please quantify the temperature variation or clarify that 'nearly isothermal' refers to the small fractional temperature change in the simulation.
  4. [Appendix A, Table 1] The high-resolution 3D convergence run (N_r,N_theta,N_phi)=(256,64,256) is integrated only for t_end=10 Omega^-1, much shorter than the fiducial 100 Omega^-1. Since the convergence claim in Fig. A.5 compares against this run, the authors should justify that this integration time is sufficient for a steady-state comparison.
  5. [Sect. 6.2] The discussion of micron-sized grains estimates settling times of order 10^3 Omega^-1 from the terminal-velocity approximation, which exceeds the simulated duration; the text should state explicitly that the abundance of such grains is an extrapolation rather than a simulated outcome.

Circularity Check

1 steps flagged · score 4.0 of 10

Central dust-depletion result is a self-contained simulation outcome; one fitted parameter (xi=0.08m) makes the 3D 1D model's normalization partially circular.

  1. fitted input called prediction [Appendix C (Eqs. C.8-C.10), applied in Sect. 4 Eqs. 31-35 and displayed in Fig. 5b]
    "u(RB)=-xi c_s,0, with xi=(0 in 2D; 0.08m in 3D). We slightly modified the original formula given by Okamura & Kobayashi (2021), where the authors set xi=0.1m. ... In 3D, we adopt xi=0.08m, chosen to match our numerical results."

    The 3D dust accretion rate (Eq. 32) enters the 3D dust density (Eq. 35) through P_col,3D, whose only adjusted parameter xi is explicitly tuned to the simulations. The resulting 1D model curve is then presented in Sect. 4 as 'the 1D analytic model, which agrees with the numerical results of the fixed-St runs in both 2D and 3D' and in Sect. 7 as 'successfully reproduce the numerical results.' The normalization of the 3D model curve is therefore partly enforced by construction rather than predicted. The radial shape (from the isothermal gas density, Eq. 28, and the terminal/free-fall velocity, Eqs. 22-24) and the entire 2D model (xi=0) involve no fitted parameters, so the fit is confined to the 3D amplitude.

full rationale

The central claim—that the dust-to-gas ratio falls by 2-4 orders of magnitude at <0.1 R_B in the simulated beta=1, convectively stable envelopes—is a direct measurement from the multifluid simulations (Sect. 3, Figs. 3, 5, 7) and does not reduce to any fitted parameter. The only constructional circularity I found is in Appendix C: the 3D collision rate in the 1D analytic model uses xi=0.08m, 'chosen to match our numerical results,' so the amplitude of the 3D model curve in Fig. 5b is partly fit rather than predicted; the 2D model (xi=0) and the radial shape (from the isothermal gas profile, Eq. 28, and terminal/free-fall velocity, Eqs. 22-24) are parameter-free. The paper is transparent about this tuning and about the idealized nature of the beta=1 setup, explicitly stating that the regime requires a restricted opacity (kappa about 1e-3 cm2/g at 10 au, KL26c) and that isothermal runs lack the buoyancy barrier (Appendix A). Whether real envelopes reach this regime is an external-applicability concern, not a circularity (hard rule 5); the self-citations (KL26a, KL26c) used to motivate it are published or in-press companion papers whose stated assumptions do not include the depletion result, so they count as real evidence. The manuscript also flags its own limitations (dust-to-gas ratios as upper limits, unresolved <0.05 R_B interior, no dust back-reaction or dust physics), which I weighed as honest caveats. On balance the main result is independent and self-contained, with one secondary fitted parameter producing a partial circularity, yielding a score of 4.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central depletion result is a direct simulation output and does not depend on fitted parameters. The main fitted quantity is xi=0.08m in the 3D analytic collision rate, which tunes the 1D model to the simulations. The prescribed cooling time beta=1 and the initial dust distribution are modeling choices that define the scope rather than fits. No new physical entities are introduced; the recycling-radiative boundary concept is inherited from earlier work by Kurokawa and Tanigawa (2018) and Kuwahara and Lambrechts (2026a).

free parameters (4)
  • beta (dimensionless cooling time) = 1
    Prescribed cooling time chosen by hand to enforce a nearly isothermal, convectively stable envelope throughout the Bondi sphere; the paper acknowledges this is an idealized radiative end-member (Section 2, Appendices A and D).
  • xi (midplane outflow coefficient in 3D collision rate) = 0.08 m
    The 3D analytic collision rate uses xi=0.08m, explicitly 'chosen to match our numerical results' (Appendix C, Eqs. C.9-C.10); this makes part of the analytic reproduction fitted to the simulations.
  • Initial dust scale height H_d,0 = 0.1 H_g,0 = R_B
    Initial condition places dust inside the envelope; it sets the reservoir available before depletion, although the depletion trend itself is robust across runs.
  • Initial dust-to-gas ratio epsilon_0 (Z_0) = 0.01
    Standard assumed solids-to-gas ratio; not fitted here, but the absolute depletion factors are measured relative to this assumed initial value.
assumptions (6)
  • domain assumption The envelope is nearly isothermal and convectively stable under beta=1 cooling, with a positive entropy gradient that blocks polar inflow.
    Section 2 and Fig. 1 impose the thermal structure by prescribing beta=1 rather than computing it from opacity and accretion heating; Appendix A shows the isothermal limit has no such buoyancy barrier.
  • domain assumption Convectively stable, nearly isothermal envelopes are relevant for outer disks and require a restricted opacity range.
    Sections 2 and 6.1 rely on the authors' companion papers KL26a and KL26c, one in press, for the claim that such envelopes occur around 10 au and demand kappa of order 1e-3 cm^2/g; this is not demonstrated in the present paper.
  • domain assumption Dust does not back-react on gas, and turbulent diffusion is negligible.
    Equations 4 and 5 and the text in Section 2 state that backreaction and diffusion are neglected; valid for low dust-to-gas ratios but not if local dust accumulates.
  • domain assumption Dust growth, fragmentation, erosion, sublimation, and ablation are neglected.
    Section 6.2 discusses these omissions; they could replenish small grains and raise the dust-to-gas ratio, as the paper acknowledges.
  • domain assumption In the 1D model, the inward dust mass flux is radially constant and the dust velocity is min(terminal velocity, free-fall velocity).
    Equations 22-24 and 29-35 use this closure; Fig. 9 verifies constant flux in the simulations, but the terminal-velocity approximation ignores rotation and non-steady effects in detail.
  • domain assumption Neglecting headwind and large-scale radial drift does not change the envelope dust-to-gas ratio, only the absolute accretion rate.
    Appendix D shows a minor effect on the ratio but states that accretion rates are overestimated and that the derived dust-to-gas ratios should be regarded as upper limits.

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Pith. "Pith review of Dust transport in envelopes of disk-embedded planets: I. Convectively stable envelopes." pith.science (2026). https://pith.science/paper/EG4NDBJR

@misc{pith2026260805661,
  author       = {Pith},
  title        = {Pith review of: Dust transport in envelopes of disk-embedded planets: I. Convectively stable envelopes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EG4NDBJR}},
  note         = {Machine review of arXiv:2608.05661}
}
abstract

Planets embedded in protoplanetary disks accrete solids through their gaseous envelopes. The spatial distribution of these dust particles inside the envelopes of disk-embedded planets is poorly known. We present high-resolution two- and three-dimensional multifluid simulations that follow the dynamics of gas and dust around planets similar in mass to Earth. Our simulations resolve an outer recycling flow and an inner convectively stable envelope that is shielded from the recycling flow. We identify strongly dust-depleted envelopes: the dust-to-gas ratio decreases radially inward and is reduced by more than two to four orders of magnitude in the deep interior ($<0.1\,R_{\rm B}$; Bondi radius) compared to its value at the outer edge of the envelope. Small grains, with a dimensionless stopping time ${\rm St}\lesssim10^{-3}$, remain entrained in the recycling flow and do not enter the envelope, whereas large grains (${\rm St}\gtrsim10^{-2}$) penetrate the envelope but settle rapidly onto the core along the midplane. The resulting dust depletion in convectively-stable envelopes implies a substantial reduction in dust opacity throughout much of the envelope, facilitating cooling and a more rapid transition to runaway gas accretion. These results further suggest that enriching the deep envelope ($<0.1\,R_{\rm B}$) with dust or volatiles requires their delivery through large pebbles that then subsequently disintegrate, or sublimate, from their host grains in the deep envelope interior.

Figures

Figures reproduced from arXiv: 2608.05661 by the authors.

Figure 1
Figure 1. Radial profiles of the gas density (blue) and temperature (or￾ange). The dashed curve assumes hydrostatic equilibrium (Eq. 28), and the dotted curve is obtained by solving for the vortensity conservation together with radial force balance (Sect. 4). as determined by the opacity, accretion luminosity, and local disk conditions (Rafikov 2006; Kuwahara & Lambrechts 2026a). Appendices A and B of KL26c present a 1D envel… view at source ↗
Figure 2
Figure 2. Gas flow field around an embedded planet in the 2D run with streamlines, showing the three distinct regions: the shear, the horseshoe, and the envelope. The orange, cyan, and green curves mark the outer envelope, outer horseshoe and inner shear streamlines, respectively. The background color shows the gas surface density. the solar mass. We set m = 0.1 in this study. The Hill radius in code units is given by RH/Hg,0… view at source ↗
Figure 3
Figure 3. Reduction of the dust-to-gas ratio towards a planetary core in the 2D runs, for particles with different Stokes numbers. Streamlines in the midplane of gas (gray) and dust (orange) originating from the first quadrant. The background color shows the column dust-to-gas ratio, Z = ρ 2D d /ρ2D g . We note that the color bar is saturated below 10−3 . with ϵ0 ≡ ρ 3D d,0 /ρ3D g,0 = 0.01 being its initial value. We note tha… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Radial infall velocity of dust (top) and azimuthal velocity of gas and dust at the midplane (bottom). The thin curves show azimuthally averaged results from 2D runs, while thick curves show shell-averaged (panel a) and azimuthally-averaged (panel b) results from 3D run…
Figure 6
Figure 6. Figure 6: Flow field of gas and dust around an embedded planet in the 3D run. Left: Gas density and velocity vector of the gas, averaged over the azimuth ϕ ∈ [−π/2, π/2]. Middle: Midplane slice with gas streamlines. The orange, cyan, and green curves mark the outer envelope, out…
Figure 7
Figure 7. Figure 7: Dust density and velocity vector of the dust, averaged over the azimuth ϕ ∈ [−π/2, π/2], for various Stokes numbers (St = 10−3 , 10−2 , and 10−1 ). The background color shows the deviation from the initial dust distribution. The brighter and darker regions correspond t…
Figure 8
Figure 8. Figure 8: Stokes number, infall velocity of dust, and column dust-to-gas ratio obtained from the 2D, fixed dust size runs. All solid curves show the azimuthally averaged values. The dotted curves are computed under the terminal-velocity approximation and are truncated at r = RB,…
Figure 9
Figure 9. Figure 9: Inward dust mass flux obtained from the 2D (top, azimuthally averaged) and 3D (bottom, shell averaged) runs. The vertical axis is normalized by the value at the outer edge of the computational domain. The dotted curves are given by Eqs. 31 (top) and 32 (bottom), repre￾…

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