How Flow Isolation May Set the Mass Scale for Super-Earth Planets

M. M. Rosenthal; R. A. Murray-Clay

REVIEW 2 major objections 5 minor 62 references

How Flow Isolation May Set the Mass Scale for Super-Earth Planets

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A growing planet's atmosphere can deflect gas and pebbles around it; once its Bondi radius exceeds the pebble-capture radius, pebble accretion stops, fixing a characteristic 'flow isolation mass' near super-Earth scales that the paper…

desk verdict A clearly argued analytic derivation of a flow isolation mass for super-Earths, with the central static-atmosphere premise still unproven against published recycling simulations. read the letter →

arxiv 1908.06991 v2 pith:IRGSCWQG submitted 2019-08-19 astro-ph.EP

M. M. Rosenthal , R. A. Murray-Clay This is my paper · ORCID

classification astro-ph.EP

keywords pebbleaccretionflowisolationmasssuper-Earthssub-NeptunesprotoplanetarydisksplanetformationStokesnumberKeplerplanets

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

This paper asks why so many close-in exoplanets stop growing at 2–10 Earth masses when pebble accretion, once started, should carry planets past this range within a few thousand years. The proposed answer is 'flow isolation': the growing planet's atmosphere acts as a static obstacle, forcing nebular gas to stream around the planet, and small pebbles coupled to that gas are carried along and miss the planet. Once the atmosphere's Bondi radius exceeds the largest impact parameter at which any available pebble could be captured, pebble accretion is shut off entirely, leaving the planet at the flow isolation mass. For a fiducial disk the paper finds a mass scale of about $6.8\,M_\oplus$, with scalings that make it nearly independent of orbital distance in the inner disk and roughly linear in stellar mass; these properties are argued to match the similar sizes of planets within systems, a characteristic mass near $8\,M_\oplus$, and the association of inner super-Earths with outer gas giants.

What carries the argument

The load-bearing comparison is between two radii. $R_{\rm stab}$ is the largest impact parameter at which pebble accretion can capture a particle, set by balancing the planet's gravity against the gas drag force, with $R_{\rm stab} = \min(R_{\rm WS}, R_{\rm shear}, R_H)$ and upper limit $R_H$, the Hill radius; $R_B = GM_p/c_s^2$ is the Bondi radius, taken as the scale of the static atmosphere that deflects the gas. Particles are characterized by the Stokes number $St = t_s \Omega$, the particle stopping time in units of the orbital time, so small particles have small $St$ and follow the gas closely. The flow isolation condition is $f R_{\rm stab}(St_{\max}) = R_B$; once the atmosphere's radius exceeds the capture radius for the largest pebble present, all available pebbles are diverted around the planet and growth ceases. The paper also shows that the companion condition, that the particle be able to respond to the deflected flow ($t_s < R_B/v_\infty$), is automatically satisfied for $St<1$ whenever $R_{\rm stab}<R_B$, so the single radius comparison carries the argument.

What would settle it

A resolved hydrodynamical simulation of a 1–10 Earth-mass planet with a luminous, accreting atmosphere embedded in a gas disk could settle the matter: if the simulation shows gas recycling through the atmosphere and pebbles of all available sizes still reaching the planet, the flow isolation mass is wrong; if streamlines divert pebbles once the Bondi radius exceeds the capture radius, the mechanism is confirmed. Observationally, a survey of super-Earth systems around stars of different masses could check the predicted near-linear scaling of the characteristic mass with stellar mass, and a broad mass distribution without such a scaling would count against the claim.

Watch

Extended reading notes

Core claim

The central claim is that a planet growing by pebble accretion halts when its atmosphere's Bondi radius, $R_B = GM_p/c_s^2$, reaches the scale at which gas-drag-assisted capture would otherwise operate, expressed as $f R_{\rm stab}(St_{\max}) = R_B$, with $St_{\max}$ the largest Stokes number of available pebbles. For linear drag the resulting mass, relative to the thermal mass $M_{\rm th}=3(H/r)^3 M_*$, is $M_{\rm flow}/M_{\rm th} = \min\left[ f^2 (c_s/3 v_{\rm gas})\,St_{\max},\, (f^{3/2}/3)\sqrt{St_{\max}},\, (f'/3)^{3/2}\right]$ (Equation 46). With $f=f'=1.75$ and the fiducial disk, the inner-region value is $M_{\rm flow} = 6.8\,M_\oplus\, St_1^{1/2} \dot{M}_8^{3/8} M_{*,\odot}^{-1/8} \Sigma_{3000}^{3/8}$ (Equation 54), a super-Earth scale nearly independent of semi-major axis. The paper further claims that this single mass scale reproduces the observed intra-system similarity of super-Earth sizes, a characteristic planet mass near $8\,M_\oplus$ growing roughly linearly with stellar mass, and the preferential association of inner super-Earths with outer gas giants, while contrasting flow isolation with the pebble isolation mass.

Load-bearing premise

The argument depends on the growing planet's atmosphere being a dense, static obstacle that forces the nebular gas to flow around it at roughly the Bondi radius; if gas instead streams through the atmosphere and carries pebbles inward, the flow isolation cutoff does not operate and the central mass-scale claim fails.

Editorial extensions

If this is right

In the inner, viscously heated disk the flow isolation mass is nearly independent of semi-major axis, so planets forming at different distances in the same disk end up with similar masses and, absent atmospheric loss, similar sizes.
Because the inner-disk scaling is $M_{\rm flow}\propto \dot{M}^{3/8}\Sigma^{3/8}$ and both disk accretion rate and surface density are taken to rise roughly linearly with stellar mass, the characteristic mass scales about linearly with stellar mass, matching the inferred super-Earth scale near $8\,M_\oplus$.
If pebble accretion is halted by flow isolation before the critical core mass for runaway gas accretion is reached, super-Earths remain a common final state instead of runaway growth into gas giants.
In the outer, passively heated disk the flow isolation mass increases with semi-major axis, so systems that produce inner super-Earths by this route should preferentially host gas giants farther out.
When the largest available pebbles are small ($St_{\max}\lesssim 0.1$), flow isolation gives a lower limiting mass than the pebble isolation mass, offering a way to distinguish the two mechanisms in outer-disk populations.

Reading between the lines

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

If the mechanism is right, the super-Earth mass function should show a pile-up whose peak tracks the disk accretion rate; comparing planet populations around stars with different accretion histories would test the predicted $3/8$ power dependence of the mass scale.
The calibration factors $f$ and $f'$, both set to 1.75, are the principal free dials: direct hydrodynamical measurement of gas deflection around an accreting, luminous atmosphere could shift the predicted mass by a factor of 2–3 while preserving its scalings.
A sharp, testable consequence left implicit in the paper is that the cutoff is size-selective: a planet approaching the flow isolation mass should stop accreting the smallest pebbles first, narrowing the pebble size distribution delivered to the planet from below.
If the static-atmosphere premise fails and nebular gas recycles through the planet's atmosphere, pebble accretion could continue past super-Earth masses and the observed characteristic scale would require another explanation, such as pebble isolation; a resolved simulation of a sub-thermal planet with an accreting envelope would discriminate the two.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, this is a real idea: the authors write down the first analytic expressions for a flow isolation mass (Equations 43 and 46), add a fragmentation-limited version (Equation 59, Figure 7), and show the mass scale lands near 6.8 M_Earth with a stellar-mass scaling close to Wu's. Second, the loudest problem is not the uncalibrated coefficient f = 1.75 or the disk parameter choices; it is whether the atmosphere acts as a static obstacle at all. The binding-energy argument in Section 3.4 shows the envelope stays bound, but the published recycling simulations (Ormel et al. 2015; Cimerman et al. 2017) say gas flows through a bound envelope. If that is right, flow isolation does not define a mass scale, and the comparisons to Weiss, Wu, and Bryan are comparisons to a cutoff that may not operate.

What is genuinely new: the closed-form mass formulas, the fragmentation-limited version, and the direct comparison to Kepler trends. The derivation is transparent and the disk model is stated clearly enough to reproduce. The paper does not hide the f = 1.75 choice; it flags it as needing numerical calibration. That honesty deserves credit. The comparison to Wu is suggestive, and the stellar-mass scaling comes from assumed disk scalings rather than a fit to the super-Earth sample, which is both a strength and a weakness.

Where it is soft: Section 3.4 is load-bearing and does not address the recycling mechanism. The kinetic-energy-to-binding-energy ratio speaks to ablation, not to whether streamlines pass through a bound convective envelope. A bound but recycled envelope is not the solid obstacle flow isolation requires. The quantitative normalization also depends on f = 1.75 and on fiducial disk parameters taken from Powell et al. and the authors' own prior work. The comparisons to Weiss and Zhu & Wu are qualitative, not fits.

This is a paper for planet formation specialists. It deserves a serious referee, not a desk rejection: the mechanism is clearly articulated and has analytic predictions that can be falsified by targeted hydro simulations and by a statistical comparison to the Kepler sample. I would send it to peer review and tell the referee to focus on Section 3.4, because the flow isolation mass is only as solid as the static-atmosphere premise.

Referee Report

2 major / 5 minor

Summary. The paper proposes that "flow isolation"—the deflection of nebular gas and gas-coupled pebbles around a growing planet's atmosphere—sets a characteristic mass scale for close-in super-Earths. Using the pebble accretion framework of Rosenthal et al. (2018), the authors define the flow isolation mass by equating the maximum pebble accretion impact parameter to the planet's Bondi radius through Equation (12), f R_stab(St_max) = R_B, with f = 1.75. They derive analytic scalings in Section 3.3, present the fiducial inner-disk result M_flow = 6.8 M_Earth St_1^(1/2) Mdot_8^(3/8) M_*^(-1/8) Sigma_3000^(3/8) in Equation (54), and argue that this scale explains the similar sizes within Kepler multi-planet systems, the Wu (2019) characteristic mass near 8 M_Earth, and the association of inner super-Earths with outer gas giants.

Significance. If the mechanism operates, it addresses a real problem in pebble accretion theory: growth timescales near super-Earth masses are so short that without a shutdown mechanism, pebble-accreting planets would either stall at sub-Earth masses or run away to gas giants. The paper gives a transparent analytic derivation, explicit disk-model prescriptions, and concrete, falsifiable scaling laws, including a predicted linear scaling with stellar mass. It also honestly flags its main free parameters. The central issue is that the existence of the cutoff depends on a static-atmosphere premise that is not established against published atmospheric recycling simulations; the quantitative normalization also rests on an uncalibrated coefficient f. The observational comparisons are suggestive but not unique, as the paper itself acknowledges for the outer-giant correlation.

major comments (2)

[§2 and §3.4, Eq. (12), Eqs. (48)–(50)] The central criterion f R_stab(St_max) = R_B presupposes that the growing planet's atmosphere acts as a static obstacle forcing nebular gas to flow around it at the Bondi scale. The only response to the recycling simulations cited in footnote 1 (Ormel et al. 2015; Cimerman et al. 2017) is the order-of-magnitude argument in Section 3.4. That argument compares the kinetic energy of intercepted gas with the gravitational binding energy of the atmosphere, so it addresses whether the atmosphere can be unbound or ablated. Recycling flows, however, exchange envelope gas on dynamical timescales through convection and shear-driven circulation without requiring the atmosphere to become unbound; a bound but recycled envelope is not necessarily the solid obstacle that flow isolation requires. If gas streamlines enter the Bondi sphere, the R_stab cutoff does not operate and the flow isolation mass scale does not exist. The manuscript needs either a direct hydrodynamical calculation of flow around a sub-thermal planet with an accreting atmosphere, or a substantially stronger argument ruling out through-flow, before the central claim can be accepted.
[§3.3.1, Eqs. (12), (46), and (54)] The quantitative normalization M_flow = 6.8 M_Earth in Equation (54), and hence the claimed agreement with Wu (2019)'s ~8 M_Earth scale, depends on the coefficient f = 1.75 introduced in Equation (12). The text states that f is undetermined and that its value should be set by comparison with numerical simulations, which is left to future work. Since M_flow scales as f^(3/2), varying f over a plausible order-unity range changes the inner-disk mass by a factor of about 2–3 (for example, f = 1 gives roughly 3 M_Earth with the same disk parameters). No calibration or sensitivity study is presented, so the agreement with the observed normalization is not an independent test of the model. The authors should either calibrate f against published or new simulations or show explicitly that the claimed observational agreement persists over a range of f.

minor comments (5)

[§2] There is a typo, "In pratice," which should read "In practice."
[§3.3.1] The term "flow isolation mass" is used with a broader meaning than in Rosenthal et al. (2018); the text notes this, but a brief explicit definition at first use would help readers avoid confusion between the two definitions.
[§5.1] The discussion of Weiss et al. (2018) mentions the Zhu (2019) detection-bias interpretation only in passing; given that this caveat directly affects the claimed explanation of intra-system size similarity, it deserves a fuller treatment in the main text.
[§5.3] The paper acknowledges that the inner-super-Earth/outer-gas-giant correlation is not unique to flow isolation and also follows from pebble isolation or classical isolation models; this non-uniqueness should be stated in the abstract or conclusions so that the claimed observational support is not overstated.
[Figure 1] The red hatched region is labeled as the region where growth cannot occur because of flow isolation, but the caption does not specify which mass scale or atmosphere size is used; adding this information would make the figure self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the flow isolation mass is derived from independent pebble-accretion and disk inputs, then compared with Kepler trends; the uncalibrated f and static-atmosphere premise are physical assumptions, not tautological inputs.

full rationale

The central derivation starts from the independently published pebble-accretion impact parameter Rstab (Eqs. 39-41), standard drag prescriptions, and the Bondi radius; the criterion fRstab(Stmax)=RB (Eq. 12) defines a new mass scale rather than importing the observed super-Earth mass or the Kepler trends. The comparison with Wu (2019) is a comparison of scaling behavior: the linear stellar-mass dependence (Eq. 60) follows from adopted empirical scalings Mdot proportional to M*^2 and Sigma proportional to M*, not from Wu's characteristic mass formula, so the match is not constructed by fitting the target observation. The coefficient f=1.75 is explicitly left uncalibrated and chosen a priori ('We leave this comparison for future work'), so it is not a fitted parameter renamed as a prediction; it shifts the normalization by an order-unity factor but does not generate the predicted scaling or the semi-major-axis independence. The paper does rely on the authors' earlier R18 model for the pebble-accretion framework and on Powell et al. for the fiducial surface density, but those are published parameter-free models with stated assumptions that do not include the target result, and the paper's own formulas are independently checkable; this is normal self-citation, not load-bearing circularity. The main physical premise, that a sub-thermal planet's atmosphere deflects the nebular flow on the Bondi scale, is supported by an external simulation (Ormel 2013) and defended against the recycling simulations by the binding-energy estimate in Section 3.4. That defense may be incomplete, since it addresses whether the atmosphere becomes unbound rather than whether gas flows through a bound, recycled envelope; however, a disputed or under-supported physical premise is a correctness risk, not a circular reduction of the derivation to its inputs. No equation in the paper is equivalent to its inputs by construction, and the Kepler comparisons are genuine postdictions of trends rather than rewritings of the assumed disk scalings.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a physical process, flow isolation, rather than a new entity. The key unproven inputs are the static-atmosphere premise, the R18 pebble accretion model, the fragmentation-limited particle size, and the chosen fiducial disk parameters. The hand-set coefficients f and f' are free parameters that set the quantitative mass scale, while the disk normalization and opacity are chosen from prior literature, some of it the authors' own work.

free parameters (5)

f (flow isolation criterion coefficient) = 1.75
Introduced in Equation (12) as an undetermined factor of order unity in f R_stab = R_B; set to 1.75 'for presenting our results' and left for future numerical calibration. M_flow scales as f^(3/2) in Equation (43), so this hand-set value directly affects the predicted super-Earth mass scale.
f' (thermal-mass regime coefficient) = 1.75
Same coefficient used in Equation (44) for the R_B = f' R_H regime; set equal to f. Affects the upper mass cutoff for flow isolation.
u_frag (fragmentation velocity) = 1 and 10 m/s (two cases)
The fragmentation-limited maximum Stokes number, Equation (59), depends on u_frag. Lab experiments give a range; results are shown for both values, spanning an order of magnitude in St_max and shifting M_flow significantly in Figure 7.
Sigma_0 (disk surface density at 1 au) = 3000 g cm^-2
Fiducial normalization taken from Powell et al. (2019), a co-authored prior paper. M_flow is proportional to Sigma_0^(3/8) in Equation (54), so this choice affects the absolute mass scale, though not the M_* or r scalings.
kappa (Rosseland mean opacity) = 0.1 cm^2/g
Assumed constant opacity used to set the viscous heating temperature profile in Equation (18). M_flow normalization depends on it through T_c.

assumptions (6)

domain assumption The growing planet's atmosphere is a static, denser obstacle that forces the nebular gas to flow around it at the Bondi radius R_B.
Central physical premise of flow isolation. Defended by an order-of-magnitude binding-energy argument in Section 3.4, but recycling simulations (Ormel et al. 2015; Cimerman et al. 2017) are cited as counter-evidence and the paper does not simulate the flow directly.
domain assumption The pebble accretion impact parameter R_stab is correctly described by the model of Rosenthal et al. (2018), including the piecewise drag law and the min(R_WS, R_shear, R_H) prescription.
All growth timescales and M_flow values inherit the R18 model, which is a cited prior model from the same authors and is not independently verified in this paper.
domain assumption The maximum particle size in the disk is set by turbulent and laminar fragmentation as modeled in Equation (59), with u_frag in the lab-measured range.
The halt of growth requires that the flow isolation cutoff exceeds the largest particles present; the fragmentation barrier model determines St_max(r).
domain assumption The fiducial disk temperature is T = max(T_visc, T_irr) with the given viscous heating and irradiation profiles, and Sigma = 3000 g cm^-2 r^-1.
This sets c_s, H/r, eta, and alpha throughout the derivation. The choices are motivated by observations but not derived in this paper.
domain assumption Planets form in situ at their observed semi-major axes; Type I migration is neglected.
Stated in Section 3.1: 'we are neglecting Type I migration effects... instead considering expected planet masses if planets form in-situ.' This simplifies the comparison to Kepler architectures.
domain assumption Gas flows are subsonic, v_gas < c_s, for planets below the thermal mass.
Used in the derivation below Equation (8) to argue that the stopping-time criterion is always satisfied when R_stab < R_B.

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Cite this review

Pith. "Pith review of How Flow Isolation May Set the Mass Scale for Super-Earth Planets." pith.science (2026). https://pith.science/paper/IRGSCWQG

@misc{pith2026190806991,
  author       = {Pith},
  title        = {Pith review of: How Flow Isolation May Set the Mass Scale for Super-Earth Planets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRGSCWQG}},
  note         = {Machine review of arXiv:1908.06991}
}

abstract

Much recent work on planet formation has focused on the growth of planets by accretion of grains whose aerodynamic properties make them marginally coupled to the nebular gas, a theory commonly referred to as "pebble accretion". While pebble accretion can ameliorate some of the issues presented by growth by purely gravitational processes, it has other issues when compared with observations of exoplanetary systems. A particular concern is the preponderance of planets that end their growth as "super-Earths" or "sub-Neptunes", with masses in the range 2-10 $M_\oplus$. Once planets reach this mass scale, timescales for growth by pebble accretion are so rapid that ubiquitously ending growth here is difficult. In this work, we highlight this issue in detail using our previously published model of pebble accretion, and also propose a possible solution: feedback between the growing planet's atmosphere and the gas disk inhibits accretion of smaller particle sizes by forcing them to flow around the growing planet instead of being accreted. For reasonable fiducial disk parameters this "flow isolation" will inhibit accretion of all available particle sizes once the planet reaches super-Earth masses. We also demonstrate that the characteristics of this "flow isolation mass" agree with previously published trends identified in the \textit{Kepler} planets.

Figures

Figures reproduced from arXiv: 1908.06991 by the authors.

**Figure 1.** A plot of the growth timescale of a planet at a = 0.5 AU undergoing pebble accretion as a function of planet mass and small body radius. The disk parameters used are described in Section 3.1. The two panels show the growth timescale for two different levels of turbulence in the disk. In the lefthand side of both panels, the red hatched region indicates where growth cannot occur because pebbles flow around the core (… view at source ↗

**Figure 2.** The mass of a protoplanet undergoing pebble accretion as a function of time, for three different values of initial mass. All particles are assumed to have Stokes number of 10−2 . The disk parameters used are given in Section 3.1. In all cases the protoplanet’s solid mass runs away to extremely large masses on timescales shorter than the lifetime of the protoplanetary disk (∼ 3 Myr). eration of the core3 , that is Rs… view at source ↗

**Figure 3.** A cartoon illustrating schematically how flow isolation operates. The planet’s (black dot) atmosphere is shown by the gray shaded region, and extends up to RB. The gas flows around the atmosphere, as shown by the dashed blue lines. The larger, green particle, has maximal impact parameter for accretion Rstab > RB, and thus can be captured at scales of Rstab before encountering the modified gas flow. The smaller red … view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The atmospheric mass of a planet accreting at the maximal pebble accretion rate as a function of semimajor axis, using mixing length theory to calculate the temperature gradient. While the atmospheric mass is slightly reduced from the fully convective value, the decr…

**Figure 5.** Figure 5: A plot of the maximal mass a planet accreting pebbles can reach as a function of semi-major axis, accretion rate, and maximum Stokes number present. bonds. Lab experiments suggest that ufrag in the range 1-10 m/s may apply, though the (unknown) material properties of t…

**Figure 6.** Figure 6: A plot of the maximal Stokes number pebbles can reach as a function of semi-major axis and particle fragmentation velocity. The maximal particle size at a given semi-major axis is given by Equation (59). In [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: A plot of the maximal mass a planet accreting pebbles can reach as a function of semi-major axis and particle fragmentation velocity. The maximal particle size at a given semi-major axis is given by Equation (59). radii, Wu (2019) explored the effects of photoevaporat…

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