Pileups and Migration Rates for Planets in Low Mass Disks

Adam M. Dempsey; Wing-Kit Lee; Yoram Lithwick

REVIEW 3 major objections 4 minor 84 references

Pileups and Migration Rates for Planets in Low Mass Disks

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In a low-mass disk, a gap-opening planet acts as a leaky dam: the same two-sided torque that creates the exterior gas pileup sets the planet's inward migration rate.

desk verdict Solid moderate-gap theory and a real pileup result, but the deep-gap Type-II migration rate rests on a fit with an excluded extreme point. read the letter →

arxiv 1908.02326 v2 pith:UBE5W4CG submitted 2019-08-06 astro-ph.EP

Adam M. Dempsey , Wing-Kit Lee , Yoram Lithwick This is my paper · ORCID

classification astro-ph.EP

keywords planet-diskinteractionsprotoplanetarydisksaccretionplanetmigrationTypeIIgapopeningsurfacedensitypileupviscoussteadystate

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 studies planets in disks so low in mass that the planet migrates more slowly than gas flows through the disk, so the disk settles into a viscous steady state while overflowing the planet's orbit. It argues that one quantity, the total two-sided torque $\Delta T$ that the planet exerts on the disk, controls both the pileup of gas outside the planet's orbit and the planet's inward migration rate. Using 2D hydrodynamic simulations with boundary conditions that let the disk reach that steady state, the paper measures $\Delta T$, quantifies the pileup, and derives a new Type-II migration rate that connects continuously to the well-tested Type-I rate. This matters because directly imaged protoplanetary disks may show the pileup, offering a diagnostic of planet mass or disk viscosity.

What carries the argument

The load-bearing object is $\Delta T$, the total two-sided torque the planet applies to the disk, defined as the integral of the excited torque density over all radii. The argument separates angular-momentum transport into wave excitation, wave propagation, and deposition: waves carry angular momentum from Lindblad resonances near the planet to where they damp, and the deposited torque density $t_{\rm dep}$ shapes the surface-density profile through the viscous-steady-state equation $F_\nu = \dot M\ell + \int^r t_{\rm dep}\,dr'$. Far from the planet this yields $\Sigma = \Sigma_Z$ inside and $\Sigma = \Sigma_Z(1+\Delta T/(\dot M\ell))$ outside, so $\Delta T$ sets the pileup height, and the same $\Delta T$ enters the migration formula. The numerical boundary conditions are chosen to match these steady-state solutions so the pileup can survive.

What would settle it

Run a viscous-steady-state simulation with the inner boundary moved to a much smaller radius, or with the inner wave-killing zone removed, and check whether $\Delta T/(\dot M\ell_p)$ and the exterior pileup level stay within the paper's quoted accuracy; if they shift significantly, the empirical fit and the derived migration rate are biased by the boundary treatment.

Watch

Extended reading notes

Core claim

In a low-mass disk, a gap-opening planet acts as a leaky dam: gas keeps accreting inward across the planet's orbit, but the angular momentum the planet deposits makes the exterior disk denser than a planet-free disk by the factor $1+\Delta T/(\dot M\ell_p)$. The paper shows that the same two-sided torque $\Delta T$ fixes the migration rate through $\dot r_p/r_p = -2\Delta T/(M_p\ell_p)$, so in viscous steady state the planet is not locked to the disk's viscous accretion but migrates at a rate set by the torque balance. For moderately deep gaps the paper derives $\Delta T$ from a gap-depth argument based on the torque cutoff, matching simulations; for deep gaps the simulations give an empirical scaling, roughly $\Delta T/(\dot M\ell_p)\sim 4(q/\alpha)$, so the deep-gap migration rate becomes nearly independent of planet mass and viscosity. These results explain why earlier 2D simulations missed the pileup (incorrect boundary conditions) and why 1D local-deposition models overpredicted it dramatically.

Load-bearing premise

The load-bearing premise is that the measured two-sided torque $\Delta T$ is set by wave excitation close to the planet, so the artificial inner boundary and angular-momentum-conserving wave-killing zone that distort the inner disk profile do not change the pileup or migration rate.

Editorial extensions

If this is right

Gas exterior to a gap-opening planet in a low-mass disk should sit above the planet-free profile by $1+\Delta T/(\dot M\ell)$, giving a directly observable pileup that traces the planet-disk torque.
The deep-gap migration rate is nearly independent of planet mass and disk viscosity, scaling mainly with the disk-to-star mass ratio, in contrast to classical Type-II 'locked to the disk' migration.
The Type-I and Type-II migration rates join smoothly near $K\sim100$, so one torque formula covers the transition from low-mass to massive planets in low-mass disks.
Published 2D simulations that held all fluid quantities at their initial values at the boundaries would not produce the pileup, while 1D local-deposition models overpredict it by factors like $e^{200}$ at $K\sim10^4$.

Reading between the lines

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

Going beyond the paper: if an observed ring or cavity in a directly imaged disk is a pileup, comparing the brightness contrast at large radius with independent estimates of $\dot M$ and $\alpha$ could break the degeneracy between planet mass and viscosity that a single gap-depth measurement carries.
Going beyond the paper: the empirical deep-gap scaling $\Delta T/(\dot M\ell)\propto q/\alpha$ suggests the pileup may grow only sub-linearly as $K$ increases, so the leaky-dam picture implies an upper bound on pileup that could be tested by pushing simulations beyond $K\sim10^4$.
Going beyond the paper: a migration rate that is nearly independent of disk viscosity would mean giant-planet migration timescales in low-mass disks are no longer tied to the disk's viscous time, a shift that would change interpretations of observed exoplanet orbits.

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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, the paper shows that the missing pileup in 2D hydro was largely a boundary-condition problem, and that once you let the disk settle into viscous steady state, the outer pileup appears naturally. Second, the moderate-gap theory — the two-sided torque from linear C± — is clean and checks out against simulation at K ≲ 100. That half of the paper is in good shape.

The deep-gap half is where I get cautious. The headline result, the Type-II migration rate in Eq. (38), is built on the empirical fit Eq. (32) to ΔT. The fit runs from K~3 to K~10^4 in a single power law, even though the moderate-gap theory says ΔT scales as q^2/α and the deep-gap points are where the exponent has to change. And the one point that would most constrain the deep end (q1x3a3x4, pileup ~10) is discarded as 'unusually large' with statistical errors only and no sensitivity to that exclusion. The stress-test concern is genuine: include that point and the q exponent could easily move past 1.1, which erodes the claimed mass-independence. Since Eq. (38) is the main quantitative novelty, this is the load-bearing flaw.

The paper is also a bit quick to discard the eccentric and non-converged simulations; that is standard, but it is a selection step with no error assigned. And the inner wave-killing truncates the inner gap at high K; the authors argue ΔT is insensitive, which is plausible for moderate gaps but less obviously true when the gap edge runs into the damping zone.

Credit where it is due: the torque bookkeeping closes (Fig. 4e), Ṁ is constant to ~10%, the resolution check is minimal but real, and the comparison with Kanagawa et al. (2018) and Dürmann & Kley (2015) is a useful step toward understanding why earlier simulations missed the pileup and overestimated ΔT.

Audience: anyone working on gap-opening planets, Type-I/II migration, or reading ALMA rings as pileups. Worth a serious referee. I would send it out, and in revision push hard on the deep-gap fit: refit with the excluded point, bootstrap the selection, and try to derive the deep-gap scaling rather than fit it.

Referee Report

3 major / 4 minor

Summary. The paper studies planet-disk interaction in the low-disk-mass limit, where the planet's migration is slower than the disk's viscous evolution and the disk can be treated as passing through a sequence of viscous steady states (VSS). The authors introduce boundary conditions that allow 2D hydrodynamical simulations to reach VSS, and they show that the two-sided planet torque ΔT controls both the exterior pileup and the planet's migration rate. For moderately deep gaps they derive a first-principles prediction for ΔT (Eqs. 21-29) that agrees with their simulations at K ≲ 100. For deeper gaps they present an empirical power-law fit (Eq. 32) and use it to obtain a new Type-II migration rate (Eq. 38) that is nearly independent of planet mass and viscosity. They also show that previous 1D local-deposition models greatly overpredict the pileup, and they identify the difficulty of constructing a complete theory of very deep gaps.

Significance. If the central claims hold, the paper provides a clean organizing framework for planet-disk interaction in low-mass disks: a single quantity, ΔT, links the observable pileup, the gap profile, and the migration rate. The torque bookkeeping is carefully checked (Fig. 4), the moderate-gap prediction matches the simulations without fitted coefficients, and the resolution study gives 10% average (30% worst) agreement in ΔT. The paper also gives a clear explanation of why previous 1D local-deposition models fail and why earlier 2D simulations missed the pileup. These are substantial contributions. However, the headline deep-gap result, the new Type-II migration rate, rests on an empirical fit whose selection and regime boundaries are not robustly justified, so the quantitative deep-gap conclusions are not yet established at the same level as the moderate-gap results.

major comments (3)

[Section 4.5 and Eq. (32); used in Section 5.1, Eq. (38)] The empirical fit underlying the deep-gap migration rate is not restricted to the deep-gap regime. Eq. (32) is fitted to all runs with q > 10^-4, which includes many K < 100 points where the moderate-gap scaling ΔT ∝ q^2/α (Eq. 29) is known to hold and has a different dependence. The deepest-gap point q1x3a3x4 (K = 1.07e4, ΔT/(Ṁ l_p) = 11.0) is excluded as 'unusually large,' yet this is the most extreme and most observationally relevant point. At the quoted fit this point lies a factor of roughly 2 above the prediction, so its exclusion is consequential. Since Eq. (38) and the claim of near-independence from q and α follow directly from the fitted exponents q^1.05 and α^-0.91, the authors should refit using only K ≳ 100 points, report the fit with and without q1x3a3x4, and show how Eq. (38) changes. As written, the central deep-gap result is not robust to reasonable changes in the fitting procedure.
[Section 4.1, Section 5.4, and Table 1] The assertion that the artificial inner wave-killing zone has 'negligible effect on the value of ΔT' is load-bearing for the deep-gap results, but it is not demonstrated by a domain-size or wave-killing-zone test. For the highest-K runs the gap extends past the inner boundary (Section 5.2 and Fig. 11), and the deepest-gap point q1x3a3x4 is also the point with the largest low-resolution deviation in ΔT (14.2 vs. 11.0 in Table 1). Because ΔT at high K is the small difference of larger one-sided torques, a test that moves ri,wkz inward (or enlarges the domain) is needed to show that the deepest-gap pileup and the fitted Eq. (32) are not biased by the inner boundary treatment.
[Section 3.2 and Section 5.1] The VSS convergence criterion is a 10% global consistency of Ṁ, and runs that become eccentric or fail to converge are omitted from the analysis. This selection could bias the fitted q-α scaling if convergence correlates with q or α, which is plausible given that the deepest-gap and lowest-α runs are the most expensive and the least converged. The authors should show that the excluded or longer-time runs do not change the fitted exponents in Eq. (32), or at least quantify how the fit depends on the adopted convergence threshold.

minor comments (4)

[Section 4.5, Eq. (32)] The displayed uncertainty on the coefficient in Eq. (32) is garbled ('4.36.6 2.8'); the statistical errors on the exponents are quoted, but the notation should be cleaned up and the covariance of the fitted parameters should be reported.
[Section 3.1 and Fig. 4] The description of the inner and outer wave-killing zones would be clearer if the radii ri,wkz and ro,wkz were marked consistently on all panels of Figure 4; currently the reader must look back and forth between the text and the figure.
[Section 5.1, Eq. (38)] The statement that the deep-gap migration rate is 'roughly independent of their mass and the disk's viscosity' is only as good as the fitted exponents in Eq. (32). A short sensitivity statement around Eq. (38), showing how the q^0.05 α^0.09 factors change under the alternative fits suggested above, would help the reader assess the strength of this claim.
[Section 7, Open Questions] The paper is commendably explicit about the limitations of the deep-gap theory and about the inner-boundary issue. These statements should be retained in the final version; they correctly frame the empirical fit as a starting point rather than a complete theory.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor non-independent consistency check in the deep-gap one-sided torque comparison, but the central VSS derivation and moderate-gap theory are not circular.

fitted input called prediction [Section 5.4, Eq. (31) and Figure 14]
"Figure 14 compares this prediction for T± with what is found in the simulations, where the “prediction” makes use of the values of x± and Σ± extracted from the simulations."

The one-sided torques T± are measured from the same hydrodynamical simulations that supply the inputs x± and Σ±. Equation (31) evaluates the analytic torque formula at those measured peak locations and peak surface densities, and Figure 14 then compares the result with the measured T±. The agreement is therefore a consistency check of the approximate torque formula, not an independent, out-of-sample prediction of the deep-gap torques. The paper itself marks the word “prediction” with scare quotes, and this step is not used to determine the deep-gap ΔT or the migration rate; those come directly from the simulation measurements and the empirical fit in Eq. (32). Hence the circularity is real but minor and non-load-bearing.

full rationale

The main derivation chain is not circular. The VSS relations, Eqs. (17)–(19), follow from angular-momentum conservation, and the moderate-gap torque prediction, Eqs. (21)–(29), is obtained from linear theory with the coefficients C± computed in Appendix B rather than fitted to the simulations; it is then checked against the simulations and agrees at K ≲ 100. The deep-gap migration rate, Eq. (38), is not presented as a first-principles prediction: the paper states “Using that fit to ΔT, the migration rate is,” and Eq. (38) is the algebraic consequence of the empirical power-law fit Eq. (32). That is an honest simulation-based result, not a fitted parameter renamed as a prediction. The only place where a “prediction” is built from the same simulation’s own outputs is the Eq. (31)/Figure 14 comparison of one-sided torques, which is non-independent but does not support the central pileup or migration claims. The self-citations (Lee 2016; Lee et al. 2019) are used for technical sub-points, not as load-bearing justification, and no uniqueness theorem is imported from the authors’ prior work. Concerns about excluding the K ∼ 10^4 point or omitting eccentric/non-converged runs are statistical robustness issues, not circularity. Overall, the central moderate-gap theory is self-contained, and the deep-gap results are transparently empirical.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The derivation relies on the VSS equations (Section 2), linear Lindblad torque theory with computed coefficients C±, and standard viscous disk theory. No new particles or forces are introduced. The free parameters are the deep-gap empirical fit coefficients (Eq. 32) and the gap-depth fit parameter K_c (Section 5.2). The main axioms are the locally isothermal 2D disk model, the fixed-planet non-accreting assumption, the applicability of the standard torque formula at x±, and the validity of the boundary/wave-killing treatment of the inner disk. The paper itself flags the EoS, 3D, and accretion effects as open questions in Section 7.

free parameters (2)

Deep-gap ΔT fit: normalization and exponents (Eq. 32) = A = 4.36 (range 2.8 to 6.6), exponent on q = 1.05 ± 0.06, exponent on α = -0.91 ± 0.04
Fitted to the VSS simulation values of ΔT for q ≥ 10^-4, with the K ≈ 10^4 point excluded (Section 4.5, Figure 10). This fit is then recast as the deep-gap migration rate in Eq. (38), so uncertainties from the fit propagate directly into the headline migration result.
Gap depth fit parameter K_c = K_c = 180
Fit parameter in the corrected gap depth scaling Σ_p = 1/(1 + 0.04K + (K/K_c)^2) in Section 5.2; this matches the 'two-step' depth behavior and is not derived from theory.

assumptions (5)

domain assumption The disk is locally isothermal, P = c_s^2 Σ with c_s = h r Ω_K, with h = 0.05 fixed.
Adopted in all simulations and in the VSS equations (Section 3). The paper notes in footnote 5 that t_dep differs for an adiabatic EoS and lists EoS tests as future work (Section 7).
domain assumption The planet is on a fixed circular orbit and does not accrete material; migration is inferred from the torque on a stationary planet.
Core to the VSS setup (Sections 2 and 5.1). The validity criterion M_d ≲ M_p is derived in Section 5.1, and planet accretion is listed as an open question in Section 7.
domain assumption The waves' angular momentum deposition (t_dep) controls the surface density profile, and the standard Lindblad torque formula with the leading asymmetry (Eq. 30) is applicable at the excitation sites x±.
Used for the moderate-gap derivation (Section 2.3.2) and the deep-gap consistency steps (Section 5.4); verified against linear calculations in Figure 7, but the deep-gap extension assumes Eq. (30) with C ≈ 2.5.
domain assumption The 2D hydrodynamical treatment with a softened planet potential (softening 0.6h) captures the relevant physics; 3D effects are neglected.
Section 3; 3D effects and planet accretion are listed as open questions in Section 7.
ad hoc to paper The imposed boundary conditions with wave-killing zones that damp only v_r (conserving angular momentum) drive the disk to the true VSS, and the artificial inner wave-killing zone does not affect ΔT.
Sections 3.1 and 4.1. The insensitivity of ΔT to the inner boundary is argued from the locality of excitation, and the paper explicitly states the Σ profile is incorrect at r < r_i,wkz.

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Cite this review

Pith. "Pith review of Pileups and Migration Rates for Planets in Low Mass Disks." pith.science (2026). https://pith.science/paper/UBE5W4CG

@misc{pith2026190802326,
  author       = {Pith},
  title        = {Pith review of: Pileups and Migration Rates for Planets in Low Mass Disks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UBE5W4CG}},
  note         = {Machine review of arXiv:1908.02326}
}

read the original abstract

We investigate how planets interact with viscous accretion disks, in the limit that the disk is sufficiently low mass that the planet migrates more slowly than the disk material. In that case, the disk's surface density profile is determined by the disk being in viscous steady state (VSS), while overflowing the planet's orbit. We compute the VSS profiles with 2D hydrodynamical simulations, and show that disk material piles up behind the planet, with the planet effectively acting as a leaky dam. Previous 2D hydrodynamical simulations missed the pileup effect because of incorrect boundary conditions, while previous 1D models greatly overpredicted the pileup due to the neglect of non-local deposition. Our simulations quantify the magnitude of the pileup for a variety of planet masses and disk viscosities. We also calculate theoretically the magnitude of the pileup for moderately deep gaps, showing good agreement with simulations. For very deep gaps, current theory is inadequate, and we show why and what must be understood better. The pileup is important for two reasons. First, it is observable in directly imaged protoplanetary disks, and hence can be used to diagnose the mass of a planet that causes it or the viscosity within the disk. And second, it determines the planet's migration rate. Our simulations determine a new Type-II migration rate (valid for low mass disks), and show how it connects continuously with the well-verified Type-I rate.

Figures

Figures reproduced from arXiv: 1908.02326 by the authors.

**Figure 1.** Illustration of the VSS solution given in Eqs. (17) and (19). The top panel shows an example steady-state Σ profile compared to the ZAM Σ profile given in Eq. (20). The bottom panel shows the corresponding Fν profile compared to the ZAM Fν profile, M ` ˙ . The constant offset between Fν and M ` ˙ at large radii corresponds to the total torque input by the planet, ∆T. For ease of reference below, we call it the “zero… view at source ↗

**Figure 2.** The parameter space we explore with FARGO3D. Filled circles indicate simulations which have converged to VSS (M˙ deviations less than 10%). Brown open squares show simulations which transitioned to an eccentric disk state, and hence will be discarded from our analysis. We omit a simulation with q = 10−3 and α = 10−4 that did not converge to VSS. We set h = 0.05 in all simulations. We have indicated where the thermal… view at source ↗

**Figure 3.** The two-dimensional surface density for our standard simulation with q = α = 10−3 . We overplot a sample of gas streamlines (white lines) and the separatrices (red dashed line) which separate the circulating streamlines from the librating streamlines [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (13 more)

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: The measured values of x± for all of the VSS simulations. The color of each point represents its α value. Simulations with K . 100 have |x±| ≈ h while larger K simulations have |x±| > h. and replaced the missing bit with dashed lines. We do so because the profiles have…

**Figure 7.** Figure 7: The specific torque profiles, tex/ hΣi, for K ∼ 10 (top row) and K ∼ 3, 000 (bottom row). The figure shows that the linear torque is always an adequate approximation at x ∼ x±. And in the high-K case, the analytic torque is a good approximation at x ∼ x±. See main text…

**Figure 8.** Figure 8: Two-dimensional maps of tdep for two simulations with K ∼ 1, 000. The colorscale is logarithmic for values greater than 100 and linear for values less than 100. The separatrices (black lines) mark the transition from librating to circulating fluid streamlines in the co…

**Figure 9.** Figure 9: One-sided Lindblad torques, T±,LR, (left) and total (two-sided) torques, ∆T, (right) for all of our VSS simulations as a function of K. In the right panel, the open points show the total Lindblad torque and the filled points show the total torque (Lindblad + co-orbital…

**Figure 10.** Figure 10: α∆T, normalized to M ` ˙ p, as a function of q for all of our VSS simulations. The lines show the result of the fit given in Eq. (32) for each of the α values, demonstrating that α∆T is roughly independent of α. includes co-orbital torques, and appears to provide a be…

**Figure 11.** Figure 11: Top: Gap depths for all of our VSS simulations. Here we define the gap depth as the minimum surface density excluding the circumplanetary disk region. Overplotted we show the literature scaling relation for moderately deep gaps, 1/(1 + 0.04K) (dotted line; Eq. 26), an…

**Figure 12.** Figure 12: compares our values of ∆T, with those from simulations by Durmann & Kley ¨ (2015) and Kanagawa et al. (2018). In contrast to our VSS boundary conditions, Kanagawa et al. (2018) set all fluid quantities equal to their initial conditions, which corresponds to the ZAM s…

**Figure 13.** Figure 13: Schematic illustration of a VSS Σ profile highlighting the four important quantities, x± and Σ±. Most of the one-sided torques are excited at x± with strengths given by Eq. (31). The Σ profile between x− and x+, as well as the locations of x± are set by the local tor…

**Figure 14.** Figure 14: One-sided Lindblad torques for simulations with K & 100 (triangles) compared to Eq. (31) for the measured values of x± and Σ± (crosses). Above K ∼ 100, the agreement between Eq. (31) and T±,LR shows that the analytic torque formula evaluated at x± is a good approxima…

**Figure 16.** Figure 16: Radial profiles of Fwave and Fν from an example hydrodynamical simulation. Left : Radial profiles of Fwave (grey line) and the three components defined in Eq. (9). The standard wave flux term ∝ [PITH_FULL_IMAGE:figures/full_fig_p024_16.png]

**Figure 17.** Figure 17: Comparison of different analytic tex profiles for Σ = const. In the outer disk, Eq. (E27) (orange solid) agrees with Ward’s tex (blue solid; Eq. (E20)) to within 3% near x ≈ 0.2, and both lie above the standard torque formula (black dotted; Eq. (E23). Similarly, in th…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

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[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix "arXiv" = new.block " " eprint * " " * new.block " " eprint * " " * if if if FUNCTION format.doi doi empty "" " " doi * " " * if FUNCTION format.pid doi empty eprint empty ""...

[3] [3]

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thebibliography [1] 20pt to REFERENCES 6pt =0pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command Each re...

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