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REVIEW 3 major objections 5 minor 41 references

Correlation between planet formation rate and gas surface density: an analog of Kennicutt Schmidt law for planet formation

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper derives a star-formation-law analog for planet formation: $PFR \propto \Sigma_g^n$, with $n$ from $4/3$ to $2$ depending on the formation mechanism.

desk verdict A useful scaling-law package, but the gas-giant exponent rests on an unstated seed-density assumption and the observational test is only two points; worth peer review with major revision. read the letter →

arxiv 2412.16278 v1 pith:BNOJLBXY submitted 2024-12-20 astro-ph.EP astro-ph.GA

Mor Rozner This is my paper · ORCID
classification astro-ph.EPastro-ph.GA
keywords planetformationrateprotoplanetarydisksgassurfacedensitypebbleaccretiongiantgravitationalinstabilitypower-lawscalingstar-formationlaw
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 tries to establish that the rate at which planets form in a protoplanetary disk is a power-law function of the gas surface density, $PFR \propto \Sigma_g^n$, in direct analogy to the star-formation scaling law measured in galaxies. The exponent $n$ is derived separately for each major formation channel, and the values are $3/2$ for terrestrial planets in the headwind pebble-accretion regime, $4/3$ in the shear regime, $2$ for gas giants in both runaway and gap-opening accretion, and $3/2$ for planets formed by gravitational instability. If the relation is correct, the gas content of a disk directly sets the pace of planet formation, giving observers a simple diagnostic for which mechanism produced a given planet population.

What carries the argument

The load-bearing estimator is $PFR \approx \epsilon_{PFR}\,\Sigma_p/t_{grow}$, where $t_{grow} = M/(dM/dt)$ is the mass-growth timescale of a typical seed. The power-law transfer from growth rate to gas density is carried by four physical inputs: the pebble-to-gas fraction $f_{peb}=\Sigma_{peb}/\Sigma_g=0.01$ that ties pebble accretion (capture of small gas-coupled particles by a growing protoplanet) to the gas reservoir, the Bondi accretion rate $\dot{M}_{RA}\propto\rho_g$ for runaway envelope growth, a gap-opening accretion rate proportional to $\Sigma_g$, and the free-fall timescale $t_{ff}\propto\rho_g^{-1/2}$ for gravitational instability. Efficiencies $\epsilon_{pro}$ and $\epsilon_{core}$ are set to $0.2$, following the pebble-flux treatment of core formation, and they set the overall normalization rather than the exponents.

What would settle it

Take a sample of protoplanetary disks with measured gas surface densities and with directly detectable embedded planets whose accretion rates give independent planet formation rates, extend the two-point comparison made here to tens of disks, and fit $\log PFR$ versus $\log \Sigma_g$ separately for terrestrial-type and giant-type systems; the central claim fails if the giant-planet slope is not close to $2$ or the terrestrial slope is not close to $3/2$ or $4/3$. A numerical simulation that varies disk gas density while holding the seed population fixed should reproduce the same slopes.

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Extended reading notes

Core claim

Defining the planet formation rate as $PFR \approx \epsilon_{PFR}\,\Sigma_p/t_{grow}$, with $\Sigma_p$ the surface density of growing seeds and $t_{grow}$ the growth timescale from each accretion mechanism, the paper finds that all channels collapse to $PFR \propto \Sigma_g^n$ with $n = 3/2$ (terrestrial, headwind), $4/3$ (terrestrial, shear), $2$ (gas giants and gap-opening gas giants), and $3/2$ (gravitational instability). The key steps are writing the pebble surface density as a fixed fraction $f_{peb}\Sigma_g$, treating runaway gas accretion as Bondi accretion proportional to gas density, using a gap-opening accretion rate linear in $\Sigma_g$, and using the free-fall time for gravitational instability. The paper further claims this local law yields a global disk-integrated relation, applies above a threshold gas density, and predicts self-regulation as the gas is depleted.

Load-bearing premise

The load-bearing premise is that the surface density of growing seeds, $\Sigma_p$, stays proportional to the gas surface density through a fixed efficiency factor; if seed production saturates or decouples from the gas reservoir, the exponents and normalizations of the planet formation rate change.

Editorial extensions

If this is right

  • In the headwind pebble-accretion regime, terrestrial planet formation should speed up as $\Sigma_g^{3/2}$, while in the shear regime the slope is shallower, $\Sigma_g^{4/3}$, so resolved disk density maps can pick out which accretion mode operates.
  • Gas giant formation should scale as $\Sigma_g^2$, making giant planets strongly concentrated in the most gas-rich disks and rare in gas-poor ones.
  • A measured slope near $3/2$ in a planet population would point to gravitational instability or headwind pebble accretion, whereas a slope near $2$ points to gas-accretion-dominated channels.
  • Integrated over a disk, the local law becomes a global relation between total planet formation rate and disk gas content, and it predicts that planet formation self-regulates downward as gas is depleted.

Reading between the lines

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

  • The paper leaves implicit that, if the relation holds, surveys measuring disk gas mass could be converted into expected planet formation rates, letting modelers predict where planets should appear before they are directly imaged.
  • A clean statistical test would compare exoplanet occurrence rates against host-disk gas mass: a break from $n\approx 4/3$-$3/2$ at low masses to $n\approx 2$ at high masses would support mechanism-dependent slopes, while a single global slope would indicate that seed densities decouple from gas density.
  • Migrating planets complicate the comparison: planets form at the local gas density but are later observed far from their birth site, so matching the predicted slopes to observed planet positions requires accounting for orbital migration and disk evolution.
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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

3 major / 5 minor

Summary. The manuscript proposes an analog of the Kennicutt-Schmidt law for planet formation, claiming a first-principles derivation of a power-law relation PFR ∝ Σ_g^n for different formation channels. It derives n = 3/2 for terrestrial planets in the headwind pebble-accretion regime, n = 4/3 in the shear regime, n = 2 for gas giants, n = 2 for gap-opening gas giants, and n = 3/2 for gravitational instability (Eq. 7). The paper then compares the predicted scaling with crude PFR estimates for the TW Hya and PDS 70 disks, concluding that the observed trend is consistent with the theory. The central result is a set of characteristic exponents that could serve as observational discriminants for formation mechanisms.

Significance. If the derivation were fully rigorous, this would be a useful conceptual bridge between planet formation and star formation, offering falsifiable predictions that could be tested with future ALMA/JWST observations of planet-forming disks. The paper is clearly written and the bottom-up exponents follow from textbook accretion rates, so the core idea is attractive. However, the gas-giant exponent rests on an unstated assumption, the observational test is only a two-point comparison with orders-of-magnitude uncertainties, and the gravitational-instability timescale is chosen without clear physical justification. The strengths are the compact formulation of a potentially testable scaling law and the explicit enumeration of channel-specific exponents.

major comments (3)
  1. [§II.B, Eq. (7)] The gas-giant exponent n=2 is not derived from first principles because the scaling of the core surface density Σ_p with gas surface density is never established. The text defines ε_core = Σ_core/Σ_p, which is tautological (it forces ε_core=1 and contradicts the adopted value 0.2), and even if the intended definition is ε_core = Σ_core/Σ_peb, the result PFR ∝ Σ_g^2 requires the additional unstated assumption that Σ_core ∝ Σ_g. In drift-limited pebble accretion, the core mass is set by the integrated pebble flux and the core spacing by isolation conditions, so a linear proportionality is not guaranteed; the authors should either derive this scaling or explicitly state it as an assumption and discuss its range of validity.
  2. [§III, Fig. 1] The observational comparison is too weak to support the claim that the data show the predicted trend. Only two disks are used, the area A = π(100 AU)^2 used to convert a single-planet accretion rate into PFR_obs is arbitrary, and the ranges for Σ_g and PFR overlap substantially between the two objects. The statement that 'this trend could be seen also from the observational data' is an overstatement; the comparison should be framed as an illustrative order-of-magnitude check, not as a validation of the derived exponents.
  3. [§II.C, Eq. (5)] The use of the free-fall time t_ff as the growth timescale for gravitational instability is not justified for a rotating disk, where the natural dynamical timescale is the orbital period Ω^{-1}. Since Ω is independent of Σ_g at fixed radius and stellar mass, adopting Ω would give PFR_GI ∝ Σ_g rather than n=3/2. The authors should justify the t_ff choice physically or acknowledge that the GI exponent is only one of several possible scalings.
minor comments (5)
  1. [Eq. (7)] There is a typo: 'terrestiral' should be 'terrestrial'.
  2. [Eq. (1)] The two-case equation for the pebble accretion rate is missing a closing brace or delimiter; the Ṁ_PA expression is not fully typeset, which hampers readability.
  3. [Eq. (5)] The free-fall time is written as sqrt(2π/32Gρ_g); the standard expression is sqrt(3π/(32Gρ_g)) or equivalent. The numerical factor does not affect the scaling, but the formula should be correct.
  4. [Table A] The fiducial value ε_PFR = 1 is not an efficiency but means no efficiency reduction; the table entry should be clarified. Some rows are also misaligned in the table formatting.
  5. [§III, Eq. (6)] The text says PFR 'measures the number of planets' but the formula ε_PFR Σ_p Γ_grow has dimensions of mass per area per time (a surface density growth rate). The wording should be aligned with the dimensional meaning.

Circularity Check

1 steps flagged · score 6.0 of 10

Partial circularity: gas-giant n=2 entries in Eq. (7) are carried by an unstated Sigma_core proportional to Sigma_g assumption hidden behind the tautological efficiency definition epsilon_core = Sigma_core/Sigma_p.

  1. self definitional [Section III, Eq. (6) and the efficiency definitions immediately following it; exponent list in Eq. (7)]
    "We estimate the PFR by PFR ≈ ϵPFRΣpΓgrow ... where ... Σp is the surface density of "growing seeds", i.e. protoplanets for the case of terrestrial planets and cores for gas giants formation ... We set the efficiency of protoplanets/cores formation by ϵpro = Σpro/Σpeb and ϵcore = Σcore/Σp correspondingly"

    For gas giants the paper has already defined Sigma_p as the core surface density, so the defining relation epsilon_core = Sigma_core/Sigma_p is an identity (epsilon_core = 1), not a physical efficiency; the quoted value 0.2 is inconsistent with the definition. Substituting the Bondi rate (3), with Gamma_grow = Mdot_RA/M_core proportional to Sigma_g, into Eq. (6) gives PFR_GG proportional to Sigma_core times Sigma_g. The exponent 2 in Eq. (7) therefore requires the extra, never-derived assumption that the core surface density itself scales linearly with gas surface density. The gas-giant 'prediction' is thus an input inserted through the efficiency notation and not a first-principles result.

full rationale

The terrestrial-planet exponents (3/2 headwind, 4/3 shear) and the gravitational-instability exponent (3/2) are self-contained given Eq. (2), constant epsilon_pro, and t_ff proportional to rho_g^{-1/2}; these branches are not circular. The gas-giant branches (n=2 for both ordinary and gap-opening giants) depend on a linear core-seed scaling that the manuscript never derives and that its own efficiency definition makes tautological. There is no load-bearing self-citation (the single self-reference, [21], is not used in the PFR derivation), and the exponents are not fitted to the two observational disks, so the circularity is partial rather than total. Because the gas-giant branch is a major element of the central law and the paper's 'derived from first principles' claim is overstated for that branch, a score of 6 reflects this branch-specific, construction-level circularity.

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

The derivation of the power-law indices relies on standard published growth rates and on a collection of adopted efficiencies and fiducial masses. The most consequential hidden input is the assumed linear scaling of seed surface density with gas surface density; without it, the gas-giant exponent changes. No new physical entities are introduced.

free parameters (6)
  • f_peb, pebble-to-gas surface density ratio = 0.01
    Adopted as a typical constant in Section II A despite Eq. (2) showing radial and time dependence. Sets the normalization of pebble accretion and, through the assumed seed density scaling, affects the PFR power law.
  • epsilon_PFR, planet formation efficiency = 1
    Chosen in Eq. (6); a free normalization for all channels.
  • epsilon_pro, protoplanet formation efficiency = 0.2
    Appendix A; converts pebble surface density to protoplanet seed surface density following reference [23].
  • epsilon_core, core formation efficiency = 0.2
    Appendix A; converts pebble surface density to core surface density for gas giant formation.
  • Typical seed masses M_pro, M_core, M_gap = 1e-3 M_earth, 10 M_earth, 1 M_Jup
    Appendix A; used to evaluate the relative growth rate Gamma_grow in Eq. (6), setting normalization but not the exponent.
  • A, assumed disk area for PFR_obs = pi (100 AU)^2
    Section III; used to convert observed accretion rates to an areal planet formation rate for both TW Hya and PDS 70; this arbitrary choice shifts the data points vertically.
assumptions (8)
  • domain assumption PFR can be modeled as epsilon_PFR * Sigma_p * Gamma_grow
    Eq. (6); the central definition of the planet formation rate, with a constant efficiency factor.
  • ad hoc to paper Seed surface density Sigma_p scales linearly with Sigma_g
    Needed for the n=2 gas giant exponents in Eq. (7); nowhere derived or explicitly justified.
  • domain assumption f_peb is constant at 0.01 independent of radius and time
    Section II A adopts a typical value despite Eq. (2) showing f_peb depends on pebble production line and drift velocities.
  • domain assumption Gas volume density is Sigma_g divided by a fixed disk scale height at the planet location
    Used implicitly in Bondi accretion and free-fall scaling to convert Sigma_g to rho_g.
  • domain assumption Bondi accretion describes runaway gas accretion onto cores
    Section II B; standard but approximate, with corrections cited that are said not to change the scaling.
  • domain assumption Gap-opening accretion rate of Tanigawa and Watanabe applies
    Section II B Eq. (4); used for the second gas-giant channel.
  • domain assumption Free-fall time is the growth timescale for gravitational instability planets
    Section II C Eq. (5); an analogy to star formation rather than a derived planet formation rate.
  • ad hoc to paper Observed single-planet accretion rate divided by an arbitrary disk area estimates the local PFR
    Section III; used for the TW Hya and PDS 70 comparison, mixing point-source accretion with an areal rate.

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Cite this review

Pith. "Pith review of Correlation between planet formation rate and gas surface density: an analog of Kennicutt Schmidt law for planet formation." pith.science (2026). https://pith.science/paper/BNOJLBXY

@misc{pith2026241216278,
  author       = {Pith},
  title        = {Pith review of: Correlation between planet formation rate and gas surface density: an analog of Kennicutt Schmidt law for planet formation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BNOJLBXY}},
  note         = {Machine review of arXiv:2412.16278}
}
abstract

The efficiency of planet formation is a fundamental question in planetary science, gaining increasing significance as observational data from planet-forming disks accumulates. Here we derive from first principles a correlation between the planet formation rate (PFR) and the gas surface density, i.e. $\rm{PFR}\propto \Sigma_g^n$. This relation serves as an analog for the well-established Kennicutt-Schmidt law for star-forming galaxies. We study the different planet formation mechanisms and the density dependence in each one of them, to finally formulate a simple relation. We find that the powerlaw ranges between $n\approx 4/3-2$, depending on the type of the forming planet, when we carry out different analyses for the formation rates of terrestrial planets, gas giants, and also planets formed by gravitational instability. We then compare our results with the available observational data. The relation we derive here aims to shed more light on the interpretation of observational data as well as analytical models, and give a new perspective on the properties of planet formation and its connection to gas.

Figures

Figures reproduced from arXiv: 2412.16278 by the authors.

Figure 1
Figure 1. FIG. 1. The planet formation rate (PFR) as a function of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.