Origin of compact exoplanetary systems during disk infall

Raluca Rufu; Robin M. Canup

REVIEW 2 major objections 5 minor 15 references

Origin of compact exoplanetary systems during disk infall

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that compact multi-planet systems form during the final infall of gas and solids onto the disk, with planet masses set by a balance between solid accretion and inward Type-I migration, yielding a common system mass ratio…

desk verdict A genuinely new physical mechanism for the compact-system mass ratio, built on a coherent accretion-migration balance, with one large unmodeled planetesimal-formation premise that the paper honestly flags. read the letter →

arxiv 2505.22806 v1 pith:ORN5GWFA submitted 2025-05-28 astro-ph.EP physics.space-ph

Raluca Rufu , Robin M. Canup This is my paper

classification astro-ph.EPphysics.space-ph

keywords compactexoplanetarysystemsplanetformationdiskinfallType-Imigrationplanetesimalprotoplanetarydiskssystemmassratiosuper-Earths

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 proposes that the compact, short-period multi-planet systems seen by Kepler are relics of planet formation that happened while the circumstellar disk was still being fed by infall, not after infall ended. Inside the centrifugal radius, a growing planet's mass is capped by the point where the rate of solid infall equals the rate of inward gas-driven migration, because larger planets migrate faster. This self-regulating balance yields similar-sized planets and, integrated over the disk, a total system mass of a few $10^{-5}$ to $10^{-4}$ times the host star's mass across a wide range of disk and infall conditions. The predicted mass ratio is essentially independent of stellar mass and weakly dependent on metallicity, matching the observed 'peas-in-a-pod' compact systems and their weak metallicity bias.

What carries the argument

The central object is the accretion-migration balance: a planet inside the centrifugal radius $r_c$ accretes infalling solids on a timescale $\tau_{\rm acc}\propto M_p/(\text{infall rate})$, while the gas disk drives inward Type-I migration on a timescale $\tau_I\propto (M_*/M_p)(\Sigma_g r^2)^{-1}(H/r)^{-2}$. Setting $\tau_{\rm acc}=\tau_I$ gives the critical mass $M_{\rm crit}$ (eqn. 2); the mass density of critical-mass planets, integrated over the infall region, gives the system mass ratio (eqn. 3); and evaluating it when migration slows ($\tau_I\sim10\tau_g$) gives the final ratio (eqn. 4). The analysis also uses a gas disk model with infall, viscous spreading, and photoevaporation to estimate $\beta\equiv \tau_g/\tau_{\rm in}$ in the range $1.3$ to $2$ for compact ($r_c<1$ au) disks.

What would settle it

A survey of Class 0/I protostellar disks showing that planetesimals do not form interior to the centrifugal radius during infall would falsify the model's premise; a testable alternative is that compact-system mass ratios should be nearly metallicity-independent, so a strong metallicity trend would rule it out.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the common mass ratio of compact exoplanetary systems is not an accident of initial conditions but a self-regulating outcome of accreting during infall. Equating the solid-infall-limited accretion timescale with the Type-I migration timescale yields a critical planet mass $M_{\rm crit}$ (eqn. 2) that varies weakly with infall rate; planets near $M_{\rm crit}$ migrate inward and are replaced by newly accreting ones, so the integrated system mass ratio $M_{\rm tot}/M_*$ (eqn. 3) stays near a narrow band. When gas dispersal makes migration slow compared with the disk lifetime, the surviving systems retain a final mass ratio (eqn. 4) of a few $10^{-5}$ to $10^{-4}$, consistent with observations. The mechanism also produces the observed uniformity of planet sizes within each system.

Load-bearing premise

The load-bearing premise is that km-sized planetesimals form inside the centrifugal radius during the final infall stage, on a timescale shorter than the infall decay; the paper argues this is plausible from high-temperature sticking, streaming instability, and particle concentration, but does not model it.

Editorial extensions

If this is right

Compact systems should show a total mass ratio of a few $10^{-5}$ to $10^{-4}$ times the host star, nearly independent of stellar mass and of the specific infall rate or disk viscosity.
The system mass ratio depends on stellar metallicity only weakly, as a power of about $0.2$ to $0.3$, so compact systems should be found around stars of widely varying metallicity.
Systems with estimated mass ratios below the predicted range may host additional, as-yet-undetected planets, since the model sets a system-wide mass budget rather than a per-planet one.
After the gas disperses, dynamical instabilities can reshape the period-ratio distribution toward the observed, non-resonant configuration while changing the system mass ratio by only a few percent.

Reading between the lines

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

If infall-era accretion is real, young Class 0/I disks should show dynamical or dust-continuum evidence of embedded massive bodies near $r_c$; searching for such signatures with ALMA would test the proposed timing of planet formation.
The model predicts that the innermost planet orbit traces the inner edge of the planetesimal formation region, so comparing minimum orbital periods with condensation-temperature radii across compact systems would provide a test.
The same accretion-migration balance may operate in any infall-fed disk, including circumplanetary disks, and the difference in the ratio of disk-dispersal to infall timescales may explain why satellite systems peak at a somewhat lower mass ratio than compact exoplanetary systems.

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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

The paper gives a physical mechanism for the observed common mass ratio of compact multi-planet systems: planets form during the final infall phase, and their mass is set by the balance between accretion of infalling solids and inward Type-I migration. That is new and worth taking seriously. The analytical derivation is transparent, and the N-body simulations reproduce the analytical predictions across a range of parameters. The weak-metallicity-dependence prediction is a nice falsifiable consequence that distinguishes the model from some alternatives.

What it does well: the derivation is internally consistent (eqns. 2-4 are the backbone, and the supplementary derivation checks out), the simulations span two stellar masses and both with and without inner cavities, and the paper is unusually honest about its own limitations. The Methods section explicitly concedes that models for grain growth up to planetesimal formation are lacking and collapse conditions are unclear, particularly for turbulent disks.

That concession points to the real soft spot. The load-bearing premise is that km-sized planetesimals form interior to rc during the few 1e5 yr of infall. The N-body simulations bypass this by injecting solids as a few 1e-8 M* bodies, and the analytical tau_acc assumes the solid infall rate limits accretion rather than the dust-to-planetesimal step. If growth to St ~ 0.01-0.1 or streaming-instability collapse takes longer than tau_in ~ 3e5-1e6 yr, the accretion-migration balance never establishes and eqn. 4's mass-ratio regulation fails. The paper argues plausibility from high-temperature sticking, vortices, and drift pile-up, but it does not demonstrate the timescale. That is a genuine unmodeled physical step, not an algebraic inconsistency, and the paper says so.

The absolute normalization also depends on (alpha epsilon/f), H/r, and Min/M*, none of which are tightly constrained. The beta range 1.3-2 comes from a disk-evolution model with its own assumptions, though the paper checks that low-alpha fast-infall cases behave differently. The code is proprietary, but some simulation outputs are archived on Zenodo, which is at least partial credit.

If the planetesimal-formation premise holds, this explains a puzzling observational pattern that has resisted prior models. The premise is identified clearly and could be tested by dedicated dust-growth simulations. I would send this to peer review: the model is coherent, novel, and the weak spot is addressable in the literature rather than fatal. A referee should push on the premise and on whether the beta range is robust, but the paper merits serious engagement.

Referee Report

2 major / 5 minor

Summary. The paper proposes that compact multi-planet systems form during the final infall phase of the circumstellar disk, rather than after infall ends. In the proposed scenario, planet masses are set by a balance between the accretion of infalling solids and increasingly rapid Type-I migration, so that the total system mass is regulated to a few times 10^-5 to 10^-4 of the stellar mass (Eq. 4). The authors derive this mass ratio analytically (Eqs. 2-4) and support it with N-body simulations that inject growing bodies into an infall-supplied disk. They argue that the weak dependence of the final mass ratio on disk and infall parameters explains the observed clustering of compact-system mass ratios, the 'peas-in-a-pod' architecture, and the weak metallicity dependence. The paper explicitly identifies as a premise that planetesimals form interior to the centrifugal radius r_c during the final infall stage, and it acknowledges that the dust-to-planetesimal growth step is not modeled.

Significance. If the central premise holds, the paper offers a novel and potentially important explanation for the observed common mass ratio of compact exoplanetary systems, one that naturally connects to the physics of infall and migration. The analytical derivation is internally consistent, the N-body simulations reproduce the analytical scalings across a wide parameter range, and the paper makes falsifiable predictions (weak metallicity dependence, a transition in mass-ratio behavior near r_c, and early accretion during infall). The authors also provide data for the simulations. However, the entire framework rests on the unmodeled assumption that planetesimals form efficiently inside r_c during the final infall stage; the paper itself states that models for this growth are lacking. This limits the current significance to a plausibility argument rather than a demonstrated formation pathway.

major comments (2)

[Results, 'A premise of our model'; Methods, 'Dust-to-planetesimal accretion'] The central claim of the paper, expressed in Eq. (4) and the abstract, depends on planetesimal formation interior to r_c during the final infall stage. The paper states this as a premise in the Results and then concedes in the Methods section 'Dust-to-planetesimal accretion' that 'models that explicitly treat grain growth up to planetesimal formation are lacking, and whether conditions for collapse are actually achieved is unclear, particularly for turbulent disks.' The analytical accretion timescale (Supplemental Discussion 2, Eq. 2.2) and the N-body injection scheme assume that the solid infall rate limits accretion; they do not include the time or efficiency of the dust-to-planetesimal step. If growth to Stokes number St ~ 0.01-0.1 and streaming-instability collapse at Zcrit ~ 0.02-0.2 for alpha = 10^-4 to 10^-3 takes longer than tau_in ~ 3 x 10^5 to 10^6 yr, the tau_acc ~ tau_I balance in Eq. (2) never establishes and Eq. (4)'s mass-ratio regulation does not follow. Please either provide a quantitative estimate of the dust-to-planetesimal timescale in the inner infall region, or show that the final mass ratio is insensitive to delayed or inefficient planetesimal formation (e.g., through N-body experiments with delayed injection of solids).
[Methods, 'Infall description' and 'N-body planet accretion simulation'] In the N-body simulations, the infalling solids are injected as bodies of mass a few x 10^-8 M_sun, which for a solar-mass star is about 3 x 10^-3 M_Earth, roughly four to five orders of magnitude more massive than km-sized planetesimals. This injection scheme bypasses not only the dust-to-planetesimal stage but also the growth from planetesimals to embryos. The simulations therefore validate the accretion-migration balance only after a population of massive embryos is assumed to exist; they do not provide independent evidence for the planetesimal-formation premise. I recommend that the authors state this limitation explicitly in the main text and, if feasible, test sensitivity to the injected mass and to the temporal and radial distribution of injection.

minor comments (5)

[Introduction, paragraph 4] The sentence 'The flux density at different wavelengths in systems undergoing infall is indicative of grain growth [15]' appears to cite reference [15] (Tychoniec et al. 2020), which is a dust mass survey; please verify the intended citation.
[Results, Eq. (2)] In the definition of chi, the normalization constants '16 Myr' and '100 days' are used without explanation; a brief statement that these are dimensionless normalizations would improve readability.
[Figures 2 and 3] Please unify the notation for the parameter combination that appears variously as alpha epsilon / f and alpha epsilon/f; in particular, check that the x-axis label in Figure 3 matches the text.
[Abstract] The phrase 'few 10^{-5} to 10^{-4}' contains a missing space and is informal; consider replacing with 'approximately 10^{-5} to 10^{-4}'.
[Methods, 'Relative timescales of gas disk dispersal vs. infall'] The two-stage viscosity treatment for initially massive disks (setting alpha = 0.1 for the initial state and then reducing to alpha = 10^-4) is described in one sentence; a brief justification of this procedure would help the reader assess its influence on the derived beta values.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mass-ratio result is an analytical scaling from an explicit accretion-migration balance, not a restatement of the observed target; the acknowledged planetesimal-formation premise is a physical gap, not a circular reduction.

full rationale

The central derivation is self-contained. Equation 2 solves tau_acc = tau_I for M_crit; Equation 3 integrates M_crit over the feeding-zone spacing to obtain M_tot(t); Equation 4 evaluates that expression at the stated survival condition tau_I = 10 tau_g. Each step is obtained by algebra from stated physical inputs (infall flux F_in, gas-to-solids ratio f, incorporation fraction epsilon, viscosity alpha, aspect ratio H/r, and beta = tau_g/tau_in), and the final mass ratio retains an explicit, weak power-law dependence on those inputs. The comparison to observed compact systems in Figure 3 is a forward prediction across a range of assumed parameter values, not an inversion or fit: the paper does not tune alpha*epsilon/f, f/epsilon, or beta to reproduce the observed M_tot/M_*; it varies them over orders of magnitude and shows that the resulting mass ratios remain in the observed range. The N-body simulations independently reproduce the analytical scaling without imposing the 10*tau_g criterion, providing a separate check. The paper explicitly identifies its load-bearing premise that planetesimals form interior to r_c during final infall, and in Methods ("Dust-to-planetesimal accretion") it concedes that "models that explicitly treat grain growth up to planetesimal formation are lacking, and whether conditions for collapse are actually achieved is unclear, particularly for turbulent disks." That is an unmodeled physical assumption and a genuine correctness risk, but it is not a circular one: the mass-ratio prediction is not equivalent to the premise by construction. The self-citations to Canup & Ward (2002, 2006) supply the accretion-timescale and feeding-zone prescriptions and the gas-disk torque treatment; those prior models were developed and tested against satellite systems, not against compact exoplanet mass ratios, so the present target is not contained in the cited fitted outputs. No step in the paper reduces the predicted quantity to an input by definition, and no fitted parameter is renamed as a prediction. Therefore no significant circularity is found.

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

The model rests on a small set of physical inputs, most of which are conventional in planet formation (Type-I migration, viscous disk evolution). The most consequential input is the premise of rapid planetesimal formation interior to rc during infall, which is not simulated. The free parameters (alpha, epsilon/f, H/r, Min/M*, beta, rc) are chosen within observationally or theoretically plausible ranges; the output mass ratio depends on them with exponents that are mostly small, so the claimed regulation is robust to order-of-magnitude variations in alpha*epsilon/f, though the absolute normalization is set by Min/M*, f/epsilon, and H/r.

free parameters (6)

alpha (disk viscosity parameter) = varied; fiducial alpha*epsilon/f = 5e-5
Controls gas surface density and migration rate; constrained by observed accretion rates but not measured directly.
epsilon/f (solid incorporation fraction over gas-to-solids ratio) = 0.01 (f/epsilon = 100)
Sets the solid accretion rate; chosen as a plausible value, enters eqns 3 and 4 with a weak exponent.
H/r (disk aspect ratio at rc) = 0.05 (1 Msun), 0.07 (0.1 Msun)
Sets the Type-I migration timescale; chosen from the disk temperature model; enters the final mass ratio with exponent about 1.4, so a factor 2 change shifts the output by about 2.6.
Min/M* (total mass delivered during final infall) = 0.03 to 0.05
Sets the normalization of the final mass ratio in eqn 4; chosen rather than derived.
beta = tau_g/tau_in = 1.3 to 2
Ratio of gas disk dispersal time to infall timescale; derived from the paper's 1D disk evolution model for small rc, but depends on assumed alpha and tau_in.
rc (centrifugal radius) = 0.6 au (1 Msun), 0.07 au (0.1 Msun)
Radial extent of the infall region; chosen to match compact systems like TRAPPIST-1 and Kepler-11.

assumptions (5)

domain assumption Planetesimals form interior to rc during the final infall stage.
Stated as a premise in Results ('A premise of our model is that planetesimals form somewhere within the region of infall'); argued from dust sticking and streaming instability but not modeled directly.
domain assumption Infall flux is radially uniform and decays exponentially; gas disk density decays exponentially.
Adopted in Results; shown to be a reasonable approximation for beta between 1.3 and 2 in Methods, but not universal (gray region in Figure 6).
standard math Type-I migration timescale follows eqn 1 with torque constant Ca of order unity.
Standard torque formula from Tanaka and Ward (2004), widely used in planet formation modeling.
ad hoc to paper Inner cavity cancels Lindblad and co-rotation torques, halting migration inside the cavity.
Assumed in Methods for cavity simulations; the paper notes this is a simplification and actual edge torques depend on cavity steepness.
domain assumption Planets do not accrete gas and remain solid by mass.
Adopted as in prior N-body models of compact system origin; reasonable for the low-mass planets considered here.

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Cite this review

Pith. "Pith review of Origin of compact exoplanetary systems during disk infall." pith.science (2026). https://pith.science/paper/ORN5GWFA

@misc{pith2026250522806,
  author       = {Pith},
  title        = {Pith review of: Origin of compact exoplanetary systems during disk infall},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ORN5GWFA}},
  note         = {Machine review of arXiv:2505.22806}
}

read the original abstract

Exoplanetary systems that contain multiple planets on short-period orbits appear to be prevalent in the current observed exoplanetary population, yet the processes that give rise to such configurations remain poorly understood. A common prior assumption is that planetary accretion commences after the infall of gas and solids to the circumstellar disk ended. However, observational evidence indicates that accretion may begin earlier. We propose that compact systems are surviving remnants of planet accretion that occurred during the final phases of infall. In regions of the disk experiencing ongoing infall, the planetary mass is set by the balance between accretion of infalling solids and the increasingly rapid inward migration driven by the surrounding gas as the planet grows. This balance selects for similarly-sized planets whose mass is a function of infall and disk conditions. We show that infall-produced planets can survive until the gas disk disperses and migration ends, and that across a broad range of conditions, the mass of surviving systems is regulated to a few 10^{-5} to 10^{-4} times the host star's mass. This provides an explanation for the similar mass ratios of known compact systems.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Protostars and Planets VI195(2014)

Dunham, M.M., Stutz, A.M., Allen, L.E., Evans, N., Fischer, W.J., Megeath, S.T., Myers, P.C., Offner, S.S., Poteet, C.A., Tobin, J.J., et al.: The evolution of protostars: Insights from ten years of infrared surveys with Spitzer and Herschel. Protostars and Planets VI195(2014)

work page 2014

[2] [2]

The Astronomical Journal164(2), 72 (2022)

Millholland, S.C., He, M.Y., Zink, J.K.: Edge-of-the-multis: evidence for a transition in the outer architectures of compact multiplanet systems. The Astronomical Journal164(2), 72 (2022)

work page 2022

[3] [3]

AJ 159(5), 188 (2020)

Zhu, W.: On the patterns observed in Kepler multi-planet systems. AJ 159(5), 188 (2020)

work page 2020

[4] [4]

In: Protostars and Planets III, pp

Lissauer, J.J., Stewart, G.R.: Growth of planets from planetesimals. In: Protostars and Planets III, pp. 1061–1088 (1993)

work page 1993

[5] [5]

Chiang, E., Laughlin, G.: The minimum-mass extrasolar nebula: in situ formation of close-in super-earths. Mon. Notices Royal Astron. Soc. 431(4), 3444–3455 (2013)

work page 2013

[6] [6]

Canup, R.M., Ward, W.R.: A common mass scaling for satellite systems of gaseous planets.Nature441(7095), 834–839 (2006)

work page 2006

[7] [7]

Canup, R.M., Ward, W.R.: Formation of the Galilean Satellites: Condi- tions of Accretion.AJ124(6), 3404–3423 (2002)

work page 2002

[8] [8]

Bae, J., Hartmann, L., Zhu, Z.: Are protoplanetary disks born with vor- tices? Rossby wave instability driven by protostellar infall.ApJ805(1), 15 (2015) arXiv:1503.02694 [astro-ph.EP]

work page Pith review arXiv 2015