Inner rocky super-Earth formation: distinguishing the formation pathways in viscously heated and passive discs

Bertram Bitsch

REVIEW 4 major objections 5 minor 81 references

Inner rocky super-Earth formation: distinguishing the formation pathways in viscously heated and passive discs

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The final masses of inner rocky super-Earths are set by the thermal structure of the protoplanetary disc, with early formation in viscously heated discs matching Kepler masses and late or passive-disc formation requiring mergers.

desk verdict A clearly-argued synthesis that maps disc thermal structure onto two super-Earth formation channels and offers an observational discriminator, but the whole viscous-heating branch rests on a premise that the paper itself admits is contested. read the letter →

arxiv 1908.08710 v1 pith:YM4TJYLO submitted 2019-08-23 astro-ph.EP

Bertram Bitsch This is my paper · ORCID

classification astro-ph.EP

keywords super-EarthformationpebbleaccretionisolationmassprotoplanetarydiscstructureviscousheatingKeplerexoplanetsmeanmotionresonancesplanetarymigration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the final mass of a close-in rocky super-Earth is largely written into the structure of the protoplanetary disc in which it grows. In the pebble accretion picture, a planet stops growing at the pebble isolation mass, the mass at which it opens a partial gap and cuts off the inward drift of pebbles; this mass scales steeply with the disc's aspect ratio, so the heating source of the disc matters. Using a viscously heated disc model, the author shows that planets reaching isolation within 1 Myr in the inner few AU land at roughly 3-10 Earth masses, matching masses inferred from Kepler radii, while planets forming later or in a purely passively heated disc stop near 2-3 Earth masses and require planetary mergers to explain the observations. The paper therefore proposes two distinguishable formation channels: early pebble growth with resonant chains broken without collisions, or later merger-dominated growth that leaves mutually inclined systems. The central contribution is diagnostic: the architecture and mutual inclinations of observed super-Earth systems can reveal which channel operated and whether the inner disc was viscously heated.

What carries the argument

The central object is the pebble isolation mass, $M_{\rm iso}$: the planetary mass at which the growing embryo opens a partial gap in the gas disc, inverting the radial pressure gradient exterior to the planet so that drifting pebbles accumulate outside and the accretion of solids halts. Its key property is $M_{\rm iso}\propto(H/r)^3$, with weak corrections from viscosity and the pressure gradient, which converts the disc's thermal structure directly into a final planetary mass. The paper computes $M_{\rm iso}$ for two disc structures: a viscously heated disc model whose aspect ratio and isolation mass decline as the disc cools over a few million years, and a purely passively irradiated disc with a much smaller inner $H/r$. Comparing these curves with Kepler masses derived from a mass-radius relation carries the argument: the early viscous curve overlaps the observed masses, while the late and passive curves fall below them.

What would settle it

Measure the midplane temperature or aspect ratio of inner protoplanetary discs at ages below about 1 Myr: if $H/r$ is as low as passive models even at early times, the early viscous pathway cannot deliver 5-10 Earth-mass planets without mergers. Alternatively, a transit and radial-velocity survey of resonant super-Earth systems could falsify the early pathway: if most resonant systems with masses of 5-10 Earth masses show high mutual inclinations or evidence of recent giant impacts, resonant chains are being reshaped by mergers rather than broken without collisions.

Watch

Extended reading notes

Core claim

The central claim is that the observed Kepler super-Earth masses, roughly 3-10 Earth masses, are the pebble isolation masses of the early viscously heated inner protoplanetary disc. The isolation mass, $M_{\rm iso}=25 f_{\rm fit}\, M_{\rm E} + (\Pi_{\rm crit}/\lambda) M_{\rm E}$, depends mainly on the aspect ratio $H/r$ through $f_{\rm fit}\propto(H/r)^3$, and in the viscously heated model $H/r$ is high enough during the first million years that $M_{\rm iso}$ reaches the inferred Kepler masses. In a purely passive disc, or in the same viscous model after about 1 Myr as the disc cools, $M_{\rm iso}$ drops to 2-3 $M_{\rm E}$, below most observed masses. The paper's simulations of single embryos growing by pebble accretion and migrating inward reproduce the cumulative Kepler mass distribution only when planets reach isolation before 1 Myr and the pebble flux is large enough; the final mass is independent of the pebble flux. The two formation pathways are distinguished by what breaks the resonant chains that migration naturally builds: early viscous formation requires a mechanism that breaks resonance without mergers and leaves low mutual inclinations, whereas late or passive formation requires collisions to raise the masses, with high mutual inclinations as a by-product.

Load-bearing premise

The load-bearing premise is that the inner protoplanetary disc is strongly heated by viscous accretion during its first million years, so its aspect ratio, and hence the pebble isolation mass, is high enough to produce 5-10 Earth-mass planets; if midplane viscous heating is weak, as recent shearing-box simulations suggest, the early viscous channel fails and mergers must explain the observations.

Editorial extensions

If this is right

If the early viscous pathway dominates, super-Earth masses in a system are effectively set by the disc aspect ratio at formation, so planets in the same system should be similar in mass and the mass distribution should not depend strongly on the initial reservoir of solids.
Most observed close-in super-Earths would have to assemble within about one million years, before the disc cools and the pebble isolation mass drops below the Kepler masses.
The resonant chains produced by migration must be broken by a mechanism that avoids collisions, such as magnetic rebound, a short disc lifetime, or weak resonance trapping in low-viscosity discs, leaving systems flat with low mutual inclinations.
If formation is late or occurs in passive discs, the observed masses require giant impacts; resonant systems then survive only if mergers happen while gas is still present, and the resulting systems should show high mutual inclinations.
Transit plus radial-velocity observations can distinguish the channels by counting hidden planets and measuring mutual inclinations, turning exoplanet system architectures into a probe of inner-disc viscous heating.

Reading between the lines

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

One extension of the paper's logic is that the near-independence of final mass from pebble flux predicts a weak correlation between a star's observed disc dust mass and the masses of its close-in super-Earths; systems forming in dust-poor discs should still reach the same isolation masses if the early viscous channel operates.
The two channels also predict different mass-radius scatter within individual systems: the early viscous channel yields similar masses set by one shared disc structure, while a merger-dominated channel should leave more diversity in mass and composition among planets in the same system.
Because $M_{\rm iso}$ grows as $(H/r)^3$, the peak of the Kepler mass distribution could in principle be inverted to estimate the inner-disc aspect ratio at the time of formation, effectively using super-Earth masses as a fossil thermometer for early protoplanetary discs.

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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

You should know two things about this paper. First, it does something genuinely useful: it takes the pebble isolation mass as a function of disc structure and directly compares it against Kepler mass estimates, producing two distinct formation channels—early growth in a viscously heated inner disc without mergers, versus late or passive-disc growth that requires mergers. The observable difference, in mutual inclinations and hidden planets, is a clean and testable discriminator. Second, the entire first channel stands or falls on whether the inner disc is actually viscously heated in the first Myr. The author knows this and says so in Section 5.1, but his rebuttal to Mori et al. (2019)—that a cold disc would make Earth-building material water-rich—is an argument about the solar nebula, not about the Kepler population. If Mori is right, the high early pebble isolation masses in Fig. 1 are not realized, and the match in Fig. 2 collapses to a single merger-dominated pathway.

What the paper does well: it is honest and well-structured. The central comparison is simple and defensible as a forward model against external data. The mass-radius conversion (Wolfgang et al. 2016) is standard. The pebble flux scaling S_peb=5 is tuned, but the authors show the result is insensitive to flux once it is high enough to reach isolation mass. The self-citations are appropriate; they are building on their own prior disc and pebble models, not hiding anything. The paper also flags its own weakest points: the unknown mechanism for breaking resonant chains without mergers, the sensitivity to disc viscosity, and the unresolved viscous-heating debate.

The soft spots are real but proportionate. The paper is a conditional argument, not a proof. It would be stronger if it presented the viscous-versus-passive distinction as explicitly contingent on the disc model, rather than sometimes speaking as if the two channels are established. But it does not overclaim in the abstract or summary; it describes them as pathways within this framework.

Who is this for? Planet formation theorists working on super-Earth origins, and observers planning RV follow-up of TESS transits. I would bring it to a reading group, and I would send it to a serious referee if I were the editor. The referee should push on the viscous-heating sensitivity and ask for a sharper statement of what would falsify each channel. This is exactly the kind of paper where peer review can help—not because it is broken, but because the central claim is interesting enough to deserve careful scrutiny.

Referee Report

4 major / 5 minor

Summary. The paper compares the pebble isolation mass predicted by a viscously heated protoplanetary disc model (Bitsch et al. 2015a) and a purely passively heated disc model against the inferred masses of Kepler super-Earths, using the Wolfgang et al. (2016) mass-radius relation. In the viscously heated model, the inner disc (0.1–1 AU) initially has a high aspect ratio, giving pebble isolation masses of several Earth masses; the paper argues that planets formed within 1 Myr by pebble accretion can therefore match the observed Kepler mass distribution without mergers. In contrast, the passive disc has low pebble isolation masses (2–3 ME), so the paper argues that mergers are required to match observations. From this comparison, the author proposes two formation pathways: early formation in viscously heated discs, where resonant chains must be broken without mergers, and late or passive-disc formation, where mergers during the gas phase are required. The paper makes testable predictions: the two pathways should differ in mutual inclination distributions and in the number of hidden planets detectable by RV follow-up of TESS targets.

Significance. If the underlying viscous-heating premise is correct, the paper provides a simple and elegant explanation for the typical masses of close-in super-Earths, tying the observed mass distribution to disc thermal structure rather than to initial solid reservoirs. The work is transparent in its assumptions, uses an external observational sample, and gives falsifiable predictions for future RV and transit observations. The comparison is a forward model against data, not a fit calibrated to the Kepler mass distribution, which is a strength. However, the significance is strongly conditional on the contested assumption that the inner regions of planet-forming discs are viscously heated during the first Myr; the paper itself acknowledges a competing simulation result that would invalidate the main pathway. The two-pathway distinction also relies on an as-yet unidentified mechanism for breaking resonant chains without mergers. These caveats make the paper an interesting and useful framework, but its central claim is more provisional than the abstract suggests.

major comments (4)

[§5.1, Figs. 1 and 2] The central mass match depends entirely on the inner disc having the high aspect ratio of a viscously heated disc in the first Myr. The paper acknowledges in Section 5.1 that Mori et al. (2019) find midplane viscous heating to be suppressed by weak turbulence, which would eliminate the high early pebble isolation masses and collapse the 'viscously heated pathway' into the merger-dominated pathway. The rebuttal offered — that a cold disc would make the Earth's building material water-rich — is an argument about the solar nebula, not about the population of Kepler stars, so it does not protect the exoplanet claim. As written, the abstract's statement that planets growing within 1 Myr in the viscously heated inner disc reach pebble isolation masses matching Kepler observations is conditional on a premise the manuscript itself identifies as contested. The manuscript should either provide direct evidence or a quantitative argument that inner discs around solar-type stars are viscously heated during the first Myr, or explicitly reframe the conclusions as conditional on that premise, with the passive/merger scenario as the null hypothesis.
[§5.2, §4.1, abstract] The viscously heated pathway requires resonant chains to be broken without planetary mergers, yet the paper states in Section 5.2 that 'it is yet unclear which mechanism could drive this breaking of the resonant chains without causing major instabilities.' Since the observed period-ratio distribution of Kepler systems is used to justify the need to break resonances, this unexplained step is load-bearing for the claim that the viscously heated pathway can reproduce the full Kepler population. The magnetic rebound effect is mentioned as a possibility but is admitted to be untested for multi-planet resonant chains. The manuscript needs to specify a plausible, tested mechanism or clearly mark this pathway as incomplete and speculative, rather than presenting it as a viable channel on equal footing with the merger pathway.
[§3, Fig. B.1, Appendix B] The paper states that final planetary masses in the pebble accretion scenario are independent of the pebble flux, but this is only true above a threshold. Fig. B.1 shows that for S_peb = 0.75 and 1.0, planets do not reach the pebble isolation mass within 1 Myr, and the match in Fig. 2 is obtained using S_peb = 5.0, a freely chosen scaling factor. The central comparison therefore depends on a tuned parameter, and this dependence should be stated explicitly in the main text (not only in the appendix). The paper should quantify the range of S_peb for which the simulated masses match the Kepler distribution, and discuss whether S_peb = 5 is physically motivated or merely chosen to produce agreement.
[Fig. 2] The claim that the simulated mass distributions 'match the Kepler observations very well' is supported only by visual inspection of cumulative distributions. No statistical test (e.g., Kolmogorov-Smirnov) or uncertainty propagation from the Wolfgang et al. (2016) mass-radius relation is provided. Given that this match is the core quantitative result of the paper, the manuscript should include a quantitative comparison and state the sensitivity of the match to the assumed mass-radius relation and to the selection cuts (3–10 ME).

minor comments (5)

[§1, Eq. (2) vicinity] The text near Eq. (2) contains a duplicated word: 'the Stokes number in the used model model varies' should read 'the Stokes number in the used model varies.'
[Fig. 1 caption] The caption has a typo: 'with cut-oﬀat planetary radii' should be 'with cut-off at planetary radii.'
[§5.1] The phrase 'current sheats' appears to be a typo for 'current sheets.'
[References] Several references are cited as 'arXiv e-prints' without numbers (e.g., Izidoro et al. 2019, Lambrechts et al. 2019, Baillié et al. 2015). These should be updated to their published or arXiv identifiers for reproducibility.
[Fig. 2 and Fig. B.1] The axis label 'Cummulative distribution' should be 'Cumulative distribution.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Kepler-mass comparison is a forward prediction from disc-structure and gap-opening models, not a fit to the data.

full rationale

The paper's central claim is that planets growing within ~1 Myr in a viscously heated disc reach the pebble isolation mass, whose value is then compared with Kepler-derived masses. The pebble isolation mass formula (Eqs. 1-2) and the disc structure are taken from Bitsch et al. (2018b) and Bitsch et al. (2015a), respectively; neither set of prior results was calibrated to the Kepler mass distribution, and the comparison in Figs. 1-2 is therefore a forward model against external data rather than a fitted reproduction. The pebble-flux scaling Speb is a free parameter, but it only determines whether growth to the isolation mass is completed before or after 1 Myr; the final mass ceiling is set by the independently computed Miso, so the headline mass match is not statistically forced by Speb. Heavy author self-citation is present, but it is normal model reuse, not load-bearing circularity: the cited disc and gap-opening results have independent content and were not defined in terms of the Kepler masses they are used to explain. The author explicitly flags in Section 5.1 that MHD shearing-box simulations (Mori et al. 2019) could suppress viscous heating and invalidate the high-Miso pathway; this is a genuine physical correctness risk, but it does not make the derivation circular.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on established but debated domain models: the pebble isolation mass formula, the viscous disc thermal structure, the mass-radius relation, and the migration prescription. The S_peb scaling factor is a numerical choice not derived from first principles.

free parameters (3)

S_peb (pebble flux scaling factor) = 5.0 (adopted; 0.75, 1.0, 2.5 tested)
Scales the pebble flux to a level where planets reach the pebble isolation mass within 1 Myr; the central match to Kepler masses requires a high enough S_peb (Fig. B.1).
alpha_disc (viscosity parameter for Miso) = 0.001 (main; 1e-4 to 0.0054 in Appendix A)
The pebble isolation mass depends on alpha; the disc thermal structure in the viscous model uses alpha=0.0054. The trend is robust to alpha, but the quantitative match is computed at alpha=0.001.
Stokes number tau_f = 0.1 (plotting; simulations 0.05-0.2)
The pebble isolation mass depends weakly on tau_f; the value is derived from drift-growth equilibrium, not fit to Kepler data.

assumptions (6)

domain assumption The pebble isolation mass formula of Bitsch et al. (2018b) (Eq. 1-2) determines the maximum mass from pebble accretion.
The paper takes this fitted formula from hydro simulations as given; it sets the final planet mass in the model.
domain assumption The disc thermal structure of Bitsch et al. (2015a) represents viscously heated protoplanetary discs with H/r ~ 0.05 in the inner regions within 1 Myr.
The central mass match depends on this disc model; it is flagged as debated in Section 5.1.
domain assumption The mass-radius relationship of Wolfgang et al. (2016) converts Kepler radii to masses.
Kepler masses in Figs. 1-2 are inferred via this relationship; systematic errors in it shift the comparison.
domain assumption Type-I migration follows Paardekooper et al. (2011) and drives planets to the inner edge where they form resonant chains.
The resonant chain formation and two-channel distinction rely on this migration prescription (Section 5.2).
domain assumption Inner super-Earths are predominantly rocky and form interior to the water ice line.
Used to restrict formation to the inner 2-3 AU; if planets form outside and migrate in, they would be ice-rich (Section 5.4).
domain assumption Gas accretion onto cores below about 10 Earth masses is inefficient, so post-isolation growth via gas is small.
Motivates the 3-10 ME comparison window and the cut at 10 ME (Section 5.3).

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Cite this review

Pith. "Pith review of Inner rocky super-Earth formation: distinguishing the formation pathways in viscously heated and passive discs." pith.science (2026). https://pith.science/paper/YM4TJYLO

@misc{pith2026190808710,
  author       = {Pith},
  title        = {Pith review of: Inner rocky super-Earth formation: distinguishing the formation pathways in viscously heated and passive discs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YM4TJYLO}},
  note         = {Machine review of arXiv:1908.08710}
}

read the original abstract

The formation of super-Earths is strongly linked to the structure of the protoplanetary disc, which determines growth and migration. In the pebble accretion scenario, planets grow to the pebble isolation mass, at which the planet carves a small gap in the gas disc halting the pebble flux and thus its growth. The pebble isolation mass scales with the disc's aspect ratio, which directly depends on the disc structure. I compare the growth of super-Earths in viscously heated discs and discs purely heated by the central star with super-Earth observations. This allows two formation pathways of super-Earths to be distinguished in the inner systems. Planets growing within 1 Myr in the viscously heated inner disc reach pebble isolation masses that correspond directly to the inferred masses of the Kepler observations for systems that feature planets in resonance or not in resonance. However, to explain the period ratio distribution of Kepler planets -- where most Kepler planet pairs are not in mean motion resonance configurations -- a fraction of these resonant chains has to be broken. In case the planets are born early in a viscously heated disc, these resonant chains thus have to be broken without planetary mergers. If super-Earths form either late or in purely passive discs, the pebble isolation mass is too small to explain the Kepler observations, implying that planetary mergers are important for the final system architecture. Resonant planetary systems thus have to experience mergers already during the gas disc phase, so the planets can get trapped in resonance after reaching 5-10 Earth masses. In case instabilities are dominating the system architecture, the systems should not be flat with mutually inclined orbits. This implies that future observations of planetary systems with RV and transits could distinguish between these two formation channels of super-Earth. (abridged)

Figures

Figures reproduced from arXiv: 1908.08710 by the authors.

**Figure 1.** Planetary masses of Kepler systems derived from the mass-radius relationship from Wolfgang et al. (2016), with cut-off at planetary radii of 4 RE (grey dots) and super-Earths detected by RV (black dots) with cut-off at 20 Earth masses. The black lines connect planets within the planetary system (for RV systems only). The red dots correspond to Kepler planets in multiple systems with period ratios of ∼2:1, 3:2 and 4:… view at source ↗

**Figure 2.** Planetary masses for Kepler systems derived from the massradius relationship from Wolfgang et al. (2016), where the grey color shows the same sample as in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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