Super-Earth masses sculpted by pebble isolation around stars of different masses

Anders Johansen; Beibei Liu; Fan Liu; Michiel Lambrechts

REVIEW 3 major objections 4 minor 159 references

Super-Earth masses sculpted by pebble isolation around stars of different masses

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

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Pith's one-line read The paper argues that the pebble isolation mass, the point at which a growing planet halts pebble accretion, sets the characteristic super-Earth mass and predicts a near-linear scaling with host-star mass.

desk verdict First pebble-accretion population synthesis across 0.08–1 solar masses, with a clean super-Earth mass–stellar mass prediction that is largely inherited from the adopted isolation-mass formula; worth refereeing, but not decisive on its own. read the letter →

arxiv 1909.00759 v1 pith:NIQMUYBX submitted 2019-09-02 astro-ph.EP

Beibei Liu , Michiel Lambrechts , Anders Johansen , Fan Liu This is my paper · ORCID

classification astro-ph.EP

keywords planetformationpebbleaccretionisolationmasssuper-EarthsMdwarfspopulationsynthesisexoplanetdemographicsprotoplanetarydiskevolution

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

Planets forming in disks around stars from 0.08 to 1 solar mass stop growing by pebble accretion when they reach the pebble isolation mass, the mass at which a planet's gravity reverses the local disk pressure gradient and halts inward-drifting pebbles. The paper builds a pebble-driven population synthesis model and shows that this cutoff sets the characteristic super-Earth mass, giving roughly one Earth mass around a late M dwarf and about 20 Earth masses around a solar-mass star. The predicted scaling, approximately $M_{\rm iso} \propto M_\star^{4/3}$, reproduces the observed trend that more massive stars host more massive super-Earths and that gas giants appear only above a threshold stellar mass. The same model also yields the observed weak metallicity dependence of super-Earth formation, the strong metallicity preference of gas giants, and water-content differences that depend on where embryos are born.

What carries the argument

The key mechanism is the pebble isolation mass, defined as the planet mass at which the planet's gravitational perturbation opens a gap and reverses the disk's local pressure gradient, stopping the inward drift of pebbles and therefore ending pebble accretion. The paper adopts a 3D hydrodynamic fitting formula, Eq. (26), for this mass and, using the disk aspect ratio at the water ice line in its viscously heated inner disk, reduces it to the simple scaling $M_{\rm iso} \simeq 25\,(M_\star/M_\odot)^{4/3}\,M_\oplus$ (Eq. 39). This scaling carries the argument: it converts the observed planet$-$stellar-mass correlation into a prediction of core-growth truncation, and it is insensitive to details such as embryo birth location or the turbulent $\alpha_t$ value, which changes $M_{\rm iso}$ by only about 25% across an order of magnitude in $\alpha_t$.

What would settle it

Run 3D hydrodynamic simulations of a growing planet in a disk around a 0.1 solar-mass star and measure the actual pebble isolation mass; if it differs substantially from the extrapolated value, the central scaling fails. Observationally, precise masses of close-in planets around late M dwarfs provide a test: the model predicts an upper envelope near 1 to 2 Earth masses around a 0.08 solar-mass star, so discovering a population of rocky planets at several Earth masses around such stars would falsify the isolation-mass truncation.

Watch

Extended reading notes

Core claim

The central claim, stated in Sect. 6, is that the characteristic core-dominated planet mass may be set by the pebble isolation mass. During growth a planet eventually opens a shallow gap and reverses the local pressure gradient, so pebbles stop drifting inward and accretion terminates; the mass at which this happens is $M_{\rm iso} = 25\,(M_\star/M_\odot)\,(h_g/0.05)^3\,M_\oplus \simeq 25\,(M_\star/M_\odot)^{4/3}\,M_\oplus$ for the adopted inner-disk structure. Using a Monte Carlo synthesis of embryos that grow by pebble accretion, with gas accretion and type I/II migration included, the paper finds that the upper mass envelope of super-Earths follows this scaling from about 1 $M_\oplus$ around a 0.08 $M_\odot$ star to about 20$-$25 $M_\oplus$ around a solar-mass star. Excluding gas giants, the simulated population reproduces the observed planet-mass$-$stellar-mass trend, the rarity of giant planets around low-mass stars, and the stronger metallicity dependence of gas giants compared with super-Earths.

Load-bearing premise

The argument stands on the assumption that the pebble isolation mass formula fitted to hydrodynamic simulations for solar-type disks remains valid for disks around stars from 0.08 to 1 solar mass, so that its dependence on stellar mass is as predicted.

Editorial extensions

If this is right

Super-Earth masses around low-mass stars should be capped near the pebble isolation mass, so systems around late M dwarfs are expected to host Earth- to few-Earth-mass planets and no gas giants.
Because $M_{\rm iso}$ stays below roughly 10 $M_\oplus$ for stars below about 0.3 $M_\odot$, such systems should rarely or never form gas giants through this channel, matching the observed absence of massive planets around very low-mass stars.
Gas giant formation should be strongly favored around stars above roughly 0.3 $M_\odot$ and in metal-rich disks, while super-Earth formation should be nearly metallicity-independent; both trends agree with current exoplanet populations.
The water content of a super-Earth depends on where its embryo formed: ice-line embryos in low-turbulence disks end up with about 10$-$15% water by mass, while embryos spread across the disk produce a bimodal mix of nearly dry and water-rich planets.

Reading between the lines

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

The authors note that giant impacts after disk dispersal are not modeled; if collisions add mass, final planet masses in multi-planet systems could exceed the isolation mass, so the cleanest test of the scaling is single-planet systems or the lower mass envelope of close-in super-Earths.
A direct observational discrimination could come from transit and radial-velocity surveys of ultracool dwarfs: if planets around 0.08$-$0.1 $M_\odot$ stars are found at masses well above about 2 $M_\oplus$, the isolation-mass truncation would need revision or an extra mass source.
Because the scaling is carried by the inner-disk aspect ratio, disks that are significantly hotter or colder than the adopted structure would shift the normalization and exponent of the planet$-$star mass relation, making the observed correlation a potential indirect probe of inner disk temperatures.

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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

Worth reading carefully. This is the first pebble-driven population synthesis that spans 0.08 to 1 solar mass, and it makes concrete predictions: super-Earth masses should scale roughly linearly with stellar mass, from about an Earth mass around late M dwarfs to tens of Earth masses around solar-type stars, with distinctive water-content signatures depending on where embryos start. That is a real contribution. The paper also does something I respect: it tests its own assumptions in Appendices A.1–A.7, including the pure-irradiation disk, early seed formation, different luminosity relations, and very low turbulence. The authors are not hiding the warts.

The soft spots are real but not fatal. The central mass scaling enters through the Bitsch et al. (2018) pebble isolation formula, Eq. (26), which already contains Miso ∝ Mstar. Equation (39) then repackages that dependence. So the headline prediction is not an independent derivation; it is an application of an existing result to a broader stellar mass range. That is fine for a population synthesis paper, but it means the paper is not a test of the isolation-mass mechanism itself. The stress-test note is right: Appendix A.2 shows that migration-limited growth in a pure-irradiation disk also produces a linear Mp–Mstar scaling, so the observed trend does not uniquely point to pebble isolation. The paper also concedes that planets in multi-planet systems can be pushed above Miso by giant impacts, making Miso a lower limit for final masses. These caveats are stated clearly, but they do reduce the force of the main claim.

The observational comparisons are qualitative. Figure 1 shows a nice visual correlation with a handful of low-mass-star systems, and the model reproduces the rough metallicity trends for super-Earths versus gas giants, but there is no statistical fit, no completeness correction, and no quantitative comparison between the simulated and observed mass distributions. The model also overproduces hot Jupiters relative to cold giants, which the authors acknowledge and attribute to missing multi-body dynamics. That is a minor blemish at this level of modeling, but worth noting.

No code or data artifacts are provided, so the Monte Carlo implementation cannot be checked independently. The equations are mostly internally consistent, and the parameter choices are reasonable and clearly stated.

Who should read this? Anyone working on super-Earth formation, pebble accretion, or M-dwarf planet populations. It is a useful framework for making predictions that TESS and SPECULOOS can test. I would send it to a serious referee. My own recommendation would be conditional acceptance with two requests: a quantitative statistical comparison to the observed sample, and a section that honestly discusses whether the observed scaling can discriminate between pebble isolation and migration-limited growth.

Referee Report

3 major / 4 minor

Summary. The paper presents a pebble-driven population synthesis model for planet formation around stars from 0.08 to 1 solar mass. The model includes viscous and irradiated disk structures, pebble accretion, gas accretion, type I/II migration, and Monte Carlo sampling of disk parameters. The authors report that the characteristic super-Earth mass is set by the pebble isolation mass and scales with stellar mass, from about 1 Earth mass around late M dwarfs to about 20 Earth masses around solar-mass stars. They also discuss water fractions, metallicity trends, and compare the simulated populations with observed exoplanet masses, orbital distances, metallicities, and water-content inferences. The paper's central claim is that the observed Mp–Mstar correlation is produced by pebble isolation, but the authors themselves show in an appendix that a migration-limited growth scenario can produce a similar scaling.

Significance. If the central claim is established, the paper would provide a physically motivated explanation for the characteristic masses of super-Earths and their stellar-mass dependence, connecting pebble isolation to a major observed demographic trend. The model is quite comprehensive for a population synthesis study: it includes a two-component disk model, a wide parameter study, and multiple robustness tests in Appendices A.1–A.7, covering alternatives such as pure irradiated disks, early embryo formation, different stellar luminosity relations, and low turbulence levels. The paper also makes concrete, falsifiable predictions for planets around very low-mass stars. However, the manuscript's main attribution of the mass trend to pebble isolation is not fully supported by its own tests, and the observational comparison is qualitative rather than statistical. The result is therefore significant conditional on a sharper causal test.

major comments (3)

[§5.2.1 and Appendix A.2] The central attribution of the Mp–Mstar scaling to pebble isolation is underdetermined by the paper's own tests. Section 5.2.1 concedes that the simulated planets are single embryos and that for observed multi-planet systems the pebble isolation mass is only a lower limit, since post-disk giant impacts can raise final masses. Appendix A.2 then shows that a model with no pebble isolation mass, in which growth is truncated by migration, also produces a linear Mp–Mstar scaling (Fig. A.2). The paper's key conclusion in Sect. 6 therefore requires a quantitative model comparison that can distinguish pebble isolation from migration-limited growth; the qualitative agreement shown in Figs. 1 and 7 does not provide that discrimination.
[Eqs. (26), (27), (39)] The predicted scaling is inherited from the adopted Bitsch et al. (2018) isolation mass formula rather than independently derived. Equation (26) already contains an explicit linear Mstar factor, and Eq. (39) evaluates that formula at the ice line using the assumed Mdot_g ∝ Mstar^1.8 relation, yielding Miso ∝ Mstar^(4/3) rather than a purely linear relation. The paper should state this inheritance explicitly and test the sensitivity of the central slope to the extrapolation of Eq. (26) below about 0.3 solar masses, because the validity of that hydrodynamic fit at very low stellar masses is a load-bearing assumption.
[§5.2.1 and Fig. 1] The observational support for the central trend is currently qualitative and does not account for selection effects. Fig. 1 mixes radial-velocity minimum masses with true masses from transit/TTV measurements, and no detection completeness function is modeled. The statement that the observed Mp–Mstar trend is not due to observational bias is asserted rather than demonstrated. A quantitative comparison, even a forward-modeled occurrence rate with a simple detection probability, would be needed to support the claim that the simulated population 'agrees well' with observations.

minor comments (4)

[Abstract and Eq. (39)] The abstract and Sect. 1 describe the relation as linear, while Eq. (39) gives Miso ∝ Mstar^(4/3); this discrepancy should be clarified, for example by stating that the superlinear exponent is weak and approximately linear over the considered range.
[Eq. (25)] The unit for the pebble internal density in Eq. (25) is written as g cm^-1 but should be g cm^-3.
[Sect. 2.1 and Sect. 5.2.2] There are minor typographical errors such as 'mangetohydynamical' in Sect. 2.1 and 'sumarrize' in Sect. 2.1.2; these should be corrected.
[Sect. 4.2, Fig. 6] The description of water fraction ranges in the text is sometimes given as ranges like '& 10% to 1%' in Sect. 6, which appears to be an inverted or incomplete interval; the intended ordering should be checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the predicted Mp–Mstar scaling is inherited from an externally calibrated pebble-isolation formula, but the observational comparison is an independent test, and no parameter is fitted to the exoplanet masses.

full rationale

The paper's central claim is that the characteristic super-Earth mass is set by the pebble isolation mass. The scaling in Eq. (39) is derived by inserting the disk scale-height scaling into the Bitsch et al. (2018) isolation-mass formula, Eq. (26). That formula already carries the linear Mstar dependence, so the present paper's predicted scaling is algebraically inherited from this input rather than newly derived. This is not circular, however, because the input was calibrated to 3D hydrodynamic simulations of gap-opening, not to the exoplanet mass–stellar mass relation used for comparison. The population synthesis does not fit any parameter to the observed Mp–Mstar trend, and the agreement with Wu (2019) and Pascucci et al. (2018) is an external falsifiable check. The self-citations (Lambrechts & Johansen 2012; Bitsch et al. 2018; Ormel & Liu 2018) provide independent simulation or analytic results that do not contain the target observational correlation, so they do not raise the circularity score. Appendix A.2 shows an alternative migration-limited channel also yields a linear scaling, but that is a uniqueness or underdetermination caveat, not a definitional reduction; the paper's own derivation is self-contained once the externally calibrated isolation-mass prescription is granted. Correctness risk about extrapolating the Bitsch formula to 0.08 Msun is a model-assumption concern, not circularity.

Assumptions & free parameters 14 free parameters · 8 assumptions · 0 invented entities

The central mass scaling rests on the adopted Bitsch et al. (2018) isolation mass formula and on sampled disk scaling relations. The free parameters are mostly chosen from observations or convention rather than fitted to the exoplanet mass data used for comparison, but the Mp-Mstar result depends directly on the Miso formula's built-in Mstar scaling.

free parameters (14)

alpha_g (global disk viscosity) = 1e-2
Chosen fiducial Shakura-Sunyaev alpha; controls disk surface density, lifetime, and gas accretion (Sect. 2.1).
alpha_t (turbulent viscosity for pebbles and migration) = 1e-3, 1e-4
Adopted from ALMA constraints on disk turbulence; sets pebble scale height and migration torques (Sect. 2.1, Table 2).
kappa0 (disk opacity coefficient) = 0.01
Fiducial opacity-law coefficient; affects inner disk structure and ice line location (Sect. 2.1.2).
xi (pebble-to-gas flux ratio) = log-uniform 0.0033 to 0.03
Randomized input mapped to stellar metallicity via [Fe/H] = log10(xi/0.01) (Sect. 5.1, Eq. 38).
Mdot0-Mstar relation = 6e-8 (Mstar/Msun)^1.8 Msun/yr, sigma 0.3
Empirical scaling of initial disk accretion rate with stellar mass; drives pebble flux and disk lifetime (Sect. 3, Table 2).
Rd0 (initial characteristic disk size) = uniform 20 to 200 AU
Assumed independent of stellar mass; controls disk lifetime through viscous spreading (Sect. 3.1, Table 2).
t0 (embryo injection time) = uniform 0.1 to 3 Myr
Determines remaining disk gas and pebbles; early formation increases masses (Appendix A.3).
Mstar sampling = log-uniform 0.08 to 1 Msun
Stellar mass is the key independent variable; sampled log-uniformly (Table 2).
embryo initial mass = 1e-2 Mearth
Starting mass of protoplanetary embryos; fixed in all runs (Sect. 2.2.1).
pebble size = 1 mm, Stokes ~0.01
All pebbles fixed at 1 mm, motivated by bouncing limits and ALMA spectral indices; no growth or fragmentation (Sect. 2.2.1).
water mass fraction in icy pebbles = 35 percent
Assumed composition of pebbles beyond the water ice line following Bitsch et al. (2019); drives water-content predictions (Sect. 2.1.3).
kappa_env (envelope opacity) = 0.05 cm2/g
Sets Kelvin-Helmholtz gas accretion timescale; assumed independent of metallicity (Sect. 2.2.2).
facc (Hill sphere gas accretion fraction) = 0.5
Parameterizes how much gas inside the planet's Hill sphere is accreted; affects gas giant growth (Sect. 2.2.2).
Lstar-Mstar exponent = p = 2
Fiducial pre-main-sequence luminosity relation; affects ice line and irradiation (Sect. 3, Appendix A.4).

assumptions (8)

standard math Standard alpha-disk and self-similar viscous evolution (Lynden-Bell and Pringle 1974) describe gas disk evolution.
Used in Sect. 2.1.1 to evolve Mdot and disk size.
domain assumption The disk has a two-component structure: viscous heating inside, stellar irradiation outside, with no MHD winds.
Sect. 2.1.2; ignores non-ideal MHD effects named in Sect. 2.1.
domain assumption Pebble accretion efficiencies from Liu and Ormel (2018) and Ormel and Liu (2018) apply to 1 mm pebbles with Stokes numbers below 10.
Sect. 2.2.1, Eqs. (20)-(21); central to core growth.
domain assumption The pebble isolation mass formula of Bitsch et al. (2018), Eq. (26), is valid across 0.08 to 1 Msun and halts pebble accretion.
Used in Eqs. (27) and (39); this is the load-bearing premise for the Mp-Mstar scaling.
domain assumption Type I migration torques (Paardekooper et al. 2011) and gap and type II prescriptions (Kanagawa et al. 2015, 2018) describe orbital evolution, with Mgap = 2.3 Miso.
Sect. 2.3; migration determines where planets stall and how much they accrete.
domain assumption Stellar metallicity maps directly to disk pebble-to-gas flux ratio via [Fe/H] = log10(xi/xi_sun), and opacity scales as kappa/kappa0 = xi/xi_sun.
Sect. 5.1, Eq. (38); without this mapping the metallicity comparison is not defined.
domain assumption Embryos grow as isolated single planets; multi-body dynamics, giant impacts, and planet-planet scattering are neglected.
Sect. 5.2.1 notes many observed systems are multiples and calls for future N-body work.
domain assumption Pebbles beyond the water ice line contain 35 percent water by mass; interior pebbles are dry silicates.
Sect. 2.1.3; drives all water-fraction predictions in Sect. 4.2.

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Cite this review

Pith. "Pith review of Super-Earth masses sculpted by pebble isolation around stars of different masses." pith.science (2026). https://pith.science/paper/NIQMUYBX

@misc{pith2026190900759,
  author       = {Pith},
  title        = {Pith review of: Super-Earth masses sculpted by pebble isolation around stars of different masses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NIQMUYBX}},
  note         = {Machine review of arXiv:1909.00759}
}

read the original abstract

We develop a pebble-driven model to study the formation and evolution of planets around stars in the mass range of 0.08 and 1 solar mass. The growth and migration of a large number of individual protoplanetary embryos are simulated in a population synthesis manner. We test two hypotheses for the birth locations of embryos: at the water ice line or log-uniformly distributed over entire protoplanetary disks. Two types of disks with different turbulent viscous parameters alpha of 1e-3 and 1e-4 are investigated, to shed light on the role of outward migration of protoplanets. The forming planets are compared with the observed exoplanets in terms of masses, semimajor axes, metallicities, and water contents. We find that gas giant planets are likely to form when the characteristic disk sizes are larger, the disk accretion rates are higher, the disks are more metal-rich and/or their stellar hosts are more massive. Our model shows that 1) the characteristic mass of super-Earth is set by the pebble isolation mass. Super-Earth masses increase linearly with the mass of its stellar host, corresponding to one Earth mass around a late M-dwarf star and 20 Earth masses around a solar-mass star. 2) The low-mass planets up to 20 Earth masses can form around stars with a wide range of metallicities, while massive gas giant planets are preferred to grow around metal-rich stars. 3) Super-Earth planets that are mainly composed of silicates, with relatively low water fractions can form from protoplanetary embryos at the water ice line in weakly turbulent disks where outward migration is suppressed. However, if the embryos are formed over a wide range of radial distances, the super-Earths would end up having a distinctive, bimodal composition in water mass. Our model succeeds in quantitatively reproducing several important observed properties of exoplanets and correlations with their stellar hosts.

Figures

Figures reproduced from arXiv: 1909.00759 by the authors.

**Figure 1.** Plot of the observed planet mass vs the stellar mass. The blue dots are the planets only detected by radial velocity surveys with a low mass limit, and the red dots are the planets with true masses either from a combined radial velocity and transit surveys, or from transit timing variation measurements. The black line is adopted from Eq. (39), indicating the pebble isolation mass (see details in Sect. 5), and the gr… view at source ↗

**Figure 2.** Type I migration coefficient fI as functions of the planet mass and disk radius. The red (blue) means the migration is outward (inward). The left and right columns show typical disk models with M˙ g = 3 × 10−8 M yr−1 around M? = 1 M stars, and M˙ g = 3 × 10−10 M yr−1 around a M? = 0.1 M , respectively. The top and bottom rows are disk turbulent viscosity αt of 10−3 and 10−4 . The black line refers to the zero-torque… view at source ↗

**Figure 3.** Evolution of disk accretion rate (black) and characteristic disk size (blue). When M˙ g < M˙ pho (Eq. (6)), the disk angular momentum is transported by viscous accretion; later the gas removal is driven by stellar X-ray evaporation. The expansion of the disk sizes are shown in solid lines due to viscous spreading. Top: three systems with different initial disk sizes (Rd0 = 30 AU, 100 AU and 160 AU) are shown around… view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Planet mass growth (top) and orbital evolution (bottom) with different initial disk sizes (panel a, run_R1 to run_R3), initial disk accretion rates (panel b, run_D1 to run_D3), disk metallicities (panel c, run_Z1 to run_Z3) and masses of central stars (panel d, run_M1 …

**Figure 5.** Figure 5: Maps for the growth and migration of planets around stars of 1 M (left) and 0.1 M (right) and at disk turbulent αt = 10−3 (top) and 10−4 (bottom). The initial time t0 and initial location r0 are shown in x and y axis. The color corresponds to the final mass of the plan…

**Figure 6.** Figure 6: Maps for the core water mass fraction of planets around stars of 1 M (left) and 0.1 M (right) and at disk turbulent αt = 10−3 (top) and 10−4 (bottom). The initial time t0 and initial location r0 are shown in x and y axis. The color corresponds to the water fraction, an…

**Figure 7.** Figure 7: Monte Carlo sampling plot of the planet mass vs the stellar mass, with the ice line planet formation model (Scenario A) in the left, the loguniform distributed planet formation model (Scenario B) in the right, αt = 10−3 in the top and αt = 10−4 in the bottom. The colo…

**Figure 7.** Figure 7: The turbulent strength αt of 10−3 and 10−4 are exhibited in the upper and lower panels. The color corresponds to the planetary water fraction. The simulations show that the masses of planets correlate with the masses of their stellar hosts. In both formation scenario…

**Figure 8.** Figure 8: Monte Carlo sampling plot of the planet mass vs the stellar metallicity, with the ice line planet formation model (Scenario A) in the left, the log-uniform distributed planet formation model (Scenario B) in the right, αt = 10−3 in the top and αt = 10−4 in the bottom. T…

**Figure 9.** Figure 9: Monte Carlo sampling plot of the planet mass vs the semimajor axis, with the ice line planet formation model (Scenario A) in the left, the log-uniform distributed planet formation model (Scenario B) in the right, αt = 10−3 in the top and αt = 10−4 in the bottom. The co…

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