A primordial radius valley as a consequence of planet formation

Anders Johansen; Caroline Dorn; Jesper Nielsen; Komal Bali

REVIEW 4 major objections 5 minor 1 cited by

A primordial radius valley as a consequence of planet formation

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper argues that the observed radius valley around 1.5–2 Earth radii can be produced by planet formation alone, with no atmospheric mass loss required: only cores reaching the pebble isolation mass accrete enough gas to be inflated…

desk verdict Serious population-synthesis case for a primordial radius valley, but the no-boil-off assumption is load-bearing and untested. read the letter →

arxiv 2502.10186 v1 pith:G7RHBHRN submitted 2025-02-14 astro-ph.EP

Jesper Nielsen , Anders Johansen , Komal Bali , Caroline Dorn This is my paper · ORCID

classification astro-ph.EP

keywords radiusvalleypebbleisolationmassaccretionatmosphericcontractionplanetformationsub-Neptunesexoplanetradiimigration

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

This paper seeks to establish that the observed dearth of close-in planets at roughly 1.5–2 Earth radii—the radius valley—can arise from planet formation itself, without requiring later atmospheric mass loss. Using a pebble-accretion formation model, the authors argue that cores which do not reach the pebble isolation mass stay small and rocky, while cores that do cross that threshold cool, accrete a few percent of gas, and are inflated to sizes above the gap. The valley then appears after the protoplanetary disc disperses, as accreted atmospheres contract on gigayear timescales but leave the two populations separated. If correct, the valley is a fossil of the formation process rather than a signature of photoevaporation or core-powered mass loss, and it should be most pronounced for close-in planets.

What carries the argument

The switch is the pebble isolation mass $M_{\rm iso}$, the core mass at which a planet's perturbation of the disc halts pebble accretion and lets the planet cool. Below this mass, the atmosphere is held thin by pebble-accretion heating; above it, Kelvin-Helmholtz contraction drives gas accretion. To follow what happens after the disc disappears, the paper builds an analytical contraction model based on a convective envelope with an effective Bondi radius, solving for the luminosity and atmospheric energy as functions of the radiative–convective boundary height, then maps that height to a transit radius.

What would settle it

Run the same population with a boil-off prescription, for example by removing the disc's outer pressure boundary at dispersal and letting the atmosphere expand to the Bondi radius, and check whether the gap survives; if most close-in planets above the isolation mass lose their atmospheres within a few million years and the valley vanishes or shifts below about 1.5 Earth radii, the claim that contraction alone can create the primordial valley is refuted.

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Extended reading notes

Core claim

The central claim is that the radius valley is primordial in the sense that it is set during the formation of planetary cores, and no mass loss after disc dispersal is needed to produce the gap. Planets below the pebble isolation mass never accrete more than about $10^{-5}$ of their mass as atmosphere and remain bare cores, while planets that reach the isolation mass accrete atmospheres of roughly 0.1–10% of the core mass and are inflated to radii above the valley. The paper further predicts that the valley's location for the intrinsic population tracks the pebble isolation mass and rises with orbital period, while for close-in planets ($P<100$ days) the fitted slope reverses, and that the valley gradually fills in for wider orbits.

Load-bearing premise

The load-bearing premise is that after the protoplanetary disc disperses, no atmospheric mass is lost: the paper assumes contraction outruns any expansion from stellar XUV heating or the planet's own thermal energy, and it omits boil-off, the rapid loss triggered by the sudden removal of disc pressure; if boil-off strips a substantial part of the accreted gas, the inflated population shrinks and the valley moves or disappears.

Editorial extensions

If this is right

Close-in planets below the valley should be predominantly rocky and nearly atmosphere-free, while those above should host H/He atmospheres of a few percent by mass.
The radius gap for the full planet population should rise with orbital period as $P^{0.1-0.17}$, whereas for $P<100$ days the slope should steepen and reverse.
The valley should appear within about a gigayear after disc dispersal and persist for at least 3 Gyr even with no mass loss.
Surveying planets with periods beyond 100 days should reveal the valley progressively filled in by bare, water-rich cores of a few Earth masses.

Reading between the lines

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

If the valley is a formation signature, its location directly encodes the pebble isolation mass, so radius-valley surveys could be inverted to measure disc scale height and turbulence at the time of planet formation—an application the paper does not develop.
Mass-loss and formation are not mutually exclusive: adding photoevaporation in the paper's own test partially fills the valley, so a realistic population may show a deeper or shifted gap depending on which process dominates at a given stellar mass and age.
A discriminating observation would be atmospheric age dating or mean-molecular-weight measurements of planets just above the valley: formation-only models predict pristine accreted H/He on young planets, while mass-loss models predict depletion signatures.
The predicted filled valley at long periods could be tested with PLATO or Roman transit samples; a valley that remains deep beyond 100 days would disfavour the primordial interpretation.

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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

Short version: this is a serious, well-documented population synthesis that makes a real case for a primordial radius valley, but the case leans on a load-bearing assumption—no boil-off—that the authors openly admit but never test. The new pieces are worth engaging with: a full pebble-accretion population synthesis, a new analytical contraction model for post-disc cooling, and a specific prediction that the valley slope reverses for close-in planets and the valley fills in at wider orbits.

The paper does several things well. The valley is not fitted; it emerges from the pebble isolation mass threshold. The authors run many variants (alpha, fragmentation velocity, opacity, disc gaps, photoevaporation) and show the valley location is robust. They also compare their analytical contraction model to the full numerical solution and quantify its self-gravity overestimate. The close-in slope reversal is a sharp, falsifiable prediction for PLATO. And they are transparent about what the model does not include.

The main soft spot is the omission of boil-off. As the stress-test note correctly points out, this is not a minor technical detail. The population that sits above the valley includes low-mass cores (1-2 Earth masses) with a few percent atmospheres—exactly the regime where sudden loss of disc pressure can strip a large fraction of the atmosphere. If that happens, the large-radius side shrinks and stripped cores can fill or shift the valley. The authors acknowledge this in Sect. 6.1 and even note that boil-off "would mimic the effects of photoevaporation," but they do not include it. Appendix F tests XUV photoevaporation, which is not the same process. So the central claim that the valley is primordial—i.e., survives without any mass loss—is not yet established.

Other issues are smaller. The analytical model deliberately overestimates self-gravity in the luminosity; the authors show it is a reasonable approximation, but it could bias contraction timescales. The varying-inner-edge model is ad hoc and produces an excess of 3-4 Earth-radius planets that the authors wave away with mass loss. And there is no released code or data, so the population synthesis is not immediately reproducible.

None of this makes the paper sloppy. It is honest, clearly written, and the central bifurcation is physically motivated. The paper is for exoplanet formation theorists and observers interpreting Kepler/PLATO demographics. It deserves a serious referee. I would send it to review, with the request that the boil-off question be either modeled or more carefully bounded before publication.

Referee Report

4 major / 5 minor

Summary. This paper presents a population synthesis of close-in planets formed by pebble accretion, in which gas accretion begins only after a core reaches the pebble isolation mass, and the accreted H/He atmosphere contracts after the disc disperses with no mass loss. The model produces a radius valley around 1.5-2 R⊕: planets below the isolation mass remain bare rocky cores below the valley, while planets that reach it accrete a few-percent atmosphere and are inflated above it. The authors show that this dichotomy appears in three migration prescriptions, derive slope fits for the full and P<100 d populations, and report that the valley is mostly a close-in feature that becomes filled at wider orbits. They also explore variations of the fragmentation velocity, alpha, opacity, photoevaporation, and steam atmospheres.

Significance. If the result holds, the paper makes an important conceptual point: the radius valley need not be carved by mass loss, but can arise from the threshold behavior of gas accretion set by the pebble isolation mass. The model is transparently documented, uses the analytic Piso-Youdin framework consistently, and is tested against several parameter variations; the authors are also explicit about what is and is not included. The prediction that the valley is a close-in phenomenon and becomes filled on wider orbits is falsifiable by PLATO/Roman transit surveys. Because the full-population slope essentially recovers the adopted pebble isolation mass scaling, the independent predictive content is concentrated in the close-in slope and composition split; those predictions should be emphasized. The main caveat is that the primordial interpretation depends on atmosphere survival immediately after disc dispersal (boil-off), which the paper explicitly omits.

major comments (4)

[Sect. 4 (first paragraph) and Sect. 6.1] The central claim that the valley is primordial requires that the atmospheres accreted above the pebble isolation mass survive the loss of pressure support when the disc dissipates. The paper omits boil-off and replaces it with the assumption that contraction is faster than expansion from internal or XUV heating; it then acknowledges in Sect. 6.1 that boil-off can remove up to 90% of an atmosphere (citing Rogers et al. 2024) and would mimic photoevaporation. Appendix F tests only XUV photoevaporation, not the immediate Bondi-regime outflow that acts on the thermal energy of the atmosphere. The low-core-mass (1-2 M⊕), few-percent-atmosphere planets that populate the large-radius side of the valley are precisely the planets for which boil-off is most efficient, so even partial stripping would shrink that population and move stripped cores into or below the valley. A quantitative boil-off treatment, or at least a physically grounded bound on the mass lost before contraction outruns the outflow, is needed before the 'primordial valley' conclusion is established.
[Sect. 4.1, Eqs. (19)-(23), Fig. 6] The post-dissipation contraction model replaces the enclosed mass m(r) with the total planet mass Mpl in the effective Bondi radius and energy integrals. The authors correctly note that this overestimates self-gravity and luminosity, and Fig. 6 shows that the overestimate is largest precisely at the high atmosphere mass fractions that define the inflated side of the valley. Because L = -dE/dt sets the contraction rate, the analytic model will make the large-radius planets shrink faster than the numerical structure equations would; this can artificially shift the inflated population toward (or into) the valley. The validation in Fig. 6 is carried out for the embedded, disc-boundary case and for one core mass; it does not quantify the effect on the final 3 Gyr radius distribution. A comparison of the analytical contraction model with the numerical model, or a corrected self-gravity treatment, for representative post-dissipation configurations is required to establish that the valley depth and location are not artifacts of this approximation.
[Sect. 5.3, paragraph after Fig. 11] The reported full-population gap slopes, Rgap ∝ P^0.17, P^0.16, and P^0.11, are, as the authors explicitly state, essentially the radius of a bare core at the pebble isolation mass, which scales as P^0.19 for the adopted Miso formula and a fixed density. This means the full-population slope is a consistency check of the input Miso prescription rather than an independent prediction of the model. The comparison with observed slopes is therefore only meaningful for the P<100 d subsample, where the slope reverses and is set by the contraction model. The paper should be reframed accordingly, and the full-population slope should not be presented as support for the model without noting this relation.
[Sect. 5, 'three sets of simulations', and Sect. 5.2] The inner-edge treatment is chosen ad hoc: gas accretion and migration are stopped at 0.1 AU in two models, and in the third each planet is randomly assigned one of seven 3:2-resonant inner edges without modeling the multi-planet dynamics that would produce such a chain. The resulting population depends strongly on this choice: the varying-inner-edge model produces an excess of 3-4 R⊕ planets that the authors themselves note is not seen in observations, and this is also the model whose close-in slope they compare to observations. Since the main observational comparison is made with a model whose stopping condition is not derived from a formation mechanism, the slope agreement is not a robust test of the theory. The inner-edge distribution should either be modeled from N-body or resonant-migration simulations, or varied systematically to show that the slope comparison is robust to the prescription.

minor comments (5)

[Sect. 2, after Eq. (1)] There is a duplicated article: 'The the disc scale height is given by H = cs/Ω'. This should read 'The disc scale height...'.
[Fig. C.1 caption] The caption refers to 'the top row in Fig. C', but the figure being compared is Fig. 7; the cross-reference should be corrected.
[Sect. 6.4] The sentence 'The details of the model are described in H' should read 'described in Appendix H'.
[Sect. 2, paragraph on alpha choice] The sentence 'We choose to adopt a value of α = 0.01, which have been needed in order to match...' has a subject-verb agreement error; it should be 'which has been needed'.
[Sect. 6.1, last paragraph] The spelling 'boil-o ff' should be 'boil-off' throughout, and the same applies to 'di fferent' in several places; a final proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the valley location is set by an external pebble-isolation-mass criterion rather than fitted to observations, and the paper explicitly labels the intrinsic slope as a recovery of that input; the close-in slope reversal is an independent model output.

full rationale

The derivation is self-contained in the relevant sense. The valley location is set by the pebble isolation mass Miso(P) from Bitsch et al. (2018), Eq. (1), which is adopted a priori from external hydrodynamical simulations and is not fitted to the observed radius valley. The synthetic core-mass distribution is produced by the pebble accretion model of Nielsen et al. (2023), and the atmosphere accretion and contraction calculations are explicit (Eqs. 10-29). The intrinsic-population gap slope indeed reduces to the Miso bare-core scaling (P^0.19), but the paper states this openly in Sect. 5.3: "we essentially recover the pebble isolation mass as the size of a bare planet core... will scale with orbital period as P^0.19." This is a transparent consequence of an input, not a disguised prediction. The close-in slope reversal and the gradual filling of the valley at wide orbits are genuine outputs of the contraction model, and no parameter is fitted to the observed valley location or slope. The only notable unmodeled process, boil-off, is explicitly acknowledged in Sect. 4 and Sect. 6.1 as an omission and discussed as a physical limitation; an assumption about an external process is not a circular reduction of outputs to inputs. Self-citations such as Nielsen et al. (2023) and the overlapping authorship of Bitsch et al. (2018) supply model ingredients, but they do not smuggle in the target result or forbid alternatives through a uniqueness claim. Therefore no load-bearing circular step was identified.

Assumptions & free parameters 8 free parameters · 9 assumptions · 0 invented entities

The central claim rests on a pebble-accretion population synthesis with several hand-chosen parameters (viscosity, turbulence, fragmentation velocity, disc mass, injection ranges) and a set of physical assumptions (hydrostatic atmospheres, fixed cores after isolation, neglect of boil-off, no core cooling). These are standard in the field and are sensitivity-tested in the appendices, but they are not derived within the paper.

free parameters (8)

Viscous alpha = 0.01
Sets gas accretion and disc evolution; chosen to match observed disc sizes and stellar accretion rates (Najita & Bergin 2018; Hartmann et al. 1998). Varied to 1e-3 in Appendix C.
Turbulence alpha_t = 1e-4
Sets maximum pebble size through the fragmentation limit; chosen from literature (Lesur et al. 2023).
Pebble fragmentation velocity = 2 m/s
Chosen to reach Stokes numbers of about 0.01 in the outer disc; sensitivity tested in Appendix A.
Initial gas accretion rate onto star = 1e-7 M_sun/yr
Typical for solar-mass stars; sets disc lifetime of about 2 Myr.
Initial disc gas mass = 0.1 M_sun
Together with the accretion rate sets the disc lifetime.
Injection time range = 1 kyr to 1 Myr, uniform
Sampled uniformly; the authors tested variations and found little effect on the final population.
Injection location range = 0.5 to 20 AU, log-uniform
Sampled log-uniformly; the authors tested variations and found little effect.
Transit pressure = 20 mbar
Standard estimate for where optical depth reaches unity in transit; adopted uniformly for all planets (Burrows et al. 2004; Lopez & Fortney 2014).

assumptions (9)

domain assumption The atmosphere is spherically symmetric, in hydrostatic equilibrium, and in pressure balance with the disc midplane.
Invoked at the start of Sect. 3 to set up the structure equations (Eqs. 4a-4c).
domain assumption The gas is an ideal gas with P = ρRT.
Used in Eq. (5) for the atmospheric structure.
domain assumption The atmosphere has an inner convective layer and an outer radiative layer, with the radiative-convective boundary where ∇rad = ∇ad.
Sect. 3.1; standard two-layer model from Piso & Youdin (2014).
ad hoc to paper After disc dissipation, the atmosphere contracts without mass loss; contraction is faster than expansion from XUV or internal thermal energy.
Sect. 4, first paragraph; this is the paper's key simplifying assumption and is explicitly acknowledged by the authors.
domain assumption The core is supercritical, convective, and nearly isothermal, so its thermal energy does not affect the contraction timescale.
Sect. 4.2 and Appendix E; based on high bottom-of-atmosphere temperatures.
domain assumption Planets stop accreting solids once they reach the pebble isolation mass, and thereafter only accrete gas.
Sect. 3 opening; standard in pebble-accretion models (Lambrechts et al. 2014).
domain assumption The pebble isolation mass formula of Bitsch et al. (2018), Eq. (1), sets the threshold for gas accretion.
Used throughout to decide which planets accrete atmospheres; formula taken from prior hydrodynamical simulations.
ad hoc to paper The atmospheric energy integral replaces the enclosed mass m(r) with the total planet mass Mpl to make the equations analytically tractable.
Sect. 4.2, around Eq. (24); the authors note this overestimates self-gravity effects.
ad hoc to paper The transit radius is approximated as RRCB + ΔRtr with a constant transit pressure of 20 mbar.
Sect. 4.2, Eq. (29); a single transit pressure is adopted for all planets.

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Pith. "Pith review of A primordial radius valley as a consequence of planet formation." pith.science (2026). https://pith.science/paper/G7RHBHRN

@misc{pith2026250210186,
  author       = {Pith},
  title        = {Pith review of: A primordial radius valley as a consequence of planet formation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G7RHBHRN}},
  note         = {Machine review of arXiv:2502.10186}
}

abstract

The radius distribution of close-in planets has been observed to have a bimodal distribution with a dearth of planets around ~1.5-2.0 $R_\oplus$ commonly referred to as the ''radius valley''. The origin of the valley is normally attributed to mass-loss process such as photoevaporation or core-powered mass loss. Recent work, however, has suggested that the radius valley may instead arise as a consequence of gas accretion by low-mass planets. In this work we therefore aim to investigate the formation of a primordial radius valley from the formation of planet cores through pebble accretion up until the dissipation of the protoplanetary disc and subsequent contraction of accreted atmospheres. The goal of this work is to explore the conditions for forming a primordial radius valley from first principles of planet formation theory, rather than attempting to explain the detailed structure of the observed valley. We use an analytical model with minimal assumptions to estimate the contraction rate of atmospheres and, indeed, find the formation of a primordial radius valley. The planets smaller than the valley did not reach the pebble isolation mass, which is required for the planets to cool down sufficiently to be able to accrete a significant amount of gas. We also estimate the slopes of the radius gap as a function of orbital period for the intrinsic population as well as for planets with orbital periods <100 days. For the intrinsic population, the radius gap follows the pebble isolation mass and increases with increasing orbital period, while for close-in planets the direction of the slope reverses and decreases with increasing orbital period. We find that planets smaller than the radius valley are predominantly rocky while the population of planets larger than the valley consists of a mixture of rocky and water-rich planets.

Figures

Figures reproduced from arXiv: 2502.10186 by the authors.

**Figure 1.** Flowchart of our procedure for calculating the accreted gas when the planet is embedded in the disc (left). We also show our method for calculating the rate of contraction after disc dissipation (right). The orange boxes show the input parameters going into each calculation. The main loops are shown as solid arrows in their respective colors. The dashed arrow indicates that we check if the calculated ∆t is within th… view at source ↗

**Figure 3.** The embedded planet luminosity as a function of atmosphere mass fraction for different distances to the host star. We also show the temperature of the surrounding protoplanetary disc for all distances. The solid lines are the luminosities for a planet with a core mass of 1 M⊕ while the dashed lines are the luminosities using a planet with a core mass of 2 M⊕. For both core masses, the luminosity increases by a facto… view at source ↗

**Figure 4.** shows our gas accretion method in comparison to the method by Piso & Youdin (2014) for a 5 M⊕ core at 60 AU using the same protoplanetary disc boundary conditions as in their work but with detailed opacities that allow metal grains and gas molecules to affect the opacity as well. Using the more complex opacity law results in slightly faster gas accretion due to the lower opacities and therefore higher luminosities.… view at source ↗

Figures from the paper (10 more)

**Figure 5.** Figure 5: Atmosphere mass fraction and core mass for four planets as a function of time. The atmosphere masses remains low during pebble accretion due to the high accretion luminosity. Once pebble accretion is halted after pebble isolation mass is reached (triangles), the gas ra…

**Figure 6.** Figure 6: Comparison of luminosity (black) and total atmospheric energy (blue), between the full numerical solution (solid) presented in Sect. 3.1 and our analytical model (dashed) for the same setup as in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: Top row: Final semi-major axes and planet masses for the planet population after disc dissipation. The color denotes the final atmosphere mass fraction. The dashed lines show the pebble isolation mass at the start and end of the disc lifetime; the variation is due to t…

**Figure 8.** Figure 8: Radius as a function of time after disc dissipation for two different core masses and two different distances to the star. The planets have atmosphere mass fractions of 1% and 5% respectively, which are typical values for these core masses based on our simulations. We…

**Figure 9.** Figure 9: Histogram of the resulting planet radii for the three growth models at three different snapshots in time after disc dissipation. The filled histograms show the entire planet population while the unfilled histograms show planets with P < 100 days. Clearly, the radius ga…

**Figure 10.** Figure 10: Planet radii as a function of their semi-major axis for all three models after 3 Gyr of evolution. The color of each point indicates the atmosphere mass fraction. The dashed lines show the core radius at the pebble isolation mass at the start and end of the disc lifet…

**Figure 11.** Figure 11: Planet radius as a function of orbital period for the three different models. Circles show planets above pebble isolation mass while triangles show planets below pebble isolation mass. The color of the points show the mass fraction of water. The solid lines show the f…

**Figure 12.** Figure 12: Mass fraction of accreted water for close-in planets (P < 100 days). In magenta, we show planets with masses below the pebble isolation mass while in teal we show planets with masses above the pebble isolation mass. We find that in all models, planets with masses abov…

**Figure 13.** Figure 13: The radius distribution after 3 Gyr of evolution, using a realistic interior structure that includes the compressibility of core and mantle material as well as steam atmospheres. The distributions should be compared to the lower panels of 9. As our more detailed inter…

**Figure 14.** Figure 14: Top: Planet radius as a function of temperature for several planets with varying masses and water mass fractions. The effect of surface temperature is small in all cases, with the exception of the planet with a mass of 1 M⊕ and a high water mass fraction (40%). Plane…

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Pith tools

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

[1] [1]

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

ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

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