Evolutionary tracks of giant planets formed by disk instability

Mirco Bussmann; Ravit Helled

REVIEW 3 major objections 5 minor 151 references

Evolutionary tracks of giant planets formed by disk instability

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Disk-instability cooling tracks reproduce the measured masses of directly imaged giant planets.

desk verdict A genuinely unified set of disk-instability tracks with a fair comparison to the four dynamical-mass planets; the adiabatic collapse assumption is the main soft spot, but the paper deserves a serious referee. read the letter →

arxiv 2608.11984 v1 pith:CIYJTCMG submitted 2026-08-12 astro-ph.EP

Mirco Bussmann , Ravit Helled This is my paper · ORCID

classification astro-ph.EP

keywords diskinstabilitygiantplanetformationplanetaryevolutioncoolingtracksdynamicalcollapsemetallicitydirectlyimagedexoplanetsage-luminosityrelation

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 builds a unified evolutionary model of giant planets born by disk instability, following a gas clump from its extended, cold pre-collapse state through dynamical collapse and into billions of years of contraction and cooling. The aim is to see whether such tracks, without tuning to observed luminosities, can match the age–luminosity–mass relations of directly imaged exoplanets. For the four planets with measured dynamical masses in the model's $1$–$12\,M_{\rm J}$ range—HR 8799 e, AF Lep b, Beta Pic b, and Beta Pic c—the inferred masses agree with the dynamical measurements. The authors also find that assumed metallicity shifts inferred masses by up to $1.5\,M_{\rm J}$ at the ages typical of directly imaged planets, and that luminosity evolution alone cannot distinguish disk instability from core accretion. If the tracks are right, disk instability remains a viable formation pathway for these objects, and composition must be included when deriving masses from luminosity.

What carries the argument

The carrying object is a set of cooling tracks for $1$–$12\,M_{\rm J}$ protoplanets computed in one numerical framework that spans three phases: a slow Kelvin–Helmholtz contraction of an extended, low-density gas clump (radii of order AU, surface temperatures below 100 K, modeled with low-temperature opacities down to 10 K); a dynamical collapse triggered when molecular hydrogen begins to dissociate at central temperatures of roughly 2000 K; and long-term contraction and cooling lasting several gigayears. The collapse itself is not simulated: the post-collapse radius is read from a precomputed table that maps mass, composition, and total entropy to radius under the assumption that the collapse is adiabatic, so the initial post-collapse entropy is fixed by the clump's pre-collapse evolution rather than chosen by hand. Metallicity enters through opacity scaling and through the resulting initial structure, and the same tracks are then inverted to turn measured age and luminosity into inferred mass.

What would settle it

A directly imaged planet in the $1$–$12\,M_{\rm J}$ range whose dynamical mass lies outside the mass range implied by its measured age and luminosity for every metallicity between $0.5$ and $2$ times protosolar would contradict the tracks; a concrete version is a benchmark planet whose dynamical mass disagrees with the inferred mass by more than the model's $1.5\,M_{\rm J}$ metallicity spread. A direct calculation that would also settle the matter is a 3D radiation-hydrodynamic collapse simulation showing that the total entropy before and after collapse differs by more than a few percent.

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Extended reading notes

Core claim

The central claim is that the initial entropy of a disk-instability planet is not an arbitrary hot-start parameter but is set by the preceding clump evolution, and that cooling tracks built on that entropy reproduce the observed luminosities of all currently known directly imaged giant planets with dynamical mass constraints. For a protosolar composition, the tracks imply masses of $8.42$–$10.90\,M_{\rm J}$ for HR 8799 e (dynamical mass $9.6^{+1.9}_{-1.8}\,M_{\rm J}$), $4.11$–$6.94\,M_{\rm J}$ for AF Lep b ($3.75\pm0.5\,M_{\rm J}$), $6.21$–$8.10\,M_{\rm J}$ for Beta Pic c ($8.3\pm1.0\,M_{\rm J}$), and above $11.85\,M_{\rm J}$ for Beta Pic b ($9.3^{+2.6}_{-2.5}\,M_{\rm J}$). Metallicity between $0.5$ and $2$ times protosolar changes the inferred mass of a given object by up to $1.5\,M_{\rm J}$, with the largest spread at ages of tens of megayears. The authors conclude that disk instability cannot be ruled out by current observations, and that luminosity is not a discriminator between formation pathways because core accretion with hot accretion, and even cold accretion with massive cores, can produce nearly identical cooling curves.

Load-bearing premise

That the dynamical collapse is adiabatic—the clump's total entropy just before collapse equals its entropy just after, so the newborn planet's radius and luminosity come from an entropy-conserving interpolation table rather than from simulating the collapse—is the load-bearing assumption; if entropy is lost or gained during collapse, the initial radii and luminosities, and with them the masses inferred in Section 6, would shift.

Editorial extensions

If this is right

The masses of directly imaged planets can be inferred from luminosity and age using disk-instability tracks, and the resulting estimates agree with independent dynamical masses for HR 8799 e, AF Lep b, Beta Pic b, and Beta Pic c.
Planetary metallicity must be treated as a source of uncertainty in mass estimates; at ages around 30–60 Myr, unknown composition changes the inferred mass by up to $1.5\,M_{\rm J}$.
Luminosity alone cannot identify the formation pathway: disk-instability tracks converge with hot-accretion core-accretion models, and with cold-accretion models when the core is massive.
The pre-collapse phase is brief but mass-dependent, lasting about $2.6\times10^5$ yr for a $1\,M_{\rm J}$ clump versus $5.6\times10^3$ yr for a $10\,M_{\rm J}$ clump, and higher metallicity roughly doubles that timescale by raising opacity.
Formation models with different initial entropies can still be distinguished at young ages, so evolutionary models remain useful for diagnosing formation physics even though luminosity is degenerate at later times.

Reading between the lines

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

If more dynamical masses arrive, a population-level test becomes possible: a systematic offset of future objects from the disk-instability mass–luminosity relation across the full metallicity range would start to disfavor one formation channel.
The entropy-conserving collapse assumption could be checked with 3D radiation-hydrodynamic simulations of collapsing clumps; substantial entropy loss during collapse would shift the inferred masses systematically lower.
Because metallicity shifts masses most at ages of tens of megayears, atmospheric or spectral constraints on the composition of young directly imaged planets could break the degeneracy and sharpen mass estimates.
The result suggests that apparent discrepancies between hot-start models and observed planets should be interpreted as composition and age effects before being attributed to formation pathway.

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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

Colleague,

Here's my take on Bussmann & Helled. The new thing is a single numerical framework, built on MESPA, that follows a gas clump from the pre-collapse phase through the dynamical collapse and into gigayear cooling tracks, for masses 1–12 MJ and metallicities 0.5–2 Zproto. Previous work treated the pre-collapse contraction and the long-term evolution separately; this paper connects them and makes the comparison to directly imaged planets with dynamical masses. That comparison—four objects, HR 8799 e, AF Lep b, β Pic b, β Pic c—is the useful part. The paper does not oversell: the conclusion is that disk instability remains viable, not that it is the pathway, and they explicitly show that luminosity alone cannot distinguish DI from hot-start core accretion.

The metallicity grid is a genuine plus. They show that for a fixed age and luminosity the inferred mass can shift by ~1.5 MJ depending on assumed Z, which matters for interpreting direct imaging surveys. That is an actionable result.

The soft spots are real but not fatal. The load-bearing assumption is that the collapse conserves total entropy: they replace the dynamical collapse with a jump to a tabulated radius at fixed entropy. The stress-test note is right that shock heating during collapse would change the post-collapse entropy and, with it, the inferred masses. The paper acknowledges the approximation in Section 2 and in the summary, but it never quantifies the sensitivity. A simple test—running the post-collapse tracks with initial entropy varied by a few percent—would have told them how robust the mass predictions are. I suspect it would not break the overall viability claim, because the sample is small and the dynamical mass uncertainties are broad, but for two of the four planets (β Pic b and AF Lep b) the agreement is already in the 1-sigma tail. For β Pic b, raising the entropy would actually lower the inferred mass and improve the agreement; for AF Lep b it would push the lower edge down, potentially weakening it. So the effect is not one-directional, and the authors should show it rather than leave it.

The EoS switch from SCvH to CMS is handled carefully in Appendix A: the 50% radius difference disappears after a few Myr, and the choice of EoS matters more than the initial radius. That is good. The simple grey atmosphere is a limitation, but the comparison with non-grey models (ATMO, Sonora) suggests it is not driving the results. On reproducibility, MESPA is public but the paper does not pin a version; a versioned release would help. The self-citations to the group's own code are appropriate given the code is open.

Overall: this is a solid, honest paper. It deserves a proper peer review. A good referee should ask for an entropy-sensitivity test and a versioned code snapshot, but neither request is grounds for rejection. I'd cite it and I'd bring it to the group.

Referee Report

3 major / 5 minor

Summary. The paper presents self-consistent evolutionary tracks for giant planets formed by disk instability, using the MESPA extension of MESA. The authors follow a gas clump from an extended pre-collapse phase, through an assumed adiabatic dynamical collapse, into long-term contraction for masses 1-12 Jupiter masses and metallicities 0.5-2 times proto-solar. They compute cooling curves, study the metallicity dependence of inferred masses, compare their tracks to established hot-start and core-accretion models, and apply the tracks to infer masses of directly imaged planets with dynamical constraints (HR 8799 e, AF Lep b, Beta Pic b, Beta Pic c). The central claim is that the disk-instability tracks, without tuning to observations, yield masses consistent with all four dynamical mass measurements, so disk instability remains a viable formation pathway; they also argue that luminosity alone cannot distinguish disk instability from core accretion.

Significance. If the tracks are substantiated, this is a valuable unified framework that connects the pre-collapse clump phase to the post-collapse long-term evolution, a step that has been largely missing for disk instability. The paper explicitly quantifies the EoS-induced initial-radius uncertainty in Appendix A, compares with multiple published hot-start models and core-accretion models, and provides falsifiable mass predictions for benchmark directly imaged planets. The finding that metallicity can change inferred masses by up to 1.5 Jupiter masses is a useful caution for mass inference in the direct-imaging community. The main caveat is that the central consistency claim rests on a model assumption (adiabatic collapse with conserved total entropy) that is not stress-tested in the manuscript, and the agreement for two of the four planets is marginal rather than strong.

major comments (3)

[Section 2, Figure 2, Table 2] The adiabatic-collapse assumption is load-bearing: the post-collapse model is constructed by conserving total entropy, and the initial entropy controls the subsequent luminosity evolution. Appendix A quantifies the radius mismatch between the SCvH and CMS equations of state at fixed entropy, but it does not test the sensitivity of the tracks to the entropy itself. A non-adiabatic collapse with modest shock heating could shift the inferred masses in Table 2 by an amount comparable to the margin of agreement for Beta Pic b and AF Lep b. Please add a quantitative sensitivity test (for example, perturb the post-collapse entropy by plus or minus 1-2 k_B per baryon and recompute the cooling tracks and inferred masses), or cite published hydrodynamical evidence demonstrating that total entropy is conserved to the required precision during clump collapse.
[Table 2, Beta Pic b row; Section 6.3] The text states that Beta Pic b is consistent with the disk-instability tracks, but the inferred mass is >11.85 MJ while the dynamical mass is 9.3 +2.6/-2.5 MJ. The 1-sigma dynamical interval extends only to about 11.9 MJ, so the overlap is at the extreme tail. Calling this 'consistent' without a quantitative confidence statement overstates the agreement and is central to the paper's main claim. Please report the probability content of the overlap (e.g., by propagating the age and luminosity uncertainties into M_evol and computing the joint overlap) or phrase the result as marginal consistency.
[Section 6.3 and Table 2] The mass ranges in Table 2 appear to reflect the metallicity variation only (plus the age interval for HR 8799 e), and the full age uncertainty for the Beta Pic group is not propagated. For Beta Pic b and c, the age is 18.5 +2.0/-2.4 Myr; a younger age within this range would lower the inferred masses, possibly improving agreement for Beta Pic b, while an older age would worsen it. The paper demonstrates the age sensitivity for AF Lep b, but does not apply the same propagation to Beta Pic. Please propagate the age (and luminosity) uncertainties into the quoted M_evol ranges, or explicitly state that the ranges include only the metallicity variation.

minor comments (5)

[Abstract and Section 2] The text says 'the MESPA code' as if it were a standalone code; MESPA is an extension to MESA. Please write 'MESA with the MESPA extension' (or similar) for precision.
[Section 3, Figure 1 caption] The caption says 'The sharp transitions in each panel corresponds to the dynamical collapse.' The verb should agree: 'correspond'.
[Table 1 caption] The table is titled 'Synthetic observations with inferred masses,' but the table lists synthetic observations and the resulting inferred masses; consider renaming it 'Synthetic observations and inferred masses' to avoid ambiguity.
[Section 6.1] The target name 'WISPIT 2' appears in the sentence about multi-planet systems; this is likely a typo or malformed name. Please verify and correct.
[Section 6.2] The phrase 'a 1.09 ± 0.06 Solar mass star' uses a capital S; for consistency with journal style, use 'solar'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity in the core comparison: inferred masses from disk-instability tracks are tested against independent dynamical masses, not fitted to them.

full rationale

The central claim is tested in Section 6, where the paper infers masses by reading its precomputed cooling tracks at the observed ages and luminosities and compares those inferred masses with astrometric dynamical masses (Brandt et al. 2021a,b; Balmer et al. 2025). No parameter is fitted to those four planets. The tracks are generated with the MESA/MESPA code using stated EoS and opacity inputs, and the initial clump conditions are relaxed from an independent 12 M_J hydrodynamical clump structure from Humphries et al. (2019), not from the observed exoplanet luminosities. The adiabatic-collapse step in Section 2 sets the post-collapse total entropy equal to the pre-collapse entropy and constructs the radius from a mass/composition/entropy table; this is an explicit physical assumption, and Appendix A tests only the EoS-dependent radius part of it, while the broader entropy-conservation assumption is a correctness risk rather than a circular input. The self-citations to MESPA (Helled et al. 2025), to the opacity scaling (Helled & Bodenheimer 2011), and to adiabatic collapse (Vazan & Helled 2012) support modeling choices and prior work; they do not feed the target result into the input. Therefore no specific equation reduces by construction to the dynamical masses, and the paper should be scored at the low, non-circular end of the scale.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The model introduces no new physical entities. The free parameters are the linear opacity-metallicity scaling and the chosen collapse trigger; the main axioms are the adiabatic collapse approximation and the EoS/opacity choices. The results depend on these assumptions, but they are stated.

free parameters (1)

Opacity metallicity scaling coefficient = 1 (linear scaling)
The pre-collapse and post-collapse opacities are assumed to scale linearly with metallicity (Section 2, after Pollack et al. 1985 and Alexander & Ferguson 1994). The slope is chosen by hand, not derived; the authors note that a different scaling would change the relation.

assumptions (6)

standard math The stellar structure equations and the Henyey solution scheme as implemented in MESA describe the thermal evolution of a gas clump and young planet.
Used throughout Section 2; the paper relies on MESA's standard stellar structure treatment.
ad hoc to paper The dynamical collapse is adiabatic, preserving the total entropy from the pre-collapse clump to the post-collapse protoplanet.
Section 2 states: 'we assumed an adiabatic collapse (Rakavy et al. 1967; Kovetz et al. 2009; Vazan & Helled 2012), and construct a new post-collapse model in which the radius is selected so that the total entropy of the object is the same as before collapse.'
domain assumption Pre-collapse opacities at temperatures below 100 K can be represented by the Pollack et al. (1985) and Alexander & Ferguson (1994) tables with a linear metallicity scaling.
Section 2; the paper extends MESA's opacity tables down to 10 K and scales by Z/Zproto.
domain assumption The SCvH EoS is valid for the pre-collapse state and the CMS EoS for the post-collapse state, and switching between them at collapse does not bias long-term evolution.
Section 2 and Appendix A; the switch causes up to 50% radius differences, but the paper argues it is short-lived.
domain assumption The time at which the gas clump forms can be identified with the age of the system.
Section 3: 'as gaseous clumps are expected to form rather early on in the disk, it is reasonable to treat the time of the cooling curves as the age of the system.'
domain assumption The heavy-element mass fraction of the planet corresponds to the assumed Z and the observed planets' metallicities lie within the 0.5-2 Zproto grid.
Used when inferring masses in Section 6; the grid covers 0.5, 1, 2 Zproto.

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Cite this review

Pith. "Pith review of Evolutionary tracks of giant planets formed by disk instability." pith.science (2026). https://pith.science/paper/CIYJTCMG

@misc{pith2026260811984,
  author       = {Pith},
  title        = {Pith review of: Evolutionary tracks of giant planets formed by disk instability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CIYJTCMG}},
  note         = {Machine review of arXiv:2608.11984}
}

read the original abstract

The evolution of giant planets depends on their formation history. While several evolutionary models self-consistently link planet formation by core accretion to long-term evolution, such models for planets formed by disk instability are lacking. We simulate the evolution of giant planets formed by disk instability and follow their evolution including the pre-collapse phase, dynamical collapse, and long-term contraction in a unified numerical framework. The evolution is simulated using the MESPA code with modifications that allow us to model gas clumps in the pre-collapse phase. We consider masses between 1 and 12 Jupiter masses and metallicities ranging from 0.5 to 2 times the protosolar value. We confirm that the pre-collapse timescale strongly depends on the planetary mass, and that after dynamical collapse the objects reach a state of long-term contraction which lasts for billions of years. We show that metallicity is a major source of uncertainty in mass estimates derived from the age-luminosity relations. For the metallicity range considered here, we find that for a given measurement of age and luminosity the difference in the inferred mass can be up to 1.5 Jupiter masses. We find that our evolution tracks predict masses that are consistent with the measured dynamical mass constraints for HR 8799 e, AF Lep b, Beta Pic b and Beta Pic c. We also show that both core accretion and disk instability can lead to very similar long-term evolutionary tracks. The agreement between our models and dynamical mass measurements suggests that disk instability remains a viable formation pathway for giant exoplanets. The luminosity evolution alone cannot distinguish between the two formation pathways. Finally, we suggest that planetary metallicity must be taken into account when inferring the masses of young giant planets from their luminosities, as it significantly affects their evolution.

Figures

Figures reproduced from arXiv: 2608.11984 by the authors.

**Figure 1.** shows the full evolutionary tracks for objects with 1, 3, 6 and 10 MJ . The evolution can be split into three phases: pre-collapse, dynamical collapse, and long-term (post-collapse) evolution. During the pre-collapse phase the gas clumps are still very extended with radii of the order of astronomical units. The gas clumps slowly contract, leading to a steady increase in the central density and temperature. Due to th… view at source ↗

**Figure 4.** Evolution of luminosity, central temperature, radius and central density throughout the pre-collapse phase for a 1 MJ gas clump. Higher metallicities lead to higher opacities and thus prolong the pre-collapse phase. The assumed metallicity influences the cooling curves in three main ways: First, an increased metallicity prolongs the precollapse phase and delays the onset of the dynamical collapse. As a result, the … view at source ↗

**Figure 3.** Complete cooling curves for different planetary masses. The evolution includes the pre-collapse evolution, the dynamical collapse and the long-term contraction. 4. The importance of metallicity Different metallicities in gaseous clumps significantly alter the thermal evolution [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: Luminosity evolution for protoplanets with masses of 1, 4 and 10 MJ assuming three different metallicities. The plot shows the end of the pre-collapse phase, the dynamical collapse and the initial phase of the post-collapse, long-term evolution. For all the masses we c…

**Figure 7.** Figure 7: Comparison of our cooling curves with Sonora Bobcat (Marley et al. 2021) and ATMO 2020 (Phillips et al. 2020) for a range of planetary masses. The comparison above demonstrates that our cooling tracks are broadly consistent with established hot-start calculations. How…

**Figure 8.** Figure 8: shows our cooling tracks for three planetary masses (1, 7 and 12 MJ) compared to evolutionary models associated with core accretion under three different assumptions: hot accretion, cold accretion with a core mass of 20 M⊕ and cold accretion with a core mass of 127 M⊕.…

**Figure 9.** Figure 9: Cooling tracks of various planetary masses with a protostellar composition. We focus on ages typical of directly imaged planets. The data points correspond to the four exoplanets with measured bolometric luminosities and (dynamical) masses. The measured dynamical masse…

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