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Convective mixing in distant and close-in giant planets -- Dependences on the initial composition, luminosity, bloating and semi-convection

T0 review · 4 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Evolution models of giant planets indicate that dilute cores—extended interiors enriched in heavy elements—rarely survive when the planet starts out brighter than a few thousand Jupiter luminosities, so most giants should end up fully…

desk verdict Careful and unusually honest parameter study, but the headline dilute-core conclusion rests on an unresolved mesh-resolution dependence that the paper itself concedes. read the letter →

arxiv 2411.18686 v1 pith:HR37OUVJ submitted 2024-11-27 astro-ph.EP

Jesse Polman , Christoph Mordasini This is my paper · ORCID
classification astro-ph.EP
keywords dilutecoresgiantplanetevolutionconvectivemixingsemi-convectionhotJupitersinitialluminositycompositionalgradientsJupiterinterior
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

Giant planets may form with an extended region of enriched heavy elements in their deep interiors, but this paper argues that such dilute cores are usually erased. In a one-dimensional evolution model that tracks convective mixing, a Jupiter-mass planet retains its dilute core only if its post-formation luminosity starts below roughly $3\times10^3$ Jupiter luminosities and its initial composition contains sharp enough steps. Standard formation models generally deliver higher starting luminosities, so the authors conclude that only a minority of giant planets should keep a dilute core through 4.5 billion years of evolution. The result matters for interpreting exoplanet atmospheres: if dilute cores are rare, atmospheric heavy-element abundances are a more direct probe of bulk composition than they would be if layered interiors were common.

What carries the argument

The mechanism that carries the argument is compositional convection governed by the Ledoux criterion in a one-dimensional planetary evolution model. Convection begins where the radiative-conductive gradient $\nabla_{\mathrm{rad}}$ exceeds the sum of the adiabatic gradient and the compositional gradient $\nabla_X$; mixing is then modelled as a diffusion process on a separate equal-mass mesh of $5\times10^4$ points using mixing-length theory for the diffusion coefficient. Steep compositional steps—the 'stairs' in the heavy-element profile—are what hold a dilute core in place, because they make $\nabla_X$ large enough to suppress mixing. The paper's key numerical control is the mesh resolution, since it sets the maximum compositional gradient that can be represented, and the key physical control is the initial luminosity, since $\nabla_{\mathrm{rad}}$ grows roughly linearly with it.

What would settle it

Repeat the paper's Jupiter-like simulation at an initial luminosity of $10^4\,L_J$ with about $3\times10^5$ equally spaced mass points, the resolution implied by the overshooting-length estimate quoted in Sect. 3.6; if a dilute core then survives, the luminosity ceiling is an artifact of the coarser mesh rather than a physical limit.

Watch

Extended reading notes

Core claim

The central claim is that dilute cores—deep interior regions where heavy elements are present at moderate enrichment rather than in a pure compact core—are difficult to preserve. The paper finds that for a Jupiter-like planet with the heavy-element profiles taken from formation models, the limiting factor is the initial luminosity, which sets the radiative-conductive gradient that drives convection in the deep envelope. Above about $3\times10^3\,L_J$, mixing destroys the compositional staircase that stabilises the dilute core; at $10^4\,L_J$ the envelope mixes completely. Because hot-start and even many cold-start formation scenarios produce luminosities above this threshold, the paper concludes that retaining a dilute core through the whole evolution is an unlikely outcome for most giant planets, at least under the assumptions and resolution of the model.

Load-bearing premise

The central conclusion depends on assuming that 50,000 equal-mass layers capture the real compositional layering, even though the paper shows the survival of the dilute core changes drastically with mesh resolution and offers no physical argument that this count matches the true layering scale.

Editorial extensions

If this is right

  • If dilute cores are as fragile as this model says, JWST measurements of atmospheric metallicity in giant planets can be interpreted with relatively simple core-plus-homogeneous-envelope structures for most planets.
  • The first billion years matter almost exclusively: nearly all mixing happens early, so the conditions set by formation, not later evolution, decide whether a dilute core survives.
  • Close-in hot Jupiters should not simply be assumed convective: bloating lowers the intrinsic luminosity and can suppress mixing, leaving slightly more dilute cores than at wide orbits, provided they formed in place.
  • Strong semi-convection can shrink a dilute core and enrich the outer envelope, but it cannot fully erase a large initial core in the paper's models.
  • The threshold of roughly $3\times10^3\,L_J$ provides a formation-property test: planets formed by hot accretion are expected to be fully mixed, and any observed dilute core would point to cold accretion or early radiative zones.

Reading between the lines

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

  • If confirmed, the strong mesh dependence raises the possibility that the qualitative conclusion—that dilute cores are rare—is an artifact of unresolved layering; a physically motivated resolution would require resolving the estimated overshoot length, roughly $3\times10^5$ mesh points, which the paper did not run at the decisive luminosity.
  • A corollary beyond the paper is that the atmospheric-versus-bulk metallicity mismatch should be systematically absent in young, bright giant planets and possibly present in old, dim ones.
  • The bloating results imply a testable orbital-distance trend: if hot Jupiters form in situ, their interior mixing should be minimal near 0.04–0.05 AU and stronger both farther out and closer in; a survey of atmospheric metallicities versus orbital distance could look for that non-monotonic signature.
  • The paper uses water for all heavy elements; if real interiors are richer in heavier molecules, the compositional gradient is larger and mixing is weaker, so the luminosity ceiling would move upward and the claim that most planets lose their dilute cores would be too strong.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The manuscript models the 4.5 Gyr evolution of giant planets with the 1D code completo21, adding Ledoux-criterion convection with composition gradients and a separate mass-mesh treatment of convective mixing. Starting from four initial heavy-element profiles (compact, extended, metal-rich, and Jupiter-like), the authors vary orbital distance and bloating, mixing length, semi-convection efficiency, opacity, mesh resolution, and initial luminosity. They find that dilute cores can be retained under some conditions, that semi-convection can shrink them, that bloating has a modest effect, and that mesh resolution strongly controls step formation and core extent. The headline claim is that dilute cores cannot persist at initial luminosities much above ~3e3 L_J for a Jupiter-mass planet, and the paper concludes that retention of dilute cores in a large fraction of giant planets is unlikely.

Significance. If the headline result were robust, it would be an important constraint for post-formation giant-planet evolution and for interpreting atmospheric versus bulk metallicities. The paper also contributes a systematic parameter study of hot Jupiters and explicitly tests the effect of an alternative luminosity distribution. The central result is not a fit: the main parameters are selected a priori, not tuned to produce dilute-core survival. However, the load-bearing claim is not numerically converged. The paper itself concedes that the appropriate number of mesh points is unclear and that step locations and sizes are mesh dependent. Because finer meshes consistently make dilute cores more persistent, the luminosity threshold in Sect. 5 could shift materially. The conclusion about the rarity of retained dilute cores is therefore conditional on an unvalidated numerical parameter; the significance is real but provisional.

major comments (4)
  1. [§3.6, Fig. 10 and §5] The central claim is not shown to be converged with respect to the mass mesh. At L_init = 1e3 L_J, going from 2e4 to 5e4 to 1e5 mesh points changes whether four initial profiles retain a dilute core, how many steps form, and how far the compact core extends (from ~6% to ~17% of the mass). The resolution study is only performed at the standard L_init = 1e3 L_J; the key threshold of ~3e3 L_J (Sect. 3.7) is never tested at higher resolution. Since finer meshes consistently make dilute cores more persistent, the threshold could shift upward at 1e5 or 2e5 points, potentially above the cold-start luminosities of ~2e4 to 6e4 L_J invoked in Sect. 3.7. The paper's own statement in Sect. 3.6 that 'It is unclear what number of mesh points is most realistic' and the Sect. 5 caveat 'Assuming that the number of mesh points of 5e4 ... provides a good approximation' make this a load-bearing unresolved issue, not a presentation issue.
  2. [§4.4, Eq. (3)] The luminosity distribution dL/dm = L/M is an ad hoc simplifying assumption, and the paper shows that the alternative dL/dm = -T dS/dt changes the early luminosity and radiative-conductive gradient by up to a factor of 3 in the 0.1-0.2 m/M region, which is precisely the usual extent of the dilute core. This alternative also reduces the number of compositional steps from 7 to 2 for the compact structure. Since the mixing criterion and the ability of a step to inhibit convection depend on the radiative-conductive gradient (Eqs. 5, 6, and 8), the simplified luminosity profile is a possible source of the reported retention threshold. At minimum, the paper should quantify how the step-size and luminosity thresholds change under a more physical entropy-based luminosity profile, or present a physical justification for why dL/dm = L/M is adequate for the central claim.
  3. [§4.2 and §2.1] The artificial 1 Myr delay in applying the radiative-conductive gradient for mixing purposes is a parameter that conditions the outcome. The manuscript states that without this delay, some initial compositions mix completely and lose their dilute core, whereas with the delay they retain it. The delay is justified qualitatively by hot-start accretion scenarios, but no sensitivity study is presented for its duration or the shape of the ramp. Because the central conclusion that dilute cores can persist at L_init = 1e3 L_J depends on this prescription for the compact and Jupiter-like cases, the paper needs to show that the retention does not hinge on the particular choice of 1 Myr or on the functional form of the delayed onset.
  4. [§3.7 and §5] The luminosity threshold itself is inferred from runs at only three initial luminosities (1e3, 3e3, and 1e4 L_J), and the paper uses the maximum luminosity reached during mixing rather than the literal initial value. The increase in luminosity during mixing is sizable (e.g., from 3e3 to up to 5.7e3 L_J for the metal-rich profile), but it is not shown how this maximum depends on mesh resolution or on the delayed-convection prescription. A combined resolution study at and above the stated threshold is needed before the conclusion 'it is unlikely that a large number of giant planets retain a dilute core' can be considered robust.
minor comments (3)
  1. [§2.1] There is a typo: 'the size of the the thermodynamic evolution timestep' should read 'the size of the thermodynamic evolution timestep'.
  2. [§4.1] The phrase 'for which be do not observe a consistent relation' should read 'for which we do not observe a consistent relation'.
  3. [Abstract and §3.7] The notation '3 x 1e3 LJ' is awkward; in the published version this should be typeset as 3 × 10^3 L_J consistently throughout the abstract and text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the dilute-core retention threshold is an emergent simulation outcome, and the numerical-resolution dependence is an acknowledged limitation, not a circular step.

full rationale

The paper's central result (dilute cores do not persist at initial luminosities much above about 3 x 10^3 L_J for the studied heavy-element profiles) is an emergent outcome of integrating Eqs. (5)-(12): the radiative-conductive gradient is proportional to luminosity (Eq. 6), the compositional gradient enters through Eq. (8), and the Ledoux criterion (Eq. 5) determines where mixing occurs. The initial luminosity is a scanned input, not a parameter fitted to the retention outcome; the threshold is found, not imposed. The only calibrated value, the 0.5% bloating efficiency in Sect. 2.2, is fitted to the present-day radius of HD 209458 b and affects only the hot-Jupiter subset, not the luminosity threshold. The mesh-size dependence (Sect. 3.6, Fig. 10) is a numerical convergence limitation that the authors explicitly acknowledge with the sentence 'It is unclear what number of mesh points is most realistic'; however serious for the headline claim, a resolution artifact is not a circularity because the claim does not reduce to its inputs by construction. The formation-luminosity comparison in Sect. 3.7 uses Mordasini (2013), a prior work by a co-author, but it is corroborated by Marley et al. (2007) and is an external premise rather than an imported uniqueness claim or a fitted prediction. No load-bearing step in the derivation is equivalent to its own input, so the analysis is self-contained apart from standard, acknowledged modeling assumptions.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced. The model relies on standard planetary structure equations plus several assumptions about mixing efficiency, opacity, equation of state, and luminosity distribution. The most consequential choices are the mesh resolution and the artificial delay of convection, both of which directly affect whether dilute cores persist.

free parameters (6)
  • Mixing length parameter alpha = 1e-3 (standard), varied to 1e-2 and 1e-1
    Sets convective diffusion coefficient D = 0.1 alpha v Hp; chosen by hand, not fitted. Generally small effect on final structure.
  • Semi-convection efficiency alpha_sc = 0 (standard), varied 1e-2, 1e-1, 1
    Controls semi-convective diffusion coefficient (Eq. 12); not fitted. Strong semi-convection reduces dilute core extent.
  • Bloating efficiency for hot Jupiters = 0.5%
    Fitted to reproduce the present-day radius of HD 209458 b (Sect. 2.2). Affects hot-Jupiter mixing results but not the central luminosity threshold.
  • Initial luminosity L_init = 1e3 L_J (standard), varied to 3e3 L_J
    Chosen as a cold-start Jupiter analog; the central threshold (3e3 L_J) is a model output, but the choice of starting value influences the accessible region.
  • Number of mesh points = 5e4 (standard), tested 2e4 and 1e5
    Controls the maximum compositional gradient and whether steps inhibit mixing; the paper shows dilute-core persistence depends on this parameter.
  • Convective initiation delay = 1 Myr ramp
    Radiative-conductive gradient slowly increased over 1 Myr to prevent instantaneous mixing; ad hoc choice discussed in Sect. 4.2.
assumptions (7)
  • domain assumption Ledoux criterion (nabla_rad < nabla_ad + nabla_X) determines convective stability.
    Used in Eq. 5; standard in stellar and planetary structure but a modeling choice for mixing.
  • domain assumption Convective mixing is diffusive with D = 0.1 (l/H_p) v H_p from the Mihalas (1978) efficiency estimate.
    Eq. 9; efficiency 0.1 taken from prior studies (Vazan et al. 2015, 2018), not derived.
  • domain assumption Opacity tables of Bell & Lin (1994) for radiative and Cassisi et al. (2007) for conductive opacity are accurate.
    Used in Eqs. 6-7; opacity is a major factor in the radiative-conductive gradient and can flip complete mixing vs retention at 10x opacity.
  • domain assumption Water EOS (Haldemann et al. 2020) represents all heavy elements; H/He EOS from Chabrier & Debras (2021).
    Sect. 2.1; using water instead of SiO2 changes the mean molecular weight gradient and thus mixing.
  • ad hoc to paper The luminosity is distributed as dl/dm = L/M in the envelope.
    Sect. 2.1; an approximation. The alternative dl/dm = -T dS/dt changes luminosity by up to factor 3 in the dilute-core region and reduces the number of steps (Sect. 4.4).
  • domain assumption The double-grey atmosphere model of Guillot (2010) gives the outer boundary condition.
    Sect. 2.1; simplified treatment of irradiation.
  • domain assumption Convective mixing can be numerically separated from thermal evolution and computed with a Crank-Nicolson scheme on a mass grid.
    Sect. 2.1; follows Kippenhahn et al. (2012) and Vazan et al. (2015), a standard but approximate approach.

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Cite this review

Pith. "Pith review of Convective mixing in distant and close-in giant planets -- Dependences on the initial composition, luminosity, bloating and semi-convection." pith.science (2026). https://pith.science/paper/HR37OUVJ

@misc{pith2026241118686,
  author       = {Pith},
  title        = {Pith review of: Convective mixing in distant and close-in giant planets -- Dependences on the initial composition, luminosity, bloating and semi-convection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HR37OUVJ}},
  note         = {Machine review of arXiv:2411.18686}
}
read the original abstract

Recent structure models of Jupiter suggest the existence of an extended region in the deep interior with a high heavy element abundance, referred to as a dilute core. This finding has led to increased interest in modelling the formation and evolution processes with the goal of understanding how and under what circumstances such a structure is formed and retained, to in turn better understand the relation between atmospheric and bulk metallicity. We modelled the evolution of giant planets, varying various parameters relevant for the convective mixing process, such as the mixing length parameter and the size of the mesh, and parameters related to the general evolution, such as the orbital distance and the initial luminosity. We in particular studied hot Jupiters and find that the effect of bloating on the mixing process is small but can in some cases inhibit convective mixing by lowering the intrinsic luminosity for a given entropy. Semi-convection can significantly lower the extent of a dilute core if it is strong enough. We find that dilute cores are unable to persist for initial luminosities much higher than 3 x 1e3 LJ for a Jupiter-like planet for the initial heavy element profiles we studied. From this we conclude that, based on our model, it is unlikely that a large number of giant planets retain a dilute core throughout their evolution, although this is dependent on the assumptions and limitations of our method. Future work should focus on improving the link between formation and evolution models so that the mixing process is accurately modelled throughout a planet's lifetime and on improving the understanding of how to model convection near radiative-convective boundaries.

Figures

Figures reproduced from arXiv: 2411.18686 by the authors.

Figure 1
Figure 1. Total luminosity (blue), bloating luminosity (orange), and intrinsic luminosity (green) as a function of pressure for a planet at 0.03 AU. The intrinsic luminosity scales as dl dm = L M , and the bloating luminosity follows a power law, L ∝ P −0.6 , between 103 and 106 bar [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Composition of the different models at the start (top) and end (bottom) of the simulation at 4.5 Gyr. The x-axis shows the normalised mass and the y-axis the fraction of high￾Z material, water in our case. The compositions ‘compact’, ‘extended’, and ‘metal-rich’ correspond to Hot_Compact_Z, Cold_extended_Z, and Cold_high_Z in Müller et al. (2020), respectively. our four different initial structures. All initial comp… view at source ↗
Figure 3
Figure 3. Luminosity (top) and radius (bottom) for the models of our standard case and when changing the initial composition to consist of a fully mixed envelope with the same average Z as the models for the standard case, which are Z=0.103, 0.149, and, 0.278 for the compact, extended, and metal-rich initial compo￾sitional structures, respectively. The inset shows a closer look at the luminosity near the end of the evolution,… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Final composition of the compact and extended initial compositional structures for orbital distances of 0.05 and 0.03 AU. The standard case at 5.2 AU is shown for comparison. For the hot Jupiters, the bloating efficiency is in both cases 0.5% 0.00 0.05 0.10 0.15 0.20 m…
Figure 5
Figure 5. Figure 5: Final composition of the compact initial compositional structure for orbital distances of 0.03 to 0.05 AU in 0.001 AU steps. a large part of the envelope being hotter at the start of our simulation, while the dilute core itself is cooler at the same entropy. This leads…
Figure 6
Figure 6. Figure 6: Final composition of the compact and extended initial compositional structures for mixing length parameter values of 10−2 and 10−1 . The standard case of α = 10−3 is shown for comparison. 0.000 0.025 0.050 0.075 0.100 0.125 0.150 m/M 0.0 0.2 0.4 0.6 0.8 1.0 Z 10 3 10 2…
Figure 7
Figure 7. Figure 7: Final composition of the compact initial compositional structure for values of α ranging from 10−3 to 10−1 . or αsc = 1, although the difference is not very large. For αsc = 1 the number of steps reduces to four, whereas there were seven without semi-convection. We als…
Figure 10
Figure 10. Figure 10: Final composition of the compact and extended initial compositional structures for 2 × 104 and 105 mesh points. The standard case with 5×104 mesh points is shown for comparison. in f = (m/M) 2/3 + c1XH − c2lnp − c3ln T T + c4 (13) given by the model (Kovetz et al. 200…
Figure 11
Figure 11. Figure 11: Final composition of the compact initial compositional structures for an initial luminosity of 3 × 103 LJ . The standard case with an initial luminosity of 103 LJ is shown for comparison. Overall the initial luminosity is shown to be a signifi￾cant factor in whether d…
Figure 12
Figure 12. Figure 12: Temperature (top) and radiative-conductive gradient (bottom) for the compact compositional structure in its original state (unmixed) or starting with its final compositional structure (mixed). The temperature is shown at both the start of the simu￾lation and after 4.5…
Figure 13
Figure 13. Figure 13: Final composition of the compact structure, compar￾ing two simplifications of determining the luminosity and an illustration of the difference in luminosities at the start of the evolution. The solid line shows the composition with the lumi￾nosity scaling linearly wit…

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