The evolution and internal structure of Neptunes and sub-Neptunes II. Convective mixing and thermal conductivity

Mark Eberlein; Ravit Helled

arxiv: 2605.30078 · v1 · pith:LZG3DFIWnew · submitted 2026-05-28 · 🌌 astro-ph.EP

The evolution and internal structure of Neptunes and sub-Neptunes II. Convective mixing and thermal conductivity

Mark Eberlein , Ravit Helled This is my paper

Pith reviewed 2026-06-29 00:20 UTC · model grok-4.3

classification 🌌 astro-ph.EP

keywords sub-Neptunesconvective mixingthermal conductivityplanetary evolutioncomposition gradientsradius evolutionNeptunes

0 comments

The pith

Convective mixing in hot-forming planets with broad composition gradients leads to converging radii after billions of years.

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

The paper models the evolution of 5, 10, and 15 Earth-mass planets with composition gradients using modified MESA code. It compares three thermal conductivity models and includes convective mixing that can erode the gradients. For planets starting hot with gradual heavy-element changes from core to envelope, mixing strongly affects radius evolution, causing convergence to similar values over time. In contrast, for colder starts or sharp gradients, the choice of conductivity model is more important for how energy is trapped. This shows that mixing and conductivity interact to shape planetary radii in ways that depend on initial conditions.

Core claim

For hot forming planets with a large composition gradient where the heavy-element mass fraction changes gradually from the core to the envelope, convective mixing has a large impact on the radius evolution and the radii converge to similar values after billions of years. The thermal conductivity is less relevant in this case. For cold forming planets or narrow composition gradients, convective mixing is less efficient and the thermal conductivity becomes critical in determining the energy transport and radius.

What carries the argument

Convective mixing eroding composition gradients, as implemented in evolutionary calculations that also vary thermal conductivity between pure water, fully ionized matter, and constant electron conductivity models.

If this is right

For hot-forming cases with large gradients, radii become similar after several billion years independent of conductivity.
High thermal conductivity can inhibit convection for intermediate initial entropies if the profile is not mixed.
Radii evolution depends on whether the composition profile is altered significantly by mixing.
Constraints on initial entropy and composition profile can reduce degeneracy in radius-time relations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Observed radii of mature sub-Neptunes may not retain memory of their formation entropy if mixing occurred.
Models ignoring mixing could predict larger spreads in radii than actually occur for hot formation scenarios.
Determining conductivity in realistic mixtures would mainly matter for planets that avoid significant mixing.

Load-bearing premise

The three thermal conductivity models apply to the high-density, high-temperature mixtures in sub-Neptune and Neptune interiors.

What would settle it

Finding that old planets formed with hot initial conditions and gradual composition gradients have radii that differ substantially according to their conductivity would falsify the dominance of mixing.

Figures

Figures reproduced from arXiv: 2605.30078 by Mark Eberlein, Ravit Helled.

**Figure 1.** Figure 1: Boundaries of the different conductivity sources in density-temperature space. The red area (krad AESOPUS) shows the tabulated region from the AESOPUS2.1 tables (Marigo et al. 2024; Eberlein & Helled 2025), the dark blue (kelec FR17) and light blue (kvib F19) area shows the region with ab initio data points for partially ionized water (French & Redmer 2017; French 2019). The green (kelec C07) area shows th… view at source ↗

**Figure 2.** Figure 2: Initial profiles for the wide composition gradient, show [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Thermal conductivity as a function of temperature and density (top row), and as a function of density along the initial and [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 5.** Figure 5: Radius over time with and without mixing. The simu [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Radius evolution for a planet with Mp = 10 M⊕ and a wide gradient. The blue, orange, and yellow lines represent the cold, warm, and hot initial entropies, respectively. Solid lines correspond to the Cond-1 model, dashed lines to the Cond-2 model, and the dotted lines to the Cond-3 model. In [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 8.** Figure 8: Heavy-element mass fraction as a function of normalized [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

read the original abstract

Sub-Neptunes and Neptunes are often modeled with distinct, fully convective layers. Yet, there are several arguments for compositions gradients that can inhibit convection. In these regions, energy transport depends on the thermal conductivity and radiative opacity. We compare three thermal conductivity models and investigate their impact on planetary evolution accounting for the possibility of convective mixing eroding composition gradients. Using a modified version of MESA, we model the evolution of planets with masses of Mp=5, 10, 15 Mearth and three initial entropies. We implement thermal conductivities for: pure water, fully ionized matter, and constant electron conductivity. Convective mixing complicates the relation between conductivity, evolution, and radius. For hot forming planets with a large composition gradient, where the heavy-element mass fraction changes gradually from the core to the envelope, convective mixing has a large impact on the radius evolution. In this case, the thermal conductivity is less relevant and the radii converge to similar values after billions of years. For cold forming planets or narrow composition gradients, convective mixing is less efficient. If the composition profile is not altered significantly, the thermal conductivity becomes critical. It determines how much energy can be trapped beneath stable composition gradients. For intermediate initial entropies, high thermal conductivity inhibits convection. Further work is required to determine the thermal conductivity for various mixtures expected in sub-Neptune and Neptunes at high densities and temperatures. In addition, further constraints on the entropy and composition profile after formation can reduce the degeneracy of the planetary evolution, in particular, the dependence of the radius with time.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Convective mixing can make thermal conductivity secondary for radius evolution in hot-forming sub-Neptunes with broad gradients, but the three conductivity models are too simplistic for real H/He/Z mixtures.

read the letter

The main takeaway is that for hot-forming planets with large composition gradients, convective mixing erodes the gradient enough that radii converge after a few Gyr regardless of which conductivity model is used. For cold starts or narrow gradients, mixing is weaker and the conductivity choice controls how much energy stays trapped below the stable layer.

The paper extends their earlier MESA work by adding convective mixing that can change the composition profile over time. They run 5, 10, and 15 Earth-mass cases across three initial entropies and compare pure-water, fully-ionized, and constant-electron conductivity. This produces a clear map of when mixing dominates versus when conductivity sets the cooling rate. That comparative result on the interplay is new and directly useful for people trying to interpret observed radii.

The modeling itself is straightforward and uses an established code with standard assumptions. The abstract spells out the regimes without overclaiming, and the call for better post-formation entropy and composition constraints is fair.

The soft spot is the conductivity prescriptions. Applying pure water, fully ionized matter, or constant electron values to high-density, high-temperature H/He plus heavy-element mixtures is an approximation at best. The paper notes that realistic conductivities for these conditions still need to be worked out. If the actual numbers sit well outside the range spanned by the three models, the distinction between mixing-dominated and conductivity-critical cases could shift. That is the main uncertainty, not a minor one.

This is for researchers who build or use interior-evolution models for sub-Neptunes. A reader who cares about how gradients and mixing affect radius tracks will find concrete cases to test against. The work is grounded enough and addresses a real modeling gap, so it deserves a serious referee even though the conductivity part will probably need more justification or sensitivity checks.

Referee Report

2 major / 1 minor

Summary. The paper models the thermal evolution of 5-15 M_earth sub-Neptunes and Neptunes that form with composition gradients using a modified MESA code. It compares three thermal-conductivity prescriptions (pure water, fully ionized matter, constant electron conductivity) and includes convective mixing that can erode the gradients. The central result is that, for hot initial entropies and broad gradients, mixing dominates radius evolution and radii converge after a few Gyr regardless of conductivity; for cold initial conditions or narrow gradients, conductivity controls energy trapping beneath stable layers and therefore the radius-time relation.

Significance. If the numerical results hold, the work demonstrates that convective mixing can erase much of the dependence on uncertain microphysical conductivities for a plausible subset of formation conditions, thereby reducing one source of degeneracy in sub-Neptune radius evolution. The use of a standard stellar-evolution code (MESA) with explicit mixing is a methodological strength that allows direct comparison with other interior models.

major comments (2)

[Abstract / conductivity implementation] Abstract and model-description section: the three conductivity prescriptions are stated to be applicable to the high-density, high-temperature mixtures inside sub-Neptunes, yet the text itself concludes that “further work is required to determine the thermal conductivity for various mixtures.” Because the claim that “thermal conductivity is less relevant” when mixing dominates rests on these three models being at least order-of-magnitude representative inside the stable layers, the lack of validation or sensitivity tests to plausible alternative conductivities is load-bearing for the mixing-versus-conductivity regime distinction.
[Results / radius-evolution figures] Evolution results (implicit in the abstract’s description of radius convergence): no quantitative radius values, time scales, or error estimates are supplied in the abstract, and the reader’s report notes the absence of verification that the reported convergence is robust to the chosen initial entropies and gradients. Without these numbers it is impossible to judge whether the claimed convergence after “billions of years” is statistically distinguishable from the no-mixing case.

minor comments (1)

[Abstract] The abstract lists planet masses (5, 10, 15 M_earth) and three initial entropies but does not specify the numerical values of those entropies or the functional form of the initial composition gradients; these should be stated explicitly for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight important points about the presentation of conductivity models and the need for quantitative details on radius evolution. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Abstract / conductivity implementation] Abstract and model-description section: the three conductivity prescriptions are stated to be applicable to the high-density, high-temperature mixtures inside sub-Neptunes, yet the text itself concludes that “further work is required to determine the thermal conductivity for various mixtures.” Because the claim that “thermal conductivity is less relevant” when mixing dominates rests on these three models being at least order-of-magnitude representative inside the stable layers, the lack of validation or sensitivity tests to plausible alternative conductivities is load-bearing for the mixing-versus-conductivity regime distinction.

Authors: We agree that the three prescriptions are simplified and that the paper already notes the need for further microphysical work on mixture conductivities. The main result—that mixing can dominate and reduce sensitivity to conductivity choice—is shown by the convergence of radii across the three models for hot, broad-gradient cases. To strengthen the claim, we will add a short sensitivity discussion (e.g., scaling one prescription by a factor of 3) and revise the abstract and model section to emphasize that the models are intended as order-of-magnitude brackets rather than precise predictions. This directly supports the regime distinction while acknowledging the limitation. revision: partial
Referee: [Results / radius-evolution figures] Evolution results (implicit in the abstract’s description of radius convergence): no quantitative radius values, time scales, or error estimates are supplied in the abstract, and the reader’s report notes the absence of verification that the reported convergence is robust to the chosen initial entropies and gradients. Without these numbers it is impossible to judge whether the claimed convergence after “billions of years” is statistically distinguishable from the no-mixing case.

Authors: We accept that quantitative anchors improve clarity. In the revised manuscript we will update the abstract with specific examples (e.g., “radii converge to within ~0.2 R_earth after ~3 Gyr for hot broad-gradient cases”). We will also add a direct comparison panel or table contrasting mixing and no-mixing runs, including approximate spreads from the three conductivity choices and the three initial entropies, to demonstrate that the convergence is distinguishable from the no-mixing evolution on Gyr timescales. revision: yes

Circularity Check

0 steps flagged

No circularity; results follow from MESA integrations with external conductivity inputs

full rationale

The paper performs numerical planetary evolution calculations in a modified MESA framework. It adopts three distinct thermal-conductivity prescriptions (pure water, fully ionized matter, constant electron conductivity) as external model inputs and evolves planets under specified initial entropies and composition gradients. The reported radius convergence for hot-forming cases with broad gradients is an output of those integrations once convective mixing is enabled; no fitted parameter is relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The explicit statement that further work is required on conductivity for realistic mixtures treats the conductivity models as provisional inputs rather than derived results. The derivation chain is therefore self-contained against the stated numerical setup.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard planetary science assumptions about composition gradients inhibiting convection and on modeling choices for initial conditions and material properties; no new entities postulated.

free parameters (2)

initial entropy
Three discrete initial entropies chosen for the evolution runs.
planet mass = 5,10,15 M_earth
Discrete masses of 5, 10, 15 Earth masses selected for modeling.

axioms (2)

domain assumption Composition gradients inhibit convection allowing conductivity to control energy transport
Invoked to justify comparing conductivity models in non-convective regions.
domain assumption Convective mixing can erode composition gradients over time
Central modeling choice whose efficiency depends on initial entropy and gradient width.

pith-pipeline@v0.9.1-grok · 5825 in / 1407 out tokens · 37437 ms · 2026-06-29T00:20:54.383796+00:00 · methodology

discussion (0)

Reference graph

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