The Evolution and Internal Structure of Neptunes and Sub-Neptunes: The importance of thermal conductivity in non-convective regions

This study demonstrates that the thermal evolution and inferred radii of Neptune and sub-Neptune exoplanets are highly sensitive to thermal conductivity assumptions in non-convective, compositionally stratified layers, revealing that current theoretical uncertainties significantly exceed observational errors and necessitate improved data on material properties and primordial states.

The Gist

Imagine a Neptune-sized planet as a giant, multi-layered cosmic onion. For a long time, astronomers thought these onions were simple: a rocky core, a thick icy middle, and a fluffy hydrogen-helium skin, all mixed together in a way that heat could flow freely from the center to the surface, like a pot of boiling water.

But this new paper suggests that picture is too simple. It turns out these planets might have "stuck" layers where the ingredients don't mix well, creating a traffic jam for heat. The authors, Mark Eberlein and Ravit Helled, wanted to see how this traffic jam changes the planet's size and how it cools down over billions of years.

Here is the breakdown of their discovery using everyday analogies:

1. The "Thermal Traffic Jam" (Composition Gradients)

Usually, we assume that inside a planet, hot stuff rises and cold stuff sinks, creating a constant mixing pot (convection). This is like a pot of soup on the stove where the heat moves easily.

However, the authors suggest that during a planet's birth, it might swallow different materials at different times, creating layers of different "densities" or "flavors" that don't want to mix. Think of it like pouring oil and water into a jar. They separate. In a planet, this separation creates a non-convective layer. Heat can't rise through this layer easily; it gets stuck.

2. The Three Ways Heat Moves (The Conductivity Problem)

When heat is stuck in a layer where it can't rise (convection), it has to move by other means. The paper looks at three ways heat travels through this "traffic jam":

• Radiation (The Flashlight): Heat moves as light (photons). This works well in the outer, thinner parts of the planet.
• Electrons (The High-Speed Train): In the deep, hot, and squished center, atoms are broken apart into electrons and ions. These free electrons zip around carrying heat very fast.
• Vibrations (The Rattle): In the middle zone, it's hot but not hot enough to break atoms apart, and it's too dense for light to travel far. Here, heat moves like a rattle in a solid object—molecules bumping into each other. This is the slow, inefficient way.

The Big Mistake: The authors found that most scientists have been using the "High-Speed Train" (electron) model for all deep layers, assuming everything is broken down into electrons. But in reality, there's a huge middle zone where the "Rattle" (vibration) is the only way heat moves, and it's very slow.

3. The Result: The "Puffy" vs. The "Shrinking" Planet

Because the "Rattle" method is so slow at moving heat, the planet's core gets trapped with its own heat.

• The Old Model (Fast Heat Flow): If you assume heat moves fast (like the High-Speed Train), the planet cools down quickly. As it cools, it shrinks. It becomes a compact, smaller ball.
• The New Model (Slow Heat Flow): If you use the correct "Rattle" model, the heat gets trapped deep inside. The outer layers stay hot and puffy because they are being warmed from below by the trapped heat. The planet stays inflated and larger for much longer.

The Analogy: Imagine a winter coat.

• Old Model: The coat has no insulation. Your body heat escapes instantly, and you get cold and shrink (metaphorically).
• New Model: The coat is a super-insulated, high-tech down jacket. Your body heat gets trapped inside. You stay warm and "puffy" for hours.

4. Why This Matters (The 20% Surprise)

The authors ran simulations for planets of different masses (5, 10, and 15 times the mass of Earth). They found that simply changing how they calculated the heat flow changed the predicted size of the planet by about 20%.

To put that in perspective: If you measure a planet and think it's the size of Earth, the "old math" might say it's a small, rocky world, while the "new math" (with the heat trapped) says it's a giant, puffy gas ball.

They also found that if we don't know exactly how hot the planet was when it was born (its "primordial entropy"), that uncertainty changes the size by another 25%.

The Takeaway

The paper concludes that we have been looking at these planets with the wrong "thermal glasses."

1. We need better data: We don't know enough about how heat moves through "rattling" molecules in high-pressure water and rock mixtures.
2. We need to rethink size: When we see a Neptune-sized planet, its size tells us less about what it's made of and more about how well it traps heat.
3. The "Puffy" Truth: Many of these planets might be much larger and puffier than we thought because their heat is trapped deep inside, unable to escape through the "traffic jam" layers.

In short: Planets aren't just boiling pots; they are sometimes insulated thermos flasks. If we don't account for the insulation, we get the size of the planet completely wrong.

Technical Summary

1. Problem Statement

Intermediate-mass planets (Neptunes and sub-Neptunes) constitute the majority of the detected exoplanet population. Standard modeling often assumes these planets possess fully convective, adiabatic interiors with distinct layers. However, formation models suggest that composition gradients (variations in metallicity) can exist within these planets due to accretion processes or migration across ice lines. These gradients can inhibit convection, creating stable, non-convective layers.

In non-convective regions, heat transport is governed by radiation, diffusion, and conduction rather than convection. A critical gap in current modeling is the treatment of thermal conductivity in these regions. Standard models often rely on simplified assumptions (e.g., assuming fully ionized matter or constant conductivity) that may be inappropriate for the partially ionized, high-pressure, and high-density environments found in Neptune-like interiors. This study aims to quantify how different thermal conductivity assumptions affect the thermal evolution, inferred radii, and internal structure of these planets.

2. Methodology

The authors utilized the Modules for Experiments in Stellar Astrophysics (MESA) code, modified to handle medium-sized planets, to simulate planetary evolution over 10 Gyr.

• Equation of State (EoS): They employed a hybrid EoS combining non-ideal H-He interactions (Chabrier & Debras 2021) with heavy-element EoS for a 50/50 rock-water (SiO$_2$-H$_2$O) mixture.
• Opacity and Conductivity:
  • Radiative Opacity: They generated custom opacity tables using AESOPUS2.1 to cover high metallicities ($Z \le 1$) and temperatures up to $\log(T/K) = 4.5$, addressing limitations in standard MESA tables.
  • Thermal Conductivity ($k_{tot}$): The total conductivity was calculated as the sum of radiative ($k_{rad}$), vibrational ($k_{vib}$), and electronic ($k_{elec}$) contributions.
• Model Variations:
  • Masses: 5, 10, and 15 $M_\oplus$.
  • Initial Entropy: Three primordial states ("Cold," "Warm," "Hot") corresponding to specific entropies of 0.5, 0.6, and 0.7 $k_B/m_H$.
  • Composition Profiles: Both wide ($\Delta q = 0.1$) and narrow ($\Delta q = 0.001$) composition gradients between the envelope and deep interior.
  • Conductivity Cases: Four distinct models were compared:
    1. Cond-1: Vibrational conductivity for H$_2$O (French 2019) + Electron conductivity for partially ionized H$_2$O (French & Redmer 2017). Recommended as the most physically accurate.
    2. Cond-2: Default MESA electron conductivity assuming fully ionized matter (Cassisi et al. 2007).
    3. Cond-3: Constant electron + vibration conductivity ($4$ W/m/K), mimicking Earth's core-mantle boundary.
    4. Cond-4: Phonon model for vibration (Stamenković et al. 2011) + partially ionized electron conductivity.

3. Key Contributions

• Implementation of Realistic Conductivity: The study integrates physically motivated models for vibrational and electronic conductivity in partially ionized, high-density fluids, moving beyond the standard assumption of fully ionized plasma.
• Custom Opacity Tables: The creation and implementation of AESOPUS2.1 opacity tables specifically tailored for high-metallicity planetary interiors, improving upon the limited range of standard MESA tables.
• Quantification of Theoretical Uncertainty: The paper demonstrates that uncertainties in thermal physics (conductivity and initial entropy) dominate over observational uncertainties in radius measurements.

4. Key Results

• Impact on Radius Evolution: The assumed thermal conductivity significantly alters the predicted radius.
  • Cond-2 (Fully Ionized): Leads to high conductivity throughout the interior, allowing efficient heat escape. This keeps the outer envelope hot and inflated, resulting in the largest radii.
  • Cond-1 & Cond-3 (Realistic/Low Conductivity): Vibrational conductivity in the intermediate, non-ionized region is inefficient. This traps heat in the deep interior, preventing the outer envelope from being heated from below. Consequently, the envelope cools and contracts more efficiently, leading to significantly smaller radii.
• Magnitude of Deviation:
  • Depending on the conductivity model, inferred radii deviate by $\sim 20\%$.
  • Depending on the primordial entropy (initial energy state), radii deviate by $\sim 25\%$.
  • These theoretical uncertainties are substantially larger than the typical observational uncertainty of $\sim 5\%$ (and potentially lower with future missions like PLATO).
• Role of Composition Gradients:
  • Wide Gradients: Create a broad non-convective zone where vibrational conductivity dominates, effectively insulating the core and slowing contraction.
  • Narrow Gradients: Result in a thinner insulating layer. While cooling is faster than in wide-gradient models, the conductivity assumptions still dictate the final radius.
• Vibrational Conductivity Importance: The study highlights that neglecting vibrational conductivity (or assuming it is negligible due to high electron conductivity) leads to erroneous radius predictions. Cond-1 and Cond-4 (which include vibrational terms) yield similar results, differing by only $\sim 1.6\%$, but both differ significantly from Cond-2.

5. Significance and Conclusions

The paper concludes that the characterization of Neptunes and sub-Neptunes is critically dependent on the modeling of thermal transport in non-convective regions.

• Oversimplification Warning: Common assumptions (e.g., fully ionized interiors or constant conductivity) are inappropriate for these planetary types and lead to inflated radius predictions that do not match the physical reality of composition-gradient-stabilized interiors.
• Theoretical vs. Observational: Theoretical uncertainties regarding thermal conductivity and primordial entropy are currently the limiting factor in interpreting exoplanet data, not measurement precision.
• Future Directions: The authors emphasize the urgent need for:
  1. Experimental and theoretical data on thermal conductivities for partially ionized materials and mixtures (H-C-N-O) under planetary conditions.
  2. Better constraints on the primordial thermal state and composition gradients of these planets.
  3. Self-consistent models that couple convection, mixing, and composition gradients.

In summary, accurate modeling of intermediate-mass planets requires a shift from adiabatic, homogeneous assumptions to models that rigorously account for composition gradients and the specific, often inefficient, thermal conductivity mechanisms in non-convective layers.