Sub-Neptune Memories I: Implications of Inefficient Mantle Cooling and Silicate Rain

This study utilizes the \texttt{APPLE} evolution code to demonstrate that inefficient mantle cooling and silicate rain in sub-Neptune envelopes can significantly inflate planetary radii at Gyr ages, offering an alternative thermal explanation for observed mean densities that challenges traditional water-world interpretations.

The Gist

The Big Idea: Planets with "Hot Memories"

Imagine you have a cup of coffee. If you leave it on a table, it cools down quickly. But if you wrap that coffee in a thick, high-tech thermal blanket, it stays hot for hours.

For decades, astronomers have been looking at a specific type of planet called a Sub-Neptune (planets slightly bigger than Earth but smaller than Neptune). They measure how big these planets are and how heavy they are to guess what they are made of.

The standard theory was: "If a planet is this big and this heavy, it must be mostly made of water (an 'Ocean World'). It can't be rock, because rock would be too heavy and small."

This paper says: "Wait a minute. What if the planet isn't made of water? What if it's just a rocky planet that forgot to cool down?"

The authors, using a super-computer code called APPLE (which is basically a time machine for planets), discovered that these planets might be holding onto a "thermal memory" of their birth. They are still hot from when they were formed billions of years ago, and that extra heat makes them puff up like a balloon, looking bigger than they really are.

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The Three Main Secrets

The paper reveals three "tricks" that keep these planets inflated:

1. The Thermal Blanket (Inefficient Cooling)

The Analogy: Imagine a house with a very thick, insulated attic. The heat from the basement (the planet's core) tries to escape, but the insulation (the boundary between the rock mantle and the gas atmosphere) is so good that the heat gets stuck.

The Science: Usually, scientists assumed the deep rock inside these planets cooled down at the same speed as the gas atmosphere. But this paper shows that the rock acts like a slow-cooling oven. Because the heat can't escape easily, the rock stays liquid and hot for billions of years. This internal heat pushes the planet's atmosphere outward, making the planet 10% larger than expected.

2. The Rain of Rocks (Silicate Rain)

The Analogy: Imagine a storm inside the planet's atmosphere. But instead of water droplets, it's raining tiny droplets of molten rock (silicates). As these heavy rock droplets fall from the upper atmosphere down toward the core, they release energy (like friction heating up your hands when you rub them together). This energy heats up the upper atmosphere, keeping the planet puffed up.

The Science: The authors found that at certain temperatures and pressures, rock and hydrogen gas don't mix well. The rock separates out and "rains" down. This process:

• Cleans the upper atmosphere of heavy rock (making it look like a pure gas planet).
• Releases heat that inflates the planet by another 5%.

3. The Layer Cake Effect (Stratification)

The Analogy: Think of a layered drink where oil sits on top of water. They don't mix. In these planets, the heavy elements (like rock or water) can get stuck in a layer that won't mix with the gas above it. This creates a "stably stratified" layer that acts like a lid, trapping heat even more effectively.

The Science: This layering stops the planet from churning its heat around efficiently. It's like putting a lid on a pot of boiling water; the heat stays trapped inside, keeping the planet inflated.

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Why Does This Matter? (The "Water World" Myth)

For a long time, we thought planets like GJ 1214 b or K2-18 b were "Water Worlds"—giant balls of ice and liquid water.

The New View:
The authors ran simulations on four famous planets. They found that you don't need water to explain their size.

• You can have a planet that is 90% solid rock and iron (like a super-Earth).
• If that rock is still hot from its birth (a "hot start"), and it has a thin layer of gas on top, it will puff up to look exactly like a "Water World."

The Takeaway: Just because a planet looks light and big doesn't mean it's made of water. It might just be a hot, rocky planet wearing a "thermal coat."

The "Memory" Concept

The title "Sub-Neptune Memories" refers to the idea that these planets haven't forgotten how they were born.

• Old Theory: Planets cool down and forget their starting temperature after a few hundred million years.
• New Theory: Because of the "thermal blanket" and "rock rain," these planets remember their hot, chaotic birth for billions of years.

What's Next?

The authors suggest that if we look at young planets with powerful new telescopes (like the James Webb Space Telescope), we might see that their atmospheres are "depleted" of rock (because the rock rained out), but the planets are still huge.

This changes how we understand the family tree of planets. It suggests that many "Water Worlds" might actually be Rocky Planets with a hot temper, and we just need to learn how to read their "thermal memories" to find out what they are really made of.

Summary in One Sentence

These sub-Neptune planets aren't giant balls of water; they are likely hot, rocky planets that are still holding onto their birth heat, making them look bigger and puffier than they actually are.

Technical Summary

1. Problem Statement

Sub-Neptune exoplanets (radii $\sim$1.5–4 $R_\oplus$, masses 3–15 $M_\oplus$) are the most common type of planet in the galaxy. Their bulk densities are often interpreted as evidence for "water worlds" (planets composed largely of volatiles like water/ices) because their densities are lower than pure rock but higher than pure gas. However, these inferences suffer from significant degeneracies:

• Thermal State Degeneracy: Standard evolution models often assume the deep interior (mantle/core) cools at the same rate as the envelope, ignoring that initial thermal energy may reside primarily in the mantle.
• Composition Degeneracy: Low densities could result from a thick volatile envelope, a water-rich interior, or a hot, liquid rocky interior that has not yet cooled.
• Phase Separation: The potential for phase separation (e.g., silicate rain) and stable stratification in the envelope is often neglected, yet these processes can trap heat and alter radius evolution.

The authors argue that current models may overestimate the prevalence of water worlds by failing to account for inefficient mantle cooling and silicate phase separation, which can keep rocky planets inflated for billions of years.

2. Methodology

The authors utilize and upgrade APPLE (a planet evolution code inspired by stellar evolution codes like MESA) to self-consistently couple thermal and compositional evolution. Key methodological upgrades include:

• Equations of State (EOS):
  • Mantle: Uses an updated ab initio liquid MgSiO$_3$ EOS (Luo & Deng 2025) for high pressures/temperatures, and solid perovskite/post-perovskite EOSs for lower temperatures.
  • Core: Uses an Fe$_{16}$Si alloy EOS (Fischer et al. 2012) to better match Earth's core density and heat capacity.
  • Envelope: Uses a mixture of H-He and heavy elements (modeled as water or silicates) with updated opacities and conductivities.
• Heat Transport Physics:
  • Conduction: Implements temperature-dependent thermal conductivities for MgSiO$_3$ and the core.
  • Convection: Uses a modified Mixing-Length Theory (MLT) that accounts for viscous convection in partially molten mantles (transitioning from inviscid to viscous limits based on melt fraction).
  • Latent Heat: Explicitly includes latent heat release during solidification and time-dependent radiogenic heating.
• Silicate Rain & Phase Separation:
  • Incorporates ab initio results on silicate-hydrogen immiscibility (Stixrude & Gilmore 2025).
  • Deploys an advection-diffusion scheme to model the sedimentation of silicates ("silicate rain") from the envelope. This drives local metal fractions ($Z$) toward equilibrium values ($Z_{low}$, $Z_{high}$) defined by the coexistence (binodal) curve.
• Boundary Conditions:
  • Uses non-gray radiative-convective atmospheric models to link the interior flux to the equilibrium temperature ($T_{eq}$) and stellar irradiation.
  • Calculates transit radii ($R_{transit}$) using an isothermal two-stream radiative transfer model.

3. Key Contributions

1. Inefficient Mantle Cooling: Demonstrates that the envelope-mantle boundary (EMB) acts as a thermal bottleneck. Due to the steep mean molecular weight gradient, convection is suppressed, and heat must transport via conduction. This allows hot, liquid mantles to retain primordial heat for Gyr timescales.
2. Silicate Rain Mechanism: For the first time in sub-Neptune modeling, they simulate silicate rain using ab initio immiscibility data. They show this creates a bifurcated envelope: an upper H-He-rich layer and a lower silicate-rich layer, separated by a stable stratified region.
3. Thermal "Memory": Establishes that sub-Neptunes retain a "memory" of their formation entropy and initial thermal state, unlike adiabatic models which forget initial conditions after $\sim$500 Myr.
4. Alternative to Water Worlds: Provides a viable alternative explanation for low-density sub-Neptunes: they may be hot, liquid rocky planets with thin H-He envelopes, rather than water-rich worlds.

4. Key Results

A. Impact of Initial Thermal State

• Radius Inflation: Models starting with hot mantles ($\sim$9,000–16,000 K) retain radii 5–10% larger than cold-start models at ages of 1–5 Gyr.
• Mechanism: The EMB prevents efficient convective cooling. Heat is trapped in the deep interior, maintaining a shallow temperature gradient in the mantle and keeping the planet inflated.
• Timescale: Even after 10 Gyr, a significant radius difference can persist depending on the initial entropy.

B. Stable Stratification

• Inhomogeneous Envelopes: Introducing a stably stratified region (due to compositional gradients) above the EMB further inhibits cooling.
• Effect: This increases radii by an additional 5–7% compared to homogeneous models. The lack of convective mixing in these layers creates a thermal bottleneck.

C. Silicate Rain Evolution

• Depletion: Silicates phase-separate from the H-He envelope, raining down toward the mantle. This depletes the outer atmosphere of silicates over time.
• Radius Effect: The release of gravitational potential energy from raining silicates heats the outer envelope, inflating the radius by an additional ~5%.
• Atmospheric Composition: Young sub-Neptunes may appear metal-rich (silicate-rich), while older ones ($\gtrsim$100 Myr) appear metal-poor (silicate-depleted) but remain physically larger than adiabatic predictions.
• Location: The rain region typically occurs in the middle-to-upper envelope (1–5 GPa), well above the EMB, creating a "bifurcated" structure.

D. Case Studies (GJ 1214 b, K2-18 b, TOI-270 d, TOI-1801 b)

• The authors modeled these four planets using hot, liquid rocky mantles comprising 87–96% of the total mass, with thin H-He envelopes (5% mass).
• Result: These models successfully reproduce the observed radii and mean densities without requiring a water-rich interior.
• Implication: Planets previously classified as "water worlds" based on mean density may actually be rocky planets with hot interiors and thin envelopes.

5. Significance and Implications

• Re-evaluating Bulk Composition: Mean density alone is insufficient to distinguish between water worlds and hot rocky planets. The thermal state of the mantle and the history of envelope mixing/phase separation are critical variables.
• Formation Constraints: Because these planets retain thermal memory, future observations (e.g., by JWST or the Habitable Worlds Observatory) of young sub-Neptunes could constrain their post-formation entropies and formation temperatures.
• Atmospheric Interpretation: Silicate rain implies that older sub-Neptune atmospheres should be depleted of silicates compared to their host stars, even if the planet started with a metal-rich envelope.
• Future Directions: The paper highlights the need for better constraints on silicate droplet physics, the mutual phase separation of water and silicates, and the diffusion of hydrogen into rocky mantles.

In summary, this work fundamentally challenges the "water world" paradigm for many sub-Neptunes, proposing that inefficient heat transport and silicate phase separation are dominant factors in shaping the observed radii and densities of these planets, allowing rocky interiors to remain hot and inflated for billions of years.