The Radius Cliff is a Waterfall: Explaining Sub-Neptune Exoplanets with Steam Worlds

The demographics of Kepler planets provide a key testbed for models of planet formation and evolution, particularly for explaining the radius valley separating super-Earths and sub-Neptunes. A primordial interpretation based on differences in bulk densities – where rocky and water-rich planets form via migration pathways – offers an alternative to atmospheric loss scenarios. Updated interior structure models of water worlds with adiabatic steam atmospheres reproduce the observed valley near $\sim2R_\oplus$ more accurately. Furthermore, migration models from our Genesis library suggest that these formation pathways can also account for the distinct period distributions of super-Earths and sub-Neptunes, as well as the emergence of the hot Neptune desert. Motivated by this, we develop a Bayesian hierarchical mixture model for close-in Kepler planets ($P<100$ days), combining rocky planets and water worlds without H/He envelopes. The inferred mass distributions of rocky and water-rich planets peak at $\sim2.6M_\oplus$ and $\sim7M_\oplus$, respectively, with the water mass fraction of water worlds peaking at $\sim41%$. Water worlds provide a good representation of the Kepler sub-Neptune population, with the radius cliff emerging as a ``waterfall" – a sharp decline in their occurrence. However, our mass-radius analysis shows that water worlds alone cannot explain planets with $R \gtrsim 3R_\oplus$, implying that at least $\sim20%$ of sub-Neptunes in the sample are enriched in H/He gas.

💡 Research Summary

The paper tackles two of the most prominent statistical features in the Kepler small‑planet population – the “radius valley” near 2 R⊕ and the “radius cliff” near 4 R⊕ – by invoking a primordial compositional dichotomy rather than post‑formation atmospheric loss. The authors first update interior‑structure models for water‑rich planets by adding a hot, super‑critical steam envelope on top of a rocky core. This “steam world” model, based on the equations of state and static interior calculations of Aguichine et al. (2021), yields significantly larger radii for a given mass than the traditional condensed‑ice models, especially at the high equilibrium temperatures typical of close‑in Kepler planets.

Using these updated mass–radius relations, the authors construct a Bayesian hierarchical mixture model that treats the Kepler sample (0.9–6 R⊕, P < 100 days) as a superposition of two sub‑populations: (i) bare rocky planets with an Earth‑like composition (32.5 % Fe, 67.5 % silicates) whose masses follow a log‑normal distribution centered at ≈2.6 M⊕, and (ii) water‑rich “steam worlds” without H/He envelopes, whose masses follow a separate log‑normal distribution centered at ≈7 M⊕. The water‑mass‑fraction (WMF) of the latter is modeled with several functional forms; the data favor a truncated normal distribution with a mean WMF of ≈0.41 (i.e., 41 % of the planet’s mass in water) and a standard deviation of ≈0.12, limited to a physically motivated upper bound of 0.5.

The orbital‑period distribution for each sub‑population is tied to the empirical broken‑power‑law fit of Rogers & Owen (2021) but modulated by the hyperbolic‑tangent transition function of Bergsten et al. (2022). This yields a higher fraction of rocky planets at short periods and a rising fraction of water worlds toward longer periods, a trend the authors interpret as a primordial imprint of formation and migration rather than a consequence of photo‑evaporation.

To compare the model with observations, the authors generate synthetic planet populations, compute a two‑dimensional kernel‑density estimate (KDE) in log‑P–log‑R space, and apply the CKS completeness map (including transit probability and detection efficiency) to obtain the expected number of detectable planets per grid cell. Assuming Poisson statistics, they maximize the likelihood and explore the posterior with MCMC sampling.

The resulting posterior reproduces the observed radius valley as a sharp drop in the occurrence of water worlds – a “waterfall” – rather than as the outcome of atmospheric stripping of H/He envelopes. The model also naturally generates the observed period dependence of the super‑Earth/sub‑Neptune split. However, the steam‑world population alone underpredicts the occurrence of planets larger than ≈3 R⊕. The authors therefore conclude that at least ~20 % of the sub‑Neptune sample must retain a modest H/He envelope, implying that atmospheric loss processes still play a role alongside primordial compositional differences.

Key implications include: (1) steam‑world interiors can explain the bulk of the 2–4 R⊕ Kepler planets without invoking large H/He envelopes; (2) the steep decline in water‑world occurrence with increasing radius (“waterfall”) provides a new diagnostic of early disk chemistry and migration pathways; (3) a hybrid scenario, where both primordial water‑rich cores and post‑formation atmospheric loss operate, is required to fully account for the full radius distribution, especially the tail beyond 3 R⊕; and (4) future spectroscopic observations (e.g., JWST, ARIEL) targeting water‑vapor signatures, together with improved mass measurements, will be crucial to test the steam‑world hypothesis and to refine the relative contributions of formation versus evolution in shaping the Kepler exoplanet demographics.