Several small dense exoplanets are now known, inviting comparisons to Earth and Venus. Such comparisons require translating their masses and sizes to composition models of evolved multi-layer-interior planets. Such theoretical models rely on our understanding of the Earth's interior, as well as independently derived equations of state (EOS), but have so far not involved direct extrapolations from Earth's seismic model -PREM. In order to facilitate more detailed compositional comparisons between small exoplanets and the Earth, we derive here a semi-empirical mass-radius relation for two-layer rocky planets based on PREM: R R ⊕ = (1.07 − 0.21 · CMF) · ( M M ⊕ ) 1/3.7 , where CMF stands for Core Mass Fraction. It is applicable to 1∼8 M ⊕ and CMF of 0.0∼0.4. Applying this formula to Earth and Venus and several known small exoplanets with radii and masses measured to better than ∼30% precision gives a CMF fit of 0.26 ± 0.07.

The radii and orbital periods of 4000+ confirmed/candidate exoplanets have been precisely measured by the Kepler mission. The radii show a bimodal distribution, with two peaks corresponding to smaller planets (likely rocky) and larger intermediate-size planets, respectively. While only the masses of the planets orbiting the brightest stars can be determined by ground-based spectroscopic observations, these observations allow calculation of their average densities placing constraints on the bulk compositions and internal structures. Yet an important question about the composition of planets ranging from 2 to 4 Earth radii (RÅ) still remains. They may either have a rocky core enveloped in a H2-He gaseous envelope (gas dwarfs) or contain a significant amount of multi-component, H2O-dominated ices/fluids (water worlds). Planets in the mass range of 10-15 MÅ, if half-ice and half-rock by mass, have radii of 2.5 RÅ, which exactly match the second peak of the exoplanet radius bimodal distribution. Any planet in the 2-4 RÅ range requires a gas envelope of at most a few mass%, regardless of the core composition. To resolve the ambiguity of internal compositions, we use a growth model and conduct Monte Carlo simulations to demonstrate that many intermediate-size planets are "water worlds". Keywords: exoplanets / bimodal distribution / ices / water worlds / planet formation Significance Statement: The discovery of numerous exoplanet systems containing diverse populations of planets orbiting very close to their host stars challenges the planet formation theories based on the Solar system. Here we focus on the planets with radii of 2-4 RÅ, whose compositions are debated. They are thought to be either gas dwarfs consisting of rocky cores embedded in H2-rich gas envelopes or water worlds containing significant amounts of H2Odominated fluid/ice in addition to rock and gas. We argue that these planets are water worlds.