New Perspectives on the Exoplanet Radius Gap from a Mathematica Tool and Visualized Water Equation of State

Recent astronomical observations obtained with the Kepler and TESS missions and their related ground-based follow-ups revealed an abundance of exoplanets with a size intermediate between Earth and Neptune (1 R _⊕ ≤ R ≤ 4 R _⊕). A low occurrence rate of planets has been identified at around twice the size of Earth (2 × R _⊕), known as the exoplanet radius gap or radius valley. We explore the geometry of this gap in the mass-radius diagram, with the help of a Mathematica plotting tool developed with the capability of manipulating exoplanet data in multidimensional parameter space, and with the help of visualized water equations of state in the temperature-density (T-ρ) graph and the entropy-pressure (s-P) graph. We show that the radius valley can be explained by a compositional difference between smaller, predominantly rocky planets (<2 × R _⊕) and larger planets (>2 × R _⊕) that exhibit greater compositional diversity including cosmic ices (water, ammonia, methane, etc.) and gaseous envelopes. In particular, among the larger planets (>2 × R _⊕), when viewed from the perspective of planet equilibrium temperature (T _eq), the hot ones (T _eq ≳ 900 K) are consistent with ice-dominated composition without significant gaseous envelopes, while the cold ones (T _eq ≲ 900 K) have more diverse compositions, including various amounts of gaseous envelopes.