Sculpting the Valley in the Radius Distribution of Small Exoplanets as a by-product of Planet Formation: The Core-Powered Mass-Loss Mechanism

Recent observations revealed a bimodal radius distribution of small, short-period exoplanets with a paucity in their occurrence, a radius `valley', around $1.5-2.0$ $R_\oplus$. In this work, we investigate the effect of a planet's own cooling luminosity on its thermal evolution and atmospheric mass-loss (core-powered mass-loss) and determine its observational consequences for the radius distribution of small, close-in exoplanets. Using simple analytical descriptions and numerical simulations, we demonstrate that planetary evolution based on the core-powered mass-loss mechanism alone (i.e., without any photoevaporation) can produce the observed valley in the radius distribution. Our results match the valley's location, shape and slope in planet radius-orbital period parameter space, and the relative magnitudes of the planet occurrence rate above and below the valley. We find that the slope of the valley is, to first order, dictated by the atmospheric mass-loss timescale at the Bondi radius and given by $\text{d log} R_p/ \text{d log} P \simeq 1/(3(1-\beta))$ which evaluates to $ -0.11$ for $ \beta \simeq 4$, where $M_c/M_\oplus = (R_c/R_{\oplus})^{\beta} (\rho_{c*}/\rho_{\oplus})^{\beta/3}$ is the mass-radius relation of the core. This choice for $\beta$ yields good agreement with observations and attests to the significance of internal compression for massive planetary cores. We further find that the location of the valley scales as $\rho_{c*}^{-4/9}$ and that the observed planet population must have predominantly rocky cores with typical water-ice fractions of less than $\sim 20\%$. Furthermore, we show that the relative magnitude of the planet occurrence rate above and below the valley is sensitive to the details of the planet-mass distribution but that the location of the valley is not.