Evidence that Core-Powered Mass-Loss Dominates Over Photoevaporation in Shaping the Kepler Radius Valley

The dearth of planets with sizes around 1.8 $\mathrm{R_\oplus}$ is a key demographic feature discovered by the $Kepler$ mission. Two theories have emerged as potential explanations for this valley: photoevaporation and core-powered mass-loss. However, Rogers et al. (2021) shows that differentiating between the two theories is possible using the three-dimensional parameter space of planet radius, incident flux, and stellar mass. We use homogeneously-derived stellar and planetary parameters to measure the $Kepler$ exoplanet radius gap in this three-dimensional space. We compute the slope of the gap as a function of incident flux at constant stellar mass ($α$ $\equiv$ $\left(\partial \log R_{\mathrm{gap}} / \partial \log S \right)_{M_\star}$) and the slope of the gap as a function of stellar mass at constant incident flux ($β$ $\equiv$ $\left(\partial \log R_{\mathrm{gap}} / \partial \log M_\star \right)_{S}$) and find $α$ = 0.069$^{+0.019}_{-0.023}$ and $β$ = $-$0.046$^{+0.125}_{-0.117}$. Given that Rogers et al. (2021) shows that core-powered mass-loss predicts $α$ $\approx$ 0.08 and $β$ $\approx$ 0.00 while photoevaporation predicts $α$ $\approx$ 0.12 and $β$ $\approx$ --0.17, our measurements are more consistent with core-powered mass-loss than photoevaporation. However, we caution that different gap-determination methods can produce systematic offsets in both $α$ and $β$; therefore, we motivate a comprehensive re-analysis of $Kepler$ light curves with modern, updated priors on eccentricity and mean stellar density to improve both the accuracy and precision of planet radii and subsequent measurements of the gap.