Sculpting the sub-Saturn Occurrence Rate via Atmospheric Mass Loss

The sub-Saturn ($\sim$4--8$R_{\oplus}$) occurrence rate rises with orbital period out to at least $\sim$300 days. In this work we adopt and test the hypothesis that the decrease in their occurrence towards the star is a result of atmospheric mass loss, which can transform sub-Saturns into sub-Neptunes ($\lesssim$4$R_{\oplus}$) more efficiently at shorter periods. We show that under the mass loss hypothesis, the sub-Saturn occurrence rate can be leveraged to infer their underlying core mass function, and by extension that of gas giants. We determine that lognormal core mass functions peaked near $\sim$10--20$M_{\oplus}$ are compatible with the sub-Saturn period distribution, the distribution of observationally-inferred sub-Saturn cores, and gas accretion theories. Our theory predicts that close-in sub-Saturns should be $\sim$50\% less common and $\sim$30\% more massive around rapidly rotating stars; this should be directly testable for stars younger than $\lesssim$500 Myr. We also predict that the sub-Jovian desert becomes less pronounced and opens up at smaller orbital periods around M stars compared to solar-type stars ($\sim$0.7 days vs.~$\sim$3 days). We demonstrate that exceptionally low-density sub-Saturns, "Super-Puffs", can survive intense hydrodynamic escape to the present day if they are born with even larger atmospheres than they currently harbor; in this picture, Kepler 223 d began with an envelope $\sim$1.5$\times$ the mass of its core and is currently losing its envelope at a rate $\sim$2$\times 10^{-3}M_{\oplus}~\mathrm{Myr}^{-1}$. If the predictions from our theory are confirmed by observations, the core mass function we predict can also serve to constrain core formation theories of gas-rich planets.