A Theoretical Framework for the Mass Distribution of Gas Giant Planets Forming through the Core Accretion Paradigm

This paper constructs a theoretical framework for calculating the distribution of masses for gas giant planets forming via the core accretion paradigm. Starting with known properties of circumstellar disks, we present models for the planetary mass distribution over the range $0.1M_J < M_{\rm p} < 10M_J$. If the circumstellar disk lifetime is solely responsible for the end of planetary mass accretion, the observed (nearly) exponential distribution of disk lifetime would imprint an exponential fall-off in the planetary mass function. This result is in apparent conflict with observations, which suggest that the mass distribution has a (nearly) power-law form $dF/dM_{\rm p}\sim M_{\rm p}^{-p}$, with index $p\approx1.3$, over the relevant planetary mass range (and for stellar masses $\sim0.5-2M_\odot$). The mass accretion rate onto the planet depends on the fraction of the (circumstellar) disk accretion flow that enters the Hill sphere, and on the efficiency with which the planet captures the incoming material. Models for the planetary mass function that include distributions for these efficiencies, with uninformed priors, can produce nearly power-law behavior, consistent with current observations. The disk lifetimes, accretion rates, and other input parameters depend on the mass of the host star. We show how these variations lead to different forms for the planetary mass function for different stellar masses. Compared to stars with masses $M_\ast$ = $0.5-2M_\odot$, stars with smaller masses are predicted to have a steeper planetary mass function (fewer large planets).