The End of Runaway: How Gap Opening Limits the Final Masses of Gas Giants

Abstract

Gas giants are thought to form by runaway accretion: an instability driven by the self-gravity of growing atmospheres that causes accretion rates to rise super-linearly with planet mass. Why runaway should stop at a Jupiter or any other mass is unknown. We consider the proposal that final masses are controlled by circumstellar disc gaps (cavities) opened by planetary gravitational torques. We develop a fully time-dependent theory of gap formation and couple it self-consistently to planetary growth rates. When gaps first open, planetary torques overwhelm viscous torques, and gas depletes as if it were inviscid. In low-viscosity discs, of the kind motivated by recent observations and theory, gaps stay predominantly in this inviscid phase and planet masses finalize at $M_{\mathrm{final}}/M_{\star }\sim (\Omega t_{\mathrm{disc}}{)}^{0.07}(H/a{)}^{2.73}(G{\rho }_0/{\Omega }^2{)}^{1/3}$, with $M_{\star }$ the host stellar mass, $\Omega$ the planet's orbital angular velocity, $t_{\mathrm{disc}}$ the gas disc's lifetime, $H/a$ its aspect ratio, and ${\rho }_0$ its unperturbed density. This final mass is independent of the dimensionless viscosity $\alpha$ and applies to large orbital distances, typically beyond $\sim$10 AU, where disc scale heights exceed planet radii. It evaluates to a few Jupiter masses at 10-100 AU, increasing gradually with distance as gaps become harder to open.

Accepted to MNRAS

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