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

## TL;DR

This paper develops a theory explaining why gas giant planets stop growing at certain masses, showing that circumstellar disc gaps limit their final size, especially at large orbital distances beyond 10 AU.

## Contribution

It introduces a self-consistent, time-dependent model of gap formation that predicts the final mass of gas giants based on disc properties, independent of viscosity.

## Key findings

- Final masses are around a few Jupiter masses at 10-100 AU.
- Gap opening halts runaway accretion in low-viscosity discs.
- Final mass depends on disc lifetime, aspect ratio, and density, not viscosity.

## 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 $M_{\rm final}/M_\star\sim(\Omega t_{\rm disc})^{0.07}(H/a)^{2.73}(G\rho_0/\Omega^2)^{1/3}$, with $M_\star$ the host stellar mass, $\Omega$ the planet's orbital angular velocity, $t_{\rm disc}$ 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.