is lower than the numerically computed “grid-level” surface density $\Sigma (r_{\rm p},t)$—lower by the Lindblad reduction factor $1/(1+B/\nu)$ (equation \[eqn:oom1\]). In other words, repulsion is encoded/enforced at a sub-grid level. We stress that equation (\[eqn:subgrid\]) is used only in our numerical simulation to capture repulsion and should not be used outside of it; contrast (\[eqn:subgrid\]) with, e.g., (\[eq:m\_dot\_ratio\]), and note that $\Sigma(r_{\rm p},t)$ is notation specific to the simulation and should not be confused with $\Sigma_{\rm p}$, the actual surface density at the planet’s position.

Our numerical procedure captures the gap depth but not the gap width, as the sub-grid modification is restricted (for simplicity) to the grid cell containing the planet. We consider this crude scheme acceptable insofar as we are more interested in the gross magnitudes for $\Sigma_{\rm p}/\Sigma_+$ and $\Sigma_{\rm p}/\Sigma_-$ and less interested in the precise surface density gradients. An untested assumption underlying our numerical procedure—and in our steady-state analytics—is that material flows radially through the gap at whatever velocity $u_r$ is needed to maintain continuity, i.e., to enforce $\dot{M}_- = \dot{M}_+ - \dot{M}_{\rm p} = -2\pi \Sigma_{\rm p} r_{\rm p} u_r$ (where $u_r < 0$ for accretion toward the star). We cannot test this assumption 