This procedure is repeated until the disc is evolved for several $t_{\nu,{\rm p}}$, long enough for the disc near the planet to achieve a quasi-steady state.

![How the surface density profile of a viscous disc responds to a planet that both consumes disc gas, and repels gas away by Lindblad torques. Surface densities are calculated from our 1D numerical simulation of a planet of fixed mass, either $M_{\rm p} = 0.3 \,M_{\rm J}$ (top panel) or $M_{\rm p} = 10\, M_{\rm J}$ (bottom panel), at $t = 3 t_{\nu,\mathrm{p}}$ when the disc near the planet at $r_{\rm p} = 10$ au has viscously relaxed. When computing the planetary accretion rate $\dot{M}_{\rm p}$, the gap is modeled as a single cell whose “true” surface density equals the grid-level $\Sigma$ lowered by a factor of $(1+B/\nu) \simeq B/\nu$; plotted here are the true sub-grid values $\Sigma_{\rm p}$. Accordingly, the planet’s gap is not spatially resolved and its width should not be taken literally from this figure. Red double-tipped arrows have lengths equal to their associated variables in dex, and demonstrate good agreement between numerics and analytics. The planet of mass $M_{\rm p} = 0.3 \,M_{\rm J}$, accreting at the Bondi rate, creates 