an asymmetric gap, with the inner disc surface density $\Sigma_-$ lower than the outer $\Sigma_+$ by $A_{\rm Bondi}/(3\pi B) > 1$; conditions are always consumption-dominated for Bondi accretion and $B$ as given by (\[eqn:kanagawa\]). The planet of mass $M_{\rm p} = 10\, M_{\rm J}$, accreting at the Hill rate, creates a symmetric gap where $\Sigma_-/\Sigma_+ \sim 1$; conditions here are repulsion-dominated as $M_{\rm p} > M_{\rm repulsion,visc}$ (equation \[eqn:repulsionmass\]).[]{data-label="fig:sigma_detailed"}](Fig2_newA.pdf){width="\linewidth"}

### Results {#sec:res222}

Figure \[fig:sigma\_detailed\] shows, for $M_{\rm p} = \{0.3, 10\} M_{\rm J}$, the numerically computed surface density profiles $\Sigma(r)$ at $t = 3 t_{\nu,{\rm p}} \simeq 5$ Myr. Overlaid for comparison is our numerical solution without a planet, which we have verified matches the analytic time-dependent similarity solution of @lp_1974. For the case with a planet, rather than plot at face value the numerically computed (grid-level) $\Sigma (r_{\rm p},t)$, we plot that value multiplied by the sub-grid reduction factor $1/(1+B/\nu)$—this is the “true” value for $\Sigma_{\rm p}$ that incorporates the repulsive Lindblad torque. Since this sub-grid correction factor is applied to only a single grid point, we cannot resolve gap widths; our focus instead is on the gross gap contrasts $\Sigma_{\rm p}/\Sigma_+$ and $\Sigma_{\rm p}/\Sigma_-$.

The surface density profiles 