of gaps and the surface densities there (cf. @gs_2018 who use the @gr_2001 wave steepening theory to lift these assumptions). This shortcoming of the $1/x^4$ prescription means that it cannot be used to compute the planetary accretion rate $\dot{M}_{\rm p}$, which depends on knowing the gas density in the planet’s immediate vicinity.

What we do instead to include the repulsive Lindblad torque when calculating planetary accretion is as follows. Within the radial grid cell at $r = r_{\rm p}$ of width $\Delta r_{\rm p}$, the surface density is reduced after every timestep $\Delta t$ according to $$\begin{aligned}
 \label{eqn:euler}
\Sigma (r_{\rm p}, t+\Delta t) = \Sigma (r_{\rm p}, t) - \frac{\dot{M}_{\rm p} (t) \Delta t}{2 \pi r_{\rm p} \Delta r_{\rm p}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, ({\rm simulation})\end{aligned}$$ where the label “simulation” reminds us that this equation applies to the numerical simulation only and should not be used outside of that context. It is in evaluating $\dot{M}_{\rm p}$ that we include, in a “sub-grid” manner, the repulsive Lindblad gap: $$\begin{aligned}
 \label{eqn:subgrid}
\dot{M}_{\rm p}(t) &= A \times \frac{\Sigma (r_{\rm p},t)}{1+B/\nu} \,\,\,\,\,\,\,\,\,\,\,\,\,\, \, \, \,\,\,\,\,\,\,\,\,\,\,\,\,\, \, \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,({\rm simulation}).\end{aligned}$$ What equation (\[eqn:subgrid\]) says is that the disc surface density the planet actually “sees” when consuming local gas 