hierarchy of length scales reverses so that $R_{\rm B} > R_{\rm H} > H$—the planet’s sphere of influence, now set by gravitational tides at radius $R_{\rm H}$, “pops out” of the disc—and arguably the planet accretes in a more 2D fashion, presenting a cross-section of order $R_{\rm H} H$ to disc gas that shears by at a velocity $\Omega R_{\rm H}$. The corresponding “Hill rate” for consumption is then $$\label{eqn:hillsimple}
\dot{M}_{\rm p,Hill} \sim \rho_{\rm p} \times R_{\rm H}H \times \Omega R_{\rm H} \sim \Sigma_{\rm p} R_{\rm H}^2 \Omega$$ whence $$A_{\rm Hill} \sim m^{2/3} \Omega r^2 \,.$$ A Hill-based scaling for consumption is commonly used in 2D disc-planet hydrodynamical simulations (e.g., @znh_2011 [@dk_2015; @dk_2017; @mfv_2019]). We have assumed in writing the above that the planet masses are large enough for accretion to be hydrodynamically-limited as opposed to cooling-limited (@gc_2019a, cf. their fig. 1).

In this paper we will calculate the growth of planets from sub-thermal to super-thermal masses, so will have occasion to use both $A_{\rm Bondi}$ and $A_{\rm Hill}$. We recognize that the 2D picture motivating our Hill scaling may not be correct; in 3D, meridional flows from gap walls can feed the planet along its poles [@smc_2014; @msc_2014; @fc_2016]. 