Relatedly, the disc density scales with height $z$ above the midplane as $\exp[-z^2/(2H^2)]$ (for an isothermal atmosphere), which implies that a considerable fraction of the disc mass resides between $|z|= H$ and $2H$; accordingly, the planet does not pop out of the disc until it is strongly super-thermal, i.e., until $m$ is a large multiple of $h^3$ (cf. equation \[eqn:thermalmass\_0\]). An isotropic version of super-thermal accretion controlled by the Hill sphere gives $\dot{M}_{\rm p,Hill,iso} \sim \rho_{\rm p} \times R_{\rm H}^2 \times \Omega R_{\rm H}$ or $A_{\rm Hill,iso} \sim m \Omega r^2 / h$. Yet another prescription for accretion is given by @tw_2002: $A_{\rm TW} \sim m^{4/3} \Omega r^2 / h^2$, an empirical relation based on their 2D numerical simulations (see also @tt_2016). To the extent that these alternative scalings increase with $m$ more steeply than our nominal $A_{\rm Hill} \propto m^{2/3}$, whatever final super-thermal planet masses we derive should be lower limits (see sections \[sec:proc\] and \[sec:sum\]).

![Sketch of the disc surface density and accretion flow in the vicinity of a planet. The planet is located at orbital radius $r_\mathrm{p}$, inside a gap having surface density $\Sigma_\mathrm{p}$. At $r > r_{\rm p}$, the disc surface density is $\Sigma_+$ and mass 