$$\label{eqn:thermalmass}
M_{\rm thermal} \equiv 3 h^3 M_\star \simeq 0.5 \left( \frac{h}{0.054} \right)^3 M_{\rm J}$$ defined by equating $H$ with $R_{\rm H} = (m/3)^{1/3} r$, with $M_{\rm J}$ the mass of Jupiter. Equation (\[eqn:kanagawa\]) is taken from the numerical 2D simulations of @kmt_2015 [see also @dm_2013 and @d_2015 who report similar results].

Note further that the expressions we used in section \[sec:oom\] for the steady disc accretion rates $\dot{M}_+$ and $\dot{M}_-$ should be amended with the numerical pre-factor $3\pi$, i.e., $\dot{M}_+ = 3\pi \Sigma_+ \nu$ and similarly for $\dot{M}_-$ (e.g., @fkr_2002). This correction is already embedded in the diffusion equation (\[eq:dsig\_dt\]). Including this pre-factor in equation (\[eqn:mass\]) implies that $A$ should be replaced with $A/(3\pi)$ in equations (\[eqn:oom2\])–(\[eqn:mdot-\_Hill\]). Putting it all together, we have $$\begin{aligned}
 \label{eq:A_B_visc}
    \frac{A_{\rm Bondi}}{3 \pi B} \simeq \frac{1.3}{h} > 1\end{aligned}$$ implying that consumption always dominates for sub-thermal masses. Furthermore, $$\begin{aligned}
 \label{eq:A_B_visc_Hill}
    \frac{A_{\rm Hill}}{3 \pi B} \simeq 5.5 m^{-4/3} h^3 \simeq 1.0 \left( \frac{m}{5 \times 10^{-3}} \right)^{-4/3} \left( \frac{h}{0.054} \right)^3\end{aligned}$$ implying that repulsion dominates for super-thermal masses exceeding a “repulsion mass”

$$\begin{aligned}
 \label{eqn:repulsionmass}
M_{\rm repulsion,visc} &\simeq 3.6 h^{9/4} M_\star \nonumber \\
& \simeq 5.3 M_{\rm J} \left( \frac{h}{0.054} \right)^{9/4} \simeq 5.3 M_{\rm J} \left( \frac{r}{10 \, {\rm au} 