as we do not resolve the flow dynamics inside the gap. We will call out this assumption in the results to follow (sections \[sec:nc10au\] and \[sec:numwind\]). Also, as a reminder, we note that while the surface density changes as a result of consumption, in this subsection we fix $M_{\rm p}$, i.e., we do not update $M_{\rm p}$ using $\dot{M}_{\rm p}$ (this assumption is relaxed in section \[sec:mass\]).

In evaluating the consumption and repulsion coefficients $A$ and $B$, we make choices similar to those in our earlier order-of-magnitude analysis (section \[sec:oom\]), except that now we include numerical pre-factors for greater precision: $$\begin{aligned}
A_{\rm Bondi} &= 
0.5 \, \Omega r^2 \frac{m^2}{h^4}  && {\rm for} \, \operatorname{sub-thermal}\,\, m \leq 3 h^3  \label{eqn:bondi}\\
A_{\rm Hill} &= 2.2 \, \Omega r^2 m^{2/3} && {\rm for} \, \operatorname{super-thermal}\,\, m > 3 h^3 \label{eqn:hill}\\
B &= 0.04 \, \Omega r^2 \frac{m^2}{h^3}
\label{eqn:kanagawa}\end{aligned}$$ where all quantities are evaluated at $r_{\rm p}$. The pre-factor of $0.5$ in equation (\[eqn:bondi\]) is calibrated using 3D simulation results for $\dot{M}_{\rm p}$ from D’Angelo et al. ([-@dkh_2003]; these are re-printed in fig. 1 of @tt_2016). The coefficient of 2.2 in equation (\[eqn:hill\]) follows from requiring that (\[eqn:bondi\]) match (\[eqn:hill\]) at the thermal mass 