onto the star at rate $\dot{M}_-$. Dropping numerical pre-factors (these will be restored in later sections), we have $$\begin{aligned}
\dot{M}_+ &= \dot{M}_- + \dot{M}_{\rm p} \nonumber \\
\Sigma_+ \nu &\sim \Sigma_- \nu + \dot{M}_{\rm p} \nonumber \\
&\sim \Sigma_- \nu + A \Sigma_{\rm p} \,. \label{eqn:mass}\end{aligned}$$ There are a number of assumptions embedded in these order-of-magnitude statements. For $\dot{M}_+$ and $\dot{M}_-$ we have substituted standard steady-state expressions for a disc of shear viscosity $\nu$ (e.g., @fkr_2002), valid asymptotically at locations far from any mass sink ($|r-r_{\rm p}| \gtrsim r_{\rm p}$). At the same time, the locations we are considering in the outer and inner discs are not so far from the planet that we need to account for spatial variations in $\nu$, which may change by order-unity factors over length scale $r$.

For the planet’s accretion rate, we have assumed in (\[eqn:mass\]) that it scales linearly with the local surface density $\Sigma_{\rm p}$ with proportionality constant $A$: $$\label{eqn:Adef}
\dot{M}_{\rm p} = A \Sigma_{\rm p} \,.$$ This assumption is satisfied, e.g., by a planet accreting at the Bondi rate (e.g., @fkr_2002): $$\begin{aligned}
 \label{eq:m_dot_bondi_simp}
    \dot{M}_{\rm p, Bondi} &\sim \rho_{\rm p} c_{\rm s} R_{\rm B}^2 \nonumber \\
    &\sim \frac{\Sigma_{\rm p}}{H} c_{\rm s} \left( 