h^3 < 1 \label{eqn:mdotp_Hill}\\
    \frac{\dot{M}_-}{\dot{M}_+} &\sim 1 - A_{\rm Hill}/B \sim 1 - m^{-4/3} h^3 \label{eqn:mdot-_Hill}\end{aligned}$$ and the planet diverts only a small fraction, $A_{\rm Hill}/B$, of the disc accretion flow onto itself.

The order-of-magnitude considerations presented here are firmed up in subsequent sections, including in Appendix \[sec:analy\_ss\], where we derive in greater analytic detail the surface density profile and mass accretion rates, drawing from @ld_2006.

Numerical simulations {#sec:num}
---------------------

### Procedure {#sec:proc}

We solve numerically for the 1D evolution of a viscously shearing disc [e.g., @fkr_2002] with a planetary mass sink. The governing equation for the surface density $\Sigma(r,t)$ in cylindrical radius $r$ and time $t$ reads $$\begin{aligned}
\pd{\Sigma}t=\frac{3}{r}\pd{}r\left[r^{1/2}\pd{}r\left(r^{1/2}\nu\Sigma\right)\right] - \frac{\dot{M}_{\rm p}(t)}{2\pi r} \delta (r - r_{\rm p}) 
\label{eq:dsig_dt}\end{aligned}$$ where $\delta$ is the Dirac delta function and $r_{\rm p}$ is the radial position of the planet (held fixed). For the viscosity $\nu$ we employ the @ss_alpha $\alpha$-prescription: $$\label{eq:ss_alpha}
\nu = \alpha c_{\rm s}^2/\Omega = \alpha h^2 \Omega r^2$$ where $\Omega$ is the Keplerian orbital frequency around a $1 \, M_\odot$ star, $c_{\rm s} = \sqrt{k_{\rm B}T/\overline{m}}$, the disc temperature is $T = 200 \, {\rm K} (r/{\rm au})^{-1/2}$, $k_{\rm B}$ is Boltzmann’s constant, $\overline{m} = 2m_{\rm H}$ is the 