accretes inward at rate $\dot{M}_+$. Downstream of the planet, at $r < r_{\rm p}$, the corresponding surface density and accretion rate are $\Sigma_-$ and $\dot{M}_-$, respectively. The difference $\dot{M}_+ - \dot{M}_-$ is the accretion rate onto the planet $\dot{M}_{\rm p}$.[]{data-label="fig:sig_cartoon"}](Fig1_newA.pdf){width="\linewidth"}

Momentum conservation provides another relation between the surface densities. It is easiest to write down downstream of the planet in the accretion flow (in the inner disc), as the flow of momentum upstream (in the outer disc) is complicated by the mass sink presented by the planet. In the inner disc there are no sinks of mass or momentum, only a steady transmission of mass inward and angular momentum outward (assuming, as we do throughout this paper, a non-migrating planet; see section \[sec:sum\] for pointers to the migrating case). The rate at which angular momentum is carried viscously outward by the inner disc equals the viscous transport rate local to the planet, plus the repulsive Lindblad torque exerted by the planet on the disc:[^2] $$\begin{aligned}
\Sigma_- \nu \Omega r^2 \sim \Sigma_{\rm p} \nu \Omega r^2 + B \Sigma_{\rm p} \Omega r^2 
\label{eqn:momentum}\end{aligned}$$ with $$\label{eqn:B}
B \sim \frac{m^2}{h^3} \Omega r^2$$ given by the standard @gt_1980 linear Lindblad torque, integrating the effects 