In section \[sec:mass\], we free up the planet’s mass and allow two-way feedback between planet and disc.

Order-of-magnitude scalings {#sec:oom}
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Consider an accreting planet embedded in a viscous disc. From Figure \[fig:sig\_cartoon\] we identify three disc surface densities: $\Sigma_{\rm p}$ at the orbital radius of the planet ($r = r_{\rm p}$), $\Sigma_+$ exterior to the planet, and $\Sigma_-$ interior to the planet. The planet depresses the local surface density because it is both consuming disc gas and repelling disc gas away by Lindblad torques. Our goal is to estimate the depth of the planet’s gap in relation to the inner and outer discs: $\Sigma_{\rm p}/\Sigma_-$ and $\Sigma_{\rm p}/\Sigma_+$. We assume a steady state where the disc has viscously relaxed: given a viscosity $\nu$, the system age $t$ is at least as long as the diffusion time $r^2/\nu$ across the disc. In addition, $t$ is at most the planet growth timescale $M_{\rm p}/\dot{M}_{\rm p}$, so that we may consider the planet mass fixed at any given moment.

Mass flows steadily inward at rate $\dot{M}_+$ from the outer disc. Part of this flow is accreted by the planet at rate $\dot{M}_{\rm p}$, with the rest feeding the inner disc which accretes 