The maximum accretion rate of a protoplanet: how fast can runaway be?

The hunt is on for dozens of protoplanets hypothesised to reside in protoplanetary discs with imaged gaps. How bright these planets are, and what they will grow to become, depend on their accretion rates, which may be in the runaway regime. Using 3D global simulations we calculate maximum gas accretion rates for planet masses $M_{\rm p}$ from 1$\,M_{\oplus}$ to $10\,M_{\rm J}$. When the planet is small enough that its sphere of influence is fully embedded in the disc, with a Bondi radius $r_{\rm Bondi}$ smaller than the disc's scale height $H_{\rm p}$ -- such planets have thermal mass parameters $q_{\rm th} \equiv (M_{\rm p}/M_{\star}) / (H_{\rm p}/R_{\rm p})^3 \lesssim 0.3$, for host stellar mass $M_{\star}$ and orbital radius $R_{\rm p}$ -- the maximum accretion rate follows a Bondi scaling, with $\max \dot{M}_{\rm p} \propto \rho_{\rm g} M_{\rm p}^2 / (H_{\rm p}/R_{\rm p})^3$ for ambient disc density $\rho_{\rm g}$. For more massive planets with $0.3 \lesssim q_{\rm th} \lesssim 10$, the Hill sphere replaces the Bondi sphere as the gravitational sphere of influence, and $\max \dot{M}_{\rm p} \propto \rho_{\rm g} M_{\rm p}^1$, with no dependence on $H_{\rm p}/R_{\rm p}$. In the strongly superthermal limit when $q_{\rm th} \gtrsim 10$, the Hill sphere pops well out of the disc, and $\max \dot{M}_{\rm p} \propto \rho_{\rm g} M_{\rm p}^{2/3} (H_{\rm p}/R_{\rm p})^1$. Applied to the two confirmed protoplanets PDS 70b and c, our numerically calibrated maximum accretion rates imply their Jupiter-like masses may increase by up to a factor of $\sim$2 before their parent disc dissipates.