To cool is to accrete: Analytic scalings for nebular accretion of planetary atmospheres

Planets acquire atmospheres from their parent circumstellar disks. We derive a general analytic expression for how the atmospheric mass grows with time t as a function of the underlying core mass $M_{core}$ and nebular conditions, including the gas metallicity Z. Planets accrete as much gas as can cool: an atmosphere's doubling time is given by its Kelvin–Helmholtz time. Dusty atmospheres behave differently from atmospheres made dust-free by grain growth and sedimentation. The gas-to-core mass ratio (GCR) of a dusty atmosphere scales as GCR $\propto t^{0.4}{M}_{core}^{1.7}{Z}^{-0.4}{\mu }_{rcb}^{3.4}$, where ${\mu }_{rcb}\propto 1/(1-Z)$ (for Z not too close to 1) is the mean molecular weight at the innermost radiative–convective boundary. This scaling applies across all orbital distances and nebular conditions for dusty atmospheres; their radiative–convective boundaries, which regulate cooling, are not set by the external environment, but rather by the internal microphysics of dust sublimation, H₂ dissociation, and the formation of H⁻. By contrast, dust-free atmospheres have their radiative boundaries at temperatures $T_{rcb}$ close to nebular temperatures $T_{out}$, and grow faster at larger orbital distances where cooler temperatures, and by extension lower opacities, prevail. At 0.1 AU in a gas-poor nebula, GCR $\propto t^{0.4}{T}_{rcb}^{-1.9}{M}_{core}^{1.6}{Z}^{-0.4}{\mu }_{rcb}^{3.3}$, while beyond 1 AU in a gas-rich nebula, GCR $\propto t^{0.4}{T}_{rcb}^{-1.5}{M}_{core}^1{Z}^{-0.4}{\mu }_{rcb}^{2.2}$. We confirm our analytic scalings against detailed numerical models for objects ranging in mass from Mars ($0.1M_{\oplus }$) to the most extreme super-Earths (10–$20M_{\oplus }$), and explain why heating from planetesimal accretion cannot prevent the latter from undergoing runaway gas accretion.