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{μ }_{rcb}^3.4, where {μ }_{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{μ }_{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{μ }_{rcb}^2.2. We confirm our analytic scalings against detailed numerical models for objects ranging in mass from Mars (0.1{M}_\oplus ) to the most extreme super-Earths (10-20{M}_\oplus ), and explain why heating from planetesimal accretion cannot prevent the latter from undergoing runaway gas accretion.