To Cool is to Accrete: Analytic Scalings for Nebular Accretion of Planetary Atmospheres
Lee et al
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 Mcore 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 ∝t0.4M1.7coreZ−0.4μ3.4rcb, where μrcb∝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 Trcb close to nebular temperatures Tout, 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 ∝t0.4T−1.9rcbM1.6coreZ−0.4μ3.3rcb, while beyond 1 AU in a gas-rich nebula, GCR ∝t0.4T−1.5rcbM1coreZ−0.4μ2.2rcb. We confirm our analytic scalings against detailed numerical models for objects ranging in mass from Mars (0.1 M⊕) to the most extreme super-Earths (10-20 M⊕), and explain why heating from planetesimal accretion cannot prevent the latter from undergoing runaway gas accretion.