TO COOL IS TO ACCRETE: ANALYTIC SCALINGS FOR NEBULAR ACCRETION OF PLANETARY ATMOSPHERES

Abstract

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 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 , where (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 close to nebular temperatures , 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 , while beyond 1 AU in a gas-rich nebula, GCR . We confirm our analytic scalings against detailed numerical models for objects ranging in mass from Mars () to the most extreme super-Earths (10–), and explain why heating from planetesimal accretion cannot prevent the latter from undergoing runaway gas accretion.