To Cool is to Accrete: Analytic Scalings for Nebular Accretion of Planetary Atmospheres

Eve J. Lee; Eugene Chiang

doi:10.1088/0004-637X/811/1/41

paper

To Cool is to Accrete: Analytic Scalings for Nebular Accretion of Planetary Atmospheres

arXiv:1508.05096 · doi:10.1088/0004-637X/811/1/41

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 $M_{core}$ 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, H $_{2}$ 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.

9 pages, 6 figures, accepted to ApJ

To Cool is to Accrete: Analytic Scalings for Nebular Accretion of Planetary Atmospheres

Abstract

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