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

TL;DR

This paper derives analytic formulas for how planetary atmospheres grow by accreting gas from circumstellar disks, considering dust effects, nebular conditions, and core mass, and confirms these with numerical models.

Contribution

The paper introduces new analytic scalings for atmospheric accretion that incorporate dust physics, nebular conditions, and core mass, extending previous models.

Findings

Gas-to-core mass ratio scales as t^{0.4} with time.

Dusty atmospheres have radiative boundaries set by internal microphysics.

Dust-free atmospheres grow faster at larger orbital distances.

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 $M_{\rm core}$ 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 $\propto t^{0.4} M_{\rm core}^{1.7} Z^{-0.4} \mu_{\rm rcb}^{3.4}$, where $\mu_{\rm rcb} \propto 1/(1-Z)$ (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 $T_{\rm rcb}$ close to nebular temperatures $T_{\rm out}$, 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 $\propto t^{0.4} T_{\rm rcb}^{-1.9} M_{\rm core}^{1.6} Z^{-0.4} \mu_{\rm rcb}^{3.3}$, while beyond 1 AU in a gas-rich nebula, GCR $\propto t^{0.4} T_{\rm rcb}^{-1.5} M_{\rm core}^1 Z^{-0.4}\mu_{\rm rcb}^{2.2}$. 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.