Interior dynamics of envelopes around disk-embedded planets

TL;DR

This study shows that envelopes around disk-embedded planets are not static, but dynamically exchange gas with the surrounding disk through recycling flows. By performing 3D hydrodynamic simulations with a beta-cooling parameter and accretion heating, it identifies three cooling regimes—fast, intermediate, and slow—each with a characteristic envelope structure (nearly isothermal with an inner radiative layer, a three-layer convective–radiative–recycling envelope, and a fully convective envelope, respectively). The authors develop analytic transport models and demonstrate that fully convective envelopes mix material on timescales of a few orbits, while radiative layers can trap tracers and volatiles on much longer timescales, with implications for volatile delivery and core/envelope growth. Their results suggest a disk-location–dependent dichotomy in planetary composition and growth, underscoring the importance of coupling gas dynamics with dust evolution, radiative transfer, and pebble accretion in planet formation models.

Abstract

In the core accretion scenario, forming planets start to acquire gaseous envelopes while accreting solids. Conventional one-dimensional models assume envelopes to be static and isolated. However, recent three-dimensional simulations demonstrate dynamic gas exchange from the envelope to the surrounding disk. This process is controlled by the balance between heating, through the accretion of solids, and cooling, which is regulated by poorly-known opacities. In this work, we systemically investigate a wide range of cooling and heating rates, using three-dimensional hydrodynamical simulations. We identify three distinct cooling regimes. Fast-cooling envelopes ($β\lesssim 1$, with $β$ the cooling time in units of orbital time) are nearly isothermal and have inner radiative layers that are shielded from recycling flows. In contrast, slow cooling envelopes ($β\gtrsim10^3$) become fully convective. In the intermediate regime ($1\lesssimβ\lesssim300$), envelopes are characterized by a three-layer structure, comprising an inner convective, a middle radiative, and an outer recycling layer. The development of this radiative layer traps small dust and vapour released from sublimated species. In contrast, fully convective envelopes efficiently exchange material from inner to outer envelope. Such fully convective envelopes are likely to emerge in the inner parts of protoplanetary disks ($\lesssim$ 1 au) where cooling times are long, implying that inner-disk super-Earths may see their growth stalled and be volatile depleted.

Figures (20)

• Figure 1: Shell-averaged density, temperature, entropy, and temperature gradient (from top to bottom) for different $\beta$ values. We set $m=0.1$ and $\dot{M}=10\,M_\oplus/$Myr. All panels are the snapshots at the end of the calculation, $t=50$. The dashed and dotted curves in panels a--c are the analytic models given by Eqs. (\ref{['eq:iso den profile']}), (\ref{['eq:ad den profile']}), and (\ref{['eq:ad temp profile']}). The horizontal dashed line in panel d is the adiabatic gradient, Eq. \ref{['eq:Schwarzschild criterion']}.
• Figure 2: Radial velocity of the gas, as a function of cooling time$\beta$. We set $m=0.1$ and $\dot{M}=10\,M_\oplus/$Myr. All panels are snapshots at the end of the calculation, $t=50$. The dashed circle denotes the Bondi radius. Top: Midplane slices, with streamlines shown in panel a. Bottom: Vertical slices where the velocity is hemispherically averaged in the azimuthal direction, $\phi\in[-\pi/2,\pi/2]$
• Figure 3: Hemispherically- and time-averaged density and velocity vector on the vertical slice. We set $m=0.1$. The physical quantities are averaged over the azimuth and the time, $\phi\in[-\pi/2,\pi/2]$ and $t\in[40,50]$. The white dash circle in the top panels is the Bondi radius of the planet.
• Figure 4: Boundary between the recycling and radiative layers (orange) and the radiative and convective layers (blue), as a function of cooling time $\beta$. The squares and the circles are the numerical results, obtained for $m=0.1$ and $\dot{M}=10\,M_\oplus/$Myr. The dashed lines are the fitting formulae (Appendix \ref{['sec:Appendix fitting formulae']}). The hatched region is not probed by our simulations.
• Figure 5: Shell-averaged mean speed for $\beta=10^{-2}\text{--}10^1$. We set $m=0.1$ and $\dot{M}=10\,M_\oplus/$Myr. This panel shows the state at the end of the simulation, $t=50$. The radiative and recycling layers are qualitatively indicated. The horizontal dotted lines correspond to Eq. \ref{['eq:vsh_shell']} with $r=0.3\,R_{\rm B}$ and $r=0.8\,R_{\rm B}$. The gray dashed-dotted curve represents Eq. \ref{['eq:vsh_shell']}.