Dust back-reaction on gas around planets modifies the cold thermal torque

TL;DR

The paper investigates how dust back-reaction on gas and the cold thermal torque from thermal diffusion shape the total torque on a low-mass planet embedded in a gas-dust protoplanetary disk. Using high-resolution 3D two-fluid simulations with Fargo3D, the authors vary dust Stokes numbers and planet masses to capture linear and nonlinear regimes, revealing a bifurcation at $St\approx10^{-2}$: for $St\le10^{-2}$ the gaseous component (including the cold thermal torque) governs migration, while for $St>10^{-2}$ the dust torque dominates and drives inward migration through growing dust lobes. The results imply possible stagnation or inward runaway migration in regions where dust is not fully coupled to the gas, with the cold thermal torque becoming relevant in strongly coupled or inner-disk regions. This work highlights the importance of incorporating dust back-reaction and thermodynamics to predict planetary migration in realistic dusty disks.

Abstract

A nascent planet in a gas disk experiences radial migration due to the different torques which act on it. It has recently been shown that the torques produced by the gas and dust density variations around a non-accreting low-mass planet, the so-called cold thermal and dust streaming torques, can surpass each of the other torque components. We investigate how the total torque acting on the planet is affected by the presence of dust grains and their aerodynamic back-reaction on gas, while taking into account the cold thermal torque produced by thermal diffusion in the gas component. We perform high-resolution local and global three-dimensional two-fluid simulations within the pressureless-fluid dust approximation using the Fargo3D code. We explore the influence of different dust species parameterized by the Stokes number, focusing on non-accreting protoplanets with masses from one-third the mass of Mars to one Earth mass. The dust feedback has substantial impact on the asymmetry of the cold thermal lobes (which produce the cold thermal torque). However, the total torque is dominated by the dust torque when St $>10^{-2}$. The dust torque becomes more negative over time due to the formation of dust lobes that resemble the cold thermal lobes that form in the gas component. Therefore, the dust streaming torque prevails over the cold thermal torque. On the other hand, when St $\leq10^{-2}$, the dust streaming torque is negligible and thus, the total torque on the planet comes from the gaseous component of the disk. Our results suggest that a planet embedded in a gas-dust disk may experience stagnant migration or inward runaway migration in regions of the protoplanetary disk where the dust is not fully coupled to the gas. However, this behaviour could change in regions with strong dust-gas coupling or in the inner transition region of the disk, where the cold thermal torque may become relevant.

Figures (9)

• Figure 1: Normalized gas torque as a function of time in the linear case ($x_p/\lambda_c\approx0.57$) for different values of $\mathrm{St}$. The horizontal dotted line shows the torque value expected in the adiabatic disk according to JM2017 which we have included as a reference value. The right vertical axis corresponds to the usual normalization of the torque.
• Figure 2: Normalized total torque (gas plus dust) as a function of time in the settling regime for global models (see Table \ref{['tab:t1']}) when $\mathrm{St}=10^{-3}$. Although $x_p/\lambda_c\approx0.45$ for all simulations, the smallest mass of the planets considered here is equal to the critical mass $M_c$ (see Eq. \ref{['eq:potential']}), so these models can be classified beyond the linear theory. The dashed gray lines overlapped on the curves represent the torque value of the gas component. For all planet masses, the gas component dominates the total torque.
• Figure 3: Normalized total torque (gas plus dust) as a function of time in the settling regime, for the global models Mp0.3AD and Mp0.3TD (see Table \ref{['tab:t1']}) considering that $\mathrm{St}=\{0.0,0.1\}$. The total torque is dominated by the dust torque.
• Figure 4: Similar to Fig. \ref{['fig:torq1']} but for a planetary mass of $M_p=1.0M_\oplus$ and $\mathrm{St}=0.1$. It can clearly be seen that the total torque is dominated by the dust torque component.
• Figure 5: Same as Fig. \ref{['fig:torq1Mp1']} but for $\mathrm{St}=0.01$. In this case, we see that the total torque is dominated by the gas torque component.