The Dynamical Interaction between Low-mass Planets and Dust Coagulation

TL;DR

This work analyzes how a low-mass planet embedded in a protoplanetary disk affects dust coagulation and, in turn, planetary migration. It employs a local 2D linear framework with coupled dust–gas dynamics and a single-size coagulation model to identify a planet-induced coagulation mode (CM) that can suppress dust growth in the co-orbital region and modify the dust-driven torque. A key finding is that outward migration requires a threshold stopping time $\tau$ of about $0.3$ under typical turbulence $\alpha$ and sticking efficiency, due to the phase-shifted interactions between dust density, stopping time, and radial velocity. This work links coagulation physics to pebble-driven migration, offering new insights into early planet formation and disk evolution, while noting limitations from the linear, 2D, single-size approximation that motivate future nonlinear, 3D, and full-size-distribution treatments.

Abstract

We investigate the impact of a low-mass planet on dust coagulation, and its consequent feedback on planetary migration, using a linear analysis of the coupled dust-gas hydrodynamic equations. Dust coagulation is incorporated via a single-size approximation. In the co-orbital region of the planet, we find that the growth of dust size is significantly suppressed by planet-induced coagulation modes (CMs). This effect are less pronounced with smaller stopping times, stronger gaseous turbulence or imperfect sticking. Regarding planetary migration, we find that CMs make outward migration require $τ\gtrsim 0.3$ ($τ$ is dimensionless stopping time) with typical turbulent strength and dust coagulation efficiency. We demonstrate that the torque variations are reasonable and arise from phase shifts between the density and stopping time perturbations in the coagulation modes.

Figures (5)

• Figure 1: Panel (a) and (b): The real and imaginary part of $k_x$ for the quasi-drift mode and CMs. The solid dots represent our numerical results. Panel (c)-(f): the wave functions normalized by their respective equilibrium values. The parameters are set as $\tau = 0.1$, $k_y = 0.3$ and $\alpha = 1\times 10^{-3}$. In those four panels, the blue, orange and gray lines represent the real part, imaginary part and amplitude, respectively.
• Figure 2: The waveforms of CM, which illustrates the mechanism of CI. The three waveforms represent the real parts of the perturbations in dust surface density $\delta \Sigma_d$, radial velocity $\delta u_{dx}$, and stopping time $\delta \tau$. Their amplitudes have been rescaled to same order. The propagation direction of the waves is inward, as indicated by the black arrow. A positive $\delta \Sigma_d$ leads to an increase in $\delta \tau$ with a phase shift of $\pi/2$ due to dust coagulation. As $\delta \tau$ increases, the aerodynamic drag exerted by the gas on the dust decreases, which in turn increases the dust radial velocity. In the figure, $\delta u_{dx}$ is approximately out of phase with $\delta \tau$. Consequently, the phase difference between $\delta u_{dx}$ and $\delta \Sigma_d$ is close to $\pi/2$, implying that dust grains tend to accumulate near the crests of $\delta \Sigma_d$ and disperse from the troughs, as indicated by the red arrows. This leads to the amplification of the initial $\delta \Sigma_d$. The positive feedback among $\delta \Sigma_d$, $\delta \tau$, and $\delta u_{dx}$ ultimately drives the CI.
• Figure 3: 2D morphologies illustrating the effects of a planet on dust coagulation, with $\tau = 0.1$, $\alpha = 1 \times 10^{-4}$ and $\varepsilon_{\mathrm{eff}}=1.0$ except for Panel (d), (e) and (f). The $x$- and $y$-axes correspond to the radial and azimuthal directions, respectively. Subtitles within each panel indicate the specific parameters used. Panel (a) show the perturbations in stopping time. Panel (b) shows the perturbations in dust size. Panels (c) and (d) provide the zoomed-in views of the dust size perturbations for $\tau = 0.1$ and $\tau = 0.05$, respectively. Panel (e) shows the dust size perturbations for $\alpha = 1 \times 10^{-3}$. Panel (f) shows the dust size perturbations with $\varepsilon_{\mathrm{eff}}=0.5$. Panel (g) shows the dust density perturbations with $\varepsilon_{\mathrm{eff}}=0.5$. Gray solid lines with arrows those Panels indicate the equilibrium streamlines. The colorbar denotes the normalized values. For a comparison, the values in Panel (b)-(f) are normalized by their common maximum value.
• Figure 4: Dusty torque on the planet with different values of $\varepsilon_{\mathrm{eff}}$ and $\alpha$. Lines with different colors correspond to different $\alpha$ as the legend labels. Dashed lines represent the results with $\varepsilon_{\mathrm{eff}} = 0$, i.e., without dust coagulation. Dot-dashed lines represent the results with $\varepsilon_{\mathrm{eff}} = 0.5$. Solid lines represent the results with $\varepsilon_{\mathrm{eff}} = 1.0$, i.e., perfect sticking
• Figure 5: Disk torque map with perfect sticking. The $y$ axis represents gaseous turbulent strength. The black solid line marks the zero-torque contour, while three gray solid line shows the zero-torque contours with different sticking efficiency.