Planet-Disk Interactions and the Convective Overstability. I. Low Mass Planets

Abstract

Rapid inward migration driven by Type I torques threatens the survival of low-mass planets in their nascent protoplanetary disks (PPDs). Positive co-rotation torques offer a potential solution, but require viscous diffusion to remain unsaturated. However, it is unclear if (magneto)-hydrodynamic turbulence provides the necessary diffusion, and disk profiles supporting such torques are often also susceptible to the Convective Overstability (COS) for suitable gas cooling timescales. To this end, we investigate torques on low-mass planets through radially global 2D (razor-thin) and vertically unstratified 3D hydrodynamic simulations of PPDs with thermal diffusion and optically thin cooling. Our 3D models with thermal diffusion, which allows COS development, show systematically different torque behavior compared to 2D models, wherein COS is absent. In 3D, the COS saturates into large-scale, long-lived vortices that migrate radially and interact gravitationally with the embedded planet. When these vortices encounter the planet, they typically provide positive torque "kicks" counteracting inward migration, as the less-massive vortices are scattered onto horseshoe orbits by the more-massive planet. We validate our simulation methods against the theoretical framework of Paardekooper et al. (2011) and demonstrate that COS-induced torque modifications can extend migration timescales by factors of approximately 10. For plausible disk models, our results suggest that COS activity can lengthen migration timescales sufficiently to overlap with, or even exceed Super-Earth formation windows (0.1-5 Myr). In contrast, simulations with optically thin cooling do not show significant torque modifications, as COS saturates in near-axisymmetric structures without producing large-scale vortices for the disk models considered here.

Figures (12)

• Figure 1: Steady-state fluid streamlines and density field in the vicinity of a low mass planet with $q_p=1.26 \cdot 10^{-5}$ and viscosity $\nu_p=10^{-7}$. The slight radial inward shift of the gas' horse shoe orbits with respect to the planets location is caused by the radial gas pressure gradient.
• Figure 2: Maps of the total torque resulting from the citecite.paardekooper2011P11-model with thermal diffusion (upper panels) and $\beta$-cooling (bottom panels). The used parameters are $h_p=0.1$, $q_p=2.52\cdot 10^{-5}$, $b/h_p=0.4$ and $\gamma=1.4$. The left panels vary the disk structure via $p$ and $q$ at fixed viscous and thermal diffusion, while the right panels vary $\chi_p$ and $\nu_p$ at fixed disk structure. The entire region between the black solid lines yields $N_r^2<0$ via Eq. (\ref{['eq:nr22']}) in Section \ref{['sec:hydro']} and is therefore susceptible to the COS.
• Figure 3: Measurements of $\gamma_{\mathrm{eff}}$ from the Lindblad torque in 2D simulations with viscosity $\nu=10^{-7}$ (red squares) and inviscid simulations (blue circles) and optically thin cooling indicated on the horizontal axis, as explained in the text.
• Figure 4: Measurements of the scaled corotation torque $\Gamma_C/\Gamma_0$ in viscous 2D simulations with $q_p=1.26\times 10^{-5}$ and $h_p=0.05$. Symbols indicate simulation results obtained by time-averaging the total disk torque over the final 100 orbits and subtracting the theoretical Lindblad torque ($\Gamma_L$, Eq. \ref{['eq:lindblad']}). Upper panels use $\beta$-cooling, varying $\beta$ at fixed $p_{\nu}=0.33$ (left) and varying $p_{\nu}$ at fixed $\beta=10$ (right). Lower panels consider thermal diffusion of varying strength at fixed $p_{\nu}=0.33$ (left) and varying $p_{\nu}$ at fixed $\chi = 3 \times 10^{-6}$ (right). Pink squares denote temperature diffusion (method C) and blue circles entropy diffusion (method A). Theoretical curves (dashed black) are based on the torque formulas (\ref{['eq:GammaC']})--(\ref{['eq:linent']}). Vertical dashed and dotted lines indicate critical values for non-linear saturation and the linear transition, respectively. These correspond to critical cooling times (Eqs. \ref{['eq:beta_crit_nl']}, \ref{['eq:beta_crit_lin']}) and thermal diffusivities (Eqs. \ref{['eq:chi_crit_nl']}, \ref{['eq:chi_crit_lin']}) in the left panels, and critical viscosities (Eqs. \ref{['eq:nucrit_final']}, \ref{['eq:nu_crit_lin']}) in the right panels. To facilitate direct comparison, the resolution for the thermal diffusion runs (lower panels) matches the grid of citecite.paardekooper2011P11 while the $\beta$-cooling runs (upper panels) utilize the same grid as our 3D simulations (see Sect. \ref{['sec:results']}).
• Figure 5: Saturation of COS in a disk with thermal diffusion (method $A$). The upper panel shows the measured turbulent radial angular momentum flux as function of thermal diffusivity at the planet's location. The lower panel shows the evolution of RMS radial and vertical velocities as the COS saturates over several hundreds of orbits in the same simulations with different $\chi_p$.