Capture and Stability of Resonant Planet Pairs in Turbulent Disk

Abstract

We present a theoretical framework for the resonance capture and stability of two-planet systems in turbulent disks. By incorporating stochastic forcing (parameterized by $κ$) alongside laminar angular momentum and eccentricity damping timescales ($τ_{\rm m}, τ_{e}$), we derive an analytical criterion for the general $j:j-1$ mean motion resonances, and validate it through N-body simulations. The outcome is mapped in $κ$-$τ_{\rm m}/τ_{e}$ parameter space, revealing two distinct regimes: resonance trapping and turbulence-induced disruption -- which occurs either directly cross or via temporary capture followed by escape through turbulent diffusion. Crucially, our analysis identifies turbulence as a universal destabilizer. It amplifies the intrinsic overstability mechanism: In laminar disks, escape requires $τ_{\rm m}/τ_{e}$ to drop below a critical limit due to excessive eccentricity excitation. We demonstrate that turbulent diffusion lowers this limit, demanding stronger damping (larger $τ_{\rm m}/τ_{e}$) for stability. Thus, greater turbulence promotes escape, and sufficiently strong diffusion precludes resonance retention irrespective of eccentricity damping.

Figures (7)

• Figure 1: Simulations of the $2$:$1$ MMR capture and stability for a planet-pair at three different disk turbulent strengths: (a) $\kappa {=} 10^{-7}$ and (b) $10^{-5}$. The planet masses are $m_{\rm i}{=}1\ M_{\oplus}$, $m_{\rm o}{=}10\ M_{\oplus}$. Solid lines represent our analytical criteria for a laminar disk, while colored dots correspond to numerical results. While weak turbulence ($\kappa = 10^{-7}$) preserves the laminar-like behavior, strong turbulence ($\kappa = 10^{-5}$) introduces dominant stochastic forcing that completely disrupts resonance stability.
• Figure 2: Time evolution of the period ratio ($P_{\rm o}/P_{\rm i}$) for a planetary pair converging towards the $2$:$1$ MMR with $\tau_{\rm m} = 6 \times 10^{5}$ yr and $\tau_{\rm m}/\tau_{e} = 10^{3}$. The colored lines represent different turbulence strengths $\kappa$, and the grey dotted line represents the nominal resonant location. As the turbulence strength increases, the resonance is disrupted more rapidly.
• Figure 3: Evolution of a two-planet system in phase space at two different disk turbulent strengths: (a) $\kappa {=} 10^{-7}$ and (b) $\kappa {=} 10^{-6}$. The resonant angle is defined as $\varphi_{\rm i} = j\lambda_{\rm o} - (j-1)\lambda_{\rm i} - \varpi_{\rm i}$, where $\lambda$ is the mean longitudes of the planet, and $\varpi$ is the relevant longitude of pericenter. The planet mass and migration parameters are $m_{\rm i} {=} 1\ M_{\oplus}$, $m_{\rm o} {=} 10\ M_{\oplus}$, $\tau_{\rm m} {=} 6 \times 10^{5}$ yr and $\tau_{\rm m} / \tau_{e} {=} 10^{3}$. Turbulence induces orbital eccentricity diffusion around the equilibrium point, with stronger diffusion observed at a higher $\kappa$ value.
• Figure 4: Analytical predictions for the convergent migration of a planet pair near the 2:1 MMR in a turbulent disk. The green region denotes stable resonance trapping, while the brown region corresponds to resonance disruption (resulting in either direct crossing or transient capture followed by escape). The solid brown line represents the turbulence-induced criterion (Eq. \ref{['eq:final_kappa']}) that separates these two regimes. The horizontal dotted line marks the laminar limit, and the vertical dotted line indicates the strong turbulence limit (Eq. \ref{['eq:kappa_crit']}). For a typical sub-Neptune system ($\sim 10$$M_{\oplus}$), this strong turbulence limit corresponds to $\kappa_{\mathrm{crit}} \approx 10^{-6}$.
• Figure 5: Simulations of 2:1 MMR capture and stability in a turbulent disk for a planet pair with a mass ratio $q=0.1$. The solid brown line represents the turbulence-induced disruption criterion (Eq. \ref{['eq:final_kappa']}). For comparison, the horizontal and vertical dotted lines mark the laminar limit (Eq. 24 in Paper I) and the strong turbulence limit (Eq. \ref{['eq:kappa_crit']}), respectively. The simulation outcomes show broad consistency with the analytical predictions