Resonance Capture and Stability Analysis for Planet Pairs under Type I Disk Migration

TL;DR

The paper develops a unified analytical framework for resonance capture and post-capture evolution of two planets undergoing convergent type I migration, deriving criteria for general $j$:$j-1$ mean-motion resonances across arbitrary mass ratios. By balancing resonant gravitational interactions against disk damping (characterized by $\tau_m$ and $\tau_e$) and employing a Hamiltonian-based stability analysis, the authors identify four dynamical regimes: no-trap, stable trap, overstable trap, and escape, and they validate these predictions with extensive N-body simulations. The key contributions include explicit trapping and escape criteria with mass-ratio dependencies via functions $h(q)$ and $f(q)$, plus a generalized treatment that extends to multiple resonances and arbitrary $q$. The framework provides a practical tool to interpret observed resonant configurations and to connect disk properties to the long-term architecture of planetary systems, while also outlining limitations related to disk turbulence, time-varying damping, and migration regimes.

Abstract

We present a theoretical framework for investigating a two-planet system undergoing convergent type I migration in a protoplanetary disk. Our study identifies the conditions for resonant capture and subsequent dynamical stability. By deriving analytical criteria for general $j$:$j-1$ first-order mean-motion resonances (MMRs) applicable to planet pairs with arbitrary mass ratios, we validate these predictions through N-body simulations. The key results are demonstrated in $τ_{\rm m}$-$τ_{\rm m}/τ_{e}$ plots, where $τ_{\rm m}$ and $τ_{e}$ are the timescales of the angular momentum and eccentricity damping, respectively. Specifically, we determine which combinations of orbital damping timescales allow for capture into resonance, showing that too fast migration or too strong eccentricity damping inhibit successful capture. After capture, the subsequent evolution can be classified into three regimes: stable trap, overstable trap and escape. Importantly, resonant capture always remains stable when the inner planet significantly outweighs the outer one. In contrast, when the mass of the inner planet is lower than or comparable to that of the outer planet, the system transitions from the stable to overstable trap, and eventually escapes the resonance, as the relative strength of eccentricity damping to migration ($τ_{\rm m}/τ_{e}$) decreases.

Figures (12)

• Figure 1: Resonance crossing of a two-planet system with convergent migration. The plot shows the time evolution of (a) the outer-to-inner planet period ratio, (b) the planet eccentricities, (c) the resonant angle, and (d) the phase curve. The planet and disk parameters are $m_{\rm i}{=}1\ M_{\oplus}$, $m_{\rm o}{=}10\ M_{\oplus}$, $\tau_{\rm m} {= }2.2 \times 10^5$ yr and $\tau_{\rm m} / \tau_{e} {=} 3 \times 10^3$. The planets directly cross the $2$:$1$ resonance at $t{\simeq}6\times 10^{3}$ yr, while the inner planet's eccentricity briefly excites to $0.01$ before rapid decay. During the crossing, the resonant angle of the inner planet $\varphi_{\rm i}$ jumps from $0$ to $\pi$ and subsequently oscillates with growing amplitude. In phase space, the planet’s trajectory is not captured by the resonant equilibrium. Instead, it circulates and eventually evolves toward the origin, corresponding to a low-eccentricity, non-resonant state. See https://github.com/llh-astro/resonance/raw/main/gif/no_trap.gif
• Figure 2: Stable resonant capture of a two-planet system with convergent migration. The planet and disk parameters are $m_{\rm i}{=}1\ M_{\oplus}$, $m_{\rm o}{=}10\ M_{\oplus}$, $\tau_{\rm m} {= }2\ \times 10^5$ yr and $\tau_{\rm m} / \tau_{e} {= }1200$. The planets get trapped in a $2$:$1$ MMR at $t{\simeq}10^{4}$ yr exciting the inner planet’s eccentricity to $0.02$. The resonant angle $\varphi_{\rm i}$ shifts from $0$ to $2\pi/3$ as the system transitions into the resonant state. It gradually decreases as convergent migration strengthens the resonant interaction, causing the libration center to drift slowly. In phase space, the trajectory converges toward the resonant equilibrium and co-evolves with it. We refer to this resonant capture at a stationary equilibrium as a stable trap. See https://github.com/llh-astro/resonance/raw/main/gif/stable_trap.gif
• Figure 3: Overstable resonance capture of a two-planet system with convergent migration. The planet and disk parameters are $m_{\rm i}{=}1\ M_{\oplus}$, $m_{\rm o}{=}10\ M_{\oplus}$, $\tau_{\rm m} {= }8 \times 10^5$ yr and $\tau_{\rm m} / \tau_{e} {= }10^{3}$. By $t{\simeq}3\times 10^{4}$ yr, the planets become trapped in a $2$:$1$ MMR. The inner planet's eccentricity rapidly excites to $0.02$, followed by oscillations with significant amplitude, while the resonant angle exhibits similar behavior. In phase space, the trajectory initially evolves toward a resonant equilibrium point (black dot) before stabilizing into a limit cycle around this point. We term this phenomenon$-$where the system remains trapped by librating around an equilibrium with the finite amplitude$-$an overstable trap. See https://github.com/llh-astro/resonance/raw/main/gif/overstable_trap.gif
• Figure 4: Level curves of the Hamiltonian (Eq. \ref{['H']}) for different $\eta$, plotted in the phase space of the conjugate variables $X{=}\sqrt{\Phi}\cos{\varphi}$ and $Y{=}\sqrt{\Phi}\sin{\varphi}$. The black (red) dots mark the stable (unstable) fixed points, the thick black line in panel (b) marks the separatrix. Each curve represents the conserved Hamiltonian in the absence of dissipitive disk forces. The color of the lines represents the values of the Hamiltonian.
• Figure 5: Escape from resonance in a two-planet system with convergent migration. The planet and disk parameters are $m_{\rm i}{=}1\ M_{\oplus}$, $m_{\rm o}{=}10\ M_{\oplus}$, $\tau_{\rm m} {= }5 \times 10^5$ yr and $\tau_{\rm m} / \tau_{e} {= }200$. The system is temporarily trapped in a $2$:$1$ MMR during $t{\sim}2{-3}\times 10^{4}$ yr, and the inner planet reaches the equilibrium eccentricity with growing amplitude. The system escapes from resonance afterward, with $e_{\rm i}$ quickly damping to a very low value. Similarly, $\varphi_{\rm i}$ features an increasing libration amplitude during the trap and eventually oscillates from $0$ to $2 \pi$. In phase space, the trajectory initially librates around the right fixed point (black dot). As the amplitude grows, it ultimately escapes from the fixed point and settles into the origin. See https://github.com/llh-astro/resonance/raw/main/gif/escape.gif