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