The Orbital Stability of Planets Trapped in the First-Order Mean-Motion Resonances

TL;DR

This study investigates the stability of planets in first-order mean-motion resonances, identifying a critical number of planets beyond which orbital instability occurs after gas depletion, explaining the diversity of observed super-Earth systems.

Contribution

It determines the critical number of resonantly trapped planets for stability and explores how this number varies with orbital separation and planetary mass, providing insights into planetary system diversity.

Findings

Critical number of planets for stability depends on resonance and system parameters.

Orbital instability occurs when the number exceeds the critical threshold.

Sharp transition in stability explains diversity in observed super-Earth systems.

Abstract

Many extrasolar planetary systems containing multiple super-Earths have been discovered. N-body simulations taking into account standard type-I planetary migration suggest that protoplanets are captured into mean-motion resonant orbits near the inner disk edge at which the migration is halted. Previous N-body simulations suggested that orbital stability of the resonant systems depends on number of the captured planets. In the unstable case, through close scattering and merging between planets, non-resonant multiple systems are finally formed. In this paper, we investigate the critical number of the resonantly trapped planets beyond which orbital instability occurs after disk gas depletion. We find that when the total number of planets ($N$) is larger than the critical number ($N_{\rm crit}$), crossing time that is a timescale of initiation of the orbital instability is similar to non-resonant cases, while the orbital instability never occurs within the orbital calculation time ($10^8$ Kepler time) for $N\leq N_{\rm crit}$. Thus, the transition of crossing time across the critical number is drastic. When all the planets are trapped in 7:6 resonance of adjacent pairs, $N_{\rm crit} = 4$. We examine the dependence of the critical number of 4:3, 6:5 and 8:7 resonance by changing the orbital separation in mutual Hill radii and planetary mass. The critical number increases with increasing the orbital separation in mutual Hill radii with fixed planetary mass and increases with increasing planetary mass with fixed the orbital separation in mutual Hill radii. We also calculate the case of a system which is not composed of the same resonance. The sharp transition of the stability can be responsible for the diversity of multiple super-Earths (non-resonant or resonant), that is being revealed by $Kepler$ mission.