Higher-Order Mean-Motion Resonances Can Form in Type-I Disk Migration

Abstract

Type-I disk migration can form a chain of planets engaged in first-order mean-motion resonances (MMRs) parked at the disk inner edge. However, while second- or even third-order resonances were deemed unlikely due to their weaker strength, they have been observed in some planetary systems (e.g. TOI-178 bc: 5:3, TOI-1136 ef: 7:5, TRAPPIST-1 bcd: 8:5-5:3). We performed $>6,000$ Type-I simulations of multi-planet systems that mimic the observed {\it Kepler} sample in terms of stellar mass, planet size, multiplicity, and intra-system uniformity over a parameter space encompassing transitional and truncated disks. We found that Type-I migration coupled with a disk inner edge can indeed produce second- and third-order resonances (in a state of libration) in $\sim 10\%$ and 2\% of resonant-chain systems, respectively. Moreover, the relative occurrence of first- and second-order MMRs in our simulations is consistent with observations (e.g. 3:2 is more common than 2:1; while second-order 5:3 is more common than 7:5). The formation of higher-order MMRs favors slower disk migration and a smaller outer planet mass. Higher-order resonances do not have to form with the help of a Laplace-like three-body resonance as was proposed for TRAPPIST-1. Instead, the formation of higher-order resonance is assisted by breaking a pre-existing first-order resonance, which generates small but non-zero initial eccentricities ($e\approx10^{-3}$ to 10$^{-2}$). We predict that 1) librating higher-order resonances have higher equilibrium $e$ ($\sim 0.1$); 2) are more likely found as an isolated pair in an otherwise first-order chain; 3) more likely emerge in the inner pairs of a chain.