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

TL;DR

The paper addresses whether higher-order mean-motion resonances can form during Type-I disk migration in Kepler-like planetary systems. Using a large ensemble of N-body simulations with an inner disk edge and observations-informed initial conditions, the authors identify two- and three-body resonances via librating angles and classify resonant chains. They find second- and third-order two-body MMRs form in roughly $10 ext{ extpm}1$% and $2 ext{ extpm}0.5$ ext% of resonant chains, respectively, and these occurrences align with observed resonant statistics; slower migration and specific mass configurations favor higher-order captures, often seeded by eccentricity excitation from a prior first-order resonance. Importantly, higher-order resonances can arise without requiring a Laplace-like three-body resonance, and the work makes testable predictions about their eccentricities and the tendency to occur in inner-chain pairs, informing interpretations of systems such as TOI-178, TOI-1136, and TRAPPIST-1.

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.

Figures (13)

• Figure 1: Cumulative distribution functions (CDFs) of stellar mass, radius ratio and period ratio between neighboring planets. The black curves show the $239$ confirmed multi-planet ($N_p\geq 3$) systems from the NASA Exoplanet Archive. In blue and red, we display the corresponding values from our simulations. Note that the period ratios are initial period ratios. The red curves denote systems that experienced some close encounters and are hence discarded (see Section \ref{['sec: 2.4']}). The blue curves represent the systems that completed disk migration.
• Figure 2: Planet multiplicities for observed systems (black), systems that experienced close encounters (red), and systems that complete disk migration with no close encounters (blue). Note that observed multiplicity is smaller than the actual multiplicity. Section \ref{['sec: 2.4']} explains why close encounters are more common in higher-multiplicity systems. The blue systems were used in our subsequent analysis.
• Figure 3: More than 70% of our simulations were discarded due to having close encounters. Among the remaining systems: $\sim 5\%$ and $\sim 0.5\%$ of resonant planet pairs were in librating second- or third-order resonances; $\sim 13\%$ and $\sim 1\%$ of planetary systems classified as three-body resonant chains and two-body resonant chains contain at least one second-order or third-order resonance.
• Figure 4: A representative set of nine systems at the end of our simulations. We label each resonant pair of planets with its integer ratio and deviation $\Delta$. Colors indicate the order of resonance: nonresonant in grey, first-order in black, second-order in blue, and third-order in red. Librating three-body resonant angles are shown in orange. Two case study systems in Section \ref{['sec: 3']} are indicated with check marks. The top three systems are 'Partial Resonant Chains' (at least one pair not in MMR); the middle three are 'Two-Body Resonant Chains'; the bottom three are 'Three-Body Resonant Chains.' In our simulations, low-mass planets occasionally pass through the disk's inner edge (see Section \ref{['sec: 2.1']}).
• Figure 5: A subset of observed resonant chains plotted like the simulated systems shown in Fig. \ref{['fig: 4']}. Unlike Fig. \ref{['fig: 4']}, these observed systems are displayed with respect to period to reduce whitespace. In most example observed systems, it is unclear if the two-body resonant angles librate. We choose to mark the most proximal MMRs. Librating triplets are indicated in accordance with observation.