Formation and disruption of resonant chains of super-Earths: Secular perturbations from outer eccentric embryos

TL;DR

This paper investigates how resonant chains of inner super-Earths form and why they often break after disk dispersal. By simulating growth and migration of embryos in a 1 au ring and contrasting fast- and slow-migration disk models, it shows that resonant chains form readily and can destabilize on ~100 Myr timescales when outer eccentric embryos are present. The authors develop a secular perturbation framework and validate it with N-body tests, identifying outer-eccentricity, outer-mass, and orbital distance as key factors for breaking resonances, aligning the final period-ratio distributions with observations. This provides a plausible mechanism for why older planetary systems commonly lie away from exact resonances.

Abstract

Recent observations have revealed the distribution of orbital period ratios of adjacent planets in multiple super-Earth systems and how these distributions change with time. The aim of this study is to clarify under what conditions the observed features of orbital period ratios of super-Earths can be explained, and to identify what causes the dynamical instability of super-Earths captured into resonant chains. We perform N-body simulations for 100 Myr that follow the formation and orbital evolution of super-Earths originating from a ring of planetary embryos at 1 au from the star. The simulations show that super-Earths undergo inward migration in the disk and are captured into mean-motion resonances with their neighbors. As a result, several resonant pairs form a resonant chain. After disk dispersal, some of these chains become dynamically unstable. In such cases, the final distribution of orbital period ratios and their time evolution can be consistent with recent observations. The instabilities of resonant chains are likely triggered by secular perturbations from embryos that remain on outer orbits beyond 1 au, indicating that not only giant planets but also small embryos can disrupt the resonances among inner super-Earths. We therefore further investigate the secular perturbations from outer embryos using analytic formulas and additional orbital calculations. We discuss the conditions required to excite the eccentricities of inner super-Earths on a timescale of about 100 Myr. These conditions include the need for large eccentricities of the outer embryos, as well as constraints on their masses and semimajor axes.

Figures (8)

• Figure 1: Gas surface density profiles and their temporal evolution for two disk models. The solid line represents the two-component power-law model with an inner edge at $r = 0.1\,\mathrm{au}$, referred to as the rapid migration case. The dashed line represents the model evolving under the influence of disk winds, referred to as the slow migration case.
• Figure 2: Simulation results for the fast migration case. (a)–(c) Time evolution of the semimajor axes. Panels (a), (b), and (c) correspond to simulations with initial embryo masses of (a) $0.05\,M_{\oplus}$, (b) $0.1\,M_{\oplus}$, and (c) $0.2\,M_{\oplus}$, respectively. (d)–(f) Summary of ten simulation runs for each setup. (d) Cumulative distribution of orbital period ratios of super-Earth pairs at $t = 100\,\mathrm{Myr}$. Vertical lines indicate the locations of mean-motion resonances, and the red line shows the distribution from Kepler observations. (e) Cumulative distribution of the offset $\Delta$ from the nearest integer period ratio for super-Earth pairs at $t = 100\,\mathrm{Myr}$. Vertical lines mark $\Delta = -0.015$ and $\Delta = 0.03$. (f) Fraction of super-Earth pairs in mean-motion resonances at $t = 3$, 10, 30, and $100\,\mathrm{Myr}$.
• Figure 3: Same as Figure \ref{['fig:100Myr_fast']}, but showing the results for the slow migration case.
• Figure 4: Properties of the systems at $t = 3\,\mathrm{Myr}$. (a) Cumulative distribution of the orbital period ratios of adjacent super-Earth pairs. (b) Cumulative distribution of the deviations from exact commensurabilities for adjacent super-Earth pairs. (c) Cumulative distribution of the number of super-Earths per system. (d) Cumulative distribution of the total mass of embryos that remain on outer orbits with $a > 1\,\mathrm{au}$.
• Figure 5: (a) Plot of the coefficient $C_{12} (=e_{1,\max}/e_{20})$ from Equation (\ref{['eq:e1max']}), evaluated using Equations (\ref{['eq:A11']})–(\ref{['eq:A22']}). (b) Approximate secular timescale obtained from Equation (\ref{['eq:Tsec']}), assuming $a_1 = 0.3\,\mathrm{au}$ and $M_1 = 3\,M_\oplus$.