Formation and dynamics of the resonant chain in the trappist-1 exoplanet system

Abstract

The TRAPPIST-1 system consists of seven Earth-sized planets hosted by an 8-Gyr-old very-low mass M-dwarf star where the period ratios of each adjacent planet pairs are close to two-body mean motion resonances. The period ratios are 8:5, 5:3, 3:2, 3:2, 4:3, and 3:2 from the innermost to the outermost planet pair. Earlier observations hint at the presence of three-body resonances between each adjacent planet triplets, rendering this system the longest resonant chain system discovered to date.

This work looks into the dynamics of the TRAPPIST-1 system from the best fit orbital parameters to the system provided by Wang et al. (2017) and Grimm et al. (2018) by means of long-term N-body integrations. More than 80% of the Wang et al. (2017) fits with high angular momentum deficit (AMD) became unstable in less than 100 Myr. For the remaining 20% with low AMD that remained stable for at least 100 Myr, it was found that the resonant arguments involving the longitudes of pericentres of planets d, and f are librating. The fits from Grimm et al. (2018) are stable up to similar timescales due to their lower AMD. The resonant arguments for the first order two-body mean motion resonances involving planets d to h are all librating for the Grimm et al. (2018) best fits. In particular, the resonant arguments involving the longitudes of pericentres of planets e and f exhibit small amplitude librations. Both fits from Wang et al. (2017) and Grimm et al. (2018) suggest that the inner planets (planets b to d) are not in two-body mean motion resonances.

With regards to the three-body resonances, the fits tell a different story: for the Wang et al. (2017) fits, they are disrupted by 120 kyr; the Grimm et al. (2018) fits indicate that three-body resonances involving planets d to h survive the maximum integration time albeit with alternating libration centres. It was also found that three-body resonances could exist independently of two-body mean motion resonances. However, the three-body resonances play a minor role in the dynamics of the TRAPPIST-1 system given the presence of first order two-body mean motion resonances among the planets.

Results from simulations modelling the migration and subsequent capture of the TRAPPIST-1 planets into a chain of two-body mean motion resonances with period ratios matching the values from the observations show that the planets in general have lower equilibrium eccentricities compared to the time-average eccentricities from the fits. The planets in the TRAPPIST-1 system are likely to have low eccentricities as reported by Grimm et al. (2018) but the planet pairs may not be deep in two-body resonances given their observed eccentricities.