Long-term tidal evolution of the TRAPPIST-1 system

The ultracool M-dwarf star TRAPPIST-1 is surrounded by seven planets configured in a resonant chain. Transit-timing variations have shown that the planets are caught in multiple three-body resonances and that their orbits are slightly eccentric, probably caused by resonant forcing. The current values of the eccentricities could be a remnant from their formation. Here, we run numerical simulations using fictitious forces of trapping the fully grown planets in resonances as they migrated in the gas disc, followed by numerical simulations detailing their tidal evolution. For a reduced disc scale height h similar to 0.03-0.05, the eccentricities of the planets upon capture in resonance are higher than their current values by factors of a few. We show that the current eccentricities and spacing of planets d to h are natural outcomes of coupled tidal evolution wherein the planets simultaneously damp their eccentricities and separate due to their resonant interaction. We further show that the planets evolve along a set of equilibrium curves in semimajor axis-eccentricity phase space that are defined by the resonances, and that conserve angular momentum. As such, the current 8:5-5:3-(3:2)(2)-4:3-3:2 resonant configuration cannot be reproduced from a primordial (3:2)(4)-4:3-3:2 resonant configuration from tidal dissipation in the planets alone. We use our simulations to constrain the long-term tidal parameters k(2)/Q for planets b to e, which are in the range of 10(-3) to 10(-2), and show that these are mostly consistent with those obtained from interior modelling following reasonable assumptions.