Resonance capture and dynamics of three-planet systems

We present a series of dynamical maps for fictitious three-planet systems in initially circular coplanar orbits. These maps have unveiled a rich resonant structure involving two or three planets, as well as indicating possible migration routes from secular to double resonances or pure three-planet commensurabilities. These structures are then compared to the present-day orbital architecture of observed resonant chains. In a second part of the paper, we describe N-body simulations of type-I migration. Depending on the orbital decay time-scale, we show that three-planet systems may be trapped in different combinations of independent commensurabilities: (i) double resonances, (ii) intersection between a two-planet and a first-order three-planet resonances, and (iii) simultaneous libration in two first-order three-planet resonances. These latter outcomes are found for slow migrations, while double resonances are almost always the final outcome in high-density discs. Finally, we discuss an application to the TRAPPIST-1 system. We find that, for low migration rates and planetary masses of the order of the estimated values, most three-planet sub-systems are able to reach the observed double resonances after following evolutionary routes defined by pure three-planet resonances. The final orbital configuration shows resonance offsets comparable with present-day values without the need of tidal dissipation. For the 8/5 resonance proposed to dominate the dynamics of the two inner planets, we find little evidence of its dynamical significance; instead, we propose that this relation between mean motions could be a consequence of the interaction between a pure three-planet resonance and a two-planet commensurability between planets c and d.