A Six-Planet Resonance Chain in K2-138?

The K2-138 system hosts six planets and presents an interesting case study due to its distinctive dynamical structure. Its five inner planets are near a chain of 3/2 two-body mean-motion resonances, while the outermost body (planet {\it g}) is significantly detached, having a mean-motion ratio of $n_f/n_g \sim 3.3$ with its closest neighbor. We show that the orbit of $m_g$ is actually consistent with the first-order three-planet resonance (3P-MMR) characterized by the relation $2n_e - 4n_f + 3n_g = 0$ and is the first time a pure first-order 3P-MMR is found in a multi-planet system and tied to its current dynamical structure. Adequate values for the masses allow to trace the dynamical history of the system from an initial capture in a 6-planet chain (with $n_f/n_g$ in a 3/1 resonance), up to its current configuration due to tidal interactions over the age of the star. The increase in resonance offset with semi-major axis, as well as its large value for $n_f/n_g$ can be explained by the slopes of the pure three-planet resonances in the mean-motion ratio plane. The triplets slide outward over these curves when the innermost pair is pulled apart by tidal effects, in a \textit{pantograph-}like manner. The capture into the 3P-MMR is found to be surprisingly robust given similar masses for $m_g$ and $m_f$, and it is possible that the same effect may also be found in other compact planetary systems.