2023ApJ...954...57C

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 g) is significantly detached, having a mean-motion ratio of n_f/n_g ∼ 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 has been found in a multiplanet system and tied to its current dynamical structure. Adequate values for the masses allow one to trace the dynamical history of the system from an initial capture in a six-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 semimajor 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 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.