A Six-planet Resonance Chain in K2-138?

Abstract

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 nf/ng ∼ 3.3 with its closest neighbor. We show that the orbit of mg is actually consistent with the first-order three-planet resonance (3P-MMR) characterized by the relation 2ne − 4nf + 3ng = 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 nf/ng 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 nf/ng, 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 mg and mf, and it is possible that the same effect may also be found in other compact planetary systems.