Global Mean-Motion Resonances: Part I—An Exceptional Multiplanetary Resonant Chain in TOI-270 and an Exact Laplace-like Resonance in HD 110067

Super-Earth b and sub-Neptunes c and d are orbiting about the M3.0V dwarf TOI-270 in that order from the star. Their global resonant chain (3:5, 1:1, 2:1) is extremely surprising because planet d appears to be the only known planet occupying the 2:1 resonant orbit without participating in a Laplace resonance (LR) or another planet intervening between the 1:1 and 2:1 orbits as in HD 110067. We do not believe that TOI-270 d is an exception to the empirical rule calling for 2:1 vacancy except in 1:2:4 LRs and Laplace-like 2:3:4 chains. Instead, a LR might exist in this system, and we searched (to no avail) the TESS light curves of TOI-270 for hints of an outer planet that would complete the LR chain. Alternative explanations would be an unknown planet more massive than planet c (Mc=6.20M⊕) establishing the actual 1:1 orbit, or planet b residing in the 1:2 Laplace orbit with a period shorter by 0.53 days. However, these possibilities are ruled out by current data. This leaves only one other option to explore: the observed orbits could be in a stable 35:1:2 resonant chain. Preliminary calculations do not preclude this possibility that should be investigated further by numerical orbit integrations. To this end, we determine two potentially resonant angles, φ and φ^, related via the Laplace phase φL by φ^=φL+2φ. In contrast, HD 110067 is shown to have planets d-e-f in a Laplace-like 1:32:2 resonance with phase φ=2φL precisely.