Global Mean-Motion Resonances: Part I—An Exceptional Multiplanetary Resonant Chain in TOI-270 and an Exact Laplace-like Resonance in HD 110067 - Dimitris M. Christodoulou; Nicholas M. Sorabella; Sayantan Bhattacharya; Silas G. T. Laycock; Demosthenes Kazanas (Galaxies)
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 (<semantics>Mc=6.20<mspace width="0.0pt"></mspace>M⊕</semantics>) 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 <semantics><mstyle scriptlevel="0" displaystyle="true">35</mstyle></semantics>: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, <semantics>φ</semantics> and <semantics><mover accent="true">φ^</mover></semantics>, related via the Laplace phase <semantics>φL</semantics> by <semantics><mover accent="true">φ^</mover>=φL+2<mspace width="0.0pt"></mspace>φ</semantics>. In contrast, HD 110067 is shown to have planets d-e-f in a Laplace-like 1:<semantics><mstyle scriptlevel="0" displaystyle="true">32</mstyle></semantics>:2 resonance with phase <semantics>φ=2φL</semantics> precisely.