(Proposal ID) S17B-079 (PI) Narita, Norio (Proposal Title) Monitoring an Overtaking Double-Transit Event in the TRAPPIST-1 System (Abstract) We propose to monitor an overtaking double-transit event of TRAPPIST-1b and TRAPPIST-1e on October 23, 2017 with Keck/MOSFIRE and Gemini-North/GMOS simultaneously in order to detect/constrain the presence of a planet-planet eclipse event. We emphasize that the next overtaking double-transit event is an extremely rare opportunity, and there will be no similar chance for the planet b and e at least within the next decade. A detection or even non- detection of the planet-planet eclipse event will give us an insight into the mutual inclination of the two planets. A resultant value or lower limit of the mutual inclination would become important information when discussing the formation/migration mechanism and the dynamical stability of this interesting planetary system. We also emphasize the necessity of TWO 8-10m class telescopes for the proposed observation, since both high photometric precision and high time-resolution are critically important to detect or constrain the presence/ absence of a planet-planet eclipse event during the overtaking double-transit event, and to avoid any misconception of a planet-planet eclipse event due to systematic effects or a starspot crossing event.