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Although conjunctions and oppositions frequently occur in planetary systems,eclipse-related phenomena are usually described from an Earth-centricperspective. Space missions to different parts of the Solar system, as well asthe mounting number of known exo-planets in habitable zones and the possibilityof sending featherweight robot spacecraft to them, prompt broaderconsiderations. Here, we derive the geometry of eclipses, transits andoccultations from a primarily exo-Earth viewpoint, and apply the formulation tothe Solar system and three types of three-body extrasolar planetary systems:with 1 star and 2 planets (Case I), with 2 stars and 1 planet (Case II), andwith 1 planet, 1 star and 1 moon (Case III). We derive the general conditionsfor total, partial and annular eclipses to occur at exo-syzygy, and implementthem in each case in concert with stability criteria. We then apply theformalism to the TRAPPIST-1, Kepler-444 and Kepler-77 systems -- the first ofwhich contains multiple potentially habitable planets -- and provide referencetables of both Solar system and TRAPPIST-1 syzygy properties. We conclude bydetailing a basic algebraic algorithm which can be used to quickly characterizeeclipse properties in any three-body system.

Eclipse, transit and occultation geometry of planetary systems at exo-syzygy