SIMBAD references

Solar system, exoplanet, and stellar science rely on transits, eclipses, and occultations for dynamical and physical insight. Often, the geometry of these configurations are modelled by assuming a particular viewpoint. Here, instead, I derive user-friendly formulae from first principles independent of viewpoint and in three dimensions. I generalize the results of Veras & Breedt by (i) characterizing three-body systems that are in transit but are not necessarily perfectly aligned and by (ii) incorporating motion. For a given snapshot in time, I derive explicit criteria to determine whether a system is in or out of transit, if an eclipse is total or annular, and expressions for the size of the shadow, including their extreme values and a condition for engulfment. These results are exact. For orbital motion, I instead obtain approximate results. By assuming fixed orbits, I derive a single implicit algebraic relation that can be solved to obtain the frequency and duration of transit events - including ingresses and egresses - for combinations of moons, planets, and stars on arbitrarily inclined circular orbits; the eccentric case requires the solution of Kepler's equation but remains algebraic. I prove that a transit shadow - whether umbral, antumbral, or penumbral - takes the shape of a parabolic cylinder, and finally present geometric constraints on Earth-based observers hoping to detect a three-body syzygy (or perfect alignment) - either in extrasolar systems or within the Solar system - potentially as a double annular eclipse.