paper

Explicit relations and criteria for eclipses, transits and occultations

arXiv:1811.05484 · doi:10.1093/mnras/sty3104

Abstract

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 generalise the results of Veras & Breedt (2017) by (i) characterising three-body systems which 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 which 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.

Accepted for publication in MNRAS

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