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

## TL;DR

This paper develops a geometric framework for analyzing eclipses, transits, and occultations from an exo-centric perspective, applying it to Solar and exoplanetary systems to understand eclipse conditions and stability.

## Contribution

It introduces a novel exo-Earth viewpoint for eclipse geometry and provides a general algebraic algorithm for characterizing eclipses in three-body systems.

## Key findings

- Derived conditions for total, partial, and annular eclipses at exo-syzygy.
- Applied formalism to Solar system and exoplanetary systems like TRAPPIST-1.
- Provided reference tables for eclipse properties in key systems.

## Abstract

Although conjunctions and oppositions frequently occur in planetary systems, eclipse-related phenomena are usually described from an Earth-centric perspective. Space missions to different parts of the Solar system, as well as the mounting number of known exo-planets in habitable zones and the possibility of sending featherweight robot spacecraft to them, prompt broader considerations. Here, we derive the geometry of eclipses, transits and occultations from a primarily exo-Earth viewpoint, and apply the formulation to the 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), and with 1 planet, 1 star and 1 moon (Case III). We derive the general conditions for total, partial and annular eclipses to occur at exo-syzygy, and implement them in each case in concert with stability criteria. We then apply the formalism to the TRAPPIST-1, Kepler-444 and Kepler-77 systems -- the first of which contains multiple potentially habitable planets -- and provide reference tables of both Solar system and TRAPPIST-1 syzygy properties. We conclude by detailing a basic algebraic algorithm which can be used to quickly characterize eclipse properties in any three-body system.