Analytic Light Curves for Mutual Transits of Two Bodies

We present a solution for the light curve of two bodies mutually transiting a star with polynomial limb darkening, where a “mutual transit” is any transit in which overlap occurs between the two bodies during their transit of a third body. The two transiting bodies could be two planets, an eclipsing binary and a planet, two stars eclipsing a third, or an exoplanet with an exomoon companion. We include analytic derivatives of the light curve with respect to the positions and radii of both bodies. Our solution is, to our knowledge, the first analytic solution written specifically for the mutual transit of two bodies, although a general solution for any number of bodies has been described by (Pál, 2012) which does not include derivatives. We couple our solution to a Kepler solver to implement a photo dynamical model for mutual transits. We include dynamics for hierarchical systems in which a secondary body orbits a larger primary (i.e. an exomoon system) and confocal systems in which two bodies independently orbit a central mass (i.e. two planets in widely separated orbits). Our code is fast enough to enable inference with a variety of MCMC algorithms, and the inclusion of derivatives allows us to make use of gradient-based inference methods such as Hamiltonian MCMC. While applicable to a variety of systems including compact multi-planet systems and transiting triply-eclipsing stars, this work was undertaken primarily with exomoons in mind. It is our hope that by making this code publicly available we may reduce barriers for the community to assess the detectability of exomoons with current and future instruments, conduct searches for exomoons, and attempt to validate existing exomoon candidates. We also anticipate that our code will be useful for studies of planet-planet transits in exoplanetary systems and eclipses in triple-star systems.