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We derive efficient, closed form, differentiable, and numerically stablesolutions for the flux measured from a spherical planet or moon seen inreflected light, either in or out of occultation. Our expressions apply to thecomputation of scattered light phase curves of exoplanets, secondary eclipselight curves in the optical, or future measurements of planet-moon andplanet-planet occultations, as well as to photometry of solar system bodies. Wederive our solutions for Lambertian bodies illuminated by a point source, butextend them to model illumination sources of finite angular size and roughsurfaces with phase-dependent scattering. Our algorithm is implemented inPython within the open-source starry mapping framework and is designed withefficient gradient-based inference in mind. The algorithm is 4-5 orders ofmagnitude faster than direct numerical evaluation methods and about 10 ordersof magnitude more precise. We show how the techniques developed here may oneday lead to the construction of two-dimensional maps of terrestrial planetsurfaces, potentially enabling the detection of continents and oceans onexoplanets in the habitable zone.

Analytic Light Curves in Reflected Light: Phase Curves, Occultations, and Non-Lambertian Scattering for Spherical Planets and Moons