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

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