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

TL;DR

This paper presents a fast, accurate, and differentiable method for modeling reflected light phase curves and occultations of spherical planets and moons, enabling advanced exoplanet surface mapping.

Contribution

It introduces a novel closed-form solution for reflected light flux that is efficient, stable, and extendable to complex scattering scenarios, integrated into an open-source framework.

Findings

Algorithm is 4-5 orders faster than numerical methods.

Achieves about 10 orders of magnitude higher precision.

Enables potential mapping of exoplanet surfaces.

Abstract

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