Closed-form ab initio solutions of geometric albedos and reflected light phase curves of exoplanets

Studying the albedos of the planets and moons of the Solar System dates back at least a century1–4. Of particular interest is the relationship between the albedo measured at superior conjunction, known as the ‘geometric albedo’, and the albedo considered over all orbital phase angles, known as the ‘spherical albedo’2,5,6. Determining the relationship between the geometric and spherical albedos usually involves complex numerical calculations7–11, and closed-form solutions are restricted to simple reflection laws12,13. Here we report the discovery of closed-form solutions for the geometric albedo and integral phase function, which apply to any law of reflection that only depends on the scattering angle. The shape of a reflected light phase curve, quantified by the integral phase function, and the secondary eclipse depth, quantified by the geometric albedo, may now be self-consistently inverted to retrieve globally averaged physical parameters. Fully Bayesian phase-curve inversions for reflectance maps and simultaneous light-curve detrending may now be performed due to the efficiency of computation. Demonstrating these innovations for the hot Jupiter Kepler-7b, we infer a geometric albedo of $$0.2{5}_{-0.02}^{+0.01}$$ , a phase integral of 1.77 ± 0.07, a spherical albedo of $$0.4{4}_{-0.03}^{+0.02}$$ and a scattering asymmetry factor of $$0.0{7}_{-0.11}^{+0.12}$$ . An ab initio closed-form analytical solution for the geometric albedo and the integral phase function, which is valid for any law of reflection provided that it depends only on the scattering angle, is proposed. Such solutions can be applied to determine the scattering properties of planetary atmospheres or surfaces.