Closed-formed solutions of geometric albedos and phase curves of exoplanets for any reflection law

The albedo of a celestial body is the frac-tion of light reflected by it. Studying the albe-dos of the planets and moons of the Solar Sys-tem dates back at least a century [1, 2, 3, 4, 5]. Of particular interest is the relationship between the albedo measured at superior conjunction (full phase), known as the “geometric albedo”, and the albedo considered over all phase angles, known as the “spherical albedo” [2, 6, 7]. Modern astronom-ical facilities enable the measurement of geomet-ric albedos from visible/optical secondary eclipses [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and the inference of the Bond albedo (spherical albedo measured over all wavelengths) from in-frared phase curves [21, 22, 23, 24, 25] of transit-ing exoplanets. Determining the relationship be-tween the geometric and spherical or Bond albe-dos usually involves complex numerical calculations [26, 27, 28, 29, 30, 31, 32] and closed-form solu-tions are restricted to simple reflection laws [33, 34]. Here we report the discovery of closed-form solu-tions for the geometric albedo and integral phase function that apply to any law of reflection. The integral phase function is used to obtain the phase integral, which is the ratio of the spherical to the geometric albedos. The generality of the solu-tions stems from a judicious choice of the coor-dinate system in which to perform different parts of the derivation. The closed-formed solutions have profound implications for interpreting obser-vations. The shape of a reflected light phase curve and the secondary eclipse depth may now be self-consistently inverted to retrieve fundamental phys-ical parameters (single-scattering albedo, scatter-ing asymmetry factor, cloud cover). Fully-Bayesian phase curve inversions for reflectance maps and si-multaneous light curve detrending may now be per-formed, without the need for binning in time, due to the efficiency of computation. We demonstrate these innovations for the hot Jupiter Kepler-7b, inferring a revised geometric albedo of 0.12 ± 0.02, a Bond albedo of 0.18 ± 0.03 and a phase integral of 1.5 ± 0.1, which is consistent with isotropic scatter-ing. The scattering asymmetry factor is 0.04±0.15, implying that the aerosols are small compared to the wavelengths probed by the Kepler space tele-scope. In the near future, one may use the closed-form solutions discovered here to extract funda-mental parameters, across wavelength, from multi-wavelength phase curves of both gas-giant and ter-restrial exoplanets measured by the James Webb Space Telescope.