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

The albedo of a celestial body is the fraction of light reflected by it. Studying the albedos of the planets and moons of the Solar System dates back at least a century. Modern astronomical facilities enable the measurement of geometric albedos from visible/optical secondary eclipses and the inference of the Bond albedo (spherical albedo measured over all wavelengths, weighted by the incident stellar flux) from infrared phase curves of transiting exoplanets. Determining the relationship between the geometric and spherical albedos usually involves complex numerical calculations and closed-form solutions are restricted to simple reflection laws. 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 fundamental physical parameters. Fully-Bayesian phase curve inversions for reflectance maps and simultaneous light curve detrending may now be performed, without the need for binning in time, due to the efficiency of computation. Demonstrating these innovations for the hot Jupiter Kepler-7b, we infer a geometric albedo of $0.19 \pm 0.01$, a Bond albedo of $0.35 \pm 0.03$, a phase integral of $1.84 \pm 0.09$ and a scattering asymmetry factor of $0.18 \pm 0.17$. These ab initio, closed-form solutions enable cloud/haze properties to be retrieved from multi-wavelength phase curves of both gas-giant and terrestrial exoplanets measured by the James Webb Space Telescope.