Diffuse and specular brightness models applied to LEO satellites. Case study: The ONEWEB constellation

Context. To better understand the observed brightness of low Earth orbit satellites, we must characterize their reflectivity, which in turn depends importantly on their bus designs. The reflectivity of a body can be described by Lambert's law, in terms of its albedo, cross-sectional area, range (distance), phase angles, and the mixing coefficient between diffuse and specular reflection components. Aims. We aim to analyze the reflectivity of more than 300 ONEWEB satellites using the diffuse Lambertian sphere, diffuse and specular Lambertian sphere, and the relative reflectance brightness models. Methods. Astrometric and photometric measurements, plus two-line elements (TLE) orbital information were used to compute the apparent and range-magnitude, as well as the relevant angles related to the orientation of the Sun, the satellites, and the observer. A differential evolution Monte Carlo algorithm was used to obtain each model's parameters that best fit the data. Results. All models can fit the mean observed brightness of the satellites but cannot describe the observed phase-angle-dependent brightness modulations. The residuals in all cases have a standard deviation of $\sim$0.6 magnitudes, while the observational photometric errors are on average $\sim$0.2 magnitudes. Conclusions. The studied brightness models, which depend on the relative Sun-body-observer position but are independent of the specific orientation of the reflecting body surface(s) with respect to the observer, cannot entirely explain the observed brightness of the ONEWEB constellation satellites. Accounting for the real shape and the changing attitude of the satellite, as well as the effect of Earth's albedo is needed to better explain satellite photometric observations