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

Abstract

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

Figures (9)

• Figure 1: Distribution of stationary magnitudes of all ONEWEB satellites studied. The sample is divided by filter and orbit type. There are 451 on-station satellites, of which 410 were observed with the V filter and 41 with the R filter. There are 175 off-station satellites observed 158 with the V filter and 17 with the R filter. The average $\mu$ and standard deviation $\sigma$ of $m_{s}$ are shown in the legend.
• Figure 2: Graphical representation of the position of the satellite and definitions of the Sun incidence angle $\theta_{0}$ (in orange), the observer angle $\theta_{1}$ (in dark blue), the Sun phase angle $\Phi$ (in green), and the solar elongation angle $\gamma$ (in brown). Notice that $\theta_{0}$, $\theta_{1}$ and $\Phi$ are not contained in the same plane but form a triangular pyramid. The distance is not to scale.
• Figure 3: Histogram of $\sigma_\text{tle}$ for all the ONEWEB satellites studied, showing the difference between the TLE-predicted and the observed equatorial coordinated of satellites.
• Figure 4: Horizontal coordinates of the ONEWEB LEOsats and the Sun projected on a polar plane. North and east are at $0^\circ$ and $90^\circ$, respectively. Both visible and non-visible sky are projected onto the same plane. The Sun is always below the horizon. Left panel: LEOsats range-corrected magnitude $H^{1200}$ distribution. The larger the symbol, the brighter the satellite. Center panel: LEOsats diffuse sphere effective albedo distribution. The larger the symbol, the larger the effective albedo $\rho_{\text{eff,ds}}$. Right panel: Same as center panel, but for the half-specular sphere effective albedo $\rho_{\text{eff,dss}}$.
• Figure 5: Results of the RR model applied to all the ONEWEB LEOsats of this study. Left and center: Observed $H^{1200}$ (gray points) and DSS model estimates $H^{1200}_\text{rr}$ (purple points) vs. $\theta_0\,$ and $\,\theta_1$, respectively. Right: Histogram of residuals $H^{1200}-H^{1200}_\text{rr}$.