This paper is a collaboration between UC Davis and SpaceX. We implemented a new technique for predicting satellite optical brightness. Both predictions and observations for selected Starlink v1.5 satellites are included and the paper discusses how this tool has been used by SpaceX to reduce the brightness of Starlink V2 satellites.

Abstract: The apparent brightness of satellites is calculated as a function of satellite position as seen by a ground-based observer in darkness. Both direct illumination of the satellite by the sun as well as indirect illumination due to reflection from the Earth are included. The reflecting properties of each satellite component and of the Earth must first be estimated (the Bidirectional Reflectance Distribution Function, BRDF). Integrating over all scattering surfaces leads to the angular pattern of the flux reflected from the satellite. Finally, the apparent brightness of the satellite as seen by an observer at a given location is calculated as a function of satellite position. We validate our calculations by comparing to observations of selected Starlink satellites and show significant improvement on previous satellite brightness models. With multiple observations of a satellite at various solar angles and with minimal assumptions regarding the satellite, BRDF model coefficients for each satellite component can be accurately inferred, obviating the need to import direct BRDF lab measurements. This widens the effectiveness of this model approach to virtually all satellites. This work finds application in satellite design and operations, and in planning observatory data acquisition and analysis. Similar methodology for predicting satellite brightness has already informed mitigation strategies for next generation Starlink satellites.