  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 the
satellite components and of the Earth must first be estimated (the
Bidirectional Reflectance Distribution Function, or BRDF). The reflecting
properties of the satellite components can be found directly using lab
measurements or accurately inferred from multiple observations of a satellite
at various solar angles. Integrating over all scattering surfaces leads to the
angular pattern of flux 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 develop an improved model for reflection of
light from Earth's surface using aircraft data. We find that indirectly
reflected light from Earth's surface contributes significant increases in
apparent satellite brightness. This effect is particularly strong during civil
twilight. We validate our approach by comparing our calculations to multiple
observations of selected Starlink satellites and show significant improvement
on previous satellite brightness models. Similar methodology for predicting
satellite brightness has already informed mitigation strategies for
next-generation Starlink satellites. Measurements of satellite brightness over
a variety of solar angles widens the effectiveness of our approach to virtually
all satellites. We demonstrate that an empirical model in which reflecting
functions of the chassis and the solar panels are fit to observed satellite
data performs very well. This work finds application in satellite design and
operations, and in planning observatory data acquisition and analysis.
