Reflection From a Sphere

Assume a distant source uniformly illuminates a mirror sphere with a radius of 1 unit. What is the brightness per steradian of the reflection at angle \theta from the light source?

If we map angular space around the sphere in spherical coordinates, with \theta as the angle of the reflection from the sphere, then the place were it reflects on that sphere is a ring with a radius 2\pi~\sin(\theta/2) and a width d\theta/2. Projected towards the light source, that makes an annular ring with width \cos(\theta/2)d\theta/2. So the total light collected is

light in = 2\pi~\sin(\theta/2)\cos(\theta/2)d\theta/2~=~2\pi\sin(\theta)d\theta/4.

That is projected onto the "sky sphere", a ring with diameter 2\pi~\sin(\theta) and a width of d\theta, for an angular area of 2\pi~\sin(\theta)d\theta. The intensity in that ring is:

intensity( θ ) = \left(2\pi\sin(\theta)d\theta/4\right)~/~\left(2\pi~\sin(\theta)d\theta\right)

intensity( θ ) = 1/4

Amazing! Projecting onto a sphere reflects the light in all directions uniformly. Scaling properly, if the incoming light is intensity I W/m² onto a sphere of radius R, the light out has an intensity of I R²/4 W/steradian in all directions.

A sphere is a good model for an ensemble of thinsats pointed randomly in all directions.