#format jsmath
= 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/m^2^ onto a sphere of radius R, the light out has an intensity of I R^2^/4 W/steradian in all directions.

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