Wouldn't brightness have a cube root relationship to energy, because it radiates into space? No. The light actually expands in a 2-D wave. So it falls off as

Wouldn't brightness have a cube root relationship to energy, because it radiates into space? No. The light actually expands in a 2-D wave. So it falls off as the square of distance. But if we consider 2 sources near each other, they both spread out the same amount by the time they get to us. Therefore there is a linear relationship between energy and apparent brightness. (On top of the inverse square relationship to distance.) Hopefully this helps!