Dark Fraction of the Moon

phase angle ϕ

The fraction of the Moon that is dark is a function of the phase angle

ϕ

. This function is

cos

2

ϕ

2

and can be found by taking an area integral over the circle. A sphere with one hemisphere black, the other hemisphere white, and rotated by

ϕ

is shown in the top-left corner. The area integral being performed is shown in the top-right corner.