Math/Science/Astronomy
posted by Anonymous .
What is the formula to calculate the illumination fraction of the Moon?

Without going into a lengthy derivation, I would assume a sinusoidal variation would be a good approximation. The period between new monns in 27.53 days. If T is the time since the previous new moon,
Illumination fraction =
0.5  0.5 cos (2 pi T/27.53)
The angle in parentheses is in radians.
A more accurate formula would have to consider the variation in the sunearthmoon angle, which is not strictly sinusoidal with time due to the eccentricity and inclination of the moon's orbit. 
You will find a better formula here, in terms of the "Phase Angle" between the sun and the moon, as see from earth. you will need an ephemeris (or a measurement) to get that angle.
http://themoon.wikispaces.com/phase