Posted by Anonymous on Wednesday, December 24, 2008 at 4:50pm.
What is the formula to calculate the illumination fraction of the Moon?

Math/Science/Astronomy  drwls, Wednesday, December 24, 2008 at 6:46pm
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. 
Math/Science/Astronomy  drwls, Wednesday, December 24, 2008 at 7:07pm
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