Posted by Anonymous on Wednesday, December 24, 2008 at 4:50pm.
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 sun-earth-moon 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://the-moon.wikispaces.com/phase
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