Quote:
Originally Posted by a humble lich
The motion of something in a circular orbit is given by
x = A sin( 2 pi t/ T + phi)
y = A cos( 2 pi t/ T + phi)
where x and y are the coordinates, A is the amplitude or radius of the orbit, t is time, T is the period of the orbit, phi is the phase and pi is 3.14159.... For this purpose the phase (phi) can be ignored---the phase will tell us where in the orbit it starts but that information is not useful to us.
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While this is correct, why on earth are you bothering? You don't need to know x and y -- the moon is full when its angle is opposite the sun. If we define towards the sun as an angle of zero, that means the decimal portion of ( t/T + phi/2pi ) is, say, between 0.45 and 0.55 (equivalent to a 3 day full moon).
We can even account for solar motion. The inner moon undergoes 4488.2 sidereal months per year; it will undergo either 4489.2 or 4487.2 solar months per year, depending on the direction of its orbit (4489.2 would be considered 'standard') and thus its solar month will be 0.08136 or 0.08140 days. Same math applies to the other moons.