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## Main Question or Discussion Point

If it takes the moon 27.4 days to complete one orbit about the Earth, why does it take 29.5 days between a full moon to full moon cycle? Any mathematical proof would be helpful, thanks.

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If it takes the moon 27.4 days to complete one orbit about the Earth, why does it take 29.5 days between a full moon to full moon cycle? Any mathematical proof would be helpful, thanks.

Integral

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The relationship can be found by

[tex]month_{synodic} = \frac{1}{\frac{1}{month_{sidereal}}- \frac{1}{year_{Earth}}}[/tex]

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A solar month can be taken to mean 1 / 12th of a solar year. Then the mean-intercalary-month-period can be calculated by a formula much like that given by Janus. In other words, how long does it take for the ( shorter ) synodic month to take a lap on the ( longer ) solar month? This would be the "blue moon" period. It's about 33 or 34 months. These notions are important in luni-solar calendars that require a 13th month added to an otherwise 12 synodic month year.

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w=w1-w2 where w1 = 1/27.5 and w2 = 1/365.2425 (w, w1 and w2 = angular velocities)

So T = 1/w = ~29.5 days.

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