CRESCENT-SIGHTING AND ISLAMIC CALENDARS
By David McNaughton, "A Universal Islamic Calendar" (Hamdard Islamicus, Karachi, January 1997)
Variations in month-length
On average, a lunar month (or lunation) is 29 days 12.73 hours, but it can sometimes be as short as 29 days 6½ hours, or as long as 29 days 20 hours. Fluctuations in length are cyclic. There is a fast cycle averaging about 412 days (which is just under 14 lunar months); this is associated with changes in the eccentricity, or shape of the lunar orbit. That rapid cycle is modified by a slower one whose mean wavelength is 8.85 years (equal to one complete revolution of the axis of the moon's elliptical orbit). In addition, there are other oscillations, some causing variations extending over many hundreds of years - when it may not even make sense to look for an average wavelength.
Other factors too contribute to the moon's complex behaviour. For instance, longer than normal lunations tend to occur between October and March because of faster movement by the Earth in its orbit round the sun. In February and March, this delays the instant when the moon appears to overtake the sun. Also, the sun lies closer to a newborn crescent than it would if Earth's speed was uniform - thus postponing the onset of New Moon visibility. In October and November, on the other hand, the sun's position on the ecliptic line is "behind schedule" - so the moon overtakes it earlier.
Occasionally, four consecutive lunations will span more than 119 days. Under those circumstances, it is possible to have four successive 30-day months. When that happens, the moon only just becomes visible at the beginning of the first month in the sequence, and just fails to do so on the 29th of the fourth month.