Full moon cycle
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The full moon cycle is a cycle of about 14 lunations over which full moons vary in apparent size. Also in the same cycle the age of the full moon (time since new moon) varies. The abbreviation fumocy was introduced by Wikipedia user Karl Palmen in the CALNDR-L mailing list in October 2002. The sequence is
- Full moon big - (perigee at full moon)
- Full moon young - (perigee at first quarter)
- Full moon small - (perigee at new moon)
- Full moon old - (perigee at last quarter)
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Explanation
The apparent size of the Moon varies because the orbit of the Moon is distinctly elliptic, and as a consequence at one time it is nearer to the Earth (perigee) than half an orbit later (apogee). The orbital period of the Moon from perigee to apogee and back to perigee is called the anomalistic month.
The appearance, or phase, of the Moon is due to its motion with respect to the Sun. It varies in a period of time called a lunation, also called synodic month. The age is the number of days since new moon. - See Meeus (1981).
The ellipticity of the orbit also causes the duration of a half lunation to depend on where in the elliptical orbit it begins, and so effects the age of the full moon. - See Sinnott (1993).
The fumocy is slightly less than 14 synodic months and slightly less than 15 anomalistic months. Its significance is that when you start with a large full moon at the perigee, then subsequent full moons will occur ever later after the passage of the perigee; after 1 fumocy, the accumulated difference between the number of completed anomalistic months and the number of completed synodic months is exactly 1.
The average duration of the anomalistic month is:
- AM = 27.55454988 days
The synodic month has an average duration of:
- SM = 29.53058885 days
The full moon cycle is the beat period of these two, and has a duration of:
- <math> FC = \frac{SM \times AM}{SM-AM} = 411.78443 d</math>
Fumocy and the year
Formulated in another way: the fumocy is the period that it takes the Sun to return to the perigee of the Moon's orbit (as seen from the Earth). So it is a kind of "perigee year", similar to the eclipse year which is the time for the Sun to return to the ascending node of the Moon's orbit on the ecliptic.
Why does a fumocy last almost 14 lunations rather than just the 12.37 lunations of a year? This would be the case, if the moon's orbit kept a constant orientation with respect to the stars, but the tidal effect of the sun causes the orbit to precess over a cycle just under 9 years. In that time, the number of fumocies passed becomes one less than the number of sidereal years passed.
Hence the fumocy can be defined such that the lunar precession cycle is the beat period of the fumocy and sidereal year. See lunar precession.
Matching synodic and anomalistic months
When tracking fumocies by counting cycles of 14 synodic months, a correction of 1 synodic month should take place after 18 fumocies:
- 18×FC = 251×SM = 269×AM, not:
- 18×14 = 252×SM
The equality of 269 anomalistic months to 251 synodic months was already known to Chaldean astronomers (see Kidinnu). A good longer period spans 55 fumocies or rather 767 synodic months, which is not only very close to an integer number of synodic and anomalistic months, but also when reckoned in synodic months is close to an integer number of days and an integer number of years:
- 767×SM = 822×AM = 22650 days = 55×FC + 2 days = 62 years + 4 days
There are 13.944335 synodic months in a fumocy, the 251-month cycle approximates the fumocy to 13.944444 synodic months and the 767-month cycle approximates the fumocy to 13.9454545 synodic months.
References
- Jean Meeus (1981): Extreme Perigees and Apogees of the Moon, Sky&Telescope Aug.1981, pp.110..111
- Jean Meeus (1991): Astronomical Algorithms, Ch.47 p.321; Willmann-Bell, Richmond, VA. ISBN 0-943396-35-2 ; based on the ELP2000-82 lunar ephemeris.
- Roger W. Sinnott (1993): How Long Is a Lunar Month?, Sky&Telescope Nov.1993, pp.76..77