Full moon cycle
From Academic Kids

The full moon cycle (the abbreviation fumocy was introduced by Karl Palmen in the CALNDRL mailing list in October 2002) 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 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)
Contents 
Explanation
The variation in apparent size of the Moon is due to the fact that 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 period of the Moon's phases, that is the motion of the Moon with respect to the Sun, is called synodic month.  See Meeus (1981).
This same 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}{SMAM} = 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. 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 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 251month cycle approximates the fumocy to 13.944444 synodic months and the 767month cycle approximates the fumocy to 13.9454545 synodic months.
Use of fumocy in predicting new and full moons
periodic corrections
Besides predicting when a full moon will be large, the fumocy cycle can be used to more accurately predict the exact time of the full moon or new moon (together called: syzygies). The Moon's phases do not repeat very regularly: the time between two similar syzygies may vary between 29.272 and 29.833 days (see new moon for a detailed account). The reason is that the orbit of the Moon is elliptic, its velocity is not constant, so the time of the true syzygy will differ from the mean syzygy. The deviations can be expressed as a series expansion of sine terms. The major terms depend on the mean anomaly at the time of (mean) syzygy, that is: the distance along its orbit from the perigeum, which is the phase of the Moon in its anomalistic cycle. As we have seen, this anomalistic cycle coincides with the synodic cycle again after 1 fumocy.
The first three terms for the computation of true phase from mean phase are (from Meeus 1991):
New Moon  Full Moon  Argument  
−0.40720  −0.40614  M'  mean anomaly of Moon 
+0.01608  +0.01614  2×M'  
+0.17241  +0.17302  M  mean anomaly of Sun 
Amplitudes in days; take the sine of the arguments.
Now instead of computing the actual value of M' and 2*M' and the sine terms for every new or full moon, we can use the fact that these approximately repeat every fumocy. So we can make do with a short table of 14 values, one for every new or full moon in a fumocy cycle. We only need to keep track of where we are in the cycle of 14 lunations.
Mean syzygy
But first we have to find the moment of mean syzygy, before we can correct it with our fumocy correction. Polynomial expressions are given on the pages of new moon and full moon. We can also implement a simpler linear expression by using an accumulator. This is similar to calculating the molad in the Hebrew calendar. It works as follows:
The period of the mean synodic month can be approximated as 29 + 26/49 days (a more accurate vulgar fraction is 29 + 451/850; the Hebrew calendar uses 29 + 12 hours + 793/1080 hours). We maintain an accumulator which essentially is the time of day that the mean syzygy falls. So for one lunation to the next, we add 29 days, and we add 26 to the accumulator. Whenever the accumulator reaches 49 or higher, a day is filled, so the syzygy falls 1 day later and we subtract 49 from the accumulator.
Because of the error in this approximation by a fraction, and because of the higherorder terms for the moment of mean syzygy, the accumulator needs to be corrected by subtracting 1 once every 65 years or so.
fumocy correction
Now using the unit of 1/49 day, we should apply the following fumocy corrections to the moment of mean new or full moon (first posted by Tom Peters to CALNDRL on February 7 2003):
Fumocy phase (× 1/14):  0  1  2  3  4  5  6  7  8  9  10  11  12  13 
Correction:  0  8  15  19  20  16  9  0  9  16  20  19  15  8 
With a proper epoch, you can correct the basic cycle after 18 fumocy's by skipping the first entry of the first fumocy (of the next large cycle of 18), i.e. use the entry with value "8" instead of "0".
Because the fumocy corrections add up to 0, and the correction after 18 fumocy's involves skipping a value of 0, it is possible to apply the fumocy correction to the accumulator directly, in combination with the linear increment of 26 (first posted by Tom Peters to CALNDRL on February 10 2003). However, if you use an accumulator then for each successive lunation you first have to subtract the fumocy correction for the previous lunation, then add the mean increment of 26, and then add the new fumocy correction. That is, you have to add differential increments to the accumulator. The cyclic table (first posted by Tom Peters to CALNDRL on February 11 2003) is:
Fumocy phase (× 1/14):  0  1  2  3  4  5  6  7  8  9  10  11  12  13 
Correction:  18  18  19  22  25  30  33  35  35  33  30  25  22  19 
At the jump after 18 fumocy's, first correct the accumulator by subtracting 8, and then apply the differential correction for the new fumocy phase (18 under entry 1 in the table above). As before, the accumulator needs to be computed modulo 49, and if it exceeded its bound, then the syzygy falls a day later.
epochs and constants
An optimum epoch for New Moons at the meridian of Jerusalem (at 35:14:03.4 deg. East of Greenwich = +0.097873 days ahead of UT) is July 29 1992. That syzygy preceded the first syzygy of the current cycle of 251 New Moons, so it had the fumocy correction phase 13 (in the cycle of 14) of fumocy 17 (in a cycle of 18), both counting from 0 . After this the 1st fumocy correction of the new cycle was dropped, and we started fumocy cycle 0 with fumocy correction phase 1 . This means that the first Dark Moon of 2000, on January 6, was phase 8 (in the cycle from 0 to 13), of fumocy 6 (in a cycle from 0 to 17). The value of the accumulator at that time was 34, the fumocy correction was +9, and the solar correction (see below) was 0. So the New Moon occurred at (34+9)/49 = 0.88 days after local midnight, or at 0.78 days UT. The true time of New Moon was 18:14 UT = 0.760 days: an error of 0.02 days = 0.5 hours.
To compute the date and time of Full Moon the same method can be used with the same tables; but because the Full Moon comes a half cycle after the New Moon, its fumocy corrections are out of phase by half a cycle from those for the New Moon. Hence its epoch is (18/2)×14+(14/2)+0.5 = 118.5 synodic months = 9 + 7/12 years earlier: at December 30 1982. The first Full Moon of 2000, on January 21, had phase 1 (in the cycle from 0 through 13) of fumocy 15 (in a cycle from 0 to 17); the value of the accumulator at that time was 23, the fumocy correction was 8, and the solar correction (see below) was +4. So the Full Moon occurred at (238+4)/49 = 0.39 days after local midnight, or at 0.29 days UT. The true time of Full Moon was 4:41 UT = 0.195 days: an error of less than 0.1 days, or 2.3 hours.
Note: there was a lunar eclipse at that time.
An alternate epoch for use with the prime meridian is January 21 1890. This epoch was chosen by looking for a date that satisfied the following criteria:
 Epoch is after switch from Julian to Gregorian calendar to avoid confusion in date references.
 Initial value of 26/49 accumulator should be zero.
 Adjustment to this accumulator by phase should be zero.
 Calculated error (difference between actual dark moon and calculated value in 49th days) should be minimal at the epoch.
January 21 1890 is the first date to match these criteria. The next date to match the criteria is January 1 2120. The former is chosen because it is in the past.
The actual dark moon for that date occurred at 23:49 UT the previous day, 11 minutes earlier than the epoch.
solar correction
The remaining error of the predicted time of the new or full moon can be halved again by taking account of the solar term (the third in the table above). The anomalistic period of the Sun (365.259636 days) can be approximated by the calendar year (365 or 366 days; 365.2425 days on average in the Gregorian calendar). Since a calendar year has 12 or 13 new and full moons, it is sufficient to evaluate the solar term for 12 representative phases of this annual cycle, and put these in another table. The mean anomaly of the Sun currently is 0 around 2 January, so the table starts with the new or full moon closest to the begin of January.
Lunar month:  I  II  III  IV  V  VI  VII  VIII  IX  X  XI  XII  XIII 
Correction:  0  4  7  8  7  4  0  4  7  8  7  4  0 
These values must be used to correct the time of syzygy, not added to the accumulator itself.
statistics
The following table lists the errors of the polynomial, the fumocy correction, and the fumocy plus solar correction, as compared to true syzygy, for a period of 372 years:
Max.err. (h)  RMS (h)  % day off  
mean new moon  14.13  7.51  26.8% 
with fumocy corr.  +6.90  3.06  11.6% 
with fumocy and solar corr.  3.86  1.11  3.9% 
mean full moon  +14.12  7.49  27.3% 
with fumocy corr.  +6.88  3.05  11.4% 
with fumocy and solar corr.  4.02  1.12  3.9% 
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; WillmannBell, Richmond, VA. ISBN 0943396352 ; based on the ELP200082 lunar ephemeris.
 Roger W. Sinnott (1993): How Long Is a Lunar Month?, Sky&Telescope Nov.1993, pp.76..77