- 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)

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

- SM = 29.53058885 days

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.

- 767*SM = 822*AM = 22650 days = 55*FC + 2 days = 62 years + 4 days

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.

The period of the mean synodic month can be approximated as 29 + 26/49 days (a more accurate 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 higher-order terms for the moment of mean syzygy, the accumulator needs to be corrected by subtracting 1 once every 65 years or so.

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 CALNDR-L on 10-Feb-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 CALNDR-L on 11-Feb-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.

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 30-Dec-1982. The **first Full Moon** of 2000, on 21 January, 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 (23-8+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 21 Jan-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.

The actual dark moon for that date occurred at 23:49 UT the previous day, 11 minutes earlier than the epoch.

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.

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% |

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