User:Tom Peters/Fumocy

The full moon cycle is a cycle of about 14 lunations over which full moons vary in apparent size and age (time since new moon). 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)

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:
 * $$ FC = \frac{SM \times AM}{SM-AM} = 411.78443 d$$

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&times;FC = 251&times;SM = 269&times;AM, not:
 * 18&times;14 = 252&times;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&times;SM = 822&times;AM = 22650 days = 55&times;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.

Use of fumocy in predicting new and full moons
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). This section describes in detail a possible method of doing this.

mean syzygy
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.

Instead of working with full polynomials, we can use a linear approximation. And instead of computing with decimals, we approximate the lunation length by a vulgar fraction. Moreover it is sufficient to keep track of just the numerator by adding once every lunation, an integer constant to a variable that is called the 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 a variable called the accumulator which essentially is the time of day that the mean syzygy falls; in our case its unit is 1/49 of a day. 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 in the polynomial for the moment of mean syzygy, the accumulator needs to be corrected by subtracting 1 once every 65 years or so.

periodic corrections
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 of the time of true new or full moon from the mean new and full moon (which repeat at regular intervals), can be expressed as a sum of a series of sine terms, i.e. are of the form:
 * C1*sin(A1) + C2*sin(A2) + C3*sin(A3) + ... ,

where the A's are arguments that vary with time and are made from combinations of 4 fundamental periods that appear in the orbits of the Moon and Earth; and the C's are amplitudes that have a constant value for a particular term. There are hundreds of terms; the two major terms depend on the mean anomaly of the Moon at the time of (mean) syzygy, that is: the distance along its orbit from the perigee, 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 three largest terms for the computation of true phase from mean phase are (from Meeus 1991):

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 basic cycle of 14 lunations. This very much simplified procedure gives much more accurate predictions of the syzygies than just using the mean values, but without computing a series of sine terms at every lunation.

fumocy correction
The first two sine terms in the table above can be evaluated together by making use of the fumocy period: using the unit of 1/49 day, we should apply the following fumocy corrections to the value in the accumulator for the moment of mean new or full moon (first posted by Wikipedia user Tom Peters to CALNDR-L on February 7 2003):

more efficiency
It is possible to simplify the computation of the approximate time of syzygy by combining the monthly linear increment to the accumulator for the mean syzygy, with the fumocy correction. When keeping a running count in 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. This can be done in one step using a single table with 14 entries like before: this is possible because the fumocy corrections add up to 0 (first posted by Tom Peters to CALNDR-L on February 10 2003). That is, you have to add differential increments to the accumulator. The cyclic table (first posted by Tom Peters to CALNDR-L on February 11 2003) is:

As before, the accumulator needs to be computed modulo 49 every lunation, and if it exceeds its bound, then the syzygy falls a day later.

long-term accuracy
The basic 14-lunation fumocy synchronization between synodic and anomalistic months is not very accurate after running a few years, so using this basic cycle to find more accurate times of the syzygies gives increasingly poorer results as time passes by. As we have seen, the Babylonian ratio of 269/251 is a much better approximation, and it spans 18 fumocy's which are equal to 18 basic 14-lunation cycles minus 1 month. So we should correct the basic cycle (of 1 fumocy ~ 14 lunations) after 18 fumocy's; with the proper epoch (starting date), this can be done by skipping the first entry of the first fumocy (of the next large cycle of 18 fumocy's), i.e. use the entry with value "-8" instead of "0" from the first basic 14-month table above.

When using an accumulator with the second, cyclic table above, then at the jump after 18 fumocy's, first correct the accumulator by subtracting 8. Then apply the differential correction for the new fumocy phase: use the value of 18 under entry 1 in the second, cyclic table above. What happens is that we skip a value of 0 for the fumocy correction (under entry 0 in the first basic table above), which preserves the cyclic nature of the tables.

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 of sine terms 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 beginning of January.

These values must be used to correct the time of syzygy, not added to the accumulator itself.

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 0 through 13) of fumocy 17 (in a cycle of 0 through 17). 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 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.

in short:

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)&times;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 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 total lunar eclipse at that time.

(*) The Full Moon occurred on 28 January 1983 in UT, but on 29 January in Jerusalem mean local time; however the fumocy+solar corrections are off from reality by about 3 hours, and put the syzygy back at 28 January in Jerusalem too.

An alternate epoch for use with the prime (Greenwich) 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.

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 = 4601 synodic = 4931 anomalistic months:


 * RMS: Root-Mean-Square error (a type of statistical average)
 * % day off: the percentage of cases that put the computed syzygy on the wrong day