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 affects the age of the full moon. - See Jawad (1993).
The full moon cycle 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 full moon cycle, 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:
The synodic month has an average duration of:
The full moon cycle is the beat period of these two, and has a duration of:
Why does a full moon cycle 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 full moon cycles passed becomes one less than the number of sidereal years passed.
Hence the full moon cycle can be defined such that the lunar precession cycle is the beat period of the full moon cycle and sidereal year. See lunar precession.
When tracking by counting cycles of 14 synodic months, a correction of 1 synodic month should take place after 18 cycles:
The equality of 269 anomalistic months to 251 synodic months was already known to Chaldean astronomers (see Kidinnu). A good longer period spans 55 cycles 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:
There are 13.944335 synodic months in a full moon cycle, the 251-month cycle approximates the full moon cycle to 13.944444 synodic months and the 767-month cycle approximates the full moon cycle to 13.9454545 synodic months.
The saros is an eclipse cycle of 223 synodic months = 239 anomalistic months = 242 draconic months. This is also equal to 16 full moon cycles. The circumstances of an eclipse depend much on the apparent size of the Moon, and therefore on its phase in its anomalistic cycle and consequently in its full moon cycle. In the duration of a saros cycle, there are about 40 eclipses. 1 saros after an eclipse, another eclipse is very likely to occur that much resembles that first eclipse. Moreover, eclipses that occur a multiple of full moon cycles apart, are also very similar. This may have been known to the ancient Greeks: in the Antikythera mechanism, the saros cycle is represented in a dial arranged as a 4-turn spiral, which also has quadrant divisors on its inside. It has been proposed (Freeth et al. 2008) that this matches a division of the saros in 16 full moon cycles, and may have been used to predict the appearance of eclipses.
Besides predicting when a full moon will be large, the full moon cycle can be used to more accurately predict the exact time of the full moon or new moon (together called: syzygies).
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.
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:
The three largest terms for the computation of true phase from mean phase are (from Meeus 1991, ch. 47 p.321):
|Amplitude for New Moon||Amplitude for Full Moon||Argument||Meaning of the argument|
|−0.40720||−0.40614||M'||mean anomaly of Moon|
|+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 full moon cycle. So we can make do with a short table of 14 values, one for every new or full moon in a full moon 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.
|Full moon cycle phase (× 1/14):||0||1||2||3||4||5||6||7||8||9||10||11||12||13|
|Correction (× 1/49 day):||0||-8||-15||-19||-20||-16||-9||0||9||16||20||19||15||8|
|Full moon cycle phase (× 1/14):||0||1||2||3||4||5||6||7||8||9||10||11||12||13|
|Correction (× 1/49 day):||18||18||19||22||25||30||33||35||35||33||30||25||22||19|
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.
When using an accumulator with the second, cyclic table above, then at the jump after 18 full moon cycles, first correct the accumulator by subtracting 8. Then apply the differential correction for the new full moon cycle 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 full moon cycle correction (under entry 0 in the first basic table above), which preserves the cyclic nature of the tables.
|Correction (x 1/49 day):||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.
|epoch||first New Moon in cycle||first New Moon in 2000|
|full moon cycle cycle||17||0||6|
|full moon cycle phase||13||1||8|
|initial accumulator||43||=43+26-49 =20||34|
|full moon cycle correction||+8||-8||+9|
|cyclic accumulator||43+8-49 =2||=2 -8 +18 = 20-8 =12||43|
|computed local Jerusalem time of syzygy||(47/49)*24 = 23h||(5/49)*24 = 2h||(43/49)*24 = 21h|
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 full moon cycle 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 full moon cycle cycle 15 (in a cycle from 0 to 17); the value of the accumulator at that time was 23, the full moon cycle 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.
|epoch||first Full Moon in cycle||first Full Moon in 2000|
|full moon cycle cycle||17||0||15|
|full moon cycle phase||13||1||1|
|initial accumulator||25||=25+26-49 =2||23|
|full moon cycle correction||+8||-8||-8|
|cyclic accumulator||25+8 =33||=33 -8 +18 = 2-8+49 =43||15|
|computed local Jerusalem time of syzygy||(33/49)*24 = 16h||(47/49)*24 = 23h (*)||(19/49)*24 = 9h|
The actual dark moon for that date occurred at 23:49 UT the previous day, 11 minutes earlier than the epoch.
|Maximum error (hours)||RMS (hours)||% day off|
|mean new moon||-14.13||7.51||26.8%|
|with full moon cycle correction||+6.90||3.06||11.6%|
|with full moon cycle and solar corr.||-3.86||1.11||3.9%|
|mean full moon||+14.12||7.49||27.3%|
|with full moon cycle correction||+6.88||3.05||11.4%|
|with full moon cycle and solar corr.||-4.02||1.12||3.9%|