WHY DOES THE MOON SKIP AROUND?
 -----------------------------
 John Pazmino
 NYSkies Astronomy Inc
 www.nyskies.org
nyskies@nyskies.org
 2015 August 14

Introduction
 ----------
    One of the curious features of the solar-lunar motions around 
Earth is the 'moon-skip- effect. Home astronomers commonly are shown 
this when eatching the perseid meteor shower, altho it occurs for any 
annual event occurring on a given calendar date.
    The Perseids happen to be one of the prime astronomy events that 
take place on the same date each year. Almost all other astronomy 
actiivity distrgards the civil calendar or occur at irregular or andom 
instances. Ther are many meteor showers but the Perseids, for many 
reasons, is the 'big one' for home astronomers. One reason is that it 
can be counted on to give at least a pleasing fall of shooting stars, 
at least a few tens per hour in the erakest years. Other showers are 
more erratic, ranging form a good show to a dud, with only lousy 
forecasts in advance.
    One item emphasized for the Perseids is the location of the Moon 
during the predawn hours of August 12 and 13, the days of the peak 
downfall of shooting stars. In some years there is a large Moon in the 
sky; on others, no Moon or a small one. The Moon adds her own luminous 
graffiti to slash the number of meteors discernible. Only the brighter 
ones are noticed, too few per hour to give the 'shower' effect. 
    The Perseids, with regard to the Moon's vicinity to them,  have 
favorable and unfavorable apparitions. 

Lunar-3
 -----
    Meteor observers sometimes speak of this effect as the 'lunar-3' 
or 'rule-of-3', the rough cycle of three years when the Perseids has 
one good and two poor apparitions. 
    The rule is really crude because the Moon does NOT closely repeat 
her location on a three-year cycle. In fact, the rule falls apart 
after three round and has to reset by indexing one year for the next 
set of rounds.
    A more descriptive term, with no connotation of regularity, s the 
'moon-skip' effect. Each year on the same date the Moon skips along 
the zodiac to cause either a favorable or unfavorable influence on the 
event on that date. 

Lunar months 
 ----------
    The Moon takes one 'month' to circle Earth but there are many ways 
to complete one lap of her orbit. The two that pertain to the Moon-
skip are the sidereal and synodic months. Other months, like the 
draconic, tropical, and anomalistic months, play only a minor role for 
this article.
    I use here mean values for the parameters of the lunar orbit, which 
is quite sufficient to show the Moon-skip effect.
    The sidereal month is the interval between successive passings of 
the Moon against the celestial sphere. Typicly it's the time between 
conjunctions of Moon and a fixed star. This is 27.32 days.
    The synodic month is the interval between successive instances of 
a given phase or age of the Moon. Conventionally it's the period 
between new Moons or full Moons. The synodic month is 29.53 days.
    The synodic month is longer because after rounding the Earth once 
for a sidereal month, the Sun moved downrange along the ecliptic. The 
Moon must run for a couple more days to catch up and regain her next 
new Moon phase and complete the synodic month.

Calendars
 -------
    Since the dawn of humankind we used the celestial activity to 
regulate daily life. The day-night cycle is our day and the cycle of 
seasons is our year. The latter works only for a temperate zone of 
latitude, in which most of human culture flourished eons ago. much 
later we learned that the seasons were tied to the apparent 
circulation of the Sun around the Earth. When one cycle of season was 
finished, the Sun made one lap around Earth. 
    There Moon gave us a third natural slice of time between the fay 
and year. One cycle of phases was a handy span of 29 or so days. 
Crudely there were 12 laps of phases in one lap of the Sun to make a 
year.
    It's the 'crudely' that throws the wrench into the neat order of 
calendars. It would be wonderful os there was an integer count of 
synodic months in a year! We would than count off synodic months, 
cycles of phases, to keep track of the year.
    This is far more secure than  watching the seasons, like for first 
frost or first bloom. Seasons do not repeat precisely year to year. 
The Moon phases do. But not in step with the year.

Leftover days
 -----------
    One culture that wrestled with the Moon-Sun inequity of motion is 
the Hebrew culture. Banking off of amazingly accurate work by 
Babylonian astronomers, the hebrews figured out how to mesh the 
sunodic months into the solar year. They first count off 12 synodic 
months and note that there were still about 10 days left over to 
finish the solar year. This Hebrew year has 12 months and the 10 extra 
days are put into a breadbox.
    After sa second solar year there is a second 10 extra days. This 
year also has 12 months and the extra days are put aside. 
    In the third year there is a third 10 extra days, amounting 
closely to a full extra synodic month. This is handled by adding a 
13th month to the year to bring the solar and lunar cycles back in 
step. 
    In the actual Hebrew calendar the insersion of a leap month is NOT 
every three years. The synodic month is 29-1/2 days and the leftover 
days each year is more like 10-1/2 days. Three of these is 32-1/2 
days, which is too much for a good fit by a synodic month. 
    The addition of leap-months follows a schedule that jiggers the 
29-1/2 days and 31-1/2 days so that after 19 years the Moon and Sun 
are almost exactly back in step. This account is NOT close to an clear 
explanation of the Hebrew calendar but is meant to show one effort to 
get the Sun and Moon to cooperate in calendar management.

Why 19 years?
 -----------
    Home astronomers are shown an other lunar rule, the 'rule-of-19', 
when they study lunar and solar eclipses. The rule states, in its 
simplest level, that given an eclipse, either kind, on a calendar 
date, there is an other of the same kind on the same date 19 years 
later and 19 years earlier. Eclipse chasers often mark their endurance 
in astronomy by the number of 19-year cycles. 
    When an eclipse takes place the Moon, for a lunar eclipse, is 
full. She also stands at a certain longitude along the ecliptic, that 
opposite from that of the Sun for that day. On the next, later or 
earlier, instance she again is full and occupies the same place in the 
sky. Remember that the calendar date in the civil calendar is 
essentially congruent with the longitude of the Sun.
    There must be an integer number of sidereal months fitting into 19 
years for the Moon to regain her longitude. There is also an integer 
number of synodic months for the Moon to regain her phase in that 19-
year span.
    We have for the sidereal months in 19 years 

    sidereal cycles = (19yr) * (365.25 day/yr) / (27.32 day/sid) 
                     = 254 sidereal months 

For the synodic cycles in the same 19 years we have 

    synodic cycles = (19yr) * (365.25 day/yr) / (29.53 day/syn 
                   = 235 synodic months 

    The rule-of-19 is more general than only for eclipses. If the Moon 
is observed at a given phase at a given longitude on a certain date, 
that apparition repeats 19 years apart. The rule is not perfect. After 
a few rounds it falls apart. At the same time a new 19-year cycle is 
assembling to replace it for several more rounds. There is a break in 
the sequence of years between the two cycles.

Within the 19 years
 -----------------
    Altho the longitude and phase come together after 19 years, they 
are way out of step year to year. The offset is amazingly regular, one 
that can be tabulated for foretelling where the Moon is in future 
years. This a like the table of Hebrew leap months,  which is used for 
each 19 year span thru the ages with a calculated adjustment to jump 
between closing and opening cycles two or three times per century. 
    We start with the sidereal months. There are 13 cycles of months 
in a calendar year with part of the 14th month left to fill out the 
year. The next year starts with this offset, the Moon running ahead of 
its place since the previous year. Her place in the ecliptic is 
displaced these extra days all thru this next year. 
    The extra days movement of the Moon into her 14 sidereal month is 

    excess movement = (360 deg/sid) * ((10.09day)  / (27.32 day/sid) 
                    = 132.96 deg
                    = 4 sign, 13 deg 

    This excess, the 'head start' for each successive year, is roughly 
1/3 of the sky downrange in the zodiac. After three years the Moon 
more or less returns to the same sign, producing the 'lunar-3'. After 
three years the Moon actually moved along three times the 4sign, 13 
degree, or 5 sign, 9 deg, The lunar-3 trick is really a poor one to 
live by.

Moon phase 
 -------- 
    An exactly parallel calculation is done to find the shift in Moon 
phase within the 19-year period. We don't need the angular 
displacement because it's simpler to state the phase as 'age' in days 
since the preceding new Moon. Full Moon, halfway thru the 29.53 day 
synodic cycle, is age 14.7 days.

    excess age = )365.25 day/yr) - (12 syn) * (29.53 day/syn) 
               = 10.89 day

    Each year on the same date the Moon is 10.89 days 'older'. The age 
wraps around new several times in 19 years, eventually returning to 
the same age at the end of the 19-year cycle. 

Moon-skip table 
 -------------
    We here step year by year and note the position of the Moon. We 
start with the Moon in the zero degree of Aries, at the vernal 
equinox. On the same date in the next year the Moon is 133 degrees or 
4 signs, 13 degrees downrange, in Leo 13. We continue this for each 
year for 19 years. Because the displacement is not exactly 132 
degrees, we must hitch a degree once in a while, which I do here every 
four years to spread the correction more evenly thru the 16-year span. 
    I start the age at new Moon, ae 00.0 days. This does NOT mean 
there is or was an actual year when the new Moon was at the vernal 
equinox. The two columns should be worked separately, once for the 
displacement and an other for the age, from which so ever values for 
each were in force at the instant starting date. A calculette is handy 
for this chore, minding the rollover after each 30 degrees for the 
sign and 29.53 days for each phase cycle. 
    It can be easier to imagine the Moon's movement with the sign-
degree notation. The table preents both longitude and sign-degree. 

    ---------------
    MOON-SKIP TABLE 
    ------------------------------
    yr | lon | sign   | age  | lap
    ---+--------+-----+------+- --
    00 | 030 | Ari 00 | 00.0 |  0 
    01 | 133 | Leo 13 | 10.9 |  0
    02 | 266 | Sgr 26 | 21.8 |  0
    03 | 039 | Tau 09 | 03.1 |  1
    04 | 171 | Vir 21 | 14.0 |  1
    05 | 304 | Aqr 04 | 24.9  |  1
    06 | 077  | Gem 17 | 06.2  |  2
    07 | 210 | Sco 00 | 17.1  |  2
    08 | 342 | Psc 12 | 28.0  |  2
    09 | 115 | Cnc 25 | 09.3  |  3
    10 | 248 | Sgr 08 | 20.2  |  3
    11 | 021 | Ari 21 | 01.6  |  4
    12 | 153 | Vir 03 | 12.4  |  4
    13 | 285 | Cap 15 | 23.3  |  4
    14 | 058 | Tau 28 | 04.7  |  5
    15 | 191 | Lib 11 | 15.6  |  5 
    16 | 323 | Aqr 23 | 26.4  |  5
    17 | 095 | Cnc 05 | 07.8  |  6
    18 | 227 | Sco 17 | 18.7  |  6
    19 | 000 | Ari 00 | 00.0  |  0
    ------------------------------

Use of the table
 --------------
    It is usually not feasible to tell just where in the zodiac the 
Moon is, by her sign and degree or longitude. This is specially true 
for twilight, large Moon, hazy sky. It is also usually not feasible to 
tell the age of the Moon, particularly when she is near full. 
    These factors make the table impractical as an absolute cycle, 
starting from a known location and age of the Moon. It is easiest to 
employ it as a relative cycle.  It is zeroed or homed at the 'year #0' 
for the situation of the instant Moon. 
    For the position and age of the Moon in future years, read the 
figures as positive, so many degrees downrange and so many days older 
than now. For past years, the figures are negative, for uprange 
position and younger age. As a sanity check, the Moon odes -- roughly 
-- skip 1/3 way round the zodiac and 183 phase cycle per year with an 
    The typical application is to find if the Moon interferes with an 
event on the same ate in various years. If, say, the Moon on the 
instant occasion is nearly full in the east, she will be well below 
the horizon, many hours before rising on the next year's event. She 
also will be a waning crescent when she does rise. For this event 
there is no Moon in the sky.
    Do mind well that diurnal rotation can displace the Moon in the 
sky, even tho during a night she moves only a few degrees thru the 
zodiac and ages only a few hours. The displacement of position and age 
are applied to the aspect of the Moon for a particular hour, like a 
predawn hour for the Perseids. Assessing the Moon's aspect by the 
Moon-skip table for the evening of an other year on the same date 
yields nonsense results. 

Conclusion
 --------
    While the Moon's movement seems at first complicated, there is a 
cycle to it. The skipping of the Moon around the sky year by year has 
a regularity that starts over again in 19-year blocks. Like other 
astronomy cycles, this Moon-skip unravels after several rounds, long 
enough to satisfy a person's lifespan but not to cover several 
generations. 
    Most of us don't keep to hand almanacs for historical and future 
years. The lunar position and age could be looked up in them for the 
given calendar date. The Moon-skip table is a handy substitute in case 
of wanting such almanacs. 
    Keeping this Moon-skip table handy helps to appreciate the lunar-
solar interactions and better bond you to the heavens. This is 
specially true if you in fact observe the Moon annually, like at a 
garden party each year to watch the 4th-of-July fireworks.