View source for Talk:Full moon cycle

Hey, we went over this in 2003 when we wrote the page.  One of the Wikipedia supervisors had the same objections and removed the article; I salvaged it for a while under my personal page.  The adversary conceded when I could show some earlier literature that discussed regularities in the size and timing of the syzygies - still quoted at the end of the article.

All this stuff is factually correct, which is increasingly rare in the Wikipedia.  Also at least 2 people (myself and Victor Engel) have been contributing and using this, so it is at least twice as big as a personal pet project.  So why remove all this?

Finally, recently someone flagged this article as sub-standard.  What exactly are the problems?

[[User:Tom Peters|Tom Peters]] 21:37, 18 April 2006 (UTC)

==Use of fumocy in predicting new and full moons==

The later part of the article is about a project presented to CALNDR-L to make a lunar calendar that takes accounts of the fumocy to get a better match with moon phases.

This is at best original research and may be seen to be a pet project. 

I'm not sure whether it should be included in wikipedia and move it here pending discussion about it.

===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: [[syzygy|syzygies]]). The Moon's [[lunar phase|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):

<table border=1>
<caption>Main correction terms for times of New and Full Moon</caption>
<tr><td>Amplitude for New Moon</td><td>Amplitude for Full Moon</td><td>Argument</td><td>Meaning of the argument</td></tr>
<tr><td>&minus;0.40720</td><td>&minus;0.40614</td><td>M'</td><td>mean anomaly of Moon</td></tr>
<tr><td>+0.01608</td><td>+0.01614</td><td>2&times;M'</td><td></td></tr>
<tr><td>+0.17241</td><td>+0.17302</td><td>M</td><td>mean anomaly of Sun</td></tr>
</table>

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.

===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]].

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.

===fumocy correction===
The first two sine terms in the table above can be evaluated together by making use of the 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]]):

<table border=1>
<caption>Basic Fumocy Correction Table</caption>
<tr align="right"><td>Fumocy phase (&times; 1/14):</td>
<td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td>
<td>7</td><td>8</td><td>9</td><td>10</td><td>11</td><td>12</td><td>13</td></tr>
<tr align="right"><td>Correction (&times; 1/49 day):</td>
<td>0</td><td>-8</td><td>-15</td><td>-19</td><td>-20</td><td>-16</td>
<td>-9</td><td>0</td><td>9</td><td>16</td><td>20</td><td>19</td><td>15</td><td>8</td></tr>
</table>

<!-- ok this is the first paragraph that I now find confusing. Alba -->
<!-- TP elaborated, rephrased, and re-organized 20060425 -->
====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:

<table border=1>
<caption>Cyclic Fumocy Correction Table</caption>
<tr align="right"><td>Fumocy phase (&times; 1/14):</td>
<td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td>
<td>7</td><td>8</td><td>9</td><td>10</td><td>11</td><td>12</td><td>13</td></tr>
<tr align="right"><td>Correction (&times; 1/49 day):</td>
<td>18</td><td>18</td><td>19</td><td>22</td><td>25</td><td>30</td><td>33</td>
<td>35</td><td>35</td><td>33</td><td>30</td><td>25</td><td>22</td><td>19</td></tr>
</table>

As before, the accumulator needs to be computed [[modular arithmetic|modulo]] 49 every lunation, and if it exceeded its bound, then the syzygy falls a day later.

====long-term accuracy====
The 14-month fumocy synchronization between synodic and anomalistic months is not very accurate after running a few years, so using the fumocy 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 its spans 18 fumocy's minus 1 month.  So we should correct the basic cycle (of 1 fumocy ~ 14 lunations) after 18 fumocy's; with the proper epoch, 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.

<table border=1>
<caption>Solar Correction Table</caption>
<tr><td>Lunar month:</td>
<td>I</td><td>II</td><td>III</td><td>IV</td><td>V</td><td>VI</td>
<td>VII</td><td>VIII</td><td>IX</td><td>X</td><td>XI</td><td>XII</td><td>XIII</td></tr>
<tr><td>Correction:</td>
<td>0</td><td>4</td><td>7</td><td>8</td><td>7</td><td>4</td>
<td>0</td><td>-4</td><td>-7</td><td>-8</td><td>-7</td><td>-4</td><td>0</td></tr>
</table>

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

===epochs and constants===
An optimum [[epoch (astronomy)|epoch]] for New Moons at the [[meridian (geography)|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 trough 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:

<table border=1>
<caption>Parameters for New Moons</caption>
<tr align="right"><th>&nbsp;</th><th>epoch</th><th>first New Moon in cycle</th><th>first New Moon in 2000</th></tr>
<tr align="right"><td>date</td><td>1992-07-29</td><td>1992-08-28</td><td>2000-01-06</td></tr>
<tr align="right"><td>fumocy cycle</td><td>17</td><td>0</td><td>6</td></tr>
<tr align="right"><td>fumocy phase</td><td>13</td><td>1</td><td>8</td></tr>
<tr align="right"><td>initial accumulator</td><td>43</td><td>=43+26-49 =20</td><td>34</td></tr>
<tr align="right"><td>fumocy correction</td><td>+8</td><td>-8</td><td>+9</td></tr>
<tr align="right"><td>cyclic accumulator</td><td>43+8-49 =2</td><td>=2 -8 +18 = 20-8 =12</td><td>43</td></tr>
<tr align="right"><td>solar correction</td><td>-4</td><td>-7</td><td>0</td></tr>
<tr align="right"><td>computed local Jerusalem time of syzygy</td><td>(47/49)*24 = 23h</td><td>(5/49)*24 = 2h</td><td>(43/49)*24 = 21h</td></tr>
</table>


To compute the date and time of [[full moon|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.

<table border=1>
<caption>Parameters for Full Moons</caption>
<tr align="right"><th>&nbsp;</th><th>epoch</th><th>first Full Moon in cycle</th><th>first Full Moon in 2000</th></tr>
<tr align="right"><td>date</td><td>1982-12-30</td><td>1983-01-28(/29) (*)</td><td>2000-01-21</td></tr>
<tr align="right"><td>fumocy cycle</td><td>17</td><td>0</td><td>15</td></tr>
<tr align="right"><td>fumocy phase</td><td>13</td><td>1</td><td>1</td></tr>
<tr align="right"><td>initial accumulator</td><td>25</td><td>=25+26-49 =2</td><td>23</td></tr>
<tr align="right"><td>fumocy correction</td><td>+8</td><td>-8</td><td>-8</td></tr>
<tr align="right"><td>cyclic accumulator</td><td>25+8 =33</td><td>=33 -8 +18 = 2-8+49 =43</td><td>15</td></tr>
<tr align="right"><td>solar correction</td><td>0</td><td>+4</td><td>+4</td></tr>
<tr align="right"><td>computed local Jerusalem time of syzygy</td><td>(33/49)*24 = 16h</td><td>(47/49)*24 = 23h (*)</td><td>(19/49)*24 = 9h</td></tr>
</table>
(*) The Full Moon occured 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:
<ul>
<li>Epoch is after switch from Julian to Gregorian calendar to avoid confusion in date references.
<li>Initial value of 26/49 accumulator should be zero.
<li>Adjustment to this accumulator by phase should be zero.
<li>Calculated error (difference between actual dark moon and calculated value in 49th days) should be minimal at the epoch.
</ul>
[[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:

<table border=1>
<caption>Error Statistics</caption>
<tr><td>&nbsp;</td><td>Maximum error (hours)</td><td>RMS (hours)</td><td>% day off</td></tr>
<tr><td>&nbsp;</td><td colspan="3">&nbsp;</td></tr>
<tr><td>mean new moon</td><td>-14.13</td><td>&nbsp;7.51</td><td>26.8%</td></tr>
<tr><td>with fumocy correction</td><td>&nbsp;+6.90</td><td>&nbsp;3.06</td><td>11.6%</td></tr>
<tr><td>with fumocy and solar corr.</td><td>&nbsp;-3.86</td><td>&nbsp;1.11</td><td>&nbsp;3.9%</td></tr>
<tr><td>&nbsp;</td><td colspan="3">&nbsp;</td></tr>
<tr><td>mean full moon</td><td>+14.12</td><td>&nbsp;7.49</td><td>27.3%</td></tr>
<tr><td>with fumocy correction</td><td>&nbsp;+6.88</td><td>&nbsp;3.05</td><td>11.4%</td></tr>
<tr><td>with fumocy and solar corr.</td><td>&nbsp;-4.02</td><td>&nbsp;1.12</td><td>&nbsp;3.9%</td></tr>
</table>

: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

---
[[User:Tom Peters|Tom Peters]] 23:46, 18 April 2006 (UTC)
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[[User:Karl Palmen|Karl Palmen]] 12 April 08:35 UT