Babylonian lunar phenomenon

Lunar Three

Lunar Three are the following parameters (as defined by A. Sachs):
The Lunar Three (including naN, ŠÚ, ME and GE6) is implemented in ARCHAEOCOSMO package.
<1 equals ~4 Min (~24*60 [Min]/360 [])>

Lunar Four

Lunar Four are the following parameters all related to near Full Moons:
The Lunar Four (including lunation length of previous month, naN and KUR) is implemented in ARCHAEOCOSMO package.

Lunar Six

Lunar Six are the following parameters: The Lunar Six (including lunation length of previous month) is implemented in ARCHAEOCOSMO package.

Summations of Lunar Four values

In interesting aspect of the Lunar Four is that if one make sums like: ŠÚ+na; ME+GE6; and ŠÚ+na+ME+GE6, these sums become almost independent of longitude, latitude and horizon altitude. This is ideal for determining a parameters independent of location. Below picture is calculated using ARCHAEOCOSMO

Different Lunar Four value
The spiky na curve (green) and smoother ŠÚ+na curve (blue)
<Remark: above na curve is slightly different then Brack-Bernsen's Figure 4 as that had a small ephemeris error [Pers. Comm. Brack-Bernsen, 2012]>

Brack-Bernsen showed e.g. that ŠÚ+na+ME+GE6 is a very good way to determine the periodicy of Lunation-length PΦ [Lis Brack-Bernsen, and Matthias Brack. "Analysing shell structure from Babylonian and modern times." In 10th Nuclear physics workshop, 1-13. Kazimierz Dolny Poland, 2003, page 10.]: ~411.8 days (being the HarmonicDifference(Lunar apse cycle, Tropical year)). (Meeus [2002], page 21-23)
From ŠÚ+na and ME+GE6 (above picture) one can relate the Tropical year with 12.368 Synodic months.

Predicting Lunar Six values: Goal-Year method

The Lunar Six can be predicted, based on earlier (223 Synodic months or a Saros cycle earlier) values, by using the following formula [Lis Brack-Bernsen. "Goal-Year tablets: Lunar data and predictions." In Ancient Astronomy and Celestial Divination, edited by N.M. Swerdlow: MIT Press, 1999, page 169]:
(naN)i=(naN)i-223 + 1/3*(ŠÚ+na)i-229
ŠÚi=ŠÚi-223 + 1/3*(ŠÚ+na)i-223
nai=nai-223 - 1/3*(ŠÚ+na)i-223
MEi=MEi-223 + 1/3*(ME+GE6)i-223
(GE6)i=(GE6)i-223 - 1/3*(ME+GE6)i-223
KURi=KURi-223 + 1/3*(ME+GE6)i-229 mod(ME+GE6)i-229

My proposed modulus function maps nicely on the additional correction rules mentioned in BM 42282+42294 (Lis Brack-Bernsen, and Hermann Hunger. "BM 42282+42294 and the Goal-Year method." SCIAMVS 9 (2008): 3-23).

The 223 (Synodic months) is the Saros cycle, and was used by the Babylonians in these formula and recorded this method around 600 BCE. The below picture is calculated using ARCHAEOCOSMO:
nai and nai-223
  Green curve is nai while Blue curve is nai-223 - 1/3*(ŠÚ+na)i-223
As can be seen in the above graph, there can be still large discrepancies (e.g. at lunation 236), but if one realises this modulus function, the difference between predicted and actual value is not that large.

Comparing Babylonian Lunar Four sums with DE406

A few Goal-Year Lunar Four values have been compared with values calculated with Swiss Ephemeris (using ARCHAEOCOSMO). The following method has been used:
121 and 122 Se Goal Year
The form of the curves for the Babylonian and calculated values are comparable
<Remark: lunation 14 seems to be somewhat off, might be a scribe error?>
[LBAT 1251+1252, Hunger, 1999, page 91-94]

175 and 186 SE Goal Year text
The form of the curves for the Babylonian and calculated values are comparable
[LBAT 1285, Hunger, 2006, page 269-275]

Difference between Babylonian measurements and calculated Lunar Four sum values

From the above graphs one can see there is a difference between the Lunar Four sum values of the Babylonians and the calculated ones:
Difference between calcuated D406
            and babylonian

The 1σ seen between Babylonian and calculated Lunar Four sums is around 1.5 [us]. As this is a sum of two Lunar Four values; the 1σ of each Lunar Four values is around 1 [us] (1.5/√2). This errors looks large compared to what is found during evaluation of eclipses (Stephenson and Fatoohi, 1994). They found that 1 hour is equivalent to 15.3 +/- 0.5 [us] (page 109) for dates after 300 BCE. This would a 1σ of around 0.1 [us] (0.5/√15). They also found no seasonal changes.
So this larger than expected error is unlikely to be explainable due to the impreciseness of the Babylonian water clocks around 200 BCE. Would there be a systematic measurement procedure difference?


I would like to thank the following people for their help and constructive feedback: Lis Brack-Bernsen, Jens Høyrup, Hermann Hunger and all other unmentioned people. Any remaining errors in methodology or results are my responsibility of course!!! If you want to provide constructive feedback, let me know.

Disclaimer and Copyright

Major content related changes: Aug. 8, 2012