Mail
NEW

Comparing actual and pre/postdicted Babylonian dates of the Moon's first visibility

The principles behind the comparison:

The date of the first visibility of the Moon is depending on several parameters (like astronomy, observation location, photometric and meteorological parameters). The calculations are based on Schaefer's criterion.
Tables with 333 actual Babylonian observations dates (Table 1, which includes most of Fatoohi's observations [1999]) and 110 Babylonian pre/postdicted observations dates (Table 2) are taken from S. Stern [2008]. The dates of Table 2 have thus not actually been observed (due to the meteorological circumstances on that date [mist, clouds, rain, etc.]).
Three observations (-206/1/13, -194/6/6 and -145/2/8) in Table 1 have been changed. The first one is a typo (-206/1/23). For the remaining two cases I choice to use the observation dates of Fatoohi [1999] (-194/6/7 and -145/2/7), as these map better the Babylonian observation descriptions (pers. comm.: S. Stern [2008], H. Hunger [2008]).
The Astronomical Extinction Coefficient (AEC: k_tot) is used to determine the meteorological sensitivity of each of the tabled Babylonian observations.
The AEC is varied within realistic boundaries: the minimum AEC evaluated is 0.15 [-] (best of excellent visibility) and the maximum AEC evaluated is 0.9 [-] (average of good visibility), in bins of 0.05 [-]
Each observation has been attached with a Range of AEC (RAEC), which predicts the particular tabled Babylonian observation date: the actual (Table 1) or the pre/postdicted (Table 2) date.
All observations where the RAEC starts with the same AEC (Start AEC: SAEC) are grouped
Starting with the group with the lowest SAEC; the observations are counted in an accumulated way for all AEC bins.
Each colored line represents the accumulated count of observations up to a certain SAEC (see legend in the graph)
The number of observations is normalized to the number of observations that have been seen with a SAEC = 0.15 [-].
Although meteorological and photometric (acuity, age and luminance of objects) parameters can have a relatively large influence (1 or in rare case 2 days difference on first visibility of the Moon: 1-2 IFA [Ideal First Appearance]); it is relatively small compared to the impact these have when looking at planets (where it can have 5-6 days IFA). So the Moon's first visibility is relatively insensitive to meteorological and photometric parameters. A comparable evaluation has been done for Mercury.

Determining the RAEC and SAEC distribution for each table

The following things can be gotten when evaluating the two tables with the above mentioned method.

Table 1 graph (positive observations with actual observation dates):

All observations can be explained by realistic RAEC.
A large RAEC is present even for observations that have a high SAEC
At the peak of accumulated observations the SAEC is ~0.55 [-]
As a test the date ordered observations have been split in three time periods, in all three cases (1 [-567 to -207], 2 [-233 to -161] and 3 [-176 to -73]) the distribution of the SAEC/RAEC stayed comparable.
A large range of weather conditions is responsible for this spread of the observations.And the results of the observation method(s) seemed to have stayed consistent over the whole time.

Table 2 graph (negative Babylonian observations with Babylonian pre/postdicted observation dates):

All observations can be explained by realistic RAEC.
For the pre/postdicted (negative) observation, it looks the pre/postdicted date have a small SAEC and a small RAEC.
The RAECs/SAECs are small compared to RAECs/SAECs seen for positive observations (Table 1)
At the peak of accumulated observations the SAEC is ~0.35 [-].
As a test the date ordered observations have been split in three time periods (labelled below as 1 [-465 to -245], 2 [-241 to -161] and 3 [-161 to -72]), and the further away the time period is the more compact the distribution of the SAEC/RAEC is.
The above could be due to the pre/postdiction criterion the Babylonians used; in the earlier periods this criterion looks to be biased towards good weather situations and might have changed over time.
The chances near low AECs are quite high (between 75%-90%), so the chance of a real first crescent a day earlier than the pre/postdicted day is smaller than in the observed crescent Table 1. This could mean that the Babylonian pre/postdicted criterion was quite accurate.

Conclusions

The use of Schaefer's criterion

For this comparison a 100% proof benchmarking of Schaefer's criterion is less important, as we do:

a comparison between two Tables that are handled in the same way and furthermore
a kind of averaging is done by looking at the RAEC distribution.

These two aspects will iron out small (not big!) deviations in the criterion from the actual behavior. Schaefer's criterion though adds an essential part to this evaluation (compared to other criterion): the meteorological parameters. These meteorological parameters are the most important for understanding first/last visibility while these parameters are omitted from all other criterions! So Schaefer's criterion hopefully will be seen as indeed being able to predict reality better.

No obvious false observations when using Schaefer's criterion

The early and late accuracy designations by Stern [2008], come out in this evaluation as observations under excellent and fairly good meteorological conditions. So they are in this evaluation not seen as false.
The validity of Schaefer's criterion needs testing not only in Babylonian times, but also in recent times; as done in this bench marking process.

Evolution of observations and pre/postdictions

Looking at the comparable distribution of the RAEC/SAEC for the split time periods in Table 1; one deducts that no obvious differences in date determination has happened when actual observations were done. So it looks that the results of the observation method(s) were consistent over time.

When doing a comparable evaluation with Table 2, there is a larger dependency per time period: the later the dates are the smaller the SEAC and REAC are.
It looks the pre/postdiction evolves closer to the distribution of the actual observations. So did the Babylonian criterion became over time more realistic and depending on e.g. meteorological parameters?

Comparing the tables

Looking at the behavior of the AEC for each Table/graphs, one can deduct that the dates in each Table are related to a different meteorological distribution. So comparing the tables will need to be done with care due to these statistical/distribution differences.

General

In general, the statement of Stern ([1999], page ) is indeed mapping the above evaluation; although the above evaluation is more directly coupled to the meteorological dependency.

It seems reasonable that sightings should sometimes be late (for instance, because of bad weather or atmospheric conditions), and rarely early (since early sightings are purely erroneous). But there is no particular reason why non-sightings, which are effectively pre/postdictions, should sometimes be early never late. This suggest only that the Babylonian astronomers' new moon visibility criteria - which we do not clearly know - were conductive to early pre/postdictions.

Warning about pre/postdictions

Don't forget to check the general conclusions.

Acknowledgements

I would like to thank the following people for their help and constructive feedback: Hermann Hunger, Sacha Stern 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

Home

Mail

Major content related changes: Jan. 29, 2009