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Comparing actual Babylonian dates of Mercury's visibility with Schaefer's criterion

64 heliacal events (morning first and morning last) have been gotten from Assar, who deducted these from Hunger H. and Sachs A. J. (Astronomical diaries and related texts from Babylon, Vienna, Volumes I [1988], II [1989], III [1996], V [2001], VI [2006]).

The data is originally in the form of:
13.X.110 SEB
The 13th, Mercury's first appearance in the east in Sagittarius; it was bright and high, rising of Mercury to sunrise: 20º, ideal first appearance on the 8th.

A few parameters can be deducted from the above:

The data in Babylonian calendar (13.X.110 SEB) of the observed heliacal event
The planet (Mercury)
The planet's location in the sky (Sagittarius)
The type of heliacal event (morning rising, first appearance)
An indication of the time (expressed in degrees) between rise of planet and sun (20º). Is also called na-su.
the Ideal First Appearance (IFA), which is a number of days before the actual heliacal event (8^th). This might be a post/prediction of the Babylonians.

The principles behind the evaluation are:

The date of the first visibility of the Mercury is depending on several parameters (like astronomy, observation location, photometric and meteorological parameters). The calculations are based on Schaefer's criterion.
A table with 64 actual Babylonian observations dates is taken from F. Assar (pers. comm. [2008]).
One date -181/1/4 was wrongly deciphered from the cuneiform signs, it should be -181/1/14 (F. Assar and H. Hunger pers. comm [2008])).
An illustration of the different start/end times in Babylonian calendar and the Julian Calendar (important when doing conversions for moon's first visibility settings):
In this evaluation the corrected conversion table of Chris Bennett has been used (that conversion is based on several sources). Five Babylonian dates (16.X.30 SEB, 10.XI.48 SEB, 27.IV.140 SEB, 24.III.141 SEB and 7.VIII.141 SEB) had a different Julian calendar date than Assar's data, the dates from the conversion table has been used.
Conversion of Babylonian dates to Julian calendar dates might have some circular argument in it as some (not all) dates are derived using Schoch's criterion.
In two cases, where Assar provides a range of two dates; the earliest date has been used (-155/9/17 and -129/9/22)
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 0.125 [-] (best of excellent visibility) and the maximum AEC evaluated is 0.55 [-] (best of good visibility), in bins of 0.025 [-]
Each observation has been attached with a Range of AEC (RAEC), which predicts the particular actual tabled Babylonian observation 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)
Although meteorological and photometric (acuity, age and luminance of objects) parameters have a relatively large influence (6 days difference: 6 days IFA [Ideal First Appearance]) on first visibility of the Mercury); it is also relatively large compared to the impact these have on the Moon (where it can have 1-2 days IFA). So Mercury's first visibility is relatively sensitive to meteorological and photometric parameters. A comparable evaluation has been done for the Moon.

Determining the RAEC and SAEC distribution

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

All observations can be explained by a realistic RAEC.
The RAECs are small (due to sensitivity of the visibility of the event on meteorological parameters)
At the peak of accumulated observations the SAEC is ~0.25 [-].
Proof is being sought to determine that the above picture maps the distribution seen for first observations of the Moon. The two pictures are the convolution between the actual AEC at the time of observation and the sensitivity (RAEC size) of the first visibility of the celestial object.

Analyzing the difference between actual observation and IFA

For several (26) of the above 64 observations the Ideal First Appearance (IFA) [days] was note by the Babylonians (pers. comm. Assar [2008]).
One might (not sure though!) see the IFA as a kind of pre/postdiction result based on optimum meteorological circumstances. It is interesting to see that the IFA in days is very close to the variability seen when varying the AEC: around 6/7 days.
So a possible correlation was determined between the calculated AEC (using Schaefer's criterion) for the actual observation and the recorded Babylonian IFA. Some results are in the below graph:

What do you see in the above picture:

the blue dots/line are the 26 IFAs recorded by the Babylonians with the AECs that match these actual observations (AEC determined with Schaefer's criterion).
for three dates (-302/2/2 [yellow], -171/6/18 [brown] and -156/9/1 [pink]) the variability of the dates (so a calculated IFA: not from the Babylonians) has been determined by changing the AEC.
One can see that the variability of the AEC gives a range of IFAs that is overlapping the (blue) Babylonian IFAs.
The sensitivity of the above behavior will not be largely depending on the specifics of the criterion, as it is using a differential.

Conclusions

Ideal First Appearance (IFA) and Object's altitude

It is important to realize that the Babylonians recognized the IFA, means they recognized more or less celestial and meteorological parameters (Swerdlow [1998], page 17).
Furthermore it is interesting to see that the meteorological parameters mostly works on the altitude the celestial object will be visible for its first appearance, not on the Sun's altitude. So the object's altitude at first visibility date is highly correlated to the IFA, and could perhaps be used to decide on the IFA!

Babylonian pre/postdictions

If the Babylonians used some pre/postdiction criterion to determine the IFA, then there is indeed a correlation between the recorded IFA and the possible AEC (as calculated with Schaefer's criterion). This is also mirrored by Swerdlow ([1998], page 52)

Warning about pre/postdictions

Don't forget to check the general conclusions.

Acknowledgments

I would like to thank the following people for their help and constructive feedback: Farhad Assar, Chris Bennett, 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.

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Major content related changes: Feb. 1, 2009