On Dating the Lunar Eclipse of Alexander and the Battle of Gaugamela: Discussion of Evidence and Use of Archaeoastronomy for Chronology

Here we propose an archaeoastronomical discussion of evidence concerning the dating of the Battle of Gaugamela. This battle was preceded by a total lunar eclipse, which is also known as the eclipse of Alexander, used to date the decisive battle of the invasion of the Persian Achaemenid Empire. This eclipse is one of the many used for chronology.

find in Wikipedia. From that battle, "Though heavily outnumbered, Alexander emerged victorious due to his army's superior tactics and his deft employment of light infantry. It was a decisive victory for the Hellenic League and led to the fall of the Achaemenid Empire." [7] In [7] it is told that "After the Macedonian army had crossed the Tigris a lunar eclipse occurred.
[W15] Following the calculations, the date must have been October 1 in 331 BC [W17]. Alexander then marched southward along the eastern bank of the Tigris. On the fourth day after the crossing of the Tigris his scouts reported that Persian cavalry had been spotted, numbering no more than 1000 men." The sentence of Wikipedia -following the calculations, the date must have been October 1 in 331 BC -is not good, because it seems referring to the lunar eclipse and not the Battle. In any case, let us consider the given notes [W15] and [W17], and investigate how the date had been determined.
[W15] refers to Arrian's Anabasis, that we can find at the following link https://en.wikisource.org/wiki/The_Anabasis_of_Alexander/Book_III/Chapter_VII . This Anabasis of Alexander was translated by E. J. Chinnock. Chapter VII -Passage of the Euphrates and Tigris.
Here the part concerning the lunar eclipse and the related notes.

Anabasis of Alexander [W15] Alexander arrived at Thapsacus in the month Hecatombaion,[a1]
in the archonship of Aristophanes at Athens; and he found that two bridges  So we see from Arrian, that more than eight days passed from the eclipse to have the two armies close each other.
The Anabasis of Alexander was composed by Arrian of Nicomedia in the second century AD, most probably during the reign of Hadrian. The Anabasis -complete, in seven books -is the history of the campaigns of Alexander the Great, between 336 and 323 BC. From the Anabasis of Alexander we have the year, 331 BC, of the Battle of Gaugamela. It is given as the year of the archonship of Aristophanes at Athens. It seems that the year in the calendar used by Athenians was starting from the summer solstice. So the Aristophanes' rule was covering half year 331 BC and half 330 BC.
However, we have another remarkable detail in Arrian's text, the lunar eclipse. And this astronomical event is giving the year. In the note [a4], we read that it was on 20 September 331 BC.

CalSKY and Stellarium
As previously told, we can use some astronomical software to simulate the sky in the past. In particular, we will use CalSKY and Stellarium. The date is given according to   If we know the astronomical event, we can properly define the time of it, given in the astronomical framework of the Julian date in the Julian Period. If we use Stellarium, we can also observe the part of the sky where it was occurring ( Figure 2). We need to be careful using Stellarium, where year "0" means 1 BC. The conventional BC/AD scheme has no year 0, thus the year before 1 AD is 1 BC. Astronomers often use a different scheme, referring to AD dates as a positive integer, and redesignating 1 BC/BCE as the year 0. Then, -330, 9, 20 gives us the moon in the same position given by CalSKY.     quia ad annum Iulianum dumtaxat accomodata est" [9]. Julian refers to Julius Caesar, who introduced the Julian calendar. In origin, the Julian Period was used to count years, but in his book Outlines of Astronomy, first published in 1849, the astronomer John Herschel added the counting of days elapsed from the beginning of the Julian Period [9,10]. Today, astronomers adopted the Herschel's "days of the Julian period" to count time, using the meridian of Greenwich. This contemporary account tells us that, when the eclipse started occurring Jupiter was setting, and that the moon was near Saturn; using Stellarium we can see Saturn (see the Figure 2) and Jupiter setting. We find also the date of October 1, 331 BC, was that we have found by means of Plutarch. Greek mercenaries. Darius was in danger of being cut off, and the widely held modern view is that he now broke and ran, with the rest of his army following him. This is based on Arrian's account: Actually, the passage concerning Darius' army break in the tablet of the Astronomical Diary has two possible translations, that we can find in [12] and [13]. A detailed discussion is given in [14]. In any case, the contemporary record tells us that, as given by Plutarch, the armies were facing each other on October 1.