One book I reviewed was Babylonian Star-lore. An Illustrated Guide to the Star-lore and Constellations of Ancient Babylonia by Gavin White. This was an interesting book, and it makes the (apparently controversial) claim that Mesopotamian astronomers/astrologers (back then there was no difference) discovered the precession of the equinoxes. I have no special knowledge of the history of astronomy, but prima facie this seems plausible, since they had been recording astrological observations for thousands of years -- plenty enough time for the difference to become apparent.
Why then do all the history books say that the precession of the equinoxes was first noticed by Hipparchus a little more than a century before Christ? Being an astrologer was a good livelihood back then, and the methods used to make their important predictions -- for kings and rich men -- would have no doubt been an ancient example of a trade secret. (How to make Damascus steel and Greek fire were somewhat later examples of such trade secrets; they were kept so secret that the knowledge eventually died out.)
The Mesopotamian astrologers certainly took some simpler steps to confuse the uninitiated, creating a system of multiple meanings in which a planet might be called by the name of a constellation, for example. Other differences in their terminology appear to be due to more fundamental cultural differences. For them, a star was any noteworthy phenomenon in the heavens, rather than our modern idea of a huge ball of gas kept glowing hot by nuclear fusion -- just like the word "fish" used to mean any animal that spent its whole life in water, including whales, shellfish, jellyfish, starfish, etc., but today we use "fish" in a much narrower sense. White's book particularly mentions halos around the moon as phenomena that the Mesopotamians considered "stars".
This has to be kept in mind when reading the account of the Star of Bethlehem and the visit of the Magi. It is a strange story for several reasons.
For one, St. Matthew's Gospel is notably oriented towards a Jewish audience. In fact, ancient sources tell us he wrote both the Greek version we have today and another version (now lost) in Aramaic. Astrology was much less acceptable among the Jews (due to numerous Old Testament warnings) than among the Gentiles, yet it is only in St. Matthew's Gospel that we read of the Magi.
For another, no one else seems to have seen anything particularly out of the ordinary, and certainly no one else seems to have drawn the same conclusions about whatever was seen as the Magi drew. On the other hand, Herod and his court seem to find the account of the "star" believable -- though the court shows no sign of having believed the conclusions of the Magi, only of being (rightly) "troubled" by Herod's possible reaction.
This makes it seem likely that one of the points made in the video below is correct: many people likely saw the same thing the Magi did, but did not find it striking, let alone attribute much importance to it.
Finally, one of the strangest parts of the whole story is Matthew 2:9:
Who having heard the king, went their way; and behold the star which they had seen in the East, went before them, until it came and stood over where the child was.This is easy to understand if the "star" was within 500 feet of the ground, as is often shown in art, but it makes no sense whatsoever for a supernova, a comet, or a planetary conjunction -- so either the star must not have been any such thing, or we have to understand "stood over where the Child was" in some different sense. For example, there is a star cluster in the constellation Cancer that was known to the Greeks as "The Manger" -- maybe the "Star" of Bethlehem appeared near "The Manger", so the Magi, combining this with what Herod's court had told them, looked for any baby in a manger in Bethlehem? (This could be if the "Star" first appeared to them much earlier, giving them time to travel. I like to think that the "Star" actually first appeared at the Annunciation, so they would have had nine months to make it to Bethlehem.)
In the end, all this is empty speculation. We do not know what astrological system the Magi were using. It may have been closely related to Greek and Roman astrology -- but again, it may not.
One last speculation. There are very few passages in the Old Testament that might give a hint regarding the timing of the birth of Christ, but one that sounds promising to me is Psalm 109/110: 3, which says (Douay-Rheims translation), "... from the womb before the day-star I begot thee." This is a messianic passage, and the day-star (the sun, or in at least one other translation, the morning star) is, in a narrow, literal sense astronomical. The Fathers of the Church did discuss this passage, but to the best of my knowledge they never linked it to what the Magi saw -- but maybe, just maybe, there is a connection here, somehow passed on through the Jewish exile into Babylon.