those present said... that the perpetrators should not be exempted from punishment. Herod therefore dealt rather mildly with these others but removed the high priest Matthias from his priestly office as being partly to blame for what had happened, and in his stead appointed his wife's brother Joazar as high priest. Now it happened during this Matthias' term as high priest that another high priest was appointed for a single day - that which the Jews observe as a fast - for the following reason. While serving as priest during the night preceeding the day on which the fast occurred, Matthias seemed in a dream to have intercourse with a woman, and since he was unable to serve as priest because of that experience, a relative of his, Joseph, the son of Ellemus, served as priest in his place. Herod then deposed Matthias from the high priesthood. As for the other Matthias, who had stirred up the sedition, he burnt him alive along with some of his companions. And on that same night there was an eclipse of the moon. (Antiquities 17.164ff, emphasis mine)The Loeb footnote is almost certainly justified to identify this as the Fast of Esther, which evidently began at sundown on March 12th, several hours before the eclipse that took place after midnight on March 13th. Josephus' language leaves little room for doubt about this. Although it is not unlike Josephus to include random information in an aside, or to render flashbacks with unspecific chronology, his asides are never completely random and their details are usually significant to the surrounding narrative. In this particular passage, the details strongly suggest a very tight correlation of events.
Here is what must have happened. Around dawn on the morning of March 12th, the chief men of Jerusalem were gathering to depart for the meeting Herod had called at Jericho. With good horses, they could easily arrive by noon. But the word must have already spread that the high priest had recused himself. The scandal itself would not have been so outrageous, but only the King could legally appoint new high priests. So when Matthias chose his own replacement, even a temporary one, the chief men were obligated to report it to Herod.
But Matthias must have had both friends and enemies in the Jerusalem embassy. His political opponents would take any opportunity to replace him as high priest, while his friends would be bound by their own safety not to withhold the facts from King Herod. As it turned out, the friends and enemies of Matthias may have compromised. Josephus does not tell us that Herod knew about Matthias' offense, but the high priest has suddenly and somehow, implausibly, come to be "partly to blame" for what happened with the golden eagle some days or weeks before.
So far as we know, Matthias had nothing to do with the two rabbis and their overzealous students. He was not arrested, bound or executed with the others whom Josephus specifically names as responsible for the assault on the eagle. If Josephus' story about the rabbis is accurate, then Matthias was innocent, and therefore someone at the Jericho assembly must have come up with their own reasons to name Matthias "as partly to blame" (ws aition tou merous).
This scenario makes Josephus' flashback/aside perfectly relevant to the content and chronology of the burning. The mention together of a fast and an eclipse naturally evokes Purim, which always occurred at the full moon one month before Passover. Josephus' strong suggestion, therefore, is that this eclipse occurred at the time of that festival - which puts Herod's death solidly in 4 BC.