Here is some information:
2. Nabon. No. 8, or the Hillah stele, was discovered in the neighborhood of Hillah, southeast of the ruins of Babylon, at the end of the 19th century. A transcription of the text was first published in 1896 and a second in 1912. The information given in this stele helps to establish the length of the whole Neo-Babylonian era from Nabopolassar to the reign of Nabonidus. This inscription, too, contains a record of astronomical observations which enables us to fix the reign of Nabonidus. The stele tells of occurrences in Nabonidus's accession year and his first full year, and contains a description of a configuration of planets and stars observed by Nabonidus in an unnamed evening during this period. It is stated that Venus, Saturn, and Jupiter were visible after dusk while Mars and Mercury were absent. Certain bright stars were also mentioned. If, as has been established, Nabonidus ascended the throne in 556 B.C. and his first full year was 555/4 B.C. (Nisan-Nisan), we should find this configuration of stars and planets during that period. The above mentioned Hildegard Lewy calculated the date for this configuration and concluded: "The only time within the given interval when this constellation occurred was the period of 3 days comprised between Simanu 2 and Simanu 6 of Nabu-na'id's first full year (May 31 to June 4, 555 B.C.), during which period, in fact, also the fixed stars enumerated by the king were visible in the evening sky." So again, we find Nabonidus's reign astronomically fixed and his seventeen years of rule confirmed.
In several of his royal inscriptions (Stelenfrgm. III,1 and XI, Nabon. H1,B and Zyl. III,2) Nabonidus says that in a dream in his accession year he was commanded by the gods Marduk and Sin to rebuild the temple e.hul.hul in Harran. In connection with this the text under discussion (Nabon. No. 8) provides a very interesting piece of information: "As to the temple e.hul.hul in Harran which was in ruins for 54 years -- through a devastation by the Manda-hordes the(se) sanctuaries were laid waste -- the time (predestined) by the gods, the moment for the appeasement (to wit) 54 years, had come near, when Sin should have returned to his place." The date when the temple e.hul.hul in Harran was ruined by the "Manda-hordes" is known to us from two different reliable sources: The Babylonian chronicle BM 21901 and the Harran inscription Nabon. H1,B (this is described below). The chronicle states that in the 16th year of Nabopolassar, in the month of Marcheswan, "the Umman-manda (the Medes), [who] had come [to help] the king of Akkad, put their armies together and marched to Harran.... The king of Akkad reached Harran and [...] he captured the city. He carried off the vast booty of the city and the temple." The Nabonidus stele H1,B gives the same information: "Whereas in the 16th year of Nabopolassar, king of Babylon, Sin, king of the gods, with his city and his temple was angry and went up to heaven -- the city and the people that (were) in it went to ruin."
Thus Nabonidus reckons the 54 years to be from the 16th year of Nabopolassar to the beginning of his own reign when the gods commanded him to rebuild the ruined temple. This is in excellent agreement with the figures for the Neo-Babylonian reigns given by Berossus and Ptolemy. As Nabopolassar reigned for 21 years, 5 years remained from his 16th year to the end of his reign. After that Nebuchadnezzar ruled for 43, Evil-Merodach for 2, and Neriglissar for 4 years before Nabonidus came to power (Labashi-Marduk's few months may be neglected). Adding up these regnal years (5+43+2+4) we get 54 years -- exactly as Nabonidus states on his stele. If, as has already been established, Nabonidus's first year was 555/4 B.C., Nabopolassar's sixteenth year must have been 610/609, his first year 625/4 and his 21st year 605/4 B.C. Nebuchadnezzar's first year, then, was 604/3, and his 18th, when he destroyed Jerusalem, was 587/6 B.C. These dates agree completely with the dates arrived at from Ptolemy's king list and Berossus's figures.
Consequently, this stele alone establishes the length of the whole Neo-Babylonian era. It fixes the reign of Nabonidus astronomically, and it gives the total length of the reigns of all the Neo-Babylonian kings prior to Nabonidus. The strength of this evidence from the Neo-Babylonian era itself can hardly be overestimated.
From: http://corior.blogspot.com/2006/02/part-2-discussion-of-historical.html
I got this by Google of "Hillah stele." Most of the information you want can be found if you just Google it.
I'm glad I checked further because the "Hillah stele" is Nabon No. 8. The document I referred to earlier was Nabon 18, a different text, so the astronomical event in that text is different than in this text. Sorry for any confusion.
LS