It's the end of an era (but how did they know?)
 

The ages of the sun all ended in great age defining events, the fall of Thangorodrim, The War of the Last Alliance and the Fall of Barad-Dur. But how did the inhabitants of Arda know when the Ages of the Stars were over since nothing comparable happened there. Like Melkor was doomed to be held captive by the Valar for two (or was it three) ages. Who determined the timespan of these ages? Did Eru decide after a random allotment of years he felt like a new age so he told Manwe or something?

After the making of the trees, the known counting of time began (the previous one is not known to the Eruhini); concerning the counting:
One year of the valar was roughly 10 Sun Years.

I'd always assumed the Ages of the Stars were much longer than the Ages of the Suns. Doesn't Tolkien mention somewhere about them being briefer?

The ages of the Sun are not based on astronomy but on history; as such, their length is variable, and, arguably, random. Tolkien speculates in the letters that the ages would accelerate; as such, the last four ages, together, may have passed in as little as 6000 years.

It's always puzzled me that the most significant event since the creation of the world was not chosen as the start of a new Age.
I refer, of course, to the complete re-shaping of the world following Numenor's rebelion against the Valar.

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