I'm really asking a general question about the study of history. Allegedly, the Babylonian Empire fell to the Persians under Cyrus the Great in 539 BCE. All articles regarding this event simply take it for granted as an established fact. How would I go about finding the precise sources and reasoning that cause historians to generally agree on historical facts, such as Babylonia succumbing to the Persians in 539BCE?
One way to go about this type of research is to simply dig deeper, one step at the time: review the bibliography of the articles you've run into, and read the citations of potential interest. Rinse and repeat until you finally locate one or more articles that argue about the precise date - historians aren't the type of scholars who take ancient texts at face value, so you'll necessarily run into such articles if you dig deep enough.
A shortcut for the above is to head to Google scholar directly, enter something like "Fall of Babylon date", and hit Search. Doing so will reveal a heap of articles that argue about the date and how it was arrived at (i.e. considerations such as astronomy, other events that were occurring at the same time documented in this or that source, and so forth).
The shortcut method might require a bit of digging deeper too, but it'll often be faster. This article from the first page of the above-mentioned Google Scholar results, for instance, includes a tasty looking citation in footnote 7. The book is out of print but you can download it from U Chicago:
R. A. Parker and W. H. Dubberstein, Babylonian Chronology 626 B.C.-A.D. 75 (Providence: Brown U, 1956), 29.