The real world star Alpha Eridani (or "Achernar") is located approximately 139 light years from the Sol system. According to Captain Jonathan Archer, in ENT: "Two Days and Two Nights", set in 2151, the farthest that Humans had traveled at that point was 90 light years. Assuming Alpha Eridani II was settled by Humans, this would suggest the colony was established some time between 2151 and 2156.
Star Trek Maps, Star Trek Concordance and an entry on Startrek.com refer to the planet mentioned in "Wolf in the Fold" as "Alpha Proxima II". The pronunciation of this planet in the episode sounded more like "Alpha Aradoni II", which was also supported by the closed captioning. The spelling of "Alpha Eridani II" can be found in the Redjac entry on Startrek.com and in the Star Trek Encyclopedia as well as the Star Trek: Star Charts. According to Star Trek Maps (pg. 22), Alpha Proxima II was one of the first extra-solar Human colonies, with major industries in mining, farming and sea-farming. By the 23rd century, the planet had a total population of 12 million with more than 3 million living in the port city Heliopolis.
Taking place in the late 24th century, the novels Articles of the Federation and Rough Beasts of Empire, as well as the Star Trek Maps and Star Trek: Star Charts place the star Achernar within the Romulan Star Empire. The latter describes Achernar as a binary star system, consisting of two B-class components with a magnitude of +1 (hundred times brighter than Sol).