1) Listen to this morning's class, on the halakhic history of when to begin Barekh Alenu, here. Follow along with the source sheets here.
2) For further research, read the classic article on this topic, "The Jewish Prayer for Rain in the Post-Talmudic Diaspora", by Arnold and Daniel Lasker, here.
Read Shlomo Brody's brief summary, in A Guide to the Complex, here.