I found this curious article from Chabad in my inbox that attempts to answer the question Why Is the Prayer for Rain Based on the Civil Calendar? I’ve had this question numerous times, basically every time I read the line "say ‘tal uMatar’ starting from September 5th" (‘tal uMatar’ refers to a small insertion to pray for rain to fall in the Amida). Why this strange date? Why not a Hebrew date.
You can read the calendar details in the Chabad article or in some other article on this topic (e.g., Arnold A. Lasker and Daniel J. Lasker: The Strange Case of December 4: A Liturgical Problem). Basically, the rabbis in Babylonia determined that the Jews (of Babylonia) should start to pray for rain starting on the sixtieth day after the tekufah (Ta’anit 10a, Rambam Hilchot Tefilah 2:16). The tekufah refers to the halachic start of autumn. This should coincide with the the equinox, September 22nd, based on some issues with the assumed length of the year, the Julian calender and so on, we end up with September 24th in the Julian calender. 60 days later would make the start of saying ‘tal uMatar’ November 22nd.
But the Julian calender has some problems (the year is a tiny bit, .0078 days, too long) and the Gregorian calender was introduced. If we translate September 24th of the Julian calendar into the Gregorian calendar, we end up with the date of October 7th (for this century roughly). Adding the 60 days, we get December 5th for starting to say ‘tal uMatar’. As the day starts the night before, start at Ma’ariv of December 4th to say the blessing. This is what most of the Jews outside of Israel on the northern hemisphere do.
So, to sum up, we start to add the prayer of rain on December 5th because it’s when rain is needed in Babylonia, based on incorrect calculations. Why don’t we just pray for rain when it is needed in the country we live in? I just don’t get the point.