Rosh Chodesh: One or two days?

A follow up to questions and discussion on Shabbat afternoon about why Rosh Chodesh Elul is two days.

The answer is entirely predictable and has nothing to do with when a crescent moon is seen. In fact, we are said to have switched from a Sanhendrin-dependent validation of the observation of the crescent moon (with some math and common sense thrown in, probably) to a mathematical, formulaic model in the 360 CE (Hillel II) timeframe.

The reason we sometimes have Rosh Chodesh for one or two days is due to the conjunction of the moon's revolution and the rotation of the earth. The new moon is off a half day from when the sun sets (connecting back to the astronomical story, below), so, as you all probably know, each lunar month is actually 29.5 days. We handle this by alternating months of 29 days (called chaser, or defective) and months of 30 days (malei, full). When a month has 30 days, its last day plus the first day of the next month are both counted as Rosh Chodesh of the following month.

Tevet, Adar I and II, Iyar, Tammuz, and Elul are chaser and thus celebrate Rosh Chodesh for two days (starting on the last day of the previous month), and Tishrei, Shevat, Nisan, Sivan, and Av are malei and have a single-day Rosh Chodesh (Tishrei's being Rosh Hashanah, which then begs the question why that holiday may be celebrated for two days. And I'm not going to ask that question right now).

For example, if you look up Rosh Chodesh Sh'vat (which is malei) in Hebcal, it says that it begins on 1 Sh'vat. And if you look up Rosh Chodesh Adar (which is chaser) in Hebcal, you'll see that it actually begins on 30 Sh'vat.

The rabbi, who announced in English that Elul is Monday and Tuesday, is right, in a way. Because Elul is chaser, it has a two-day Rosh Chodesh, so its Rosh Chodesh starts on 30 Av, which is indeed Monday night; this Rosh Chodesh does go through Tuesday and Wednesday (until sunset), so the cantor, who chanted those days in Hebrew, was right on that front. However ... the English of our siddur says, "The new month begins on...." For a chaser month like Elul, its Rosh Chodesh begins on Monday night, but Elul does not begin until Tuesday night. (That a month can have possession of a Rosh Chodesh that is outside of its own dates is kind of cool in an anthropomorphic kind of way.)

Kislev and Cheshvan fluctuate -- separately  -- in order to enable us to follow the "four rules of postponement." (I'm not even going to read that article now....) They are related to manipulating the calendar to ensure that Rosh Hashanah is scheduled in such a way so that the rest of the the High Holy Days do not fall on certain days of the week in relation to Shabbat.

Maimonides goes into all of this as well as all the calculations in the Mishneh Torah, Sefer Zemanim, Kiddush HaChodesh (19 chapters!), as you can imagine. The Wikipedia "Hebrew Calendar" entry has a ton of mathematical detail as well.

There is also a bit of an astronomical story, which  I would like to think of as the reason we continue to have a two-day Rosh Chodesh. An astronomical new moon reflects no sunlight and is not visible; the religious new moon is the moment when a crescent becomes visible, which is at the end of the phase of the astronomical new moon.  And, as I remember from astronomy class, the new moon rises in the morning and sets in the evening. You'll never have a new moon at 3 AM (and you'll never have a full moon at 3 PM).  The crescent may or may not first appear just around sunset, just as the new moon is setting, and whether or not sunset occurs before the crescent is seen and whether we would have to wait until the following day's moonrise would be the source of the Sanhendrinic debate.

To apply this, on August 25, 2014, in Jerusalem, the (astronomical) new moon will rise at 5:55 AM, begin to crescent at 5:13 PM, and set at 6:55 PM. The sun will rise at 6:10 AM and set at 7:12 PM, so the moon will set before it is dark.  I suspect that if we were looking for a crescent moon, we might not see it until August 26 (when, in fact, moonset is after sunset, so if we can't see it in the morning we'd definitely see it that night).