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It is known that there is a slight discrepancy in the average length of a year according to the Hebrew calendar and the average length of an astronomical solar year which causes a seasonal drift in the Hebrew calendar of about 6.5 minutes per year. My question is, does the fixed calendar slightly drift from the astronomical lunar month or is the average length of a fixed Hebrew-calendar month exactly the same as the average length of an astronomical lunar month? Assuming there is some amount of drift, how much is there?

From my amateur perspective, it seems that if there is a lunar drift, it is much smaller than the solar drift as Rosh Hodesh continues to correspond with the new moon and if there was a significant lunar drift, the new month would begin to occur farther away from the new moon. I understand that the molad often does not fall exactly on the day before Rosh Hodesh, but I don't have much in-depth knowledge of how the molad works and how it relates to Rosh Hodesh.

  • As the molad period is fixed and not expressed with infinite precision, it's impossible for there not to be some systematic drift over time in one direction or the other. The only question is how much. I'm sure the answer to that is available. Regarding your last sentence, note that the lunar period is variable, with its average being very close to the molad period, so molads defined by the fixed period will naturally fall in a distribution around the actual time of lunar conjunction. – Isaac Moses Aug 26 '16 at 17:35
  • This has nothing to do with the fixed calendar, btw. The calculation of Moladot was used by Beit Din to evaluate witnesses. – Double AA Aug 26 '16 at 17:49
  • In terms of your last sentence, the Molad only matters for determining Rosh Chodesh Tishrei. All other month lengths are just fixed so that the next Rosh Chodesh Tishrei works out right, even if they themselves are off. – Double AA Aug 26 '16 at 20:49
  • Possible duplicate of How accurate is the Molad? – Yishai Dec 29 '16 at 3:41
  • @Yishai As I mentioned in my question, I don't know a whole lot about how the molad works at all; however, I don't believe the amount of time between the molad of two months is exactly the same as the average length of the fixed-calendar month. Therefore I believe the questions are different albeit related. Thanks for pointing that question out! – Daniel Dec 29 '16 at 14:55
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Wikipedia says:

This [the molad] value is as close to the correct value of 29.530589 days as it is possible for a value to come that is rounded off to whole parts (1/18 minute). The discrepancy makes the molad interval about 0.6 seconds too long. Put another way, if the molad is taken as the time of mean conjunction at some reference meridian, then this reference meridian is drifting slowly eastward. If this drift of the reference meridian is traced back to the mid-4th century, the traditional date of the introduction of the fixed calendar, then it is found to correspond to a longitude midway between the Nile and the end of the Euphrates. The modern molad moments match the mean solar times of the lunar conjunction moments near the meridian of Kandahar, Afghanistan, more than 30° east of Jerusalem.

Furthermore, the discrepancy between the molad interval and the mean synodic month is accumulating at an accelerating rate, since the mean synodic month is progressively shortening due to gravitational tidal effects. Measured on a strictly uniform time scale, such as that provided by an atomic clock, the mean synodic month is becoming gradually longer, but since the tides slow Earth's rotation rate even more, the mean synodic month is becoming gradually shorter in terms of mean solar time.

  • To be explicit, the mean synodic month is 29d 12h 44m 2.802s, while the Molad period is 29d 12h 44m 3.336s. So it drifts very slowly. Just as a rough estimate: 1750*12*(3.336-2.802)=3.1 hours which is about 45 degrees longitude. Kandahar is about 30.5 degrees longitude off from Jerusalem. – Double AA Aug 26 '16 at 17:51
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As discussed by the Shvili D'Rakia (sections ד and ז), each synodic period (the time between two consecutive moladim) is 29 days, 12 hours, and 793 chalakim. However, we can't split a single day between two months. Therefore, generally speaking, months alternate between 29 and 30 days, with a loss of 793 chalakim.

Now, as discussed further by the Tiferes Yisrael (sections יא and יב), this didn't always happen as planned. (In fact, the Tur notes that the months only alternate about 29.5% of the time.)

Thus, over the 247-year cycle, the calendar runs short by 905 chalakim. In order for the calendar to repeat itself exactly, it will take 609,872 years.

  • I don't see how this answers the question. The calendar is fixed, but it drifts. In 180000 years, Rosh Chodesh will fall on the new moon if we keep using the current value of 29/12/793. That means after the 689472 years, we'll have drifted about 60 days ie two months. I guess we should add an extra Adar II every 300000 years :) – Double AA Aug 26 '16 at 20:50

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