Arachin 9b ~ The Problem of the Moving Molad

To understand a passage in today’s page of Talmud requires a deep dive into lunar mechanics. Take it slowly and it all makes sense. We will get back to the Talmud in a moment. But first, some astronomy.

The Molad and the lunar conjunction

Each Jewish lunar cycle begins on the molad (lit. birth). This is the moment at which the sun, the moon and the earth line up. At that time the moon is completely invisible because no sunlight can reach the side of the moon that faces us. In astronomy, this moment is called the lunar conjunction.

The precise moment of the lunar conjunction. It should be the same as the calculated time of the molad. But it isn’t.

The precise moment of the lunar conjunction. It should be the same as the calculated time of the molad. But it isn’t.

The Jewish lunar month (based on Babylonian measurements) has a fixed period of 29 days, 12 hours, 44 minutes and 3+1/3 second (or 29 days, 12 hours, and 793 chalakim, a chelek being 1/1080 of an hour). So if you know the time of the molad of one Jewish month, the next one will be precisely 29 days, 12 hours, 44 minutes and 31/3 seconds later. On the Shabbat before the start of every Jewish month (except one), the time of the molad is announced in the synagogue. The first day of that new month follows the molad, though the interval of time between the two varies. So far so good.

Here’s the Problem - and you could even see it

From  here .

From here.

The problem is that the length of the calculated molad and the actual length of time between one conjunction and the next is not the same. There was a visual demonstration of this in North America during the total solar eclipse of 2017.

A total solar eclipse occurs when the moon gets directly between the sun and the earth and they are all on the same plane. That time is of course by definition, the same as the lunar conjunction for that month. The 2017 eclipse began on August 21 at 15:46 GMT which was 5:58pm in Jerusalem. That was when the moon started to move across the face of the sun. The total eclipse - when the moon directly covered the sun, occurred at 18:25 and 35 seconds GMT, which was 8:25:35pm in Jerusalem. That is the true astronomic lunar conjunction. But the molad for that month (which was Rosh Chodesh Elul) was announced as “Tuesday, August 22, at 10:44 a.m. and 15 chalakim” (Jerusalem time) — about 12 hours and 20 minutes hours later.

That solar eclipse (which I was lucky enough to witness on the beach in Charleston South Carolina - it was amazing-) visibly demonstrated two things. First, that molad we announce on the Shabbat preceding Rosh Chodesh represents a theoretical time only, and has absolutely no relationship to any astronomical phenomenon. And second, that the molad and the lunar conjunction are often several hours apart.

Why the molad time is not astronomically correct

The length of the Jewish lunar month is very precise, but alas, not accurate. Or at least not accurate enough. This is because the actual length of a lunar month varies from month to month and from year to year. It is affected by the speed of the earth’s orbit around the sun which changes as the earth gets closer or further away from the sun, and the changing distance of the moon from the earth (neither orbit being perfectly circular). NASA kindly provided a histogram which shows the length of the lunar month over 5,000 years. As you can see, thousands and thousands of lunar months have longer or shorter lengths compared to the mean length of the month.

Length of lunation over 5,000 years.png

The actual lunar month is longer than the molad month when the earth is moving at its slowest (the aphelion) and the moon is moving at its fastest (the perigee). In addition, the average length of our solar day (fixed at 24 hours) is getting longer as the tides imperceptibly slow the spin of the earth on its axis. The moon also slows the spin of the earth; every 100 years the day is about 2 milliseconds longer.

Today the average difference between the traditional moladot and the true mean lunar conjunctions, (referred to the meridian of Jerusalem) range from 2+1/2 hours early (Nisan) to 5 +2/3 hours late (TishreiCheshvan), but the maximum differences range from 12 hours early to 16 hours late.

Sometimes then, the Jewish lunar month is shorter or longer than the actual lunar month, but over time it is not all averaging out. Something remarkable is happening: more and more of the calculated molad lunar months are falling behind the actual new moon. You can see this graphically below, thanks to the clever work of Dr. Irv Bromberg at the university of Toronto. The chart depicts the average relationship between the molad and the actual time of the lunar conjunction. The difference between the red line and the blue curve represents the error of the molad with respect to the actual lunar conjunction.

Dots above the thick horizontal line at zero indicate months in which the  molad  was or will be  after  the actual new moon. Dots below the thick horizontal line at zero indicate months in which the  molad  was or will be  before  the actual new moon.The difference between the red line and the blue curve represents the error of the  molad  with respect to the actual lunar conjunction. From  here .

Dots above the thick horizontal line at zero indicate months in which the molad was or will be after the actual new moon. Dots below the thick horizontal line at zero indicate months in which the molad was or will be before the actual new moon.The difference between the red line and the blue curve represents the error of the molad with respect to the actual lunar conjunction. From here.

Now we can tackle today’s daf

Since some Jewish months have 29 days and others have 30, the average should be 29 1/2 days, But what do we do with that extra 44m and 3secs (or 793 chalakim) left over each month? Eventually they will add up to an entire 24-hour day, and so every three years or so an additional day must be added into the length of one of the Jewish months to bring it back into synchrony with the true lunar conjunction. In such a Jewish year (if it is a regular year) there will then be 355 days instead of the usual 354.

But even this adjustment is not enough. This three-year tweek still leaves an additional 48 minutes each year which adds up to just over one day every 30 years.* So every 30 years an additional day must be added to one of the Jewish months, and (if it is a regular year) there will then be 355 days instead of the usual 354.

ערכין ט,ב

והאיכא יומא דשעי ויומא דתלתין שני

This extra fraction beyond twenty-nine and a half days is compounded month after month and year after year, and as a result, eventually there is an additional day that must be added to the calendar every three years due to the accumulation of hours, and an additional day that must be added every thirty years due to a further accumulation of the extra parts of an hour.

But as we have seen even this fix does not get us back on track over the long run. In about another 3,000 years every calculated molad will be after the actual lunar conjunction, and the difference will grow over time (following the path of the blue curve in the graph above). Who will fix the problem then?

ואין לנו לדאוג כ”כ יותר כי בודאי בעת ההיא וגם הרבה קודם יהיה הגאולה ונקדש ע”פ הראיה
There is no need to worry [about any future problems with the calendar] for certainly by that time the Redemption will have occurred and we will go back to sanctifying the new moon through the testimony of witnesses.
— Biur Halacha 427.

For the calendrically inclined:

1 Jewish lunar month= 29d 12 h 793ch

Those extra 793ch are a problem.

Over a year of 12m they add (793ch x 12=) 9,516ch or 8h 48s to the length of the lunar cycle.

So over 3 years that is an additional 24 hours = 1 day PLUS 876 ch or 48m 40 secs

Over 30 years that additional 876ch is 26,280ch or just over one day (one day is 25,920ch).

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December 4th, Shmuel and the Pope

If you live in Israel, you need not read on.

December 4th is a significant day in the Jewish liturgical calendar. It is the day on which those who live outside Israel begin to add the words ותן טל ומטר - give us dew and rain. Here is the talmudic source for this change:

תענית דף י, א

משנה. בשלשה במרחשון שואלין את הגשמים. רבן גמליאל אומר: בשבעה בו, חמשה עשר יום אחר החג, כדי שיגיע אחרון שבישראל לנהר פרת

גמרא. אמר רבי אלעזר: הלכה כרבן גמליאל. תניא, חנניה אומר: ובגולה עד ששים בתקופה.

Mishnah: On the third of Marcheshvan we ask for rain [that is, we insert the words ותן טל ומטר into the Amidah prayer]. Rabban Gamliel said: it is started on the seventh.

Talmud: Said Rabbi Elazar, the law is like Rabban Gamliel. It was taught in a Beraita: Chaninah said: outside of Israel, it is inserted sixty days after the fall equinox.

This is the only Jewish ritual that is tied to a real solar event. That event is the autumnal equinox, the day on which the sun is directly overhead on the equator, and on which the lengths of day and night are (almost) equal.(Please do not write to tell me about Birkat Hachamah. It is not tied to any real solar or other event anywhere in the cosmos. See here for more details.)

This year the autumnal equinox was on September 22nd (at 9.54am EST to be precise). The Talmud tells us to begin saying ותן טלֹ sixty days later. So you take out the calendar and count. That will bring us to November 21st. But in every siddur with instructions you will read that we start to recite the addition at Maariv on the evening of December 4th. Which is tonight (or was last night if you are reading this in Australia). That’s a full seventy-three days after the equinox. How did that happen?

It is really not difficult to understand, and requires no skill in observational astronomy or non-Euclidian geometry. Here is the explanation in four easy steps.

  1. The Julian Calendar

In the Julian calendar the solar year is exactly 365 1/4 days. That is, 365 days plus an additional 6 hours. This is also the length of the year according to the great talmudic sage of the second century, Samuel of Nehardea. And it is the assumed length of the solar year that is still used today in Jewish calculations. The actual length of the tropical year (i.e. one complete cycle of seasons) is slightly shorter: 365 days, 5 hours, 48 minutes and 46 seconds. That’s a difference of less than twelve minutes. Not much you say. And you’d be right, except that over a century that difference adds up to about 3/4 of a day. So every century the Julian calendar (365.25 days) falls behind the actual length of the year by three-quarters of a day.

2. The Gregorian Calendar

Over hundreds of years that difference adds up to many days, which became a real problem for the Church, when Easter was slipping further back in the calendar. The vernal equinox (a real solar event) had fallen on March 21st in the Julian calendar, but over the centuries had slipped back to March 11th. This threw the calculation of Easter into turmoil. To correct this (and other problems) Pope Gregory removed ten days from the calendar. By papal decree, the last day of the Julian calendar was on October 4, 1582. The next day became the first day of the Gregorian calendar, October 15, 1582. Now the spring equinox would fall on March 21, and everyone would be happy.

3. Jewish law stuck to the old Shmuel - Julian calendar

Rather than go along with this change, it was ignored by rabbis of the early modern period. They would rather have been wrong with Shmuel, than right with the Pope. They added back in the ten lost days. But by 1900 an additional three days of slippage had built up. The calendar of Shmuel is now some 13 missing days behind the Gregorian.

4. So add back in the missing days

If you add an additional thirteen days to the sixty days prescribed by the Talmud, you arrive at….December 4th! I told you it was easy. Here, try it for yourself:

Screen Shot 2018-12-04 at 8.27.21 PM.png

Happy December 4th from Talmudology

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Sanhedrin 11b ~ The Tekufot

The Jewish leap year

The Jewish year usually contains twelve lunar months. In today's page of Talmud we learn the reasons that the rabbis may add an extra month to the Jewish calendar, and proclaim a thirteen month leap year:

סנהדרין יא, ב

תנו רבנן על שלשה דברים מעברין את השנה על האביב ועל פירות האילן ועל התקופה על שנים מהן מעברין ועל אחד מהן אין מעברין

The rabbis taught in a Baraisa. We may intercalate a month and declare a leap year on account of three things: On account of the grain's ripening after the month of Nissan, on account of the fruit ripening after Shavuot, and on account of a season [tekufah] that is scheduled to begin in the wrong calendar month. We intercalate the year if two of these reasons are present, but for just one of these reasons we do not intercalate.

Let's look at the last reason. If the season is scheduled to being in 'the wrong month', the rabbis adjust the length of the year to bring it back into line.  Pesach must always fall in the Spring, so if the 15th of Nissan is part of the season of winter, we add another lunar month to the previous year and so push Pesach into Spring. 

The Solar Year, according to Shmuel

Here is the thinking behind this.  The rabbis of the Talmud, led by Shmuel, believed that a solar year was 365.25 days long, or 365 days and 6 hours. No more and no less. (This is also the length of the Julian calendar, adopted by Rome and all of Europe, including the Catholic Church.  It was the prevailing calendar in Europe, so Shmuel was in good company. ) Shmuel also believed that each of the four seasons was precisely one quarter the length of the solar year, or 91 days and 7.5 hours. Jewish tradition taught that the Sun was created on what would have been a Tuesday at 6pm. This time was also considered by the rabbis to have been the vernal equinox and the start of  Spring in the first year of creation.  Summer would therefore have started 91 days and 7.5 hours later, and fall another 91 days and seven hours after that.  The cycle repeats itself every 4 years, at which time the start of Spring will once again be at precisely 6pm, though not on a Tuesday. (It takes 28 years for the start of Spring to once again fall on a Tuesday at 6pm - and the next day is when we celebrate the rare event of Birkat Hachama. But I digress.)

Two Problems with Shmuel's Jewish Calendar

The first problem is that the solar year is not exactly 365 days and 6 hours in length. In fact it is 365 days, 5 hours, 48 minutes and 46 seconds - or 11 minutes and 14 seconds shorter than the Julian/Talmudic year. That's a small difference to be sure, but it adds up to three days every 400 years.  As a result of this difference, Spring, as measured by the day of the equinox, kept falling behind. In the days of Julius Caesar it had started on March 21, but by 1582 it had fallen behind and coincided with March 11. To bring the calendar back on track, Pope Gregory XIII knocked ten days out of the calendar, declaring the day after October 4, 1582 to be October 15th.  He also instituted a leap year of one extra day every four years, (except for years divisible by 100).  Shmuel's calendar had no similar fix, and the start of Spring in that calendar is about two weeks away from the actual, measurable Spring equinox.  

Path of the sun as seen over the northern hemisphere. On June 21, the start of summer the sun is at its most northern position. On Dec 21, the start of winter, it is at its most southern position.    

Path of the sun as seen over the northern hemisphere. On June 21, the start of summer the sun is at its most northern position. On Dec 21, the start of winter, it is at its most southern position.    


A second problem with Shmuel's calculation of the four seasons is that he attributed the same length to each, when in fact the seasons are not of equal length. This is because the earth does not revolve around the sun in a perfect circle, but instead has an elliptical orbit.  When the earth is closer to the sun it speeds up, and when it is further away it slows down.  In fact not one of the four seasons are 91 days and 7.5 hours in length. Here are the actual lengths:

Spring - about 92 days

Summer  - about 93 days

Fall - about 90 days

Winter - about 89 days

Shmuel's orderly system has no astronomical basis, and his seasons do not follow an orderly and predictable pattern. If you were to ask, "what happens in the sky at the beginning of Spring according to Shmuel?"the correct answer is nothing.  If you were to ask the same question of the actual day and time on which Spring begins, the answer would be "on that day and at that precise moment, the tilt of the earth's axis is exactly perpendicular to the sun."

Here are the actual start days of the seasons for this year, and their start according to the calendar of Shmuel.  

Start of the Seasons
Season Astronomical Shmuel
Times are for Jerusalem
Spring March 20
6:29 A.M. EDT
April 7
Summer June 21
12:24 A.M. EDT
July 8
10.30 AM
Fall September 22
4:02 P.M. EDT
Oct 7
Winter Dec 21
11:28 A.M. EST
Jan 6

Shmuel's calendar is pretty accurate, but over time, small discrepancies become larger.  Pope Gregory instituted a fix, but we have yet to do so. By not tweaking our Jewish calendar, the average date of the first day of Pesach is one day later every 216 years compared to the Sun, and one day every 231 years compared to the Gregorian date. This may not seem like a problem now.  But around the year 15,000 CE [sic] the first day of Pesach will occur on June 22, which is clearly not in the Spring. But I guess by then it will be somebody else's problem to fix.

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