by Dr. Irv Bromberg, University of Toronto, Canada
[Click here to go back to the Hebrew Calendar Studies home page]
I used astronomical algorithms and calendrical calculations to prove the following:
|A special "Thank You!" to Yehoshua Honigwachs, author of The Unity of Torah, Feldheim Publishers (May 1991, 350 pages): When I first documented the non-linear relationship between the traditional molad and the actual astronomical lunar conjunctions, I posted the information here in August 2004 merely as a curious finding, without attempting to explain its cause or significance. Soon thereafter, Yehoshua Honigwachs started to correspond with me about it, and, over the course of dozens of email messages over the next 3 months, he prodded me to explain the finding and its significance, and then to work out how to adjust the molad moments and shift the reference meridian, and further to validate and document how the study was done and its underlying arithmetic. He persistently wouldn't let me "off the hook" until he was satisfied that the matter had been appropriately accounted for and explained. He also brought the existence of the Israeli New Moon Society and its web site to my attention, as well as many papers written by F. Richard Stephenson and others about historical eclipses.|
Common names for lunar conjunction are "New" Moon, or "Dark" Moon.
Contrary to popular misconceptions, the moment of the lunar conjunction is not when Moon reaches its the minimum angular distance from Sun, nor is it the moment when the actual Sun-to-Moon distance is minimal, nor is it the moment of first visibility of the new lunar crescent at the beginning of a lunar month. It also has nothing to do with Moon or Sun rising or setting.
Astronomically, the lunar conjunction is the moment when the celestial longitudes of Moon and of Sun are equal when projected onto the ecliptic (the celestial path of the plane of Earth's orbit around Sun), from the prespective of the center of Earth. Alternatively, one could describe it as the moment when the centers of Sun, Moon, and Earth, in that order, are all in a plane that is perpendicular to the plane of Earth's orbit (the ecliptic plane). Since it doesn't in any way depend on any Earth surface location, the moment of the astronomical lunar conjunction occurs at the same instant everywhere, but people in different time zones may choose to reckon that moment according to their local clocks or according to some agreed-upon clock.
It is only rarely that at a lunar conjunction the centers of Sun, Moon, and Earth, in that order, are aligned in the same line. Such a perfect alignment only occurs during the maximum of a total solar eclipse at the moment that the center of the lunar umbra (darkest part of the lunar shadow) crosses the terrestrial latitude that equals the solar declination.
At the time of a lunar conjunction, Moon rises or sets at nearly the same time as Sun, but Moon is not visible except for a few minutes if there is a solar eclipse.
In the present era, according to the observational records of the Israeli New Moon Society as posted on-line at <http://sites.google.com/site/moonsoc/>, without optical aids the last visibility of the old lunar crescent before sunrise is at least 24 hours before the lunar conjunction, and the first visibility of the new lunar crescent after sunset is at least 24 hours after the lunar conjunction.
The following photographs at the NASA Astronomy Picture of the Day (APOD) web site show some examples of thin lunar crescents, a few of which were exceptionally "young" and very difficult to see (less than 24 hours after the lunar conjunction, seen either with optical aids or time-exposure photography):
The "world record" for youngest new moon photography is at <http://astrophoto.fr/new_moon_2010april14.html>, which was taken in full daylight at the moment of the lunar conjunction, made possible only because the lunar latitude was near maximal (almost 5°N) and the near infrared imaging system was equipped with a screen to block out interfering sunlight.
For information, images, and movies depicting the lunar phases over the course of the full lunar cycle, see the USNO web site at:
Regarding the dark moon interval, the Talmud Bavli tractate Rosh HaShanah page 20b says:
R. Zeira said in the name of Rav Nachman: Moon is covered (invisible) for 24 hours. To us (in Babylonia) it is invisible for 6 (hours) of the old moon and 18 (hours) of the new moon. To them (in the land of Israel) it is invisible for 6 (hours) of the new (moon) and 18 (hours) of the old (moon).
The actual difference in local mean time between Israel and Babylonia is less than 30 minutes, so the assertion of R. Zeira that there is such a large difference between Babylonia and Israel is astronomically impossible.
Rabbi Moshe ben Maimon ("Rambam"), also commonly known by his Greek name, (Moses) Maimonides, wrote a book entitled Hilchot Kiddush haChodesh (title translated as "Sanctification of the New Month" or alternatively as "Sanctification of the New Moon") around the Julian year 1178 or Hebrew year 4938, which is the eighth treatise in Zmanim (The Book of Seasons), the third book of his Mishneh Torah collection (code of Jewish Law). Quoting from that source:
|Moon becomes hidden each month and is invisible for about 2 days: about one day before its conjunction with Sun at the end of the month, and about one day after its conjunction with Sun, before it can be seen in the west in the evening.
"Sanctification of the New Month", chapter 1, paragraph 3.
|הלבנה נסתרת בכל חודש ואינה נראית כמו שני ימים או פחות או יותר מעט
כמו יום אחד קודם שתדבק בשמש בסוף החודש
וכמו יום אחד אחר שתדבק בשמש ותראה במערב בערב
הלכות קידוש החודש פרק א הלכה ג
The minimum period of lunar invisibility (the dark moon interval) is a physical phenomenon, not a matter of opinion, nor of law or tradition.
Using modern astronomical algorithms it is possible to estimate whether Moon will be visible during the twilight before a specified sunrise or during the twilight after a specified sunset at a given locale in the Middle East. The criteria for reckoning the visibility of the lunar crescent as recommended by Dershowitz & Reingold in Calendrical Calculations: 3rd Edition (CC3, see <http://www.calendarists.com/>) are:
Click here to view charts based on the above criteria calculated for Jerusalem, Israel 178 KB, 7 pages
The charts span one saros cycle before and after the beginning of the year 2000 AD (a total of two saros cycles). I also generated similar charts for the era that was 2000 years earlier, but they are not shown because they were not significantly different.
Since we observe (from the ground and without optical aids) the dark moon interval as spanning from the last visible old lunar crescent in the twilight prior to sunrise until the first visible new lunar crescent in the twilight after sunset, it is impossible for that interval to equal an integer number of days. Rather, it always equals at least one day plus the length of the daytime (measured from the dawn twilight until the dusk twilight), which seasonally varies fractionally in length (longer in summer, shorter in winter).
The dark moon interval varies greatly with latitude, with the least variations at the equator. It tends to be shorter when the lunar conjunction is near perigee (Moon closest to Earth) because Moon moves faster near perigee. For the northern hemisphere it tends to be shorter when the lunar latitude near the conjunction is north of the ecliptic, whereas for the southern hemisphere it tends to be shorter when the lunar latitude near the conjunction is south of the ecliptic. There is also a weak tendency for the dark moon inteval to be shorter when the conjunction is near the Earth orbital aphelion (Earth furthest from Sun and moving slowest).
At Jerusalem the duration from dawn twilight to dusk twilight (solar depression = 4°30' in both cases) seasonally varies over a nearly 4 hour range from about 10+3/4 hours to almost 15 hours.
For Jerusalem, a dark moon interval as short as one day plus the daytime length is very rare, and occurs only when Moon is near perigee, the lunar latitude is near its maximum north deviation (about 5°N), Earth is near its orbital aphelion, and in such cases both the last old and first new lunar crescents are barely visible. The most common dark moon interval at Jerusalem is two days plus the daytime length, occurring in about 55% of cases, and in such cases Moon is most often north of the ecliptic and near perigee. In about 40% of cases the dark moon interval at Jerusalem is three days plus the daytime length, and in such cases Moon is most often south of the ecliptic and near apogee. Very rarely, the dark moon interval at Jerusalem reaches four days plus the daytime length, and such cases happen when Moon is very close to apogee and the lunar latitude is near its maximum south deviation (about 5°S).
Astronomically, the lunar opposition or "Full" Moon is the moment when the celestial longitude of Moon is 180° away from Sun when projected onto the ecliptic. Alternatively, one could describe it as the moment when the centers of Sun, Earth, and Moon, in that order, are all in the plane that is perpendicular to the plane of Earth's orbit (the ecliptic plane).
At the moment of a lunar opposition, somewhere in the nightime world Moon sets in the west at nearly the same time as Sun rises in the east, while simultaneously at nearly antipodal locations in the daytime world Moon rises in the east at nearly the same time as Sun sets in the west. Moon will appear full, and when it occasionally passes through Earth's shadow it will appear dull red, an event referred to as a lunar eclipse.
The lunar orbital plane is tilted about 5° 9' (mean value) relative to the ecliptic plane, therefore most of the time the lunar latitude is either north or south of the ecliptic. If the lunar latitude at the moment of a lunar conjunction is greater than the apparent solar angular diameter north or south of the ecliptic (32 arcminutes, nearly equal to the mean apparent lunar angular diameter) then Moon passes north or south of Sun, respectively, without any solar eclipse.
The Hebrew word molad (plural moladot) means "birth", and it is also a generic term for New Moon, in the sense of that Moon is "born" again at the beginning of each lunar cycle. Depending on the context, however, the word molad could have several alternative meanings:
The Hebrew calendar is nominally supposed to stay aligned with both the solar year (the seasons) and the lunar month.
Originally each Hebrew calendar month started after the observation in Israel of the first visible crescent after the new moon. Jewish communities outside Israel depended on messengers to communicate calendar decisions. In Hebrew year 4119 (Julian 358 AD), after Roman Emperor Constantius II outlawed New Moon announcements, Hillel ben Yehudah, the second-last President of the ancient Sanhedrin, promulgated a fixed arithmetic calendar (probably developed a century earlier by Amora Shmuel "the Astronomer" of Nehardea, Babylonia) based on the Metonic cycle of leap years (235 months in 19 years, with 7 leap months per 19 year cycle) and using simple arithmetic to approximate the moment of the mean lunar conjunction (the "molad", or in plural "moladot"). Since then, the molad of the Hebrew calendar has been nothing more than a fixed arithmetic cycle that determines the provisional date of Rosh HaShanah (the Hebrew New Year Day), subject to possible postponement of 0, 1 or 2 days (depending on the weekday and timing of the molad moment).
For centuries Jews have followed a tradition of announcing the moment of the molad for the coming month in their synagogues during the morning services on the last Sabbath before the start of each month except Tishrei. All of these announced moladot are of no relevance to the Hebrew calendar! The only molad that matters in Hebrew calendar arithmetic is the molad of Tishrei, yet that molad is never announced!
Furthermore, if 1/4 day is added to the molad moment, a shortcut originally proposed by German Mathematician Johann Carl Friedrich Gauss (Gauß), or if a 1/4 day is permanently included as an offset at the epoch, then even for the molad of Tishrei its time-of-day becomes irrelevant, all that matters is the day that the molad lands on.
There is nothing "mystical" about the molad, it is just simple arithmetic, 100% predictable and exactly calculated.
Some people seek to know the clock time of the molad moment, perhaps to choose an auspicious moment for starting a significant life event after a new lunar cycle begins. In the case of the traditional molad, such thinking is at best misguided, because there is nothing particularly "auspicious" about such a moment, due to the drift of the traditional molad with respect to the mean lunar cycle, as will be shown below, and due to the superimposed periodic variations of the duration of the lunar cycle. In addition, there is no valid method to convert a traditional molad moment to a clock time in any time zone. For such purposes everyone should refer to reliable sources for the actual astronomical moment of the lunar conjunction, properly adjusted to the local time zone, taking into account Daylight Saving Time, if applicable.
Our freeware calendrical calculator, Kalendis, can export all of the moladot for a full Hebrew calendar year in a variety of formats, click here for an example of its built-in web page format. Note that the right-most column on that report shows the moment of the actual lunar conjunction, expressed in terms of the time zone of the currently selected locale (in the example, Jerusalem). Looking at the numbers, it is hard to see any relationship between the announced traditional molad moments and the actual lunar conjunction moments, but rest assured that there is a relationship, which we will examine in detail and quantify here. Kalendis also shows the molad moment to announce (for the coming month) at the bottom of each monthly zmanim (ritual times) export report, click here for examples of zmanim reports for an entire Hebrew calendar year.
Some hold that the molad time is based on Temporal Hours (sha'ot zmaniot) , but that seems inconsistent with the simple arithmetic used to calculate the molad, which adds a constant 29+13753/25920 days per month. In our time, Hebrew year 5766, the Molad minus Mean Lunar Conjunction difference varies from the molad being early by 2+1/2 hours ahead of the Mean New Moon in Nisan to the molad being late by 5+2/3 hours after the Mean New Moon in Tishrei and Cheshvan. This more than 8-hour span dwarfs the few minutes of variance that could be accounted for by the use of Temporal Time. Furthermore, the modest differences in temporal hour length at the latitude of Israel average out during the full solar year cycle.
Nevertheless, the Talmud and many other rabbinic sources regard the molad moment as if it were in terms of Temporal Time, so that a molad moment that is between 00:00h and 05:59h is regarded as between sunset and midnight, between 06:00h and 11:59h is regarded as between midnight and sunrise, between 12:00 to 17:59h is regarded as between sunrise and mid-day, and between 18:00h to 23:59h is regarded as between mid-day and sunset. Thus rabbinic sources freely relate the molad moment to the time-of-day without actually calculating the moments of sunset, midnight, sunrise, or mid-day.
Evidently, for the purposes of Jewish Law (halachah), this approximation has a long tradition of acceptance, and I used it to advantage in simplifying my astronomical Hebrew calendar arithmetic during development of the rectified Hebrew calendar.
The reader can skip this section if the details for calculating the traditional molad moment are not of interest.
In the traditional method of reckoning molad moments, as it often appears in rabbinic literature, one simply adds or subtracts 29 days 12 hours and 793 parts from a known molad moment to obtain the molad moment of the next or prior month, respectively. Although simple, this classical method is often inconvenient, because it requires starting from a known molad moment that is reasonably near to the desired molad. It is inefficient when one needs to jump from a given molad to another that is an arbitrary number of months into the future or past, because that requires repeated steps. Any serially repeated addition or subtraction calculation is also risky because if a mistake is made somewhere along the way then that introduces an offset error that may be carried through indefinitely. A shortcut commonly employed in classical calculations is to ignore the day count, simply tracking the weekday instead. This shortcut can lead to ambiguities when the ignored days add up to more than a week, and although it does yield sufficient information for the purpose of announcing a molad moment, the weekday with time of day is not enough information for calendrical calculation purposes. Instead, the following is a direct method for the unambiguous calculation of any molad moment.
A molad moment has an integer part, which represents the calendar day and weekday of the molad, and a fractional part that represents the time-of-day of the molad. The ideal way to carry out molad calculations is using using exact computing engines capable of arbitrary-precision arithmetic, such as Mathematica, or the computer programming language "LISP", which can calculate the molad moment to the exact fraction or whole number of parts. It is also possible to obtain correct results using floating point arithmetic or a digital calculator, provided that the user takes care to round intermediate results at appropriate points, as will be shown below, but the calculation must support at least Double Precision floating point, because Single Precision is often insufficient to resolve moments to better than one second accuracy.
For the purposes of announcing the molad moment or determining the molad of Tishrei for Hebrew calendar arithmetic, the moment must be calculated exactly. For the purpose of comparing moladot to astronomical lunar conjunction moments, however, the small errors introduced by the limitations of floating point arithmetic are negligible.
In a computer program, non-variable values can be declared as constants, for optimal performance.
Calendrical calculations make frequent use of dividing a number and keeping only the remainder, for example, dividing by 7 to determine the weekday, as will be done below. Many programming languages have a MOD operator or function intended for this purpose, but in many languages MOD handles negative or real numbers improperly (the MOD operator of Microsoft Visual Basic is defective on both counts). To avoid the risk of such errors, herein I will use the solution recommended by Dershowitz & Reingold in Calendrical Calculations: 3rd Edition (CC3, see <http://www.calendarists.com/>):
modulus( x, y ) = x – y × floor( x / y )
Not being limited to integer division, the CC3 modulus function also works properly with floating point (real number) parameters provided both the x and the y parameter and the function return value are declared as Double Precision.
The traditional epoch or first day of the Hebrew calendar, on the first day of Tishrei of the first year, was a Monday (Yom Sheini, or second day of the week), starting at the sunset at the end of the daytime on Sunday. Assign day number one to that epoch:
HebrewEPOCH = 1
(A good case could be made to assign HebrewEPOCH = 0 instead, to count elapsed days relative to the epoch, but that would require changes to some of the arithmetic below, such as the weekday expression. Suffice it to say that the elapsed day count is always one day less than any calculated day number.)
To compute a traditional molad moment, start by calculating the Lunation number since the traditional epoch of the Hebrew calendar, which depends on the given Hebrew year number and month number:
Lunation = ElapsedMonths( hYear, hMonth )
Within the ElapsedMonths function, adjust the year number to account for the traditional month numbering that starts from Nisan, storing the adjusted year number in a local variable:
IF hMonth < Tishrei THEN TheYear = TheYear + 1 ELSE TheYear = hYear
where the Hebrew months are numbered from Nisan=1 to Tishrei=7 to Adar=12, etc.
In 19 years there are 19 × 12 = 228 regular months, plus 7 leap months, a total of 235 lunar months per cycle. The following expression returns the number of elapsed months:
RETURN hMonth – Tishrei + quotient( 235 × TheYear – 234 , 19 )
For example, for Cheshvan, the 8th month of traditional Hebrew calendar year 5766:
= 8 – 7 + quotient( 235 × 5766 – 234 , 19 ) = 1 + quotient( 1354776 , 19 ) = 1 + 71304 = 71305
The assumed initial conjunction time (called "BaHaRad", or PartsAtEpoch below) of the first molad (Lunation=0) was 5 hours and 204 parts (each part = 1/18 minute) after the epoch of the Hebrew calendar:
HoursPerDay = 24
MinutesPerHour = 60
MinutesPerDay = MinutesPerHour × HoursPerDay = 1440
PartsPerDay = MinutesPerDay × 18 = 25920
PartsPerHour = PartsPerDay / HoursPerDay = 1080
PartsPerMinute = PartsPerDay / MinutesPerDay = 18
PartsAtEpoch = 5 / HoursPerDay + 204 / PartsPerDay = 5/24 + 204/25920 = 5604/25920 = 467/2160 = 0.2162037...
(the overscored digits of the decimal fraction repeat forever)
The constant traditional molad interval is 29 days, 12 hours, and 44+1/18 minutes. Separate the 44+1/18 minutes in order to improve the accuracy of floating point calculations:
Parts793 = 44 / MinutesPerDay + 1 / PartsPerDay = 793/25920 of a day
TwentyNineAndHalf = 29 + 12 / HoursPerDay = 29.5 days
Ari Meir Brodsky of Toronto has pointed out that in mental molad arithmetic for the purpose of announcing the molad moment it is easier to disregard whole weeks and calculate using the molad interval in excess of 4 weeks as 1+1/2 day, plus 3/4 of an hour, minus 1 minute, plus 1 part. For more information click here to see Ari's mental molad method web page.
The assumed molad epoch was traditionally derived as follows (Tosefot on Talmud Bavli tractate Rosh HaShanah page 8b):
The following expression yields the molad moment as the number of days since the epoch of the traditional Hebrew calendar, plus the fraction of a day elapsed since a mean sunset time that is 6 hours before civil midnight:
MoladMoment = HebrewEPOCH + PartsAtEpoch + ( Lunation × TwentyNineAndHalf ) + ( Lunation × Parts793 )
If the calculation to this point was carried out using floating-point arithmetic then it is necessary to round the molad moment so that its fractional portion corresponds to a whole number of parts (this must be carried out before doing anything else with the molad moment):
MoladMoment = round( MoladMoment × PartsPerDay ) / PartsPerDay
To determine the weekday of any molad moment, divide the number of whole days since the epoch by 7, and add one. This yields a number from 1=Sunday (Yom Rishon, or first day of the week) to 7=Saturday (Shabbat, or seventh day of the week):
MoladWeekday = modulus( floor( MoladMoment ) , 7 ) + 1
For this weekday expression to yield the correct result, modulus( HebrewEPOCH, 7 ) must equal 1, which will be the case if HebrewEPOCH was set equal to 1 as recommended above. If for some reason you set HebrewEPOCH to some other value, then you must modify the expression appropriately.
When announcing the molad weekday, state it in Hebrew, because Hebrew calendar days begin at sunset, not midnight.
Only for the purpose of announcing the molad time, separate it into hours and parts (alternatively, not recommended because it is not the traditional way: hours, minutes, and parts). Extract the fractional portion of the molad moment and convert it to an exact whole number of parts:
MoladFractionAsParts = [ MoladMoment – floor( MoladMoment ) ] × PartsPerDay
Then calculate the separate time components for the announcement:
MoladHour = floor( MoladFractionAsParts / PartsPerHour )
MoladParts = modulus( MoladFractionAsParts , PartsPerHour )
Alternatively, to separate the minutes and residual parts of a minute:
MoladMinute = modulus( floor( MoladFractionAsParts / PartsPerMinute ) , MinutesPerHour )
MoladParts = modulus( MoladFractionAsParts , PartsPerMinute )
When announcing the molad time, state that the time is counted from mean sunset, not midnight.
Some organizations subtract 6 hours to convert the molad time to "civil" time, but that is a misleading and futile adjustment because the traditional molad has an undefined and undefinable time zone, as will be demonstrated herein.
If a "civil" time adjustment is applied to the molad moment itself then the molad will land on the incorrect weekday 25% of the time.
Never apply Daylight Saving Time to a molad moment, otherwise the molad of Tishrei will sometimes land on or be postponed to the wrong weekday.
Engineer Yaaqov Loewinger of Tel Aviv, Israel published an essay in Hebrew on the subject of the announcing of the molad moment in synagogues. It is a good review and cites an impressive collection of traditional sources on the subject. The default format of the molad moment as reported by Kalendis is as recommended by Loewinger, and is also the same format as just outlined above. Please see Loewinger's essay, which is freely available on-line in PDF format at <http://www.hakirah.org/Vol 6 Loewinger.pdf> 244 KB in the Summer 2008 issue of Hakirah, The Flatbush Journal of Jewish Law and Thought.
As an example, by the above arithmetic the traditional molad moment for the month of Cheshvan in Hebrew year 5766 was at 2105680.2310571 days or exactly 2105680 days and 5989 parts. Dividing the integer portion by 7 yields a remainder of 3 to which we add one to obtain the molad weekday of 4 = Yom Rivii. We don't translate this weekday as Wednesday, because it is a Hebrew weekday and the molad moment is 27 minutes and 5 parts before midnight, so in terms of "civil" time it would be late on "Tuesday evening" (placed in quotes because the molad time zone is undefined, as explained in the previous paragraph). The 5989 parts separate to 5 hours and 589 parts (or 5 hours 32 minutes and 13 parts) after mean sunset.
Some people use the molad moment to determine the earliest and latest time for saying the Kiddush Levanah prayer (sanctification of the Moon) once during the waxing half of each lunar cycle, when Moon is clearly visible, but again because of the ritually undefined time zone problem such calculations are of doubtful validity. Key sources for earliest and latest time rules are Shulchan Aruch: Orach Chaim 426:3-4 and the commentary upon it Mishneh Berurah 426:17-20, but neither source is reconcilable with astronomical reality.
Why not determine the Kiddush Levanah time limits relative to the actual astronomical lunar conjunction and lunar opposition moments, respectively, expressed according to the local clock time? These astronomical moments can be calculated by our freeware computer program, Kalendis, see <http://individual.utoronto.ca/kalendis/kalendis.htm>. Alternatively, there is the simple observational method: if the local moonrise is before sunset then the waxing half has not yet ended.
For comparison with astronomical lunar conjunction moments, further adjustment is necessary, because astronomical moments are usually calculated as Universal Time (from midnight at the Prime Meridian), and usually relative to one of the following calendar epochs:
Finally, adjust for the time zone difference. For example, if the astronomical moment is computed as usual for the Prime Meridian, to evaluate if the molad moment refers to Jersualem Local Mean Time add 2 hours and 21 minutes to the astronomical moment, because Jerusalem is about 35+1/4° east of the Prime Meridian, and each degree corresponds to 4 minutes of local mean time difference, so Jerusalem Local Mean Time is 35+1/4 × 4 = 141 minutes = 2 hours and 21 minutes ahead of time at the Prime Meridian.
If the molad is the moment of the mean lunar conjunction, then it can't be so for every location on Earth. The mean lunar conjunction can occur at that moment only for a single reference meridian of longitude on Earth. At the moment of a mean astronomical lunar conjunction, a clock placed at that longitude and set to display local mean time will indicate the same time as the corresponding molad moment (less 6 hours, because the molad moment is relative to mean sunset whereas clock time is relative to civil midnight).
The molad reference meridian has nothing to do with direct observation of lunar conjunctions, which can never be observed except during a solar eclipse when Sun is above the local horizon. Even during solar eclipses it is technically difficult to determine the maximum eclipse moment. Furthermore, astronomical lunar conjunctions are computed for a geocentric "observer", even though an observer at the center of Earth couldn't possibly make any astronomical observations, let alone get there and survive! Geocentric calculations eliminate variations due to locale-specific parallax.
The explicit specification of a reference meridian is standard for any astronomical calendar. For example, the modern Persian astronomical solar calendar uses 52.5° E, which corresponds to Iran Standard Time (UT+3.5h), and the traditional oriental lunisolar astronomical calendars use Standard Time in Beijing (UT+8h) for the Chinese, Standard Time in Tokyo (UT+9h) for the Japanese, Standard Time in Seoul (also UT+9h) for the Korean, and Standard Time in Hanoi (UT+7h) for the Vietnamese calendar.
I could not find, however, any rabbinical or classical or even a modern source that authoritatively specified which longitude is the reference meridian for the traditional molad of the Hebrew calendar, but there were many internet web sites and books that casually mentioned a commonly-held assumption that the molad refers to the meridian of Jerusalem. For example this assumption is mentioned in many places at the web site of the Orthodox Union at <http://www.ou.org>, and in the translation of the "Blessing of the New Month" ceremony of the very popular prayer book entitled The Complete ArtScroll Siddur, published by Mesorah Publications Ltd., Brooklyn, New York, it says "Announcement of the Molad: It is customary – but not obligatory – to announce the precise time at which the new cycle of the moon will begin in Jerusalem." I also found a minority of sources (web sites and books) that assumed or even presented evidence that the molad referred to somewhere in Iraq/Babylonia/Persia, or Afghanistan, or the "center of the civilized world", or the East China Sea between China and Japan (90° east of Jerusalem), or in the Pacific Ocean an hour to the east of Japan (120° east of Jerusalem).
It seemed to me that it ought to be straightforward, using modern astronomical algorithms, to prove which of the alternatives, if any, was the correct meridian, or at least the historically correct meridian. We know today that the mean interval between lunar conjunctions is steadily getting shorter in terms of the mean solar days that are appropriate for calendar calculations. We also know today that the Earth rotation rate is more-or-less steadily slowing down due to Earth-Moon tidal interactions. Back in the era when the fixed arithmetic traditional Hebrew calendar was established, however, they did not know about tidal or even basic gravitational interactions. Therefore if we want to know which meridian was the original reference meridian for the molad, we must include the era of Hillel ben Yehudah in our calculations, as well as dates in the remote past and distant future. Furthermore, to end up at the correct meridian, our calculations must as accurately as possible take into account the historical changes in the lunar revolution rate and the Earth rotation rate.
Limitations: The astronomical algorithms that I initially used for lunar position are based on a truncation by Jean Meeus of the semi-analytical ELP-2000 lunar theory of Jean Chapront et al. The parabola used to estimate Delta T (to take changes in the Earth rotation rate into account) was based on the approximation, derived mainly from historical records of solar eclipses, that the mean length of the solar day changes by about 1.75 milliseconds longer each century (for more information about Delta T see this page). Relativistic effects are not accounted for. (These limitations were essentially eliminated in my repeat calculation, to be discussed further down this page, but the conclusions were nearly the same.)
I started with the most widely held assumption, that the molad referred to the mean lunar conjunction in terms of Jerusalem Local Mean Time (≈ Israel Standard Time + 21 minutes). For all Hebrew years from the traditional year of Creation to the year 10000, I compared the moments of the actual lunar conjunctions, computed using modern astronomical algorithms and expressed in terms of Jerusalem Local Mean Time, with those of the corresponding molad moments. Using that data, I generated a collection of graphs depicting how their average differences and their month-specific differences change over the years.
This first chart (click here 741 KB) shows the difference between the molad moment and the actual astronomical lunar conjunction (in terms of Jerusalem Local Mean Time) for the first 10000 years of the Hebrew calendar. I expressed the x axis as the number of months elapsed since the epoch of the Hebrew calendar for convenience in relating that directly to the molad moment. (I used Microsoft Excel to produce this chart, but that program can't plot more than 32000 points as a single series, therefore I limited the amount of data by sampling every 4th year.) This chart essentially depicts the average relationship between the molad and the actual lunar conjunction.
The horizontal red line is the zero difference line. If the molad truly referred to the Jerusalem Local Mean Time of the actual mean lunar conjunction then all of the plotted points would be distributed as a straight band horizontally across the chart, vertically centered upon that zero difference line, with slightly more than ±14 hours of vertical spread due to the natural variations in the duration of the lunations.
Instead, we see that the points actually follow a curved band, and please note the blue line through the middle of that band, which is a quadratic least-squares statistical regression line (hard to see on a printout, even in color, but easily seen on a computer display screen). The equation for that line is given below the title of the graph — don't panic about all the numbers, it is simply a quadratic equation whose coefficients could have been rounded to perhaps 6 significant figures.
The difference between the red line and the blue curve represents the error or drift of the molad with respect to the actual lunar conjunction. The blue line almost touches the red line near 51000 elapsed months, which was the era of Hillel ben Yehudah (Tishrei 4119 was lunation number 50933). This implies that the molad interval was essentially equal to the actual mean synodic month in that era, and that the molad reference meridian was close to Jerusalem, which has a longitude of 35° 14' 4" East. The closest that the blue line approaches the red line is +23 minutes, which implies that the original molad reference meridian was 23 mean solar minutes east of Jerusalem. Earth rotates 360° in 24 hours, so the corresponding longitude difference is 360° × FractionOfDay = 360° × (23 minutes) / (1440 minutes per day) ≈ 5 3/4°, which corresponds to eastern Jordan — why would the molad refer there? What if we doubled the longitude difference to ≈ 11 1/2° ? The historical site of origin of the Chaldeans, renowned astronomers of Babylonia, was around the City of Ur, which was at a longitude of about 46° 5' East, a longitude difference of almost 11° east of Jerusalem. In the era of Hillel ben Yehudah the majority of Jews lived in the region of that meridian in Babylonia, mostly north of Ur and extending up to Bahgdad. If we subtract 23 minutes from all points on the chart, they all shift down by that amount, and the blue line will then touch the red line in the era of Hillel ben Yehudah.
This evidence suggested that the original reference meridian of the molad was midway between Israel and Babylonia. Could this have been chosen as a compromise between the two major Jewish population regions of that era? Does that make sense? Where can one find a Torah or Talmud source to justify or even suggest such a compromise? Are we to learn from this that if it were done today then the molad meridian would be placed halfway between Israel and New York (at the longitude of the western tip of Africa, Canary Islands, Iceland)? There had to be a better explanation. Due to limitations of the astronomical algorithms and Delta T approximation that I initially used, it seemed worthwhile to independently and more accurately confirm the calculations.
At the beginning of January 2007 a new version 9.1 of SOLEX was released. This program performs high accuracy astronomical calculations using numerical integration, the "gold standard" for celestial mechanics, with documented excellent agreement against the world reference Development Ephemeris (DE) algorithms of NASA's Jet Propulsion Laboratory. After I requested this feature, the new SOLEX version introduced the ability to automatically search for lunar conjunction moments, logging those moments to better than one-second accuracy. I used SOLEX 9.1 to develop very accurate arithmetic for calculating mean lunar conjunction moments, as given on my "Length of the Lunar Cycle" web page at <http://individual.utoronto.ca/kalendis/lunar/>. SOLEX takes the most important relativistic corrections into account. For more information about SOLEX and to download the program please see its web page at <http://www.solexorb.it/>.
Furthermore, near the end of January 2007 the NASA Eclipses web site published a new 5-millennium canon of solar eclipses, and the authors Fred Espenak and Jean Meeus also posted at <http://eclipse.gsfc.nasa.gov/SEcat5/deltatpoly.html> newly updated expressions for approximating Delta T to take into account historical changes in the Earth rotation rate, based on recently revised and extended analyses of historical lunar and solar eclipses records, published by F. Richard Stephenson et al.
For the first 10500 years of the Hebrew calendar (>130000 lunar months), I used SOLEX to compute the mean lunar conjunction moments in terms of TT (Terrestrial Time), converted those moments to UT (Universal Time) by subtracting the new NASA Delta T approximation, shifted them to Jerusalem Local Mean Time by adding 2 hours and 21 minutes, then added 6 hours to count time from mean sunset instead of civil midnight, and finally subtracted those astronomical moments from the corresponding traditional molad moments to generate the following plot (click here or on the chart to open a high-resolution PDF version 91KB):
This numerical integration independently confirmed that the molad minus Jerusalem mean lunar conjunction difference was around 23 minutes in the era of Hillel ben Yehudah (the close-up inset graph shows it at about 22+1/2 minutes). However, back in the era of the Maccabees and of the ancient Greek astronomer Hipparchus, there was another minimum that was even closer to Jerusalem time, just under 15+1/2 minutes, corresponding to an original molad reference meridian of longitude that was slightly less than 4° east of Jerusalem.
Since its origin in the era of the Maccabees the molad reference meridian has been drifting eastward at an accelerating rate, because the traditional molad interval was fixed at a length of 29 days 12 hours and 44+1/18 minutes whereas the astronomical mean synodic month has been getting about 25 microseconds shorter with each passing month. Around two centuries before the era of Hillel ben Yehudah, the Greek astronomer Ptolemy of Alexandria published his multi-volume treatise of astronomy, known today as Almagest, in which he gave the the length of the mean synodic month in sexagesimal format as 29 days 31' 50'' 8''' 20'''', that is 29 days 31 minutes 50 seconds 8 thirds and 20 fourths, numerically exactly equal to the traditional molad interval (see "Why Divide Hours into 1080 Parts?" at <http://individual.utoronto.ca/kalendis/hebrew/chelek.htm>), but Ptolemy cited Hipparchus as the source for that mean synodic month. We no longer have the original writings of Hipparchus on this topic, but in the 20th century a bronze astronomical calculator mechanism dating back to the 2nd century BC was discovered near Antikythera on a shipwreck laden with goods from the Isle of Rhodes (where Hipparchus used to live and teach). The Antikythera Mechanism is thought to have been constructed by Hipparchus or Posidonius or others associated with his school of astronomy. Its clockwork multi-geared mechanism quite accurately calculated the lunar position and phase, including a clever mechanical accounting for the major periodic variations of the lunar cycle that are due to the eccentricity of the lunar orbit. Perhaps such a calculator was available to the Maccabees? Another authority who lived around the era of Ptolemy was Rebbe Eliezer ben Hurcanus (אליעזר בן הורקנוס), traditional author of Pirkei D'Rebbi Eliezer, who described the mean lunar cycle in considerable detail (chapters 6 and 7), including specification of the traditional molad interval that is still used today, although his teachings weren't assembled and published until several centuries after Hillel ben Yehudah.
The earlier minimum difference of 15 minutes 27 seconds corresponded to an original molad reference meridian that was 3° 51' 45" east of Jerusalem. By adding Jerusalem's longitude of 35° 14' 4" E (from the Prime Meridian that runs through the Greenwich Observatory, UK) one obtains 39° 5' 49" E as its original longitude (2h 36m 23s ahead of UT), which is between Israel and Babylonia, but not halfway, rather it is closer to Israel than Babylonia. Although this independent recalculation confirmed that the molad reference meridian was never in Jerusalem or even in Israel, and never will be in Jerusalem, a "compromise" between the two historical major Jewish centers can't explain why the original meridian was about 2 degrees closer to Israel than the halfway point.
Perhaps the answer comes from the promise of HaShem to Abram in the Torah, Genesis chapter 15 verse 18: "On that day HaShem made a covenant with Abram, saying To your descendants have I given this land, from the river of Egypt to the great river, the Euphrates River." This territory also corresponds to the full range of the patriarch's travels during his lifetime, as described in the Torah, from Ur to Egypt. The Torah later specified in Genesis chapter 21 verses 12-13 that this promise referred only to those who descended through Isaac (Yitzchak), and in Genesis chapter 28 verses 1-5 through Jacob (Yaacov).
The Nile River passes through Cairo at 30° 3' N, 31° 22' E. The Euphrates River joins the Tigris River near the town of al-Qurnah at 31° 1' N, 47° 25' E, continuing as the Arvadrud River to the Persian Gulf. In the era of the patriarch, however, the Euphrates and Tigris together formed a delta which drained directly into the inland extension of the gulf, and Ur was a gulf coastal city on the west bank of the Euphrates. Four millennia of mesopotamian silt deposition caused the gulf waters to recede about 200 km to the south, despite globally rising sea levels. The meridian of longitude that is halfway between Cairo and al-Qurnah is at 39° 23' 30" E, which is 4° 9' 26" E of Jerusalem, corresponding to (1440 minutes per day) × 4.157° / 360° = 16 minutes and 38 seconds ahead of Jerusalem Local Mean Time (click here to see map). This agrees with the more accurate molad meridian confirmation to within 71 seconds of time, or less than 17 arcminutes of longitude (one minute of time = 15 arcminutes of longitude, or one degree of longitude = 4 minutes of time).
The arcseconds and arcminutes in the preceding paragraph are not really significant, due to the uncertainties of the Delta T approximation and of the original Nile and Euphrates reference meridians of longitude. It suffices to round the minimum time difference to +16 minutes relative to Jerusalem Local Mean Time, corresponding to a longitude that is 4° east of Jerusalem, or a Local Mean Time that is 2 hours and 37 minutes ahead of UT or that is 37 minutes ahead of Israel Standard Time. That meridian passes through the junction of the modern borders of Jordan, Iraq, and Saudi Arabia.
Although I am aware of no specific Jewish record of this event, there was a total solar eclipse that crossed central ancient Israel shortly before the apparent era of origin of the traditional molad calculation, and its maximum occurred at the Nile-Euphrates midpoint! There are two images at NASA that portray this interesting eclipse: a map of eclipses during that era, which is at <http://eclipse.gsfc.nasa.gov/SEatlas/SEatlas-1/SEatlas-0339.GIF> (the eclipse of interest is the blue band labeled as "-0335 Jul 04" [NASA includes a year zero] with an asterisk marking the maximum point in today's eastern Jordan). There is also a schematic diagram at <http://eclipse.gsfc.nasa.gov/5MCSEmap/-0399--0300/-335-07-04.gif>, which details this specific eclipse. The date was Julian July 4, 336 BC (no year zero) = Julian Day 1598883.895 = traditional Hebrew 1 Av 3425, which happened to be less than a week after the north solstice (hence essentially as high in the sky as an eclipse can possibly be for any location in Israel), and the eclipse totality path crossed Jerusalem shortly before noon.
The Talmud Bavli, tractate Sukkah 29a declared: "If it is in eclipse in the east, it is a bad omen for those who dwell in the east; if in the west, it is a bad omen for those who dwell in the west; if in the midst of heaven it is bad omen for the whole world." Perhaps this eclipse comprised the basis for that statement, because in Israel it was "in the midst of heaven" as it happened shortly before noon, and because it occurred in the year of ascent of Alexander the Great to the throne of Macedonia, heralding 12 years of military compaigns during which he conquered essentially all of the territory that had been promised to the descendents of Abraham!
Is it possible that that solar eclipse served as the initial demonstration that a lunar conjunction is a specific, calculable moment, whereas the first visible new lunar crescent is not, initially leading to development of a calculation for the mean lunar conjunction, and ultimately to the switch in criteria for determining the start of calendar months? Perhaps it also provided a very tangible demonstration of the principle that lead to the molad zakein rule (that the molad is postponed to the next calendar day if the calculated moment is on or after noon), because that solar eclipse / lunar conjunction moment occurred just prior to noon, yet the new lunar crescent couldn't have been visible at the next sunset, by which time Moon was below the western horizon. At the sunset after that, however, the new lunar crescent should have been easily visible throughout Israel, nearly 1+1/3 days after the eclipse.
For more information, please see the heading "The MoladMoment Function" as well as the subsequent sections about the molad on the rectified Hebrew calendar web page at <http://individual.utoronto.ca/kalendis/hebrew/rect.htm>.
Some have claimed that the molad seems to refer to a meridian in Afghanistan, whose solar time is about 2 hours ahead of Jerusalem. The most likely reason for this "Afghanistan mistake" is that in the present era if one looks at a clock showing Jerusalem Local Mean Time then the moladot are an average of about 2 hours later than the astronomical mean lunar conjunction moments. Although this 2-hour delay is entirely accounted for by the accumulated progressive shortening of the Mean Synodic Month since Hebrew year 4119 (as documented in the PDFs above, especially the elapsed months version) plus the difference in minutes for referring the calculation to Jerusalem, it could be misinterpreted to imply that the molad refers to a meridian 2 hours to the east, which indeed today would correspond to the meridian of Qandahar, Afghanistan.
Another possible explanation for the "Afghanistan mistake" is neglecting to subtract Delta T when calculating the moments of all lunar conjunctions, especially for ancient times. The necessity for this correction has only been understood since the beginning of the 20th century, especially since the advent of Atomic Time (1955) and Laser Lunar Ranging (1969). The accuracy of the Delta T correction has been progressively refined over the decades since 1970 as the LLR observations have continued with improved instrumentation, and as progressively more historical lunar and solar eclipse records have been analyzed. The Delta T value that the NASA Eclipses web site polynomial approximation generates for Rosh HaShanah of Hebrew year 4119 is 7107 seconds ≡ 1h 58m 27s ≡ almost 2 hours, which agrees nicely with this explanation.
In fact, my Average Molad Adjustment is proportional to Delta T 47 KB, which should not surpise anybody because they have the same physical basis (Tidal Acceleration), and as a corollary both Delta T and the Average Molad Adjustment are proportional to the Mean Synodic Month 52 KB. For more information about Delta T see this page. The accuracy of the algorithm for estimating Delta T is critical to this investigation of the relationship between the molad and the actual lunar conjunctions, espeically for remote past or distant future differences.
Others have asserted that the molad refers to a meridian that is 6 or 8 hours ahead of Jerusalem, either in the East China Sea between China and Japan (90° east of Jerusalem), or in the Pacific Ocean an hour to the east of Japan (120° east of Jerusalem). A likely explanation for this mistake is the failure to add 6 hours to the astronomical moment to count time from mean sunset instead of civil midnight. In combination with the 2 hour error explained above, this can account for a +8 hour offset. Each hour of time difference corresponds to 360° / (24 hours per day) = 15° of longitude.
In the era when the traditional molad calculation was established the average lunar cycle (Mean Synodic Month, or MSM) was assumed to be eternally constant.
How was the traditional molad interval length originally determined? The MSM was probably first measured by the Babylonians, using records of eclipses, most likely lunar eclipses, which occur much more frequently and can be much more easily continuously observed than solar eclipses, and can be simultaneously seen from many locations on Earth's night time hemisphere. One could estimate the MSM by carefully determining the elapsed time between the maxima of two lunar eclipses and then dividing by the number of elapsed lunar months. For best accuracy, the eclipses should have been separated by as many years as records were available for, preferably spanning several centuries. Even so, basing this calculation on only one pair of eclipses would be misleadingly erroneous unless each eclipse happened to represent near average lunation periods, or they were at equivalent points in the lunar periodical variability cycles. To improve accuracy, it would have been necessary to average many eclipse pairs.
Around 147-148 AD, the Greek astronomer Ptolemy of Alexandria published his multi-volume treatise of astronomy, known today as Almagest, in which he gave the the length of the MSM in sexagesimal format as 29 days 31' 50'' 8''' 20'''', that is 29 days 31 minutes 50 seconds 8 thirds and 20 fourths, numerically exactly equal to the traditional molad interval (see "Why Divide Hours into 1080 Parts?" at <http://individual.utoronto.ca/kalendis/hebrew/chelek.htm>), but Ptolemy cited Hipparchus as the source for that MSM.
Whatever the source of the traditional molad interval, the next question must be: How was the molad epoch, called BaHaRad, determined? This is not a trivial question, because if an error had been made in the choice of epoch, then that error would have persisted in the calculation forever, and it is very difficult to choose an appropriate epoch on astronomical grounds, because the lunar conjunction itself is not observable. Even if the maximum of a solar eclipse were used to guide selection of the epoch, that particular eclipse might have occurred anywhere within ±14 hours of the true mean lunar conjunction. The selection of epoch could not have been based on accurate astronomy back-calculated to the epoch, because the traditional moladot near the epoch average 7+1/2 hours late, as shown in the chart above. Rather, they must have positioned the epoch at the moment that minimized the astronomical error in the era when the molad calculation was developed. I am suggesting that the molad calculation was developed in the era of the Second Temple because that is the era when the differences between moladot and actual astronomical lunar conjunctions were minimal. Alternatively, it could have been developed a few centuries later but based on historical astronomical records that originated near the era of the Second Temple.
In my own numerical integration based development of mean lunar conjunction polynomials, as discussed on my web page "The Length of the Lunar Cycle" at <http://individual.utoronto.ca/kalendis/lunar/>, I found that the mean lunar conjunction in January 2000 AD, which I set as the epoch for my "J2K" lunation count, was almost 4 hours before the actual lunar conjunction on January 6th. Accurately establishing that epoch involved literally hundreds of millions of calculations, thankfully mostly carried out automatically by computer!
In other words, I used an accurate lunar ephemeris to establish my "J2K" epoch. The ancients could have done the equivalent with a reasonably accurate lunar ephemeris that was available at the time. If the calculation was established in the era of Shmuel about a century after Ptolemy or in the era of Hillel ben Yehudah about two centuries after Ptolemy, they could have relied upon the published Ptolemaic algorithms. Alternatively, if the calculation was established much earlier, in the era of the Second Temple, then they could have relied upon a lunar ephemeris obtained from Hipparchus or from the Babylonians.
The web page "The Length of the Lunar Cycle" at <http://individual.utoronto.ca/kalendis/lunar/> also includes a section explaining a simple method for "Estimation of Fixed Lunar Cycle Calendar Drift", using the drift of the molad of the Hebrew calendar as an example, which arrives at a result that is closely comparable with the evaluations documented on this page.
In the present era the median length of the lunar cycle is about 29 days 12 hours and 30 minutes, the MSM is slightly more than 29 days 12 hours and 44 minutes, the shortest lunations are about 29 days 6 hours and 30 minutes, and the longest are about 29 days and 20 hours. Thus the length of the synodic month varies over a range spanning about 13 hours and 30 minutes! These variations were greater in the past and will diminish in the future:
Centile trends, per group of 4657 lunar months, based on SOLEX 9.1β numerical integration
(the image below is linked to a higher resolution full-page PDF version)
Although the individual lunation length variations span about 13 hours and 30 minutes, compared to the traditional molad the periodic variations appear to be substantially greater, spanning about 28 hours, because of series of several short lunations in a row (before and after Earth's orbital aphelion) alternating with several long lunations in a row (before and after Earth's orbital perihelion). Click here or on the heading or any of the next 3 charts to open up a higher-resolution PDF version containing all 3 of the following periodic variation charts 384 KB:
The rather large periodic variations of the length of the lunar cycle make it impossible to accurately calculate the mean length of the lunar cycle based on the separation between any two well-established lunar conjunctions, such as total solar eclipses, even when that separation spans many centuries.
The only molad moment that is of relevance to the traditional Hebrew calendar is the molad of Tishrei, which determines the provisional date of Rosh HaShanah, subject to the traditional Rosh HaShanah postponement rules. The variation of the molad of Tishrei relative to the actual lunar conjunctions is not subject to the seasonal variations that affect the moladot overall (discussed below), so it spans "only" about 20 hours, as shown in this chart 315 KB.
For a published explanation of periodic lunar cycle variations see "Chapter 4: The Duration of the Lunation" on pages 19-31 in "More Mathematical Astronomy Morsels" by Jean Meeus, published in 2002 by Willmann-Bell, Inc., Richmond, Virginia.
Once again referring to the chart showing the difference between the molad moment and the Actual Lunar Conjunction (Jerusalem Local Mean Time) 741 KB for the first 10000 years of the Hebrew calendar, note that the further back in time that one back-calculates the molad prior to the era of Hillel ben Yehudah, the later it was with respect to the actual mean lunar conjunction, to a maximum of about 9 1/3 hours late at the epoch of the Hebrew calendar. Going forward in time from the era of Hillel ben Yehudah, the molad drifts later and later at an accelerating rate, being about 2 hours late in our time (Rosh HaShanah 5766 is lunation number 71305 since the epoch), and more than 20 hours late by the year 10000. This implies that although the molad interval was appropriate for the era of Hillel ben Yehudah, it was too short for earlier ages, and too long for later ages. In other words, the molad interval has a constant value, but the mean interval between lunar conjunctions has been steadily getting shorter. The value of the equation for the blue line for any given lunation number represents the average error of the molad with respect to the actual mean lunar conjunction. This value, when subtracted from the corresponding molad moment, "adjusts" it by an amount that restores the relationship that existed in the era of Hillel ben Yehudah, but it includes a shift of the reference meridian to Jerusalem because the astronomical moments were converted to Jerusalem Local Mean Time.
By reconciling the average deviation of the molad, indicated by the curved blue regression line, I calculated the Apparent Mean Synodic Month, and graphically depicted its trend in relation to the Gregorian year, Hebrew year, and Hebrew calendar elapsed months. The following charts show those trends, and each includes a linear equation that can be employed, as it is in Kalendis, to estimate the Mean Synodic Month for any date:
The Kalendis estimate of the Mean Synodic Month is computed in time units of Mean Solar Days. This is quite different from either of the two formulae widely available on the internet and attributed to the ELP2000 Lunar Theory (Ephemerides Lunaires Parisiennes) of Chapront et al, which calculate the length of the Mean Synodic Month in uniform time units of Ephemeris Days (essentially equivalent to days measured in atomic time):
where T is the number of Julian Centuries (intervals of 36525 ephemeris days) relative to J2000.0 (January 1, 2000 AD at Noon, Terrestrial Time).
Mean Solar Days are getting longer over time due to tidal deceleration of Earth's rotation, whereas Ephemeris Days are by definition of constant duration over the millennia. In terms of Ephemeris Days the durations of lunations are getting longer, because Moon is moving further away, according to on-going daily Laser Lunar Ranging measurements, at an average rate of about 38 mm per year, due to tidal acceleration. However, in terms of Mean Solar Days (that is, calendar days), durations of lunations are getting shorter, because solar days are getting longer at a greater rate than the lunations.
CT Scrutton and RG Hipkin, in "Long-term changes in the rotation rate of the Earth", Earth-Science Reviews 1973 (9): 259-274, claim that if one back-calculates the history of the lunar orbit, allowing for larger tidal acceleration when Moon was closer, it implies a catastrophic period (incompatible with the survival of any forms of life) in the mid-Precambrian era, during which there would have been tremendenous tectonic upheavals, volcanic eruptions, and heat generation caused by the close proximity of Moon to Earth, at a distance of only a few Earth-radii.
After conversion of Ephemeris Days to Mean Solar Days, Chapront's linear formula corresponds to a Mean Synodic Month length that is essentially identical to and an important validation of the Mean Synodic Month formula of Kalendis, as shown in the following charts:
All of this evidence points to the conclusion that the molad interval was essentially a perfect fit to the actual mean synodic month in the time of Hillel ben Yehudah, but today it is slightly too long. The Mean Synodic Month is currently about 3/5 second or 9/50 of a part shorter than the traditional molad interval, and is continuing to get shorter at a steady rate of about 1/3 mean solar second per thousand years. The present-era 3/5 second or 9/50 part discrepancy may seem negligible in comparison with the 29+13753/25920 days of the molad interval, but those fractions inevitably accumulate as each and every month passes, so that over the years it progressively adds up to minutes, then hours, and so on. The total accumulated drift of the molad relative to the average alignment that it had in the era of Hillel ben Yehudah is currently about 1 hour and 37 minutes. If one adds 23 minutes to shift the reference meridian to Jerusalem, then the average molad today is more than 2 hours late with respect to the actual mean lunar conjunctions in terms of Jerusalem Local Mean Time.
Many with rabbinic education traditionally hold that the molad interval was given to Moses at Mount Sinai.
Many historians have noted that the Hebrew calendar molad interval matches the mean lunation period that was used by Babylonia, Hipparchus, and Ptolemy.
At the time of the Exodus from Egypt, traditionally given as Nisan (Aviv) of Hebrew year 2448 (counting the first year of the life of Adam haRishon as year zero), the traditional molad interval was about 2/5 second shorter than the Mean Synodic Month. (Anyone who is designing a fixed cycle lunar calendar for use today and for as long as possible into the future would be well advised to choose a mean lunation period that is intentionally slightly too short, to maximize the duration that the calendar will serve with reasonable accuracy.)
The era of the Babylonians (most specifically the Chaldeans, who were renowned for their astronomic observations) was around the 6th century BC. For the month of March in 600 BC on the proleptic Julian calendar the Mean Synodic Month was almost 1/4 second longer than the molad interval.
The era of Hipparchus was around 190 BC to 120 BC. For the month of March in 120 BC on the proleptic Julian calendar the Mean Synodic Month was essentially equal to the traditional molad interval (within 25 milliseconds).
Ptolemy's Almagest was published around 150 AD. For the month of March in 150 AD on the Julian calendar the Mean Synodic Month was very close to the traditional molad interval (within 1/10 second).
Taking Rosh HaShanah 4119 as the date that Hillel ben Yehudah started the fixed arithmetic traditional Hebrew calendar = September 21, 358 AD, the Mean Synodic Month was again nearly equal to the traditional molad interval (within 25 milliseconds).
Out of sheer curiosity I separately analyzed the molad minus actual new moon relationship separately for each month of the traditional Hebrew calendar. This revealed a periodic annual variation. In the present era, the molad of the month of Nisan is earliest (about 2 1/2 hours early), then the moladot drift later until the months of Tishrei and Cheshvan, whose moladot are equally the latest (about 5 2/3 hours late), after which the moladot drift earlier until the next month of Nisan. Currently the months of Tammuz and Shevat are close to the annual average, which itself is about 2 hours late (referred to Jerusalem). Click here to view the chart for Nisan alone 309 KB, Tishrei alone 315 KB, or all of the Hebrew calendar months in one large file 3.8 MB. Each of these month charts has its own quadratic least-squares regression line that could be used to implement an adjustment of the molad moment tailored for that month's typical seasonal variation, but to do so would be going considerably beyond restoration of the original state that the molad had in the era of Hillel ben Yehudah, and anyhow the only molad moment that matters to the Hebrew calendar is the molad of Tishrei.
In each era there is one month that has the earliest moladot, another month 1/2 year later that has the latest moladot with respect to the actual mean lunar conjunctions, and there are two months in-between whose moladot are closest to the average for the year:
Variations of the Molad Minus Mean New Moon Across the Months of the Calendar Year (hours)
These variations shift rightward (later in the calendar year) as the centuries pass. From the table above one can see that at the calendar epoch (year 1) the earliest molad was in Shevat, by year 4119 it had shifted later to Adar 2 and Nisan, by 5766 (the present era) Nisan alone was earliest, and by 10000 it will have shifted later to Sivan. Meanwhile the latest molad was in Av in the year 1, Tishrei in 4119, both Tishrei and Cheshvan in the present era, and then rightward to Tevet by 10000. As shown in the row for year 4119 and for year 5766, when the earliest molad reaches the last month of the year, it cycles back around to the first month of the year.
Seasonal variations in the timing of the New Moon were noted by Rambam, without explanation, in fact most of chapter 17 in his book deals with such seasonal adjustments to the lunar longitude. These variations, however, have nothing to do with the climatic seasons, nor are they related to precession of the equinoxes.
This shifting pattern follows a perpetual cycle, which turns out to be due to the long-term Earth orbital perihelion cycle. As explained by Jean Meeus (cited above), when Earth is near its orbital perihelion it moves more rapidly, prolonging the durations of the lunations because it takes longer for Moon to move fully into conjunction with Sun (giving the fixed molad arithmetic a chance to "catch up"). When Earth is near its orbital aphelion it moves more slowly, reducing the durations of the lunations because it takes less time for Moon to move fully into conjunction with Sun (so the fixed molad arithmetic "falls behind"). The Earth orbital perihelion gradually shifts later in the calendar year, variably taking 20000 to 25000 years to make a full cycle through all of the seasons. Perihelion advances more rapidly as the Earth orbital eccentricity declines. Perihelion is currently in early January / Tevet and aphelion is currently in early July / Tammuz.
I created a spreadsheet to depict these perihelion-related variations of the molad in a modestly animated way in this Molad minus Mean New Moon spreadsheet 45KB. This spreadsheet requires Microsoft Excel — if you don't have a compatible program then you can click here to download the free Microsoft Excel Viewer 2003 for Windows. At the top right of the spreadsheet page the user can enter any desired Hebrew year number from 1 to 10000, or can use the arrow buttons beside the year number to scan backward or forward through the years (these buttons only work in the full version of Excel, not the free Excel Viewer), then observe the effect on the displayed table and chart. Notice how the peak and valley of the plotted curve shift rightward, and observe the way that the annual average rises later and later as the years progress into the future (or further into the remote past approaching the epoch). Go to the year 4119 and observe the effect of toggling the checkbox option in the top line "Molad refers to halfway between Ur, Babylonia and Jerusalem, Israel." on and off, noting that the annual average line is superimposed upon the zero difference line only when that option is checkmarked.
If you are a pure mathematician with an irrational revulsion against spreadsheets then you can instead open this limited PDF version that covers a few selected years 333KB, but believe me, you will miss a lot!
I was able to generate a 3-dimensional chart that depicts the Hebrew calendar month on the x-axis (horizontal), the Molad minus Mean New Moon on the y-axis (vertical), and the Hebrew Calendar Elapsed Months on the z-axis (depth). The image looks like a saddle, with a hump (due to the latest months) gradually shifting diagonally across the saddle as the years pass (due to the perihelion cycle). Click here to see the Molad minus Mean New Moon as a 3-dimensional surface 43 KB.
If the moladot were adjusted to eliminate the average drift, then that 3-D surface would be a flat plane vertically centered at the zero difference level, with a perihelion-related hump and valley gradually moving diagonally across.
If the moladot were adjusted to match the seasonal variations in the actual mean lunar conjunctions, then the surface would be a hump-free flat plane at the zero difference level.
Looking carefully at any of the Molad minus actual New Moon charts, one can see that the breadth of the curved band is broader at the calendar epoch (year 1), tapering to a slightly narrower breadth by the year 10000. That effect is not an optical illusion, it really is so, and is due to the steadily declining Earth orbital eccentricity over the first 10000 years of the Hebrew calendar. One can see this effect more obviously in the Molad minus Mean New Moon spreadsheet 45KB (click here to download the free Microsoft Excel Viewer 2003 for Windows) — note how the span or full swing of the annual cycle steadily declines from year 1 through to year 10000, along with the Earth orbital eccentricity, which is shown further down the page.
As explained by Jean Meeus, the variability of the duration of lunations decreases as the Earth orbital eccentricity declines, a trend which will continue for many millennia into the future. If Earth's orbit were perfectly circular then there would not be any seasonal variations in the duration of lunations. These seasonal variations have essentially no relationship to Earth's well-known axial tilt nor to precession of the equinoxes, which are involved in most other seasonally-variable effects, although the perihelion cycle does have an effect on the lengths of the seasons. See also "Chapter 33: Long-period variations of the orbit of the Earth" on pages 201-205 in "More Mathematical Astronomy Morsels" by Jean Meeus, published in 2002 by Willmann-Bell, Inc., Richmond, Virginia.
The variations in the Earth orbital eccentricity contribute to the variations in the time that it takes for each perihelion cycle. As the orbital eccentricity declines the perihelion advances more rapidly. That is why it is not valid to state the duration of the perihelion cycle more precisely than "20000 to 25000 years". For further insight, please see my web page entitled "The Lengths of the Seasons" at <http://individual.utoronto.ca/kalendis/seasons.htm>.
An "average" molad adjustment would symmetrically align the Molad minus Mean New Moon annual curve above and below the zero difference line, allowing it to shift rightward with time along with the Earth orbital perihelion. Individual moladot would vary by several hours from the zero difference. The distribution band of the Actual New Moons would gently "wave" around an average level close to the adjusted molad that would be characteristic for each month of the Hebrew calendar. Click here to view an average-adjusted Molad minus Actual New Moon chart for Nisan 301KB and for Tishrei 299 KB.
By using a separate adjustment formula for each month of the Hebrew calendar year, each molad could be locked (vertically centered) at the Molad minus Mean New Moon zero difference line for all Hebrew calendar years from 1 to 10000, with the Actual New Moon always within slightly beyond ±10 hours of the adjusted molad. Click here to view an month-specific adjusted molad minus Actual New Moon chart for Nisan 302KB and for Tishrei 302 KB. All of the other month-specific adjusted molad charts look the same.
A month-specific molad adjustment might be worth considering for the molad of Tishrei, because it is the only molad that affects the arithmetic of the Hebrew calendar, by determining the provisional date of Rosh HaShanah. In the present era the moladot of Tishrei and of Cheshvan have the latest average molad moments. Specifically adjusting the molad of Tishrei would have the probably undesirable side-effect of making it less likely that the new lunar crescent will be visible on the eve of Rosh HaShanah.
Adjustment of the molad moment should ideally be a nearly linear polynomial function to account for the progressive shortening of the Mean Synodic Month combined with a cyclic (periodic) function to account for the perihelion cycle, preferably adjusting for the date of the molad relative to the date of perihelion rather than by the Hebrew month, and the swing of the cyclic function should be controlled by an expression that is a function of the Earth orbital eccentricity. The period of the cyclic component itself ungoes long-period variation, however, so it can't be reduced to a single trigonometric term that will remain in sync for more than a few centuries.
Therefore the best long-term adjustment that is practical with simple arithmetic is an "average" molad adjustment, which ought to be acceptable or perhaps even preferable anyway, since it would restore the molad vs actual lunar conjunction relationship to the state that it had back in the era of the Second Temple, resetting the reference meridian to its original longitude at the midpoint between the Nile river and the end of the Euphrates river, tracking the progressively shorter mean lunar conjunction interval, discussed next.
On my separate web page, "Length of the Lunar Cycle" web page at <http://individual.utoronto.ca/kalendis/lunar/>, I show how to calculate the mean lunar conjunction moment for the Prime Meridian. It is easy to use that arithmetic to calculate the mean lunar conjunction moment for the meridian that is midway between the Nile and the Euphrates. The arithmetic there works in terms of the lunation number relative to January 2000 AD (J2KL), but for Hebrew calendar arithmetic purposes we need arithmetic that works in terms of Hebrew calendar elapsed months (HCEM), that is the lunation number starting from zero at the Hebrew calendar epoch. For interconversion of J2KL and HCEM we only need to know the HCEM at January 2000 (Rosh Chodesh Shevat, 5760):
HebrewLunationAtJ2000 = 71233
J2KL = HCEM – HebrewLunationAtJ2000
HCEM = J2KL + HebrewLunationAtJ2000
Note that Shevat 5760 was month number 71234 since the Hebrew calendar epoch, but it didn't start until the 8th of January 2000 AD, whereas the mean lunar conjunction functions need to have the count of months elapsed (completed).
So to calculate the desired moment of the mean lunar conjunction, use the ElapsedMonths function defined above (or better still, the Traditional or Rectified mode of the Rectified Hebrew calendar ElapsedMonths function, as desired) to calculate the HCEM, then subtract HebrewLunationAtJ2000 to obtain the J2KL to pass to the MeanNewMoonUT function defined on the "Length of the Lunar Cycle" web page, and then add 2 hours and 37 minutes = 157/1440 of a day to convert that moment to Local Mean Time at the Nile-Euphrates midpoint (in other words, Jerusalem Local Mean Time +16 minutes). That, however, is the moment referred to Civil Midnight, whereas the molad moment is traditionally quoted as the time since mean sunset, so we should add 6 hours = 1/4 of a day for the purpose of announcements or relating the moment to the Hebrew weekday. Finally, to use such a molad moment for the purpose of calculating the provisional date of Rosh HaShanah, add another 6 hours = 1/4 of a day to refer the moment to the prior Local Mean Noon and thereby include the Gauss shortcut, which bypasses the molad zakein postponement rule. Obviously this list of additions can be combined into one operation, if desired.
The interval between such moladot will grow progressively shorter, presently at a rate of about –25 mean solar microseconds per month, which would be incompatible with the fixed cutoff limits traditionally applied to the 3rd the 4th Rosh HaShanah postponement rules. The solution is very simple and fully equivalent to the traditional way that those rules are applied, as explained under the heading "The First Day of Tishrei (Rosh HaShanah) — the HebrewNewYear Function" on the rectified Hebrew calendar web page at <http://individual.utoronto.ca/kalendis/hebrew/rect.htm>.
At the time of writing the rectified Hebrew calendar employs a progressive molad based on a quadratic molad adjustment that was derived using older astronomical algorithms and Delta T approximation, referred to the meridian of Jerusalem. The Nile-Euphrates mean lunar conjunction calculation documented above would be a better choice. My current development efforts, however, are directed at obtaining equivalent accuracy (same calendar dates) while working strictly with integers, through the use of yerm era arithmetic, discussed below. Stay tuned...
From the information presented here, you may have the impression, as many others have written, that the drift of the molad is minor and can be neglected. Not so.
For example the calculated moment of the molad of Tishrei 5768 is 10 hours and 26 minutes after mean sunset on Yom Rivii on the 29th of Elul 5767. Rosh HaShanah can't start on Yom Rivii so it is postponed to the next day, Yom Chamishi. The corresponding astronomical mean lunar conjunction at the meridian midway between the Nile River and the end of the Euphrates River is on the same day but 8 hours 41 minutes and 15 seconds after mean sunset. On that day the calculated molad moment is therefore 1 hour 44 minutes and 15 seconds too late = 1/24 + 44/1440 + 15/86400 = 139/1920 or about 0.0724 of a day. This fraction represents the probability, as of year 5768, that the molad of Tishrei will land one day later than the sages originally intended. Expressed as a percentage, it means that already in almost 7+1/4% of years the molad of Tishrei lands one day late.
However, the resultant proportion of years in which Rosh HaShanah and the High Holy Days of Tishrei land on the "wrong" day is considerably greater, because of interactions with the Rosh HaShanah postponement rules, which in turn are subject to interactions between adjacent years.
As the drift of the molad accelerates, due to the growing discrepancy between the traditional molad interval and the astronomical mean synodic month, the percentage of years which will start on the "wrong" day will increase proportionately.
I placed "wrong" in quotation marks above because although the dating seems astronomically incorrect, according to Jewish law the dates of the calendar are not determined in Heaven, nor by astronomy, but rather the "correct" days for observing Rosh HaShanah and the other High Holy Days etc. are whichever days the Jewish people agree to observe them on. This principle is known in Hebrew as Torah lo ba'shomayim hi.
Even more importantly, one must understand that the molad interval sets the Hebrew calendar mean year length, which traditionally equals 235 lunar months times the duration of the molad interval divided by 19 years = 365 days 5 hours 55 minutes and (25+25/57 seconds or 7+12/19 chalakim) ≈ 365.2468222 days, which is presently 6 minutes and 25+25/57 seconds too long per year. Most of that excess length is due to the excess length of the traditional 19-year Hebrew calendar leap cycle, but a significant portion is due to the increasingly excessive length of the molad interval.
I have developed the rectified Hebrew calendar, see <http://individual.utoronto.ca/kalendis/hebrew/rect.htm>, based on a leap cycle that has 130 leap years per 353-year cycle (353 years with 12 months = 4236 months plus 130 leap months = 4366 months per cycle). It has fractionally fewer leap months than the traditional leap cycle — although it will take 6707 years for the difference to amount to one full month, that is just the right amount to restore the calendar mean year to the length of the actual present era mean northward equinoctial year. Using the traditional molad, the rectified Hebrew calendar will maintain excellent equinox alignment until perihelion reaches the northward equinox in the 11th Hebrew calendar millennium (for an explanation please see my "Lengths of the Seasons" web page at <http://individual.utoronto.ca/kalendis/seasons.htm>). However, it so happens that applying the progressive molad instead of the traditional molad will allow the rectified Hebrew calendar to maintain its long-term alignment with the northward equinox for three extra millennia (until the 14th millennium), because the adjustment of the molad interval will progressively shorten the rectified Hebrew calendar mean year length at the rate of about 3/2 seconds per 353-year leap cycle or about 17/4 seconds per millennium, thus helping the rectified leap cycle to "hold on" to the equinox much longer (until the rate of change of the length of the northward equinoctial year accelerates beyond range, which will happen when perihelion advances beyond the equinox).
With use of the traditional molad interval the mean year of the rectified Hebrew calendar will equal the traditional molad interval × 4366 months per cycle / 353 years per cycle = 365 days 5 hours 49 minutes and 5+25/1059 seconds, which is much more accurate than the traditional 19-year leap cycle, being only about 5 seconds longer than the mean northward equinoctial year. If, however, the molad interval is adjusted to match the actual secular mean synodic month (the word secular, derived from the Latin saeculum, in this context refers to a variation that spans centuries, that is the progressively shorter mean synodic month), then in the present era the mean year of the rectified Hebrew calendar will be about 365 days 5 hours 48 minutes and 58 seconds, which is about 2 seconds shorter than the northward equinoctial year. By the 11th Hebrew millennium, the progressive molad will cause the rectified Hebrew calendar mean year to shorten to 365 days 5 hours 48 minutes and 40 seconds, but in that future era the advance of perihelion will have passed the northward equinox and the mean northward equinoctial year will be even shorter, about 365 days 5 hours 48 minutes and 32 seconds.
This chart shows the mean year of the rectified Hebrew calendar 130/353 leap cycle using the traditional and adjusted molad, in comparison with the actual mean northward equinoctial year 28KB. With regard to the periodic variability of the northward equinoctial year, from year-to-year the dominant cause of variability is Moon, but I reduced the variations in this chart by plotting the average for every one hundred years and smoothing the lines connecting the points.
Note that all of the mean years were equal in the era of Hillel ben Yehudah, Hebrew year 4119. Over the next few millennia the Rectified mean year with progressive molad will get steadily shorter while the mean equinoctial year gets slightly longer. During that time the progressive molad-based calendar will on average pull slightly ahead (by a small fraction of a day). Around the Hebrew year 9000 the progressive molad-based calendar mean year will nicely match the mean equinoctial year length. During the subsequent millennia the mean equinoctial year length will shorten faster. Beyond the 10th millennium the rectified Hebrew calendar with traditional molad will fall progressively behind the mean equinoctial year at an accelerating rate, but the progressive molad-based calendar will be able to "hold on" to the equinox until near the end of the 13th millennium, because of its having pulled slightly ahead during the 8th to 10th millennia.
I have also carried out analyses of the relationship between the first day of Hebrew months, the actual lunar conjunction moments, and the Lunar Phase. Some examples are shown below for the Hebrew months of Nisan and Tishrei.
|First Day of Month vs. Actual New Moon||116 KB||114 KB|
|First Day of Month vs. Lunar Phase||116 KB||115 KB|
The charts of first day against actual New Moon and against Lunar Phase have a curved band pattern that relates to the curved bands seen above in the Molad minus Actual New Moon charts.
Rebbe Eliezer ben Hurcanus, traditional author of Pirkei D'Rebbi Eliezer, described the molad cycle in terms of a "short cycle" of 3 lunar years, after which the molad day would be one weekday earlier than where it started, and a "long cycle" of 21 lunar years, after which the molad day would be on the same weekday that it started (having stepped backwards by one day for each of 7 elapsed short cycles). These cycles are approximations, for example after two long cycles the molad day will fall one weekday later.
The traditional molad can, nevertheless, be expressed to absolute accuracy in terms of cycles, but to perceive its inherent repetition pattern one must let go of the concept of a 12-month "lunar year" and any relationship to the solar year, as well as the idea that the cycle began at the traditional year of Creation.
Traditionally the molad day is the day upon which the molad moment lands, but if the calculated moment is after noon then the molad day is considered to have advanced to the following day (based on the molad zakein postponement rule). Referring to the calculation expressions defined above, the molad day can be calculated as follows:
MoladDay( Lunation ) = floor( MoladMoment( Lunation ) + 1/4 )
where the 1/4 day offset takes care of the molad zakein postponement rule,
and Lunation is the number of months elapsed from the epoch of the traditional Hebrew calendar.
The number of days from one molad day to the next can only be either 29 or 30 days, corresponding to a "deficient" (D) or a "full" (F) month, respectively. The following are the molad month lengths near the present era, from the molad of Nisan 5726 until the molad of Elul 5818, inclusive:
FDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFD FDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDF DFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFD FDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDF DFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFD FDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDF DFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFD FDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDF DFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFD FFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDF DFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFD FDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDF DFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFD FDFDFDFFDFDFDFDFDFDFDFDF
It is hard for the eye to discern any particular pattern in the above sequence, which contains mostly alternating full and deficient months, but it actually does contain a very simple repetition. Focus on the places where consecutive full months occur. A simple repeating pattern becomes clear if one inserts a space between consecutive full months and a new line after every third group:
FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF FDFDFDFDFDFDFDF FDFDFDFDFDFDFDFDF
In summary, the above sequence of months, which is only one "era" of the full molad sequence, contains 23 repeats of a 49-month sequence, each having two 17-month groups and one 15-month group, completed by a final stand-alone 17-month group, having a total of 1144 months in the era.
Each 17- or 15-month group starts with a full month, then has deficient alternating with full months until the end of the group. K.E.V. (Karl) Palmen of the Rutherford Appleton Laboratory in the United Kingdom (retired in late 2018) discovered these groups and patterns in mean lunar month sequences, invented the word "yerm" for them, and used them as the basis for his yerm lunar calendar, see <http://www.hermetic.ch/cal_stud/palmen/yerm1.htm>. Each yerm has exactly one more full month than its count of deficient months, so the basic yerms are:
17-month yerm = FDFDFDFDFDFDFDFDF
15-month yerm = FDFDFDFDFDFDFDF
The average or mean month of any sequence of alternating full / deficient months, where the number of full months equals the number of deficient months, is exactly 29+1/2 days. If the sequence contains any consecutive full month pairs, however, then its mean month increases accordingly. In any sequence of full and deficient months, if NFM is the number of full months and NDM is the number of deficient months, then the mean month length equals the number of days in the cycle divided by the number of months in the cycle = (NFM × 30 + NDM × 29) / (NFM + NDM) days. This expression can be simplified because the mean month is always 29 days plus a fraction equal to the number of full months divided by the total number of months in the cycle = 29 + NFM / (NFM + NDM) days.
The mean month of the more frequent 17-month yerm is (9 × 30 + 8 × 29) / (9 + 8) = 29+9/17 days = 29 days 12 hours 42 minutes and 21+3/17 seconds, which is exactly 30+11/17 parts shorter than the molad interval. Any longer yerm has a shorter mean month.
The mean month of the less frequent 15-month yerm is (8 × 30 + 7 × 29) / (8 + 7) = 29+8/15 days = exactly 29 days 12 hours and 48 minutes, which is exactly 71 parts longer than the molad interval. Any shorter yerm has a longer mean month.
In the overall sequence 17-month yerms occur about twice as frequently as 15-month yerms because the deviation of the 17-month yerm mean month from the traditional molad interval is about half that of the 15-month yerm.
The sequence inherently contains larger groups having subcycles of 17+17+15 = 49 months, occasionally followed by an extra 17-month yerm. Each 49-month subcycle contains 3 yerms, having 3 more full months (2×9+8=26) than deficient months (2×8+7=23), and has a mean month of (26 × 30 + 23 × 29) / 49 = 29+26/49 days = 29 days 12 hours 44 minutes and 4+44/49 seconds, which is only 23/49 parts longer than the molad interval. The extra 17-month yerm that is occasionally inserted compensates for the slight mean month excess of each series of 49-month subcycles.
[According to unpublished manuscripts composed around 1700 AD, Sir Isaac Newton recognized the 17- and 15-month subcyles, and called the 49-month subcycle a "Great Lunar Cycle", although he apparently had difficulty deciding where to begin the latter. Inexplicably, he thought that the mean month of the 49-month group was too short, so he proposed adding one extra day, even though he otherwise relied heavily on the lunar algorithms of Jeremiah Horrocks, which had an impressively accurate mean month of 29 days 12 hours 44 minutes 3.16 seconds, almost 1+3/4 seconds shorter than that of the 49-month group. See: Notes & Records of The Royal Society 2005; 59: 223-254, especially the discussion under the heading "Lunar Part", on page 230.]
The present traditional molad era has 70 yerms, but the next molad era from the molad of Tishrei 5819 until molad of Adar Sheini 5907, inclusive, is an example of the alternative type of era that will have one fewer 49-month subcycle = 67 yerms, so it will have only 22 repeats of the 49 month subcycle plus the final 17-month yerm, for an era total of 1095 months. Each 1095-month era contains 3 yerms for each of the 22 subcycles that have 49 months plus 1 yerm for the extra 17 months = 22×3+1 = 67 yerms, having 67 more full months than deficient months, so it has (1095-67) / 2 = 514 deficient months and 514+67 = 581 full months. The mean month of this 1095-month type of era is (581 × 30 + 514 × 29) / (581 + 514) = 29+581/1095 days, which is only 1/73 part shorter than the traditional molad interval.
Each 1144-month era, like the present era, contains 3 yerms for each of the 23 subcycles that have 49 months plus 1 yerm for the extra 17 months = 23×3+1 = 70 yerms, having 70 more full months than deficient months, so it has (1144-70) / 2 = 537 deficient months and 537+70 = 607 full months. The mean month of such an 1144-month era is (607 × 30 + 537 × 29) / (607 + 537) = 29+607/1144 days, which is only 1/143 part longer than the molad interval. The 1144-month era type occurs almost twice as frequently in the complete molad cycle because its mean month distance from the target molad interval length is almost half that of the 1095-month era type.
A complete cycle with mean month exactly equal to the traditional molad interval has 25920 months with 13753 full months and 12167 deficient months, comprising exactly 2160 twelve-month lunar "years" or about 2095+2/3 solar years. It has 15 eras of 70 yerms interleaved with 8 eras of 67 yerms. Thus the full cycle has 15 × (23 × 2 + 1) + 8 × (22 × 2 + 1) = 1065 of the 17-month yerms plus 15 × 23 + 8 × 22 = 521 of the 15-month yerms, for a total of 1586 yerms. The complete interleaved traditional molad yerm era sequence, from the molad of Elul 4724, which was the most recent cycle starting point prior to the present era, is:
67, 70, 70 yerms
67, 70, 70
67, 70, 70
67, 70, 70
67, 70, 70
67, 70, 70
67, 70, 70
In summary, the complete traditional molad cycle has 7 repeats of (67, 70, 70) followed by final 67- and 70-yerm eras. That complete pattern repeats forever in the fixed arithmetic cycle of the traditional molad. The simple arithmetic reason for regarding the final 67- and 70-yerm eras as the "end" of the complete cycle is so that the first era of each triplet as well as the final partial triplet has 67 yerms, and all other eras have 70 yerms.
Confirming the previous calculation, the total number of yerms in the complete cycle = 8×67 + 15×70 = 1586 yerms, which is the required excess of full months over deficient months in the complete sequence of 25920 months. Thus the full cycle has (25920-1586) / 2 = 12167 deficient months and 12167+1586 = 13753 full months. The mean month of the full cycle is (13753 × 30 + 12167 × 29) / (13753 + 12167) = 29+13753/25920 days, which is exactly equal to the traditional molad interval.
Click here to view a complete report of the traditional molad yerm era analysis covering the first 10000 years of the Hebrew calendar (includes 5 complete molad yerm era cycles, in which era zero is defined as the era that started prior to and ended after the Hebrew epoch). The following formula, due to Karl Palmen, calculates the molad day using only the era and lunation numbers:
Determine the lunation number by using the ElapsedMonths function defined above.
Determine the era number by comparing the given lunation number with the "HCEM" column on the traditional molad yerm era list.
molad day = EraZeroStartDay + 30 + floor( [ (Lunation – SeriesStartLunation – 1) × 1447 – EraNumber ] / 49)
where EraZeroStartDay is 5286 days before the Hebrew epoch, 30 is the number of days in a full lunar month, SeriesStartLunation is -179 (as shown in the HCEM column of the era zero row), 1447 is the number of days in a 49-month subcycle, EraNumber is determined by comparison of Lunation with the yerm era list, and 49 is the number of months in a 49-month subcycle.
By inserting constants from the yerm era list and rearranging, the above expression simplifies to:
molad day = HebrewEPOCH – 5256 + floor( [ 1447 × Lunation + 257566 – EraNumber ] / 49)
I have programmatically verified that the above expression yields the correct molad day for every molad of the full 689472-year repeat cycle of the traditional Hebrew calendar.
The actual astronomical mean lunar cycle contains similar patterns of yerms and 49-month subcycles, although fewer yerms are required per era because the appropriate number of yerms per fixed arithmetic lunar month cycle is inversely proportional to the mean month and the present era mean lunar cycle is about 2/3 second shorter than the traditional molad interval. As the mean lunar month becomes slightly shorter, the necessary yerms per era rapidly decreases. This is evident in this progressive yerm eras analysis of mean lunar conjunctions, in which era zero is defined as the era in which the astronomical mean synodic month decreased to less than 29+26/49 days, and which shows that although 70- and 67-yerm eras were appropriate around the time of the Second Temple, since the era of Rambam 55 or 52 yerms would have sufficed. As shown by the yellow highlighted row in the report, the present era is the last 52-yerm era and the next era will be the first ever 49-yerm era in history!
Although these simple yerm patterns exist in lunar month sequences based on the mean lunation interval, sequences of actual astronomical lunar month lengths are not so simple, because typically several longer lunations occur consecutively near Earth's orbital perihelion (not always long enough to be considered a 30-day month), and several shorter lunations occur consecutively near aphelion (not always short enough to be considered a 29-day month), as discussed and charted above.
Amos Shapir, of Israel, was the first to develop a progressive yerm era series, but he generated a long string of F and D codes based on calculated actual lunar conjunction moments relative to midnight at the meridian of Jerusalem. Then he "had to clean up the sequence, occasionally replacing FD by DF in order to avoid DD sequences." He then used a text editor to group runs of 17 and 15 months into yerms, and replaced them with "7" or "5" accordingly. Finally, he used the text editor to search and replace strings of "775" and "7775" to generate the yerm eras.
My freeware calendrical calculator, Kalendis [version 9.558(1278) or later], has a built-in yerm eras report generator, developed in consultation with Karl Palmen and Amos Shapir, that automates this entire process for more than a quarter million mean lunations by computing the mean lunar conjunction (or quadrature or opposition) moments for any selected locale, optionally including an offset value, automatically finding the yerm era starts and the number of consecutive 49-month subcycles in a matter of a few seconds of computer time. After the report is generated Kalendis launches the report file so that the user can view it, and also uses that progressive yerm era series as the basis for its yerm lunar calendar dates, as displayed in its "Lunar" window.
Karl Palmen determined that the beginning of each yerm era having a mean month that is shorter than 29+26/49 days is uniquely marked by the sequence "FDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFD", which includes the last 17-month yerm of the prior era, so the era itself starts at the 18th character position, as highlighted in boldface red text.
For the very remote past, well before the traditional epoch of the Hebrew calendar, when yerm eras had a mean month that was longer than 29+26/49 days, such eras started with a 15-month yerm that Karl Palmen calls a "yerm zero". Karl determined that the sequence "FFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFDFFDFDFDFDFDFDFDFFD" is a unique marker for the start of that era type, where the era itself started at the 34th character position, at which the 15-month yerm zero is highlighted in boldface red text. The era in which the first walled settlement in Israel was founded at Jerico, which Encyclopaedia Judaica gives as 6850 BCE ±210 years, corresponding to 6850 – 3760 ≈ 3090 years before the epoch of the traditional Hebrew calendar, was the last yerm era that started with a yerm zero.
The simplest way to generate a 25920-month sequence having a mean month exactly equal to the traditional molad interval is by executing a loop 25920 times, where the full or deficient status of any month in the sequence is given by:
IF ( MonthNumber × 13753 ) MOD 25920 < 13753 THEN Full ELSE Deficient
where MonthNumber starts at zero.
Although the generated sequence has the correct mean month, its sequence of month lengths is not the same as the traditional molad because the traditional molad epoch didn't start at moment zero on day zero of month zero. If we define the molad day as the day upon which the traditional molad moment lands after including the +1/4 day offset due to Gauss (thereby including the molad zakein rule), and if we take the length of the month as the difference from one molad day to the next, then the month length status of any month in the molad sequence is given by:
IF [ ( ElapsedMonths + 21589 ) × 13753 ] MOD 25920 < 13753 THEN Full ELSE Deficient
where ElapsedMonths starts at zero and is the number of Hebrew calendar months elapsed since the epoch of the calendar, as per the arithmetic given above. This means that the molad day sequence starts 4431 months before or 21589 months after the start of the zero-based sequence. Rearranging the expression to make its coefficients less obscure we obtain:
IF ( MonthNumber × 13753 + 12084 ) MOD 25920 < 13753 THEN Full ELSE Deficient
where MonthNumber is the Hebrew calendar month number since the epoch, starting at month one = elapsed months + 1. The coefficient 13753 is the number of parts that the traditional molad interval is in excess of 29 days, and it is also the number of full months in the complete cycle. The coefficient 12084 includes the 5 hours 204 parts = 5604 parts, which was the traditional moment of the first molad after the first sunset plus +1/4 day offset due to Gauss to bypass the molad zakein postponement rule = 25920/4 = 6480 parts, so we have 5604+6480=12084 parts. The coefficient 25920 is the number of parts per day and is also the total number of months in the complete cycle.
An interesting sequence is generated by:
IF [ ( ElapsedMonths + 2256 ) × 13753 ] MOD 25920 < 13753 THEN Full ELSE Deficient
This sequence starts 2257 months after the zero-based sequence or 6588 months after the molad day sequence, and its complete molad cycle as given above started from the first molad at Creation, however after Creation it generates molad days that are always one day later than the traditional molad day.
Thanks to K.E.V. (Karl) Palmen, developer of the yerm lunar calendar, for his comprehensive assistance with the above yerm and subcycle arithmetic.
In section 8.3 starting on page 125 of Calendrical Calculations: The Ultimate Edition (4th edition, which I will abbreviate as "CCUE"), authors Edward M. Reingold and Nachum Dershowitz (see also their Calendarists.com web site) derived direct arithmetic expressions for determining the Hebrew year and month that corresponds to a given molad Hebrew weekday and time. Their discussion was terse, leaving readers to refer to multiple other places in their book to figure out how they obtained their exceptionally large numeric coefficients.
I wondered whether this was necessary — how long would it take to execute a "brute force" search, starting from the molad epoch and checking each molad until a match is found? The worst case is the last molad in the cycle. There are 7 possible weekdays, 24 hours per day, and 1080 parts (chalakim) per hour, so the total number of unique moladot equals 7 × 24 × 1080 = 181440 months or almost 14670 years! I implemented such a "brute force" search in Microsoft Excel 2003 using the built-in VBA (Visual Basic for Applications) macro programming language, and it turned out that even on my old Core 2 Duo PC running 32-bit Windows XP the worst case molad is found in about one second.
A friend pointed out that my search could go faster if it only checked moladot that are on the required weekday. Obviously this enhancement was non-essential, but the challenge was on. One might have expected that this strategy might be 7 times faster, but actually it's considerably faster than that: same-weekday moladot can occur after 4-, 5- or 9-month intervals, but the majority are 9 months, and an additional performance gain is realized by not having to actually check the weekday of each molad, so altogether the worst case jump same weekday search is about 10 times faster than the brute force method.
The CCUE direct arithmetic expression is derived from the following equation, whose coefficients the authors glossed over but are detailed below:
n = (56930610241×b — 1425770202875604) mod 138880163520
The extremely large coefficients of this equation and its result, with all digits significant, is beyond the arithmetic capability of VBA. Even if one divides through the coefficients by 765433 to calculate the lunation number (Hebrew calendar elapsed months) the operation overflows VBA's built-in MOD operator, but it does work with the CCUE authors' modulus function as explained above.
The final direct expression that almost instantaneously yields the required lunation number is:
Lunation = modulus[765433 × (b — 31524), 181440]
All that remains to do is to execute a relatively simple algorithm that uses the lunation number to calculate the Hebrew year and month.
As a demonstration of all 3 of these molad inversion methods (brute force, jump same weekday, direct expression) as well as my algorithm for converting the lunation number to the Hebrew year and month, along with supporting functions and a procedure for validating the full 181440-molad cycle for each of the 3 inversion methods, please see this Excel workbook, which contains the functions and procedures as open source VBA:
Click here to download the Invert-Molad.xlsm Excel workbook 70KB
For the calculations to work, you have to enable VBA macro execution, and to view the open source VBA macro statements you have to enable the "Developer" menu and then go into the Visual Basic Editor. The instructions for enabling these vary with the version of Excel that you have.
The worksheets allow a molad time to be specified in terms of hours and parts, or as hours minutes and parts. Invalid out-of-range entries are highlighted as dark red text on a light red background. Separate worksheets within the workbook can invert the molad even when the weekday is unknown, or when the hour is unknown, or when the residual parts of a minute are unknown. Although multiple solutions are obtained when a component is unknown, usually most can be excluded because they are in the future or remote past.
Updated 29 Av 5779 (Traditional) = 29 Av 5779 (Rectified) = August 32, 2019 (Symmetry454) = August 30, 2019 (Symmetry010) = August 29, 2019 (Gregorian)