Proposed Molad Adjustment Equations

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The Molad adjustments documented herein have been superseded by the simpler and more accurate Progressive Molad of the rectified Hebrew calendar. This web page is retained as an archive, and because the month-specific (seasonal) adjustments are not otherwise documented or used (but may be revisited).

Rosh Hashanah Postponement Rules

Note that if the Molad is adjusted then the Rosh Hashanah postponement rules can't be handled in the classical manner because the adjusted Molad interval is not constant (gradually getting shorter).

Instead the method to use is as described by Dershowitz & Reingold, computing the date of the previous and next Rosh Hashanah, postponing if Sunday, Wednesday, or Friday, and then checking for allowed year lengths compared to the previous and next Rosh Hashanah dates.

For more information please see my web page about the Hebrew Calendar Rosh HaShanah Postponement Rules.

Calculating the Hebrew Calendar Elapsed Months (HCEM)

HCEM = Number of Hebrew Calendar Elapsed Months, that is the count of all months prior to the Molad of the desired month of the desired Traditional Hebrew Calendar year, calculated as follows, where Month is the number of the Hebrew Calendar month ( Nisan = 1, Tishrei = 7 ):

IF Month < Tishrei THEN Year = HebrewYear + 1 ELSE Year = HebrewYear

HCEM = MonthTishrei + INT [ (235 × Year – 234) / 19 ]

For example for the number of months elapsed on the Hebrew Calendar prior to the Molad of Cheshvan 5765:

Month of Cheshvan is 8, so we take Year = 5765

HCEM = 8 – 7 + INT [ ( 235 × 5765 – 234 ) / 19 ] = 1 + INT [ ( 1354775 – 234 ) / 19 ]
= 1 + INT [ 1354541 / 19 ] = 1 + INT ( 71291.63) = 1 + 71291 = 71292

The Traditional Molad Calculation

In the traditional Molad calculation HCEM is multiplied by the Molad period (29 days, 12 hours, 44 minutes, 1 part) and then BeHaRad is added (5 hours, 204 parts) as the assumed initial conjunction to derive the moment of the Molad as the number and fraction of days elapsed since the epoch of the Hebrew Calendar, counting from sunset as the beginning of the Hebrew day ("Talmudic Temporal Time"). For comparison with Moon times it is necessary to subtract 6 hours to convert the Molad to "Civil" Time, counting the hours from midnight.

The Length of the Mean Synodic Month in Any Given Hebrew Month

This formula shows how to progressively track the length of the Mean Synodic Month, in days, for any given Hebrew month, as per this PDF:

MeanSynodicMonth_at_Epoch = 29.53061 days

MeanSynodicMonthSlope = 3.102144E-10 days per elapsed month (to be subtracted from Mean Synodic Month at Epoch)

Compute HCEM as per the formula given above for the desired HebrewYear and Month

MeanSynodicMonth for that HCEM = MeanSynodicMonth_at_EpochMeanSynodicMonthSlope × HCEM

An Annually Averaged Molad Adjustment

This simple adjustment corrects the drift of the Molad moment relative to the actual mean lunar conjunction, accounting for the long-term change in the Mean Synodic Month. It restores the timing of the Molad to the annual cycle that existed in Hebrew year 4119, when the fixed arithmetic calendar was started, as illustrated by the lavender curve in this Molad minus Mean New Moon spreadsheet 31KB (click here to download the free Microsoft Excel Viewer 2003 for Windows) [click here to download a PDF version that covers a few selected years 333KB]:

Note that Tishrei has to be the target month for this average adjustment algorithm because the Molad of the fixed Hebrew Calendar directly affects only the date of Rosh Hashannah — the Rosh Chodesh of every other month is fixed relative to Rosh Hashannah:

Compute HCEM as per the formula given above for Tishrei of the desired HebrewYear

MoladInterval = 29 days + 12 hours + 44 minutes + 3 1/3 seconds = 29.530594 days
[ this is the same as the MeanSynodicMonth for HCEM = 50934 (Hillel ben Yehudah) ]

Compute the AverageMeanSynodicMonth for the interval from Tishrei 4119 ( Hillel ben Yehudah, FixedHCEM = 50934 ) to the HCEM computed above:

AverageMeanSynodicMonth = MeanSynodicMonth_at_Epoch – ( FixedHCEM + HCEM ) × MeanSynodicMonthSlope / 2

MonthDifference = HCEMFixedHCEM

AverageMoladAdjust in minutes = MinutesPerDay × MonthDifference × ( MoladIntervalAverageMeanSynodicMonth )

The adjustment is to be subtracted from the traditionally calculated Molad moment.

Up to this point, the adjustment corrects the Molad moment for the progressive shortening of the lunar cycle.
However, without further adjustment the Molad still refers to the meridian that is halfway between Babylonia and Israel.

To reset the meridian to Jerusalem Mean Solar Time, subtract another 23 minutes from the Molad moment.

For example, for the Hebrew year 5765:

HCEM for Tishrei 5765 as per the formula given above = 71291 months

AverageMeanSynodicMonth = 29.53061 – ( 50934 + 71291 ) × 3.102144E-10 / 2 = 29.53059067 days

MonthDifference = 71291 – 50934 = 20357 months

AverageMoladAdjust in minutes = 1440 × 20357 × ( 29.530594 – 29.53059067 ) = 101.6 minutes = 1 hour 42 minutes.

To reset the meridian to Jerusalem: 101.6 + 23 = 124.6 minutes = 2 hours 5 minutes to be subtracted from the traditional Molad moment.

The above are the actual equations that Kalendis uses, as of version 7.17(433), for computing the "Average Proposed Adjustment" shown at the bottom of exported Moladot reports.

The advantage of this simpler average Molad adjustment is that it ought to remain reasonably accurate indefinitely, as long as Moon continues its historically very steady rate of change in the Mean Synodic Month. This adjustment restores the annual cycling of the moments of the Molad, relative to the actual mean lunar conjunctions, back to the general pattern that existed at the advent of the fixed arithmetic Hebrew Calendar.

More Advanced Month-Specific Molad Adjustment

This adjustment eliminates the traditional cyclic variation whereby the Moladot of months after perihelion were a few hours earlier than the actual mean lunar conjunction and the Moladot of months after aphelion were a few hours later than the actual mean lunar conjunction. In other words, this adjustment makes the computed moment of the adjusted Molad equal to the actual moment of the mean lunar conjunction for each month of each year.

It corrects for the progressive change in the Mean Synodic Month, the month-to-month differences in the timing of the Molad relative to the actual mean lunar conjunction, and the currently diminishing eccentricity of Earth's orbit.

Some hold that the Molad time units are Temporal Hours, but that seems inconsistent with the simple arithmetic used to calculate the Molad, which adds a constant 29 days + 12 hours + 44 minutes + 3 1/3 seconds per lunar cycle. For the Hebrew year 5765 the Molad minus Mean Lunar Conjunction difference varies from the Molad being 2 hours ahead of the Mean New Moon in Nisan to being late by 6 hours after the Mean New Moon in Tishrei and Cheshvan. This 8 hour range dwarfs the few minutes of variance that could be accounted for by the use of Temporal Time. Furthermore, the small temporal differences average out during the full solar year cycle.

Each Molad Adjustment equation below is specific for a certain month of the Hebrew Calendar year. They are all quadratic equations (parabolic curves), but I use "HCEM × HCEM" instead of "HCEM ^ 2" to indicate the square of HCEM, because usually computers multiply faster than they raise a number to a power.

These are the actual equations that Kalendis uses, as of version 6.6(371), for computing the rightmost "Proposed Adjustment" tabulated column on exported Moladot reports.

The value calculated is the number of minutes to subtract from the moment of the Traditional Molad.

Derivation: Each individual month equation is simply the second-order polynomial obtained by least-squares regression of "Molad minus Actual New Moon" versus "Hebrew Calendar Elapsed Months" for the first 120,000 lunations of the Hebrew Calendar (10000 years) 3.8 MB

I have carried through the precision of the coefficients to the 15 significant figure limit of Microsoft Windows / Visual Basic, but they could be rounded to about six significant figures with negligible effect on the calculated correction.

Nisan = HCEM × HCEM × 2.60571110766646E-07 – HCEM × 2.66848709499308E-02 + 459.733981958199

Iyar = HCEM × HCEM × 2.58045601478965E-07 – HCEM × 2.82798810954184E-02 + 602.613646722384

Sivan = HCEM × HCEM × 2.47436539938585E-07 – HCEM × 2.84308587281714E-02 + 739.45859065704

Tammuz = HCEM × HCEM × 2.31521759919978E-07 – HCEM × 2.71527221585827E-02 + 838.661835188114

Av = HCEM × HCEM × 2.1372693269941E-07 – HCEM × 2.47163302893743E-02 + 875.723168374493

Elul = HCEM × HCEM × 1.9776368538253E-07 – HCEM × 2.16233269966286E-02 + 838.834835036855

Tishrei = HCEM × HCEM × 1.87421489154775E-07 – HCEM × 1.85882945343438E-02 + 734.333759456244

Cheshvan = HCEM × HCEM × 1.85731162681391E-07 – HCEM × 0.016396901469117 + 586.895817822247

Kislev = HCEM × HCEM × 1.94119804074587E-07 – HCEM × 1.57422001314177E-02 + 436.719151346648

Tevet = HCEM × HCEM × 2.10896269743342E-07 – HCEM × 0.016883189899286 + 325.43829209883

Shevat = HCEM × HCEM × 2.31193785102518E-07 – HCEM × 1.94832635929029E-02 + 281.94752164862

Adar or AdarRishon = HCEM × HCEM × 2.48853647133124E-07 – HCEM × 2.27488822887766E-02 + 314.646199528858

AdarSheini = HCEM × HCEM × 2.56850690652121E-07 – HCEM × 2.48814040457114E-02 + 372.819958045515

The valid range of years for which I have evaluated and confirmed the month-specific Molad adjustment equations above is Hebrew years 1 through 10000, although it is probably valid until the Hebrew year 17000, around which time the Moladot that are the most number of hours late relative to the actual mean lunar conjunctions will be Adar / Adar 2. To extend the valid range further into the future it would be necessary to incorporate an explicit function for the eccentricity of Earth's orbit (this already exists in Kalendis), a cyclic function related to that eccentricity, plus the progressive change in the Mean Synodic Month as described above. Otherwise, the adjustment could revert to the average Molad adjustment described above, which ought to be reasonably accurate indefinitely.

Updated 17 Av 5766 (Traditional) = 17 Av 5766 (Rectified) = August 12, 2006 (Symmetry454) = August 11, 2006 (Gregorian)