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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.

*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* = *Month* – *Tishrei* + 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**

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.

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_Epoch* – *MeanSynodicMonthSlope* × *HCEM*

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* = *HCEM* – *FixedHCEM*

*AverageMoladAdjust* **in minutes** = *MinutesPerDay* × *MonthDifference* × ( *MoladInterval* – *AverageMeanSynodicMonth* )

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.

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.**

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)