Analysis by Dr. Irv Bromberg, University of Toronto, Canada

[Click here to go back to the Symmetry454 / *Kalendis* home page]

**Overview****Glossary of Terms****Celestial Mechanics by Numerical Integration****Reference Material****Short-Term Periodic Variations of the Lunar Cycle****A Close Look at the Short-Term Periodic Variations of the Lunar Cycle****Fixed Arithmetic Lunar Calendars****Constant Interval New Moon Estimate****Periodic Lunar Cycle Variations Relative to a Fixed Lunar Calendar****Which Point in the Lunar Cycle has the Greatest Amount of Periodic Variations?****Lunar Half-Cycles: Triple the Periodic Variations!****Periodic Earth-Moon Distance Variations****New Moon minus Estimate****The Mean New Moon Moment****Estimation of Fixed Lunar Cycle Calendar Drift****Moment to Lunation Number Conversion****The Mean Synodic Month****Mean Lunar Angular Motion (Mean Sidereal Month)****The Rate of Change of the Mean Synodic Month****Is There a Long-Term Effect of the Mean Earth Orbital Eccentricity?****The Role of Earth Axial Tilt (Obliquity)****More Things That Would be Nice To Do**

This page examines the periodic variations in the length of the lunar cycle, and its long-term trend over a 19000-year period.

Expressions will be presented for computing the moment of the mean lunar conjunction, the mean synodic month, and the rate of change of the mean synodic month.

All of the charts are in GIF format but can be viewed in more detail and higher resolution by clicking the chart, which will open the corresponding Adobe Acrobat PDF (Portable Document Format) file in a new window. To obtain the freeware Acrobat Reader, click here.

**Terrestrial Time (TT)**- Uniform time base running at the same rate as International Atomic Time at the Prime Meridian (longitude 0°) on the surface of the rotating geoid (surface of Earth, at sea level). Astronomical calculations are typically carried out in terms of TT in order to avoid uncertainties due to irregular, periodic and long-term variations of the Earth rotation rate.
**Universal Time (UT)**- Mean Solar Time at the Prime Meridian, running at a rate such that each mean solar day has exactly 86400 mean solar seconds.
**Delta T = ΔT**- ΔT = TT – UT, UT = TT – ΔT, and (approximately, but adequate for the next 10000 years) TT = UT + ΔT.
- Typically, astronomical calculations are carried out in terms of TT, then converted to UT by subtracting ΔT.
**In order for Delta T to work optimally for the purposes suggested herein, its approximation should be a continuous function of the time elapsed relative to J2000.0, without any annual or monthly stepwise granularity.**Suitable expressions (published January 2007) can be found at the NASA Eclipses web site at <http://eclipse.gsfc.nasa.gov/SEhelp/deltatpoly2004.html>, except that to avoid monthly granularity in the Delta T approximation use the following expression when calculating*y*(the fractional year number):

*y*= 2000 + (*TTmoment*– J2000.0 ) /*MARY*

- where
*TTmoment*is the Terrestrial Time moment and J2000.0 is as defined below, both in terms of the number of days and fraction of a day elapsed relative to a specified ordinal day numbering epoch, and*MARY*(Mean Atomic Revolution Year) = 365+^{31}/_{128}atomic days, as explained on "The Lengths of the Seasons" at <http://individual.utoronto.ca/kalendis/seasons.htm>. Over the entire range from 500 BC to 2050 AD, this modification never causes more than^{4}/_{5}second of difference compared to the unmodified arithmetic of the NASA algorithm. - The studies documented herein were carried out before NASA published the new Delta T approximation.
- Click here for
**Delta T comparison charts**of Dershowitz & Reingold (used herein)*vs.*Meeus Espenak (new at NASA) 42KB.

Click below to access Robert Harry van Gent's web page explaining Delta T and the history of Delta T approximations:

<http://www.staff.science.uu.nl/~gent0113/deltat/deltat.htm>. **J2000.0**- January 1, 2000 AD at
**Noon**, Terrestrial Time = Julian Day 2451545.0 =*Rata Die*730120.5 (days since Gregorian epoch). **Ecliptic**- Path of Earth's orbit around Sun, projected onto the celestial sphere.
**Earth Axial Tilt = Obliquity**- The angle between the Earth's equator and the ecliptic, or the angle between Earth's axis of rotation and a line perpendicular to the ecliptic.
**Lunar Conjunction = New Moon = Dark Moon**- Moment when the celestial longitudes of Moon and of Sun are equal when projected onto the ecliptic. It is only rarely that at a lunar conjunction the centers of Sun, Moon, and Earth, in that order, are aligned in a straight line. Such an 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 is equal to the solar declination. The lunar orbital plane is tilted about 5° 9' relative to the ecliptic (mean value), therefore most of the time the lunar latitude is either north or south of the ecliptic. If the lunar latitude is >32 arcminutes (32' = the solar angular diameter ≈ lunar angular diameter) north or south of the ecliptic at the moment of a lunar conjunction then Moon passes north or south of Sun, respectively, without any solar eclipse. Nevertheless, one could properly describe the lunar conjunction as the moment when the centers of Sun, Moon, and Earth, in that order, are all in the
__plane__that is perpendicular to the plane of Earth's orbit (the ecliptic plane). **Mean Lunar Conjunction**- Average lunar conjunction moment, assuming that Moon's orbit is a circle with Earth at its center, and that Earth's orbit is a circle with Sun at its center.
**Length of the Lunar Cycle = (Duration of the) Lunation = Synodic Month = Lunar Month**- Time elapsed from one lunar conjunction until the next.
**Mean Lunar Cycle = Mean Synodic Month (MSM)**- Average time elapsed from one mean lunar conjunction until the next. Its long-term variations will be examined herein.
**Lunation Number**- A count of lunar months before or after a specified epoch. There are approximately 12368 lunations per millennium.
- The study presented here counts mean lunations relative to the mean lunar conjunction nearest to J2000.0, which was on January 6th, 2000 AD, taken as lunation number zero.
- Lunations prior to that epoch are negative, and lunations after that epoch are positive lunation numbers.
- Fractional values of 0, 0.25, 0.5, and 0.75 refer to the Mean New Moon, Mean 1st Quarter, Mean Full Moon, and Mean 3rd Quarter, respectively.
- Add 71233 to convert J2000 lunation numbers to traditional Hebrew calendar lunation numbers (count of months elapsed since the Hebrew calendar epoch), such as are used to calculate the traditional
*molad*. - Add 24724 to convert J2000 lunation numbers to
*nth-new-moon*lunation numbers, as used in the algorithms of the book Calendrical Calculations, by Reingold & Dershowitz. - Add 953 to convert J2000 lunation numbers to Brown lunation numbers, for example as quoted by the Royal Astronomical Society of Canada (RASC)
in their annual Observer's Handbook, which are relative to January 1923 AD, based on the series described by
Ernest W. Brown in
*Planetary Theory*(1933). - Add 17038 to convert J2000 lunation numbers to the Islamic lunation number since the epoch of the Observational Islamic calendar. This conversion is approximate, because the J2000 lunation number refers to the mean lunar conjunction at the Prime Meridian whereas the Islamic lunation number refers to the first visible new lunar crescent, whose timing is always later and also varies from locale to locale.
- Add 37105 to convert J2000 lunation numbers to Goldstine lunation numbers, as used by Herman Goldstine in his 1973 book
*New and Full Moons 1001 B. C. to A. D. 1651*. **Lunar Apogee and Perigee**- Apogee is the point in Moon's elliptical orbit where it is furthest from Earth.
- Perigee is the point in Moon's elliptical orbit where it is closest to Earth.
- Both points advance eastward (with some oscillations) around Earth (in the same direction as Moon revolves), completing a full cycle
__with respect to Sun__in an average of about 411.8 days or almost 14 lunar months. One complete revolution of perigee__relative to the northward equinox of date__takes an average of 3231.50 days or about 8.85 tropical years, or__relative to the stars__an average of 3232.61 days (sidereal period). NASA has posted a picture showing the apparent lunar diameter at apogee and perigee, see: <http://antwrp.gsfc.nasa.gov/apod/ap071025.html>. **Aphelion and Perihelion**- Aphelion is the point in Earth's elliptical orbit where it is furthest from Sun, currently near July 3rd.
- Perihelion is the point in Earth's elliptical orbit where it is closest to Sun, currently near January 3rd.
- Both points advance through the entire calendar year, completing a full cycle in about 21000 years (faster as orbital eccentricity declines).
- The season that contains aphelion is longer because Earth moves slower near aphelion. When aphelion is near an equinox or solstice, the two adjacent seasons are both relatively long.
- The season that contains perihelion is shorter because Earth moves faster near perihelion. When perihelion is near an equinox or solstice, the two adjacent seasons are both relatively short.
- NASA has posted a picture showing the apparent solar diameter at perihelion and aphelion, see: <http://antwrp.gsfc.nasa.gov/apod/ap070709.html>.

To obtain the accurate moments of the actual astronomical lunar conjunctions, as the basis for evaluating the lengths of the lunar cycle and how it changes over time, and to prepare the collection of charts offered below, I used **numerical integration**, which is arguably the "gold standard" for celestial mechanics, and which is easy to do using **SOLEX** version 9.1β (or later), a free computer program written by Professor Aldo Vitagliano of the Department of Chemical Sciences at the University of Naples Federico II, Italy. Version 9.1β was the first version which could automatically find lunar conjunction moments, logging those moments to a "MINDISH.OUT" text file. **The latest version is available at the SOLEX web site.**

The SOLEX integration was carried out in terms of Terrestrial Time (usually abbreviated TT but indicated as TDT within SOLEX), with Delta T switched off and the geographic locale set to the Equator at the Prime Meridian, starting from date January 1, 2000. SOLEX integrated forward at 1-day intervals to beyond the year 12000 AD, and then starting again from year 2000 SOLEX integrated backward at 1-day intervals to before the year 7000 BC.

According to the SOLEX documentation, its numerical integration takes into account:

- Starting conditions of the Jet Propulsion Laboratory DE409 ephemeris.

(I also tried the older DE406 settings but rejected those because I found that DE409 was obviously more precise.) - Sun, Mercury, Venus, Earth, Moon, Mars, Ceres, Vesta, Pallas, Jupiter, Saturn, Uranus, Neptune, and Pluto.
- Osculating orbital elements, precession, nutation, aberration, light time.
- First order relativistic effects (see Special Relativity, General Relativity).
- Solar oblateness, and solar mass loss (due to nuclear fusion, and the solar wind)

SOLEX Limitations:

- For the evaluated range of dates (within ±10000 years of the present era), SOLEX calculates precession and Earth axial tilt (obliquity) using formulas published by J. G. Williams in "Contributions to the Earth's Obliquity Rate, Precession and Nutation."
*Astronomical Journal*1994 Aug;**108**(2): 711-724. - SOLEX assumes a constant
*J*_{2}parameter for the Earth, so it makes no attempt to model long-term variations in Earth's oblate shape (polar : equatorial flattening) by taking into account tectonic plate movements or the mass of the polar ice caps. - SOLEX ignores planetary satellites and rings, except for our Moon.
- By default, SOLEX also ignores comets, asteroids, and minor planets other than those mentioned above. The user can optionally include thousands of additional objects in the integration, but doing so will slow down the analysis accordingly.

**For more information about SOLEX and to download the program please see its web page at <http://www.solexorb.it/>.**

For those charts that are in terms of mean solar days, I assumed steady tidal slowing of the Earth rotation rate such that mean solar days get longer by 1.75 milliseconds per century. Of course the actual Earth rotation rate does not slow down at such a perfectly steady rate, but goes through short-term fluctuations and long-term periodic cycles, many of which are unpredictable with our present state of knowledge. Tidal slowing was probably greater in the past when the polar ice caps were more massive with lower sea levels and axial tilt was greater than in the present era, and tidal slowing will probably diminish over the coming several millennia due to global warming (reduction of polar ice mass, rising sea levels) and due to declining axial tilt.

Jean Meeus published and excellent chapter entitled "**The Duration of the Lunation**" on pages 19-31 (chapter 4) in "**More Mathematical Astronomy Morsels**" by Jean Meeus, published in 2002 by Willmann-Bell, Inc., Richmond, Virginia. Herein, when I mention Jean Meeus without further qualification, it refers to what he wrote in that chapter.

Another relevant chapter by Jean Meeus is chapter 33, "Long-period variations of the orbit of the Earth" in More Mathematical Astronomy Morsels, pages 201-205, published in 2002 by Willmann-Bell, Richmond VA, in which he presented his adaptation of the 18-term astronomical algorithm of Pierre Bretagnon (1984) for determining the **mean** Earth orbital eccentricity over a ±million year range. For more information and charts of the Earth orbital eccentricity, based on Meeus' adaptation of the Bretagnon algorithm, please see my "Lengths of the Seasons" web page at <http://individual.utoronto.ca/kalendis/seasons.htm>.

In the present era the median length of the lunar cycle is about 29d 12h 30m, the average (MSM) is slightly more than 29d 12h 44m, the shortest lunations are about 29d 6h 30m, and the longest are about 29d 20h. **Thus the length of the synodic month varies over a range spanning about 13h 30m.** These variations were greater in the past and will diminish in the future:

- The longest lunar cycles occur when Moon is moving slowest (near apogee) and Earth is moving fastest (near perihelion).
- The shortest lunar cycles occur when Moon is moving fastest (near perigee) and Earth is moving slowest (near aphelion).
- The declining mean Earth orbital eccentricity tends to reduce the range of lunar cycle variations.
- The average lunar cycle (mean synodic month) has miniscule long-term change compared to short-term periodic variations.

**Centile trends, per group of 4657 lunar months, based on SOLEX 9.1β numerical integration**

I will explain later why I chose the "magic" number 4657 for averaging the lengths of lunar cycles.

The chart above happens to be in terms of Terrestrial Time (TT), but there would not be any discernible difference if the data had been calculated in terms of Universal Time (UT), because of the very wide range of the *Y*-axis scale.

The following chart shows the variations obtained when the actual length of each lunar cycle is subtracted from the actual length of the mean synodic month, as calculated by SOLEX in terms of Terrestrial Time. Each plotted point is a lunar conjunction, with more than 18000 plotted in total:

- There is a weak periodic pattern that repeats about every 2277 lunations (almost 184 years), which according to Jean Meeus is the time required for the lunar orbital nodes to regress westward 180° with respect to the Earth orbital perihelion.
- The points are more densely clustered at about ±3 hours because those points occur more frequently, whereas other points are more uniformly spread. The reason for this clustering will become evident in the next chart.
- The variations range over about ±7 hours, a span of about 14 hours!

The next chart takes a closer view at the periodic variations by zooming into ±333 lunations relative to the J2000 epoch:

- If your eyes are telling you that the lunar cycle appears to spend more time "above the zero line" than below it, rest assured that that is not an illusion. As mentioned above, in the present era the median length of the lunar cycle is about 29d 12h 30m but the average (MSM) is slightly more than 29d 12h 44m, so indeed lunar cycles are more likely to be long than short.
- There is a strong periodic pattern that on this near-present-era chart repeats about every 111 lunations (almost 9 years), but over the centuries the period slowly varies, averaging about 8.85 years or 109.5 lunations. This period corresponds to what Jean Meeus described as the time required for the lunar orbital perigee to advance eastward 360° with respect to the Earth orbital perihelion.
- It is now clear why the points are more densely clustered at about ±3 hours: peaks of reduced height often occur in pairs. Those
lesser peaks

occur when the lunar conjunction is nearly midway between perigee and apogee. - As Earth's orbit becomes more circular (declining mean eccentricity), as is the case for the present era and for many millennia into the future, the amplitude of the tallest peaks will diminish and at the same time the amplitude of the lesser peaks will progressively increase, until they all merge into a nearly uniformly spread distribution when Earth's orbit becomes nearly circular.
- This explains why over the long-term the centiles near the median on the trend chart above gradually moved outward towards the converging minimum and maximum lines.

The following chart, adapted from a similar chart received from **Jonathan Jay** of Hawaii, is a histogram of the durations (rounded to the nearest minute) of more than 60000 lunar cycles over a 5000-year period, showing that the lesser peaks

are indeed the most common. The most common durations are at 29d 11h 0m and 10h 52m, with lots of nearby similarly high frequency durations, and the secondarily most common durations are at 29d 15h 11m and 15h 16m, again with lots of nearby similarly high frequency durations. **Click here** or on the chart to open a high resolution PDF version.

The next chart takes an even closer view at the periodic variations by zooming into ±70 lunations relative to the J2000 epoch:

- There is a strong periodic pattern that repeats about every 14 lunations (about 412 days), which according to Jean Meeus is due to the cyclic eastward advance of the lunar orbital perigee.
- It is now clearly evident that series of several short lunations in a row alternate with several long lunations in a row. This is due to Earth's eccentric orbit, which causes several short lunations then several long lunations to occur in alternating 7-lunation series.
- The maximum positive peaks occur when Earth is near perihelion (moving fastest) and the lunar conjunction is near apogee (Moon moving slowest).
- The maximum negative peaks occur when Earth is near aphelion (moving slowest) and the lunar conjunction is near perigee (Moon moving fastest).
- The shortest positive peaks occur when Earth is near perihelion (moving fastest) and the lunar conjunction is nearly midway between perigee and apogee (Moon moving near average velocity).
- The shortest negative peaks occur when Earth is near aphelion (moving slowest) and the lunar conjunction is nearly midway between perigee and apogee (Moon moving near average velocity).

A variety of arithmetic lunar calendars are in use that are based on a repeating fixed lunar cycle duration. Some examples, sorted in descending order of the assumed lunar cycle length, are listed below, along with a good contemporary estimate of the mean synodic month:

Description | Assumed Lunar Cycle Length (exact days) |
Decimal Month (days) overscored groups repeat |
Time in excess of 29 days |
---|---|---|---|

Orthodox Easter computus |
(365+^{1}/_{4}) × ^{19}/_{235} = 29+^{499}/_{940} |
≈ 29.530851 | 12:44:25+^{25}/_{47}s |

Hebrew calendar molad |
29+^{12}/_{24}+^{44}/_{1440}+^{1}/_{25920} = 29+^{13753}/_{25920} |
29.530594135802469... | 12:44:03+^{1}/_{3}s |

Yerm Lunar calendar | ^{25101}/_{850} = 29+^{451}/_{850} |
29.530588235294117647... | 12:44:02+^{14}/_{17}s |

Hindu Surya calendar (modern) |
29+^{7087771}/_{13358334} |
≈ 29.530587946 | ≈ 12:44:02.8 |

Mean Synodic Month (2000 AD, mean solar days) |
29+^{82517}/_{155520} |
≈ 29.53058770576 | 12:44:02+^{7}/_{9}s |

Tibetan Phugpa calendar |
29+^{3001}/_{5656} |
29.530586987270155... | 12:44:02+^{506}/_{707}s |

Gregorian Easter computus |
^{2081882250}/_{70499183} = 29+^{37405943}/_{70499183} |
≈ 29.53058690056 | ≈ 12:44:02.7 |

Cycle of 49 yerms | ^{25101}/_{801} = 29+^{425}/_{801} |
29.530588235294117647... | 12:44:02+^{62}/_{89}s |

Cassidy-Dee Easter computus |
^{48091470}/_{1628531} = 29+^{864071}/_{1628531} |
≈ 29.530583083773 | ≈ 12:44:02.4 |

25 Saros cycle | 29+^{2958}/_{5575} |
≈ 29.5305829596412556 | 12:44:02+^{82}/_{223}s |

Hindu Arya calendar (old) |
^{1577917500}/_{53433336} = 29+^{2362563}/_{4452778} |
≈ 29.53058180758 | ≈ 12:44:02.3 |

Fixed Islamic calendar | ^{(30×6×59+11)}/_{(30×12)} = 29+^{191}/_{360} |
29.5305... | 12:44:00 |

Those that are listed above the year 2000 AD mean synodic month have assumed lunar cycle lengths that are too long and are drifting late, and those that appear below it are too short and are drifting early, relative to the present era mean lunar cycle. For optimizing the long-term drift of a fixed interval lunar calendar, however, it is better to choose a mean month that is slightly too short, because the mean lunar cycle is getting progressively shorter in terms of the mean solar days that are relevant to calendars.

**Regardless of the assumed length of the lunar cycle, the short-term periodic variations of actual lunar conjunction relative to mean lunar conjunctions calculated using any of the above fixed intervals will be about double the periodic variations of the length of the actual lunar cycle**, as will be shown next.

To evaluate the variations of the lunar cycle relative to a fixed lunar calendar cycle, we will examine those variations relative to estimated New Moon moments that are uniformly spaced at constant time intervals of 29 days 12 hours 44 minutes and 2+^{7}/_{8} seconds (atomic time) per lunation, counting the lunations relative to zero near the J2000,0 epoch on Gregorian January 6, 2000 AD at 14:20:44 TT. (The **actual** lunar conjunction was at 18:14:42 TT, according to SOLEX, but our estimate is intended to relate to **mean** lunar conjunction moments.)

The choice of J2000.0 as the lunation epoch is arbitrary, but is convenient because today most modern astronomical algorithms are calculated relative to J2000.0, and because the use of a near-present-era epoch optimizes the floating point arithmetic accuracy by maximizing the number of significant digits to the right of the decimal point.

To compute the fixed day number of the New Moon estimate, relative to J2000.0, in terms of Terrestrial Time:

NewMoonEstimateTT(Lunation) =MeanNewMoonTT_J2000+MSM_TT_J2000×Lunation+J2000

where *Lunation* is the lunation number relative to zero = January 6, 2000 AD, and the following are constants:

MeanNewMoonTT_J2000= 5 –^{1}/_{2}+^{14}/_{24}+^{20}/_{1440}+^{44}/_{86400}

(The^{1}/_{2}day is only deducted so that the time components can refer to midnight instead of noon.)

When this value is added to J2000 as above it represents the epoch mean lunar conjunction moment on January 6, 2000 at 14:20:44 TT.

MSM_TT_J2000= 29 +^{12}/_{24}+^{44}/_{1440}+ (2+^{7}/_{8}) / 86400

This is the approximate MSM at J2000.0, but it doesn't need to be exact for our purposes because any error will be cancelled out later.

Fractional *Lunation* values of 0, 0.25, 0.5, and 0.75 refer to the Mean New Moon, Mean 1st Quarter, Mean Full Moon, and Mean 3rd Quarter, respectively.

The following chart shows the variations obtained when the constant interval New Moon estimates are subtracted from the corresponding the actual lunar conjunctions, as calculated by SOLEX in terms of Terrestrial Time. Each plotted point is a lunar conjunction, with more than 18000 plotted in total:

- There is a weak periodic pattern that repeats about every 2277 lunations (almost 184 years), which according to Jean Meeus is the time required for the lunar orbital nodes to regress westward 180° with respect to the Earth orbital perihelion.
- The points are more densely clustered at about ±6 hours because those points occur more frequently, whereas other points are more uniformly spread. The reason for this clustering will become evident in the next chart.
**The variations range over ±14 hours, a span of 28 hours, more than double the actual short-term variations of the lunar cycle!**This is due to Earth's eccentric orbit, which causes several short lunations then several long lunations to occur in alternating 7-lunation series, thus accumulating larger deviations relative to our constant interval estimate. These series will be clearly evident in the next two charts.

The next chart takes a closer view at the periodic variations by zooming into ±333 lunations relative to the J2000 epoch:

- There is a strong periodic pattern that on this near-present-era chart repeats about every 111 lunations (almost 9 years), but over the centuries the period slowly varies, averaging about 8.85 years or 109.5 lunations. This period corresponds to what Jean Meeus described as the time required for the lunar orbital perigee to advance eastward 360° with respect to the Earth orbital perihelion.
- It is now clear why the points are more densely clustered at about ±6 hours: peaks of reduced height often occur in pairs.

Those "lesser peaks" occur when the lunar conjunction is nearly midway between perigee and apogee. - As Earth's orbit becomes more circular (declining mean eccentricity), as is the case for the present era and for many millennia into the future, the amplitude of the tallest peaks will diminish and at the same time the amplitude of the lesser peaks will progressively increase, until they all merge into a nearly uniformly spread distribution when Earth's orbit becomes nearly circular.
- This explains why over the long-term the centiles near the median on the trend chart above gradually moved outward towards the converging minimum and maximum lines.

The next chart takes an even closer view at the periodic variations by zooming into ±70 lunations relative to the J2000 epoch. This chart shows the difference between the actual lunar conjunction and our constant interval estimate (fixed lunar calendar, black line with "x" symbols), and for comparison also shows the difference between the actual length of each lunar cycle and the mean synodic month (blue line with solid dots):

- Note that the periodic variations of the fixed lunar calendar are double those of the lunar cycle relative to the mean synodic month, and the curves are 90° "out of phase": each time that the blue line crosses zero the black line reaches a maximum and reverses its direction. This shows how the series of 7 short lunations in a row (causing the constant interval estimate to fall behind) alternates with 7 long lunations in a row (allowing the constant interval estimate to pull ahead) and accumulates to double the periodic variations.
- There is a strong periodic pattern that repeats about every 14 lunations (about 412 days), which according to Jean Meeus is due to the cyclic eastward advance of the lunar orbital perigee.
- The maximum positive peaks occur when Earth is near perihelion (moving fastest) and the lunar conjunction is near apogee (Moon moving slowest).
- The maximum negative peaks occur when Earth is near aphelion (moving slowest) and the lunar conjunction is near perigee (Moon moving fastest).
- The shortest positive peaks occur when Earth is near perihelion (moving fastest) and the lunar conjunction is nearly midway between perigee and apogee (Moon moving near average velocity).
- The shortest negative peaks occur when Earth is near aphelion (moving slowest) and the lunar conjunction is nearly midway between perigee and apogee (Moon moving near average velocity).

As impressive as the periodic variations at the lunar conjunction are, that is actually the point in the lunar cycle that has the *least* periodic variations! The *most* variations are actually found at the first quarter Moon (lunar quadrature, lunar phase = 90°).

To investigate this, I used the newer version 11 of SOLEX, which has the ability to automatically report the moments of each lunar quarter (actually the moments of conjunctions, quadrature, and opposition of any object). I ran SOLEX 11 in DE421 mode with extended 80-bit precision, 18th order integration, and forced solar mass loss, with automatic searching for lunar conjunctions, quadratures, and oppositions, integrating backward and forward from the present era over the date range from November 30, 1815 through September 17, 3288 AD with Delta T switched off (atomic time), a total of 18216 lunar months, during which time the lunar orbital nodes retrogressed four times 360° with respect to the Earth orbital perihelion. Then I calculated the duration of the lunation measured from conjunction-to-conjunction, quadrature-to-quadrature, and opposition-to-opposition, and found the maximum, minimum, and range of variation in each case:

Lunar Quarter and Lunar Phase | Maximum >29 days | Minimum >29 days | Range (max-min) | Comment |
---|---|---|---|---|

(degrees of ecliptic elongation from Moon to Sun) | (HH:MM:SS) | (HH:MM:SS) | (HH:MM:SS) | |

Conjunction (New Moon = 0°) | 19:54:52 | 06:33:40 | 13:21:12 | least variations, about 13+^{1}/_{3} hours |

First Quadrature (1st Quarter = 90°) | 22:12:09 | 04:13:11 | 17:58:58 | almost 18 hours of variations |

Opposition (Full Moon = 180°) | 19:57:48 | 06:34:19 | 13:23:29 | a tad more variable than conjunctions |

Last Quadrature (3rd Quarter = 270°) | 22:12:54 | 04:14:04 | 17:58:51 | almost 18 hours of variations |

The variations at oppositions are only 00:02:17 greater than at conjunctions.

The variations at first and last quadrature are essentially equal (within 7 seconds) but are almost 4+^{2}/_{3} hours greater than at conjunctions.

These findings are in agreement with those reported by Jean Meeus, although he evaluated the lunation variations for only the 20th and 21st centuries (see chapter 2, "About the extreme durations of the lunation" in __Mathematical Astronomy
Morsels V__, published by Willmann-Bell, Inc., 2009).

A sampling of these variations are graphically depicted below (click on the chart title or the chart image to view a high quality PDF version):

In the chart above, the height of the tallest peaks is about an hour greater than the depth of the deepest valleys.

When peaks are tallest the opposing valleys are shallowest. Conversely, when valleys are deepest the opposing peaks are shortest.

In connection with the intercalation of the Hebrew calendar, some traditional commentators in the Babylonian *Talmud* stated that the year must be intercalated by insertion of a winter leap month if otherwise the spring equinox will fall later than the waxing or "renewal" half of the lunar cycle, in other words later than the lunar opposition or full moon moment. To examine the astronomical practicality of such an intercalation criterion, I investigated the periodic variations of the lunar half-cycles, that is the duration of the waxing half-cycle from lunar conjunction to opposition (new moon to full moon) and the duration of the waning half-cycle from lunar opposition until the next conjunction (full moon to new moon).

For this evaluation I used the lunar phase algorithms that were published by Jean Meeus in "**Astronomical Algorithms**", second edition, published in 1998 by Willmann-Bell, Richmond, Virginia, USA and the NASA Espenak-Meeus Delta T polynomials as referenced above, to find sequentially each lunar conjunction and opposition for ±1000 lunations relative to January 2000 AD (that is March 2, 1919 to November 11, 2080 AD) and then calculated the length in days of the waxing and waning half-cycles and full cycles.

The following chart shows the distribution of half-cycle and full cycle lengths obtained, expressed in terms of length in days versus the cumulative centiles. Click here or on the chart to open a high-resolution PDF version (which also includes the next chart). The left and right *y*-axes both span a 2-day variation range so that it is easy to compare half-cycles to full cycles:

- Near the present era, the length of each half of the lunar cycle varies over a range of about 41 hours, from a minimum of about 13 days and 21+
^{2}/_{3}hours to a maximum of about 15 days and 14+^{2}/_{3}hours, with an average or median of about 14 days and 18+^{1}/_{3}hours. - The distributions of waxing and waning variations are essentially the same (the red and black lines are superimposed).
**Amazingly, the periodic variation range of the lunar half-cycles span**__triple__the 13+^{1}/_{2}hour variation range of the full cycle!- As a proportion of the mean full lunar cycle, its 13+
^{1}/_{2}hour variation range amounts to ±1.9% from the mean, whereas as a proportion of the mean half lunar cycle, its 41-hour variation range amounts to ±11.6% from the mean, which is proportionately more than__six__times greater!

The following chart, focussing on only ±98 lunations relative to January 2000 AD, shows why this is so: as each half-cycle gets longer the other half gets shorter by a similar about, cancelling about ^{2}/_{3} of the full-cycle variations. Click here or on the chart to open a high-resolution PDF version (which also includes the previous chart). Here the left and right *y*-axes both span a 3-day variation range, again so that it is easy to compare half-cycles to full cycles. The points for the lengths of the waxing half-cycles and full cycles are plotted against whole lunation numbers, but the points for the waxing half-cycles are plotted against half-lunation numbers:

- The full cycle lengths equal the sums of the half-cycle lengths.
- The waxing and waning half-cycles are nearly equal and of average length when the lunar orbital perigee or apogee is near the lunar conjunction or opposition.
- The waxing half is maximal when apogee is near its midpoint, and in the same cycle the waning half is minimal because perigee is near its midpoint.
- The waxing half is minimal when perigee is near its midpoint, and in the same cycle the waning half is maximal because apogee is near its midpoint.
- With such large periodic variations, the idea of employing the length of the waxing or waning half of the lunar cycle as an intercalation criterion for any calendar seems astronomically dubious.
- Although it may be astronomically valid to use mean lunar oppositions or average lengths of the waxing or waning half-cycles for calendrical purposes, those are very "fuzzy" targets!

In addition, Jean Meeus has shown that *on average* the first and last lunar quarters are each about 10 minutes longer than each of the second and third lunar quarters, in other words Moon spends on average 20 more minutes over the day side of Earth than it does over the night side of Earth, per lunar cycle (see pages 12-13 in chapter 1 of Jean Meeus' "**About the phases of the Moon**" in "**Mathematical Astronomy Morsels IV**", published by Willmann-Bell, Inc., 2007). This means that although the use of fractional mean lunation numbers as suggested in several places on this page are a good estimate of the mean lunar conjunction (fraction = 0) and opposition (fraction = 0.5), the mean waxing quadrature (fraction = 0.25) tends to be about 10 minutes premature and the mean waning quadrature (fraction = 0.75) tends to be about 10 minutes late.

The following chart shows the variation in Earth-Moon distance over 3 present-era years, with the distances at syzygy events (lunar phase 0° with Earth-Moon-Sun approximately aligned at lunar conjunctions = New Moon, or lunar phase 180° with Moon-Earth-Sun approximately aligned at lunar oppositions = Full Moon) highlighted with colored symbols and dashed curves. Click here or on the chart to open a high-resolution PDF version (which also includes the next 2 charts). The *y*-axis shows the lunar distance in millions of metres and the *x*-axis shows the Gregorian date in YYYY-MM-DD format, with major tick-mark interval of 59 days (close to 2 lunar cycles) and minor tick-mark interval of 14.75 days (close to one half lunar cycle). Keep in mind that currently Earth passes perihelion (closest to Sun) on January 3rd or 4th, and passes aphelion (furthest from Sun) on July 4th or 5th.

- The closest Earth-Moon distances are about 87+
^{2}/_{3}% of the furthest. - The maximum Earth-Moon distances occur at those lunar conjunctions where Moon is near apogee (furthest from Earth)
__and__Earth is near perihelion (closest to Sun), so that the gravitational effect of Earth on Moon is minimal and the gravitational effect of Sun on Moon is maximal. - The Earth-Moon distances at lunar oppositions reach maxima that are almost as great when Moon is near apogee (furthest from Earth)
__and__Earth is near aphelion (furthest from Sun), such that the gravitational influences of both on Moon are minimal. - The minimum Earth-Moon distances occur at those lunar oppositions where Moon is near perigee (closest to Earth)
__and__Earth is near perihelion (closest to Sun), such that the gravitational effects of both Sun and Earth on Moon are maximal. - The Earth-Moon distances at lunar conjunctions reach minima that are almost as close when Moon is at perigee (closest to Earth) and Earth is at aphelion (furthest from Sun), such that the gravitational effect of Earth on Moon is maximal and the competing gravitational effect of Sun on Moon is minimal.
- In other words, the instantaneous lunar elliptical orbital eccentricity varies periodically, reaching a maximum whenever its major axis is aligned with the Sun-Earth line and a minimum whenever its minor axis is aligned with the Sun-Earth line (major axis perpendicular to the Sun-Earth line).
- The closest lunar oppositions are sometimes called "supermoon", and the furthest lunar oppositions are sometimes called "micromoon". Although the difference in apparent lunar diameter is significant, it doesn't deserve such exaggerated terms, and these aren't rare events: see this picture and description at NASA's Astronomy Picture of the Day web site.
- A supermoon is followed 7 lunar months later by a micromoon, then 7 lunar months later a supermoon, and so on, because the lunar orbital perigee advances around Earth in the same direction that Moon revolves, completing a full cycle
__with respect to Sun__in about 411.8 days or almost 14 lunar months.

The following chart again shows the variation in Earth-Moon distance over the same 3 present-era years, but with the distances at the first and last quarter moon events (lunar phase 90° or 270°) highlighted with colored symbols and dashed curves. Click here or on the chart to open a high-resolution PDF version (which also includes the previous and next chart). Although Moon looks half-illuminated at these events, we call it a quarter Moon because they mark the end of the first quarter or the beginning of the last quarter of the lunar cycle.

- The greatest distances at quarter moons reach about 99+
^{1}/_{3}% of the most extreme distances that can occur at syzygy events. - The closest distances that can occur at syzygy events reach about 96+
^{1}/_{4}% of the closest distances that can occur at quarter moon events. - Maximum quarter moon distances occur when Moon is near apogee.
- Minimum quarter moon distances occur when Moon is near perigee.
- These extremes usually alternate at intervals of 7 lunar months.

The following chart __combines__ the previous two charts, making it easy to see that although most of the lunar distance extremes occur near syzygy or quarter events, a few don't (about twice as many far as close extremes). Click here or on the chart to open a high-resolution PDF version (which also includes the previous two charts).

In the above lunar distance charts instances of closest distances seem to occur in groups of 3 or 4 events, and likewise instances of furthest distances seem to occur in groups of 3 or 4 events. This is a hint that such extreme events occurs more commonly. To check whether that is correct, the following chart shows the distribution of Earth-Moon distances during the 21st century, including separate cumulative centile curves for all lunar conjunctions, first quarters, oppositions, and last quarters. Also plotted, as dashed curves, is the first derivative (point-by-point slope) of the conjunction and first quarter curves, showing the relative frequencies of the various lunar distances. Click here or on the chart to open a high-resolution PDF version.

- If these were Gaussian distributions then each curve would be "S"-shaped, with tails ending horizontally, which would indicate that there are few events near the tails, but these curves all have nearly vertical tails, suggesting that the extremes occur with relatively high frequency. This is demonstrated by the derivative curves, whose steep increases near their ends confirm that the extreme distances do indeed have the highest frequencies.
- The distributions of lunar distances at conjunctions and oppositions are almost identical, and likewise the distributions of lunar distances at quarter moon events are almost identical.
- At the short distance end of the conjunctions and oppositions curves there is a small gap, with oppositions having a slightly shorter minimum distance, and at the far end a similar gap indicates that conjunctions have a slightly further maximum distance.

What about the lunar distance when Moon is actually at perigee or apogee? The following chart summarizes the distributions of those distances. The distances for perigee are intentionally plotted "backwards" to show the true relationship with apogee: the closest perigee distances go together with the furthest apogee distances (maximum lunar orbital eccentricity, major axis parallel to the Earth-Sun line), and the furthest perigee distances go together with the closest apogee distances (minimum eccentricity, major axis perpendicular to the Earth-Sun line). Click here or on the chart to open a high-resolution PDF version (which also includes the next two charts).

The next two charts zoom in on perigee and apogee separately, each plotted as an ascending cumulative centile curve along with its first derivative (slope), showing yet again that the extreme distances occur more frequently than intermediate distances. This implies that the lunar orbit alternates relatively rapidly between minimum and maximum eccentricity, spending relatively little time at intermediate eccentricities. Click here or on the charts to open a high-resolution PDF version (which also includes the previous chart).

The perigee distance varies over a much greater range (almost 14 × 10^{6} metres) than does the apogee distance (less than 3 × 10^{6} metres). The higher frequency of the perigee derivative curve at the closest distances indicates that these occur most frequently.

The apogee derivative curve, which resembles an *inverted* Gaussian distribution curve, indicates that the closest and furthest apogee distances occur most frequently and with almost equal frequency.

I calculated the moments of these events using the *new-moon-at-or-after* or *lunar-phase-at-or-after* functions, and then calculated the distances between the centers of Earth and Moon using the *lunar-distance* function as published in the book **Calendrical Calculations**, 3rd edition, by Nachum Dershowitz and Edward M. Reingold, Cambridge University Press, 2008.

For more discussion and charts about lunar distances and instantaneous lunar orbital elements, see pages 11-25 (chapters 1-4) of Jean Meeus' **Mathematical Astronomy Morsels**, published by Willmann-Bell Inc., 1997. For calculating the moments of perigee and apogee, I started with the mean perigee from Jean Meeus' **Astronomical Algorithms**, 2nd edition, published by Willmann-Bell Inc., 1999, chapter 45 "Position of the Moon", page 343, based on Chapront ELP2000-82B, 1998 (mean apogee is the same but add 180° then modulo 360°), then used a binary search within ±2 days to find the actual minimum perigee or maximum apogee distance to within 1 metre precision. The mean-to-actual differences for the 21st century were within ±25.1° for perigee and ±5.6° for apogee.

To calculate mean lunar conjunction moments we need to average out (cancel) the short-term periodic variations. I tried averaging in groups corresponding to the strongest beat frequencies of 14 × 111 = 1554 lunations (this is also the least common multiple of 14 and 111), but the results were not as smooth as desired. Eventually I found that tripling the group sizes yielded excellent smoothing, optimal when the lunations were in groups of **4657** lunar months, which is also very close to twice the approximately 2277-lunation period of the weak periodicity.

The following chart shows the difference in days between the actual astronomical lunar conjunction (SOLEX New Moon) minus the Constant Interval New Moon Estimate as was defined above, in Terrestrial Time, for dates from 7000 BC to 12000 AD, a total of 220,000 lunar cycles, averaged in 49 groups of 4657 lunations, along with the minimum and maximum differences in each group. Each average is plotted as a "+" symbol at the middle lunation number of that group:

The polynomial near top center gives the least squares statistical regression line to the averages. Although it appears to be parabolic, over this range of dates a quartic (4th order) polynomial provides a superior fit. Although it is not very obvious, the span between *Earliest* to *Latest* gradually tapers from the past to the future, due to the decline of Earth's mean orbital eccentricity over the evaluated date range.

The polynomial above can be used to define an accurate function returning the Mean New Moon Moment in Terrestrial Time:

NewMoonAdjustTT(L) = 3.5962433E-22L^{4}– 7.799103E-17L^{3}+ 1.005115E-10L^{2}+ 2.867010E-08L+ 8.945687E-05

MeanNewMoonTT(L) =NewMoonEstimate(L) +NewMoonAdjustTT(L)

where *L* is the lunation number relative to J2000 (as above, fractional *L* values of 0, 0.25, 0.5, and 0.75 refer to the Mean New Moon, Mean 1st Quarter, Mean Full Moon, and Mean 3rd Quarter, respectively). There may be a performance and numerical stability advantage in rearranging the *NewMoonAdjustTT* polynomial, in fact any of the polynomials presented herein, into monomial form according to Horner's Rule, if the programming language will allow it, by factoring out the powers of the lunation number, replacing exponentiation with nested multiplication:

NewMoonAdjustTT(L) = { [ (3.5962433E-22L– 7.799103E-17)L+ 1.005115E-10]L+ 2.867010E-08}L+ 8.945687E-05

One could combine the arithmetic of *NewMoonEstimate*( ) and *NewMoonAdjustTT*( ) into a single polynomial expression, but only the constant and linear terms are combinable, and for clarity I felt is best to retain separate functions.

**This polynomial and the others presented on this web page are valid over the range of lunation number included in this study, that is -100500 to 123500. They can probably be pushed an extra 20000 lunations further to the past or future, but don't rely on them beyond that.**

To convert the TT New Moon moment to UT for use with a calendar, simply subtract Delta T:

MeanNewMoonUT(L) =ThisMeanNewMoon–DeltaT(ThisMeanNewMoon)where

ThisMeanNewMoon=MeanNewMoonTT(L)

Optionally one can convert the UT New Moon moment to any desired time zone or reference meridian by adding the offset as the appropriate fraction of a day relative to the Prime Meridian.

The following chart confirms that the *MeanNewMoonTT* function generates accurate mean lunar conjunction moments across the full range of lunations evaluated in this study. It is quite apparent that over many millennia the heights of the *Earliest* and *Latest* extreme peaks gradually converge toward intermediate heights (due to declining Earth orbital eccentricity):

Of course the *MeanNewMoonUT* function performs equivalently, provided that the same Delta T function is used for the astronomical lunar conjunction moments.

It is easy to use the *MeanNewMoonUT* function to estimate the drift of a fixed arithmetic lunar or lunisolar calendar with respect to the mean lunar cycle over any range of lunations:

MeanLunarCycleDrift(MSM,FromLunation,ToLunation)

where *MSM* is the assumed constant lunation interval, and the lunation numbers are relative to zero = January 6, 2000 AD.

StartMeanConjunction=MeanNewMoonUT(FromLunation)

EndMeanConjunction=MeanNewMoonUT(ToLunation)

ElapsedMonths=ToLunation–FromLunationRETURN

MSM×ElapsedMonths–EndMeanConjunction+StartMeanConjunction

The result is returned in terms of the integrated total number of days and fraction of a day drift accumulated from the starting to the ending lunation numbers. This drift is over and above any drift that had already accumulated up to the moment of the starting lunation, and it is valid regardless of the selected time zone or reference meridian of the lunar conjunction moments.

For example, we can use this function to calculate the lunar cycle drift inherent in the *molad* (mean new moon estimate) of the traditional fixed arithmetic Hebrew calendar from its inception (era of Hillel ben Yehudah, in Hebrew year 4119 = 358 AD) until the present era (say Hebrew year 5768 = 2007 AD). First we define a constant offset for converting traditional Hebrew elapsed month numbers to J2000 lunation numbers, based on the Hebrew lunation number for *Shevat* 5760, which was the Hebrew month that started shortly after the January 6, 2000 lunar conjunction:

HebrewLunationAtJ2000= 71233

Next, set the starting and ending lunations to *Tishrei* 4119 and *Tishrei* 5768, respectively, converted to J2000 lunation numbers:

FromLunation= 50933 –HebrewLunationAtJ2000= –20300

ToLunation= 71328 –HebrewLunationAtJ2000= 95

For the fixed interval *MSM* use the traditional *molad* interval of the Hebrew calendar, which is 29 days, 12 hours, and 44+^{1}/_{18} minutes per month:

MoladInterval= 29 +^{12}/_{24}+ ( 44+^{1}/_{18}) / 1440 =^{765433}/_{25920}= 29.530594135802469...

The *MoladInterval* can't be represented as an exact decimal number because the above overscored digits repeat forever, but its double precision floating point representation is certainly adequate to calculate the drift to an accuracy of a second:

Drift=MeanLunarCycleDrift(MoladInterval,FromLunation,ToLunation)

In this case *ElapsedMonths* = 20395 and the *Drift* = about 0.0682385 days or about 1 hour 38 minutes and 16 seconds. This implies that the *molad* reference meridian has been drifting eastward. Each 4 minutes of time drift corresponds to 1° of longitude drift, but it is easiest to multiply our fractional day *Drift* by the 360° of Earth rotation per full 24-hour period = 24.57° or almost 24° 34' of eastward drift relative to wherever the *molad* reference meridian of longitude was in the era of Hillel ben Yehudah. Each 360°/24 hours per day = 15° corresponds to one standard time zone, so the *molad* has drifted by nearly two time zones!

By choosing several ending lunation numbers at appropriate intervals, one can chart how the drift rate changes over time, from which it will be evident that the *molad* drift is accelerating quadratically. For further information, see my full analysis of the *molad* of the Hebrew calendar at <http://individual.utoronto.ca/kalendis/hebrew/molad.htm>.

A convenient feature to add to the *MeanLunarCycleDrift* function is to allow the user to pass the *MSM* in terms of the number of seconds in excess of 29 days, 12 hours and 44 minutes. If so, then for the *molad* example presented above the *MSM* could be passed as ^{60}/_{18} or reduced to ^{10}/_{3} seconds.

I used the arithmetic explained in this section to generate a Microsoft Excel spreadsheet that dynamically depicts lunar calendar drift rates for a wide array of fixed arithmetic lunar cycles from 8000 BC to 12000 AD. The spreadsheet employs a VBA (Visual Basic for Applications) macro that calculates the lunar calendar drift relative to a user-specified zero reference year, displaying the results graphically, optionally allowing the user to alter a Delta T multiplier. In order for the macro to run, you must have the full version of Excel, it won't run in the Excel Viewer environment. In addition, your Excel security settings must allow the macro to run, with or without your confirmation, as you prefer. To enable macros, use the Excel "Tools" menu, choose "Macro", slide over to "Security...", then choose the desired macro security level.

**Click here to download the "Lunar Calendar Drift" spreadsheet **668KB

The shorter cycles in that spreadsheet are not particularly useful today, but in the future they could each take turns for a few centuries in sequence as the mean lunation interval gets progressively shorter, as can be seen in this **progressive yerm era analysis for lunar conjunctions at the Prime Meridian**, hence the reference years for the shorter cycles are preset to the distant future.

Given any moment in time, it can be useful to find the corresponding J2000-relative lunation number.

I plotted every 1000 lunations from -100000 to +100000 against the mean lunation moment expressed in terms of J2000-relative Mean Atomic Revolution Years (MARY = 365+^{31}/_{128} atomic days per year, for more information see "The Length of the Seasons" at <http://individual.utoronto.ca/kalendis/seasons.htm>), and then used least squares statistical regression to generate a quadratic polynomial for converting any given TT moment to the corresponding J2000-relative lunation number:

TTmomentToLunation(TTmoment) = -5.367946E-10MARYs^{2}+ 12.3682665MARYs– 0.172522where

MARYs= (TTmoment– J2000.0 ) /MARYand

MARY= 365+^{31}/_{128}atomic days

The result is the lunation number relative to zero = January 6, 2000 AD. Any fractional component indicates the portion of that lunation number elapsed up to the given moment (the mean lunar phase), but one should not rely on more than 2 or 3 decimal points. Values near whole numbers are returned at each TT mean lunation moment. Actually the quadratic coefficient is very small, so one could omit it and adjust the constant coefficient slightly to make a linear expression instead, especially if the intention is to round the result to 2 or 3 decimal points anyway:

TTmomentToLunation(TTmoment) = 12.3682665MARYs– 0.184336

To find the lunation number given a UT moment, simply add Delta T and then pass the sum to the *TTmomentToLunation* function:

UTmomentToLunation(UTmoment) =TTmomentToLunation(UTmoment+DeltaT(UTmoment) )

Given methods to compute mean New Moon moments, the average length of the lunar cycle, or mean synodic month (MSM) at any given mean lunar conjunction can be calculated as the average of the current and previous mean lunation interval:

*MSM_Atomic*( *L* ) = ( *MeanNewMoonTT*( *L* + 1 ) – *MeanNewMoonTT*( *L* – 1 ) ) / 2

*MSM_Solar*( *L* ) = ( *MeanNewMoonUT*( *L* + 1 ) – *MeanNewMoonUT*( *L* – 1 ) ) / 2

where *L* is the lunation number relative to J2000, and for the *MSM_Solar* function to work properly the underlying Delta T approximation must return "non-granular" Delta T values that vary as a continuous function of the time elapsed relative to some specified epoch.

Instead of using these MSM functions, to parallel the 4657-lunation averaging groups, I used the elapsed time between the starting mean lunar conjunction of one group and the start of the next group, divided by 4657, thus computing the MSM corresponding to the middle lunation number of each group. I then subtracted 29 days 12 hours and 44 minutes from each MSM value, and finally multiplied the residual fraction by 86400 (the number of seconds in a day), thus obtaining only the few excess fractional seconds for plotting.

The following chart shows the MSM as the number of seconds in excess of 29 days 12 hours 44 minutes in terms of both **atomic time** (getting **progressively longer** due to tidal forces accelerating Moon further away from Earth) as well as **mean solar time** (getting **progressively shorter** due to tidal forces slowing Earth's rotation more than the slowing of the lunar revolutions, a natural consequence of the gravitational transfer of angular momentum from Earth to Moon):

The cubic (3rd order) polynomials shown for each MSM regression line give the MSM as the number of seconds in excess of 29 days 12 hours and 44 minutes for any desired lunation number within the evaluated range.

Excess = LunationToMSM_Atomic(L) = 1.242862E-16L^{3}– 2.021546E-11L^{2}+ 1.7369075E-05L+ 2.877432

Excess = LunationToMSM_Solar(L) = 1.2434254E-16L^{3}– 2.021679E-11L^{2}– 2.51203947E-05L+ 2.777861

where *L* is the lunation number relative to J2000.

To convert the *Excess* to days, simply divide it by the number of seconds in a day and add 29 days 12 hours and 44 minutes as follows:

MeanSynodicMonth= 29 +^{12}/_{24}+^{44}/_{1440}+^{Excess}/_{86400}

My Delta T function simplistically assumes an essentially steady rate of tidal slowing such that the length of the mean solar day will get longer by 1.75 milliseconds per century, but the actual remote past tidal slowing was probably substantially greater than that due to the historically greater mass of the polar ice caps with lower global sea levels and due to the greater Earth axial tilt that existed several millennia ago. The two MSM curves cross in the year 1810 AD, which was a few decades prior to the conventional minimum Delta T value, because the mean solar time curve is dominated by my remote past and distant future parabolic Delta T approximations.

If you prefer a different Delta T approximation then use it to derive your own mean solar time MSM polynomial instead of relying on the above.

Although the two lines seem to be reciprocals of each other, they aren't quite. In terms of atomic time, Earth's rotation rate is slowing down more than Moon's, as should be obvious by the fact that the red mean solar time trend line has a steeper slope, going all the way from the top left to the bottom right corner of the graph whereas the atomic time trend rises by only about ^{3}/_{4} of the same scale. The tidal transfer of angular momentum varies with the arrangement of the continents, the depths of the oceans, the oblateness of Earth's not-quite spherical shape, the tilt of Earth's axis, the tilt of Moon's axis, and the inclination and eccentricity of Moon's orbit, as well as other factors.

The inverse function, returning the lunation number that has a given MSM, can be obtained by mathematical rearrangement of the above polynomials, but that yields a rather unwieldy equation, so instead it is much simpler to derive a direct polynomial by swapping the axes of the above chart and then similarly obtaining the inverse polynomial by least squares statistical regression. The user can pass either the desired MSM or the number of seconds in excess of 29 days 12 hours and 44 minutes (as written, the function considers any passed MSM value <29 to be just the excess seconds):

MSM_AtomicToLunation(MSM) = -849.9472Excess^{3}+ 10174.37Excess^{2}+ 20156.62Excess– 121610.9

MSM_SolarToLunation(MSM) = -413.7623Excess^{3}+ 1822Excess^{2}– 40469.25Excess– 107460.75where, in both cases, IF

MSM> 29 THENExcess= (MSM– 29 –^{12}/_{24}–^{44}/_{1440}) / 86400 ELSEExcess=MSM

As above, the mean solar time polynomial is of uncertain reliability beyond the valid date range of the underlying Delta T function (typically from 1600 AD to the present era).

The length of the mean synodic month (MSM) divided into 360° yields the mean daily change in lunar phase, which is the angular difference between the lunar and solar ecliptic longitudes. This indicates the mean rate at which the selenographic colongitude of the lunar sunrise terminator progresses across the lunar surface, or the sunset terminator, which equals the selenographic colongitude +180°.

For example, based on the above arithmetic the MSM at *Lunation* zero on January 6, 2000 AD was about 29.5305877 mean solar days. Dividing that into 360° yields a mean change in lunar phase of about 12.19° or 12° 11' 27" per mean solar day.

With respect to the distant stars, however, for example the stars of the zodiac constellations, Moon's angular motion, or sidereal angular motion, is slightly greater. During the course of a solar year it amounts to 360° for each elapsed lunation, plus 360° to account for the motion of Sun through the entire zodiac:

MeanLunationsInYear=SolarYearLength/MeanSynodicMonth

MeanSiderealMotionInYear= (MeanLunationsInYear× 360° ) + 360°

MeanSiderealAngularMotionPerLunation=MeanSiderealMotionInYear/MeanLunationsInYear

MeanSiderealAngularMotionPerDay=MeanSiderealAngularMotionPerLunation/MeanSynodicMonth

For example, using the *Lunation* zero MSM and dividing it into the present era mean northward equinoctial year length of 365 days 5 hours and 49 minutes there were about 12.36827268 lunations per year, so the total annual lunar mean sidereal angular motion was about 4812.578°, and dividing that by the MSM yields about 389.107°, indicating that on average Moon moved slightly more than 29.1° in excess of a full 360° orbit per lunation. This explains the long known rule-of-thumb which says that the zodiac sign that is the background for the first visible new lunar crescent at sunset is the same as sign that is the background at the end of the month when the old lunar cresent is last seen in the morning before sunrise, as each zodiac sign spans about 360° / 12 signs of the zodiac = 30° and in the interim Sun advances one zodiac sign eastward. Dividing 389.107° by the MSM we find that the mean sidereal angular motion was about 13.1764° or 13° 10' 35" per day.

Calculating the length of the mean sidereal month is simply a matter of dividing the daily sidereal angular motion into 360°:

MeanSiderealMonth= 360° /MeanSiderealAngularMotionPerDay

Continuing our example, we have 360° / 13.1764° = about 27.3216 days or 27 days 7 hours 43 minutes and nearly 5 seconds.

If one wishes to calculate the mean sidereal month as directly as possible without going through all of the above steps, then the simplified expression is:

MeanSiderealMonth= (MeanSynodicMonth×SolarYearLength) / (MeanSynodicMonth+SolarYearLength)

Taking the derivative of each *LunationToMSM* polynomial gives quadratic (2nd order) polynomials for the rate of change of the mean synodic month (MSM), in terms of a miniscule fraction of a second per lunation. Multiplying by 1000000 gives the rate of change in microseconds per lunation:

LunationToMSMrate_Atomic(L) = 1000000 ( 3.728585E-16L^{2}– 4.043092E-11L+ 1.736907E-05 )

LunationToMSMrate_Solar(L) = 1000000 ( 3.730276E-16L^{2}– 4.043358E-11L– 2.5120395E-05 )

where *L* is the lunation number relative to J2000.

Although the curves are parabolic over the evaluated range of 19000 years, the trends actually appear gently sinusoidal, with a period more than double as many years. The obvious correlation to check is the Earth axial tilt (obliquity) cycle, which has a period of about 41000 years, shown as a secondary *y*-axis in the chart below:

It has long been held by authors such as Jean Chapront and D.G. Izotov that Earth orbital eccentricity has a long-term secular effect on the length of the lunar cycle, for which they included terms in their analytical expressions for lunar longitude.

This concept originated with Pierre-Simon Laplace, who "proved" it in his early 19th century

Mécanique Célestetreatise. His work, however, was based on comparatively crude contemporary observational data, mean lunar position and mean Earth orbital eccentricity expressions of limited accuracy, he and his contemporaries dismissed ancient Babylonia records of higher Earth orbital eccentricity as biased by excess observations near syzygy, he considered only a few ancient eclipses, and he didn't consider variations in the Earth rotation rate (unrecognized at the time). Furthermore his proof concluded with the "explanation" that Moon is graduallyapproachingEarth, rather than the opposite actual reality.

The following chart shows the direct correlation between the rate of change of the mean synodic month (atomic time) and the **mean** Earth orbital eccentricity as calculated according to the 18-term astronomical algorithm of Pierre Bretagnon (1984), as given by Jean Meeus in chapter 33, "Long-term variations of the orbit of the Earth" in More Mathematical Astronomy Morsels, pages 201-205, published in 2002 by Willmann-Bell, Richmond VA:

The long-term variations of mean Earth orbital eccentricity has maxima at intervals of around 100000 years, much longer than the 41000-year period that was suggested for the MSM rate of change chart in the section above. For most of the lunations plotted prior to January 2000 AD there appears to be a correlation, but from the present era until 12000 AD two widely separated eccentricity values are associated with the same MSM rate of change. If declining mean orbital eccentricity is truly the explanation for the progressive reduction in MSM rate of change in the past then that trend ought to continue into the future as the mean eccentricity will continue to decline until Earth's orbit becomes nearly circular (29000-30000 AD, according to the Bretagnon algorithm).

Therefore, although mean Earth orbital eccentricity has a well-known effect on the short-term periodic variation of the length of the lunar cycle (find the word "eccentricity" on this page), it doesn't plausibly account for its long-term secular variation.

Finally, as evidence supporting a relationship to Earth axial tilt, the following chart shows the direct correlation between the rate of change of the mean synodic month (atomic time) and the obliquity angles that are actually used within SOLEX 9.1β:

Indeed there was an exponentially greater rate of change in the lunar revolution time in the remote past when Earth axial tilt was near its periodic maximum. With substantially more angular momentum transferred from Earth to Moon during that era, tidal slowing of the Earth rotation rate must have been much greater than it is today, even more so when one considers the probably much greater polar ice cap masses and lower global sea levels that must also have existed at that time.

The "hook" at the bottom left of the chart does not have the same significance as the broader hook that is seen on the Earth orbital eccentricity chart, because this one begins much further into the future and could be due to inaccuracy of the polynomial used to project obliquity into the future, or it could be due to inherent hysteresis in the Earth-Moon system, possibly due to a reactive change in the lunar orbital inclination. The important point is that the turning back of the curve does coincide with the reversal of the Earth axial tilt cycle.

According to SOLEX, near the present era the lunar orbital inclination varies annually from 5° to 5.3° relative to the ecliptic plane of date. Also according to SOLEX, the mean lunar orbital inclination relative to the Earth equatorial plane of the date is always equal to the mean Earth axial tilt, periodically varying by an amount equal to plus or minus the lunar orbital inclination relative to the ecliptic plane of the date, with one period equal to the time it takes for the lunar orbital nodes to regress westward 360° (in the present era 6798.38 taking days with reference to the northward equinox of date or about 18+^{2}/_{3} years, or 6793.48 days with reference to the stars). The instantaneous nodal regression rate varies with a principal period of 173.3 days, and the nodes are temporarily stationary whenever the major axis of the lunar elliptical orbit is nearly aligned with the Sun-Earth line. During each lunar orbital node westward regression cycle Earth's axis passes through all possible orientations relative to the equatorial lunar orbital inclination, therefore the mean equatorial lunar orbital inclination per lunar orbital node westward regression cycle must nearly equal the Earth axial tilt. More precisely, because of the small annual variation of the ecliptic lunar orbital inclination the mean equatorial lunar orbital inclination more exactly matches the mean Earth axial tilt when averaged over an integral number of years, such as 10 lunar orbital node westward regression cycles (≈186 years).

Lunar secular accleration expressions that employ terms for Earth orbital eccentricity instead of Earth axial tilt most likely appear to work because for several millennia around the present era there happens to be a non-causal but almost perfectly linear correlation between them, as shown below:

In conclusion, it is an oversimplification to describe the tidal acceleration of Moon as a constant rate "secular" change, because it more likely varies as a periodic function of the Earth axial tilt cycle. Earth's oblateness is also a very important factor, both in the short-term and in the long-term, but the most accurate observational data in that regard is based on satellite laser ranging, only available since the 1980s, which is too limited to interpret with regard to long-term trends.

SOLEX can output orbital osculating elements for the planets and our Moon, for any range of dates. It would be nice to experiment with this capability to evaluate long-range trends in:

- Lunar orbital inclination
- Lunar orbital eccentricity (near the present era SOLEX shows it varies through a monthly cycle that can range from about 2.6% to more than 7.6%)
- True Earth orbital eccentricity (rather than the
**mean**eccentricity of the Bretagnon algorithm and all limited-date-range polynomials), which SOLEX shows varies through a monthly cycle that can range from less than 1.6% to more than 1.75% near the present era.

It would be nice to extend the range of lunations further into the remote past and distant future to show at least one full Earth axial tilt cycle (41000 years), but that would require improved expressions for:

- Axial tilt (obliquity)
- General precession of the equinoxes
- Earth oblateness (polar : equatorial flattening) and its variation over the historical past (to the extent known)
- Quantitative relationship between lunar angular momentum and Earth angular momentum (transfer function of angular momentum from Earth to Moon) [it would be nice to know what component of the progressively slowing Earth rotation rate is due to Earth oceanic ridge expansion]

Updated Feb 20, 2022 (Symmetry454) = Feb 18, 2022 (Symmetry010) = Feb 19, 2022 (Gregorian)