Lunar month

This article is about the definitions of a 'month' that are mainly of significance in astronomy. For other definitions, including a description of a month in the calendars of different cultures around the world, seemonth.

Inlunar calendars,alunar monthis the time between two successivesyzygiesof the same type:new moonsorfull moons.The precise definition varies, especially for the beginning of the month.

Animation of theMoonas it cycles through its phases, as seen from the Northern Hemisphere. The apparent wobbling of the Moon is known aslibration.

Variations

InShona,Middle Eastern,andEuropeantraditions, the month starts when theyoung crescent moonfirst becomes visible, at evening, afterconjunctionwith the Sun one or two days before that evening (e.g., in theIslamic calendar). Inancient Egypt,the lunar month began on the day when the waning moon could no longer be seen just before sunrise.Others run fromfull moonto full moon.

Yet others use calculation, of varying degrees of sophistication, for example, theHebrew calendaror theecclesiastical lunar calendar.Calendars count integer days, so months may be 29 or 30 days in length, in some regular or irregular sequence.Lunar cyclesare prominent, and calculated with great precision in the ancient HinduPanchangamcalendar, widely used in the Indian subcontinent.In India, the month from conjunction to conjunction is divided into thirty parts known astithi.Atithiis between 19 and 26 hours long. The date is named after thetithiruling at sunrise. When thetithiis shorter than the day, thetithimay jump. This case is calledkṣayaorlopa.Conversely atithimay 'stall' as well, that is – the sametithiis associated with two consecutive days. This is known asvriddhi.

In Englishcommon law,a "lunar month" traditionally meant exactly 28 days or four weeks, thus a contract for 12 months ran for exactly 48 weeks.In the United Kingdom, the lunar month was formally replaced by thecalendar monthfor deeds and other written contracts by section 61(a) of theLaw of Property Act 1925and for post-1850 legislation by theInterpretation Act 1978(Schedule 1 read with sections 5 and 23 and with Schedule 2 paragraph 4(1)(a)) and its predecessors.

Types

See also:Orbital period § Related periods

There are several types of lunar month. The termlunar monthusually refers to thesynodic monthbecause it is the cycle of the visiblephases of the Moon.

Most of the following types of lunar month, except the distinction between the sidereal and tropical months, were first recognized inBabylonian lunar astronomy.

Synodic month

Further information:Synodic period

See also:Lunar dayandSynodic day

"Lunation" redirects here. For the number, seeNew moon § Lunation number.

Thesynodic month(Greek:συνοδικός,romanized:synodikós,meaning "pertaining to a synod, i.e., a meeting"; in this case, of the Sun and the Moon), alsolunation,is the average period of the Moon's orbit with respect to the line joining the Sun and Earth: 29 (Earth) days, 12 hours, 44 minutes and 2.9 seconds.This is the period of thelunar phases,because the Moon's appearance depends on the position of the Moon with respect to the Sun as seen from Earth. Due totidal locking,the same hemisphere of the Moon always faces the Earth and thus the length of alunar day(sunrise to sunrise on the Moon) equals the time that the Moon takes to complete oneorbit around Earth,returning to the samelunar phase.

While the Moon is orbiting Earth, Earth is progressing in its orbit around the Sun. After completing its§ Sidereal month,the Moon must move a little further to reach the new position having the same angular distance from the Sun, appearing to move with respect to the stars since the previous month. Consequently, at 27 days, 7 hours, 43 minutes and 11.5 seconds,the sidereal month is about 2.2 days shorter than the synodic month. Thus, about 13.37 sidereal months, but about 12.37 synodic months, occur in aGregorian year.

SinceEarth's orbitaround the Sun isellipticaland notcircular,thespeedof Earth's progression around the Sun varies during the year. Thus, theangular velocityis faster nearerperiapsisand slower nearapoapsis.The same is true (to an even larger extent) for the Moon's orbit around Earth. Because of these two variations in angular rate, the actual time betweenlunationsmay vary from about 29.27 to about 29.83 days.The average duration in modern times is 29.53059 days with up to seven hours variation about the mean in any given year.(which gives a mean synodic month as 29.53059 days or29 d 12 h 44 min 3 s)A more precise figure of the average duration may be derived for a specific date using thelunar theoryofChapront-Touzé and Chapront (1988):
29.5305888531 + 0.00000021621T−3.64×10⁻¹⁰T²whereT= (JD − 2451545.0)/36525andJDis theJulian day number(andJD = 2451545corresponds to 1 January AD 2000).The duration of synodic months in ancient and medieval history is itself a topic of scholarly study.

Sidereal month

The period of theMoon's orbitas defined with respect to thecelestial sphereof apparentlyfixed stars(theInternational Celestial Reference Frame;ICRF) is known as asidereal monthbecause it is the time it takes the Moon to return to a similar position among thestars(Latin:sidera):27.321661days (27 d 7 h 43 min 11.6 s).This type of month has been observed among cultures in the Middle East, India, and China in the following way: they divided the sky into 27 or 28lunar mansions,one for each day of the month, identified by the prominent star(s) in them.

Tropical month

Just as thetropical yearis based on the amount of time between perceived rotations of the sun around the earth (based on the Greek word τροπή meaning "turn" ), the tropical month is the average time between correspondingequinoxes.It is also the average time between successive moments when the moon crosses from the southern celestial hemisphere to the northern (or vice versa), or successive crossing of a givenright ascensionorecliptic longitude.The moon rises at the North Pole once every tropical month, and likewise at the South Pole.

It is customary to specify positions of celestial bodies with respect to theFirst Point of Aries(Sun's location at theMarch equinox). Because of Earth'sprecession of the equinoxes,this point moves back slowly along theecliptic.Therefore, it takes the Moon less time to return to anecliptic longitudeof 0° than to the same point amid thefixed stars.This slightly shorter period,27.321582days (27 d 7 h 43 min 4.7 s), is commonly known as thetropical monthby analogy with Earth'stropical year.

Anomalistic month

See also:Lunar precessionandApsidal precession

TheMoon's orbitapproximates an ellipse rather than a circle. However, the orientation (as well as the shape) of this orbit is not fixed. In particular, the position of the extreme points (the line of theapsides:perigeeandapogee), rotates once (apsidal precession) in about 3,233 days (8.85 years). It takes the Moon longer to return to the same apsis because it has moved ahead during one revolution. This longer period is called theanomalistic monthand has an average length of27.554551days (27 d 13 h 18 min 33.2 s). Theapparent diameterof the Moon varies with this period, so this type has some relevance for the prediction ofeclipses(seeSaros), whose extent, duration, and appearance (whether total or annular) depend on the exact apparent diameter of the Moon. The apparent diameter of thefull moonvaries with thefull moon cycle,which is the beat period of the synodic and anomalistic month, as well as the period after which the apsides point to the Sun again.

An anomalistic month is longer than a sidereal month because the perigee moves in thesame directionas the Moon is orbiting the Earth, one revolution in about 8.85 years. Therefore, the Moon takes a little longer to return to perigee than to return to the same star.

Draconic month

See also:Lunar precession,Nodal precession,andNodal period

Adraconic monthordraconitic monthis also known as anodal monthornodical month.The namedraconicrefers to a mythicaldragon,said to live in thelunar nodesand eat the Sun or Moon during aneclipse.A solar or lunar eclipse is possible only when the Moon is at or near either of the two points where its orbit crosses theecliptic plane;i.e., the satellite is at or near either of itsorbital nodes.

The orbit of the Moon lies in a plane that isinclinedabout 5.14° with respect to the ecliptic plane. The line of intersection of these planes passes through the two points at which the Moon's orbit crosses the ecliptic plane: theascending nodeand thedescending node.

The draconic or nodical month is the average interval between two successive transits of the Moon through the samenode.Because of thetorqueexerted by the Sun's gravity on theangular momentumof the Earth–Moon system, the plane of the Moon's orbitgradually rotateswestward, which means the nodes gradually rotate around Earth. As a result, the time it takes the Moon to return to the same node is shorter than a sidereal month, lasting27.212220days (27 d 5 h 5 min 35.8 s).The line of nodes of the Moon's orbitprecesses360° in about 6,798 days (18.6 years).

A draconic month is shorter than a sidereal month because the nodes precess in theopposite directionto that in which the Moon is orbiting Earth, one rotation every 18.6 years. Therefore, the Moon returns to the same node slightly earlier than it returns to meet the same reference star.

Cycle lengths

Regardless of the culture, all lunar calendar months approximate the mean length of the synodic month, the average period the Moon takes to cycle throughits phases(new,first quarter,full,last quarter) and back again: 29–30days.The Moon completes one orbit around Earth every 27.3 days (a sidereal month), but due toEarth's orbitalmotion around the Sun, the Moon does not yet finish a synodic cycle until it has reached the point inits orbitwhere the Sun is in the samerelative position.

This table lists the average lengths of five types of astronomical lunar month, derived fromChapront, Chapront-Touzé & Francou 2002.These are not constant, so a first-order (linear) approximation of thesecular changeis provided.

Valid for theepochJ2000.0(1 January 2000 12:00TT):

Month type	Length in days
draconitic	27.212220815+4.14×10−6×T
tropical	27.321582252+1.82×10−7×T
sidereal	27.321661554+2.17×10−7×T
anomalistic	27.554549886−1.007×10−6×T
synodic	29.530588861+2.52×10−7×T

Note:In this table, time is expressed inEphemeris Time(more preciselyTerrestrial Time) with days of 86,400SIseconds.Tis centuries since the epoch (2000), expressed inJulian centuriesof 36,525 days. For calendrical calculations, one would probably use days measured in the time scale ofUniversal Time,which follows the somewhat unpredictable rotation of the Earth, and progressively accumulates a difference with ephemeris time calledΔT( "delta-T" ).

Apart from the long term (millennial) drift in these values, all these periods vary continually around their mean values because of the complexorbital effectsof the Sun and planets affecting its motion.

Derivation

The periods are derived from polynomial expressions forDelaunay'sarguments used inlunar theory,as listed in Table 4 ofChapront, Chapront-Touzé & Francou 2002

W1 is the ecliptic longitude of the Moon w.r.t. the fixed ICRS equinox: its period is the sidereal month. If we add therate of precessionto the sidereal angular velocity, we get the angular velocity w.r.t. the equinox of the date: its period is the tropical month, which is rarely used. lis the mean anomaly, its period is the anomalistic month. Fis the argument of latitude, its period is the draconic month. Dis the elongation of the Moon from the Sun, its period is the synodic month.

Derivation of a period from apolynomialfor an argumentA(angle):

$A=A_{0}+(A_{1}\times T)+(A_{2}\times T^{2})$;

Tin centuries (cy) is 36,525 days from epoch J2000.0.

The angular velocity is the first derivative:

$\operatorname {d} \!A/\operatorname {d} \!t=A'=A_{1}+(2\times A_{2}\times T)$.

The period (Q) is the inverse of the angular velocity:

$Q={1 \over A'}={1 \over A_{1}+(2\times A_{2}\times T)}={1 \over A_{1}}\times {1 \over 1+(2\times {A_{2} \over A_{1}}\times T)}={1 \over A_{1}}\times (1-2\times {A_{2} \over A_{1}}\times T)={1 \over A_{1}}-(2\times {A_{2} \over (A_{1}\times A_{1})}\times T)$,

ignoring higher-order terms.

A₁in "/cy; A₂in "/cy²; so the resultQis expressed in cy/ "which is a very inconvenient unit.

1 revolution (rev) is 360 × 60 × 60 "= 1,296,000"; to convert the unit of the velocity to revolutions/day, divide A₁by B₁= 1,296,000 × 36,525 = 47,336,400,000; C₁= B₁÷ A₁ is then the period (in days/revolution) at the epoch J2000.0.

For rev/day²divide A₂by B₂= 1,296,000 × 36,525²= 1,728,962,010,000,000.

For$A_{2}\div (A_{1}\times A_{1})$the numerical conversion factor then becomes 2 × B1 × B1 ÷ B2 = 2 × 1,296,000. This would give a linear term in days change (of the period) per day, which is also an inconvenient unit: for change per year multiply by a factor 365.25, and for change per century multiply by a factor 36,525. C₂= 2 × 1,296,000 × 36,525 × A₂÷ (A₁× A₁).

Then periodPin days:

$P=C_{1}-C_{2}\times T$.

Example for synodic month, from Delaunay's argumentD: D′ = 1602961601.0312 − 2 × 6.8498 × T "/cy; A₁= 1602961601.0312 "/cy; A₂= −6.8498 "/cy²; C₁= 47,336,400,000 ÷ 1,602,961,601.0312 = 29.530588860986 days; C₂= 94,672,800,000 × −6.8498 ÷ (1,602,961,601.0312 × 1,602,961,601.0312) = −0.00000025238 days/cy.

See also

• Lunisolar calendar
• Chinese calendar
• Hebrew calendar
• Babylonian calendar
• Hindu calendar
• Islamic calendar
• Tibetan calendar

References

Sources

• Angell, Joseph Kinnicut (1846).A Treatise on the Limitations of Actions at Law and Suits in Equity and Admiralty.Boston: Charles C Little and James Brown. p. 52.
• Chapront, Jean; Chapront-Touzé, Michelle; Francou, George (2002)."A new determination of lunar orbital parameters, precession constant and tidal acceleration from LLR measurements".Astronomy & Astrophysics.387(2): 700–709.Bibcode:2002A&A...387..700C.doi:10.1051/0004-6361:20020420.
• Chapront-Touzé, M; Chapront, J (1988). "ELP 2000-85: a semi-analytical lunar ephemeris adequate for historical times".Astronomy and Astrophysics.190:342.Bibcode:1988A&A...190..342C.
• Goldstein, Bernard (2003)."Ancient and Medieval Values for the Mean Synodic Month"(PDF).Journal for the History of Astronomy.34(114). Science History Publications: 65.Bibcode:2003JHA....34...65G.doi:10.1177/002182860303400104.S2CID121983695.
• Lang, Kenneth (2012).Astrophysical Data: Planets and Stars.Springer.p. 57.
• Law, Jonathan, ed. (1983).A Dictionary of Law.Oxford University Press.ISBN978-0198802525.
• Linton, Christopher M. (2004).From Eudoxus to Einstein: a history of mathematical astronomy.Cambridge University Press.p. 7.Bibcode:2004fete.book.....L.
• Parker, Richard A. (1950).The Calendars of Ancient Egypt.Chicago: University of Chicago Press.
• Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac.London: Her Majesty's Stationery Office. 1961. pp.107,488.
• Seidelmann, P. Kenneth, ed. (1992).Explanatory Supplement to the Astronomical Almanac.p. 576.

Further reading

• Bishop, Roy L., ed. (1991).Observer's handbook.The Royal Astronomical Society of Canada. p. 14.