Lunar month

Last updated June 07, 2024

In lunar calendars, a lunar month is the time between two successive syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month.

Variations

In Shona, Middle Eastern, and European traditions, the month starts when the young crescent moon first becomes visible, at evening, after conjunction with the Sun one or two days before that evening (e.g., in the Islamic calendar). In ancient Egypt, the lunar month began on the day when the waning moon could no longer be seen just before sunrise.^[1] Others run from full moon to full moon.

Yet others use calculation, of varying degrees of sophistication, for example, the Hebrew calendar or the ecclesiastical lunar calendar. Calendars count integer days, so months may be 29 or 30 days in length, in some regular or irregular sequence. Lunar cycles are prominent, and calculated with great precision in the ancient Hindu Panchangam calendar, widely used in the Indian subcontinent.^{[ citation needed ]} In India, the month from conjunction to conjunction is divided into thirty parts known as tithi . A tithi is between 19 and 26 hours long. The date is named after the tithi ruling at sunrise. When the tithi is shorter than the day, the tithi may jump. This case is called kṣaya or lopa. Conversely a tithi may 'stall' as well, that is – the same tithi is associated with two consecutive days. This is known as vriddhi.

In English common law, a "lunar month" traditionally meant exactly 28 days or four weeks, thus a contract for 12 months ran for exactly 48 weeks.^[2] In the United Kingdom, the lunar month was formally replaced by the calendar month for deeds and other written contracts by section 61(a) of the Law of Property Act 1925 and for post-1850 legislation by the Interpretation Act 1978 (Schedule 1 read with sections 5 and 23 and with Schedule 2 paragraph 4(1)(a)) and its predecessors.^[3]^[4]

Types

There are several types of lunar month. The term lunar month usually refers to the synodic month because it is the cycle of the visible phases of the Moon.

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

Synodic month

The synodic month (Greek : συνοδικός, romanized: synodikós, meaning "pertaining to a synod, i.e., a meeting"; in this case, of the Sun and the Moon), also lunation, 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.^[5] This is the period of the lunar phases, because the Moon's appearance depends on the position of the Moon with respect to the Sun as seen from Earth. Due to tidal locking, the same hemisphere of the Moon always faces the Earth and thus the length of a lunar day (sunrise to sunrise on the Moon) equals the time that the Moon takes to complete one orbit around Earth, returning to the same lunar 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,^[5] 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 a Gregorian year.

Since Earth's orbit around the Sun is elliptical and not circular, the speed of Earth's progression around the Sun varies during the year. Thus, the angular velocity is faster nearer periapsis and slower near apoapsis. 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 between lunations may vary from about 29.18 to about 29.93 days.^{[ citation needed ]} The average duration in modern times is 29.53059 days with up to seven hours variation about the mean in any given year.^[6] (which gives a mean synodic month as 29.53059 days or 29 d 12 h 44 min 3 s)^{[lower-alpha 1]} A more precise figure of the average duration may be derived for a specific date using the lunar theory of Chapront-Touzé and Chapront (1988):
$29.5305888531 + 0.00000021621 T - 3.64 \times 10 -10 T 2 where T = (JD - 2451545.0)/36525 and JD is the Julian day number (and JD = 2451545 corresponds to 1 January AD 2000). [8] [9] The duration of synodic months in ancient and medieval history is itself a topic of scholarly study. [10]$

Sidereal month

The period of the Moon's orbit as defined with respect to the celestial sphere of apparently fixed stars (the International Celestial Reference Frame; ICRF) is known as a sidereal month because it is the time it takes the Moon to return to a similar position among the stars (Latin : sidera): 27.321661 days (27 d 7 h 43 min 11.6 s).^[11]^[5]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 28 lunar mansions, one for each day of the month, identified by the prominent star(s) in them.

Tropical month

Just as the tropical year is 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 corresponding equinoxes.^[5] 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 given right ascension or ecliptic longitude.^{[ citation needed ]} 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 the First Point of Aries (Sun's location at the March equinox). Because of Earth's precession of the equinoxes, this point moves back slowly along the ecliptic. Therefore, it takes the Moon less time to return to an ecliptic longitude of 0° than to the same point amid the fixed stars.^[12] This slightly shorter period, 27.321582 days (27 d 7 h 43 min 4.7 s), is commonly known as the tropical month by analogy with Earth's tropical year.^[5]^[11]

Anomalistic month

The Moon's orbit approximates 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 the apsides: perigee and apogee), 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 the anomalistic month and has an average length of 27.554551 days (27 d 13 h 18 min 33.2 s). The apparent diameter of the Moon varies with this period, so this type has some relevance for the prediction of eclipses (see Saros), whose extent, duration, and appearance (whether total or annular) depend on the exact apparent diameter of the Moon. The apparent diameter of the full moon varies with the full 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 the same direction as the Moon is orbiting the Earth, one revolution in nine years. Therefore, the Moon takes a little longer to return to perigee than to return to the same star.

Draconic month

A draconic month or draconitic month^{[lower-alpha 2]} is also known as a nodal month or nodical month.^[13] The name draconic refers to a mythical dragon, said to live in the lunar nodes and eat the Sun or Moon during an eclipse.^[14] A solar or lunar eclipse is possible only when the Moon is at or near either of the two points where its orbit crosses the ecliptic plane; i.e., the satellite is at or near either of its orbital nodes.

The orbit of the Moon lies in a plane that is inclined about 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: the ascending node and the descending node.

The draconic or nodical month is the average interval between two successive transits of the Moon through the same node. Because of the torque exerted by the Sun's gravity on the angular momentum of the Earth–Moon system, the plane of the Moon's orbit gradually rotates westward, 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, lasting 27.212220 days (27 d 5 h 5 min 35.8 s).^[15] The line of nodes of the Moon's orbit precesses 360° in about 6,798 days (18.6 years).^{[ citation needed ]}

A draconic month is shorter than a sidereal month because the nodes precess in the opposite direction to 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 through its phases (new, first quarter, full, last quarter) and back again: 29–30^[16] days. The Moon completes one orbit around Earth every 27.3 days (a sidereal month), but due to Earth's orbital motion around the Sun, the Moon does not yet finish a synodic cycle until it has reached the point in its orbit where the Sun is in the same relative position.^[17]

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

Valid for the epoch J2000.0 (1 January 2000 12:00 TT):

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 in Ephemeris Time (more precisely Terrestrial Time) with days of 86,400 SI seconds. T is centuries since the epoch (2000), expressed in Julian centuries of 36,525 days. For calendrical calculations, one would probably use days measured in the time scale of Universal Time, which follows the somewhat unpredictable rotation of the Earth, and progressively accumulates a difference with ephemeris time called ΔT ("delta-T").

Apart from the long term (millennial) drift in these values, all these periods vary continually around their mean values because of the complex orbital effects of the Sun and planets affecting its motion.^[18]

Derivation

The periods are derived from polynomial expressions for Delaunay's arguments used in lunar theory, as listed in Table 4 of Chapront, 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 the rate of precession to 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. l is the mean anomaly, its period is the anomalistic month. F is the argument of latitude, its period is the draconic month. D is the elongation of the Moon from the Sun, its period is the synodic month.

Derivation of a period from a polynomial for an argument A (angle):

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

T in 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.

A₁ in "/cy ; A₂ in "/cy²; so the result Q is 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 A₁ by 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/day² divide A₂ by 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 period P in days:

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

Example for synodic month, from Delaunay's argument D: 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.

Related Research Articles

A month is a unit of time, used with calendars, that is approximately as long as a natural orbital period of the Moon; the words month and Moon are cognates. The traditional concept of months arose with the cycle of Moon phases; such lunar months ("lunations") are synodic months and last approximately 29.53 days, making for roughly 12.37 such months in one Earth year. From excavated tally sticks, researchers have deduced that people counted days in relation to the Moon's phases as early as the Paleolithic age. Synodic months, based on the Moon's orbital period with respect to the Earth–Sun line, are still the basis of many calendars today and are used to divide the year.

A lunar phase or Moon phase is the apparent shape of the Moon's directly sunlit portion as viewed from the Earth. In common usage, the four major phases are the new moon, the first quarter, the full moon and the last quarter; the four minor phases are waxing crescent, waxing gibbous, waning gibbous, and waning crescent. A lunar month is the time between successive recurrences of the same phase: due to the eccentricity of the Moon's orbit, this duration is not perfectly constant but averages about 29.5 days.

In astronomy, the new moon is the first lunar phase, when the Moon and Sun have the same ecliptic longitude. At this phase, the lunar disk is not visible to the naked eye, except when it is silhouetted against the Sun during a solar eclipse.

A year is the time taken for astronomical objects to complete one orbit. For example, a year on Earth is the time taken for Earth to revolve around the Sun. Generally, a year is taken to mean a calendar year, but the word is also used for periods loosely associated with the calendar or astronomical year, such as the seasonal year, the fiscal year, the academic year, etc. The term can also be used in reference to any long period or cycle, such as the Great Year.

<span class="mw-page-title-main">Eclipse cycle</span> Calculation and prediction of eclipses

Eclipses may occur repeatedly, separated by certain intervals of time: these intervals are called eclipse cycles. The series of eclipses separated by a repeat of one of these intervals is called an eclipse series.

The saros is a period of exactly 223 synodic months, approximately 6585.321 days, or 18 years plus 10, 11, or 12 days, and 8 hours, that can be used to predict eclipses of the Sun and Moon. One saros period after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, a near straight line, and a nearly identical eclipse will occur, in what is referred to as an eclipse cycle. A sar is one half of a saros.

The orbital period is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars. It may also refer to the time it takes a satellite orbiting a planet or moon to complete one orbit.

A lunar day is the time it takes for Earth's Moon to complete on its axis one synodic rotation, meaning with respect to the Sun. The lunar day is therefore the time of a full lunar day-night cycle. Due to tidal locking, this equals the time that the Moon takes to complete one synodic orbit around Earth, a synodic lunar month, returning to the same lunar phase. The synodic period is about 29+1⁄2 Earth days, which is about 2.2 days longer than its sidereal period.

A lunar node is either of the two orbital nodes of the Moon, that is, the two points at which the orbit of the Moon intersects the ecliptic. The ascending node is where the Moon moves into the northern ecliptic hemisphere, while the descending node is where the Moon enters the southern ecliptic hemisphere.

Lunar precession is a term used for three different precession motions related to the Moon. First, it can refer to change in orientation of the lunar rotational axis with respect to a reference plane, following the normal rules of precession followed by spinning objects. In addition, the orbit of the Moon undergoes two important types of precessional motion: apsidal and nodal.

A synodic day is the period for a celestial object to rotate once in relation to the star it is orbiting, and is the basis of solar time.

Lunar theory attempts to account for the motions of the Moon. There are many small variations in the Moon's motion, and many attempts have been made to account for them. After centuries of being problematic, lunar motion can now be modeled to a very high degree of accuracy.

A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby obscuring the view of the Sun from a small part of Earth, totally or partially. Such an alignment occurs approximately every six months, during the eclipse season in its new moon phase, when the Moon's orbital plane is closest to the plane of Earth's orbit. In a total eclipse, the disk of the Sun is fully obscured by the Moon. In partial and annular eclipses, only part of the Sun is obscured. Unlike a lunar eclipse, which may be viewed from anywhere on the night side of Earth, a solar eclipse can only be viewed from a relatively small area of the world. As such, although total solar eclipses occur somewhere on Earth every 18 months on average, they recur at any given place only once every 360 to 410 years.

The Moon orbits Earth in the prograde direction and completes one revolution relative to the Vernal Equinox and the stars in about 27.32 days and one revolution relative to the Sun in about 29.53 days. Earth and the Moon orbit about their barycentre, which lies about 4,670 km (2,900 mi) from Earth's centre, forming a satellite system called the Earth–Moon system. On average, the distance to the Moon is about 385,000 km (239,000 mi) from Earth's centre, which corresponds to about 60 Earth radii or 1.282 light-seconds.

<span class="mw-page-title-main">Solar eclipse of March 20, 2034</span> Total eclipse

A total solar eclipse will occur on Monday, March 20, 2034. Totality will be visible in 13 countries: from east to west, Benin, Nigeria, Cameroon, Chad, Sudan, Egypt, Saudi Arabia, Kuwait, Iran, Afghanistan, Pakistan, India, and China.

<span class="mw-page-title-main">Solar eclipse of February 4, 1981</span> 20th-century annular solar eclipse

An annular solar eclipse occurred at the Moon's descending node of the orbit on February 4–5, 1981. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. An annular solar eclipse occurs when the Moon's apparent diameter is smaller than the Sun's, blocking most of the Sun's light and causing the Sun to look like an annulus (ring). An annular eclipse appears as a partial eclipse over a region of the Earth thousands of kilometres wide. This annular solar eclipse was large because the Moon covered 99.4% of the Sun, with a path width of only 25 km . It was visible in Australia, crossing over Tasmania and southern Stewart Island of New Zealand near sunrise on February 5 (Thursday), and ended at sunset over western South America on February 4 (Wednesday). Occurring only 4 days before perigee, the moon's apparent diameter was larger.

A tropical year or solar year is the time that the Sun takes to return to the same position in the sky – as viewed from the Earth or another celestial body of the Solar System – thus completing a full cycle of astronomical seasons. For example, it is the time from vernal equinox to the next vernal equinox, or from summer solstice to the next summer solstice. It is the type of year used by tropical solar calendars.

<span class="mw-page-title-main">Solar eclipse of October 4, 2089</span> Total eclipse

A total solar eclipse will occur on October 4, 2089. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. A total solar eclipse occurs when the Moon's apparent diameter is larger than the Sun's, blocking all direct sunlight, turning day into darkness. Totality occurs in a narrow path across Earth's surface, with the partial solar eclipse visible over a surrounding region thousands of kilometres wide. The tables below contain detailed predictions and additional information on the Total Solar Eclipse of 4 October 2089.

An eclipse season is a period, roughly every six months, when eclipses occur. Eclipse seasons are the result of the axial parallelism of the Moon's orbital plane, just as Earth's weather seasons are the result of the axial parallelism of Earth's tilted axis as it orbits around the Sun. During the season, the "lunar nodes" – the line where the Moon's orbital plane intersects with the Earth's orbital plane – align with the Sun and Earth, such that a solar eclipse is formed during the new moon phase and a lunar eclipse is formed during the full moon phase.

The Hindu calendar is based on a geocentric model of the Solar System. A geocentric model describes the Solar System as seen by an observer on the surface of the Earth.

References

Notes

↑ In 2001, the synodic months varied from 29 d 19 h 14 min in January to 29 d 7 h 11 min in July. In 2004 the variations were from 29 d 15 h 35 min in May to 29 d 10 h 34 min in December.^[7]
↑ In medieval times, the part of the Moon's orbit south of the ecliptic was known as the 'dragon' (which devoured the Moon during eclipses) and from this we get the terminology 'dragon's head' for the ascending node and 'dragon's tail' for the descending node. … The periods between successive nodes has, over time, been termed the dracontic, draconic and draconitic month, the words deriving from the Greek for 'dragon'.^{[lower-alpha 3]}
Linton 2004, p. 7.

Citations

↑ Parker (1950), pp. 9–23.
↑ Angell (1846), p. 52.
↑ Law (1983), p. 405.
↑ Halsbury's Laws of England , volume 27: "Time", paragraph 866 (1st edition)
1 2 3 4 5 Supplement (1961), pp. 107, 488.
↑ Seidelmann (1992), p. 577: For convenience, it is common to speak of a lunar year of twelve synodic months, or 354.36707 days.
↑ "Length of the Synodic Month: 2001 to 2100". astropixels.com. 8 November 2019.
↑ Chapront-Touzé & Chapront (1988).
↑ Seidelmann (1992), p. 576.
↑ Goldstein 2003, p. 65.
1 2 Lang 2012, p. 57.
↑ John Guy Porter, "Questions and Answers: What does the period "tropical month" represent?", Journal of the British Astronomical Association, 62 (1952), 180.
↑ Lockyer, Sir Norman (1870). Elements of Astronomy: Accompanied with Numerous Illustrations, a Colored Representations of the Solar, Stellar, and Nebular Spectra, and Celestial Charts of the Northern and Southern Hemisphere. American Book Company. p. 223 . Retrieved 10 February 2014. The nodical month is the time in which the Moon accomplishes a revolution with respect to her nodes, the line of which is also movable.
↑ Linton 2004, p. 7.
↑ "Draconic month". Encyclopedia Britannica.
↑ Espenak, Fred. "Length of the Synodic Month: 2001 to 2100" . Retrieved 4 April 2014.
↑ Fraser Cain (24 October 2008). "Lunar Month". Universe Today . Retrieved 18 April 2012.
↑ "Eclipses and the Moon's Orbit". NASA.

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