Lunar month
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
Yet others use calculation, of varying degrees of sophistication, for example, the
In English
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
Sidereal month
The period of the
Tropical month
Just as the
It is customary to specify positions of celestial bodies with respect to the
Anomalistic month
The
An anomalistic month is longer than a sidereal month because the perigee moves in the
Draconic month
A draconic month or draconitic month
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
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
Month type | Length in days |
---|---|
draconitic | 27.212220815 + 0.000000414 × T |
tropical | 27.321582252 + 0.000000182 × T |
sidereal | 27.321661554 + 0.000000217 × T |
anomalistic | 27.554549886 − 0.000001007 × T |
synodic | 29.530588861 + 0.000000252 × T |
Note: In this table, time is expressed in
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):
;
T in centuries (cy) is 36,525 days from epoch J2000.0.
The angular velocity is the first derivative:
.
The period (Q) is the inverse of the angular velocity:
,
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 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:
.
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.
See also
- Lunisolar calendar
- Chinese calendar
- Hebrew calendar
- Babylonian calendar
- Hindu calendar
- Islamic calendar
- Tibetan calendar
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'.[14]
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)
- ^ a b c d e 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.
- ^ a b 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.
- ^ a b 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.
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. .
- 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. S2CID 121983695.
- Lang, Kenneth (2012). Astrophysical Data: Planets and Stars. Springer. p. 57.
- Law, Jonathan, ed. (1983). A Dictionary of Law. Oxford University Press. ISBN 978-0198802525.
- Linton, Christopher M. (2004). From Eudoxus to Einstein: a history of mathematical astronomy. 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.