Lunar 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 m 11.6 s).[4] 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.

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) is the average period of the Moon's orbit with respect to the line joining the Sun and Earth. 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 the Earth.

While the Moon is orbiting the Earth, the Earth is progressing in its orbit around the Sun. After completing a 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. Therefore, the synodic month takes 2.2 days longer than the sidereal 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 rate is faster nearer periapsis and slower near apoapsis. The same is so for the Moon's orbit around the Earth. Because of these variations in angular rate, the actual time between lunations may vary from about 29.18 to about 29.93 days. The long-term average duration is 29.530587981 days[5] (29 d 12 h 44 m 2.8016 s). The synodic month is used to calculate eclipse cycles.[6]

It is customary to specify positions of celestial bodies with respect to the vernal 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: 27.321582 days (27 d 7 h 43 m 4.7 s). This slightly shorter period is known as the tropical month; compare the analogous tropical year.

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 m 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.

A draconic month or draconitic month[7] is also known as a nodal month or nodical month.[8] The name draconic refers to a mythical dragon, said to live in the lunar nodes and eat the Sun or Moon during an eclipse.[7] 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, where the Moon enters the Northern Celestial Hemisphere, and the descending node, where the Moon moves into the Southern.

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 m 35.8 s). The line of nodes of the Moon's orbit precesses 360° in about 6,798 days (18.6 years).

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.

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):[11]

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):

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

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

The angular velocity is the first derivative:

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

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

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

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" = 1296000" ; to convert the unit of the velocity to revolutions/day, divide A₁ by B₁ = 1296000 × 36525 = 47336400000; C₁ = B₁ ÷ A₁ is then the period (in days/revolution) at the epoch J2000.0.

For rev/day² divide A₂ by B₂ = 1296000 × 36525² = 1728962010000000.

For ${\displaystyle A_{2}\div (A_{1}\times A_{1})}$ the numerical conversion factor then becomes 2 × B1 × B1 ÷ B2 = 2 × 1296000. 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 36525. C₂ = 2 × 1296000 × 36525 × A₂ ÷ (A₁ × A₁).

Then period P in days:

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

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 = 47336400000 ÷ 1602961601.0312 = 29.530588860986 days; C₂ = 94672800000 × −6.8498 ÷ (1602961601.0312 × 1602961601.0312) = −0.00000025238 days/cy.