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length of solar day = length of sidereal day / 1 − length of sidereal day / orbital period . When calculating the formula for a retrograde rotation, the operator of the denominator will be a plus sign (put another way, in the original formula the length of the sidereal day must be treated as negative).
The formal 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 29.53 Earth days, which is about 2.2 days longer than its sidereal period.
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.
A synodic day (or synodic rotation period or solar 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. The synodic day is distinguished from the sidereal day, which is one complete rotation in relation to distant stars [1] and is the basis of sidereal time.
27.321661 days [7] (equal to sidereal orbital period due to spin-orbit locking, a sidereal lunar month) 27 d 7 h 43 m 11.5 s: 29.530588 days [7] (equal to synodic orbital period, due to spin-orbit locking, a synodic lunar month) none (due to spin-orbit locking) Mars: 1.02595675 days [3] 1 d 0 h 37 m 22.663 s: 1.02749125 [8] days: Ceres: 0.37809 ...
In the late 19th century it was suspected, and in the early 20th century it was established, that the rotation of the Earth (i.e. the length of the day) was both irregular on short time scales, and was slowing down on longer time scales. The suggestion was made, that observation of the position of the Moon, Sun and planets and comparison of the ...
Combining these two effects, the net acceleration (actually a deceleration) of the rotation of the Earth, or the change in the length of the mean solar day (LOD), is +1.7 ms/day/cy or +62 s/cy 2 or +46.5 ns/day 2. This matches the average rate derived from astronomical records over the past 27 centuries.
The traditional lunar year of 12 synodic months is about 354 days, approximately eleven days short of the solar year. Thus, every 2 to 3 years there is a discrepancy of 22 to 33 days, or a full synodic month. For example, if the winter solstice and the new moon coincide, it takes 19 tropical years for the coincidence to recur.