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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).
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 ...
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.
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.
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.
The sidereal year is 20 min 24.5 s longer than the mean tropical year at J2000.0 (365.242 190 402 ephemeris days). [ 1 ] At present, the rate of axial precession corresponds to a period of 25,772 years, [ 3 ] so sidereal year is longer than tropical year by 1,224.5 seconds (20 min 24.5 s, ~365.24219*86400/25772).
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.
Following this, an astronomical conference held in Paris in 1950 recommended "that in all cases where the mean solar second is unsatisfactory as a unit of time by reason of its variability, the unit adopted should be the sidereal year at 1900.0, that the time reckoned in this unit be designated ephemeris time", and gave Clemence's formula (see ...