Synodic month

Perhaps the most common astronomical phenomenon of horological interest is representing of the phases of the Moon on clock and watch dials. The period of the phases of the Moon is the synodic month, the interval between two successive full Moons, about 29.53 days, so the Moon phase display should move from full Moon back to full Moon in that period.

However, there is more to this period than meets the eye. The current recognized authority and reference for descriptions of motions and aspects of astronomical objects in the solar system is Seidelmann [1]. On page 576, he credits Chapront-Touzé and Chapront with the following expression for the mean (average) length of the synodic month in days:

29.5305888531 + 0.00000021621 T - 3.64 x 10-10 T2,

where T = (JD - 2451545.0)/36525.

To explain the terms, "JD" is the astronomical "Julian Date", the number of days since 4713 B.C. January 1, Greenwich noon, Julian proleptic calendar, for which the length of the synodic month is being determined. The "Julian proleptic calendar" is a calendric system employing the rules of the Julian calendar, but extended and applied to dates preceding the introduction of that calendar. "2451545.0" corresponds to 2000 January 1, Greenwich noon. "Greenwich noon", equivalently represented as "12:00 UTC" (Coordinated Universal Time), is noon, standard time, at the Greenwich Observatory outside London, England, which also is the location defining the Prime Meridian. The time JD 2451545.0 is often called "J2000", and is the current standard reference time, or epoch, from which astronomical events are reckoned. "36525" is the number of days in 100 Julian years, or a Julian century. Thus, "T" is the time, reckoned in Julian centuries, after the J2000 epoch.

The thing to note is that the mean length of the synodic month is changing (as do the periods of all other near-periodic motions in the solar system), here increasing slowly with time, from its value of 29.5305888531 at 2000 January 1 12:00 UTC. On 2100 January 1 12:00 UTC, the synodic month will be 0.0000002158 days (0.0186 seconds) longer than it is now.

Thus, in principle, the designer of a gear train to represent the synodic period of the Moon should consider the time interval over which the train is to be used in choosing the target value to attempt to match with a gear train. For example, if the gear train is expected to be used between 2000 and 2500 (clocks have been known to last 500 years), the formula above could be used to determine the (projected) length of the synodic month in 2250 and a gear train chosen which best represents that value. In practice, the change over time can usually be ignored, for two reasons: 1) the length of any particular synodic month can vary from the mean value by up to seven hours, and 2) the usual methods of displaying the phase of the Moon cannot be interpreted conveniently more closely than the nearest day. Further discussion of this subject can be found in the gear train example.

Further discussion of the accuracy of mathematical representation of astronomical phenomena of nearly constant period or ratio, such as the length of the synodic month and tropical year or of the ratio of the lengths of the mean solar day and mean sidereal day, and the related accuracy of calculation of gear trains representing chosen values representing these (and other) astronomical phenomena may be found in Mathematical considerations.

Reference:

[1] Seidelmann, P. Kenneth "Explanatory Supplement to the Astronomical Almanac", The Nautical Almanac Office [now: Astronomical Applications Department], U.S. Naval Observatory; University Science Books, Sausalito, CA 1992