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So after 19 years the epact must be corrected by +1 in order for the cycle to repeat over 19 years. This is the saltus lunae ("leap of the moon"). The sequence number of the year in the 19-year cycle is called the golden number. The extra 209 days fill 7 embolismic months, for a total of 19 × 12 + 7 = 235 lunations.
The average duration in modern times is 29.53059 days with up to seven hours variation about the mean in any given year. [7] (which gives a mean synodic month as 29.53059 days or 29 d 12 h 44 min 3 s) [a] A more precise figure of the average duration may be derived for a specific date using the lunar theory of Chapront-Touzé and Chapront (1988):
If at every point in the cycle the system is in thermodynamic equilibrium, the cycle is reversible. Whether carried out reversible or irreversibly, the net entropy change of the system is zero, as entropy is a state function. During a closed cycle, the system returns to its original thermodynamic state of temperature and pressure.
An eclipse cycle constructed by Hipparchus is described in Ptolemy's Almagest IV.2: . For from the observations he set out he [Hipparchus] shows that the smallest constant interval defining an ecliptic period in which the number of months and the amount of [lunar] motion is always the same, is 126007 days plus 1 equinoctial hour.
The principal change was to space leap years ... The 19-year cycle used for the lunar ... The following equation gives the number of days that the Gregorian calendar ...
Each complete cycle of phases is called a "lunation". [7] The approximate age of the Moon, and hence the approximate phase, can be calculated for any date by calculating the number of days since a known new moon (such as 1 January 1900 or 11 August 1999) and reducing this modulo 29.53059 days (the mean length of a synodic month).
An example is the Meeus smoothing formula, [7] with related solar cycles characteristics available in this STCE news item. [8] The start of solar cycle 25 was declared by SIDC on September 15, 2020 as being in December 2019. [9] This makes cycle 24 the only "11-year solar cycle" to have lasted precisely 11 years.
The best known of these is the Tabular Islamic calendar: in brief, it has a 30-year cycle with 11 leap years of 355 days and 19 years of 354 days. In the long term, it is accurate to one day in about 2,500 solar years or 2,570 lunar years. It also deviates from observation by up to about one or two days in the short term.