Editing Eclipse cycle (section)

== Numerical values ==

These are the lengths of the various types of [[month]]s as discussed above (according to the lunar [[ephemeris]] ELP2000-85, valid for the [[Epoch (astronomy)|epoch]] J2000.0; taken from (''e.g.'') Meeus (1991) ):

:    SM =  29.530588853 days   (Synodic month)<ref>Meeus (1991) form. 47.1</ref>
:    DM =  27.212220817 days   (Draconic month)<ref>Meeus (1991) ch. 49 p.334</ref>
:    AM =  27.55454988  days   (Anomalistic month)<ref>Meeus (1991) form. 48.1</ref>
:    EY = 346.620076    days   (Eclipse year)

Note that there are three main moving points: the Sun, the Moon, and the (ascending) node; and that there are three main periods, when each of the three possible pairs of moving points meet one another: the synodic month when the Moon returns to the Sun, the draconic month when the Moon returns to the node, and the eclipse year when the Sun returns to the node.  These three 2-way relations are not independent (i.e. both the synodic month and eclipse year are dependent on the apparent motion of the Sun, both the draconic month and eclipse year are dependent on the motion of the nodes), and indeed the eclipse year can be described as the [[Beat (acoustics)|beat period]] of the synodic and draconic months (i.e. the period of the difference between the synodic and draconic months); in formula:

:<math>\mbox{EY} = \frac{\mbox{SM} \times \mbox{DM}}{\mbox{SM}-\mbox{DM}}</math>

as can be checked by filling in the numerical values listed above.

Eclipse cycles have a period in which a certain number of synodic months closely equals an integer or half-integer number of draconic months: one such period after an eclipse, a [[Syzygy (astronomy)|syzygy]] ([[new moon]] or [[full moon]]) takes place again near a [[lunar node|node]] of the Moon's orbit on the [[ecliptic]], and an eclipse can occur again.  However, the synodic and draconic months are incommensurate: their ratio is not an integer number. We need to approximate this ratio by [[common fraction]]s: the numerators and denominators then give the multiples of the two periods&nbsp;– draconic and synodic months&nbsp;– that (approximately) span the same amount of time, representing an eclipse cycle.

These fractions can be found by the method of [[continued fractions]]: this arithmetical technique provides a series of progressively better approximations of any real numeric value by proper fractions.

Since there may be an eclipse every half draconic month, we need to find approximations for the number of half draconic months per synodic month: so the target ratio to approximate is: SM / (DM/2) = 29.530588853 / (27.212220817/2) = 2.170391682

The continued fractions expansion for this ratio is:

 2.170391682 = [2;5,1,6,1,1,1,1,1,11,1,...]:<ref>2.170391682 = 2 + 0.170391682 ; 1/0.170391682 = 5 + 0.868831085... ; 1/0.868831085... = 1 +5097171...6237575... ; etc. ;  Evaluating this 4th continued fraction: 1/6 + 1 = 7/6; 6/7 + 5 = 41/7 ; 7/41 + 2 = 89/41</ref>
 Quotients  Convergents
            half DM/SM     decimal      named cycle (if any)
     2;           2/1    = 2            [[Lunar month|synodic month]]
     5           11/5    = 2.2          pentalunex
     1           13/6    = 2.166666667  semester
     6           89/41   = 2.170731707  hepton
     1          102/47   = 2.170212766  octon
     1          191/88   = 2.170454545  [[tzolkinex]]
     1          293/135  = 2.170370370  [[tritos]]
     1          484/223  = 2.170403587  [[saros (astronomy)|saros]]
     1          777/358  = 2.170391061  [[inex]]
    11         9031/4161 = 2.170391732  selebit
     1         9808/4519 = 2.170391679  square year
   ...

The ratio of synodic months per half eclipse year yields the same series:

 5.868831091 = [5;1,6,1,1,1,1,1,11,1,...]
 Quotients  Convergents
            SM/half EY  decimal        SM/full EY  named cycle
     5;      5/1      = 5                          pentalunex
     1       6/1      = 6              12/1        semester
     6      41/7      = 5.857142857                hepton
     1      47/8      = 5.875          47/4        octon
     1      88/15     = 5.866666667                [[tzolkinex]]
     1     135/23     = 5.869565217                [[tritos]]
     1     223/38     = 5.868421053   223/19       [[Saros cycle|saros]]
     1     358/61     = 5.868852459   716/61       [[inex]]
    11    4161/709    = 5.868829337                selebit
     1    4519/770    = 5.868831169  4519/385      square year
   ...

Each of these is an eclipse cycle.  Less accurate cycles may be constructed by combinations of these.