Saros, Inex and Eclipse cycles
Felix Verbelen
© 2001  Mira Public Observatory, Grimbergen (Belgium)
In his remarkable book "Periodicity and variation of solar (and lunar) eclipses" [1] Prof. dr. G. van den Bergh shows that the time interval T between any two solar (or lunar) eclipses can be found by means of the simple formula
T = aI + bS (1)
where T = interval in days
I = the Inex period of 10571.95 days (358 lunations)
S = the wellknown Saros period of 6585.32 days (223 lunations)
a and b are integers (negative, zero or positive).
Van den Bergh arranges all 8000 solar eclipses listed in Theodor von Oppolzer’s "Canon der Finsternisse" [2] in a large array, his "SarosInex Panorama".
Each column of this array includes a Saros family of eclipses, while each row lists an Inex family.
One step down in the panorama means 1 Saros later and 1 step to the right is 1 Inex later.
Van den Bergh numbers both the columns and rows of his array and so we have for the columns the numbers of the Saros families and for the rows those of the Inex families.
To which Saros and Inex family does a given eclipse belong?
Formula (1) makes it easy to find the number of both the Saros and the Inex families of any eclipse, if one starts from an at random chosen eclipse for which are known the julian day number, the number of the Saros family and the number of the Inex family.
For example, let's start  arbitrarily  from the total solar eclipse on April 22nd, 1716.
Neglecting the decimals, this (Gregorian) calendar date corresponds to JDN 2347927.
In van den Bergh's table the eclipse belongs to Saros family (= column) 124 and to Inex family (= row) 56.
Suppose we now want to know both the numbers of the Saros and Inex families of the solar eclipse on August 11th, 1999 (JDN 2451401).
The interval T between the two eclipses is 2451401  2347927 = 103474 days.
Since we know the values of I and S, the problem consists in finding the smallest possible integer values for a and b. In most cases it will prove impossible to find an exact solution, but if one realises that the minimum interval between 2 solar eclipses is 1 lunation (29.53 days), it will be clear that a difference of 1 or 2 days between the given interval and a possible solution can be neglected.
So, using a simple computer program, it is easy to find the values for a and b and to show that
103474 days =~ (21 * Inex) + (18 * Saros)
Using the values for Saros and Inex given above, one finds in fact that
21 * Inex  18 * Saros
= 21 * 10571.95 days  18 * 6585.32 days
= 103475.19 days,
but this result is sufficiently close to the given interval of 103474 days.
So, in van den Bergh's array, the August 11th 1999 eclipse belongs to Saros family 124 + 21 = 145 and to Inex family 56  18 = 38.
The same operation can be performed for any other eclipse in the past and in the future.
Eclipse periodicity
Since formula (1) allows finding the interval between any two given eclipses as a function of I and S, it does of course also allow to find periodicities in solar (or lunar) eclipses.
It is sufficient to replace the integers a and b by freely chosen values in order to obtain a period that will allow to find, starting from a given eclipse, another eclipse, and from this second eclipse another one and so on.
In his book, van den Bergh gives a list of interesting periods, some of which were already known at that time. A number of these periods received a special name. Others have later extended this list.
Perhaps the best known periods are the following [3]:
Formula (1) 
Days 
Lunations 
Name 
5I  8S 
177.19 
6 
Semester 
3I + 5S 
1210.75 
41 
Hepton 
2I  3S 
1387.94 
47 
Octon 
1I  1S 
3986.63 
135 
Tritos 
0I + 1S 
6585.32 
223 
Saros 
10I  15S 
6939.70 
235 
Meton’s cycle 
1I + 0S 
10571.95 
358 
Inex 
3I  3S 
11959.89 
405 
Maya 
From formula (1) it will be clear that in principle an infinite number of periods can be deduced.
The only limitation is set by the physical durations of the individual Saros and Inex families.
A Saros family contains from 69 to 86 eclipses and therefor lasts from 1226 to 1532 years.
An Inex family contains much more eclipses, but it isn't eternal either.
So, to be of a real practical use and to produce longlasting periodicities, the constants a and b in formula (1), and especially constant b, should remain as small as possible.
For example, the intervals of 1 or 5 lunations between solar (or lunar) eclipses can very well be expressed by means of formula (1):
Formula (1) 
Days 
Lunations 
(proposed) Name 
38I  61S 
29.57 
1 
Lunation 
33I + 53S 
147.62 
5 
Pentalunex 
but due to the large value of coefficient b these intervals cannot be considered to be practical periodicities.
On the other hand, many long lasting periodicities were not listed, as far as I know, by previous authors.
This is just a short selection:
Formula (1) 
Days 
Lunations 
(proposed) Name 
remark 
1I + 2S 
2598.69 
88 
Tzolkinex 
(a) 
1I + 3S 
9184.01 
311 
Semanex 
(b) 
2I  1S 
14558.58 
493 


1I + 1S 
17157.27 
581 


3I  2S 
18545.21 
628 


4I  3S 
22531.84 
763 


1I + 2S 
23742.59 
804 


3I  1S 
25130.53 
851 


1I  3S 
30327.91 
1027 


4I  1S 
35702.48 
1209 


3I  1S 
38301.17 
1297 


(a) this period is equal, with a difference of 1 day, to 10 tzolkins, i.e. the sacred 260day period of the Mesoamerican calendars
(b) at the end of this interval, a solar eclipse occurs on the same day of the week
Which node?
A Saros period of 6585.32 days equals 223 lunations or synodic revolutions, which is equal to 242 draconic revolutions.
Therefor, after each Saros interval the Moon is again in the same node of its orbit.
An Inex period of 10571.95 days however equals 358 lunations or synodic revolutions, which equals 388.5 draconic revolutions.
So, after 1 Inex the Moon will be at the opposite node and only be at the initial node after 2 Inexes.
In order to know the Moon’s position after an interval derived by means of formula (1) it is sufficient to examine coefficient a.
If a equals 0 or if it is a even number, then at the end of each interval an eclipse will occur at the same node. If coefficient a is an odd number, then after each interval the eclipse will occur at the opposite node.
Bibliography
[1] van den BERGH G.  Periodicity and Variation of Solar (and Lunar) eclipses (H.D.Tjeenk Willink & Zoon, Harlem, 1955)
[2] von OPPOLZER Th.  Canon der Finsternisse (Wien, 1887)
[3] MEEUS Jean  Mathematical Astronomy Morsels (WillmannBell Inc, Richmond, 1997)
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