This web document explores some of the characteristics of rarely occurring eclipse intervals of 34, 101 and 257 lunations. The 34 lunation interval is a short eclipse series of 3 eclipses, each separated by 17 lunations. The 101 and 257 lunations intervals are closely linked to the 34 lunations interval. Another finding is that the 101 lunation periods are closely linked to the rare eclipse duos where one of the eclipses is total.
Introduction
According to G. van den Bergh the interval P in lunations between two solar eclipses can be described by P = a I + b S, where I is the Inex period of 358 lunations, and S the Saros period of 223 lunations and a and b are integer numbers (positive, negative or zero). Off course, simple mathematics show that these periods can also be described by P = n A + m B, where all numbers are integer, and A and B can be anything, provided A is not a multiple of B or vice versa. However, using I and S gives some special insights used further in this article, see e.g. figure1.
Next to Saros (223) and Inex (358), various authors (Van Gent, Verbelen) have mentioned a number of well known other eclipse cycles. Some of these are shown in table 1.
|
Name |
Lunations |
Formula |
Max # of eclipses in cycle |
|
Semester |
6 |
5I 8S |
10 |
|
Hexon |
35 |
8I + 13S |
6 |
|
Hepton |
41 |
3I + 5S |
16 |
|
Octon |
47 |
2I 3S |
25 |
|
Tzolkinex |
88 |
I + 2S |
36 |
|
Tritos |
135 |
I S |
70 |
|
Metonic Cycle |
235 |
10I 15S |
5 |
|
Semanex |
311 |
I + 3S |
27 |
|
Table
1: Some eclipse cycles |
|||
The interval between two eclipses
The interval between two successive solar eclipses is either 1, 5 or 6 lunations. Looking at eclipses in general many intervals are possible. Examining the 5 millenium eclipse catalog (11,898 solar eclipses) for eclipse pairs with interval P=aI+bS ≤ 1000, versus the sum of the absolute value of a and b, one finds the points displayed in figure 1.
|
|
The most frequent interval is not unexpected the Saros. Intervals (P ≤ 1000) that are rare are listed in table 2.
|
Interval P (lunations) |
a |
b |
abs(a)+abs(b) |
# of pairs |
|
34 |
-46 |
74 |
122 |
34 |
|
101 |
47 |
-75 |
122 |
10 |
|
257 |
-46 |
75 |
121 |
4 |
|
324 |
47 |
-74 |
121 |
43 |
|
682 |
48 |
-74 |
122 |
9 |
|
838 |
-45 |
76 |
121 |
19 |
|
905 |
48 |
-73 |
121 |
37 |
|
Table 2: Some intervals with low frequency for P ≤ 1000 |
||||
Several combinations of these rare intervals lead to the well know eclipse cycles such as the Tritos, Saros and Inex.
34 + 101 = 135 = I S (Tritos)
257 34 = 324 101 = S (Saros)
324 + 34 = 101 + 257 = 358 = I (Inex)
It should therefore be no surprise that these rare intervals are interrelated. Below some more trivia on the 34, 101 and 257 lunation intervals.
34 Lunations a Double Trion
In a 3 lunar year period (36 lunations) the period between any two solar eclipses 1, 5, 6, 7, 11, 12, 17, 18, 23, 24, 29, 30, 34, 35 or 36 lunations.
An interesting interval is 34 lunations. This is a combination of 2 successive 17 lunation periods. As the 17 lunations period is three eclipse seasons, Trion would seem to be an appropriate name for it. This would be in line with the names Hexon, Hepton and Octon for 6, 7 and 8 eclipse seasons. While in the 5 Millennium Catalogue there are 5507 Trion pairs, there are only 34 Trion cycle of 2 subsequent Trion pairs.
These rare Trion cycles start with a small partial eclipse near one of the poles, followed by a central eclipse, annular or total, near the tropics, and they end with a small partial near the opposite pole. Both the first and last eclipse are always part of a duo of solar eclipses.
The 101 lunations interval is always linked to a 34 lunations interval to form a Tritos. The 101 period always starts and ends with a duo eclipse. It is remarkable that a member of one of these duos is either total or almost total (0.999). The first eclipse of a 101 lunation interval is the last of a Saros series, and the second is the first of a new Saros series. All 257 lunation intervals are linked to a 101 lunation period to form a Inex period.
Very rarely a Trion cycle is followed one Saros period later by another Trion cycle. This happens only 4 times in times in the 5 millennia considered, and leads to the rare 257 lunation period between two solar eclipses. With the 257 lunation interval the first is always the beginning of new Saros and the second is the end of a Saros series.
The tables listing all 34, 101 and 257 lunations period can be found below.
References and suggested reading:
Bergh, prof. mr dr G. van den Regelmaat en Wisseling
bij Zonsverduisteringen, 1951, Tjeenk Willink, Haarlem
Espenak, Fred and Meeus, Jean
Five Millennium Canon of Solar Eclipses: -1999 to +3000, NASA Technical
Publication TP-2006-214141
Meeus, Jean Mathematical Astronomy Morsels, 1997,
Willmann-Bell
Meeus, Jean More Mathematical Astronomy Morsels,
2000, Willmann-Bell
Meeus, Jean Mathematical Astronomy Morsels III,
2004, Willmann-Bell
Meeus, Jean Mathematical Astronomy Morsels IV,
2007, Willmann-Bell
Van Gent,
Robert H. A Catalogue of Eclipse Cycles, update 2003, (http://www.phys.uu.nl/~vgent/eclipse/eclipsecycles.htm)
Verbelen,
Felix Saros, Inex and Eclipse cycles, 2001, (http://users.online.be/felixverbelen/cycles.htm)
Quaglia,
Luca and Tilley, John Saros Inex Panorama, MS-Excel sheet,
(http://sunearth.gsfc.nasa.gov/eclipse/SEsaros/SEpanorama.html)