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I just made an observation that is very puzzling. The observation concerns total solar eclipses and their spacing in time.

I know that the saros cycle dictates recurrence of same saros eclipses roughly every 18 yrs 11 days apart. However, the two American eclipses (Aug 21, 2017 and Apr 8, 2024) are spaced by 2422 days. These are saros 145 and 139 respectively - seemingly unrelated eclipses.

There is another pair of eclipses that are upcoming (Aug 2, 2027 and Mar 20, 2034), and they are also spaced exactly 2422 days apart. These are saros 136 and 130 respectively. Again, they are unrelated eclipses from each other - OK expected. The only thing that these two pairs of eclipses have in common is that their paths CROSS each other 2422 days apart. So, here is my question:

Why are these two pairs of totally unrelated eclipses "dancing" with each other with exactly the same spacing of 2422 days? What mathematical phenomena could account for this?

Thank you for your time.

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  • $\begingroup$ I'm willing to bet right up front that a Fourier decomposition of the full prediction equation will show that period (2422 days). $\endgroup$ – Carl Witthoft Apr 4 '18 at 15:20
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Since a solar eclipse is a coincidence of a new moon and an ascending or descending lunar node, the interval between solar eclipses must be a common multiple of a whole synodic month (or lunation, 29.53 days between same lunar phases) and half a draconic month (27.21 days between same lunar nodes). These line up around eclipse seasons, 173.3 days apart on average.

The saros cycle of 18.03 years is 223 synodic months, 242 draconic months, or 239 anomalistic months (27.55 days between lunar perigees or apogees). This multiple coincidence of phase, node, anomaly, and time of year makes the saros especially useful for prediction. However, there are other relationships between eclipses.

Between consecutive saros series there is another interval called the inex: 358 synodic months, 388.5 draconic months, or about 28.94 years. George van den Bergh showed in 1955 that the interval T between any two eclipses can be expressed as T = aI + bS, where I and S are the inex and saros periods, and a and b are integers. Some such combinations have names:

                   Synodic  Draconic  Anomalistic  Eclipse
Name      Formula  Months    Months     Months     Seasons

Semester  5I - 8S      6      6.511       6.43       1.02
Hepton    5S - 3I     41     44.493      43.94       6.99
Octon     2I - 3S     47     51.004      50.37       8.01
Tritos     I - S     135    146.501     144.68      23.00
Saros      S         223    241.999     238.99      38.00
Inex       I         358    388.500     383.67      61.00

Source: F. Verleben

Between the eclipse pairs in question, 2422 days = 82 lunations = 10S - 6I or 2 heptons. The two pairs also happen to be 3 heptons apart. The same hepton family includes southern hemisphere eclipses in Dec 2020, Nov 2030, and Jul 2037. Tabulated in a saros-inex panorama, all seven eclipses would fit on a single diagonal line of slope 5/3.

            Greatest eclipse  Sun      Moon
   Date      UT     Lon Lat   Decl  EcLat Node  Saros  Inex

2017-08-21  18:26   88W 37N   +12   +0.43 asc    145    39
2020-12-14  16:13   68W 40S   -23   -0.29 desc   142    44
2024-04-08  18:17  104W 25N    +7   +0.35 asc    139    49
2027-08-02  10:06   33E 25N   +18   +0.14 desc   136    54
2030-11-25  06:50   71E 44S   -21   -0.39 asc    133    59
2034-03-20  10:17   22E 16N     0   +0.29 desc   130    64
2037-07-13  02:39  139E 25S   +22   -0.71 asc    127    69

Sources: F. Espenak, JPL HORIZONS

From a Moon-centric point of view, the umbra of a solar eclipse moves straight along the ecliptic, but several factors shape its path on the Earth's surface. The terrestrial longitude of a path is directly related to the time of day (UT). A path's terrestrial latitude is very sensitive to the Moon's ecliptic latitude and is also affected by the Sun's declination. Due to the tilt of the Earth's axis, an eclipse path slants southwest-northeast near the March equinox, bends southward near the June solstice, slants northwest-southeast near the September equinox, or bends northward near the December solstice. The Moon's motion in ecliptic latitude also contributes a slight slant, northward at ascending node or southward at descending node, as the umbra passes west to east.

Both the 2017-2024 and 2027-2034 eclipse pairs occur near opposite equinoxes and at similar times of day, so their paths naturally have opposite slants and similar ranges of longitude. The 2030 and 2037 eclipse paths have similar ranges of latitude but barely cross because they occur 4 hours apart in the day. Eclipses in the same saros series, if not widely separated in longitude, occur at the same time of year and have nearly parallel paths with few opportunities to cross.

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