Are Saros series separated by "40" in numbering related in any way?

Question

Can anyone help with this question? My question arose from an eclipse related observation that I made while looking into lunar Saros series 129. It is the Saros series number for the lunar eclipse that occurred on this past July 27, 2018.

First I saw that this Saros 129 series was "born" June 10, 1351 and the series will end June 24, 2613 (70 saros cycles of 18 years = roughly 1260 years). So, I became curious to see if another Saros series had ended just before Saros 129 began. I saw that Saros 89 (less by 40 in series numbering) had ended just four years earlier on July 23, 1347. Investigating further, I learned that there are roughly 40 different Saros series that are "active" at any one given time. So then I looked back further to lunar Saros 49 and observed that there does indeed seem to be a pattern involving roughly "40" less in the numbering (i.e. yes indeed, series 49 also ended just as series 89 began). These observations do seem to me as more than mere coincidences.

So, my question now is - Are these particular lunar Saros series (separated by "40" in numbering) related in any way? - series 49, 89, and 129. Is there another cycle or a "greater" cycle also at work here that I am unaware of?

Please see the comments below if what I am asking seems unclear - my comments might rephrase the question for clarity.

Edit - I checked several other Saros series numbers separated by 40. I am adding additional information below - hoping that it might help find an answer to this question about relatedness of the "40 apart" series.

Checking other series spaced by 40 numbering, these Saros series (that are numbered 40 apart – i.e. 40, 80, 120 etc.) always contain eclipses that are separated by ~1418 days between eclipses in each respective series (i.e. exactly 4 lunar years/48 lunar months).

Small Saros series (those containing fewer cycles - i.e. 70-74 saros cycles in the series) contain saros cycles that overlap minimally (Saros 129 which is only 70 eclipse cycles in length has no overlap and began just after Saros 89 came to an end 1418 days earlier). Note - zero eclipse cycles between them overlap.

Large Saros series (those containing more cycles – i.e. 80-84 saros cycles in the series) contain saros cycles that overlap accordingly (Saros 120 which is 83 cycles long thus began 13 cycles before Saros 80 came to an end with the overlapping eclipses each gapped by 1418 days). Note - 13 eclipse cycles overlap accordingly - again about 70 eclipse cycles with no overlap between these series.

It seems like about 70 eclipse cycles (constituting ~1262 solar years or ~1300 lunar years) is the standard length of each adjacent Saros series – considering the non-overlapping portion only – each non-overlapping portion separated by 4 lunar years.

Edit - Thanks to anyone who can shed any additional light on this.

Additional Edit - In response to the given answer concerning the "puzzling" 1418 days observed between every "standard saros series of 70 cycles", I have also noted that the observed "1418 day difference" between them appears to be the same difference that exists between every "4 Saros cycles of approx 18.03 years and 4 Metonic cycles of approx 19.00 years" - i.e (4 x 19.00 x 365.25 days) - (4 x 18.03 x 365.25 days) = 1417.17 days - equaling four lunar years difference (which is the actual basis for reconciling the solar and lunar cycles). I suspect that these two distinct observations are therefore somehow related.

Answer 1

It's been 3 years and I wondered if you found further answers on this. It's sad the answers here don't relate to your question. I too wanted to find this out.

On the non overlapping, the 12 cycles are deducted from saros 120, so 83-12 = 71 eclipses or 70 cycles of 1260 years. thus these same 12 eclipses will not be deducted within saros 80 since it's "accounted" for in saros 120. For Saros 80, Saros 40 last till -71; so the first 2 eclipses in saros 80 is overlapped and not counted so it'll be 74-2 = 72 eclipses.

I wondered if there's a way to exclude/include "overlapped periods" between eclipses to make it constant at 70 cycles of 1260 years for each saros.

This is because whenever a saros series is only 71 eclipses/70 cycles of 1260 years, no overlapping occur, so it is sort of "standalone"/constant.

It still puzzles me why the saros numbering with interval of 40 able to lead to this linkage of saroses via 1418 days. there must be an obscure cycle in the interval of rising of saros series.

Answer 2

The Saros series scheme for numbering eclipses is based on the Saros–Inex Panorama.

The Key idea is this:

• After one Saros (223 months) the moon returns nearly exactly to the same node
• After one Inex (358 months) the moon returns very nearly exactly to the opposite node.
• Any other eclipse period (or month interval) can be expressed as a linear combination of the Saros and the Inex. This is because the numbers 223 and 358 are relatively prime.

We can express the relation as follows: $$T = 223n + 358m $$ Where $T$ is some interval of months between eclipses, and $n$ and $m$ represents the number of Saros or Inex we have to jump forwards or backwards (positive or negative) to arrive at our chosen interval.

This equation is what is referred to in mathematics as a Linear Diophantine Equation, and its can be solved quite easily for $n$ and $m$ using the Extended Euclidean Algorithm. We can make use of an online calculator which implements this algorithm.

Some example solutions are: $$6 = 223(-8) + 358(5)$$ $$5 = 223(53) + 358(-33)$$ $$1 = 223(-61) + 358(38) $$

These are the solutions for 6-month, 5-month, and 1-month intervals respectively.

If two eclipses are separated by an interval $T$, the Saros number is defined in such a way that the Saros number for the next eclipse is given by adding the value of $m$ to the current Saros number.

So, if the next eclipse occurs 6 months after the current eclipse, we add $m=5$ to find the current Saros number. For example, the lunar eclipse of April 24,2005 is numbered 141 in the Saros scheme. The next eclipse, of October 17,2005, is separated from it by a 6-month interval, so its Saros number is $141 + 5 = 146$. If they were separated by a 5-month interval instead then we use $m =-33$, or for 1-month interval, $m=38$ (according to the solutions).

Now, regarding your question: Why does the particular pattern appear between eclipses separated by Saros number 40.

The first thing to note is that we can combine different intervals to arrive at new solutions. In particular we can take the 6-month interval and multiply all values by $8$. This will give us a new solution: $$48 = 223(-64) + 358(40)$$

This immediately gives us an explanation for why the eclipses separated by $40$ Saros are found to be ~1418 days apart, because an interval of 48 months is given by $48\times29.53 \approx 1417.44$ days $\approx 3.88$ years, which in turn corresponds to a value $m=40$

The next thing to note is that the value $n=-64$. Now $n$ indicates the amount we jump backwards or forwards between eclipses in the same Saros series. However, eclipses in different saros series do not all start at the same Inex. This can be easily seen if we take a look at a diagram of the Saros–Inex Panorama.

The following diagram of the Saros–Inex Panorama for solar eclipses from the NASA GSFC. Each Saros series run along the vertical axis. We can see that each series starts or end at different points. In fact, the starting and end points appear to vary in a sawtooth wave-like pattern!

A saros series lasts on average 72 cycles; this means the value $n=-64$ should roughly correspond to jumping from the end of one Saros series, 64 steps backwards, to the beginning of another. There is variability however due to the different lengths of Saros Series and different starting and end points. So different saros series have different values for $n$ corresponding to the different start an finish times.

For example:

• between Lunar Saros 129, and Lunar Saros 89 there are 48 months ~ 4 years, corresponding to the value $n= 64$.
• between Lunar Saros 89 and 49 $-175$ months ~ $-14$ years, corresponding to $n=-62$; here the Series overlaps.
• between Lunar Saros 49 and Lunar Saros 9, there are $-621$ months ~ $-50$ years, corresponding to $n=-67$

In general, we can check how different values of $n$ would affect the resulting interval of time between Saros Series separated by $40$ by substituting them in the expression, or equivalently, adding or subtracting 223 months from our initial solution with $n=-64$. We get the following

• $n=-63$ would give $T=271$ months or ~$22$ years.
• $n=-64$ would give $T=48$ months or ~$4$ years.
• $n=-65$ would give $T=-175$ months or ~$-14$ years, which would mean the series overlaps.
• and the pattern continues…

In the end, there is nothing particularly special about the number $40$ between Saros numbering here, other than the fact that $40$ Inexes ~ $64$ Saroses, and $64$ in turn is nearly the same as the length of an average Saros Series. In fact, we can find a better approximation in $43$ Inexes ~ $69$ Saroses. This corresponds to the solution:

$$7 = 223(-69) + 358(43)$$

As for the apparent relation between 4 Metonic cycles and 4 Saros, this might just be a coincidence: $(235-223)\times 4 = 48$