Return to main pageRemarkably, the difference in the series numbers that would conventionally be given to the re-starts of an "eternal" series is always exactly 223 i.e. Saros series X, X+223, X+446 etc. are in fact the same series in different guises. The name "Era" has been given by Herr Leingaertner to this period of 223 Saros series, so the series numbers given above represent the re-appearance of the same series in different Eras. The proof of this is somewhat convoluted but the way the final conclusion suddenly pops out of a whole slew of apparently random numbers is truly amazing so I include it in full here:-
1) Given that (by definition) the time between successive eclipses in a series is one Saros, the interval between the start of a series and the beginning of its re-start is the number of Saroses needed to move the eclipse position on the Moon's orbit by 180deg. This is 180/D, where D is the shift in position per Saros (which we know is currently about 0.48deg). The time interval between the start of a series and its next re-start is thus (180/D) times the Saros period of 18.03yrs.
2) The time between these events is also the number of Saros series in an Era (which I shall call E) multiplied by the mean time between the birth of new series. The mean time is given by the average number of eclipses in a series (which I shall call N) divided by the average number of eclipses per year (which I shall call A).
3) We thus see that (N / A) x E = (180 / D) x 18.03. Rearranging to get E on its own we find that E = [(180 / D) x 18.03] / (N / A) which, with values of D, N and A correct for the current epoch taken from Morsels IV Table 24A and page 124 [0.4779, 72.4, 2.3865], has a value of 223.85: encouraging, but not yet a full answer.
It was not at all obvious how these apparently unconnected (and variable) values could possibly give a constant result of 223 until I re-arranged the terms in the second equation to produce E = (180 x 18.03 x A) / (D x N). Now of course D x N [value = 34.6] - the shift per eclipse relative to the [mean] node times the average number of eclipses in a Saros series - is just the width of the solar eclipse window at one node [+/- 17deg 18min]. I shall call this W. Using the definition of the Saros interval as 223 synodic months, we may expand the 18.03yrs figure to 223 x S/Y (where S is the Moon's synodic period in days and Y is the number of days in a year). A further re-grouping of the terms gives us E = 223 x (S/Y) x A x (180/W).
We now observe that Y/S is the number of lunations in a year, which I shall call L. The term (S/Y) x A can thus be written as (A / L). In words, this is the number of eclipses in a year divided by the number of lunations in a year, which is of course the fraction of all New Moons which are involved in a solar eclipse. Similarly, (W / 180) [or, more obviously, (2 x W) / 360 ] is the total angular extent of the eclipse window(s) divided by the full orbit circumference. Assuming that New Moons occur with equal frequency at all points on the orbit [a reasonable assumption, given the relatively rapid precession rate of the anomalistic and draconic axes], this expression gives us the ratio of the number of New Moons within the eclipse window (i.e. those that will be involved in a solar eclipse) to all New Moons. It thus turns out that the two expressions are calculating exactly the same thing and so must give exactly the same answer! That is, independent of any specific values we might give the parameters, (A / L) = (W / 180) by definition! We now see that, by means of this equality, the terms (S/Y) x A and (180/W) in the expression for E will cancel themselves out, resulting in an (invariant) value for E of exactly 223. QED, as my old maths master used to say!!
There is no reason for the re-starts of a series after intervals of an Era to have any particular similarity, as much will have changed in the close to 7,000yrs between them including, critically, the parameter that has the greatest effect on the composition of Saros series - delta-gamma. However, during the period covered by the part of the Saros-Inex Panorama normally shown in diagrams they will in fact be quite similar due to the pure coincidence (and I'm sure that is all it is) that 223/12 = 18.58, which just happens to be close to the mean periodicity of this part of the Panorama (19 series towards the left side of the Panorama and slowly decreasing as you move to the right). The Panorama will thus have gone through about 12 full cycles after 223 series, resulting in the series from that point onwards having a very similar pattern to those at the starting point i.e. if series X has a large number of eclipses relative to its neighbours then series X+223 will tend to as well. The absolute numbers will be smaller (as delta-gamma decreases with time) but the relative numbers will be similar. For example, series 14 has 85 eclipses and sits in a group of series having large total numbers (76,86,85,85,75). The totals around series 14+223 (237) are 69,69,76,70,69. The series around number 32 have totals 74,84,84,86,84,73 and around series 32+223 (255) the totals are 70,67,73,71,68. We thus see that both members of each pair contain larger numbers of eclipses than their neighbours. The "12 period" relationship is just a temporary phenomenon though, as the Panorama periodicity is 20 or more for series with large negative series numbers and decreases to as low as 16 for series numbers around +600.
The reader will note from the above that 223 is also the number of synodic months in a Saros period. This is not just a numerical coincidence born out of the way the formulae fall out but is in fact indicative of a "deeper meaning". Because the orbital parameters of the Moon return to the same configuration after 223 lunations (this fact being the whole basis of the Saros period) there are really only 223 unique New Moons. Under the "eternal Saros" way of thinking all New Moons have a Saros series number, not just those associated with eclipses, so if there are only 223 unique New Moons there can only be 223 unique Saros series numbers. QED again? Well, sort of. Surely the fact that the parameters only approximately return to the same configuration after 223 lunations is going to mess things up? To deal with this problem, consider the situation that would pertain if the correspondence were to be exact.
In this case, there would indeed be just 223 Saros series but because the new Moons constituting each one would be fixed at a certain position on the Moon's orbit (relative to the nodes), only those series whose fixed position fell within the eclipse window would be "active". Note also that because there would be no shift in position from eclipse to eclipse these series would be permanently active and thus have an infinite "life": all the others would be permanently dormant. The number of active series would be in the same ratio to the total number of series as the total width of the eclipse windows (one at each node) would be to the total circuit of the Moon's orbit. Thus, N = 223 x (W x 2)/360 = 42 [the actual numerical value is 42.87, but of course only whole numbers of series can be counted].
Now let's consider what would happen to this totally static picture in the [actual] case where the correspondence is not exact. Because of the slight mis-match between the synodic month and draconic month versions of the Saros interval, the orbital position (relative to the nodes) of the new Moons of each of the 223 Saros series would slowly "drift" round the Moon's orbit. Crucially though, because all series are in existence simultaneously, they would all be affected to the same extent and thus would all drift at the same [mean] rate. The number of series active at any given moment would still be limited to those which, at that time, had their new Moon positions within the eclipse window. However, instead of it being the same 42 series all the time the total would lose series which became dormant (by drifting out of the window) but gain those which became active (by drifting into the window). This of course means that series would acquire a finite life-time as they drifted through the window, the value of which would be the window width divided by the shift per eclipse. Given that the mis-match (and thus the drift rate and so the shift per eclipse) is slowly increasing with time, the average life-time of Saros series should be gradually decreasing. As all series drift at the same rate however (whatever that rate is), the number of series becoming active would always be the same as the number becoming dormant and so the overall average would remain at 42. Both these predictions are true - from the main "Saros Series" page we know that the average number of eclipses per series (which determines their average lifetime) is decreasing, and as the number of active series is also given by the number of eclipses in a Saros interval minus 1 (which is 18.03*2.3865 - 1 = 42) not only is this number clearly constant but it is also the value calculated above.
In other words, as long as the mis-match in orbital parameters is the same for all series (which of course it is), the fact that a mis-match exists does not alter the conclusion about there being 223 unique New Moons and thus only 223 unique Saros series. The uniqueness lies in the fact that, at a given time, there are only 223 possible configurations arising from the orbital parameters then applying, not that there can only ever be 223 configurations. An analogy might be the cars on a Ferris Wheel, such as the London Eye. These are a fixed distance apart and so there can clearly only be a certain number of them around the rim of the wheel, and this will be true whether the wheel is stationary or spinning. The cars will always retain the same relationship relative to each other even though their relationship to an external observer will change continuously.
This analogy can in fact be extended further. Imagine that the "station" below the Ferris Wheel has space for three cars. If the wheel is stationary the same three cars will be in the station all the time and so there will be an infinite period when they are available to take passengers. If the wheel is rotating there will still be three cars in the station but which exact cars this is will change from moment to moment as cars enter and leave the station area (at the same rate, clearly!). Each individual car will now only be able to take passengers for a short period of time - when it is in the station - but every car will become "active" eventually. The correspondence should be obvious: for Ferris Wheel read Moon's orbit, for car read Saros series, for station read eclipse window and for "able to take pasengers" read "capable of causing an eclipse".
One shouldn't take things too far, however, because in the real world the position of the eclipses constituting a series does not steadily drift through the eclipse window but speeds up and slows down to some extent because of the influence the aphelion/perihelion effect has on delta-gamma. The average drift rate is the same for all series though, so although this effect will cause a variation in the number of series active at any one time (as is found in practice), it will not change the overall conclusion as this is based on mean values.
So, yes it is "QED again" after all!