The mathematics of Saros series

How much is the shift between Saros series?

The shift per series has two components: vertical and horizontal. The vertical component is the difference in gamma - this is equal to the distance between two plateaux on the gamma-value graphs divided by the number of series this distance represents. The horizontal component is the difference in time between corresponding eclipses in adjacent series.

To determine the first value, consider the following:- because the plateaux indicate where the calendar date of the eclipses is close to the time of perihelion (around January, using precession-corrected dates), it must follow that the x-axis [horizontal] distance between the centres of two plateaux sections represents one anomalistic (i.e. perihelion-to-perihelion) year. Given that the interval between eclipses in a series is one Saros period, of 18.0292 anomalistic years, the mean calendar date difference between two eclipses in the same series is 0.0292 "anomalistic days". There are thus, on average, 1/0.0292 = 34.30 eclipse intervals per anomalistic year and thus per plateau-to-plateau cycle of the gamma value graph. Multiplying this value by the average delta-gamma [0.0408 at the time of the Saros series considered in these analyses] gives the y-axis (vertical) distance between two plateaux - 1.40 "gamma units". Now we already know that the periodicity of the Saros-Inex Panorama is about 19 series, and the fact that the Panorama is periodic at all is due to the existence of the plateaux in the gamma-value graphs. It therefore follows that the number of series needed to go from one plateau to the next is also 19. We can thus say that for this part of the Panorama the average vertical shift per series is 1.40 / 19, or 0.074 "gamma units".

Given that the maximum delta-gamma is only about 0.09, this value is perhaps larger than one might have expected. Note particularly that the shift per series is considerably greater than the mean shift - this will be important later. The explanation for the large value of shift per series is that all the eclipses on the plateaux, whose delta-gamma values are reasonably close to zero (and which would thus normally keep the average value down), clearly hardly contribute to the vertical distance between plateaux. This is therefore primarily determined only by eclipses with large values of delta-gamma and so, effectively, the average is taken over this much smaller total number of large-gamma-value eclipses.

Dealing now with the difference in time, recollect that when we move from an eclipse in one series to the corresponding eclipse in the adjacent series (where "corresponding" means "on the same row of the Panorama but in the next column"), the calendar date gets earlier by 20.3 days, but moving between eclipses in the same series (i.e. moving down the same column) shifts the date later but by only 10.8 days [taking mean values]. This means that if we want to find an eclipse in the next-higher-numbered series that has about the same calendar date as one in the current series we must move "down two rows" on the Panorama before we go "across one column": in this way the 2 x 10.8 approximately cancels out the 1 x 20.3. Recalculating using the anomalistic year gives a much more exact result: the number of days then become 10.65 and 20.58, meaning that the number of Saros periods needed for cancellation is 1.933 i.e. very close to 2.

These processes can be seen in operation in the section of the Panorama given at the left [from Meeus] which has eclipse dates marked instead of the types of eclipse. It is clear that as we move down a series the calendar dates advance by 10 or 11 days whereas if we move from series to series at the same level the dates go backwards by about 20 days. Therefore, starting from the eclipse at top left (12th October 1624), successive eclipses two rows down and one column to the right [shaded] have quite small calendar date differences: between 0 and 3 days in fact (with an average difference of 1.75 days). While the various date differences are not precise, the total variation of just 14days in 500yrs is sufficiently small that the magnitude of the aphelion/perihelion effect will be very similar at all the eclipses marked. [Note that the dates shown are in Gregorian years and not corrected for precession. The use of the anomalistic year and corrected dates would, as shown above, give an even more accurate result]

The difference in time between two series can thus be removed by shifting the entire earlier series by two eclipses, in the direction of later dates. While this is "down two" in terms of the Panorama, it is of course "two to the right" in terms of the graph of gamma values, as time runs from left to right here.

Click/tap on the diagram to see the "time shift" in operation for series 150. Observe that moving the entire series down by two eclipses makes the calendar date of all the eclipses (almost) the same as those of series 151.

So, we now have both components of the shift between series:- if we move the gamma value graphs "right by 2 data points and then down by about 0.074" per Saros series difference between them and a reference graph, all graphs should fall exactly on top of one another. But is this true? Time for the electronic graph paper! Here we have Fig.7 from the main page - click/tap on it to see whether the "shift correction" works.

  Fig.7a

And, lo and behold, it does indeed work! And to a pretty good degree of precision - well within the error introduced by the slight variation between the initial gamma values for these series (+/-0.01 gamma units). The differences in the middle section are due to the slopes of the various parts of each graph not being quite the same. This is due to the influence of the apogee/perigee status of the Moon, a topic I discuss on the main page.

I have thus demonstrated that the effect of different start dates on the gamma value graphs for Saros series is to shift the basic graph shape up & left or down & right, and thus to vary the exact positions of the flat and steep parts. Now return to the main page to see why this is important.