I've long been intrigued by lengthy cycles of natural phenomena that overlay shorter ones. Nature has countless such extended cycles ranging from hundreds to hundreds of thousands of years. Here in the Pacific Northwest, I think of the pulsing of our great volcanoes and the rhythm of large earthquakes, including the much anticipated “big one,” which after about three centuries is now apparently due to surprise us at any time. Longer range, there’s the magnetic flipping of the poles (they say it’s sooner than you think!), and the ebb and flow of the ice ages (not exactly of immediate concern, though I recall reading that we are soon due to start a “worrisome” nineteen-thousand-year decline of temperature.)

Such overlaid cycles also characterize many non-earth-shaking phenomena – in the heavens. Perhaps because we’re in a particularly rich season of eclipses, I’ve been looking at the moon. Among its many rhythms is a periodic phenomenon lasting twelve to fifteen centuries: individual Saros cycles begin and end over such spans of time, each containing seventy or more eclipses. These are not jaw-dropping geological measures of time, of course, but ones that nevertheless cross big fault lines of human history. The world at the beginning of a Saros is absolutely nothing like the world at the end of one.

To understand why these cycles exist on such a long scale, we’ll revisit a few facts about eclipses known from antiquity, and some things more recently known. We’ll focus on lunar eclipses. The ancients could see more of them from Mesopotamia than solar eclipses, and with them could perfect their mathematical techniques of prediction that were later transmitted throughout the ancient world.

*Eclipse possibilities*

You may recall from our earlier discussions that the *nodal line *runs along the joint formed by the intersection of two planes: the first plane is the ecliptic (defined by the sun and earth’s orbit) and the second is the plane of the moon’s orbit [1]. The moon's orbit crosses the ecliptic at the two endpoints of the nodal line: these are the *nodes*. Oddly, under the influence of gravitational perturbations from mainly the sun, this nodal line rotates clockwise, like a platter dropped on a table wobbling around in exceedingly slow motion, performing a full westerly revolution every 18.6 years, its effects discernable in the ocean’s tides. This is the moon’s *orbit*, not the moon, that is turning in this ‘nodal’ or ‘draconic’ circle. Because of this westward movement of the nodes, the eastward-moving moon will most of the time be out of sync with the nodes for an eclipse to occur. But occasionally the nodal line will be aimed like a rifle at the sun, earth, and moon, and it is then that an eclipse (solar or lunar, depending on the location of the moon, as new or full) will occur. This lineup happens quite precisely in intervals when the moon’s synodic month (of about 29.5 days) and the node’s draconic month [2] (of about 27.2 days) are in integral relationship with each other, e.g., most famously where 223 synodic months happen to match 242 draconic months: the result is the remarkable *Saros* match-up every 6585⅓ days [3].

There will however be quite a few ‘eclipse possibilities’ even when things are not in perfect alignment. These happen mainly because there are two nodes and because of the large size of the earth’s projected shadow in relation to the moon’s size. According to John M. Steele, historian of the exact sciences in antiquity at Brown University, “If we assume that eclipses do not occur in consecutive months, as it is apparent that the Babylonian astronomers did, it is possible to define an ‘eclipse possibility’ as the syzygy at which the Earth’s shadow or the sun is closest to the node* *every time it passes by that node. The average interval between successive eclipse possibilities is equal to about 5;52,7,44 [5.87] months. [4]”

Why exactly are there are lunar eclipse possibilities even when there is not a perfect lineup of shadow, node, and moon? The key is that the orbital plane of the moon is tilted from the ecliptic plane by about 5.1⁰. The planes are not flat with each other but are just barely crossed, jointed along the nodal line like an elongated x on its side. Because of this angling of the planes, there’s a wedge on each side of the line of nodes within which the projected shadow of the earth (in a lunar eclipse) can completely or partially fit. Within that space lies a variety of eclipse possibilities as the moon travels through it from West to East. An eclipse can thus occur even when the nodal line aims to the right or left of the shadow target. If the node is pointed too far away from the target, though, there will be no possibility of an eclipse.

The drawing below pictures only the right side of the wedge and I’ve exaggerated the tilt of the lunar orbit. As you look at the picture, you face a node and earth and sun are behind you on the ecliptic plane. Earth casts its slender 1.4-million-kilometer umbral shadow on the small red disk in the picture, surrounded by the pale penumbral shadow [5]. The earth’s shadow is shown in two places, representing its hypothetical configuration at two different times: the first in sync with the node, the other offset to the right from it. The umbral shadow in space is about 9,000 kilometers in diameter, which is more than 2½ times the lunar diameter (the exact ratio varies with varying lunar and solar distances) [6]. Because the node precesses clockwise, its transits through the center of the earth’s shadow are brief, but the moon on its celestial roadway can actually swing by a good way out from the node on either side and still get nicked by the earth’s big penumbral shadow for at least a partial penumbral eclipse. But there’s a limit: if the moon is too far away from the shadow – up or down orbit, so to speak – it will clear it altogether. This limit works out to roughly 17 ½° on each side of node, or a full 35⁰ range from side to side where there may be eclipse possibilities [7].

*The moving node*

Different eclipse configurations arise mainly because of the westward, longitudinal movement of the nodal line. The brace of nodes (called *ascending* on one side of the orbit, *descending* on the other) stride along the ecliptic from East to West at the healthy pace of about 19.3° per year [8]. It catches up with the sun every 346.6 days; or 173.3 days for the opposite node to come round. There is a theoretical box wherein eclipses of the moon can occur as the earth’s shadow approaches a node or recedes from it. It is shaded gray in the (not-to-scale) illustration below. The top line on the schematic is the first pass of the moon in its orbit. As the node shifts to the right, the moon’s orbital path is displaced from where it appeared to be before. The second line is the next pass of the moon (perhaps three actual orbits later); the plane of the moon’s orbit on this pass is displaced somewhat more to the right from the first pass. The third and fourth lines similarly represent displaced orbits intersecting ecliptic plane each time a little farther to the right.

The schematic below illustrates how the rightward shifting node determines the type of lunar eclipse. Umbral (total) lunar eclipses are pictured in the first and second orbital tracks. The second track, making a smaller slice through the umbra than the first, means a briefer total eclipse than the first. Over time the node (each denoted by a vertical arrow) continues to slip to the right in relation to the earth’s shadow. The moon's track will eventually slide under the umbra (third line from the top); then only a penumbral eclipse will occur. Once the node has regressed sufficiently to the right, however, the moon (now on the bottom path) ducks completely below the penumbra, escaping eclipse altogether. Here, the rightmost node in the diagram is too far away from the shadow (more than about 17 ½° along the ecliptic from it) to permit another lunar eclipse.

*Long cycles and gamma*

There is a much longer game where the centuries-long periods I first mentioned now come into play. And here we look not at the big wedge of rather frequent eclipse possibilities, but closer in, to the geometry of the moon’s passage within the circle of the earth’s shadow itself, and to a subtle movement of the moon’s apparent path through it discernable only after stacking an entire series of Saros eclipses together.

If you see a total lunar eclipse, it means again that the node has lined up with the sun-earth shadow, and the phenomenon will repeat (though at a different longitude) about every 18.03 years, or 6,585.32 days, which is the Saros cycle. We saw in my August 30, 2022 post (https://www.douglasmacdougal.com/post/what-is-saros-~-using-babylonian-astronomy-to-track-eclipse-paths) how three of those Saros cycles equals an almost exact 54-year *exeligmos* cycle of 19,756 days. Exeligmos is more accurate than the Saros cycle since tripling it rids the Saros of its extra third of a day. But these multiplying adjustments are not perfect. Even multiple exeligmos cycles still accumulate error. (The triple Saros of 669 synodic months differs from the 726 draconic months’ tripling by about 2 ½ hours). In Ptolemy’s words (from his *Almagest* IV.2) “Hipparchus already proved, by calculations from observations made by the Chaldaeans and in his time, that the above relationships [the exeligmos cycle] were not accurate.” And this inaccuracy is why each Saros has a beginning and an end. The inaccuracy is revealed geometrically by a gradual displacement of the apparent moon in relationship to earth's shadow as each Saros event progresses. We can see how by looking at a convenient measure of this displacement called *gamma*.

NASA defines gamma for lunar eclipses as the “distance from the center of the shadow cone axis to the center of the moon (in units of Earth's equatorial radii) at the instant of greatest eclipse.” https://eclipse.gsfc.nasa.gov/SEcat5/SEcatkey.html. All you need to know is that it is the best measure of where the disk of the moon is in relation to the shadow's center. The closer gamma (γ) is to zero, the more dead-on center the eclipse is with the center of the earth’s shadow. When it is positive or negative it means the moon is some distance above or below the shadow’s center, respectively. But the *changes* of gamma are what we seek, for as we’ll see, that will define the lifespan of any Saros series.

Let’s take a specific example. We’ll examine the lunar Saros 45 discussed in my post of August 30, 2022 [9]. If you’ll look at the chart of eclipses in that article, you’ll notice how gamma gradually grows from nearer zero (when the moon was almost exactly centered for a perfect total eclipse) to increasingly more positive values [10]. But that was a list of only six of the 85 eclipses in Saros 45. Below is a graph of the snaking progression of gamma over the __entire__ series, from August 29, -1351 to February 25, 164 CE, that I pulled from NASA files and graphed:

This Saros series starts in the southern hemisphere of the penumbra on August of -1351 (being the *astronomical* date, which is actually August 29, 1352 BCE *calendar* date), which is Sequence Number 1 on the graph. Gamma is -1.5496. For this almost non-eclipse, the moon just nicks the bottom side of the penumbral shadow of the earth as the moon sails on eastward during the course of the eclipse:

Three and a half dozen Saros cycles later and about midway through the series (Sequence Number 43 on the chart) we land on November 27 – 594. Gamma is very near the sweet spot of zero (.0011). Our moon eyes the bullseye and glides right through center of the shadow for a long totality:

You can notice here as in the prior image the slight downward slant of the lunar path (looking at the lunar disks from right to left) against the dashed line of the ecliptic in this NASA-derived image: the depicted lunar orbit slopes downward because the moon is passing through the descending node at the point of the + in the center. Incidentally, odd Saros numbers, like this Saros 45, are always *descending node* eclipse series, and the track of each successive eclipse moves *northward* through the course of such series as the value of gamma progressively increases. The opposite is true for even-numbered series.

Leapfrogging through time to the end of this 85-event Saros series, it is now February 5th, 164 CE. Having watched gamma slide upwards through the series, we’ve reached the right end of the gamma chart (Sequence Number 85); it is positive 1.528:

The moon lightly grazes the penumbral shadow like a stone skipped on pond, and we know that this, alas, must be the swansong of this Saros series. The next pass will miss the penumbra altogether, but we shouldn’t mourn: it was a good 1,515 year run for this long-ago series of Saros 45. Only about a third of the eclipses were total in the whole span of time [11].Why does this happen? Why does the moon slowly float northward in this Saros series over a whole millennium and a half? To answer that we need to go back to the numbers.

*Mathematical explanation*

This ‘gamma drift’ makes sense mathematically and is easily quantified for the general case. If we take the synodic (full moon to full moon) month of 29.530589 days, and the draconic (node to node) month of 27.212221 days, and multiply each by the respective cycles known from the Babylonian era of 223 and 242, they come out *nearly* equal at 6585.321347 days and 6585.357482 days, respectively. The story is in the word ‘nearly’ because they are not equal: they differ by + 0.036135 days per Saros, which translates to a nodal counterclockwise displacement of about 0.478 degrees per eclipse of the Saros series [12]. This conforms to NASA’s comment that, “*the Moon's node shifts eastward by about 0.48° with each eclipse in a series*.” This displacement is due to the imperfect match-up of the synodic and draconic cycles after each18-year Saros. The displacements are cumulative, amounting to a snail-like migration of the node that continues over the whole series.

The nodal shift along the ecliptic path over the 85-eclipses in our Saros 45 means that as the node migrates eastward, the moon’s orbit path ventures slowly – *very* slowly – upward. The rough illustration below may help clarify this. (It should have 85 tracks along a much longer and shallower slope to be literal, but I’ve used sample orbital paths of about 300 years apart to illustrate the idea.) The series starts with a big negative gamma as the moon begins to slice through the lower part of the penumbra. Then over the many years of Saros 45 eclipses, it inches up, like a knife slicing thinly through a ham, until it reaches the umbra for a series of total eclipses. There, gamma becomes zero as the moon cuts through the center of the shadow. As the node continues to drift eastward over the centuries, the intersections with the earth’s shadow cut northward of the ecliptic into positive gamma territory. Thence, about 1,500 years after the moon first entered the southern hemisphere of the penumbra, it departs out of the northern penumbra and the party is over for this Saros series.

For the even numbered-Saros series, the orbital arrows would point from the lower right to the upper left, representing ascending node passages, and the progression of gamma is then positive to negative as the orbital slices through the shadow descend from north to south.

__NOTES__
[1] The two points where the lunar orbital plane intersects the plane of earth's orbit are called the *nodes*. The moon moves from south to north of earth's orbit at the ascending node, and from north to south at the descending node.
[2] There is inconsistency in the literature in the use the term ‘draconic’ vs ‘draconitic’ which mean the same thing. I use the former because it has fewer syllables.
[3] The ‘anomalistic’ motion of the moon’s perigee and apogee also have a neat match-up in this cycle.
[4] Steele, John M. “Eclipse Prediction in Mesopotamia.” *Arch. Hist. Exact Sci.* 54 (2000): 423. We’ll not delve into the notion of ‘eclipse seasons’ here, suffice to say they are well explained in Fred Espenak’s https://eclipse.gsfc.nasa.gov/LEsaros/LEperiodicity.html. Of importance there is the idea that, “The time interval between any two successive lunar eclipses can be either 1, 5, or 6 lunations (synodic months).” Steele discusses how well-known these short cycles were to the Babylonians. But, as Espenak points out, “[T]hese short periods are of no value as predictors of future eclipses because they do not repeat in a recognizable pattern.”
[5] The earth’s diameter is less than 1% of the length of its shadow.
[6] The penumbral shadow diameter is much bigger, averaging about 16,300 kilometers.
[7] The actual amount slightly varies. We will use the 17.5⁰ term for the purposes of our discussion. With a little trigonometry, we can confirm the limit on how far along the ecliptic the node can be from the earth’s shadow to encompass (theoretically) the full disk in the middle and partial disks top and bottom. That is, to cover at least the tip of the moon on entry and the last tip on exit. Accommodating the shifting distances between earth and moon in the moon’s elliptical orbit, and some minor fiddling, we informally find that the number of lunar disks is 2.76 for the penumbral radius to have the ranges be between 17.1° (lunar perigee) and 15.3° (lunar apogee) on each side away from the center of the sun-earth shadow. This squares well with the physical interpretation based on shadow sizes.

[8] The node regresses at – .053 degrees per day, a value known from Babylonian times. See e.g., Neugebauer, O. 1975. *A History of Ancient Mathematical Astronomy.* 3 vols. Studies in the History of Mathematics and Physical Sciences 1, 312. Berlin: Springer. The sun advances at about 0.98562 degrees per day. Their daily separation is 1.03862 degrees per day. The node thus completes one revolution with respect to the sun in 346.6143 days.

[9] For a listing of all the lunar Saros cycles and a host of related information see https://eclipse.gsfc.nasa.gov/LEsaros/LEsaroscat.html. My data for the gamma progression in lunar Saros 45 came from https://eclipse.gsfc.nasa.gov/LEsaros/LEsaros045.html. [10] Here is the chart from that post; gamma values for each eclipse are in the fifth column:

[11] Different Saros series have different numbers of eclipses, but they are all over a millennium in duration: “A series may last 1,226 to 1,587 years and is composed of 69 to 89 eclipses.” https://eclipse.gsfc.nasa.gov/LEsaros/LEperiodicity.html. [12] The draconic period is time taken for one revolution of the node, moving 360 degrees in 27.212221 days or 13.229350 degrees per day. Thus 0.036135 days’ nodal shift in one Saros cycle means a shift of (13.229350 x 0.036135 =) 0.4780425677 degrees of longitudinal nodal shift in one Saros cycle.

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