The eclipse that repeats a third of a world away

An eclipse is a coincidence of three things, and each of them has its own period.

The Moon must be new, which happens every 29.530589 days — the synodic month. It must be at a node of its orbit, where its tilted path crosses the ecliptic, which happens every 27.212221 days — the draconic month. And if the kind of eclipse is to repeat, it must be at the same distance, which happens every 27.554550 days — the anomalistic month.

Three periods, none of them a whole number of the others, none of them commensurate with any of them. And yet eclipses repeat, on a cycle of 18 years and 11 days that was known in Babylon by the seventh century BC and has been used ever since.

The reason is arithmetic, and it is very nearly a coincidence:

$$223 \times 29.530589 = 6585.3213\ \text{days},$$

$$242 \times 27.212221 = 6585.3575\ \text{days},$$

$$239 \times 27.554550 = 6585.5375\ \text{days}.$$

Three different whole numbers of three different periods, landing within a fifth of a day of each other. That is the saros.

Why three periods and not one #

The Moon’s orbit is tilted 5.15° to the ecliptic, which is why there is not an eclipse every month. At most new moons the Moon passes above or below the Sun; only when a new moon happens near a node does the alignment close.

That gives the first two periods. The synodic month sets the phase; the draconic month sets the node crossing. Their beat produces the eclipse season: a window about 34 days long, twice a year, in which the Sun is close enough to a node for an eclipse to be possible. Two eclipse seasons per eclipse year of 346.62 days — which is shorter than a calendar year because the Moon’s nodes regress westward, completing a circuit in 18.6 years — the same nodal regression that wobbles the celestial pole.

The third period is more subtle and it is what separates a real prediction from a rough one. The Moon’s distance varies by 13% over the anomalistic month, and with it the angular size of the Moon’s disc: 33.5 arcminutes at perigee, 29.4 at apogee, against the Sun’s 31.6 to 32.7. So whether a central solar eclipse is total — the Moon’s disc covering the Sun’s — or annular, with a ring of photosphere left visible, is decided entirely by where the Moon is in its anomalistic cycle. A repeat that gets the phase and the node right but the distance wrong produces an eclipse at the right time and place and of the wrong kind. Including the anomalistic month in the cycle is what makes successive eclipses of a saros series similar, and that similarity is the property the cycle is valued for.

The leftover third of a day #

The saros is 6,585.32 days. It is not 6,585 days, and the 0.32 is the most visible feature of the whole cycle.

After one saros the Sun, Moon and node have returned to the same configuration — but the Earth has turned an extra 0.32 of a rotation, which is 115.7°. The eclipse therefore happens over a point about a third of the way round the world to the west, at roughly the same local time of day and at a similar latitude.

Three saroses — 54 years and about 33 days, called an exeligmos, “the turning of the wheel” — bring the total back to within a few hours of a whole number of days, and the eclipse returns to nearly the same longitude. That is why the Greeks knew about the triple cycle at all: over 18 years an eclipse recurred somewhere unhelpful, and over 54 it came back.

A series has a beginning and an end #

The saros is not exact, and the residual is what gives each series a lifetime.

Because 223 synodic months exceed 242 draconic months by 0.0362 days, the Moon is not at exactly the same position relative to the node at each repeat: it is displaced by about 0.48° along its orbit, always in the same direction. Successive eclipses of a series therefore march steadily across the node.

The consequence is a life cycle. A saros series begins with a tiny partial eclipse visible only near one of the Earth’s poles, as the Moon’s shadow just clips the planet. Successive members march toward the equator, becoming deeper partials, then annular or total central eclipses, then partials again at the other pole, and finally miss entirely. A series runs about 70 to 80 eclipses over 1,200 to 1,500 years.

Saros series 145, to take the best-known one, began on 4 January 1639 with a partial eclipse near the North Pole. Its first total eclipse was in 1927. The total eclipse of 11 August 1999 across Europe was its 21st member; the eclipse of 21 August 2017 across the United States was its 22nd; 2 August 2027 will be its 23rd, and the series will end on 17 April 3009. The 2017 eclipse and the 1999 one were the same eclipse, one saros apart, displaced by 116° of longitude — which is very nearly the difference between Cornwall and Oregon.

At any time about 40 saros series are running concurrently, which is why there are two to five solar eclipses in a year rather than one every eighteen.

What one saros does to the shadow track #

The 116° of longitude is the visible part of the displacement. There are two more, smaller, and together they are what makes a saros series march.

Latitude. The 0.48° drift along the Moon’s orbit at each repeat moves the shadow track steadily in latitude, by roughly 300 km per saros for a series near the middle of its life. That is the mechanism by which a series migrates from pole to pole over 1,300 years, and it is why two eclipses one saros apart are similar but not identical: the 1999 track crossed Cornwall, Munich and Romania; the 2017 track crossed Oregon, Missouri and South Carolina — the same eclipse, three hours later in the day and eight degrees further south.

Duration. The anomalistic residual is the largest of the three, 0.22 days, so the Moon is not quite at the same distance either. Its angular diameter drifts slowly through a series, and with it the duration of totality. Saros 145’s totality has been lengthening: 2 minutes 10 seconds in 1927, 2 minutes 23 in 1999, 2 minutes 40 in 2017, and it will peak at 7 minutes 12 seconds in 2522 — one of the longest total eclipses of the millennium, and a member of the same series as the one people flew to Wyoming for.

There is a fourth displacement, and it is the one that gives the series its life cycle. Because the eclipse conditions are met a little further from the node each time, the alignment is progressively less perfect, so the shadow passes progressively further from the Earth’s centre. A series that begins with the shadow cone missing the Earth entirely except at one pole ends with it missing at the other, and the total eclipses are the middle third.

What was actually measured #

The saros was found from records, not from theory, and the distinction matters because it is a case of a periodicity being extracted from a list.

Babylonian astronomers kept eclipse observations on tablets from at least the eighth century BC, and by the time of the Neo-Babylonian period they were arranging them in an 18-year scheme. The surviving “Saros Canon” tablets list eclipse possibilities in a grid of 38 columns and 24 rows — 38 eclipse possibilities per 18-year cycle, arranged so that each row is one saros. The scheme predicts when an eclipse can happen; it does not predict whether it will be visible from Babylon, and the tablets are careful to distinguish observed eclipses from computed possibilities.

What they had was a table of dates. What they extracted was the observation that the interval 223 months recurred, and the striking thing is that this is derivable from lunar eclipse records alone — lunar eclipses being visible from an entire hemisphere, so a single site accumulates a usable series in a few decades, wherever on the sphere it stands. Solar eclipses, visible along a track a hundred kilometres wide, would take centuries at one location to reveal the same pattern.

The three month-lengths themselves were measured with remarkable precision by the same tradition. The Babylonian value for the synodic month, preserved in the System B lunar theory and quoted by Ptolemy as Hipparchus’s, is 29 days 12 hours 44 minutes 3⅓ seconds — 29.530594 days, against the modern 29.530589. The error is half a second, or two parts in ten million, obtained from eclipse intervals spanning several centuries: a long baseline divided by a large integer, which is the same trick used to measure any period to a precision far beyond a single observation, and the same one that fixes a planet’s semi-major axis from its period.

The modern check runs the other way. Eclipse records are now used to measure something else: the slowing of the Earth’s rotation. A Babylonian eclipse of 15 April 136 BC, recorded as total at Babylon, would have been total in the western Mediterranean had the Earth kept a constant rotation rate — a discrepancy of about 48° of longitude, or 3.2 hours accumulated over 2,100 years. That accumulated error, $\Delta T$, is measured from ancient eclipse records and from nothing else at that baseline, and it gives a mean lengthening of the day of about 1.8 milliseconds per century. An eclipse record, in other words, is a timestamp of extraordinary precision attached to a place. It fixes where the Earth’s surface was pointing at a known instant, which is what a clock in the sky is for, and 2,000 years of them measure a change in the length of the day of two milliseconds.

Following one series far enough shows what the third-of-a-day residual does over the long run.

Fifty-four years and about a month is long enough that the exeligmos is of no practical use to an observer and was of considerable use to an astronomer with records. A single lifetime covers one return; a set of records covering three centuries covers six, which is enough to establish that the pattern is real and to measure the drift in latitude that the draconic residual produces. The Babylonian tablets that record eclipse observations span rather more than that, which is why the cycle was known there and not derived from a theory of the Moon’s motion.

The drift in latitude is the reason a series is finite. Each eclipse of a series occurs slightly further from the node than the last, so after some twelve or thirteen centuries the alignment fails entirely and the series ends — having begun, twelve centuries earlier, as a partial eclipse near one pole and worked its way across. That progression is visible in the longitudes above only as a slow poleward creep, and it is the single most useful thing the saros predicts that a simpler cycle would not. A cycle built on the synodic month alone repeats the phase and says nothing about whether the Moon is near a node; one built on the synodic and draconic months together repeats the eclipse and says nothing about whether it will be total; the saros does all three, and the price of doing all three is that its residual in each is non-zero.

The generalisation: near-commensurability is the useful case #

The saros is an instance of something that runs through the whole subject: three incommensurable periods that nearly share a multiple, and the near-ness is what makes them useful.

Exact commensurability would be worse. If the synodic and draconic months were in exact ratio, eclipses would repeat forever with no drift, which sounds ideal until one notices that the same condition would have locked the system into a resonance and probably driven the eccentricity somewhere unhelpful. Exact incommensurability would be worse still: no repeat at all, and no prediction without a full dynamical theory.

The same structure appears everywhere periods are compared. The Metonic cycle — 235 synodic months equal 19 tropical years to within two hours — is the arithmetic behind every lunisolar calendar, including the date of Easter. The 8-year cycle in which five Venus synodic periods nearly equal eight Earth years, to within two days, produced the Venus tables of the Dresden Codex and the pentagram Venus traces on the sky. Jupiter and Saturn’s near-5:2 commensurability produces the “great inequality” that took Laplace to explain. The Galilean moons’ 1:2:4 Laplace resonance is an exact one, held there by tides, and is the exception that shows what the difference means: exact resonance is dynamically maintained, near-commensurability is a coincidence of initial conditions. The general rule is that a near-commensurability of $p$ and $q$ periods produces a beat of period $|pP_1 - qP_2|^{-1}$, and the smaller the residual the longer the cycle stays good and the rarer such a cycle is. The saros residual is 0.036 days against 6,585 — five parts in a million — and no shorter cycle does nearly as well.

Where the model stops #

The three periods are not constant. All three are slowly changing: the synodic month is lengthening as tidal friction pushes the Moon outward at 3.8 cm/year. The saros of ten thousand years ago was not the same length, and the arithmetic above is an arithmetic for the present epoch.

The saros predicts possibility, not visibility. Knowing that an eclipse occurs on a date says nothing about whether it is visible from a given place until the geometry is computed properly, and it is the longitude displacement that makes this so.

Nothing here is a dynamical calculation. The three periods are inputs, taken from observation. Deriving them requires lunar theory, which is famously the hardest problem in classical celestial mechanics and which occupied Newton, Euler, Clairaut, Laplace, Delaunay and Hill in turn.

The figures cannot show what they are about. An eclipse is a coincidence in time, and every picture here is either a plot of a residual — which shows the arithmetic and not the sky — or a schematic of a geometry that is drawn at a size and a scale nothing like the real one. The Earth–Moon–Sun system at true scale is three specks separated by a page and a half of nothing, and the umbra whose length decides everything is a cone 373,000 km long and 100 km wide at its base.

The other cycle, which does the opposite #

The saros is not the only near-commensurability in the same three periods, and the second one is worth knowing because the two together generate the whole catalogue.

Three hundred and fifty-eight synodic months come to 10,571.95 days — twenty-nine years less about twenty days — and 388.5 draconic months come to 10,571.95 as well. The half is the point: after that interval the Moon is at the opposite node.

So an eclipse repeats after twenty-nine years, at the other node, at a longitude displaced by about a third of a turn in the other direction. That cycle is called the inex, and its residual is much smaller than the saros’s — about eight hundredths of a day against the saros’s third of one — so it holds for far longer.

The two behave in opposite ways and the contrast is instructive. A saros series drifts steadily across the node and lasts twelve or thirteen centuries. An inex series barely drifts at all, because its residual is tiny, and lasts tens of thousands of years — but successive members are so far apart in the Moon’s anomaly that they are not similar to each other, so the cycle predicts that an eclipse happens and says nothing about what kind.

Combining them is what modern eclipse catalogues do. Every eclipse can be labelled by which saros series and which inex series it belongs to, and the two indices together place it on a grid — one axis stepping by eighteen years and eleven days, the other by twenty-nine years less twenty. Every solar eclipse in history and for the next several millennia sits at one point of that grid, and the grid’s structure is entirely the two near-commensurabilities.

Two cycles with opposite virtues — one that preserves the character of an eclipse and drifts, one that preserves the node and does not — are between them a complete indexing of the phenomenon, and neither is a dynamical result. Both are properties of three numbers.

Extending the beat far past the saros shows both why 223 is the answer and why it is not the only one.

The reason 223 wins is that both residuals happen to be small there at once. The draconic condition alone is satisfied at many values of n and the anomalistic condition alone at many others; what makes a usable cycle is a value where the two coincide, and coincidences of that kind are rare and are not predictable from either period alone. The saros is the best such coincidence under about fifty years, which is why it was the one found by people watching the sky rather than computing it.

Longer cycles exist and are better. The inex, at 358 synodic months, has a much smaller draconic residual and a much larger anomalistic one, so it preserves the eclipse’s type poorly and its position in the node well; combining saros and inex generates the whole family of series in a two-dimensional lattice, which is how modern eclipse catalogues are organised. That construction is entirely a piece of arithmetic about three real numbers, and it predicts every solar eclipse for the next several thousand years.

The ladder from here #

Later rungs on this anchor: the eclipse season, and the 34-day window the node’s regression opens. Total against annular, decided by two angular diameters that nearly match. The umbral track computed on a rotating oblate Earth. Saros series 145 followed from 1639 to 3009. Lunar eclipses, and why their saros behaves differently. $\Delta T$ from ancient records, and the secular acceleration of the Moon. The Metonic cycle and the calendars built on it.

The word “saros” is a mistake. It is a Babylonian term, šār, meaning 3,600 — a large round number with no connection to eclipses — which Edmond Halley picked out of a garbled entry in the Byzantine lexicon Suda in 1691 and applied to the 18-year cycle. The name has been wrong for three hundred and thirty years and is not going to change now.