Eclipse seasons

Why is there not a solar eclipse at every new moon, when the Moon passes between us and the Sun each month?

Twelve or thirteen times a year the Moon arrives at new — which is to say, on the Sun's side of us. If it is between us and the Sun, why is the Sun not blocked? A solar eclipse ought to be a monthly event, and instead we get a handful a year, clustered oddly, on dates that wander.

So the naive picture makes a prediction and the sky refuses it. That refusal is useful: it tells us which assumption was wrong.

The assumption is that "new moon" means aligned. It does not. It means the Moon has the same direction as the Sun measured around the sky — the same longitude — and says nothing about whether it is on the same line.

The Moon's orbit is tilted to the plane of Earth's orbit by about 5.1 degrees (a recalled figure; the usual value is 5 degrees 9 minutes, and it wobbles a little). Two tilted circles sharing a centre meet at exactly two points, on opposite sides. Those are the nodes, and the line joining them is the line of nodes. Away from them the Moon simply passes new moon high or low, and misses.

How near a miss still counts? Work it from the sizes. The Sun's disc is about half a degree across, and so is the Moon's, so from Earth's centre the two centres would have to come within roughly half a degree to touch. But we watch from the surface, and Earth is not small at the Moon's distance: Earth's radius seen from the Moon subtends about 0.95 degrees — the Moon's horizontal parallax, near 57 arcminutes, again recalled. Somewhere on the near face of Earth, then, the shadow can graze even when the alignment is off by roughly 0.95 + 0.27 + 0.26, call it 1.5 degrees.

Now convert that tolerance into a distance along the orbit. The Moon's height above the Sun's path, when it is a small angle from a node, is about 5.1 degrees times the sine of that angle. Set 5.1 sin(x) = 1.5 and you get sin(x) = 0.29, so x is about 17 degrees. The Moon must be within roughly 17 degrees of a node — not the fraction of a degree the naive picture implies, but a generous window. (Tables give a range near 15 to 18.5 degrees, because the distances of both bodies vary; 17 is the middle of it.)

Here is where the arithmetic becomes decisive. The Sun creeps along its path at about 0.986 degrees a day. The nodes are not fixed — solar tugging drags them backwards at about 19.3 degrees a year, or 0.053 degrees a day, into the oncoming Sun. So the Sun closes on a node at 1.039 degrees a day, and it spends 34 divided by 1.039, near enough 33 days, inside the 17-degree window. That window is the eclipse season.

Compare 33 days with the 29.53 days between new moons. The window is longer than the month. So a new moon cannot miss it: at least one solar eclipse must happen in every eclipse season, and occasionally two fall inside. With two nodes there are two seasons a year, which sets a floor of two solar eclipses annually and, when the edges catch extras, as many as five.

The same closing rate gives the period: 360 divided by 1.039 is about 347 days, the eclipse year. That is 18 or 19 days short of the calendar year, which is why eclipse seasons creep steadily earlier through the calendar and repeat their place only after 360/19.3, about 18.6 years.

One folk account dies here: that eclipses are rare because the alignment is delicate. Solar eclipses are common — several a year, somewhere. What is rare is standing under the umbra, a track only a couple of hundred kilometres wide, so any one town waits centuries.

What would falsify all this? The account predicts eclipses confined to two windows about 173 days apart, drifting 19 days earlier each year. Find solar eclipses scattered evenly across the calendar, or two seasons six months apart holding station year after year, and the nodal mechanism is dead.

Two honest qualifications. First, I treated the lunar orbit as a rigid tilted plane that merely rotates. It is not: the same solar pull that regresses the nodes also makes the inclination breathe by a few arcminutes, and the eccentricities of both orbits move the 1.5-degree tolerance around. So the season has fuzzy edges, and near them whether an eclipse occurs depends on where on Earth you happen to be standing.

Second, nobody predicts real eclipses this way. Serious prediction integrates the motions numerically. What the arithmetic above buys is not a forecast but an explanation — and one testable consequence: because 223 synodic months (about 6585.3 days) come to very nearly 242 returns to a node, eclipses repeat in a near-identical family after that interval, the Saros.

• Why some solar eclipses are total and others annular, and what that says about the Moon's changing distance.
• How the Saros lets a family of eclipses be traced for a thousand years, and why each one shifts a third of the way around the globe.

Every term the collection defines is gathered in the glossary.