The Geometry of Shadows
Eclipses happen when Sun, Earth, and Moon fall into one straight line — and the Moon's tilted orbit makes that alignment rare. · 12 min
Every month has a new Moon and a full Moon. If shadows were the whole story, every month should deliver a solar eclipse and a lunar one, a fortnight apart, forever. Yet whole years can pass without an eclipse visible from where you live. This folio is about two cones of shadow — and the small tilt that keeps them from striking home each month.
Guess before you learn
Why isn't there a lunar eclipse at every full Moon?
The Moon's orbit is tipped about five degrees to Earth's orbit around the Sun. Five degrees sounds small, but Earth's shadow at the Moon's distance is only about a degree and a half wide — most full Moons sail clear over or under it. If you guessed the shadow falls short: a reasonable thought, but Earth's shadow cone actually reaches nearly four times the Moon's distance. The tilt does the deciding.
9–12
3–5
There are two kinds of eclipse, one for each shadow. When Earth stands exactly between Sun and Moon, Earth's shadow darkens the full Moon: a lunar eclipse, visible to everyone on the night side of Earth. When the Moon stands exactly between Sun and Earth, the Moon's much smaller shadow touches Earth: a solar eclipse, visible only inside one small moving spot.
The Moon's path around Earth is tilted a little compared with Earth's path around the Sun. So at most new and full Moons, the three bodies almost line up — and almost is a miss.
6–8
A shadow has two parts. The umbra is the cone of full shadow, inside which the Sun is entirely hidden; the penumbra is the surrounding partial shadow. A total lunar eclipse is the full Moon crossing Earth's umbra. A total solar eclipse is the tip of the Moon's umbra sweeping a narrow track across Earth at new Moon.
The Moon's orbit is tilted 5 degrees to the ecliptic. The two points where the orbit crosses the ecliptic plane are the nodes. An eclipse needs a new or full Moon while the Moon sits near a node — which happens during two eclipse seasons each year, each about a month long.
9–12
Scale explains the two eclipses' different manners. Earth's umbra at the Moon's distance is about 1.4 degrees wide — nearly three Moon diameters — so lunar eclipses are leisurely, running up to a few hours, and the entire night hemisphere watches the same event. The Moon's umbra barely reaches Earth: a spot at most about 270 kilometers wide, racing eastward faster than sound. Totality at any one town lasts minutes.
When the Moon is near apogee, its umbra falls short of Earth's surface entirely, and a perfect alignment yields an annular eclipse: a ring of Sun around the Moon's silhouette. That totals and annulars both occur is a coincidence of our era — the Moon's disk and the Sun's are each about half a degree across.
K–2
Stand a lamp, a ball, and a marble in a straight line. The ball's shadow falls on the marble. That is a lunar eclipse: Earth's shadow falling on the Moon.
Most months the line is not quite straight. The Moon passes a little too high or a little too low, and the shadow misses. When the line is perfect, the full Moon slowly turns red.
Undergrad
The node line is not fixed: solar torque on the Moon's orbit regresses it westward with an 18.6-year period, so the eclipse year — successive solar passages of the same node — is 346.6 days rather than 365.25. Beating the synodic month against the eclipse year yields near-commensurabilities; the cleanest is 223 synodic months, about 6585.3 days: the saros.
One saros after any eclipse, a near-twin recurs — same node, same season, similar lunar distance — displaced a third of the globe westward by the leftover 8 hours. Eclipses therefore arrive in saros series, each running for a dozen centuries from partial births to partial deaths.
Postgrad
Prediction reduces to Besselian elements: project the shadow axis onto the fundamental plane through Earth's center, tabulate the intersection point, the cone radii, and their time derivatives, then recover local circumstances by rotation to the observer. Run the machinery backward and historical eclipse timings constrain ΔT — the accumulated divergence of terrestrial time from Earth's slowing rotation.
The arrangement is temporary. Lunar laser ranging measures a 3.8-centimeter-per-year recession; the Moon's mean angular diameter shrinks accordingly, and in roughly 600 million years the umbral tip will no longer reach Earth's surface at all — annulars only, the era of total solar eclipses closed.
umbra
The cone of complete shadow behind a lit body, inside which the Sun is wholly hidden. Around it lies the penumbra, where the Sun is only partly covered.
During totality the Moon does not vanish. It turns a deep copper red. Earth blocks the direct sunlight, but our atmosphere bends some light inward around the planet's edge, and air scatters blue light away far more readily than red — the same filtering that reddens a low Sun. What reaches the eclipsed Moon is sunlight that has skimmed through Earth's ring of atmosphere, from every place on Earth where the day is just beginning or just ending. The Moon reflects that filtered light back to you.
Now flip the geometry. At new Moon, the Moon's own umbra points at Earth — and just barely reaches it. Where the tip lands, day turns briefly to night: a total solar eclipse, confined to a track a few hundred kilometers wide racing across the globe. Outside the track, observers see only a partial bite taken from the Sun. And when the Moon is at the far end of its slightly elliptical orbit, the umbra runs out before touching ground: the Moon then covers all but a rim of the Sun, leaving a brilliant ring. That is an annular eclipse.
Those two crossing points are the nodes, and an eclipse needs two coincidences at once: the Moon at a node, and the Moon new or full — which requires the Sun to lie along the node line too. The node line holds nearly still in space while Earth circles the Sun, so the Sun lines up with it only about twice a year. Each alignment opens an eclipse season roughly 35 days long, in which any new Moon casts a solar eclipse somewhere on Earth and any full Moon slides into shadow. Two seasons, most years: not twenty-four eclipses annually, but four to seven.
Why is this true?
Why is a total solar eclipse rare at any given town, while a lunar eclipse is seen by half the planet at once?
Earth's umbra at the Moon's distance is nearly three Moon-widths across, so the eclipsed Moon sits in shadow for hours and everyone facing it sees the same event. The Moon's umbra tip on Earth is at most a few hundred kilometers wide; you must stand inside that racing spot, and for any one spot it returns only every few centuries on average.
Shadow geometry, then: two cones, one small tilt, two crossings, two seasons a year. When the next eclipse reaches your sky, you will know exactly which alignment produced it — and why the full Moon before it passed by untouched. Next folio we leave the Moon behind and go hunting the five bright wanderers that share its lane.
Practice — new ink and old, interleaved
1.A friend plans to watch a total lunar eclipse and asks whether it is safe to look at. In one sentence, what do you tell them, and why?
2.The terminator you observe on a waxing Moon is —
3.The Moon rises at sunset tonight, so it is full. Roughly how many days until third quarter?
4.Earth's umbra at the Moon's distance is about 1.5 degrees wide; the Moon's disk is about 0.5 degrees. Roughly how many Moon-widths fit across the shadow?
5.Roughly how many times brighter is the full Moon than the first-quarter Moon?
6.Match each term to its meaning.
7.Without looking back: why does the eclipsed Moon turn red?
Earth's atmosphere refracts sunlight into the shadow and scatters the blue away; the surviving red light falls on the Moon and reflects back to us.
How close were you? Grade yourself honestly — it sets your review date.
8.A solar eclipse is forecast for the 14th. What phase must the Moon be that day?
9.Over time, about what percentage of the Moon's total surface can be seen from Earth?