The geometry of a shadow in space

In Discover you met the idea: an eclipse is a shadow falling somewhere that somebody can see it. That is true, and it is enough to tell solar from lunar and to watch one safely.

But it leaves a pile of sharper questions unanswered.

• Why does a shadow have a dark middle and a grey edge at all?
• The Moon's shadow cone is about 374,000 km long and the Moon is about 384,400 km away. Those numbers do not fit. So how do we ever get a total eclipse?
• Why is the path of totality so absurdly narrow — 160 km on a planet 12,742 km across?
• Why is the eclipsed Moon red rather than simply dark, and why is it never the same red twice?
• How tilted is "tilted", exactly, and how tilted would be too tilted?

This layer answers all of them with numbers you can check. Nothing here needs more than multiplication, division and the idea of similar triangles.

If the Sun were a single glowing point, every shadow in the world would have a knife-sharp edge. Light would either reach a place or it would not, and there would be nothing in between.

The Sun is not a point. It is a disc about half a degree wide. That single fact is responsible for the whole structure of an eclipse.

Stand somewhere behind a blocking object and ask: how much of the Sun's disc can I see from here?

• None of it. You are in the umbra. The blocker covers the whole disc. This is total darkness as far as that source is concerned.
• Some of it. You are in the penumbra. Part of the disc peeps past one edge of the blocker. There is real light here, just less of it.
• All of it. You are outside the shadow entirely.

That is the entire definition, and it works for a hand in front of a lamp, for the Moon in front of the Sun, and for Earth in front of the Sun.

A solar eclipse needs three things to be true at the same moment. Miss any one and nothing happens.

1. It must be new moon. The Moon has to be on the Sun's side of Earth. This is not a coincidence you hope for; it happens every 29.53 days like clockwork.

2. The Moon must be near a node. Its tilted orbit must be crossing the plane of Earth's orbit right now, so that "between Earth and the Sun" really means in front of the Sun and not 34,000 km above it. This is the condition that fails most months.

3. You must be standing where the shadow lands. Even when conditions 1 and 2 are met, the umbra's footprint is a spot roughly 160 km across on a planet 12,742 km across. Almost everybody is outside it.

The third condition is the one people forget. A total solar eclipse is happening somewhere on Earth about every 18 months. A total solar eclipse is happening over your house about once in 375 years.

Everything about which kind of solar eclipse you get comes down to a single comparison: is the Moon's disc bigger or smaller than the Sun's disc, as seen from where you stand?

Both discs change size, because both orbits are slightly oval.

The Sun's apparent width ranges from 31.5 arcminutes (early July, when Earth is furthest out) to 32.5 arcminutes (early January, when Earth is closest). An arcminute is one sixtieth of a degree.

The Moon's apparent width ranges much more: from 29.5 arcminutes at apogee to 32.9 arcminutes at perigee.

Those two ranges overlap. So:

• Moon bigger than the Sun → the disc is fully covered → total.
• Moon smaller than the Sun → a ring of Sun is left → annular.
• Only the penumbra reaches you → partial, whatever the distances.

And there is a fourth, rare case. Earth is round, so someone standing directly beneath the shadow is about 6,371 km closer to the Moon than someone near the edge of the illuminated face. If the cone's tip falls in between, the same eclipse is total in the middle of its track and annular at both ends. That is a hybrid eclipse, and only a few per century happen.

Turn around and look at the other shadow.

Earth is 3.67 times wider than the Moon, so its shadow cone is far longer and far fatter. Its umbra runs 1,381,300 km into space — about 3.6 times the Moon's distance from us. So where the Moon crosses it, the cone is nowhere near tapering out. It is still enormous.

How enormous? Use the taper formula. At 384,400 km along a 1,381,300 km cone, the umbra's radius has shrunk from Earth's own 6,371 km to:

6,371 × (1 − 384,400 ÷ 1,381,300) = 6,371 × 0.7217 = 4,598 km

That is a circle 9,196 km across — about 2.6 Moon-widths. The Moon, only 3,475 km wide, drops into it with room on either side.

Around that sits the penumbra, about 4.7 Moon-widths across.

This single fact explains everything that is different about lunar eclipses: they last hours instead of minutes, they can be total for a long stretch, the Moon can even pass through slightly off-centre and still be entirely swallowed — and because it is the Moon that changes rather than a small patch of Earth, everyone on the night side sees the same thing at the same instant.

If Earth had no atmosphere, a totally eclipsed Moon would simply go out — a black gap among the stars. It does not, and the reason is a two-step process. Both steps happen every day, right over your head.

Step 1: refraction bends sunlight into the shadow. Air is denser at the bottom than the top, and light slows down slightly in denser air, so a ray grazing Earth's edge is bent inwards — by up to about half a degree. That bending is enough to curl sunlight around the rim of Earth and pour it into the shadow behind. So the umbra is not truly empty; it is filled with a dim, bent light.

Step 2: scattering strips out the blue. That light has travelled a very long, slanting path through the whole depth of the atmosphere. Along the way, the air molecules scatter short-wavelength light — blue and violet — sideways, far more strongly than long-wavelength red. This is the same effect that makes our sky blue and our sunsets red. By the time the beam emerges on the far side, most of the blue has been thrown away and what is left is red and orange.

Put those together and you get the real explanation:

The red light landing on an eclipsed Moon is the light of every sunrise and every sunset happening on Earth at that moment, gathered into one beam.

If you were standing on the Moon during a lunar eclipse and looked back, you would not see a bright Earth. You would see a black disc, four times wider than the Sun looks to us, ringed by a thin band of blazing orange — the whole world's terminator, all at once.

"How big does it look?" is a different question from "how big is it?", and astronomy runs on the first one.

The answer is an angle: the angle between the two edges of an object as seen from your eye. It is called the angular size or apparent size, and it depends on both the real size and the distance.

Useful units:

• 1 degree (°) — about the width of your little finger held at arm's length.
• 1 arcminute (′) — one sixtieth of a degree.
• 1 arcsecond (″) — one sixtieth of an arcminute, one 3,600th of a degree.

Now the numbers that make eclipses possible. The Sun's average angular width is 0.533°, or 32.0′. The Moon's is 0.518°, or 31.1′.

They differ by less than one part in thirty. Out of everything in the sky, the two brightest objects happen to be almost exactly the same apparent size — and there is no physical reason at all. The Moon is not tuned to the Sun; it is simply where it is.

Now the central question: why not every month?

The Moon's orbit is tilted by 5.145° to the plane of Earth's orbit around the Sun (the ecliptic). Compare that with what it is trying to hit — a Sun only 0.53° wide. The tilt is nearly ten times the target's own width.

Put it in kilometres and it is starker. At the Moon's distance, a 5.145° offset is:

384,400 × tan(5.145°) = 34,600 km

That is 2.7 Earth-diameters. At most new moons the Moon is tens of thousands of kilometres above or below the Sun–Earth line, and the shadow cone flies past Earth into empty space.

But a tilted circle must cut a flat plane in exactly two points, on opposite sides. Those are the nodes: the ascending node, where the Moon crosses from south to north, and the descending node, where it crosses the other way. Near a node, and only near a node, the Moon is back in the plane where eclipses can happen.

In classical Indian astronomy those two points are called Rahu and Ketu. They are treated as the two halves of a severed body that chases and swallows the Sun and Moon — which is a rather good description of what geometry says actually goes on there.

So an eclipse needs the Sun to be near a node, and the Moon to arrive there at the right phase on the same trip.

Here is the crucial piece of arithmetic. The Sun's window around a node is about 33.3° wide, and the Sun creeps along the ecliptic at 0.986° a day. But the nodes themselves are not fixed — they slide backwards around the ecliptic at 19.34° a year, or 0.053° a day, completing one full circuit in 18.6 years. (This slow wobble is caused by the Sun's pull on the tilted orbit, and it is the same physics that makes a spinning top's axis circle around.)

Because the Sun and the nodes move in opposite directions, they close on each other at 0.986 + 0.053 = 1.039° a day. So:

• Eclipse season length = 33.3 ÷ 1.039 = about 32 days.
• That is longer than one synodic month (29.53 days), so at least one new moon must fall inside every eclipse season. There is always at least one solar eclipse per season, and so at least two a year.
• Time from one node to the next = 180 ÷ 1.039 = 173.3 days, so seasons come about every 5.7 months, not every 6.
• One full circuit — an eclipse year — is 360 ÷ 1.039 = 346.6 days, which is 18.6 days shorter than an ordinary year. That is why eclipse seasons drift steadily earlier through the calendar.

Because a season is only just longer than a month, most seasons contain two eclipses (one solar, one lunar) and some squeeze in three. Add them up and the total across a year comes to between 4 and 7, of which 2 to 5 are solar.

Every rule below has a reason. Knowing the reasons is what makes you keep the rules under pressure, when the sky is doing something amazing and there is a strong temptation to take one quick look.

Reason 1: the retina has no pain nerves. The back of your eye can detect light beautifully and cannot feel anything at all. Sunlight focused on it cooks the light-sensing cells silently. You get no warning, no flinch, no ouch. The symptoms — a blurred or blank patch in the middle of your vision — usually turn up hours later, often the next morning. This condition is called solar retinopathy, and it can be permanent.

Reason 2: your pupil opens as the eclipse deepens. As more of the Sun is covered, the world dims, and your pupils widen to let more light in. The Sun's remaining sliver has lost none of its brightness per square millimetre — but now a wider pupil is aiming more of it at your retina. A deep partial eclipse is genuinely more dangerous than an ordinary sunny day, not less.

Reason 3: the eye is a lens. Your eye's whole job is to gather light over the area of your pupil and focus it to a tiny point on the retina. That is a concentration of hundreds of times. Put a camera lens, binoculars or a telescope in front of it and the concentration multiplies again, by thousands. This is why the telescope rule is absolute.

Reason 4: filters are not about comfort. The Sun also emits infrared and ultraviolet that you cannot see. A dark piece of glass or plastic that makes the Sun look comfortable may pass plenty of both. Comfort is not the test. Certification is.