The shadow that climbs the zenith

The three definitions of twilight are angles — six, twelve and eighteen degrees of solar depression — and what they stand for is a brightness that the angles do not compute. The sky after sunset is lit by sunlight scattered in the air overhead, and how bright it is at a given depression is a question about where that sunlight can still reach. The angle is the proxy; the scattering is the thing.

The scattering can be computed from one piece of geometry and one assumption, and the result is instructive twice over. The geometry predicts a sky that darkens faster and faster, lit by a thin layer of air that climbs as the Sun sinks. The assumption — that each photon is scattered exactly once — predicts that the sky is as dark as a moonless night by nine degrees of depression. The real sky is not dark until eighteen, and the gap between those two numbers turns out to be the most important thing the calculation measures.

The shadow over the zenith #

After sunset the Earth stands between the observer’s sky and the Sun, and it casts a shadow into its own atmosphere. Directly overhead that shadow has a floor that is easy to compute. A point at height $z$ above the observer is in sunlight if the straight line from it towards the Sun clears the Earth. With the Sun a depression $d$ below the horizon, that line passes closest to the Earth’s centre at a distance $(R+z)\cos d$, so the point is lit only if that exceeds $R$ — only above

$$z_s = \frac{R}{\cos d} - R \;\approx\; \frac{R\,d^2}{2}.$$

Everything in the zenith column below $z_s$ is in the shadow and contributes nothing. Everything above it is sunlit and scatters some of that light down.

The quadratic is the whole story in miniature. At three degrees of depression the shadow over the zenith is 8.7 kilometres up, below the height of the highest mountains’ summits; at six degrees, 35.1 kilometres, above nearly all the air; at twelve, 142 kilometres, where the atmosphere is a few billionths of its density at the ground. Doubling the depression quadruples the shadow’s height, and the air thins exponentially with height, so every degree of depression switches off more of the lit air than the degree before.

The Earth’s shadow is not what makes the Moon’s phases, but it is what makes twilight, and the dark blue band that rises from the eastern horizon after sunset — the shadow seen edge-on against the air — is the same surface, viewed from the side rather than from underneath. The pinkish band that sits just above it, the Belt of Venus, is air still in sunlight on the far side of the observer, lit by rays that have crossed so much atmosphere on the way that little but the red end of the spectrum is left to scatter back.

The same formula decides what else overhead is still sunlit, and two of its consequences can be seen without an instrument. Noctilucent clouds, the thin ice clouds of the upper atmosphere, float at around 83 kilometres, and directly overhead the shadow does not reach that height until the Sun is about nine degrees down — which is why they are seen glowing against a darkening sky, well into nautical twilight, long after every weather cloud below them has gone grey. A satellite is higher again. The shadow does not reach the International Space Station’s 420 kilometres over the zenith until the Sun is about twenty degrees down, past the end of astronomical twilight, so a pass overhead on a summer evening can be sunlit and brilliant against a sky that is already fully dark.

How the brightness is computed #

The number on the vertical axis is an absolute surface brightness, and getting it takes three ingredients and no adjustable constant.

The first is the air itself: its density at every height, taken from the standard atmosphere, and its scattering cross-section, which for molecules much smaller than the wavelength is Rayleigh’s, falling as the fourth power of the wavelength. The second is the path. Each lit point on the zenith line receives sunlight that has travelled past the Earth along a grazing line, and the attenuation of that sunlight is the optical depth integrated along the actual line — dense near its tangent point, thin at both ends. The light scattered from the point then travels straight down to the observer and is attenuated again by the air below. The third is the zero point. The ratio of the sky’s radiance to the Sun’s own is the scattering integral multiplied by the solid angle the Sun subtends, divided by the full sphere it scatters into, so the Sun’s surface brightness, −10.59 visual magnitudes per square arcsecond, turns the integral directly into a magnitude on the same scale as the night sky.

Adding the contributions of every lit point on the line gives the sky’s brightness, and the same sum, kept point by point, gives where that brightness comes from. Nothing in it is fitted to a twilight observation, which is what allows the result to be compared with one.

Why the dimming accelerates #

If the air thinned by a factor of $e$ every scale height $H$, and nothing attenuated the light on its way, the brightness of the zenith would be proportional to the air above the shadow, $e^{-z_s/H}$. Written in magnitudes that is $1.086\,z_s/H$, and since $z_s \approx Rd^2/2$ the magnitude grows as the square of the depression. Its slope — magnitudes per degree — therefore grows in proportion to the depression itself.

The computed curve shows exactly that. Between three and six degrees it falls 1.4 magnitudes a degree; between seven and ten, 2.7. A constant rate of dimming would be a straight line on the figure; single scattering draws a curve that bends ever more steeply down.

That shape is the first place the calculation can be held against the sky, and it does not match. The conventional description of the real twilight sky is a dimming by about a factor of a hundred for every six degrees — five magnitudes in six degrees, close to a straight line and much less steep than the single-scattering curve at the depressions where astronomers care. Something is keeping the real sky brighter as the Sun goes further down, and doing so more and more as the directly lit layer climbs out of the air.

Where the light comes from #

The same calculation can report not only the total but its source: how much each kilometre of the zenith column contributes.

The layer has a floor and a ceiling for two different reasons. Just above the shadow the air is densest, but the sunlight reaching it has skimmed the Earth through the thickest part of the atmosphere on the way, and most of it has been scattered out or absorbed before it arrives. Well above the shadow the light arrives almost unattenuated, but there is little air left to scatter it. Between the two the product peaks, about a dozen kilometres up — a little more than one scale height above the shadow, which is where a grazing ray’s optical depth has fallen to about one.

So twilight is not the whole sky slowly dimming. It is a thin, bright sheet of air riding a dozen kilometres above a rising shadow, and the brightness at any moment is the density of the air that sheet happens to occupy.

The sheet keeps climbing after civil twilight, and past a hundred kilometres the air it moves through changes character.

Above about a hundred kilometres the atmosphere is heated by the Sun’s ultraviolet, its temperature climbs to many hundreds of degrees, and its scale height stretches from eight kilometres to tens. The directly lit layer spreads out accordingly. That is the thermosphere, and it is the same thin gas whose density decides how fast a satellite’s orbit decays — and whose density a working model can get wrong by a factor of two when the Sun is active. The last single-scattered light of evening comes from the air that drags on satellites.

Nine degrees, not eighteen #

The dashed line on the first figure is the brightness of the natural night sky, 21.9 magnitudes per square arcsecond at a dark site — a floor made of airglow, zodiacal light and unresolved stars that no amount of waiting lowers. Twilight is over, for any practical purpose, when the scattered sunlight falls below it.

Single scattering puts that moment at 9.25 degrees of depression. The sky is observed to reach its night-time floor at about eighteen, which is why eighteen is the definition of astronomical twilight. The calculation is missing half of twilight.

The missing half is light scattered more than once. After the zenith column has dropped into the Earth’s shadow, the air towards the western horizon — along the line to the Sun, at much lower altitude — is still in sunlight and still bright. That bright glow is itself a light source, and some of its light is scattered a second time by the shadowed air overhead and sent down to the observer. It is weaker than the direct sunlight was, by roughly the fraction of light the first scattering sent in the right direction, but it does not care about the shadow’s height over the zenith, because it arrives sideways. As the directly lit sheet climbs into air too thin to matter, the secondarily lit air below it takes over, and the dimming slows to the near-straight line the real sky follows.

The eight or nine degrees between where single scattering ends and where twilight actually ends is a measurement of the atmosphere’s multiple scattering, made with nothing but a clock and a photometer. It is also why the curve’s shape is the wrong test of the model and its end point is the right one: a model that includes only once-scattered light cannot be made to last eighteen degrees by adjusting any density, because at nine degrees there is almost no air left above the shadow for it to use.

What the angles mean in minutes #

A depression angle becomes a time of evening through the latitude and the season, and the first essay on twilight computed that conversion directly.

At 52 degrees on the equinox the Sun reaches six degrees of depression 34 minutes after its apparent setting and twelve degrees 40 minutes after that, so single scattering’s nine and a quarter degrees arrive roughly fifty-five minutes after sunset. The sky is not dark for another hour. Everything after the first fifty-five minutes of an equinoctial evening at this latitude is multiply scattered light — the whole of nautical twilight past its midpoint and the whole of astronomical twilight.

What ozone does to the brightness #

The ozone layer, concentrated around twenty-odd kilometres, absorbs light across the middle of the visible spectrum in a broad, weak band. For a vertical ray it removes a few per cent. For a grazing ray the story is different, because the sunlight that lights the sheet has crossed the ozone layer near its tangent point, along a path dozens of times longer than a vertical one.

A magnitude at six degrees is a factor of two and a half, removed by a layer whose total amount would be three millimetres thick if it were compressed to the pressure at the ground. The grazing geometry multiplies its effect, and it does so in a way that depends on colour, because ozone’s absorption band sits in the orange and red.

The blue sky at six degrees is seven tenths of a magnitude brighter than the red. That is a colour, and it is not the colour Rayleigh scattering alone would produce: scattering favours blue by the fourth power of the frequency, but a long grazing path strips blue from the incoming sunlight by the same law, and the two very nearly cancel. What tips the balance is the ozone, and why the twilight zenith is blue at all is a question the colour of the sky answers better than its brightness.

A screen below the shadow #

Real air is not clean all the way down. Cloud, haze and dust in the lowest dozen kilometres can be opaque to a grazing ray, and the effect is to lift the shadow: a ray must now clear the top of the opaque layer rather than the ground.

A twelve-kilometre screen changes the sky at six degrees by two tenths of a magnitude and brings single scattering’s end forward by a quarter of a degree. That is much less than lifting the shadow by twelve kilometres would suggest, and the reason is in the profile figure: the rays that grazed below twelve kilometres were already so attenuated that the brightness hardly depended on them. The screen removes light that was mostly lost anyway. The same logic explains why the refraction that lifts the setting Sun matters so little to the twilight sky once the Sun is a few degrees down: the rays it bends are the ones that have been all but extinguished.

What the calculation leaves out #

Multiple scattering, as its central omission. Every number above is once-scattered light, which is why it ends twilight at nine degrees. The eighteen-degree twilight of the real sky is the calculation’s own measure of what it lacks.

Aerosols. Particles comparable in size to the wavelength scatter strongly forward, weakly sideways and with little dependence on colour. They brighten the sky near the Sun, redden sunsets, and add a component whose height distribution varies by the hour. The model is clean air.

The Sun’s size and the view direction. The Sun is a disc half a degree across, so the shadow’s edge is a penumbra rather than a line, and the model looks only straight up. The sky near the western horizon, where most twilight light is, is a different and more complicated calculation.

And extinction on the way down is only the vertical part. The scattered light reaching the observer from the zenith crosses the whole atmosphere once; an observer looking lower crosses more, and every surface brightness here would be fainter in any other direction.

What single scattering is good for #

The calculation is wrong about when twilight ends, and it is right about everything that makes it wrong. It says where the light comes from, at every depression, to within a kilometre or two: a sheet riding a dozen kilometres above a shadow that climbs as the square of the angle. It says why the directly lit sky must collapse by nine degrees, because by then the sheet is in the thermosphere. And by collapsing where the real sky does not, it isolates the one process — light scattered more than once — that carries the second half of every evening.

The same kind of calculation, applied to another piece of physics in the same geometry, answers a question the brightness curves raised in passing: why the evening sky overhead is blue when the path the light has taken should have made it anything but.

Still open: why the zenith stays blue #

The grazing ray that lights the twilight sheet crosses the ozone layer almost tangentially, and ozone absorbs orange and red. Without it, the two effects of Rayleigh scattering — blue scattered in, blue stripped out on the way — nearly cancel, and the sky overhead would be a pale, colourless glow. With it, the zenith is blue. Past that is the same geometry on worlds whose scattering layers are taller, thinner or made of dust, where the angle over which the sky darkens is set by the height of the scatterers against the size of the planet.