Ozone keeps the twilight zenith blue

The daytime sky is blue because air scatters short wavelengths more strongly than long ones, in proportion to the inverse fourth power of the wavelength. The same explanation offered for the sky overhead after sunset runs into a difficulty. Once the Sun is below the horizon, no sunlight reaches the zenith directly through the air above the observer. It reaches the high air overhead only by grazing the Earth at a shallow angle, and on that grazing path it crosses hundreds of kilometres of atmosphere — enough to dim the setting Sun by ten magnitudes and turn it red. The light that is then scattered down from the zenith starts from sunlight that has already lost much of its blue.

In the calculation of how the twilight sky darkens, every photon scatters exactly once in a spherical atmosphere, and the same calculation done at two wavelengths gives the colour. The result is that the two effects of scattering — putting blue into the zenith and stripping it from the sunlight on the way — very nearly cancel. A sky made only of air would fade to a colourless grey after sunset. The blue that is actually seen is put there by ozone, and the explanation was given by Edward Hulburt in 1953.

Two fourth powers that cancel #

The colour on this figure is a difference of magnitudes at two wavelengths, a colour index of the kind used to take a star’s temperature, measured relative to sunlight’s own. Zero means the zenith has the same colour as the Sun; negative means bluer.

Scattering alone, with nothing removed on the way in or out, favours 450 nanometres over 650 by $(650/450)^4$, a factor of 4.35, which is 1.60 magnitudes. That is the dotted line, and it is roughly the colour of the daytime sky overhead relative to sunlight. At twilight the calculation starts from the same number and then removes, from each ray, the light lost on its way to the scattering point. Extinction by air follows the same inverse fourth power as scattering: along a path of optical depth $\tau$ at 550 nanometres, the transmission at wavelength $\lambda$ is $e^{-\tau(550/\lambda)^4}$, and blue light, with the larger exponent, is removed faster.

The two effects are not tied to cancel exactly. That they come so close is a coincidence of the geometry: the layer of air that dominates the zenith brightness at each depression is the one just thick enough, along the grazing ray that lights it, to have lost roughly as much blue as it scatters. Without ozone the figure’s zenith is 0.01 magnitudes from sunlight’s colour at sunset and 0.06 redder at a depression of 6 degrees, and it turns noticeably bluer only past 10 degrees, where the lit layer has risen above most of the air.

Where each colour comes from #

The cancellation has a structure that the brightness alone hides. The zenith column is lit from the shadow’s edge upward, and different colours come from different heights in it.

For blue light the brightest layer sits well above the shadow — eighteen kilometres above it with the Sun 2 degrees down, and twenty above it at 6 degrees — because the grazing rays that light the air just above the shadow have crossed so much atmosphere that almost no blue survives in them. The blue in the twilight zenith comes from air where the sunlight’s path was short enough to keep some.

Red light, which is extinguished less, comes from much lower: the brightest layer is only seven kilometres above the shadow at 2 degrees and at 6. The red part of the zenith light is scattered from denser air than the blue, and there is more air to scatter it. The zenith at twilight is two skies at once, a red one close above the shadow and a blue one higher up, and without ozone their brightnesses come out almost equal.

A band at the wrong end of the spectrum #

Before 1953 the blue of the twilight zenith was taken to be the ordinary blue of the sky, lasting a little into the evening. Hulburt did the calculation for an atmosphere of air alone and found that it predicted the opposite: a zenith after sunset that should turn a pale, yellowish grey, which is not what anyone sees. Adding the ozone layer, with the absorption bands James Chappuis had measured in 1880, brought the predicted colour back to blue — and it made the twilight sky one of the few everyday sights whose colour is set by a trace gas rather than by the main constituents of the air.

Ozone absorbs strongly in the ultraviolet, which is why it matters to life, and weakly in the visible, in a broad feature called the Chappuis band that stretches from about 500 to 700 nanometres and peaks near 600. At visible wavelengths it absorbs the orange and red end and hardly touches the blue.

Overhead, in daylight, that absorption is almost invisible. A typical ozone column, 300 Dobson units, has a vertical optical depth at 600 nanometres of about 0.04: it removes four per cent of the orange light and is not noticed. But ozone is concentrated in a layer centred about 22 kilometres up, and the grazing ray that lights the twilight air crosses that layer almost tangentially, on a path through it tens of times longer than the vertical one. Along that path a layer that is nearly transparent overhead becomes optically thick in the red, and the red light that would have balanced the blue in the zenith is removed before it arrives.

The effect is to break the cancellation in one direction only. With ozone the zenith is 0.34 magnitudes bluer than sunlight at sunset, 0.67 at 6 degrees, and 0.72 magnitudes bluer at 6 degrees than the same sky without ozone. The deepening blue of the zenith through civil twilight is not Rayleigh scattering’s blue. It is ozone’s absorption of everything else.

In ordinary terms 0.72 magnitudes is a factor of 1.94: at six degrees of depression the ozone layer nearly doubles the ratio of blue to red light arriving from overhead. At noon the same layer, crossed nearly vertically by the sunlight and by the scattered light, changes the colour of the zenith by only a few hundredths of a magnitude. The colour it gives the twilight sky is a matter of the grazing geometry almost entirely.

The profile figures say why the effect is largest in the middle of civil twilight. A ray that lights the air at height $z$ over the observer, with the Sun at depression $d$, passes lowest at a height of $(R+z)\cos d - R$. With the Sun six degrees down, the blue light overhead comes mostly from about 56 kilometres, and the rays lighting that layer bottom out near 21 kilometres — almost exactly at the peak of the ozone layer, where they run tangentially through its densest part. Earlier in twilight the grazing rays pass below the layer’s peak, and later they pass above it; in between, the geometry puts the longest possible path through the ozone into the light that makes the zenith’s colour.

The first figure carries the evidence. The difference between the curves with and without ozone is 0.35 magnitudes at sunset, 0.72 at six degrees, and back down to 0.63 at ten, where the rays lighting the blue part of the column have begun to pass over the top of the layer. The ozone’s effect on the colour rises and falls through twilight because the height at which the sunlight skims the atmosphere sweeps up through the ozone layer and out of it.

None of this needs much ozone. A Dobson unit is the amount that would make a layer a hundredth of a millimetre thick if it were brought to sea-level pressure and temperature, so the entire ozone column over a typical place, 300 units, would be a film three millimetres thick lying on the ground.

This is the same physics as interstellar reddening with the sign of the wavelength dependence reversed: a medium that removes light selectively changes the colour of what passes through it, and a long enough path turns a weak selectivity into a strong one. The grazing path of twilight is also the geometry of transmission spectroscopy of exoplanets, where starlight crosses a planet’s atmosphere tangentially at the limb and absorption too weak to detect from above becomes a measurable change in the planet’s apparent size.

How much ozone it takes #

If ozone makes the blue, the blue should scale with the ozone.

Halving the ozone column halves the blue it adds: 0.37 magnitudes at 6 degrees rather than 0.72. A column of 150 Dobson units is below anything that occurs over most of the Earth, but it is a fair description of the deepest ozone hole over Antarctica in spring, where the column falls below 220 and in some years well below that.

At 450 Dobson units, a value reached at high northern latitudes in late winter and spring, the blue added at 6 degrees is 1.05 magnitudes. Across the three columns the extra blue is 0.37, 0.72 and 1.05 magnitudes — almost exactly proportional to the ozone, as it should be while the absorption along the path is still modest enough to act as a small correction on a scattering calculation that does not change. The colour of the zenith at twilight is, to a first approximation, an ozone meter.

That is not only a figure of speech. The oldest method of measuring how ozone is distributed with height, the Umkehr method devised by Paul Götz in the late 1920s, uses exactly this geometry: it records the ratio of two ultraviolet wavelengths in light from the zenith as the Sun sets, and because the grazing ray samples the ozone at a height that changes with depression, the way the ratio changes through twilight reveals the vertical profile.

A screen on the horizon #

The calculation lets one more thing be varied: whether the grazing rays can reach the air at all. An opaque layer along the line to the Sun — a distant cloud deck, a range of mountains, a thick aerosol layer — removes the lowest grazing rays.

A screen makes the zenith bluer, with or without ozone. The rays it removes are the ones that had crossed the most air and lost the most blue, and they were the ones supplying the red part of the zenith light from low in the column. Without them the zenith is lit only by higher rays that kept more of their blue, and the sky overhead is 0.60 magnitudes bluer than sunlight at 6 degrees even with no ozone at all. The ozone’s contribution, 0.76 magnitudes, is almost unchanged: the grazing rays that survive the screen still cross the ozone layer tangentially.

The blue hour, and the band opposite it #

Civil twilight, from sunset until the Sun is six degrees down, is the interval over which the zenith deepens from −0.34 to −0.67 in the first figure, and it is the stretch photographers call the blue hour. How long it lasts depends on latitude and season, because the Sun’s path crosses the horizon at an angle that is steep at the equator and shallow near the poles: a little over twenty minutes at the equator, half an hour or more at middle latitudes, and in high-latitude summer, when the Sun barely dips, the whole night.

Looking away from the sunset at the same time shows the other half of the calculation. Low in the opposite sky a band of dark blue-grey rises from the horizon — the Earth’s shadow, cast on its own atmosphere — and above it lies a pink band called the belt of Venus. The pink is sunlight that has grazed the Earth, lost its blue, and been scattered back towards the observer from the air just above the shadow: the low, red-lit layer of the second profile figure, seen from the other side. As the Sun sinks the shadow climbs, the pink band rises above it and fades, and the two meet the blue zenith overhead.

The same cancellation along the line of sight #

The daytime sky shows the cancellation too, in a different place. Overhead at noon the path of the sunlight into the air is short and so is the path of the scattered light back out, extinction is small, and the zenith keeps most of scattering’s preference for blue. Towards the horizon the observer’s own line of sight grows long, and light scattered from far along it is dimmed on the way back by the same inverse fourth power that favoured it. The sky near the horizon is paler, approaching white, for the reason the twilight zenith would be colourless without ozone; the difference is only which leg of the path is long — at twilight the sunlight’s, at the horizon the observer’s.

The same selective extinction fixes the colour of the last light of the Sun itself. Refraction by the air spreads the Sun’s image into its colours, the blue image displaced slightly above the red, so the topmost sliver of a setting Sun should be the last to go and should be blue or violet. Along the horizontal path those colours have been scattered out, and the shortest wavelength that survives in strength is green: the green flash.

What single scattering leaves out #

Light scattered more than once. Once the Sun is more than a few degrees down, light scattered first in the bright sky near the horizon and then again towards the zenith becomes important, and it is redder than the singly scattered light overhead. It softens the blue of the late twilight zenith without removing it.

Particles. Aerosols scatter with a much weaker dependence on wavelength than air molecules, and a hazy atmosphere is less selective both in what it scatters and in what it removes. After large volcanic eruptions a layer of sulphate aerosol in the stratosphere produces the purple light seen a few degrees above the sunset point, where the forward scattering of reddened light mixes with the blue of the sky.

Polarisation. Light scattered by air at right angles is strongly polarised, and at twilight the zenith is almost exactly at right angles to the sunlight, so it is the most polarised part of the sky, and a polarimeter pointed at it has to tell its own polarisation apart from the sky’s. The single-scattering calculation here tracks brightness and ignores polarisation, which does not change the colour.

And the ozone cross-sections are approximate. The calculation uses a smooth interpolation of the Chappuis band, correct to within tens of per cent. The proportionality of the blue to the column does not depend on that, and its size does only in proportion.

The same sky, seen from a telescope #

The colour of the twilight sky is not only a question for someone looking up. Observers who calibrate detectors on the twilight sky, and who measure faint objects against whatever the sky behind them is doing, are working with a background whose colour changes through twilight in the way these figures describe: bluer as the Sun sinks, bluer still when there is more ozone, and bluest when something low on the horizon is cutting off the grazing light.

Still open: twilight on worlds with other air #

On the Earth the sky darkens over about ten degrees of depression, because the air that scatters is about eight kilometres thick against a planet six thousand kilometres in radius. On a world whose scatterers are dust lifted tens of kilometres high, or haze on a small moon with a tall atmosphere, the ratio of those two lengths is different, and the depression over which twilight lasts, and the hours it takes, change with it.