Why is the sky blue and the sunset red?

A common answer is that the sky is blue because it reflects the ocean. It doesn’t: the sky is just as blue over the Sahara, and the sea is blue partly because it reflects the sky. Another is that air is “a blue gas”, the way a glass of cola is brown. That isn’t right either. Air absorbs almost no visible light. What it does is scatter it: knock a little of the light out of its straight path and send it off in a new direction.

Small molecules prefer blue

Light is an electromagnetic wave. When it passes a nitrogen or oxygen molecule, the wave’s electric field shakes the molecule’s electrons back and forth, and the shaking electrons give off light of their own, in all directions. That is scattering. Because a molecule (about 0.3 nanometres across) is more than a thousand times smaller than the wavelength of visible light (400 to 700 nanometres, nm), it is called Rayleigh scattering, after John William Strutt (later Lord Rayleigh), who worked out its law in 1871.

The law is steep. A faster-shaking charge radiates much more strongly, and the strength goes as one over the wavelength to the fourth power:

scattering ∝ 1 / λ⁴

Here λ (lambda) is the wavelength. Blue light at 450 nm has a wavelength 0.64 times that of red light at 700 nm, and 0.64⁴ is about 1/6. So blue is scattered roughly six times as strongly as red. (The measured ratio, ×6.1 in Figure 1, is a little higher than 5.9 because the air’s refractive index also changes slightly with colour.)

The readout under the figure uses real numbers for the whole atmosphere. Looking straight up, there is enough air above you to scatter about 20% of violet-blue light at 450 nm out of a sunbeam, 9% of green at 550 nm, and under 4% of red at 700 nm. (These come from the measured optical depth of the atmosphere, a number that says how much a beam is dimmed: a beam crossing an optical depth τ keeps a fraction e^−τ of its light. At sea level, τ is about 0.22 at 450 nm and 0.097 at 550 nm.)

That is why the sky is blue. Look anywhere away from the Sun and the only light reaching you is light that air has scattered toward you, and that light is rich in short wavelengths. Without air, as on the Moon, the sky is black even at noon.

The long way through the air

When the Sun is overhead, its light crosses the least possible air: one atmosphere’s worth. As the Sun sinks, the slanting path gets longer. Astronomers call the length of the path, in units of the overhead path, the air mass. At 30° above the horizon it is 2. At 10° it is 5.6. At the horizon, where the path skims along the curved Earth, it is about 38.

Every extra length of air removes more blue from the direct beam. A beam crossing 38 atmospheres keeps e^{−0.22 × 38} ≈ 0.02% of its 450 nm light but e^{−0.037 × 38} ≈ 25% of its 700 nm light. What reaches you is orange and red.

Two things in Figure 2 are worth a second look. First, the Sun stays nearly white until it is quite low: at 30° it has lost about a third of its blue, which the eye barely notices. The colours of sunset are crowded into the last 10° or so, because that is where the air mass shoots up. Second, the sky overhead gets paler as the Sun drops, since the light being scattered has itself already lost blue on its way in. (The real twilight sky stays bluer than this, for a reason the model leaves out; see the end of the page.)

The air in Figure 2 starts out perfectly clean, which real air never is. It also carries aerosols: tiny particles of dust, sea salt, smoke and droplets, mostly a few hundred nanometres across. They dim light too, and still somewhat more at short wavelengths, though far less steeply than molecules do (roughly as 1/λ^1.3 instead of 1/λ⁴). Their effect is measured by the aerosol optical depth: about 0.01 in extremely clean air, 0.1 to 0.15 on an average day in the United States, and 0.4 or more in thick haze. Raise the haze slider with the Sun low. The Sun dims dramatically and turns a deeper red, and the sky turns from blue toward white, because the particles scatter all colours more evenly than molecules do.

Why isn’t the sky violet?

If shorter means more scattered, violet light at 400 nm should beat blue. It is scattered 1.6 times more strongly than blue at 450 nm. So why is the sky not violet?

Three reasons, and Figure 3 lets you see each. Sunlight contains less violet than blue to begin with. Your eyes are much less sensitive to violet: the three curves in the figure, called colour-matching functions, are the standard measure of how the eye responds to each wavelength (they are closely tied to the eye’s three kinds of colour receptor, the cones), and they fall off steeply below about 430 nm. And the most important: the sky is not one wavelength but a mix of all of them, with extra weight at the short end. Your eye sees that mix as white light with some blue added. That is pale blue, not violet.

The dominant wavelength readout names the single wavelength whose colour, mixed with white, matches what you see; the purity says how much of it is in the mix. For Rayleigh-scattered sunlight the model gives about 476 nm at about 40% purity: a sky blue, clearly paler than a pure spectral colour. Glenn Smith’s 2005 analysis, which treats the eye in more detail, found the same: skylight looks like blue light of 474–476 nm mixed with white.

Why clouds are white

The 1/λ⁴ law holds only for particles much smaller than the wavelength. A cloud droplet is about 10 micrometres across, some 20 times larger than a wavelength of light. Light bends and reflects through it much as through a tiny raindrop, and every colour is scattered about equally: the exponent n is close to 0. Scatter sunlight equally at every wavelength and you get sunlight back: white. Milk, sea foam and sugar are white for the same reason. Haze and smoke fall in between, with particles comparable to the wavelength, and make a sky that is milky rather than deep blue.

John Tyndall saw the in-between case in his laboratory in 1869: a mist of very fine particles lit by a white beam glowed blue from the side, and the blue light was polarised. Rayleigh explained it two years later with the law in Figure 1, and later showed that the air molecules themselves, not dust, are enough to make the sky blue.