Why the sky is blue and the sunset is not, from one exponent

The sky is blue and the setting sun is red, and those are usually offered as two facts with two explanations. They are one fact with one explanation, and the explanation is a single exponent.

Air molecules are far smaller than the wavelength of light — a nitrogen molecule is about a third of a nanometre across, against 500 nanometres for green light, a ratio of fifteen hundred. In that regime the amount of light a molecule redirects out of a beam goes as

$$\sigma \propto \frac{1}{\lambda^4},$$

and there is nothing else in the physics that distinguishes one colour from another. Blue is thrown sideways; red mostly goes straight on. Looking away from the sun shows the light that was thrown sideways, and looking at the sun through a long path shows what was left after the throwing.

Where the fourth power comes from #

The exponent is not a fitted parameter, and it is worth deriving because it is short and because the same argument produces several other results on this site.

Light arriving at a molecule drives its electrons into oscillation at the frequency of the light. An oscillating charge radiates, and the power it radiates depends on its acceleration rather than on its displacement. For an oscillation of fixed amplitude, acceleration goes as the square of the frequency; radiated power goes as the square of the acceleration, hence as the fourth power of frequency, hence as the inverse fourth power of wavelength.

Two squarings and nothing else. The first is the derivative taken twice, the second is that power is the square of a field.

That derivation also says what would happen if the assumption failed. The amplitude of the driven oscillation is only independent of frequency well below the molecule’s own resonances, which for air sit in the ultraviolet. Visible light is comfortably below them, which is why the clean fourth power holds across the whole visible range and why it starts to fail in the near ultraviolet, where the sky’s blue does not continue rising the way the formula alone would predict.

Why the sky is not violet #

The fourth-power law has an obvious problem: violet at 400 nm is scattered $(650/400)^4 = 7.0$ times as strongly as red, more strongly than blue, and the sky is not violet.

Three things are missing from the curve, and together they settle it.

The sun is not white. Its output peaks in the green-yellow and falls off toward the violet end, so there is less violet arriving to be scattered in the first place.

The atmosphere absorbs some of the far violet, mostly through ozone, before it can be scattered anywhere useful.

The eye is the last filter and the decisive one. Human colour vision has three receptor types, and the short-wavelength one peaks around 420 nm but is much less sensitive overall than the other two. A spectrum containing a great deal of violet, a great deal of blue and progressively less of everything else is reported by the eye as a desaturated blue, not as violet. The sky’s colour is a fact about a spectrum and about the observer, and the spectrum on its own does not fix it.

This is worth stating plainly because it is the one place where the honest answer to “why is the sky blue” is not physics alone. The scattering explains the shape of the spectrum. The colour is what that spectrum looks like to a particular three-channel detector, and a bee — whose vision extends into the ultraviolet — sees a different sky.

The sunset, from the same exponent #

Nothing new is needed for the red sun. What changes is the length of the path.

Overhead, sunlight crosses one atmosphere’s worth of air. At the horizon it crosses about thirty-eight times as much, because it enters at a grazing angle and travels a long way through the dense lower layers. The fraction of light surviving a path is exponential in the amount of scattering along it, $e^{-\tau}$, and $\tau$ carries the same fourth power.

The exponential is what turns a modest preference into a dramatic one. At one atmosphere, blue is attenuated to 80 per cent and red to 95 — a ratio of 1.18, which is barely visible and is why the midday sun looks white. At thirty-eight atmospheres the same ratio has been raised to the thirty-eighth power: blue is down to a few parts in ten thousand while red is still at 15 per cent, a ratio of nearly six hundred.

So the sunset is not a separate phenomenon requiring dust, pollution or humidity. It is the blue sky’s own explanation applied along a different line of sight, and the reddening happens over clean air in the middle of an ocean exactly as the arithmetic says. What dust and haze do is add scattering that is much less wavelength-selective — particles the size of the wavelength have no colour preference left — which reduces the colour saturation while dimming everything — which is why the most vivid sunsets follow rain rather than smog.

Why clouds are white #

The fourth-power law applies to scatterers much smaller than the wavelength. Cloud droplets are not: they are typically 10 to 20 micrometres across, twenty to forty times larger than the wavelength of visible light.

In that regime the scattering is essentially independent of wavelength. Every colour is scattered about equally, the light emerging has the same spectrum as the light going in, and a cloud is white. The transition between the two behaviours happens when the scatterer is comparable to the wavelength, and the full treatment — Mie scattering — is a considerably harder calculation with no simple power law in the middle.

That single fact resolves a common confusion. A cloud is made of water, and so is a clear humid sky, and they look completely different; the difference is not the substance but the size of the pieces it is in. It also explains milk, fog and the blue of cigarette smoke seen against a dark background versus its greyness seen against a light one — all size effects, all with the same physics as the sky.

The same ratio decides the behaviour elsewhere on this site. Diffraction through a slit is dramatic when the gap is a few wavelengths and negligible when it is many, and scattering makes the division at the same place for the same reason: what matters is whether the field is uniform across the object at any instant. Below a wavelength it is, and the object responds as one dipole; above, it does not, and the phases across the object disagree.

The sky is polarised, and by the same calculation #

Scattered light is polarised, and the pattern is a direct consequence of the mechanism.

The molecule’s electrons oscillate along the electric field of the incoming light, which is perpendicular to the direction the light came from. An oscillating charge radiates nothing along its own axis of oscillation. So light scattered through 90° can only be shaking in the one direction perpendicular to both the original beam and the new one — it is completely polarised, in principle.

A polarising filter applies $\cos^2\theta$ to whatever it is given, so a band of sky ninety degrees from the sun — which is strongly polarised — darkens sharply when the filter is turned to cross it while the rest of the sky barely changes. That is why the effect is most striking overhead at sunrise and sunset, when the strongly polarised band is directly above, and why a photographer’s polariser does more to a clear sky at those hours than at noon.

In practice the polarisation reaches about 75 per cent rather than 100, because multiple scattering mixes directions. The pattern is stable enough to navigate by: several insects use it, and the Vikings are argued — on thin evidence — to have used a birefringent crystal for the same purpose under overcast skies.

What the picture cannot show #

Every figure here computes single scattering: light is scattered once and then travels freely to the eye. That is a good approximation for a clear sky, where the optical depth is around a tenth, and it is a poor one for a cloud, where the optical depth is tens and a photon is scattered hundreds of times before it emerges.

Multiple scattering is what makes the underside of a cloud grey, what makes fog luminous rather than merely obstructive, and what fills the sky with light in directions where single scattering predicts almost none. It also puts a floor on how dark a clear sky can look near the horizon, and it is the reason the calculated polarisation of 100 per cent is never observed.

The figures also assume the atmosphere is a single slab of uniform properties characterised by an airmass. The real atmosphere thins exponentially, with a scale height of about 8.4 km, and the airmass of 38 quoted above already contains that structure along with refraction, which bends the grazing ray and lengthens the path further. What the figure treats as one number is a whole integral through a stratified medium.

The same exponent, at other wavelengths #

Because the law is about the ratio of size to wavelength and about nothing else, it predicts what happens for radiation the eye cannot see, and those predictions are the ones that get engineered against.

Radio passes through what light cannot. A centimetre-wave radar signal has a wavelength twenty thousand times longer than green light, so molecular scattering is down by a factor of $2\times10^{17}$ and is simply absent. It is why an aircraft can be tracked through cloud and why a satellite dish works in fog. The scattering that does affect radar comes from raindrops, which are comparable to the wavelength — and weather radar is that scattering used deliberately, mapping precipitation by the return from the drops.

Infrared sees further through haze. Photographing a distant landscape at 800 nm rather than 450 nm reduces the scattering by a factor of ten, which is why infrared landscape photographs show ranges of hills that are invisible in an ordinary exposure, and why long-range surveillance cameras work in the near infrared.

Distance is read as blueness. Light from a distant hillside is dimmed by scattering along the path, and the sky’s own scattered light is added on top of it. Both effects grow with distance and both are stronger at the blue end, so remote objects appear paler and bluer — the effect painters call aerial perspective and used to convey depth for centuries before anyone could explain it. It is a genuine measurement of range: the amount of blueing is an optical depth, and it is how visibility is defined and reported at airports.

The one case that is often quoted and is not this law is the yellow fog lamp, which does not penetrate fog better — fog droplets are large and scatter every colour alike. What yellow light does is reduce glare and improve contrast for the eye, which is a fact about vision rather than about scattering, and is worth separating for the same reason the violet sky had to be.

Tyndall, Rayleigh, and the dust that was not there #

The nineteenth century’s version of this explanation involved dust, and the correction is instructive.

Tyndall showed in the 1860s that a beam through a tube of fine particles scatters blue light sideways and transmits red — the experiment now named after him — and concluded that the sky’s colour came from suspended particles in the air. Rayleigh worked out the fourth-power law shortly afterwards and initially agreed.

The problem was quantitative. The scattering predicted for the density of dust the atmosphere plausibly contained was far too small, and worse, the sky’s blue does not vary the way a dust explanation demands: it is the same over the Atlantic as over a city. Rayleigh eventually attributed the scattering to the air molecules themselves, and Einstein completed the argument in 1910 by showing that a perfectly uniform medium would not scatter at all — the scattering comes from fluctuations in density, from the fact that the number of molecules in any small volume varies statistically. A gas scatters because it is grainy, and the graininess is thermal.

That is a satisfying place for the argument to end up, because it makes the blue of the sky a visible consequence of the atomic nature of matter, and lets the number of molecules in a mole be estimated by looking upward — which is what Rayleigh’s calculation, run backwards, achieved.

Making the fluctuations enormous #

Einstein’s resolution — that a gas scatters because the number of molecules in any small volume fluctuates — has a consequence that can be arranged in a sealed tube and watched.

The strength of the scattering is proportional to how large those fluctuations are, and how large they are is proportional to how easily the fluid can be compressed. For an ordinary gas that is a modest quantity and the scattering is the faint blue of the sky. But the compressibility of a fluid diverges at its critical point, where the distinction between liquid and vapour disappears — so the density fluctuations grow without bound, and so does the scattering.

The result is visible and abrupt. Take a sealed tube containing a fluid at its critical density — carbon dioxide is the usual choice — and warm it slowly. It is transparent, with a meniscus between liquid and vapour. Within a fraction of a degree of the critical temperature the meniscus fades away and the whole tube turns milky white, so opaque that a lamp behind it is hidden. Cool it again and the cloudiness clears and the meniscus reappears.

Nothing has been added. The fluid is the same substance at the same density, and what has changed is that its density fluctuations have grown from the scale of a few molecules to the scale of a wavelength of light. That is critical opalescence, and Einstein’s 1910 paper on it is the same calculation that explains the sky, evaluated where the compressibility blows up.

The colour changes too, in the way this page’s argument predicts. Far from the critical point the fluctuating regions are tiny and the scattering carries the fourth power, so the tube looks faintly blue. Close in, the regions grow to a wavelength and beyond, the wavelength dependence weakens, and the tube goes white — the same transition from a blue sky to a white cloud, produced by warming a fluid rather than by condensing one.

The twilights that made people call the fire brigade #

The essay’s remark that haze reduces the saturation of a sunset needs one exception, and it is a spectacular one.

An ordinary aerosol sits in the lower atmosphere and is washed out within days, and it does what the page says: scatters every colour much alike, dimming and greying. A stratospheric aerosol behaves differently for two reasons. It sits above the weather and lasts for years. And the sulfate droplets a large eruption produces are a few tenths of a micrometre across — comparable with the wavelength of light rather than much smaller or much larger — which puts them in the awkward middle of the size range and gives them a strongly forward-directed scattering with a real colour preference.

The consequence is a glow in the sky after the sun has set, in the direction of the sun, lasting far longer than an ordinary twilight and coloured a distinctive purple: the residual blue of the high-altitude sky, overlaid with the reddened direct light that has crossed a very long path to reach the aerosol layer and been scattered down.

Krakatoa in 1883 produced them worldwide for months. Fire brigades in several American cities were called out to non-existent fires on the strength of the glow, and the effect was reported and puzzled over everywhere from the Caribbean to Norway. Pinatubo did the same on a smaller scale in 1991, and the brightness of the resulting twilights is still used as an indirect record of how much material an eruption put into the stratosphere.

So the qualification is not that particles ruin a sunset. It is that particles of the wavelength’s own size, in a layer above the weather produce a different phenomenon, which happens where this page’s approximation has stopped being valid and its cleanest results have run out.

What the airmass of 38 is hiding #

The number that does the work in the sunset figure deserves inspection, because it is the one quantity on this page that is not a straightforward consequence of the fourth power.

Airmass is defined as the path length through the atmosphere relative to the vertical path. The naive formula is $\sec z$, the secant of the angle from overhead, which comes from treating the atmosphere as a flat slab. It works well down to about $75°$ and then fails badly: at the horizon the secant is infinite, and the true value is finite and around 38.

Three things save it. The Earth is curved, so a grazing ray eventually leaves rather than travelling forever through air. The atmosphere thins exponentially with height, so most of the path is through thin air that scatters little. And refraction bends the ray downward, lengthening it and also lifting the apparent position of the sun by about half a degree — which is roughly the sun’s own diameter, so the whole disc is geometrically below the horizon at the moment it appears to touch it.

The figure takes 38 as given and computes everything else. That is a deliberate division: the wavelength dependence is the argument being made and is computed exactly, while the airmass is a single number quoted from the geometry of a stratified sphere, which is a different subject. Anyone reading the curves as a complete model of a sunset should know which half is derived and which is imported.

Where the ladder goes next #

The rungs from here: Mie scattering and the size parameter, which fills the gap between the two regimes on this page; the blue of deep water, which is absorption rather than scattering and is a genuinely different mechanism reaching a similar colour; atmospheric refraction and the flattened sun; the green flash; extinction and optical depth as an astronomical tool; Raman scattering, where the light changes frequency and reports on the molecule; and the colour of the sky on Mars, which is the same calculation with dust that is not negligible and comes out butterscotch by day and blue at sunset — the reverse of the arrangement here, from the same equation with different parameters.

The claim to carry forward is the exponent. One number, arrived at by differentiating twice and squaring once, produces a blue sky, a red sunset, a polarised band overhead and a white cloud — and the differences between those four are differences of path length and particle size, not of physics.