How far the air lets a mountain be seen

Why the sky is blue and the sunset is not traced both colours to one exponent: air molecules scatter light in proportion to the inverse fourth power of its wavelength, so blue is scattered out of the sunbeam into the sky, and the sunbeam that has crossed a long path is left red. That essay looked up, at the sky, and along the sunbeam, at the sun. Look sideways instead, along the ground towards a distant mountain, and the same scattering does something less obvious and more practical. It sets how far anything can be seen.

The answer is not limited by how faint a distant object’s light is. A dark ridge sends almost no light, and it is seen perfectly well across tens of kilometres, as a dark shape against a bright sky. It disappears when it stops being darker than the sky, and what fills it in is the air in front of it.

The glow that fills in a dark ridge #

Between the eye and a distant ridge lies a long column of air, lit from the side and above by the sun. Every part of that column scatters a little sunlight in every direction, including towards the eye. The light reaching the eye from the direction of the ridge is therefore the ridge’s own light, dimmed by the scattering that removes it from the path, plus the airlight: sunlight scattered into the path by all the air along it, itself dimmed by the air between where it was scattered and the eye.

Harald Koschmieder worked out the consequence in 1924. For a uniform atmosphere with an extinction coefficient β — the fraction of a beam’s light removed per unit distance — a black object at distance x shows, against the sky at the horizon, an apparent brightness of $1 - e^{-\beta x}$ times the sky’s. Its contrast against the sky, the fractional difference between them, is therefore

$$C(x) = e^{-\beta x},$$

falling exponentially with distance whatever the brightness of the day, because the airlight and the sky’s brightness at the horizon both scale with the same sunlight. Everything a scatterer removes found that extinction is fixed by what a particle does in the forward direction; here it appears twice, once as the light lost from the ridge and once as the light gained from the sky, and the two halves of the same exponential are the whole story.

A threshold and a range #

The eye cannot see contrast below about two per cent against a bright background. Koschmieder took that threshold as defining how far an object can be seen: the distance at which a black object’s contrast falls to two per cent, $V = \ln 50/\beta = 3.912/\beta$. It is called the meteorological visual range, and it is what weather stations report as visibility.

The first figure computes it for three kinds of air. Perfectly clean air scatters only by its molecules, at 0.0116 per kilometre in green light at sea level. A black ridge in such air stays visible to 337 kilometres, a distance at which the Earth’s curvature hides all but the tallest mountains. Add a light haze of particles scattering 0.05 per kilometre and the range falls to 64 kilometres; a thicker haze of 0.3 per kilometre brings it to 13. The air’s own molecules set an absolute limit that real air almost never approaches.

Clean air’s limit and the Earth’s curvature turn out to be of the same size. From an eye two metres above the sea the horizon is five kilometres away; from a summit four thousand metres high it is about 225 kilometres, and two such summits can see each other across about 450 kilometres if nothing else intervenes. The longest views on Earth are therefore set jointly by the curvature, which demands height, and by the air, which demands that the path be clean — and height helps with both, because the air at altitude is thinner and carries less haze.

A range that counts molecules #

The number 0.0116 per kilometre is not a fitted constant. It follows from two measured properties of air: its refractive index, 1.000278 for green light at sea level, and the number of molecules in a cubic metre of it, 2.5 × 10²⁵. Rayleigh’s formula gives each molecule a scattering cross-section of about 4.5 × 10⁻³¹ square metres, set by how strongly the molecule polarises — which the refractive index measures — and by the fourth power of the light’s frequency; multiplied by the number of molecules it gives the extinction, and 3.912 divided by the extinction gives the 337 kilometres.

The way the number of molecules enters is the surprise. The refractive index measures the air’s total polarisability, the sum over all its molecules; for a fixed index, more molecules means each is smaller and polarises less. Scattering goes as the square of each molecule’s polarisability, so for the same index more molecules scatter less in total — in proportion to one over their number. Air made of infinitely many infinitely small molecules, a true continuum with the same refractive index, would scatter nothing and would be perfectly transparent: every mountain on the visible Earth would stand out black against the sky. That the view ends at a few hundred kilometres is a direct consequence of air being made of a finite number of molecules, and Rayleigh pointed out at the end of the nineteenth century that the clarity of a distant view, or the brightness of the blue sky, could be read the other way: as a measurement of how many molecules there are. The estimates of Avogadro’s number made that way agreed with the others to within the accuracy of the observations.

Why near ridges are blue #

The airlight has a colour. Because molecules scatter blue light more than red, as the inverse fourth power of the wavelength, the airlight in front of a moderately distant ridge is mostly blue. The second figure computes the ridge’s apparent brightness, colour by colour, as a fraction of the sky’s. At twenty kilometres the ridge already glows with forty-one per cent of the sky’s blue but only eleven per cent of its red: it is a bluish shape against a sky that is, by comparison, whiter. At a hundred kilometres the blue has nearly reached the sky’s and the red is half way. At three hundred the ridge has become the sky in every colour.

The third figure distils that into one number: how much bluer than the sky the ridge looks. For a near ridge in clean air the airlight is the scattering coefficients’ own ratio, 4.5 times as blue as the sky’s light at the horizon, which has itself crossed so much air that its colour is diluted. The excess fades steadily with distance as the redder light catches up. This is aerial perspective, the receding blues of a mountain landscape that Leonardo da Vinci described and painters have used to suggest distance ever since: near ranges blue, far ranges pale, the farthest indistinguishable from the sky. The figure shows why it is not a fixed rule. In haze, whose particles are comparable in size to the wavelength and scatter all colours more nearly alike, the blue excess is weak from the start and gone within a few tens of kilometres. Hazy hills are grey.

How far each colour can see #

The visual range itself depends on colour. Molecular scattering falls so steeply with wavelength that clean air at sea level allows 149 kilometres in blue light and 667 in red, the fourth figure shows. Thinner air allows more: three kilometres up, where the air’s density has fallen by thirty per cent, every range is longer in proportion. Photographs of summits more than four hundred kilometres away, across the Alps or from the Pyrenees, have been taken along paths high in clean air near sunrise or sunset, when the light reaching the camera is reddened and the line of sight passes over the densest air. They are not exceptions to the arithmetic but its best case.

Haze flattens the curve. Particles comparable in size to the wavelength scatter all visible colours roughly alike, with a weak dependence on wavelength measured by an Ångström exponent of about 1.3 rather than the molecules’ 4; when the particle is the size of the wave found the change of exponent in Mie’s theory of scattering. With a light haze the range in red is only twice the range in blue, and the reddened dawn light that helps on a clean day helps much less.

How much haze it takes #

The last figure turns the formula round. Real air carries particles — sea salt, mineral dust, droplets of sulphate formed from pollution, smoke, pollen — and a few of them scatter far more than the molecules around them. Particles that add even 0.01 per kilometre, less than the air’s own scattering, cut the range by nearly half. Ordinary clear air over land, with 0.1 per kilometre of particles, allows about 35 kilometres; a hazy summer day, or smoke drifting from distant fires, 1 per kilometre and under 4 kilometres.

That makes visibility a measure of pollution, and it has been used as one. Records of visibility at airports go back decades, kept for aviation, and the long decline of visibility over industrial regions in the mid-twentieth century and its recovery as sulphur emissions were cut are written in them; the eye’s two-per-cent threshold, applied to dark objects at known distances, is in effect an instrument for the air’s extinction. Modern weather stations measure the same quantity directly, with instruments that send a beam of light through a short path of air and record how much is scattered sideways, and report the result as the distance a person would see.

Why the threshold is two per cent #

The two per cent is borrowed from the eye. Measurements of how much contrast a viewer needs to detect a target against a bright, uniform background, made in the 1940s with many observers and targets of many sizes, found that the threshold for large, sharp-edged targets in daylight is about one to two per cent, and rises steeply for small ones: a target that subtends only a few minutes of arc needs several times more contrast to be seen. Koschmieder’s choice of two per cent describes a large dark object — a range of hills, a forest edge, a building — seen by a practised observer. A small, distant object vanishes sooner. A ship on the horizon may be lost while the coast behind it is still visible, not because the air is different but because the ship is smaller and needs more contrast.

The threshold enters the visual range only through its logarithm, which makes the range forgiving of the exact choice. A threshold of five per cent rather than two shortens the range by nearly a quarter, from 3.912/β to 3.0/β; a threshold of one per cent lengthens it by a sixth. What dominates the range is the extinction, which in practice varies by factors of a hundred from one day to the next.

Fog, and the end of colour #

A fog is a cloud at the ground, and its droplets, around ten micrometres across, are far larger than the wavelength of light. Such particles remove light from a beam over twice their geometric cross-section, the extinction paradox that everything a scatterer removes explained by the light diffracted round their edges, and they do it equally for every colour. A fog holding a tenth of a gram of water in each cubic metre, in droplets of that size, has about twenty-four million droplets per cubic metre and an extinction of fifteen per kilometre: a visual range of 260 metres. Meteorologists call it fog when the range falls below a kilometre.

Because fog’s extinction does not depend on wavelength, fog has none of the colour effects of clear air. A dark tree half-hidden in fog is grey, not blue, and fog lamps gain nothing from their colour in seeing through it: amber and white light are scattered alike. What helps in fog is the geometry of the light rather than its colour — a lamp mounted low and aimed down, so that the driver’s eye does not look along the brightly lit column of scattered light that a headlamp’s beam makes in front of it. That column is airlight again, the same glow that fills in the distant mountain, lit this time by the car rather than the sun.

Where the formula holds #

Koschmieder’s formula assumes a uniform atmosphere along the line of sight, a black object, a horizontal path and an observer whose threshold is two per cent. Each assumption bends in practice. Real paths cross air of varying density and haze, and a path rising towards a mountain summit passes through ever cleaner air, which is why summits are seen further than the sea-level formula allows. Real objects are not black: a snowy ridge is brighter than the sky, its contrast starts negative and passes through zero, and it can vanish at a shorter distance than a dark one and reappear further out as a brighter shape. The two-per-cent threshold is itself a convention, close to the eye’s performance for a large, sharp-edged object in daylight and pessimistic for some conditions; some services use five per cent, which shortens every range by nearly a quarter. And the Earth’s curvature limits sight-lines independently of the air: an observer and a summit must both be high enough for the line between them to clear the bulge.

What the curves cannot show #

The curves show the contrast of a uniform dark object along a uniform path, and they cannot show what makes real distant views so variable from one day to the next. The air is lit unevenly. Clouds shade parts of the path and the airlight there drops; the sun’s position decides whether the air in front of a mountain is lit from behind, which makes it glow brightly and hides the mountain, or from in front, which leaves it dark. The same mountain range visible at dawn can be invisible at noon for that reason alone. Nor can the curves show mirages and the bending of light near the ground, which can lift a distant horizon into view or drop it out of sight, or the turbulence that blurs fine detail long before contrast is lost.

Still open: the haze that the formula needs #

The visual range depends almost entirely on particles, and how many particles there are, how large and of what kind, is one of the hardest things in the atmosphere to predict. Particles grow by taking up water as the humidity rises, so the same air can halve its visual range between a dry afternoon and a damp evening, and how strongly a given mixture of sulphate, organic matter and sea salt swells depends on its chemistry in ways that are measured more readily than they are computed. Satellites now map the atmosphere’s aerosol by how much light it scatters back towards space, and ground stations measure extinction directly, but connecting the two to predict visibility at a given place and time — needed for aviation, for road safety in haze and fog, and for estimating how much sunlight the haze sends back to space and so how it cools the climate — remains an active and incompletely solved problem.

The habit worth carrying away is to ask what an object is being compared with. A dark ridge is seen by its contrast against the sky, and the air in front of it adds sunlight scattered into the line of sight until that contrast falls as e^(−βx) — two per cent, the eye’s threshold, at 3.912/β: 337 km in clean air at 550 nm, 149 km in blue and 667 km in red, but 35 km once particles add 0.1 per km — and because the added light is blue, near ridges look bluer than the sky, up to 4.5 times in clean air, and far ones fade to its colour. Distance is drawn on a landscape by the glowing air in front of it.