The night is late because the light bounces twice

Twilight has three official ends, at six, twelve and eighteen degrees of the Sun’s depression below the horizon, and the three definitions grew out of three different uses — reading outdoors, seeing the sea horizon, and doing astronomy. The last is the one with a physical claim attached: once the Sun is eighteen degrees down, the sky overhead is as dark as it will get, and anything fainter than the night sky’s own glow can be measured without the twilight getting in the way. Observatories schedule their nights by it.

The geometry of the Earth’s shadow explains most of what twilight looks like. After sunset, sunlight still reaches the air above the observer, but only the air above a certain height, the edge of the shadow, which rises as the square of the Sun’s depression. The air thins by a factor of e every eight kilometres, so as the shadow climbs, the sunlit air overhead becomes rapidly thinner and the zenith darkens. That calculation, carried out for light that is scattered exactly once — sunlight in, one scattering at some height overhead, down to the observer — gives the zenith sky a steadily falling brightness, and the rate at which it falls is set by one geometrical ratio, the square root of twice the height of the air over the radius of the planet.

The same calculation has a flaw that it announces itself. It puts the zenith sky at the brightness of the natural night sky — about 21.9 magnitudes per square arcsecond, the combined glow of the upper atmosphere, starlight and zodiacal light — at about nine degrees of depression. Twilight does not end at nine degrees. It is observed to last, measurably, to eighteen. Something brightens the sky for nine more degrees that single scattering leaves out, and the obvious candidate is light that has been scattered more than once.

Two hops round the planet #

Light scattered once reaches the zenith line only where the zenith line is in sunlight — above the shadow. Light scattered twice has another route. Sunlight strikes the air somewhere that is still sunlit, low in the sky towards the Sun and hundreds of kilometres away, and is scattered there in every direction; some of it travels back towards the observer, low, and is scattered again by the air above the observer’s head — which may be deep in the Earth’s shadow — down into the observer’s eye. The second scattering needs no sunlight at the place it happens. It needs only a bright sky nearby.

The calculation behind the figures follows exactly that route. For each height on the zenith line, it adds up, over every direction, the light arriving from sunlit air in that direction — each piece scattered once, attenuated on the way in from the Sun and on the way from the first scattering to the second — and scatters a share of it down to the ground. It uses the same atmosphere as the single-scattering curve — the standard densities, Rayleigh scattering by the molecules of the air, a layer of ozone absorbing in the middle of the visible — so the two orders are on the same scale and add directly. It stops at the second order.

The two orders behave very differently. At sunset the twice-scattered light is a minor contribution, about one part in seven of the total — the zenith is lit mostly by sunlight scattered once, in the dense air directly overhead. As the Sun sinks, the once-scattered light falls steeply, because the shadow is climbing into thinner and thinner air; the twice-scattered light falls much more slowly, and by seven degrees the two are equal. From there on the zenith sky is lit almost entirely by light that has been scattered twice, and its brightness follows the slower curve.

That changes when twilight ends. The sum reaches the brightness of the night sky at about thirteen degrees rather than nine. And the more useful criterion — the point at which the twilight’s contribution falls to a hundredth of the night sky, five magnitudes fainter than it, about the smallest excess a careful measurement of sky brightness can detect — moves from under twelve degrees for single scattering to seventeen and a half for the two orders together. The traditional eighteen degrees is where the second order puts the end of measurable twilight.

The share that has bounced twice #

The change of regime is abrupt, and its position has a simple reason. The Earth’s shadow over the zenith reaches the bulk of the atmosphere — the lowest few scale heights — at a depression of about six or seven degrees, where it stands thirty or forty kilometres up. Above that, the once-scattered light has only thin air to come from. The twice-scattered light is not constrained by the shadow over the observer at all: its second scattering can happen anywhere on the zenith line, including in the dense air near the ground, which is in shadow but not in darkness, because it is lit by the bright sky towards the Sun. The switch from one regime to the other is the moment when the thin sunlit air overhead becomes a weaker source than the dense shadowed air lit from the side.

That is also why twilight looks different in its second half. In early twilight the zenith is blue because once-scattered sunlight overhead has passed through the ozone layer on a long grazing path, and ozone absorbs the red and orange; in late twilight the zenith is lit by light that has travelled even further, round the curve of the Earth twice, and its colour reflects both paths. The deepening blue of late twilight, fading to the grey of the night sky, is the second order’s colour.

Where the light was first scattered #

With the Sun twelve degrees down, the air over the observer is in shadow to 140 kilometres, and the source of the zenith’s light is somewhere else entirely. It is a band of sunlit air hundreds of kilometres towards the Sun, from about six hundred to a thousand kilometres away and thirty to sixty kilometres up, hugging the edge of the shadow. That band is what an observer sees, looking towards the sunset, as the twilight arch — the glowing band low over the horizon where the Sun went down, bright and coloured long after the rest of the sky has darkened. The arch is lit directly by the Sun, because the Sun has not set for the air there. The observer’s zenith is lit by the arch.

The chain has a pleasing symmetry. The twilight arch seen low in the west from one place is the zenith sky of places several hundred kilometres further west, whose own twilight is several degrees younger; and those places’ arch is the zenith of places further west still. Late twilight is the evening sky of other people’s earlier evenings, relayed.

Scattered again, low down #

The second scattering happens overwhelmingly in the dense air near the ground and in the lower troposphere, far below the shadow’s edge, which by ten degrees of depression is a hundred kilometres up. That is a striking reversal. The once-scattered light of early twilight comes from the air at and above the shadow’s edge, and so from ever higher and thinner air as the evening proceeds, which is why it fades so fast. The twice-scattered light comes from the same dense, low air all evening; what dims it is only that the source — the distant arch — is sinking further below the observer’s horizon and is harder to reach through the air between. As the Sun goes deeper, the lowest layers are shielded from the arch by the air in front of them, and the peak of the second scattering moves up to around ten kilometres.

The practical consequence is that late twilight depends on the lowest part of the atmosphere, not the highest. A layer of haze or thin cloud near the ground, which would barely touch the once-scattered light of early twilight, sits exactly where the second scattering happens and can brighten or dim the late twilight sky measurably. It is one reason the observed end of twilight varies from night to night and site to site by a degree or two, and one reason mountain observatories, with the densest air below them, reach true darkness a little sooner than sea-level ones.

A second twilight half as wide again #

The argument for the second order’s slower fall is geometrical, and if it is geometrical it should hold on any world.

The once-scattered light reaches the zenith by one grazing path past the limb of the planet, and the angle over which it survives scales, as the argument for the first order showed, as the square root of twice the height of the scattering air over the planet’s radius. The twice-scattered light makes two such hops — Sun to the arch, arch to the zenith — and if each could use the whole of that angle, the second-order twilight would be twice as wide as the first. The computed ratio is about one and a half, on the Earth and on worlds whose twilights are twice and three times as wide. It falls short of two because each hop costs optical depth: the longest hops, which would make the twilight widest, pass through the most air and are the most attenuated. That the ratio stays near one and a half across a threefold change in the geometry is the geometrical part of the argument surviving the attenuation.

For the Earth this turns nine degrees into thirteen for the twilight to reach the night sky’s brightness, and into seventeen or eighteen for it to fall to the level at which it can no longer be measured. For Mars, whose dusty air stands high relative to the planet’s radius, the single-scattering twilight is already much wider; the second order extends it further, and the long Martian twilights photographed by landers — blue near the setting Sun, where the fine dust scatters blue light forward — last up to two hours after sunset. For Titan, whose haze stands hundreds of kilometres up on a small moon, the first order alone gives a twilight tens of degrees wide, and the second extends it to a large fraction of the sky.

A test the polarisation can make #

The two orders can be told apart without computing either, because they carry different polarisations. Sunlight scattered once by molecules through a right angle is almost completely polarised, perpendicular to the plane containing the Sun, the scattering molecule and the observer. The zenith at sunset is at ninety degrees from the Sun, so once-scattered zenith light is strongly polarised; measured at the start of twilight, the zenith’s polarisation is typically seventy to eighty per cent, short of complete only because some light has already been scattered more than once and because the molecules are not perfect spheres.

Twice-scattered light is much less polarised. Its first scattering happens at many different angles, in many different places along the arch, and its second scattering, overhead, happens between directions that are close to horizontal and a direction that is vertical — ninety degrees again, but for light that arrives already partly polarised in a range of orientations. The sum of many differently polarised contributions is weakly polarised. So as the share of twice-scattered light rises through the evening, the zenith’s degree of polarisation should fall, and it should fall at the depression where the share crosses one half.

That is what is observed. Measurements of the zenith’s polarisation through twilight, made since the early twentieth century with polarimeters of steadily increasing precision, show it roughly constant for the first few degrees and then falling, over a few degrees centred near six or seven, to a much lower level. The fall is the signature of the change of regime in the share figure, and it is visible in data taken by instruments that know nothing about the geometry of the arch. Polarimetry measures a direction that a simple photon count throws away, and here the direction is the one that separates the two routes the light took.

The same logic runs the other way for an instrument that is itself more polarised than the sky: a measurement of the twilight sky’s polarisation is only as good as the instrument’s calibration, and the steep change of the sky’s own polarisation through twilight has been used as a known, slowly varying source for exactly that calibration.

Light that is moved rather than lost #

The twilight calculation is a small instance of a general principle about scattering media, and it recurs across astronomy. Light scattered out of one beam is not destroyed; it goes somewhere else, and in a medium that is lit unevenly, the somewhere else can be the dark part. Dust between the stars scatters starlight out of the line of sight, dimming the star, and scatters it back in from other directions, producing reflection nebulae and a diffuse galactic light that fills the sky between the stars. The twilight zenith is lit by the same moved light, from a source — the twilight arch — that is itself only moved sunlight.

The Moon’s earthshine is the same thing on a larger scale. The dark part of a crescent Moon is lit, faintly, by sunlight reflected once from the Earth and then again from the Moon’s surface, and its brightness is a measurement of the Earth’s reflectivity — how much of the sunlight falling on it the Earth sends back into space. The second bounce makes visible a region the Sun cannot reach, and the brightness of that region is a measurement of the first. The twilight sky after nine degrees is lit by the arch in exactly that sense, and its brightness is, in principle, a measurement of how bright the arch is — which depends on the aerosols and clouds far to the west that nobody overhead can see.

What the calculation leaves out #

The calculation stops at the second order, so its sum is a lower bound: third and higher orders add more light, most of it late in twilight, when the second order itself has become faint. How much they add depends on the optical depth, which for the Earth’s clean air is small — about a tenth at visible wavelengths vertically — and each additional order costs roughly a factor of that; so the third order is a few per cent of the second at most depressions and matters most at the very end. It ignores aerosols, which scatter more strongly forward than the molecules do and would change both the brightness of the arch and how much of its light is sent sideways towards the zenith. It treats the ground as black, when real ground reflects a tenth to a third of the light that reaches it, and snow reflects nearly all; a snowfield under the arch is a second, lower source for the zenith, and winter twilights over snow are measurably brighter. It computes one wavelength at a time and represents the Sun as a point, and the refraction of the grazing rays, which lifts the shadow’s edge slightly and lets sunlight reach the arch a little longer, is left out.

The night sky’s own brightness, the floor against which the end of twilight is measured, is not a constant either. The glow of the upper atmosphere varies by a factor of two through the solar cycle and from night to night, and a darker night sky makes twilight appear to last longer, by the amount it takes the second-order curve to fall a further factor of two — about a degree. The eighteen degrees is a convention fixed long ago; the calculation shows it to be a good one for an average night sky and the second order of scattering, and no better than that.

Why the convention was right before anyone knew why #

The eighteen-degree limit is old. It appears in medieval Islamic astronomy, where the end of evening twilight fixed the time of the night prayer and the start of morning twilight the beginning of the fast, and it was measured, not derived — by watching the sky and noting the Sun’s depression, computed from tables, at the moment the last glow disappeared. Eighteen degrees, and in some traditions seventeen or nineteen, came from generations of such observations, made by eye, in dry climates with dark skies. Single-scattering theory, when it was worked out, gave a much earlier end and could not explain the tradition; the traditional number was sometimes regarded as a cautious overestimate, padded to make sure the sky really was dark.

The second-order calculation takes the padding away. The eye at its most sensitive, adapted to the dark, can just detect a sky brightness a per cent or two above the night background over a large area; the depression at which twice-scattered light falls to that level is within a degree of eighteen. The old observers were measuring the second order of scattering without knowing there was one, and their number is what that physics gives.

The modern use of the limit is more demanding. A faint star is measured against the sky, and any excess of sky brightness is noise that cannot be subtracted perfectly; an observatory that wants the faintest objects waits for full darkness. Wide-field surveys, which can measure the sky’s own brightness across the field to a fraction of a per cent, see the residual twilight gradient — brighter towards the Sun — well past eighteen degrees on some nights. That residual is the second and third orders, and its gradient points at the arch.

Still open: how much the higher orders add, and on which worlds #

For the Earth’s thin air the second order is enough to put twilight’s end in the right place. For thicker atmospheres it is not. On Titan, with a vertical optical depth of several, light is scattered many times before it reaches the observer, and the twilight is closer to a diffusion problem than to a sequence of hops; the one-and-a-half rule is a statement about the first two orders in thin air and should fail there. On Mars, during a dust storm, the optical depth rises to several as well and the twilight changes character from night to night. How the width of twilight scales between the thin-air limit, where two hops dominate, and the thick-air limit, where light diffuses round the planet, has not been worked out as a single law, and the landers now photographing twilights on Mars — and the probe that will fly through Titan’s — will supply the measurements to test one against.