The air's height against the planet sets the twilight

Civil, nautical and astronomical twilight end when the Sun is 6, 12 and 18 degrees below the horizon. The numbers are conventions, but the fact that they are a handful of degrees rather than a fraction of one or most of a right angle is not. When the brightness of the sky after sunset is actually computed, as single scattering in a spherical shell of air, the zenith fades by ten magnitudes over about nine degrees of the Sun’s descent, and the reason it is nine and not ninety is a single ratio: the height of the air against the size of the planet.

That ratio can be changed, and every other world with a scattering atmosphere changes it. Mars has dust lifted higher than the Earth’s air, on a planet half the Earth’s size. Titan and Pluto have hazes tens of kilometres tall on worlds a fraction of the Earth’s radius. The same calculation, with the Earth’s air replaced by a simple layer of scatterers of a chosen height and optical depth, gives each of them a twilight, and the twilights differ by much more than the atmospheres do.

The shadow rises as the square of the depression #

Once the Sun has set, the air directly overhead is lit only above the height at which the Earth’s shadow crosses the zenith. That height is $R(\sec d - 1)$ for a depression $d$ and a planet of radius $R$, and for the small angles of ordinary twilight it is very nearly

$$z_s \approx \frac{R\,d^2}{2}.$$

The shadow climbs as the square of the depression, slowly at first and then faster: on the Earth, 1 kilometre at 1 degree, 35 kilometres at 6, 142 at 12.

An atmosphere that thins exponentially with height, with scale height $H$, has a fraction $e^{-z_s/H}$ of its scatterers above the shadow. If the layer is thin enough that the sunlight reaching it is not much dimmed on the way, the brightness of the zenith falls in exactly that proportion. In magnitudes the dimming is $2.5\log_{10}e \times z_s/H$, and substituting the shadow’s height,

$$\Delta m \approx 1.086\,\frac{R\,d^2}{2H} = 1.086\,x^2, \qquad x = \frac{d}{\sqrt{2H/R}}.$$

The whole of twilight is measured in one natural unit of angle, $\sqrt{2H/R}$ — the depression at which the shadow over the zenith has climbed one scale height. In that unit the dimming is the same parabola on every world, and ten magnitudes is reached at $x = 3.03$.

For the Earth, with a scale height of 8.5 kilometres, the unit is 2.96 degrees and ten magnitudes arrives at about nine degrees. The full calculation in the first figure gives 8.6, a little faster than the parabola, because the real layer is not quite thin and the sunlight reaching the lowest lit air has been dimmed on its way.

Why the thickness of the atmosphere hardly matters #

The first surprise in the formula is what is absent from it. The optical depth — how much scattering material the atmosphere contains in total — does not appear.

A hundredfold change in the amount of scattering material moves the angle by a third of a degree. That is because the dimming is measured relative to the sky at sunset, and at small optical depths every layer’s contribution is proportional to the optical depth, so it cancels. A dense atmosphere makes a brighter twilight than a tenuous one, but not a longer one: the rate at which the shadow switches off the scatterers is set by how quickly their numbers fall with height, and that is the scale height.

The optical depth does matter a little once it approaches one. The sunlight lighting the lowest layers above the shadow has then been dimmed on its grazing path, so the brightest part of the column sits a little higher and is switched off a little sooner — the 0.3 degrees between the thinnest sky and the other two. That is the same effect that, done at two wavelengths, decides the colour of the twilight zenith. It changes the colour far more than it changes the timing.

A smaller planet has a longer twilight #

The planet’s radius sits in the denominator of the unit. Keep the air and shrink the world, and twilight stretches.

The scaling is exact to the precision drawn. A planet with the radius of Mars has an angle larger by the square root of the ratio of the radii, 1.37, and 8.6 degrees becomes 11.8. A planet with the radius of Pluto has an angle larger by 2.32, and 8.6 becomes 19.7, a little short of the 19.9 the scaling predicts, for a reason that returns below.

The geometry is easy to see. On a small planet the ground curves away more sharply, so the Sun has to sink further below the local horizon before the Earth’s — or the world’s — shadow has climbed a given height over the observer. The same height of shadow takes a larger angle of the Sun, and the air is switched off more slowly.

Taller air has a longer twilight #

The scale height sits in the numerator. Keep the planet and raise the air, and twilight stretches as its square root.

Drawn in their own units the three curves lie almost on top of each other and close beside the parabola, which is the sense in which the unit is the right one: a scale height of 40 kilometres gives a twilight more than twice as long in angle as the Earth’s, and yet in units of $\sqrt{2H/R}$ it is the same twilight. An Earth whose air were ten times taller would reach ten magnitudes of dimming with the Sun eighteen and a half degrees down, and astronomers would have defined the end of astronomical twilight somewhere near thirty-five.

The scale height is itself a ratio — of the thermal energy of a gas molecule to the pull of gravity on it — so warm, light gas on a world with weak gravity stands tallest. The same quantity decides how far a planet’s atmosphere extends past its limb when it is seen against its star, and in the extreme how easily a planet loses its lightest gases altogether.

Four worlds #

The first figure put the scaling to work on four real skies, with representative values for the layer that does the scattering on each.

Mars. The Martian atmosphere is thin, but it carries fine dust, lifted high, with an optical depth of about half on an ordinary day. With a scale height of 11 kilometres on a planet of 3,390 kilometres the unit is 4.64 degrees, and the sky dims by ten magnitudes at 13.4 degrees of depression, 55 minutes after sunset at the equator. Twilight on Mars was observed from its surface to linger for up to two hours, which is what a taller layer on a smaller planet predicts, and the dust gives it a colour the Earth’s twilight does not have: fine particles scatter blue light forward more efficiently than red, so the sky around the setting Sun is blue while the rest of the Martian sky is butterscotch.

Titan. Saturn’s largest moon is wrapped in an orange photochemical haze tens of kilometres in scale height, on a body smaller than Mercury. Its unit is 11.29 degrees, and the sky does not dim by ten magnitudes until the Sun is 29.8 degrees down. Because Titan keeps one face towards Saturn, its day is its orbit, sixteen Earth days long, and the Sun takes 31.7 hours to sink that far. A twilight longer than an Earth day is the direct consequence of a tall haze and a slow Sun.

Pluto. Pluto’s atmosphere is extremely tenuous, but images taken as the New Horizons spacecraft looked back at the night side showed a haze layered to a couple of hundred kilometres. With a scale height of about 50 kilometres on a world of 1,188 kilometres, the unit is 16.62 degrees, the largest of the four, and the sky dims by ten magnitudes at 43.9 degrees of depression — half-way to the nadir — which takes the Sun 18.7 hours of Pluto’s six-and-a-half-day rotation.

And the Earth, at 2.96 degrees a unit and 35 minutes, is the world with the shortest twilight of the four, because it is the largest and its scatterers are the lowest.

The conventions, carried to other skies #

The Earth’s three twilights are two, four and six of its own units. Carried to the other worlds in their units, they become different angles entirely. On Mars, civil twilight would end with the Sun 9.4 degrees down, nautical at 18.8 and astronomical at 28.2. On Titan the three would be 23, 46 and 69 degrees. On Pluto they would be 34 and 67 degrees, and astronomical twilight would end at 101 — a depression greater than a right angle, past the point directly underfoot, which the Sun can never reach. That is not a statement that Pluto’s sky never darkens. It is the scaling announcing that it has left its range, which the last figure below shows directly.

A world with no air, and a world that never turns #

The rule has two limiting cases worth stating. A world with no atmosphere has a scale height of zero and a twilight of zero degrees: on the Moon the Sun sets into immediate darkness. Astronauts orbiting the Moon before lunar sunrise sketched faint streamers above its horizon, and for a time these were attributed to dust lofted above the surface; measurements from a spacecraft built partly to look for that dust found far too little of it, and the streamers are now thought to be the Sun’s corona and the zodiacal light, which have nothing to do with the Moon.

The other limit is a world whose day never ends. A planet orbiting close to its star is usually locked to it, with one face permanently lit, and on such a world the Sun does not set at all; it stands still in the sky, low near the line between day and night. Twilight there is not an interval of time but a place: a ring round the planet, a few units of $\sqrt{2H/R}$ wide on the night side of that line, in permanent dusk. For a hot giant planet, with a scale height of a couple of hundred kilometres on a radius of seventy thousand, the unit is about four and a half degrees and the ring about thirteen degrees of longitude wide — a band of sky lit from below the horizon by a star that never moves, on a world whose extended atmosphere is itself visible against its star.

Where the scaling breaks #

Drawn in their own units, the Earth and Mars fall onto the parabola closely. Titan and Pluto both reach ten magnitudes early, at 2.64 units rather than 3.03, and they do so despite having optical depths that differ by a factor of two hundred. The optical depth is not the cause. The angle is.

The unit was built on the small-angle form of the shadow’s height, $R\,d^2/2$. At the depressions that matter on Titan and Pluto — thirty and forty-four degrees — the true height, $R(\sec d - 1)$, has pulled well ahead of it: at 44 degrees it is a third higher. A shadow that climbs faster than the parabola switches the haze off sooner, and the square of the ratio of the two unit counts, $(3.03/2.64)^2$, is exactly that third. The same effect is why the smallest planet in the radius sweep reached ten magnitudes a fraction of a degree before the scaling predicted.

So the rule has a stated range. For any world on which $\sqrt{2H/R}$ is a few degrees, twilight lasts about three of those units, whatever the atmosphere is made of and however much of it there is. On worlds where the unit itself is tens of degrees, the geometry of a round planet cuts twilight shorter than the rule, and the full expression for the shadow’s height is needed.

The same geometry, seen from outside #

A twilight is sunlight crossing an atmosphere on a grazing path, seen from the ground. The same path seen from the other end is an occultation: a distant star passing behind a planet’s limb, its light crossing the atmosphere tangentially on its way to a telescope. As the star sinks through the atmosphere its light fades, not suddenly as it would behind an airless edge but over a time set by how quickly the atmosphere thickens with depth — which is to say, by its scale height. For a thin atmosphere most of the fading is refraction rather than scattering, the lower layers bending the starlight more strongly than the upper ones and spreading it out, but the length scale that controls it is the same.

That is how Pluto’s atmosphere was found. Observations of a star disappearing behind Pluto in 1988 showed it fading gradually rather than blinking out, and the shape of the fade gave the scale height of a gas no one had yet seen directly. Spacecraft use the same geometry with their own radio signals, passing behind a planet as seen from the Earth, and the way the signal bends and fades gives the profile of the atmosphere they have just flown past. The quantity that fixes how long the sunset glows on a world is the quantity every one of these measurements is built to extract.

What the layer model leaves out #

Real atmospheres are not isothermal. The Earth’s air changes its scale height with temperature, most sharply at the stratosphere, and the full calculation with a standard atmosphere, rather than a single scale height, gives the end of single-scattering twilight at 9.25 degrees instead of 8.6.

Light scattered twice. The observed end of astronomical twilight on the Earth, at 18 degrees, is twice what single scattering gives, because light scattered first in the bright sky near the horizon and then again towards the zenith keeps the sky faintly lit long after the singly scattered light is gone. On Titan, with a haze of optical depth four, multiple scattering dominates everything.

Particles are not isotropic scatterers. Dust and haze scatter strongly forward, so the brightness of a Martian or Titanian twilight depends much more on where in the sky one looks than the Earth’s does, and the zenith is not the whole story.

And the Sun does not set vertically. The times quoted are for an observer on the equator, where the Sun descends straight down. At higher latitudes the Sun’s path meets the horizon at a slant, the same depression takes longer to reach, and on any world with a tilted axis there are places and seasons in which twilight never ends.

One ratio, and everything it controls #

Nothing in the Earth’s twilight convention of 18 degrees is fundamental, and nothing in the physics of twilight on other worlds is exotic. A shadow climbs as the square of an angle; the air above it thins by a factor of $e$ per scale height; and the angle over which a sky goes dark is the square root of twice the one length divided by the other. It is 3 degrees on the Earth and 17 on Pluto, and the sky on each goes dark over about three of them.

How long that takes is a second, independent factor: the angle, divided by 360 degrees, times the length of the solar day. The two multiply, and a world can have a long twilight for either reason. Titan’s is long in angle and long in day. Venus has a scattering layer above its clouds and a planet nearly the Earth’s size, so its twilight angle is Earth-like, but its solar day is 117 Earth days long, and a twilight no wider than the Earth’s ten degrees would take the Sun more than three days to cross.

Still open: the light that keeps the sky lit to eighteen degrees #

Every calculation here scatters each photon once, and on the Earth that puts the end of twilight near nine degrees of depression. The sky is observed to stay measurably lit to eighteen. The difference is light scattered twice — first in the bright low sky towards the sunset, where the sunlight still reaches the air, and then again up to the zenith — and the question is how that second scattering depends on the same ratio of lengths: whether a world whose single-scattering twilight is long also has a second, fainter twilight twice as wide, and how much of the Earth’s astronomical twilight is the geometry of a second bounce rather than of the shadow.