Astronomy & Celestial Mechanics
the sky, computed
The observed sky

Three definitions of night

Twilight ends at three different depression angles, and each threshold is a statement about what can no longer be done. Its length is not a duration but a rate — how fast the Sun goes down — and above one latitude the deepest of the three never arrives at all.

Assumes Seasons and Refraction.

17 min read 8 figures The sky as a clockMeasuring the unreachable

Sunset is an event and night is not. Between them lies an interval with no natural boundary at its far end, since the sky’s brightness declines smoothly through six orders of magnitude and stops at whatever floor the airglow and the zodiacal light and the stars themselves provide — a floor that is itself a background nobody can subtract exactly.

So the end of twilight has to be defined rather than observed, and it has been defined three times, by three trades with three different requirements — an instance of the rule that runs through every quantity the sky is measured in, that a threshold is a decision before it is an observation. Each definition is a depression angle — how far the Sun’s centre is below the horizon — and each marks the point at which a particular task becomes impossible.

Twilight at latitude 52°. Solar altitude through the second half of the day at 52°, on June solstice, equinox, December solstice, with the four thresholds that define twilight marked. Sunset is at −0.833° rather than at 0° because the Sun's own semi-diameter and the atmosphere's refraction together lift it by that much when it is geometrically already down. Civil twilight ends at −6°, when the brightest stars appear and outdoor work stops; nautical at −12°, when the horizon can no longer be seen against the sky and a sextant becomes useless; astronomical at −18°, when the sky stops contributing to a photometric measurement. Their durations at equinox here are 34 min, 40 min, 42 min — twilight is not a fixed length, it is the reciprocal of how steeply the Sun descends, and the Sun descends at an angle of about 90° − φ to the horizon. At this latitude the curve for June solstice never reaches −18° at all: astronomical twilight does not end, and there is no astronomically dark night from about 48.6° upward.
Fig. 1 The three thresholds, and the Sun’s altitude falling through them at one latitude. Sunset itself is at −0.833°-0.833°, not at zero, because the Sun’s semi-diameter and the atmosphere’s refraction together hold it up by that much when it is geometrically already down. Civil twilight ends at −6°-6°, when artificial light becomes necessary and the brightest stars appear. Nautical at −12°-12°, when the sea horizon can no longer be made out against the sky and a sextant becomes useless. Astronomical at −18°-18°, when the sky stops contributing measurably to a photometric measurement. The thresholds are angles, and the durations are what a latitude makes of them.

The one geometrical fact

The Sun’s altitude, at declination δ\delta seen from latitude φ\varphi at hour angle HH, is

sin⁡a=sin⁡φsin⁡δ+cos⁡φcos⁡δcos⁡H,\sin a = \sin\varphi\sin\delta + \cos\varphi\cos\delta\cos H,

and every statement in this essay comes from that expression and from nothing else. It gives the length of the day, the moment of sunrise, and — solved for the HH at which aa equals −6°-6°, −12°-12° or −18°-18° — the length of twilight.

The rearrangement is worth having explicitly, because the two cases where it fails are the interesting ones:

cos⁡H=sin⁡a−sin⁡φsin⁡δcos⁡φcos⁡δ.\cos H = \frac{\sin a - \sin\varphi\sin\delta}{\cos\varphi\cos\delta}.

If the right-hand side is less than −1-1, the Sun never gets that low: the threshold is never reached and twilight of that kind never ends. If it is greater than +1+1, the Sun never gets that high. Neither is an error condition; both are places.

The Sun's altitude through the day at latitude 52°. Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day.
Fig. 2 The same function in its more familiar use. Solar altitude through the day at one latitude, for the two solstices and the equinox: where a curve crosses the horizon is sunrise or sunset, and the width between crossings is the length of the day. The tilt is what separates the three curves — the Sun’s declination runs from +23.44°+23.44° to −23.44°-23.44° and back once a year — and the same tilt is what will separate the three twilight curves below. Twilight is not a different phenomenon from the seasons; it is the same curve read a few degrees lower down.

Twilight is a rate, not a duration

The most useful way to think about the length of twilight is to stop thinking about it as a length. The Sun descends along a path inclined to the horizon at an angle of roughly 90°−φ90° - \varphi, and it covers 15°15° of hour angle per hour. What decides how long it takes to fall from 0°0° to −18°-18° of altitude is therefore the steepness of that descent.

At the equator the descent is nearly vertical: eighteen degrees of altitude takes eighteen degrees of hour angle, and twilight is about seventy minutes. At high latitude the path is oblique, the Sun slides sideways more than it drops, and the same eighteen degrees of altitude take much longer.

How long twilight lasts, through the year and up the globe. The duration of evening twilight — sunset to the Sun reaching −18° — against day of the year, at 0°, 30°, 45°, 52°, 60° of latitude. Day 0 is the March equinox. At the equator it is 1.14 h and barely varies: the Sun goes down perpendicular to the horizon and crosses 18° of altitude in the time it takes to turn 18° of hour angle. Away from the equator the descent is oblique, twilight lengthens, and at 60° the June curve reaches 4.44 h — where the curve is cut off, the Sun never reaches −18° at all and twilight lasts until dawn. The quantity being drawn is a rate rather than a place: how fast the Sun descends, which is what the sky's brightness at a given clock time actually depends on. What this cannot show is the sky itself — how bright twilight is depends on scattering in a layer this figure has no model of, and only on the geometry through the depression angle.
Fig. 3 The consequence, through a whole year and up the globe. The duration of evening twilight — sunset to the Sun reaching −18°-18° — at five latitudes. At the equator it is 1.14 hours and barely varies from January to December, because the Sun’s path meets the horizon at nearly a right angle whatever the season. Away from the equator the descent is oblique, twilight lengthens, and the seasonal variation appears — longest at the solstices, shortest near the equinoxes, because the declination is changing fastest at the equinoxes and the Sun is therefore moving obliquely for less of the time. At 60° the June curve leaves the top of the frame: there, twilight lasts until dawn.

The latitude where the deepest threshold stops arriving

The condition for astronomical twilight never to end at midsummer has a one-line answer. At local midnight the Sun is at its lowest, and its altitude there is

amin⁡=φco−(90°−δ)⇒amin⁡=δ−(90°−φ) in the north,a_{\min} = \varphi_{\text{co}} - (90° - \delta) \quad\Rightarrow\quad a_{\min} = \delta - (90° - \varphi)\ \text{in the north},

so at the June solstice, with δ=23.44°\delta = 23.44°, the Sun fails to reach −18°-18° once

φ>90°−23.44°−18°=48.56°.\varphi > 90° - 23.44° - 18° = 48.56°.

That is a latitude that runs through northern France, the tip of Nova Scotia, and just north of Vancouver. Above it, there is no astronomically dark night in June at all — and above 60.5°60.5°, none for four months of the year. Above 54.5°54.5°, nautical twilight also fails to end at midsummer, which is why a sextant is useless at sea north of Denmark in June.

Twilight at latitude 66°. Solar altitude through the second half of the day at 66°, on June solstice, equinox, December solstice, with the four thresholds that define twilight marked. Sunset is at −0.833° rather than at 0° because the Sun's own semi-diameter and the atmosphere's refraction together lift it by that much when it is geometrically already down. Civil twilight ends at −6°, when the brightest stars appear and outdoor work stops; nautical at −12°, when the horizon can no longer be seen against the sky and a sextant becomes useless; astronomical at −18°, when the sky stops contributing to a photometric measurement. Their durations at equinox here are 51 min, 63 min, 75 min — twilight is not a fixed length, it is the reciprocal of how steeply the Sun descends, and the Sun descends at an angle of about 90° − φ to the horizon. At this latitude the curve for June solstice never reaches −18° at all: astronomical twilight does not end, and there is no astronomically dark night from about 48.6° upward.
Fig. 4 And the case in which the horizon threshold itself fails. At 66° — just inside the Arctic Circle — the June solstice curve does not cross −0.833°-0.833° at all: the Sun does not set. The other two curves behave normally. What this figure shows and the standard account of the midnight Sun does not is that the interesting boundary comes far earlier: the astronomical threshold fails at 48.6°, seventeen degrees of latitude before the Sun itself stops setting, so most of Europe and most of Canada have already lost their dark nights before anybody has seen a midnight Sun.

The 0.833 degrees at the top

The first threshold is the one with physics in it rather than definition. Sunset is not when the Sun’s centre reaches altitude zero; it is when its upper limb disappears, which is 16′16' of semi-diameter lower, and the atmosphere has bent the whole thing upward by about 34′34' at the horizon. The two together are 50′=0.833°50' = 0.833°.

Refraction against apparent altitude. How far the air lifts an object, in arcminutes, against where the object appears to be. The lift is 34.5′ at the horizon — larger than the Sun's own diameter — and falls below one arcminute above 45°. At the horizon it is also the least reliable number in positional astronomy, because it depends on the temperature profile of the air the sight line crosses.
Fig. 5 The bending, against altitude. Refraction is negligible overhead and rises steeply near the horizon, reaching about 34 arcminutes — more than the Sun’s own diameter — at altitude zero. So the Sun that appears to be sitting on the horizon is entirely below it, and every sunset is watched after it has happened. The curve is steep enough that the lower limb is lifted more than the upper, which is why the setting Sun is visibly flattened.
The Sun's disc at the horizon, before and after the air. The Sun as geometry places it, left, and as it is seen, right, at the instant the lower limb appears to touch the horizon. Refraction lifts the lower limb by 34.5 arcminutes and the upper limb by 29.3, so the disc is squashed vertically to 26.8′ — 84 per cent of its true height — while its width is untouched. Both limbs of the true disc are below the true horizon at this moment: the Sun has already set, and is still being watched.
Fig. 6 The same effect on the shape of the disc. The lower limb is refracted more than the upper because refraction varies so sharply with altitude, so the Sun is squashed vertically by about a fifth at the horizon while keeping its full width. That distortion is a direct measurement of the gradient of the refraction curve — it is not an atmospheric curiosity but the derivative of the previous figure, visible to the naked eye.

The refraction figure carries a caution the others do not. It is computed for a standard atmosphere, and the real value at the horizon varies by several arcminutes with temperature and pressure — enough to move sunset by a minute or two, and in extreme inversions to produce mirages in which the Sun is seen after it should have set by a considerable margin. Of the four altitudes in this essay, only the first depends on the weather, and it is the only one anybody notices.

The definitions, and who made them

Each of the three angles has an owner, and the reasons are worth separating because only one of them is about the sky.

Civil twilight, −6°-6°. The working definition is that ordinary outdoor activity can continue without artificial light, and it is a legal quantity in many jurisdictions: it appears in lighting-up times for vehicles, in the hours during which certain kinds of hunting are permitted, and in the definition of when an aircraft’s navigation lights must be shown. The angle was chosen because the illumination at −6°-6° is around 3 lux, roughly the level at which reading outdoors stops.

Nautical twilight, −12°-12°. This one has a genuine measurement in it. A sextant altitude is measured from a body down to the sea horizon, so the horizon has to be visible at the same moment as the star. Before sunset the horizon is clear but the stars are not out; after −12°-12° the stars are out but the horizon has gone. The window between the two is the only time a celestial fix can be taken at sea, and it is about half an hour at the equator and rather longer further north. Every navigator’s evening was organised around it.

Astronomical twilight, −18°-18°. The most recent and the most arbitrary of the three, and the only one defined by what an instrument cannot tolerate rather than by what a person cannot do.

Twilight at latitude 10°. Solar altitude through the second half of the day at 10°, on June solstice, equinox, December solstice, with the four thresholds that define twilight marked. Sunset is at −0.833° rather than at 0° because the Sun's own semi-diameter and the atmosphere's refraction together lift it by that much when it is geometrically already down. Civil twilight ends at −6°, when the brightest stars appear and outdoor work stops; nautical at −12°, when the horizon can no longer be seen against the sky and a sextant becomes useless; astronomical at −18°, when the sky stops contributing to a photometric measurement. Their durations at equinox here are 21 min, 24 min, 24 min — twilight is not a fixed length, it is the reciprocal of how steeply the Sun descends, and the Sun descends at an angle of about 90° − φ to the horizon. The critical latitude is 48.56° — 90° − 23.44° − 18° exactly — above which the June solstice has no astronomically dark night anywhere.
Fig. 7 The trades’ window at low latitude, where it is narrowest. At 10° the Sun falls almost vertically and the whole of twilight — sunset to −18°-18° — takes about seventy-five minutes, of which the navigator’s usable half is around twenty-five. That is the geometry a sailing ship’s evening routine was built around, and the reason it is a routine at all: there is no waiting for better conditions, because the interval does not lengthen. At 52° the same window is roughly twice as long and at 60° in June it does not close.

What the geometry cannot say

The three thresholds are defined in terms of the Sun’s position, which is exactly computable. What they are for is the brightness of the sky, which is not.

The sky during twilight is lit by sunlight scattered in the upper atmosphere — a shell of illuminated air whose lower boundary rises as the Sun sinks, so that later in twilight the light comes from higher and thinner layers. That is why the brightness falls roughly exponentially with depression angle, by about a factor of 100 for every six degrees, and why the three thresholds are roughly evenly spaced in the logarithm rather than in anything else.

But the constant of proportionality is not a matter of geometry. It depends on aerosol content, on ozone, on the state of the upper atmosphere, and on the direction being looked at; the sky is not uniformly bright during twilight, and the western sky at −12°-12° can be a magnitude brighter than the eastern. A depression angle is a proxy, and the thresholds are conventions chosen so that the proxy is usually good enough. Nothing in the figures above is a statement about how bright the sky actually is on any given evening.

Why an observatory cares

The astronomical threshold exists because of a specific measurement. Sky background adds noise to every photometric measurement in proportion to its brightness, and −18°-18° is roughly where the residual scattered sunlight drops below the natural airglow — so pushing further buys nothing, and stopping earlier costs signal-to-noise on faint objects.

That is a threshold with an instrument in it, and it moves. For a large telescope working at the sky-background limit on faint galaxies, the practical end of twilight is nearer −15°-15° in the red and past −18°-18° in the blue, because the scattered component is bluer than the airglow. For bright-object work it can be −10°-10° and nobody minds.

Dawn is not the reverse of dusk

The figures above are all drawn for the evening, and the geometry is symmetric about local midnight, so morning twilight is the mirror image of evening twilight to the precision of the declination’s change over one night. That is a fraction of a minute near the solstices and about four minutes near the equinoxes, where the Sun’s declination moves fastest — enough that at the equinox in northern latitudes the morning twilight is measurably shorter than the previous evening’s.

The asymmetry that anybody actually notices is not geometric. The atmosphere is coldest just before dawn and warmest in the late afternoon, so refraction near the horizon differs between the two, and the aerosol load is generally lower in the morning because a night of settling has removed some of it. Both shift the first threshold and neither touches the other two.

There is a third asymmetry with no atmosphere in it at all: the eye. Adaptation to darkness takes about thirty minutes for the rods to reach full sensitivity, so an observer who has been outside through the whole of evening twilight is dark-adapted by the end of it, and one who steps outside at the end of morning twilight is not. The astronomical threshold assumes an instrument and says nothing about an observer, and every naked-eye description of when the Milky Way becomes visible is a description of the observer as much as of the sky.

The clock underneath

One assumption has been made silently throughout and is worth naming, because this site’s other essays about the sky spend their time on it. Every calculation above uses the hour angle as a uniform measure of time — 15°15° per hour — which is true of sidereal time and not of the Sun.

What is left when the Sun is gone

The eighteen-degree threshold is not arbitrary, and the reason it sits there is a statement about what the night sky is made of.

Below the horizon by that much, the Sun’s scattered light has fallen below the sky’s own natural emission — so going deeper buys nothing, because something else is now the floor. That floor has four components and only one of them is astronomical in origin.

Airglow dominates. The upper atmosphere emits its own light, chiefly from hydroxyl molecules formed during the day and radiating in the near infrared through the night, with contributions from atomic oxygen at 557.7 nanometres and from sodium. It is the reason the night sky is never black, and it is variable — brightening and fading by tens of per cent over hours, in waves that cross the sky, driven by atmospheric dynamics.

Zodiacal light is sunlight scattered by interplanetary dust, concentrated along the ecliptic and visible as a faint cone after evening twilight. It contributes a substantial fraction of the total away from the Milky Way.

Integrated starlight is the light of the stars too faint to resolve, and diffuse galactic light is starlight scattered by interstellar dust. Together they make the Milky Way a visible structure rather than a collection of points.

The proportions matter for what a dark site is worth. At a good observatory the natural sky in the visual band is about 21.9 magnitudes per square arcsecond, and the airglow is a third to a half of it — so no site anywhere is darker than the atmosphere above it, which is one of the arguments that put telescopes in orbit and is why the deepest surveys of faint diffuse structure are done from space.

The night that has stopped arriving

The geometry in this essay computes when the Sun is far enough below the horizon. It says nothing about whether the sky is dark then, and over most of the inhabited world it is not.

Artificial light escaping upwards is scattered by air molecules and aerosols back down, producing a skyglow that is brightest towards the nearest town and present overhead. It falls off with distance from a source roughly as the inverse two-and-a-half power, which is slow enough that a city of a million people brightens the sky measurably a hundred kilometres away.

The scale of it was mapped in 2016 from satellite radiance data calibrated against ground measurements: about eighty per cent of the world’s population, and over ninety-nine per cent of the population of Europe and North America, lives under a sky brighter than the natural one, and about a third cannot see the Milky Way at all.

There is a wavelength dimension that made the recent situation worse. Rayleigh scattering goes as the inverse fourth power of the wavelength, so blue light is scattered far more efficiently than yellow — and the conversion of street lighting from sodium lamps, whose emission is concentrated in two yellow lines, to broadband white light-emitting diodes with a strong blue component increased the skyglow produced per lumen even where the total light output fell.

So the astronomical definition of night is now a statement about the Sun and not about the sky, in most of the places people live. The three thresholds still mark the same geometry they marked when they were codified, and only in a few remaining places do they still describe what an observer would see.

There is a measurable consequence for the thresholds themselves. Under a sky brightened by a factor of several, scattered sunlight drops below the ambient level well before the Sun reaches eighteen degrees — so an observer in a city reaches their own darkest sky at nautical twilight and gains nothing from waiting. The threshold that matters is where the Sun’s contribution falls below whatever the local floor is, and the eighteen degrees is the answer for a floor made of airglow alone.

Night for something that is not an astronomer

The three thresholds were written for people looking up, and the biological night they approximate belongs to organisms with quite different sensitivities.

Many animals navigate by light levels far below anything the definitions distinguish. Dung beetles orient by the Milky Way; moths and other nocturnal insects fly by patterns of polarised moonlight; and several species of bird calibrate a magnetic compass against the pattern of the sky at dusk, which requires that the relevant part of the twilight sky be visible at the right moment.

The transitions that matter to them are therefore not at six, twelve and eighteen degrees but wherever their own thresholds fall, and those are spread across the whole range and below it. A light level a person would call complete darkness is a working illumination for a species whose eyes are two orders of magnitude more sensitive.

That is the reason skyglow is treated as an ecological question rather than only an astronomical one. An added floor of scattered light removes the cues at the bottom of the range first — the faintest stars, the polarisation pattern, the difference between a moonlit night and a dark one — and those are precisely the cues that the animals using them cannot replace.

The geometry in this essay is exact and the thresholds drawn on it are conventions, chosen by one species for one purpose; the underlying quantity is a continuous fall in illumination through five orders of magnitude, and where it is divided depends entirely on who is doing the dividing.

That continuity is worth holding onto, because the thresholds are so often quoted as though they were features of the sky. Nothing happens at twelve degrees. What happens is that the illumination passes a value somebody found useful, and the usefulness was about seeing a horizon from a ship.

The same is true of the other two thresholds, which were fixed by what a sailor and a soldier needed rather than by anything in the atmosphere.

Both are still in the almanacs, and both are still useful for the purposes they were invented for, which is a reasonable fate for a definition.

One more latitude covers the case where all three definitions nearly coincide.

Twilight at latitude 30°. Solar altitude through the second half of the day at 30°, on June solstice, equinox, December solstice, with the four thresholds that define twilight marked. Sunset is at −0.833° rather than at 0° because the Sun's own semi-diameter and the atmosphere's refraction together lift it by that much when it is geometrically already down. Civil twilight ends at −6°, when the brightest stars appear and outdoor work stops; nautical at −12°, when the horizon can no longer be seen against the sky and a sextant becomes useless; astronomical at −18°, when the sky stops contributing to a photometric measurement. Their durations at equinox here are 24 min, 28 min, 28 min — twilight is not a fixed length, it is the reciprocal of how steeply the Sun descends, and the Sun descends at an angle of about 90° − φ to the horizon. The critical latitude is 48.56° — 90° − 23.44° − 18° exactly — above which the June solstice has no astronomically dark night anywhere.
Fig. 8 Twilight at thirty degrees. The Sun descends nearly vertically, so the three thresholds are crossed within about an hour and a half of one another and the distinction between civil, nautical and astronomical twilight is a matter of minutes — which is the opposite of the high-latitude case, where it is a matter of months.

Where the ladder goes next

The obvious next rung is the brightness itself: a computed twilight sky, with single-scattering in a spherical shell, which turns the depression angle into a magnitude per square arcsecond and shows why the sequence of colours occurs in the order it does. Past that is the same geometry applied elsewhere — twilight on Mars is longer and much brighter, because the dust extends higher than air does, and the depression angles that define it would have to be redefined.

Where this goes next

The Sun's altitude through the day at latitude 52°. Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day. The Sun's path, and the tilt that makes the seasons Summer is not when the Earth is closest to the Sun — that happens in January. It is when the Sun climbs higher and stays up longer, and both come from a 23.4° tilt. Refraction against apparent altitude. How far the air lifts an object, in arcminutes, against where the object appears to be. The lift is 34.5′ at the horizon — larger than the Sun's own diameter — and falls below one arcminute above 45°. At the horizon it is also the least reliable number in positional astronomy, because it depends on the temperature profile of the air the sight line crosses. The Sun sets before it sets At the moment the Sun's lower edge appears to touch the horizon, the whole of it is already below. Refraction lifts it by more than its own diameter, squashes it while it is there, and makes every sunrise and sunset time in every almanac a statement about the air rather than about the sky. Resolution stops improving at 10 cm of aperture. Angular resolution against aperture at 500 nm, both logarithmic. The falling line is diffraction alone, 1.03 λ/D, which is what a telescope in vacuum delivers and has no floor. The curve is the same telescope under an atmosphere of Fried parameter r₀ = 10 cm, combining diffraction and seeing in quadrature: it follows the diffraction line while D < r₀ and then bends onto a plateau at 0.98 λ/r₀ = 1.01″. An amateur's 100 mm at 0.1 m would resolve 1.062″ above the air and delivers 1.47″ through it; a metre at 1 m would resolve 0.106″ above the air and delivers 1.02″ through it; the VLT at 8.2 m would resolve 0.013″ above the air and delivers 1.01″ through it; the ELT at 39 m would resolve 0.003″ above the air and delivers 1.01″ through it. At 39 m the atmosphere is costing a factor of 371: the aperture is 390 coherence lengths across and every one of the 152,100 patches it collects arrives with a phase of its own. What the extra aperture still buys is photons and speckles, and those two are what adaptive optics and speckle interferometry respectively spend to get the falling line back. A ten-metre mirror that resolves like a ten-centimetre one The atmosphere delivers a wavefront in patches about ten centimetres across, and an aperture larger than a patch collects patches rather than detail. Resolution stops improving at that size — and what the extra aperture keeps buying is photons and speckles, which is why there are two entirely different ways out.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Celestial sphereDepression angleHorizonHour angleLatitudeObliquityRefractionSky brightnessSolar altitudeTwilight