Astronomy & Celestial Mechanics
the sky, computed
Concept

Sky brightness — where it appears

The surface brightness of the sky itself, in magnitudes per square arcsecond. On a moonless night at a dark site it is about 22 in visible light; through twilight it changes by more than ten magnitudes, and it sets the noise against which faint sources are measured.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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.

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.

sky · Twilight The zenith sky after sunset, as single scattering predicts it. The brightness of the zenith sky at 550 nm, in magnitudes per square arcsecond with brighter upward, against the Sun's depression below the horizon, computed by single scattering in a spherical atmosphere: US Standard Atmosphere densities, Rayleigh scattering, a 300 Dobson-unit ozone layer, and every photon scattered exactly once. The Earth's shadow climbs the zenith as R(sec d − 1) — 8.7 km at 3°, 35.1 at 6°, 142 at 12° — through air that thins by a factor of e every eight kilometres or so, so the brightness does not fall steadily: it falls 1.4 magnitudes a degree between 3° and 6°, and 2.7 a degree between 7° and 10°. The model gives 14.3 at the end of civil twilight and 27.0 at the end of nautical. The dashed line is the natural night sky, 21.9 magnitudes per square arcsecond. Single scattering reaches it at 9.25° of depression — 8.8 degrees before the 18° at which astronomical twilight is observed to end. The difference is not an error in the arithmetic; it is the light this model leaves out, scattered more than once.

The shadow that climbs the zenith

After sunset the sky overhead is lit only above the Earth's own shadow, and the shadow climbs as the square of the Sun's depression. Scattering computed once from that geometry predicts a sky that dims faster and faster and is as dark as night by nine degrees. The real sky takes eighteen, and the difference is light that has been scattered more than once.

sky · Twilight The zenith sky after sunset, with light scattered twice. The brightness of the zenith sky at 550 nm in magnitudes per square arcsecond, brighter upward, against the Sun's depression: light scattered once (dashed), light scattered twice (dash-dot), and their sum (solid), in a spherical atmosphere with US Standard densities, Rayleigh scattering and 300 Dobson units of ozone. The dashed horizontal line is the natural night sky, 21.9 magnitudes per square arcsecond. Once-scattered light reaches it at 9.2°; the sum, dominated beyond about seven degrees by light scattered twice, reaches it at 13.2°, and it falls to a hundredth of the night sky — five magnitudes fainter, about the smallest excess a careful measurement of the sky can detect — at 17.5°, where once-scattered light alone would have reached that level at 11.9°. The second order falls more slowly than the first because it can reach the zenith by two grazing hops — sunlight to the lit air low towards the Sun, and that air to the shadowed air overhead — each of which needs only part of the depression a single hop would. Orders beyond the second are left out, so the solid curve is a lower bound on the sky.

The night is late because the light bounces twice

Light scattered once can reach the zenith only while the Earth's shadow over the observer is still low in the air, and by that reckoning the sky should reach the darkness of night about nine degrees after sunset. It does not until eighteen. The difference is light scattered twice — first by the sunlit air hundreds of kilometres towards the sunset, then again by the shadowed air overhead — and adding it to the calculation moves the end of twilight almost exactly to where observers have always put it.

sky · Twilight Where in the sky satellites still shine. The whole sky above the horizon seen from below, zenith at the centre, north at the top and the Sun 20° below the western horizon. Shaded: directions in which a satellite at 550 km is sunlit — 71 per cent of the sky by solid angle. The lit region is a cap pressed against the western horizon and reaching up past the zenith, because satellites seen low in the west are far away and high above the shadow; the eastern sky is dark. Ignoring the penumbra moves its edge by under a degree.

The shadow that also switches off the satellites

After sunset the sky overhead darkens as the Earth's shadow climbs through the air, and the same shadow, climbing on past the top of the atmosphere, decides which satellites still catch sunlight. For a satellite five hundred and fifty kilometres up the zenith stays lit until the Sun is twenty-three degrees down — five degrees past the end of astronomical twilight — and at the latitudes of most large observatories' summers there is no hour of the night when it is not.

sky · Twilight

Named alongside it

The objects these essays reach for when they reach for this one.

TwilightAstronomical twilightDepression angleEarths shadowLatitudeMultiple scatteringRayleigh scatteringScale heightSolar depressionCelestial sphereEarth shadowHorizon

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