Astronomy & Celestial Mechanics
the sky, computed
Concept

Earths shadow — where it appears

The cylinder of darkness the Earth casts away from the Sun. After sunset it rises through the sky above an observer, darkening the air layer by layer and then switching off satellites as it climbs past their altitude.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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 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.

Sky brightnessTwilightAstronomical twilightDepression angleLatitudeLow earth orbitMultiple scatteringObserving conditionsOzoneRayleigh scatteringSatellite constellationsScale height

All concepts