Astronomy & Celestial Mechanics
the sky, computed
Concept

Horizon — where it appears

The great circle ninety degrees from the observer's zenith, below which nothing is visible and near which refraction is largest. Its position depends on the observer rather than on the sky, so a horizon coordinate is valid for one place and one instant and must be converted before it means anything.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

sky · Refraction 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 How far across the ground a satellite can be seen from, at 0, 10, 30° masks. The footprint of a satellite — the largest angle at the Earth's centre between the point beneath it and a station that can still see it above an elevation mask — against altitude, on a logarithmic axis. λ = arccos(R cos ε / (R + h)) − ε. At low altitude it grows as the square root of the height, λ ≈ √(2h/R), drawn dashed for the horizon mask, so doubling a low orbit's altitude widens its footprint by only about forty per cent; far out it saturates, and no altitude sees past 90° − ε. At 420 km (a space station) the 0° footprint is 20.2°, a circle 2,254 km in radius holding 3.1 per cent of the Earth's surface. At 20,180 km (a navigation satellite) the 0° footprint is 76.1°, a circle 8,472 km in radius holding 38.0 per cent of the Earth's surface. At 35,786 km (geostationary) the 0° footprint is 81.3°, a circle 9,050 km in radius holding 42.4 per cent of the Earth's surface. Raising the mask costs most at low altitude: at 420 km a 10° mask takes 38 per cent off the footprint's radius and 62 per cent off its area, because most of what a low satellite could see is near the horizon, while at 35,786 km the same mask takes 20 per cent of the area.

The circle a station can see

A ground station can talk to a satellite only while the satellite is above its horizon, and the region it can do that from is a circle drawn round the point beneath the spacecraft. The circle grows as the square root of the altitude and then stops growing, the time spent inside it diverges at one altitude, and the stations that get the most contact are not where anyone would first put them.

spaceflight · Ground tracks

Named alongside it

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

AirmassApparent altitudeAtmospheric extinctionAtmospheric refractionCelestial sphereDepression angleElevation maskFootprintGeostationary orbitGreen flashGround stationGround track

All concepts