Astronomy & Celestial Mechanics
the sky, computed
Concept

Hour angle — where it appears

The angle westward from the meridian to a body, measured along the equator, which advances at fifteen degrees an hour and serves as a clock. It is what a telescope's right-ascension axis is driven in, and the difference between it and a star's right ascension is the local sidereal time.

Named by 5 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 Why the solar day is 3m 55.91s longer than the sidereal one. Two positions of the Earth one day apart, with the direction of a fixed star and the direction of the Sun marked at each. After one sidereal day — 23h 56m 4.0905s — the Earth has turned through exactly 360° and the star is back on the meridian; the Sun is not, because the Earth has moved 0.9856° along its orbit, and the further 0.9856° of turning takes 3m 55.91s. That is the whole of the difference: 366.2422 turns against the stars in the 365.2422 solar days of a year, one more turn than the 365.2422 against the Sun. Nothing here is to scale. The orbital arc is drawn at 40°, which exaggerates the real 0.9856° by 41 times, and the Earth's disc is about 4073 times too large for its orbit. The two star sight lines are drawn parallel because they are: a star's distance cannot be put on the same page as an orbit.

The day that is four minutes short

A star crosses the meridian 3 minutes 55.91 seconds earlier each night, and the same star slips 3 minutes 56.56 seconds a day against a civil clock. Those are two different numbers, and the gap between them is the one extra turn the Earth makes against the stars in every year.

sky · Sidereal time The extreme sunset and sunrise, neither of them on the December solstice. Sunset and sunrise in local mean solar time through the ninety days around the December solstice, at latitude 52°, computed from Kepler's equation and the tilt. Each is the solar noon — which is 12:00 minus the equation of time — plus or minus half the day arc, and the two ingredients have different stationary points. The day length is stationary at the solstice, exactly, because the declination is. The equation of time is not: it is changing at about 0.50 minutes a day there, moving solar noon steadily. So the sum is still changing after the day length has stopped, and the earliest sunset comes on 14 Dec — 9 days before the solstice on 23 Dec — while the latest sunrise comes on 31 Dec, 9 days after it. The shortest day is the solstice, and neither of its ends is extreme there. Nothing in this is an approximation or a correction: it is a sum of two functions with different stationary points, and the stationary point of a sum is not the stationary point of either.

The earliest sunset is not the shortest day

A sunset time is solar noon plus half the day length, and the two have different stationary points. The day length stops shortening at the solstice; the equation of time does not stop moving, so the sum keeps falling — and the earliest sunset comes days before the shortest day.

sky · Equation of time A Sun that runs backwards for 8.1 days. The rate at which the Sun moves across the sky of a planet with a 3:2 spin–orbit ratio and eccentricity 0.2056, in degrees of hour angle per Earth day, against days from perihelion, over one 87.97-day orbit. The rate is the spin rate minus the rate at which the Sun's direction turns because the planet moves along its orbit, and by Kepler's second law that orbital rate peaks at perihelion, at 1.551 times its mean. The spin is 1.5 times the mean orbital rate, so for 8.1 days around perihelion, from −4.0 to +4.0 days, the orbital rate wins, the rate is negative, and the Sun moves backwards across the sky by 1.11 degrees before resuming. At perihelion it is moving at 0.21 degrees a day in the wrong direction. Away from perihelion the Sun crosses the sky at up to 3.4 degrees a day, and a whole solar day, noon to noon, takes 175.9 days — two orbits.

A Sun that stops and runs backwards

On Mercury the equation of time is not a correction but a reversal. The planet turns three times for every two orbits, and near perihelion its orbital motion briefly outruns its spin, so the Sun halts, backs up by a degree over eight days and resumes. From one longitude that is three noons in a week; from another, a sunrise, a sunset and a second sunrise — and a slightly rounder orbit would have stopped it happening at all.

sky · Equation of time Solar noon by the clock, in four places on ordinary time. The clock time at which the Sun crosses the meridian, through a year, for London (0.13°W, clocks on UTC+0), Madrid (3.70°W, clocks on UTC+1), Vigo (8.72°W, clocks on UTC+1), Kashgar (75.99°E, clocks on UTC+8), with summer time where it is kept, from the end of March to the end of October. In London noon falls between 11:44 (4 November) and 13:07 (27 July); in Madrid noon falls between 12:58 (4 November) and 14:21 (27 July); in Vigo noon falls between 13:18 (4 November) and 14:41 (27 July); in Kashgar noon falls between 14:40 (4 November) and 15:10 (12 February). The smooth wave on each curve is the equation of time, the same ±16 minutes everywhere. The steps are summer time, a whole hour. And the vertical offset of each curve is where the place sits inside its time zone, which for a city in the far west of a wide zone is larger than both: noon near three in the afternoon, by the clock, is an ordinary consequence of one time zone spanning a large country.

The smallest term between a clock and the Sun

A clock and a sundial disagree for three reasons, and the one astronomy supplies — the equation of time, sixteen minutes at most — is usually the smallest of them. Where a place sits inside its time zone can put noon three hours after twelve, and summer time adds a whole hour on top. The equation of time is visible in ordinary life only where the other two happen to vanish.

sky · Equation of time

Named alongside it

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

Equation of timeMean solar timeCelestial sphereLatitudeObliquityRefractionSolar dayAnalemmaAxial precessionChronometerDay lengthDaylight saving time

All concepts