Astronomy & Celestial Mechanics
the sky, computed
Concept

Sidereal time — where it appears

Time reckoned by the rotation of the Earth relative to the stars, running four minutes a day ahead of solar time. It is what a telescope is pointed by, since a star's hour angle is the sidereal time minus its right ascension.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

The sky from latitude 52°. The celestial sphere seen from latitude 52 degrees. The pole stands 52 degrees above the horizon, the celestial equator meets the horizon due east and west, and a star at declination 20 degrees traces the drawn circle once a day. Everything below the horizon is drawn faint.

The sphere that is not there, and why it is still the right model

The stars are at wildly different distances and the celestial sphere is a fiction. It is also the most useful fiction in observational astronomy, because for pointing at things, distance is exactly the information to throw away.

sky · Celestial sphere One star, two coordinate systems, at latitude 52°. The equatorial grid and the horizon grid drawn on the same sphere for an observer at latitude 52°. The star marked has declination 20° and hour angle -40° in the first, and altitude 45.5° and azimuth 239.4° in the second. The two frames differ by a single rotation through the co-latitude 38°, which is why the celestial pole stands 52° above the northern horizon.

Where a star is depends on who is asking

The sky needs two coordinate systems because two different things stay still in it — the observer's horizon and the stars themselves. One rotation converts between them, and the angle of that rotation is the time.

sky · Celestial sphere 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 A clock that has lost 5.7 hours in 2720 years. ΔT = TT − UT1, the accumulated difference between a uniform time scale and the Earth's own rotation, from 700 BC to 2020, on a logarithmic scale. The points are the published record; the smooth curve is the parabola 31.94·u² fitted to the entries at or before 1000, with u in centuries from 1820. It is a parabola and not a line because the Earth is not merely slow, it is slowing: a day lengthening at a constant rate makes a clock fall behind by the integral of the lag. That coefficient is a length-of-day rate and nothing else — 31.94 = ½ × r × 36525 days per century gives r = 1.75 ms per century, against the 1.78 ms computed from tidal angular momentum with no eclipse anywhere in the derivation. At 700 BC the offset is 5.7 hours, which is 85° of the Earth's rotation, and that is why the ancient measurements are records of the place a total eclipse was seen from rather than of the hour: the hour was never written down accurately enough to matter, and the shadow's track on the ground was.

Six kinds of second

The Earth is a clock that loses, and it has lost five and a half hours since 700 BC. That number was measured from the places ancient eclipses were seen from, not the times they were seen at — because the record carries a longitude and no clock.

sky · Timescales A sidereal clock that runs up to 1.15 seconds ahead or behind, with the Moon's node. The equation of the equinoxes from 1990 to 2030: apparent sidereal time, counted from the true equinox, minus mean sidereal time, counted from an equinox that only precesses. It is the nutation in longitude times the cosine of the obliquity, and it is drawn here from the four largest nutation terms, which carry it to about a hundredth of a second. The dominant term follows the Moon's node round its 18.61-year cycle with an amplitude of ±1.052 s; riding on it are a half-yearly term of ±0.081 s from the Sun and a fortnightly one of ±0.014 s from the Moon. Over this span the sum runs from −1.147 to 1.146 s. None of it is the Earth's rotation: it is the zero point of the clock moving, because the zero point is the intersection of the equator with the ecliptic and the equator nods.

A clock whose zero is moving

Sidereal time is counted from the equinox, and the equinox does not stay put. Its steady drift makes the sidereal day eight milliseconds short, its acceleration puts a quadratic term into the formula for sidereal time, and the Moon makes it nod by a second every nineteen years. The Earth rotation angle removes all three by counting from a point defined not to move along the equator.

sky · Sidereal time When the Orion Nebula can be observed from latitude 52°, through a year. Every night of 2027, from local noon to the following noon, for a station at latitude 52°. The shaded cells are the times at which the Orion Nebula (right ascension 5.59 h, declination −5.4°) stands at least 20° above the horizon while the Sun is more than 18° below it. The two outer curves are the start and end of astronomical darkness, which never comes on 64 nights around midsummer; the diagonal line is the object's transit, which arrives four minutes earlier each night and wraps through the whole day once a year. It crosses local midnight on 15 Dec, when the object stands opposite the Sun. The shaded season is the stretch of the year either side of that date in which the transit falls inside the dark hours. From this latitude the object transits at 32.6° altitude; it is observable on 181 nights for at least an hour, for 6.3 hours on the best of them (3 Jan), and for 860 hours in the year.

A right ascension is a date

An object can be observed only when two clocks agree — the sidereal clock that brings it high in the sky and the solar clock that makes the sky dark. The two drift apart by one turn a year, so every right ascension has a season, every latitude gives that season a different length, and an object that never sets is best observed at the opposite time of year from the one its right ascension names.

sky · Sidereal time

Named alongside it

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

EquinoxCelestial sphereDeclinationAxial precessionCircumpolarEpochEquation of timeHorizon coordinatesMeridian transitNutationRight ascensionSidereal day

All concepts