Astronomy & Celestial Mechanics
the sky, computed
Generator

The sidereal-day generator

Why the solar day is 3m 55.91s longer than the sidereal one
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.

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.

4 essays call sidereal-day. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

Integrating the measured recession back: the Moon reaches the Earth 1.54 Gyr ago. The Earth–Moon separation and the length of the Earth's day, integrated backwards from the measured present recession rate of 3.83 cm per year. Constant-Q tidal friction makes a^(13/2) linear in time, so the history is a single line in a variable nobody plots, and it is calibrated to the laser-ranging measurement rather than to a modelled k₂/Q — the k₂/Q it implies is 0.0257, or Q = 11.6 for the Earth's k₂ of 0.299, which is a startlingly dissipative Earth. Run back at that rate the separation reaches zero 1.54 Gyr ago and crosses the Roche limit at 2.88 Earth radii only 4 years before it, so the drawing is cut off there rather than extrapolated. The Moon is 4.5 Gyr old, so this is a refutation and not a date: the present rate cannot have been the rate, and a mean Q of 34 — drawn dashed, reaching 4.51 Gyr — is the sort of value the age requires. Tidal rhythmites at 620 Myr put the day at 21.9 h and the Moon at 96.5 per cent of its present distance, and this history reads 20.1 h and 92.4 per cent — too fast and too close, which is the same failure the zero crossing is. Day length follows from total angular momentum, 23.93 h today, 9.84 h at half the present lunar distance and 4.97 h at the Roche limit, and depends on the separation alone: it is the same curve whatever Q is. The rate of lengthening the recession requires is 2.10 ms per century, against a tidal total of about 2.3 including the Sun's tide, which slows the Earth without moving the Moon, and an observed 1.75 from ancient eclipses and occultations — the shortfall being the Earth's moment of inertia falling as the mantle rebounds from the last glaciation. Gravitation

A day five hours long

The tidal bulge leads, so the Earth's spin is being paid into the Moon's orbit. Run the measured payment backwards and two curves come out of one integration — a timeline that is refuted by the Moon's own age, and a day length that is refuted by nothing.

 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 observed sky

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.

 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. The observed sky

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.

 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. The observed sky

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.

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