The day that is four minutes short

Point a telescope at a star, clamp it, and come back at the same clock time tomorrow. The star will have moved — westward, by about a degree, two fields of a low-power eyepiece. A month later it is fifteen degrees away. The sky does not keep the clock’s time, and the celestial sphere it is convenient to imagine it painted on turns at a rate measurably not one turn a day.

The discrepancy is small and it is exact. The Earth turns once with respect to the stars in 23 hours 56 minutes 4.0905 seconds, and once with respect to the Sun in 24 hours: the sidereal day and the mean solar day, and the difference between them is the whole subject here.

The claim: a year holds one more day than it holds days #

The tropical year contains 365.2422 solar days. It contains 366.2422 sidereal days. Not approximately one more — exactly one more, as an identity rather than a measurement.

That is stated plainly enough to be wrong. It would be wrong if the extra turn were two, or a half, or a function of the orbit’s eccentricity. It is one, it comes from the fact that the orbit closes, and a planet whose year were forty days long or forty thousand would have exactly one more sidereal day in it than solar days.

Everything else here follows from that single unit: the four minutes, the fact that a constellation is a season, the gear ratio of a telescope drive, the Moon’s two months, and — at the far end — an eight-millisecond disagreement about the length of a day that is the only everyday appearance of the precession of the equinoxes.

One extra turn, and the reciprocal that carries it #

The derivation is three lines, so here it is in full.

Rotation carries the observer’s meridian round at $\omega_{\rm rot} = 2\pi/T_{\rm sid}$, and orbital motion carries the direction of the Sun round in the same sense at $\omega_{\rm orb} = 2\pi/T_{\rm year}$. The Sun returns to the meridian when the meridian has gained a whole turn on it, so the solar day is set by the difference of the two rates:

$$\frac{2\pi}{T_{\rm sol}} = \frac{2\pi}{T_{\rm sid}} - \frac{2\pi}{T_{\rm year}}, \qquad\text{that is,}\qquad \frac{1}{T_{\rm sid}} = \frac{1}{T_{\rm sol}} + \frac{1}{T_{\rm year}}.$$

Multiplying through by $T_{\rm year}$ turns that into a statement about counting:

$$\frac{T_{\rm year}}{T_{\rm sid}} = \frac{T_{\rm year}}{T_{\rm sol}} + 1.$$

The “+1” is not an approximation, a correction, or a small term — it is the extra turn, and it is why the reciprocals add rather than the periods. Rearranged once more, the same relation gives the difference of the two days, $T_{\rm sol} - T_{\rm sid} = T_{\rm sid}T_{\rm sol}/T_{\rm year}$, which is 235.91 seconds for the Earth.

The sign is worth testing, and the cheapest test is to reverse the rotation. A retrograde planet’s meridian and the Sun’s direction move in opposite senses, so the rates add rather than subtract, the “+1” becomes a “−1”, and the sidereal day becomes the longer.

Two numbers, both about four minutes, and neither a rounding of the other #

Almost every account gives the drift as “about four minutes a day”. Two quantities answer to that description, they differ by two thirds of a second, and which is correct depends on what is being counted.

From one transit of a star to the next, the interval is one sidereal day and the clock advances by less than a whole day: the slip is $T_{\rm sol} - T_{\rm sid} = 235.91$ seconds, or 3 minutes 55.91 seconds.

Per civil day, the slip is $T_{\rm sol}/T_{\rm year} = 236.56$ seconds, or 3 minutes 56.56 seconds — larger, because a year holds 366.2422 transits and only 365.2422 days, so the same 24 hours of drift is divided among fewer intervals.

Both are the same 24 hours shared out, and each is one whole day divided by the days in a year — but they are different days: the first is $T_{\rm sid}/365.2422$, the second $T_{\rm sol}/365.2422$. Multiplying each by its own count returns 86,400 seconds exactly, a closure the figure refuses to be drawn without.

Their ratio is the prettiest thing here: $236.5556/235.9096 = 1.002737909$, which is $T_{\rm sol}/T_{\rm sid}$, the sidereal ratio itself, arriving unbidden. The two four-minute numbers stand in exactly the proportion the two days do, 0.2738 per cent apart, so quoting one where the other belongs is an error of precisely that size in the only quantity here. Accumulated over a year, that 0.65-second gap comes to 236 seconds — one whole four-minute step, which is the extra transit.

What a meridian transit actually measures #

The sidereal day is not read off a dial. What is observed is a star crossing a wire. The archetype is the Airy Transit Circle at Greenwich, in service from 1851 and for a century afterwards the origin of longitude on Earth. The measurement is austere: the observer notes the clock time at which the star’s image crosses each of several vertical wires, and averages them. Right ascension is then defined as the sidereal time of that transit, which is why it is quoted in hours rather than degrees — a clock reading promoted to a coordinate.

Timing successive transits of one star gives the sidereal day, and what the observation produces is a ratio: ticks of the observatory’s clock between two returns of the star, against ticks between two returns of the Sun. Both count the same artefact, so the ratio survives a bad clock, provided the clock is bad uniformly.

Turning that ratio into 86,164.0905 seconds requires something no transit circle can supply — a second defined independently of the Earth. Before 1967 there was none: the second was itself a fraction of the Earth’s rotation or its orbit, so the sidereal day’s length in seconds was nearer arithmetic than measurement. Only with the caesium standard did the day acquire an external unit, and it promptly turned out to be variable. The modern determination is different again: very-long-baseline interferometry measures the Earth’s orientation against a few hundred quasars — a direction with no detectable motion of its own.

The number a telescope drive is geared to #

The relation is also why equatorial mounts exist.

A drive tracking at the solar rate accumulates 15 arcseconds of trailing error — plainly visible on any night of decent seeing — after 365 seconds. The figure asserts that rather than reporting it, because it is an identity: the rate difference is $1296000''/T_{\rm year}$, so the time to build up $15''$ is $T_{\rm year}$ in seconds divided by 86,400, which is the number of days in a year. A count of days and an interval of seconds have come out as the same number, because 15 arcseconds is what the sky turns through in one second, so an error growing at one part in 365.2422 of the rate takes 365.2422 seconds to accumulate one second’s worth of sky.

Hence the gearing: a sidereal drive turns once in 23 hours 56 minutes 4 seconds, not once a day. The 0.27 per cent is no tolerance anybody could ignore — over a five-minute exposure the untracked drift is 12 arcseconds.

The solar day is not constant either #

The drift figure’s vertical axis is civil clock time, and that choice is doing quiet work. Against mean solar time the line is straight, because mean time is uniform by construction. Against a sundial it would not be.

Two things make the true solar day vary: the Earth’s orbital motion is not uniform, so the extra arc changes through the year, and the Sun moves along the ecliptic rather than the equator, so only its projection onto the equator counts. The slope of the drawn curve is the day’s excess over 24 hours, and it runs from 21.3 seconds short in mid-September to 29.9 seconds long in late December; the accumulation is the sixteen minutes by which the Sun is a bad clock. Neither effect touches the sidereal day. Of the two, the day nobody uses is the good clock.

The Moon keeps two months, for exactly this reason #

The best evidence that the extra turn is arithmetic rather than a fact about the Earth is that the Moon does it too, on a wholly different timescale.

The Moon returns to the same place among the stars in 27.32166 days — the sidereal month, $T_{\rm mon}$ — and to the same phase in 29.53059 days, the synodic month. The relation is the same reciprocal, with the sign that says both motions run the same way:

$$\frac{1}{T_{\rm syn}} = \frac{1}{T_{\rm mon}} - \frac{1}{T_{\rm year}}.$$

Substituting gives 29.5306 days, and the counting statement is that in a year the Moon makes 13.3694 circuits against the stars and 12.3694 against the Sun — one fewer, exactly. In both cases the reference is the Sun, and one turn a year is transferred from one count to the other. The saros sets three such months against each other rather than two, and phases are the observable that defines the longer one. One detail does not carry over, and it is the hinge of the next section. The Moon’s relation needs the sidereal year, 365.25636 days; the Earth’s needs the tropical year, 365.2422 days. Those are not the same year.

The frame the sidereal day is measured against is turning #

Sidereal time is not reckoned from a star. It is the hour angle of the equinox — the intersection of equator and ecliptic — and the equinox moves, regressing along the equator at 46.12 arcseconds a year as the pole swings round its 25,772-year circle. Hence the tropical year in the relation above: the Sun’s return to the equinox, not to a star. Feed the sidereal year into the identical relation and a different day comes out. The interval between two returns of the meridian to a genuinely fixed direction — the stellar day, which is what a quasar measures — is 86,164.099 seconds, longer than the sidereal day by 8.4 milliseconds. The clocks figure computes that from the 46.12 arcseconds a year, checks its sign, and checks it against the first-order form; the sign looks wrong at first, because a regressing equinox comes to meet the meridian and is reached sooner than a fixed star. The sidereal day is the shorter of the two, so the usual gloss of it as the rotation period with respect to the stars is false by 8.4 milliseconds a day, three seconds a year.

The two published rotation ratios — 1.002737909 against the equinox, 1.00273781 against the extragalactic frame — differ in the eighth decimal place, and the difference is entirely axial precession: eight milliseconds is what a 26,000-year wobble looks like divided into 86,400-second pieces.

The rotation is also slowing, a third thing again: tidal friction lengthens the day by about 1.75 milliseconds a century, so the sidereal day quoted here is a mean over an epoch and not a constant.

What the picture cannot show #

Nothing in the geometry figure is to scale. The orbital arc is drawn at 40° where a day’s motion is 0.9856°, and the Earth’s disc is about 4073 times too large for its orbit. Drawn honestly, the two positions of the Earth would be indistinguishable and the angle the figure exists to display a hairline.

The two sight lines to the star are drawn parallel, which erases a real measurement. Over one day the Earth moves 2.6 million kilometres, which for the nearest star shifts the apparent direction by about 0.013 arcseconds — far above the tens of microarcseconds Gaia reaches, so unplottable rather than negligible. Drawing the lines converging would put the parallax on the page some twenty orders of magnitude too large.

A rotation rate is drawn as though it were constant. The length of day fluctuates by about a millisecond over a year with the atmosphere’s seasonal angular momentum, wanders further on decadal timescales through core–mantle coupling, and steps after a great earthquake. The drift line is straight to a part in $10^{8}$, which is generous, and not exact.

Nutation is absent. The equinox’s motion is drawn as a smooth regression, but superposed is an 18.6-year nod of 9.2 arcseconds driven by the Moon’s node, which sidereal time carries as the equation of the equinoxes — up to about 1.1 seconds, 130 times the 8.4 milliseconds above. The interesting small term is not the largest.

And the central quantity is a difference of two times, which nothing in the sky displays. No observation shows four minutes: what is observed is a wire crossing and a clock reading, and every number here is a subtraction between an astronomical event and an artefact.

The clock that had to be made from the sky #

Everything above treats sidereal time as something a clock displays, and for most of the period in which it mattered the arrangement ran the other way: the sky was the standard and the clock was the thing being corrected.

An observatory’s pendulum clock does not keep sidereal time to any useful precision. What it does is run at a nearly constant rate, and the transit observations measure how far it has departed. A star of known right ascension crosses the meridian at a sidereal time equal to that right ascension by definition, so the difference between the clock’s reading at that moment and the star’s right ascension is the clock’s error — determined afresh every clear night, for a set of standard stars, and interpolated between.

That is the shape of the whole enterprise. The clock’s rate is measured from the change in its error from one night to the next; its reading is corrected by the error itself; and neither is ever trusted for longer than the interval between observations.

The consequence is that time was a product of observatories rather than of laboratories, and distributing it was a substantial problem in itself. Before electrical communication a chronometer had to be carried, which is why a marine chronometer’s rate mattered more than its accuracy — a clock known to gain three seconds a day is as good as a perfect one and a clock that gains an unpredictable amount is useless.

The telegraph changed that. From the middle of the nineteenth century a time signal could be sent along a wire, and observatories began distributing time to railways, to ports and to the public — the dropping time ball, visible from a harbour, being the version of it a ship’s navigator could use without going ashore.

Radio finished the job. A time signal broadcast from a transmitter reaches everywhere at once, to within the propagation delay, so from the 1920s onward the world’s clocks were set from a handful of observatories, and the observatories’ clocks were set from transit observations of stars.

The whole arrangement was inverted in 1967. Once the second was defined by a caesium transition, the laboratory became the standard and the Earth’s rotation became a quantity to be measured against it — which is when the rotation turned out to be irregular, and when the arrangement described in this essay stopped being a definition and became an observation.

For about a century the length of a day was a fact about the sky, and it is now a fact about an atom, and the four minutes in this essay’s title are the one part of it that did not change.

One more reading applies the same arithmetic to a planet with a much longer year.

Where the ladder goes next #

Later rungs on this anchor: hour angle and local sidereal time as coordinates, and the conversion that turns a right ascension into a pointing. Greenwich mean sidereal time as a polynomial in Julian date, and why that polynomial needs a quadratic term. UT1, UTC and the leap second. The Earth rotation angle, which replaced sidereal time in the 2006 conventions precisely because the old definition was tied to a moving equinox. And the apparent motions this drift is the simplest of, where the moving reference is another planet rather than the Sun.

The word “sidereal” has held two different frames for two centuries, and the fix was to stop using it. The eight milliseconds between them are the last trace of an argument about what counts as fixed.