The smallest term between a clock and the Sun

A sundial and a clock disagree by up to sixteen minutes, and the disagreement is traced to two fictitious suns: one that removes the unevenness of the orbit and one that removes the tilt of the axis. That is the whole of the astronomy, and it describes a sundial against a clock set to the mean solar time of the place where the sundial stands.

Almost no clock in the world keeps that time. The clock on a wall keeps the mean time of some meridian chosen by a government, often hundreds of kilometres away, and for part of the year it keeps a time an hour ahead of that. The equation of time is real and it is still there, but between a clock and the Sun it is one of three terms, and in most places it is the smallest.

Three terms and a sum #

The clock time at which the Sun crosses the meridian is twelve o’clock plus three corrections, each of which can be written down separately.

The first is longitude. A clock keeps the mean time of its zone’s reference meridian, and the Sun takes four minutes to move one degree westward across the sky. A place a degree west of that meridian sees noon four minutes after the meridian does, and the clocks agree with the meridian. The correction is four minutes for every degree between the place and the meridian its clocks follow, later to the west and earlier to the east.

The four minutes are nothing more than the day divided by the circle: the Earth turns through 360 degrees in 24 hours, 15 degrees an hour, one degree in four minutes. On the ground the scale depends on latitude, because meridians converge towards the poles. At the equator a degree of longitude is 111 kilometres, and noon moves by a minute for every 28 kilometres east or west; at London’s latitude a degree is 69 kilometres, and noon moves by a minute for every 17. Two towns a morning’s walk apart have noticeably different noons, which is precisely the difference that mean time kept and zone time erased.

The second is summer time: sixty minutes added while it is in force, because the clocks have been moved an hour ahead of the zone’s own time.

The third is the equation of time, the amount by which apparent solar noon runs ahead of or behind mean solar noon on that date. It is the same everywhere on the Earth on a given day, because it belongs to the planet’s orbit and axis rather than to any place.

Kashgar is the extreme case in the world’s most populous single time zone. China has kept one time, the mean time of the meridian near Beijing, across its entire width since 1949, and its westernmost cities lie more than two thousand kilometres west of that meridian. Solar noon in Kashgar comes at three in the afternoon by the official clock, and in practice much of daily life in the region runs on an unofficial local time two hours behind it. The equation of time adds or removes a few minutes to a displacement of nearly three hours; its whole annual swing, from 14:40 to 15:10, is a sixth of the offset the time zone imposes.

A sawtooth, and what borders do to it #

Time zones fifteen degrees wide were designed so that no place would be more than half an hour from its own mean solar time. Drawn against longitude, that ideal produces a sawtooth: noon comes thirty minutes early at the eastern edge of each zone, on time at its meridian, and thirty minutes late at its western edge.

Real zones follow borders, and the borders are wider than fifteen degrees or sit off their meridians for political reasons. Spain lies almost entirely west of the Greenwich meridian and keeps the time of the meridian fifteen degrees east of it — Central European Time — because in 1940 its government moved its clocks to match Germany’s and never moved them back. The whole country sees noon between three quarters of an hour and an hour and a half late before summer time is added, and the famously late Spanish mealtimes are, in clock terms, at ordinary solar hours.

The shaded band on the figure is the equation of time’s entire range. Across most of the world’s population the political term lies outside it, often far outside, which inverts the order in which the terms are usually taught. A description of noon that starts with the equation of time starts with its smallest correction.

Summer time on top #

The second term adds an hour for seven months of the year in most of Europe and North America, and on those days it is typically the largest term of all.

In Vigo in late July all three terms point the same way. The city is at the western edge of a zone whose meridian it is far from, summer time is in force, and the equation of time is near its late-July extreme, when the Sun is running six and a half minutes slow. The result is solar noon at twenty to three by the clock, sunset after half past nine, and a noon two hours and forty minutes after the one a sundial would call twelve.

Summer time was introduced in Germany and Austria-Hungary in 1916, to save fuel for lighting during the war, and Britain followed within weeks; it had been campaigned for in London a decade earlier by a builder who disliked seeing blinds drawn on summer mornings. Its effect on the sky is to move every solar event an hour later by the clock, which is a transformation of the labels on the time axis and changes nothing astronomical at all. The European Parliament voted in 2019 to end the seasonal change, and left the choice of which time to keep permanently to member states, which have not made it.

Where the equation of time shows #

The equation of time becomes the visible term only where the other two disappear: at a place close to its zone’s meridian, on a date when no summer time is in force.

That London should be the place where the equation of time is plainest is not a coincidence. The zero of longitude runs through Greenwich because the Royal Observatory there produced the tables by which ships found their longitude, and an astronomical clock set to the meridian of the observatory is exactly the clock against which the equation of time is defined. On an autumn day in London the Sun crosses the meridian sixteen minutes before twelve by every clock in the city, and nothing else intervenes.

The same condition shows up in the earliest and latest sunsets. The earliest sunset of the year falls before the shortest day because sunset is noon plus half the day length, and noon is moving by the equation of time while the day length has stopped changing. That offset is the same whatever the time zone does, so it shows in every city’s sunset table; the zone and summer time shift the whole table up or down without changing which date holds the earliest sunset.

How the equation of time left public life #

For most of history the equation of time was not a correction to anything, because clocks followed the Sun. Towns set their public clocks by a sundial at noon, and those clocks kept local apparent time, gaining and losing through the year exactly as the Sun did.

Clocks good enough to keep time better than the Sun reversed that. A pendulum clock of the late seventeenth century kept a steadier rate than the Sun’s own, and a household that set its clock by a sundial found the two drifting apart by a quarter of an hour over a few months; almanacs began printing the equation of time so that the clock could be set correctly from the dial. Cities then switched their official time from apparent to mean: Geneva in 1780, London in 1792, Berlin in 1810, Paris in 1816. At that point the equation of time left public time. It survived as a table in the almanac and a figure-of-eight engraved on some sundials, converting what the Sun said into what the clock said.

Before apparent time there had been something further still from a clock: hours that stretched with the season, a twelfth of the daylight each, which a sundial read without any correction because the Sun defined them. The move to mean time was the second of three steps away from the Sun, and each was made because a better clock had become common. Observatories, which needed uniform time before anyone else, did not set their clocks by the Sun at all by then. They timed the transits of stars across the meridian, which repeat every sidereal day, four minutes shorter than the solar one and free of any equation of time, and converted a sidereal clock whose zero moves through the year into mean solar time by arithmetic.

The second step was the railway. Every town’s mean time differed from its neighbour’s by four minutes per degree of longitude, which was harmless while travel took days and intolerable once a timetable connected towns an hour apart. The Great Western Railway ran its trains on London time from 1840, and within fifteen years most British public clocks followed. Some towns resisted: the clock on Bristol’s exchange still carries two minute hands, ten minutes apart, one for Bristol’s own time and one for London’s. The United States and Canada divided themselves into zones in 1883, and an international conference in Washington in 1884 fixed Greenwich as the prime meridian from which the world’s zones would eventually be counted.

The order matters. Mean time removed the equation of time from public clocks a century before zone time removed longitude, and zone time was the larger change. The difference between a town’s mean time and its zone time was, on average, far larger than the equation of time had ever been.

What the clock on the wall keeps now #

The time a zone keeps is no longer, strictly, the mean solar time of its meridian either. Civil clocks follow Coordinated Universal Time offset by a whole number of hours, and that time is kept by atomic clocks, whose second is fixed by a transition in caesium and has nothing to do with the Sun. It stays close to the mean Sun only because it is adjusted: whenever the Earth’s slightly irregular rotation has let the two drift apart by approaching nine tenths of a second, a leap second is inserted. The Earth is a clock that loses, irregularly, and the leap second is how the atomic time scale is kept honest to it.

That arrangement is scheduled to end. The international bodies responsible agreed in 2022 to stop inserting leap seconds by 2035 and to let the atomic time and the Earth’s rotation drift apart, probably by around a minute over a century, before any correction is made. When that happens a fourth term will join the three in this essay — a slowly growing difference between the clock’s mean Sun and the Earth’s — and for the first time since the town clocks of the eighteenth century the reference itself will move away from the Sun, by a quantity that a sundial will eventually see. A second is not even the same length everywhere at the precision atomic clocks reach; the civil day has become a count of those seconds, loosely tied to a planet’s spin.

The same choices have already been made once more, on another planet. Mars has a prime meridian defined by a small crater and a coordinated time that is the mean solar time on that meridian, and missions there keep local times offset from it by longitude, exactly as the Earth’s railways did — with an equation of time three times as large waiting between those clocks and the Martian Sun.

A clock error that is a longitude error #

The equation of time kept one essential job long after it left the town clock, and the job shows why the longitude term and the astronomical term are really the same kind of quantity.

A navigator finding longitude by chronometer did it by timing local noon. The chronometer kept Greenwich time; the Sun’s highest point gave local apparent noon; the difference in time, at four minutes to the degree, was the ship’s longitude. But the Sun’s highest point gives apparent noon, and the chronometer keeps mean time, so the equation of time had to be applied before the subtraction. Sixteen minutes of time is four degrees of longitude, or about 440 kilometres at the equator, and a navigator who forgot the correction in November would have placed the ship that far off. The equation of time and a displacement in longitude are interchangeable in the arithmetic; each is a number of minutes between a local noon and a reference clock, and only their causes differ.

That equivalence is the thread through this whole essay. Longitude inside a zone, summer time and the equation of time are three numbers of minutes added to twelve o’clock, and what makes one of them astronomy and the others law is only where the minutes come from.

What was actually measured #

Of the three terms, only the equation of time is a measurement, and it is known to a precision far beyond anything a sundial can use: the Earth’s orbit and orientation are determined to a few milliseconds of time. The longitude term is exact once a place’s coordinates and its zone are given, and the zone is a matter of statute. Summer time is a matter of statute too, and its dates move from year to year; the figures use a single approximate span, from late March to late October, and the clock times near the changeover dates are correct only to the day.

The clock times of solar noon themselves are therefore computed rather than observed, and they are checkable by anyone with a vertical stick. The shortest shadow of the day falls at the moment in the figures, to within the minute or so by which the shadow’s length hardly changes near its minimum.

What the figures leave out #

Latitude. Solar noon does not depend on latitude at all, which is why a place’s longitude and its zone are all the figures need. Sunrise and sunset depend on latitude strongly, through the Sun’s daily path at that latitude, and a sunrise table combines these noon times with the length of the day — and with the refraction that lifts the Sun at the horizon, which adds minutes at both ends of every day and matters not at all at noon.

The real extent of zones. The sawtooth and the two segments are simplifications: Spain’s Canary Islands keep a different time, China’s zone ends at borders that are not meridians, and many zones have half-hour or quarter-hour offsets that break the sawtooth into irregular steps.

Local practice. The clock a household actually lives by is not always the legal one; the unofficial time used across western China is the clearest example, and the figures follow the law.

Still open: which hour a country should keep for good #

Abolishing the seasonal clock change means choosing between permanent summer time and permanent winter time, and the geometry decides who pays for either. Permanent summer time moves every sunrise an hour later; in a city at the western edge of its zone, where noon is already an hour and a half late in winter, it would put midwinter sunrise after ten by the clock. Permanent winter time moves summer sunsets an hour earlier, which is what the clock change was invented to avoid. How to weigh morning light against evening light — for sleep, for road safety, for the energy use that justified the change in the first place — is argued with evidence that is contested, and the answer depends, before any of that, on where in its zone each country sits.