The Sun is a bad clock, by up to sixteen minutes

A sundial and a clock disagree, and the disagreement is not a defect in either. On 3 November the Sun crosses the meridian about sixteen and a half minutes before a well-regulated clock says noon. On 11 February it crosses about fourteen minutes after. Twice a year the two agree exactly, and in between they run through a fixed annual cycle.

The quantity is called the equation of time — “equation” in the old sense of a correction to be applied — and it exists because the Sun does not move uniformly along the celestial equator. Two entirely separate effects see to that, and they have different periods, so their sum is a curve with two unequal humps rather than a simple oscillation.

What a day is, and why the question is not trivial #

A day is one rotation of the Earth with respect to the Sun. The trouble is buried in “with respect to the Sun”.

The Earth turns once with respect to the stars in 23 hours 56 minutes 4 seconds — the sidereal day. During that time it has also moved about a degree along its orbit, so it must turn about a degree further to bring the Sun back to the meridian. That extra degree takes about four minutes, which is where the 24-hour solar day comes from.

If the Earth’s orbital motion were uniform and the orbit lay in the equatorial plane, that extra rotation would be the same every day and the solar day would be a constant. Neither condition holds.

The orbit is not circular. Kepler’s second law says the Earth moves faster near perihelion in early January and slower near aphelion in July. So the angle it covers in a day varies, and so does the extra rotation required.

The orbit is not in the equatorial plane. The Sun’s apparent motion is along the ecliptic, tilted 23.4°, but a day is a rotation about the polar axis, so what matters is the projection of the Sun’s motion onto the celestial equator. Near the solstices the Sun’s ecliptic motion is nearly parallel to the equator and projects almost fully; near the equinoxes it is inclined and the projection is shortened by $\cos\varepsilon$. The angle in question is the same 23.4° that makes the seasons.

Both effects change the length of the solar day by seconds, and seconds accumulated over months become minutes.

The mean Sun, which does not exist #

The fix is a fiction, and it is one of the more successful ones in the subject.

Define a mean Sun that moves uniformly along the celestial equator, completing one circuit in a tropical year, arranged to coincide with the real Sun on average. Mean solar time is the hour angle of that fictitious body, and it advances at a perfectly constant rate. A clock keeps mean time. A sundial keeps apparent time — the hour angle of the real Sun.

The equation of time is the difference:

$$\text{EoT} = \text{apparent solar time} - \text{mean solar time}.$$

Making that concrete requires two fictions in sequence, and it is worth naming both. The first fictitious body moves uniformly along the ecliptic, so it removes the eccentricity effect and keeps the tilt. The second moves uniformly along the equator, so it removes the tilt as well. The difference between the real Sun and the first is the eccentricity term; between the first and the second is the obliquity term; and the sum is what the figure plots.

The two components, in detail #

The eccentricity term has one maximum and one minimum a year, because the orbit has one perihelion. It is largest around 3 April and 3 September, at about ±7.7 minutes, and it vanishes at perihelion in early January and aphelion in early July. Its amplitude is set directly by the eccentricity: to first order the term is $2e\sin M$ expressed in time, and with $e = 0.0167$ that gives 7.66 minutes.

The obliquity term has two maxima and two minima, because the projection effect repeats twice per circuit — once at each equinox and once at each solstice. It reaches about ±9.9 minutes, and it vanishes at all four of those points. Its amplitude is $\tan^2(\varepsilon/2)$ expressed in time, which for 23.44° gives 9.87 minutes.

The obliquity term is the larger of the two, which is not what most people guess. The tilt matters more than the shape of the orbit, and it would still produce a substantial equation of time on a planet with a perfectly circular orbit.

Their sum has four zeros, at roughly 15 April, 13 June, 1 September and 25 December, and four extrema of quite different sizes — the November maximum at +16.4 minutes and the February minimum at −14.2. The asymmetry between the two humps is the eccentricity term adding to one and subtracting from the other.

The analemma #

Plotting the equation of time against the Sun’s declination, rather than against the date, produces a figure that has been photographed and can be drawn on a wall.

The shape is a consequence of the two coordinates having different periods: the declination runs through one cycle a year, the equation of time through a superposition of one and two. A one-cycle quantity plotted against a mixed one-and-two-cycle quantity gives a figure-of-eight, and the crossing point falls where the equation of time returns to a value it has already had at the opposite declination.

Two features of it are worth reading off. The lower lobe is larger, because the southern-hemisphere summer coincides with perihelion, so the eccentricity term is at its strongest when the Sun is at its lowest declination. And the whole figure is not symmetric about the vertical, because the four zeros of the equation of time are not evenly spaced through the year.

The analemma is also the shape a sundial’s shadow tip traces at a fixed clock time, which is why an accurate sundial carries either a table of the equation of time or an analemma engraved on it. Analemmatic corrections appear on dials from the seventeenth century onward, well before mechanical clocks were good enough to need them.

What was actually measured, and when it started to matter #

For most of history the discrepancy was invisible, because clocks were worse than it.

A medieval mechanical clock lost or gained a quarter of an hour a day, and setting it by a sundial was straightforwardly correct: the dial was more accurate than the clock, and a sixteen-minute annual cycle was far inside the clock’s own error. The equation of time existed as a piece of theory — Ptolemy discusses it — and was of no practical consequence whatever.

The pendulum changed that. Huygens’s clock of 1656 reached about ten seconds a day, and within a few years the accumulated equation of time was many times the clock’s own error. It became visible as a fact about clocks: a clock set by the Sun in November and checked against the Sun in February would appear to have lost half an hour, though it had kept perfect time throughout.

The resolution took most of a century to become standard. Britain adopted mean time for civil purposes only in 1792, France in 1816, and the last public sundial-regulated clock in Europe survived rather later. Before that, “noon” meant the Sun on the meridian and the equation of time was a correction applied to the clock, in the direction opposite to modern usage.

The measurement itself is old and good. The equation of time is obtained by timing the Sun’s meridian transit against a clock, and by the eighteenth century that could be done to a second or two — sufficient to determine the curve to well within a minute. Its two components were separated theoretically rather than observationally: the eccentricity term follows from Kepler’s equation, and the obliquity term from the spherical trigonometry of the projection, and both were computable to better than the observations before anyone thought to check them.

The modern application is not timekeeping. It is solar-power modelling and satellite geometry, where the Sun’s exact position for a given clock time is needed continuously, and it is the reason every solar tracking algorithm carries an implementation of the same two terms — computed against the celestial sphere like everything else about pointing.

The earliest sunset, which is not the shortest day #

The equation of time produces one consequence people notice without knowing what they are noticing, and it is a good demonstration that a correction to noon propagates into everything.

In the northern hemisphere the shortest day is the December solstice, around the 21st. The earliest sunset is around 12 December, and the latest sunrise around 1 January. All three are different dates, and the offsets are about ten days either side.

The reason is that sunrise and sunset are not symmetric about clock noon; they are symmetric about solar noon, which is drifting. Through December the equation of time is changing rapidly — solar noon is arriving about 30 seconds later each day — so the whole solar day is being dragged later against the clock at the same time as it is shortening and then lengthening.

Adding the two effects gives the observed pattern. Sunset is the sum of a term that reaches its minimum at the solstice and a term drifting steadily later, so their sum bottoms out before the solstice; sunrise is the same sum with the opposite sign on the first term, so it peaks after. The gap between the two dates is about three weeks, and it is entirely produced by the rate at which the equation of time is changing rather than by its value.

The same asymmetry exists in June and is much smaller, because the equation of time is nearly flat there.

Each of the three parameters is worth moving on its own, because the shape of the curve is a sum of two terms and each parameter belongs to one of them.

The generalisation: any tilted, eccentric orbit does this #

The equation of time is not a property of the Earth. It is what happens to any rotating planet whose orbit is eccentric or whose axis is tilted, and the other cases are more extreme in ways that are worth having.

Mars has an eccentricity of 0.093, five and a half times the Earth’s, and an obliquity of 25.2°. Its eccentricity term is correspondingly enormous, and the resulting equation of time runs from about −51 to +40 minutes — a range of an hour and a half on a day only 40 minutes longer than the Earth’s. Its analemma is not a figure-of-eight at all but a teardrop, because the eccentricity term dominates so heavily that the double-humped obliquity term cannot produce a crossing.

Jupiter, with an obliquity of 3.1° and an eccentricity of 0.049, has almost no obliquity term and a small eccentricity one; its analemma is a narrow ellipse.

Mercury is the extreme case in a different direction. Its 3:2 spin–orbit resonance combines with an eccentricity of 0.206 to produce a solar day twice as long as its year, and near perihelion the Sun’s apparent motion across the sky reverses for several Earth-days before resuming — the angular speed of the orbit briefly exceeds the rotation rate. There are longitudes on Mercury where the Sun rises, stops, sets again, and rises a second time.

The general statement is that a planet’s apparent solar time and its uniform time differ whenever the projected angular motion of the Sun is non-uniform, and the two available mechanisms are an eccentric orbit and a tilted axis. Both are the rule rather than the exception, so a planet on which the Sun is a good clock would be the anomaly.

The clock that stopped being the Earth #

Everything in this essay concerns the difference between apparent solar time and mean solar time, both of which are the Earth’s rotation with the Sun’s motion handled differently. There is a further step, taken in the twentieth century, which is that neither is the time anybody now keeps.

Mean solar time assumes the Earth’s rotation is uniform, and it is not. Tidal friction lengthens the day by about 1.8 milliseconds per century, and superimposed on that are irregular changes of a millisecond or two over decades, caused by the exchange of angular momentum between the core, the mantle, the oceans and the atmosphere. Those are not predictable, and they are large compared with what a caesium standard can measure.

So the second was redefined in 1967 in terms of an atomic transition and detached from the Earth altogether. Coordinated Universal Time counts atomic seconds; the Earth’s actual rotation angle is a separate quantity, measured by very long baseline interferometry against quasars and published as UT1; and the two drift apart at roughly a second a year and a half.

The drift was patched with leap seconds — an extra second inserted, always at the end of a month, to keep the two within 0.9 seconds of each other. Twenty-seven have been added since 1972 and none since 2016, because the Earth has recently been rotating slightly faster and the next insertion may have to be a removal.

That patch is being abandoned. A resolution adopted in 2022 will let the difference grow to at least a minute before any correction, from 2035 — because a discontinuity in a timescale is far more dangerous to the systems that now depend on it than a slow divergence from the Sun.

The sequence is worth seeing as one line. A sundial reads the Sun; a mean clock reads an idealised Sun; a civil clock reads the Earth’s rotation averaged; and an atomic clock reads none of them, with the sky reduced to a quantity that has to be measured and applied as a correction. The equation of time is the first term in that progression, and it is the only one anybody can see.

Where the model stops #

Constant elements. The eccentricity, the obliquity and the longitude of perihelion all drift — on Milankovitch timescales — so the equation of time is slowly changing shape. In about four thousand years the two humps will be nearly equal; in ten thousand the asymmetry will have reversed.

A uniform rotation. The Earth’s rotation is not quite uniform: it fluctuates by milliseconds seasonally and is slowing from tidal friction. Mean solar time is therefore itself slightly irregular, which is why civil time is now kept by atomic clocks and reconciled with the Earth by leap seconds.

A point observer. Refraction lifts the Sun near the horizon and the Sun has a finite disc, so the moment of “sunrise” involves conventions the equation of time does not address.

Meridian transit, not sunrise. The equation of time corrects the transit; the times of sunrise and sunset also involve the declination and the observer’s latitude, and their earliest and latest dates do not coincide with the solstice. That last discrepancy is the one people notice — the earliest sunset in the northern hemisphere is in early December, a fortnight before the shortest day — and it is the equation of time and the changing declination acting together.

The figures have a limitation worth naming. Both plot a difference of two times, which is an abstraction with no direct appearance in the sky: nothing is sixteen minutes wide, and no observation shows the quantity directly. What is observed is a transit time compared against a clock, so every point on these curves is a subtraction between an astronomical event and a human artefact. The analemma is the closest thing to a picture of it, and even that requires a year of photographs taken at exactly the same clock time.

And the two remaining degrees of freedom, one of which is not a shape at all.

The ladder from here #

Later rungs on this anchor: the two components derived, from Kepler’s equation and from the spherical projection. The mean Sun, and the two fictitious bodies in sequence. The analemma’s geometry. Sundial design, and analemmatic corrections. Earliest sunset and latest sunrise, and why neither falls on the solstice. Local apparent time, local mean time, and the arrival of time zones. Universal time, ephemeris time, and atomic time. Leap seconds and the irregular rotation. And the equation of time on other planets, where Mars’s teardrop analemma and Mercury’s retrograde sunrise both fall out of the same two terms.

The word “equation” here is the medieval one — from aequatio, a making-equal — and survives in exactly two places in modern English: this quantity, and the equation of a curve. They are unrelated uses that happen to have arrived at the same word.