The earliest sunset is not the shortest day

The previous rung separated the equation of time into two reductions and derived both amplitudes. This one is what the machinery is for, and it is the most familiar consequence of it that almost nobody attributes to it.

The shortest day of the year in the northern hemisphere is the December solstice. The earliest sunset is not on that day. It is about a week earlier at British latitudes and nearly a month earlier in the tropics, and the latest sunrise is the same interval afterwards.

Two slopes, one sum #

The whole of it is one line of arithmetic. A sunset in mean solar time is

$$t_{\rm set} = 12{:}00 - E + \frac{H_0}{15}$$

where E is the equation of time in hours and H₀ is the half-day arc in degrees, set by the declination and the latitude. Sunrise is the same with the sign of H₀ reversed.

H₀ is stationary at the solstice, exactly. It depends on the declination and the declination is at its extreme there — that is what a solstice is. So dH₀/dt is zero on that day, and to first order the day length does not change across it.

E is not stationary at the solstice. It happens to be climbing through zero in late December at about half a minute a day, and there is no reason it should not be: the equation of time’s own extremes fall in early November and mid-February, nowhere near either solstice.

So at the solstice the sunset time is changing at −dE/dt and nothing else, which is half a minute a day and not zero. The sum’s stationary point is displaced to wherever the two slopes cancel — where dH₀/dt = dE/dt — and that is days away.

It is worth doing the cancellation numerically at one latitude, because the numbers are small and it is easy to disbelieve that half a minute a day moves anything by a week.

At 52° near the December solstice the day length changes by about 0.4 minutes a day at nine days out and by nothing at the solstice itself — the declination’s cosine dependence makes dH₀/dt vanish quadratically, so the day length’s rate falls off linearly as the solstice is approached. The equation of time’s rate is roughly constant at 0.5 minutes a day across the same interval. The two are equal where the day-length rate has climbed back to 0.5, which is about nine days from the solstice, and that is the answer.

The structure of that argument is worth noting: one rate is proportional to the distance from the solstice and the other is a constant, so the crossing is at a distance proportional to the ratio. Doubling the equation of time’s slope would double the offset, and it is why the June and December offsets differ.

Why the offset shrinks with latitude #

The size of the displacement is a ratio of two rates, and only one of them depends on where the observer is.

That curve is a good demonstration of a general point about extrema. The position of the extreme is set by a competition of derivatives, so it can be enormously sensitive to a quantity that contributes almost nothing to the value itself. At the equator the sunset time varies by only about twenty minutes across the whole year — a nearly flat function — and the date of its minimum moves by six weeks from where an intuition about “the shortest day” would put it.

A flat function’s extremum is badly determined and easily displaced. That is why the effect is largest exactly where it is least noticed.

The same effect in June, with the signs reversed #

Nothing in the argument is specific to December, and the June case is worth drawing because it tests the reasoning rather than repeating it.

The factor of two between the June and December offsets is the equation of time’s own slope at the two dates, and it comes from the alignment the previous rung identified: perihelion sits thirteen degrees past the December solstice, so the eccentricity term is near its steepest there and the two terms reinforce. In June they partly cancel.

That is a testable prediction rather than a description. A planet with perihelion at the June solstice would have the larger offsets in June, and one on a circular orbit would have equal offsets at both solstices from the obliquity term alone — a prediction that costs nothing to state and that the analemma’s symmetry already contains.

The quantity that really is extreme at the solstice #

It is worth naming what is extreme on the solstice, because the confusion this rung is about comes from conflating three things that are usually said interchangeably.

The declination is extreme on the solstice, by definition — the solstice is the moment the Sun reaches its greatest southern declination, and everything else follows from that.

The day length is extreme on the solstice, because it is a function of the declination alone at a fixed latitude. Sunrise to sunset is shortest on 23 December at every place on Earth where the Sun rises at all.

The sunset time and the sunrise time are not extreme on the solstice, because each is the day length plus a clock offset that is still moving.

So “the shortest day” and “the earliest sunset” refer to different things and are correctly on different dates. There is no paradox and nothing to reconcile; the popular puzzlement about it comes from treating “shortest day”, “earliest sunset” and “latest sunrise” as three names for one event.

What is actually observed #

The measurement behind the figures is the least exotic in this collection: somebody looks at a clock when the Sun’s upper limb touches the horizon.

Two things about that are worth stating, because both enter the computation and neither is astronomy. The “horizon” is not the geometric one. The Sun’s upper limb is used rather than its centre, which is a semi-diameter of 16 arcminutes; and atmospheric refraction lifts the whole disc by about 34 arcminutes at the horizon. So sunset is defined at a solar altitude of −0.833°, which is those two numbers added, and every published table uses it.

The refraction figure is a mean. Actual refraction at the horizon varies with temperature and pressure by several arcminutes, which at mid latitudes in December is a minute or two of time — comparable to the whole effect this rung is about over a few days near the extreme. So the date of the earliest sunset at a given place is well determined by the geometry and the time of it on any particular evening is not, and an observer trying to detect the effect by eye is fighting the weather.

The analemma says the same thing #

Read that way the whole question becomes geometric. Sunset time is a linear combination of the two coordinates of the analemma — a weighted sum of the equation of time and a function of the declination — so the extreme sunset is where a line of constant sunset time is tangent to the loop. The weight depends on the latitude, which is why the tangent point moves with it, and at the equator the weight on the declination goes to zero and the tangent point moves to the loop’s leftmost extreme.

That reformulation is worth having because it makes the latitude dependence obvious without any calculation. A line whose slope is set by the latitude, rolled around a fixed closed curve: the tangent point sweeps from one end of the loop to the other as the slope changes.

What the picture cannot show #

The figures compute a local mean solar time, which is what a clock at the observer’s own longitude would read if the mean Sun were overhead at noon. Nobody’s clock reads that.

A civil clock reads a zone time, which differs from local mean time by the observer’s longitude relative to the zone meridian — up to half an hour within a properly drawn zone and considerably more within a badly drawn one. Western Spain runs on central European time and is about eighty minutes from local mean solar time. Adding summer time doubles the offset for half the year.

None of that changes the dates of the extremes, because a fixed offset shifts every time equally and moves no stationary point. It changes the times, and it is by far the largest term in the difference between a computed table and a clock on a wall. The definition of the day itself is a separate matter again, and one this collection keeps deliberately apart from this anchor.

Two smaller omissions. The figures use a fixed obliquity and a fixed eccentricity, both of which drift; over a century the dates move by less than a day. And the declination is computed for the Sun’s centre as seen from the Earth’s centre, with no correction for the observer’s own position on the globe or for the Sun’s parallax — both negligible here and both included in a published almanac.

What this looks like on the other side of the equator #

The southern hemisphere has the same arithmetic and a different arrangement, and it is worth stating because the usual accounts are written for one hemisphere and read as universal.

In the south the December solstice is the longest day, so the quantity with a stationary point there is the same one and the extremes bracket it in the same way — but they are the latest sunset and the earliest sunrise rather than the earliest sunset and the latest sunrise. At Melbourne’s 37.8° south the latest sunset falls in early January, about a fortnight after the solstice, and the earliest sunrise in early December, about a fortnight before it.

The offsets are larger in the south in December than in the north, and the reason is the one the previous rung identified. The equation of time’s slope is the same everywhere on Earth on a given day; what differs is the day length’s slope, which depends on the latitude’s magnitude and not its sign. So the offsets are the same at 52° north and 52° south and the labels on them are exchanged — and Melbourne’s larger offset than London’s is a fact about 37.8 against 52, not about the hemisphere.

The date the effect is usually blamed on #

There is a competing explanation in circulation and it is worth dismissing explicitly, because it is plausible and wrong.

The story is that the earliest sunset is displaced because the Earth is near perihelion in early January, so it is moving fastest then, so the solar day is longest, so the clock and the Sun drift apart. Every clause of that is true. None of it is the mechanism.

What perihelion does is make the eccentricity term of the equation of time steep in December — and it is the steepness, not the perihelion, that displaces the sunset. A planet with the same eccentricity and perihelion at the equinox would have the same day-length behaviour, a differently shaped equation of time, and a different offset. The offset tracks dE/dt at the solstice and nothing else about the orbit.

The distinction matters because the two accounts make different predictions. The perihelion story says the effect should be largest when perihelion is nearest the solstice; the slope account says it should be largest when the equation of time is steepest at the solstice, which is a different alignment. Both happen to be nearly satisfied at present, which is why the wrong story survives.

The habit #

The result generalises past sunsets, and stating it generally is the point of the rung.

A sum of two functions has its extremes where their derivatives cancel, not where either vanishes. That is trivial as a statement and is routinely forgotten in practice, because it is natural to attribute the extreme of a sum to whichever term dominates the value. The day length dominates the sunset time — it accounts for nearly two hours of variation at 52° against half an hour from the equation of time — and it does not determine when the sunset is earliest.

This collection meets the same structure repeatedly. The peak of a light curve with a parallax distortion is not at the time of closest approach, because the magnification and the trajectory offset peak at different moments. The maximum of a transit’s duration is not at the smallest impact parameter once the orbit is eccentric. And the least total loss on an ascent is not where either the gravity loss or the drag loss is least.

The diagnostic in every case is the same: identify the terms, differentiate each, and ask where they cancel rather than where they are smallest.

A check anybody can make #

The claim is unusually easy to test, which is worth saying because most of the results in this collection are not.

Take a published sunrise and sunset table for a place at a known latitude — a newspaper, an almanac, a tide table — and read the sunset column through December. At 52° it will show the same time, to the minute, for about a week either side of 14 December, then start climbing while the sunrise column is still climbing too. The two columns move in the same direction for about a fortnight in late December, which is the whole of the effect: the day is getting longer at both ends only after the last of the three dates has passed.

The interval over which the sunset time is constant to the minute is itself informative. A stationary point is quadratic, so a quantity within a minute of its extreme is within a minute over an interval proportional to the square root — which at these rates is about a fortnight. That is why the effect is easy to see in a table and impossible to see by watching, and it is the same reason the position of the extreme is so sensitive.

The published tables are computed rather than observed, so this is a check of the arithmetic against the arithmetic. Doing it against an actual horizon requires a fixed observing position, a clear sky on the right evenings, and patience with refraction — which is why the effect was noticed by almanac-makers rather than by observers.

Twilight, which does the same thing at a different threshold #

One extension is worth a paragraph because it is the same calculation with one number changed and it has a different answer.

Sunset is defined at a solar altitude of −0.833°. The three definitions of twilight are defined at −6°, −12° and −18°, and each of them has its own earliest date computed exactly the same way: solar noon plus the half-day arc above that altitude, with the extreme where the two slopes cancel.

The half-day arc above a deeper threshold varies more steeply with declination than the one above the horizon, because the Sun’s path is crossing the threshold at a shallower angle. So the day-length term is larger relative to the equation of time, and the extreme moves closer to the solstice. At 52° the end of civil twilight is earliest about four days before the solstice against the sunset’s nine, and the end of astronomical twilight is earliest about two days before it.

That ordering is a small, checkable prediction of the same arithmetic, and it is the sort of thing a table will confirm and no intuition supplies.

Where this ladder goes next #

Three rungs have taken the equation of time from a sixteen-minute discrepancy to its two components to a consequence a reader can check against a newspaper’s tide table.

What remains on the anchor divides into the practical and the exotic. On the practical side: sundial design and the analemmatic correction, which is a figure-of-eight engraved on the dial’s gnomon so that the shadow reads civil time directly — a device that solves this whole subject mechanically and was in common use for two centuries. And local apparent time, local mean time and the arrival of the railway, which is why time zones exist at all.

On the other side: the same two terms at values no planet in this system has. Mars, whose eccentricity makes its analemma a teardrop rather than a figure-of-eight, and whose sunsets are displaced by weeks. Mercury, whose 3:2 spin–orbit resonance makes the Sun reverse in its sky near perihelion — the equation of the centre exceeding the mean motion, which is this rung’s arithmetic taken past the point where the sum has a single extreme at all.