Two fictitious suns in sequence

The first rung of this anchor established that solar noon and twelve o’clock differ by up to sixteen minutes, that the discrepancy runs through a fixed annual cycle, and that it has two causes — one from the shape of the orbit and one from the tilt of the axis.

Naming two causes is not deriving them, and the naming hides a structure that is worth having. The two terms are not two influences on one quantity. They are two separate reductions, applied one after the other, and each introduces an imaginary body of its own.

The distinction sounds pedantic and is not. Read as “two causes”, the equation of time is a phenomenon with two contributing effects, and there is no reason the two should be of comparable size or have any particular relationship. Read as “two reductions”, it is a chain of conversions between four different clocks, and each link in the chain is a well-defined quantity that appears elsewhere in celestial mechanics under its own name.

The chain the quantity actually is #

A sundial reads the hour angle of the true Sun. A clock reads the hour angle of a fictitious body that moves uniformly. Getting from the first to the second takes two steps, and each step is a different kind of correction.

Step one: the true Sun to the first mean Sun. The Earth’s orbit is an ellipse, so the Sun’s apparent motion along the ecliptic is not uniform — it is fastest at perihelion in early January and slowest at aphelion in July. Replace it with a fictitious body moving uniformly along the ecliptic at the average rate, and the difference between the two is the equation of the centre: the true anomaly minus the mean anomaly, which is what Kepler’s equation exists to compute and which every orbit in this collection needs before it can say where a body is at a given time.

Step two: the first mean Sun to the second. The first fictitious body moves uniformly along the ecliptic, and a sundial does not care about the ecliptic — a sundial reads an hour angle, which is measured along the equator. Projecting a uniform motion along a tilted great circle onto the equator does not produce a uniform motion, because the projection compresses near the equinoxes and stretches near the solstices. So a second fictitious body is introduced, moving uniformly along the equator, and the difference between the two mean suns is the reduction to the equator — the same conversion between ecliptic and equatorial coordinates that every ephemeris performs, applied to a uniform motion.

And a third body, which is the one a clock reads. The second fictitious body — moving uniformly along the equator — is what “mean solar time” means, and a civil clock is a device for tracking its hour angle. So four objects appear in the chain: the true Sun, a mean Sun on the ecliptic, a mean Sun on the equator, and the clock that follows the last. Only the first is visible and only the last is legislated.

The equation of time is the sum of those two differences. Each has its own geometry, its own period and its own fictitious body, and the reason there are two terms is that the reduction takes two steps rather than that one effect has two contributions.

Why one has a period of a year and the other half a year #

The periods follow from the geometry without any calculation.

The equation of the centre depends on where the Earth is in its orbit relative to perihelion, and there is one perihelion a year. It is zero at perihelion and aphelion, positive in between on one side and negative on the other, so it has one cycle a year with an amplitude set by the eccentricity, which for e = 0.0167 gives 7.7 minutes through the first-order form 2e sin M.

The reduction to the equator depends on where the Sun is relative to the equinoxes, and there are two of those a year. It vanishes at both equinoxes and at both solstices, and reverses sign between them, so it has two cycles a year with an amplitude set by the obliquity — to first order it is (ε²/2) sin 2λ in radians, which for 23.44° is 9.9 minutes.

Neither number is fitted. Both come out of the same integration the figure runs, and both are checkable: the eccentricity term must cross zero twice a year and the obliquity term four times, which is what the generator asserts before it draws anything.

The zeros are worth listing because they are the memorable half of the structure. The eccentricity term vanishes at perihelion (about 3 January) and aphelion (about 4 July). The obliquity term vanishes at the two equinoxes (about 20 March and 22 September) and the two solstices (about 21 June and 21 December). The sum vanishes four times a year at dates that are none of those — around 15 April, 13 June, 1 September and 25 December — because it vanishes where the two terms happen to cancel rather than where either is zero.

There is a subtlety in the second amplitude worth stating, because the usual account gets the mechanism backwards. It is tempting to say that the Sun “moves faster along the equator near the solstices”, and that is true of its right ascension but not for the reason a picture suggests. Near a solstice the ecliptic runs parallel to the equator, so the Sun’s whole motion is in right ascension and none of it in declination — a full ecliptic degree becomes 1/cos ε degrees of right ascension. Near an equinox the ecliptic crosses the equator at 23.4°, so an ecliptic degree becomes cos ε degrees of right ascension and the rest goes into declination. The projection compresses at the equinoxes and stretches at the solstices, and the factor is cos ε either way.

What each driver is worth #

The two amplitudes are set by two unrelated numbers — an eccentricity and an angle — and there is no reason for them to be comparable. On the Earth they happen to be, within thirty per cent, and the sum is therefore a shape neither term would produce alone.

The quadratic dependence on the obliquity is worth noting because it is the reason the Earth’s two terms are comparable at all. A tilt of 12° would give a projection term of 2.6 minutes and the eccentricity would dominate; a tilt of 35° would give 22 and the projection would. The Earth sits in the narrow range where neither wins.

The alignment, which is a separate parameter #

There is a third quantity in the problem and it is easy to miss, because it does not change either amplitude. It is where perihelion is relative to the equinoxes.

That figure is the one that separates this rung from the last. The Earth’s equation of time has a larger November maximum than its February minimum — the source of the four-minute difference between a sidereal and a solar day is a different effect entirely, and the two are routinely confused — and the usual explanation — “the two terms are out of phase” — is correct and unilluminating. The reason they are out of phase by the amount they are is that perihelion sits 13° past the December solstice, and that angle is precessing: it was at the solstice about eight hundred years ago and will be at the March equinox in about five thousand.

So the shape of the equation of time is not a constant of the solar system. It is a snapshot, and the precession of the equinoxes is slowly redrawing it.

It is worth asking what the equation of time would look like on the other bodies this collection visits, since both parameters vary widely.

Mars has an eccentricity of 0.093 and an obliquity of 25.2°, so its eccentricity term is five and a half times the Earth’s and its projection term is only slightly larger. The result is dominated by the eccentricity, has one large cycle a year, and swings over about fifty minutes of Mars’s own time — which is why its analemma is a teardrop rather than a figure-of-eight, the smaller lobe having been swallowed.

Mercury is stranger still. Its eccentricity is 0.206 and its 3:2 spin–orbit resonance means the Sun’s apparent motion across its sky reverses near perihelion — the orbital angular rate briefly exceeds the rotational one — so the Sun rises, stops, goes backwards, and rises again. That is the same equation of the centre this rung derived, at an eccentricity where it exceeds the mean motion.

The analemma is the same two numbers, plotted differently #

The analemma is the equation of time and the declination plotted against each other, so everything above is visible in its shape. The vertical extent is twice the obliquity, exactly. The horizontal extent is the range of the equation of time. The crossing point of the eight sits where the two terms cancel, which is in late April and late August. And the asymmetry between the lobes is the alignment of perihelion. Read as a diagram it is the Sun’s declination against the correction a sundial needs, and nothing else.

Switching the eccentricity off entirely does something similar and for a different reason: the eight becomes symmetric because the only remaining term is the projection onto the equator, which treats the two halves of the year identically. What survives is the pure obliquity analemma, 19.7 minutes wide and 46.9 degrees tall, produced by a circular orbit on a tilted axis. Two quite different changes — moving perihelion, and removing it — produce symmetric figures for unrelated reasons.

The number that decides which term is bigger #

The two amplitudes are 7.7 and 9.9 minutes, and it is worth writing down what would have to change for the ordering to reverse, because the answer is a single ratio.

To first order the eccentricity term has amplitude 2e (in radians of longitude) and the obliquity term ε²/2. Setting them equal gives e = ε²/4, which for the Earth’s tilt is 0.0418 — two and a half times the actual eccentricity. So the Earth’s orbit would have to be considerably more eccentric, or its tilt considerably smaller, for the eccentricity to dominate.

That relation is worth having as a rule because it applies to any body. Mars, at e = 0.093 and ε = 25.2°, has e well above ε²/4 = 0.048 and is eccentricity-dominated, which is why its analemma is a teardrop. Venus, at e = 0.007 and ε = 177°, is a case where the framing breaks entirely. And a hypothetical planet on a circular orbit has a pure obliquity analemma at any tilt, symmetric and closed, of the kind the previous section described.

What the picture cannot show #

Three approximations run through every figure here, and none of them changes a conclusion.

The orbit is a fixed Keplerian ellipse. It is not: the Moon pulls the Earth about a common barycentre, so the Earth’s own ecliptic longitude wobbles by a few seconds of time with a monthly period, and the planets perturb the orbit on longer ones. The published equation of time includes those and they contribute at the level of seconds against sixteen minutes.

The obliquity and the eccentricity are constants. Over the span of a human life they are; over ten thousand years neither is, and the equation of time was a visibly different function in the Bronze Age.

And the quantity computed is a difference of hour angles, which assumes the observer is at a fixed longitude on a uniformly rotating Earth. The rotation is not uniform at the millisecond level, and the difference between UT1 and atomic time accumulates — but that is a matter for a different anchor and it is six orders of magnitude below anything on these axes.

The one thing worth saying is what the figures deliberately do not attempt: a sundial correction table. Converting a sundial reading to civil time needs the equation of time, the observer’s longitude relative to the time-zone meridian, and any summer-time offset, and the last two are far larger than the first. A dial in western France reads nearly an hour from the clock for reasons that have nothing to do with celestial mechanics.

The word, and the two things it has meant #

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

The medieval sense is the useful one for understanding the structure, because it names the correction rather than the relation. The equation of the centre is the correction applied to a mean anomaly to get a true one. The equation of time is the correction applied to mean time to get apparent time. Both are quantities to be added, and in both cases the thing being corrected is a uniform fiction and the thing corrected to is what is observed.

That framing also explains a sign convention that trips people up. The equation of time is defined as apparent minus mean — sundial minus clock — so a positive value means the sundial is ahead, and the sundial is ahead in early November. Almanacs before about 1830 used the opposite sign, because the correction was tabulated as what to add to a sundial reading rather than as the difference itself, and historical tables have to be read with care.

What is actually observed #

Everything above is a computation, and it is worth saying what the observation behind it is, because it is unusually direct.

The quantity is a difference of two times, and both are measurable to a fraction of a second with equipment that has existed for centuries. Apparent solar time is read from a meridian transit: the moment the Sun’s centre crosses the local meridian, obtained by timing the two limbs and halving. Mean time is read from a clock. The difference is the equation of time, measured rather than derived, and the measurement was made routinely at every observatory from the seventeenth century onward because it was how clocks were checked.

That gives the subject an unusual history among the topics in this collection: the phenomenon was measured accurately for a century and a half before it was correctly decomposed. Ptolemy knew the equation of time existed and gave a table for it. The separation into an eccentricity term and an obliquity term needed Kepler’s ellipse for the first and a clear statement of the equatorial projection for the second, and the modern form dates from the eighteenth century. It is the reverse of the usual order in this collection, where a prediction generally precedes its confirmation.

The observation that anchors the figures here is therefore not one measurement but a long baseline of them, and the check that the calculation is right is that a computed table and an observed one agree to the second — which they do, once the Moon’s pull on the Earth is included.

Where this ladder goes next #

The two terms are now separated, their amplitudes derived, and their alignment identified as a third parameter that changes the shape without changing either. What that machinery is for is the next rung.

The most familiar consequence of the equation of time is not a sundial correction. It is that the earliest sunset does not fall on the shortest day. A sunset time is the solar noon plus half the day length, and the two have different stationary points: the day length is stationary at the solstice, exactly, and the equation of time is not — so the sum keeps falling after the day length has stopped shortening. The earliest sunset comes about a week before the December solstice at British latitudes and six weeks before it near the equator.

Beyond it: the mean Sun’s own definition and the two conventions that have held it; earliest sunrise and latest sunset in June, which are the same effect with the signs reversed; sundial design and the analemmatic correction engraved on a dial to remove all of this; 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 at different values.