*The solstice on June 22nd marked the shortest duration of sunlight (or day length) during this year. One might have expected that from that date sunrise would have started getting earlier; and prior to that date sunset to have been consistently getting earlier (as the daylight duration was shortening). In fact the latest sunrise did not occur until almost a fortnight after the solstice, whereas the earliest sunset happened around a week before the solstice and so began progressing later before the solstice was reached. The reason for these surprising facts are all tied up with the equation of time, a graph of which is shown on many good sundials. But what causes these apparent discrepancies, to which we are largely oblivious? *

It had been my intention to write this blog post on July 5th, for reasons that will become apparent. Pressures of work and other things meant that I then needed to aim for July 12th (that perhaps being Julius Caesar‘s birth date in 100 BCE, and also the anniversary of the Battle of the Boyne in 1690, still causing ructions in Northern Ireland each year; it’s also my mother’s birthday, but I think this is unrelated to those historical events).

Anyhow, it is now *Le Quatorze Juillet*, and that is Bastille Day, so best regards to all those of French extraction wherever they might be around the globe; happens that I am off to France for a few weeks from next Saturday, and especially looking forward to the *fromage* and *vin*.

Let us return, though, to July 5th, although by way of a small diversion so as to set things up. A few weeks ago I wrote about the June solstice, when the Sun reaches its northernmost point, an event that marks the start of summer in the northern hemisphere, and winter here in the southern. As everyone knows, one hopes, here in NZ the June solstice marks the shortest day, by which we understand the briefest interval between sunrise and sunset. And that is the case: if one consults a tabulation of the day-length for, say, Auckland one will find that it lasted for about 9 hours and 48 minutes at the beginning of June, minimised at 9 hours 38 minutes on June 22nd, and increased to reach the former 9 hours and 48 minutes again on July 12th.

One might expect, therefore, that the trend in sunrise times to be that they come consistently earlier from June 1st to June 22nd, and then get later thereafter on the clock; while sunset times ‘should’ get earlier as the solstice is approached, then trending later again after June 22nd has passed. This, however, is not the case. In the past I have been asked about this many times, by percipient people who discover from the sunrise and sunset times appearing in newspapers that the trend that one might naïvely anticipate is not followed at all.

Let us again use Auckland as an example; similar trends will be found for any other location in NZ, although the absolute sunrise and sunset times will differ, because they are latitude-dependent (also with consistent offsets due to the longitudes in question). The table below shows, to the nearest minute, sunrise and sunset times, plus the difference between them (the day length), on a weekly basis. I could give more granularity (daily values, to the nearest second – then the apparent discrepancy for July 20’s numbers would not appear), but it is just the general trends that I want to show, without confusing with too many digits.

So, the day length (right-hand column) decreases through until June 22nd (the solstice date) before it increases again, as is to be expected. The sunrise time, though, gets later until early July, before it then reverses its trend. In contrast, the sunset time reaches its earliest instant on a date around June 15th, a week prior to the solstice.

Something is going on that is more complicated than one might initially have imagined to be the case. Let me explore what it is.

Something special happened on July 5th, astronomically-speaking, though it takes place every year so it is not a rare event. Indeed this occurrence provides one way of defining a ‘year’.

The Sun moves up and down the eastern horizon in terms of its rising point, the southernmost rising being at the December solstice, the northernmost at the June solstice. This is due to the tilt of Earth’s spin axis.

A quite distinct thing is when Earth reaches its furthest and closest distances from the Sun in its annual orbit: the former is termed *aphelion*, and currently occurs around July 4th, while the latter is called *perihelion*, and takes place precisely half an orbit away, on about January 4th.

This year aphelion occurred late on July 4th Universal Time, but this was July 5th NZST (in fact at 10:10 a.m., although below I will dispute the utility of the terms ‘a.m.’ and ‘p.m.’ in various situations). Similarly, the recent solstice was on June 22nd at 03:54 a.m. NZST (still June 21st UTC), and so aphelion currently occurs twelve or thirteen days post-solstice.

Both these factors – the tilt of Earth’s spin axis, and the shape and orientation of Earth’s orbit around the Sun – affect when sunrise and sunset occur.

Let’s think first about the influence of the axis tilt, and assume for the time being that Earth’s orbit is circular. The planet spins at a consistent rate, rotating once relative to the distant stars in a length of time known as the sidereal day, which lasts almost four minutes short of a mean solar day of 24 hours; a sidereal day lasts for a fraction near 365.25/366.25 of a mean solar day, because in a one-year trip around the Sun the Earth actually rotates one time more than the number of days in the year.

If Earth spins at a consistent rate (which it does, to the precision required in this examination), and the Earth’s heliocentric orbit were circular (which it is not, but we will come to that later), then the rate at which the Sun appears to cross the sky would be uniform. Fine. But that solar motion across the sky is not purely from sunrise in the east to sunset in the west. That is obviously the case because it has a north-south movement, sunrise (and sunset) shifting northwards between the December and June solstices, and in the opposite direction over the following half-year. Because there is a north-south angular velocity component for the Sun, its east-west component must vary. The outcome is that the instants at which the Sun transits (say) the noon meridian generally are not spaced by 24 hours. They are only spaced by 24 hours on four occasions during each year: at the solstices and the equinoxes.

The graph below (actually Figure A3 from my book *Marking Time*) shows how the day length – noon meridian crossing to noon meridian crossing – varies across the year; note that ‘day length’ means, here, not the time between sunrise and sunset as we were using that term earlier, but the length of a ‘day and a night’. (The English language clearly has some inadequacies in that ‘day length’ can take two quite different meanings. Personally I think we should use the lovely word nycthemeron for a 24-hour interval, but I may be in a minority.)

Obviously the graph is a bit complicated. So far we have been looking only at the effect of our axial tilt, and that produces the oscillating curve marked with monthly open symbols. All will eventually become clear to the reader (I hope).

Now let’s consider the effect of the non-circularity of Earth’s orbit. Only at two times during a year is our planet moving through space in a direction that is perpendicular to the direction of the Sun: at perihelion (in early January) and at aphelion (in early July). If our orbit were precisely circular then at all times Earth’s direction of motion would be at right-angles to the Sun, but in reality our elliptical orbit and small eccentricity (currently 0.0167) results in the planet’s direction sometimes being almost as much as one degree greater than/less than 90 degrees compared to the solar direction.

From perihelion to aphelion (essentially the first half of a calendar year) Earth’s direction is just greater than 90 degrees away from the Sun’s position, and so we need to spin by slightly more than previously so as to bring the Sun back to the noon meridian. Between aphelion and perihelion (the latter half of the calendar year) the contrary is true, and the angle the planet needs to spin through is reduced.

The effect of this, due to Earth’s orbital non-circularity, is indicated by the single-sinusoid in the graph above, with black-filled symbols being plotted monthly.

There are other implications of the non-circular shape of the terrestrial orbit. Our speed varies, being highest at perihelion (about 30.3 km/sec, or around 109,100 kph), and slowest at aphelion (29.3 km/sec or 105,500 kph). Consequently, with perihelion being during austral summer, that season is the shortest in the southern hemisphere; similarly, aphelion (on July 5th in 2019) occurs during austral winter, and that is the longest season in NZ because Earth is moving slower. On the other hand, because we are nearer to the Sun in January/summer here Down Under, southern hemisphere summers are generally hotter though shorter than in the north, where they are longer but cooler. I will leave you to work out the other permutations (long/short; cooler/hotter) for yourselves. I gave an explanation of the differing durations of the seasons in my recent blog about the June solstice.

It is the complications due to the two independent effects described above that have led to us using, in the modern era, the concept of *mean solar time*. In the distant past people would use the time according to the Sun wherever they were – *(local) apparent solar time* – to chart their lives, but since the invention of clocks, and then transportation requiring coordination across a country, something more-technical has been necessary. This is mean solar time, which is the time according to the *mean Sun*, a fictitious entity which may be as much as eight solar diameters away from the location of the Sun in the sky. The mean Sun essentially describes an average position for the Sun with the effects plotted in the preceding graph having been ironed out.

*Greenwich Mean Time* (GMT) is the time according to the mean Sun on the Greenwich meridian. GMT is not a term that should nowadays be much used because we no longer use astronomy/the Sun for timekeeping, but rather atomic clocks, which give us *Coordinated Universal Time* (UTC). Leap seconds are inserted every so often in order to keep UTC within 0.9 seconds of GMT, the gradual deviation being due to the fact that the atomic second is defined in terms of the spin rate of the Earth in the late 19th century, and tidal drag largely due to the Moon is very slowly increasing the duration of a day, which I previously discussed in these columns.

So, how might we use a sundial to tell the time? Well, the two effects plotted in the preceding graph can be added together, and the result is called the *equation of time*. As a matter of fact, the equation of time is not an ‘equation’ in the sense we use that word nowadays, but rather implies the reconciliation of a difference, that difference being the amount given by subtracting the mean solar time from the apparent solar time on any date. The equation of time (EoT) can be plotted across the year, and the resultant curve appears as below (again a plot from my book Marking Time, in this case Figure A6). If the Sun in the sky is ahead of our clocks, then the EoT is positive; if it is behind, then the EoT is negative.

Taking a look at the graph above, one can see that the EoT was positive from mid-April, before reducing to become zero on about June 9th. Since then the EoT has been becoming progressively more negative, implying that the Sun’s position has been getting further behind our clocks. And that is why sunrise was still getting later for a couple of weeks after the shortest day, June 22nd: the Sun was behind our clocks (and still is, as I write on July 14th).

Similarly, when our clocks said noon on June 22nd, the Sun had yet to reach the meridian; that is what a negative EoT means. It was clock-noon, but not yet solar-noon. This is why I wrote above that the use of ‘a.m.’ and ‘p.m.’ should be avoided. The former is an abbreviation for the Latin *ante meridiem*, the latter for *post meridiem*. But the Romans used apparent time, not mean solar time. Because of the equation of time, for about half of the year noon comes before the Sun has reached the overhead meridian, and midnight before the anti-solar point has reached it, so that the use of those abbreviations is technically incorrect. Perhaps. In any case, many people (well, me at least) get confused as to whether 12 a.m. implies noon or midnight. Why not just use those simple words ‘noon’ and ‘midnight’? If 12:00 a.m. means noon, as is usually the case, how can a minute later be 12:01 p.m., especially at those times of year when the equation of time places the Sun well away from the meridian at clock-noon? Confused? You should be — because I am.

Referring back to the table given above, we noted that the earliest sunset occurred this year about a week prior to the solstice, and almost three weeks before aphelion passage. How come? There are various ways of looking at what is going on. Perhaps the simplest is to consider the EoT in the preceding graph (Figure A6), and note that one of the places where the complicated curve has its maximum negative slope is in the middle of June. That is, the duration of the full day is reducing at near its fastest rate, and so while sunrise is getting slightly later, the time of sunset is pulling back faster such that earliest sunset occurs around June 15th.

I posed the rhetorical question above of how we might use a sundial to tell the time. A decent sundial will have the equation of time inscribed on it in some way, else the time according to the Sun cannot be corrected so as to give the equivalent of a clock time.

Here is a pair of photographs of the sundial in Wellington Botanic Garden, on the hill just north of the Lady Norwood Rose Garden; a close-up of the top of it is shown in the header to this blog post. It was a cloudy day, so no shadow cast by the gnomon can be seen.

Looking at the graph moulded into the bottom of the square plate, one can see a wiggly line that looks like the equation of time as plotted in my preceding diagram (Fugure A6), except that it is the inverse. This is due to the normal sign convention for the EoT: on the sundial it is easier simply to say “add [this] time correction”.

The other thing to note is that in the graph on the sundial, all values are positive: this is because the line of longitude used to define NZ Standard Time is 180 degrees/12 hours away from the Greenwich Prime Meridian, whereas the longitude of Wellington is near 174.78 degrees east. The difference is equivalent to almost 21 minutes of time, putting Wellington consistently this much away from the solar time on the 180-degree meridian.

Does it all add up? The greatest positive value in my EoT plot is about +16 minutes at the start of November, the plus sign meaning that the Sun would be ahead of our mechanical/electronic clocks, and this has been inverted in the sundial coding so as to become effectively –16 minutes. We just saw above, though, that Wellington’s longitude implies that we need to add 21 minutes to that, resulting in an expectation of a net value of +5 minutes for the start of November. Take a gander at the close-up photograph at the head of this post and you will see that to be the correction plotted in the sundial’s graph, to within the thickness of the brass-moulding line (and in any case one cannot expect sundials to render local time values to better than a minute, given that the angular width of the solar disk is equivalent to about two minutes of time). Doing the same sort of calculation for mid-February, when the EoT has its greatest negative value of about –14 minutes in my plot above, in the sundial graph this becomes +14 minutes, we add the 21 minutes for Wellington’s longitude, and thus get a total correction of +35 minutes for that time of year, just as the sundial’s graph indicates.

Will this sundial in Wellington’s Botanic Garden work forever? What I mean is, if it were still there in 500 years’ time, would it still give a valid correction to convert from the apparent solar time (what the sundial shows) to the mean solar time (what your watch shows, more-or-less)? The answer, I am afraid, is no, despite some sundials having inscriptions containing sentiments revolving around time being short for humans, but going on forever.

The equation of time itself is time-dependent. Consider again my first plot above (Figure A3), where the two distinct factors comprising the EoT were graphed. If one defines a year in terms of the solstices and equinoxes then the double-sinusoid for the effect of the tilt of Earth’s spin axis remains almost the same. (The only change is a slight variation in shape due, in fact, to the next thing I am going to describe.) This is tantamount to the year used to define our calendar being the *mean tropical year* (as, broadly, is the case).

The *other* curve in that plot, however, actually shifts slowly to the right. That curve is that delineated by the large black dots, showing the effect of Earth’s orbit being non-circular. The cycle time for that curve to be completed is slightly longer than a mean tropical year, because it depends on Earth’s movement from perihelion to aphelion and back to perihelion again, an interval we call the *anomalistic year*. That type of year being a little longer causes it to slip to the right, as noted, and the rate at which it does so is by one day every 57.5 years.

And that is why it is very appropriate that I write this blog post today, on July 14th – in fact I am rushing to get this finished before midnight so that I can put it up whilst it is still Bastille Day, at least here in NZ. (I hope you realise this means I am missing the final of the Cricket World Cup, or at least the first half of the Black Caps’ innings.)

Likely the reader is now bewildered, so I will explain. Today is the 230th anniversary of the storming of the Bastille, on July 14th in 1789. That makes it 230 years ago. If you multiply the 57.5 years I mentioned above by four, you get 230. This is a triviality, but it means that the shift in the dates of perihelion and aphelion is by four days since the French Revolution.

As aforementioned, perihelion nowadays occurs on about January 4th, aphelion on July 4th; they do skip around a bit, due to two entirely-different influences. The first is the leap-year cycle, a human imposition which is rather poor compared to what could be done. The second is the effect of planetary gravitational perturbations on both the Earth directly, and also the solar system barycentre (the centre of mass of the solar system, which most often is not even within the Sun).

Regardless, there is a general trend in the dates of perihelion and aphelion progressively coming later, by one day every 57.5 years. Stepping back 230 years, perihelion passage was occurring around New Year, and aphelion passage on June 30th/July 1st. Over 500 years, twice as long and then a bit, the perihelion/aphelion dates will shift by a little over eight days, and the black-dot curve in Figure A3 will shift to the right by the same amount; and it will also change shape slightly as our orbital eccentricity alters. (As a matter of fact, I plotted that graph based on calculations for the situation at the start of the year 2000; now, 19 years later, perihelion and aphelion have shifted later by just about one-third of a day. I hope you noticed.)

As a consequence of the perihelion/aphelion dates moving relative to the solstice/equinox dates, the equation of time changes shape. A sundial with a graph on it to show how to correct for the equation of time (and the longitude of the site) is a wonderful thing, but it cannot be perennial. Then again, who depends on a sundial to tell the time nowadays?

## 2 Responses to “The Equation of Time”

It is very interesting that the geometry behind the second contribution, that due to the Earth’s tilt, is exactly the same as the mathematics behind the rotation of the ‘Universal Joint’, much used in mechanical systems.

https://sciblogs.co.nz/physics-stop/2016/03/10/the-universal-joint/

I first got interested in the Equation of Time as a 12-year-old, wondering why the (northern hemisphere) November evenings were so very, very dark, whereas the November mornings less so. I’ve also heard that the equation of time is one reason why the UK waits till the end of October to put its clocks back (but puts them forward end of March – thus meaning 7 months of daylight savings and 5 months of ‘normal’ time a year as opposed to the more equal distribution used in NZ.)

Thanks Marcus, all good points worth discussing.

On the asymmetry of the UK’s daylight savings, of course one/we could look into the actual Parliamentary discussions recorded in Hansard, but another possibly-contributory reason comes from what might term the thermal inertia of the planet/locality. That is, the warmest time of day is not local solar noon, but a couple of hours later. Similarly, the warmest time of year is not (in the northern hemisphere) around the June solstice, but rather about six weeks later. Thus if one thinks about temperatures as well as sunlight time, one can see that there is an argument for daylight savings not to be symmetric with reference to the June solstice.