Sun Motion 3: Analemma

Equation of time and location

In the previous post, the equation of time is derived from the angle difference from the mean sun to the true sun, after projection of the true sun on the equator plane. This angle is then converted to a time difference by proportion with the (constant) Earth's rotation speed.

The equation of time is based on the position of the true sun on the celestial sphere, which is independent of the observer's location on Earth. The equation of time is the same irrespective of the position of the observer. So knowing the equation of time, an observer knows when to look up and record the altitude of the sun, and subsequently deducts the latitude of his location.

But reversing the problem, if an observer looks up at 12:00pm sharp, the azimuth of the sun (in local coordinates) may vary significantly depending on his latitude. So if you define the analemma as the position of the sun (azimuth, altitude) in the sky at 12:00pm every day of the year, you expect significantly different aspects for different latitudes.

Below I have plotted the analemma for a number of locations (only the latitude matters), taking into account the obliquity and eccentricity of the Earth's orbit, and using the (quite precise) Gregorian duration of the year (365.2425 days), going through 366 days from 20mar12 (included) to 20mar13 (included), the 2012 spring equinox being 20mar12 05h14mn, and the perihelion date being 2jan13 04h38mn.

Note that there is a slight difference between the position of the sun on the 20mar12 and 20mar13. This is due to the extra 0.2425 day (beyond 365) in the duration of the year. This offset would grow from 2013 to 2015, and then return to almost zero on 20mar2016, because 2016 is a leap year.

Why almost ?
Because (i) the duration of the year is not exactly 365.25 but 365.2425, and (ii) even beyond Gregorian calendar precision, there are second/third order influences on the duration of the year:

• Precession of the equinoxes
• Chandler wobble, nutation
• Tidal acceleration
• Milankovitch cycles
• Length of day variation

There are a number of small order variations. It is difficult to understand more than basic concepts about them and even more to put them in perspective as they have very different periods. This short video from the Casssiopeia project is an excellent tutorial. Anyway, in the long run, the Earth orbit, and consequently duration of the year, is chaotic, meaning it cannot be accurately predicted a long in time in advance (here, say some million years). The smallest (inevitable) measurement error would become a prevailing factor in the long run.

All this can safely be and is neglected here.

Sun Motion 2: Equation of Time

Have you ever heard of the 'Equation of Time' ?

I had not until I came across a Wikipedia article about it while I was reading some stuff with respect to my previous post. I loved the words: The 'Equation of Time' sounds biblical, like a formula containing a precious secret of the Universe handed over by God to Man. The first men who discovered it must have considered it divine !

But what is it ?

If I am like my fellow citizens, few people have even heard about it, as today, and by and large since the beginning of the 20th century, we rely on man made mechanical clocks, as opposed to derive the hour from the position of the sun in the sky.

The equation of time is the measurement of the time difference between the time given by a sundial and a mechanical clock. You have always been told that the sun is exactly South at solar 12:00pm (which may be different from civil 12:00pm; in France for example, civil time is solar time+1h in autumn winter and solar time+2h in spring/summer) but this is generally not correct at your clock 12:00pm.

In order to visualize why, let us imagine an individual standing on the Equator, equipped with a reliable watch. Every day of the year, that individual looks up at 12:00pm and records the position of the position of the sun, azimuth and altitude.

Here are the definitions of azimuth and altitude for a ground observer.

Obliquity of the Earth orbit

From a ground perspective the sun travels throughout the sky on a circle called the ecliptic. If we call celestial equator is the projection of the Earth equator on the celestial sphere, then the ecliptic is the circle obtained by tilting the celestial equator in the same way (direction and angle) as the Earth is tilted (23.45°) with respect to its orbital plane. The celestial equator and the ecliptic intersect on two points, which are called the vernal and autumnal points.

Precision: All objects in the sky are projected to the celestial sphere (with respect to the Earth center), the distant stars as well and the nearby sun (and the planets, etc). As a consequence, the point on the celestial sphere represents a direction, not a location. The stars and the vernal/autumnal points are considered fixed when we analyse the sun motion over the course of the year.

Sun Motion 1: Sunset in Paris

A long time ago, before digital cameras at the end of the 20th century, on my first trip to the Americas, I was trekking Peru with a group of friends. After several weeks we were casually strolling around Lake Titicaca, near Puno, in the 'Temple of Fertility' (a pagan temple from before Christinization ) as I remember. We had spend the night with local people on the Island Amantani, had eaten sweet potatoes and were pretty relaxed, our young selves silently smiling and breathing in the high altitude air.  I was proud and happy to have travelled to the distant land of the Mysterious Cities of Gold, and seen so many wonders, like the Machu Picchu.

Then I had this strange thought of a TV advertisement that showed on French television at that time. It was about people drinking Schweppes to celebrate the day when the sun sets under the Arc de Triomphe, exactly in the axis of the Champs Elysees.

Then I wondered, hey is there such a day or is it just a special effect for the ad ? And I started to try and figure out what the direction of the sun is at sunset. I knew that this earth orbits around the sun almost on a circle. Not exactly but it is a very decent approximation. Also that the earth rotation axis is slightly tilted from the perpendicular to the orbit plane. Also that the earth goes around the sun in one year which is 365 days. Not exactly because there are leap years, but it is a decent approximation. And of course that it completes a full rotation around its axis in one day, which is 24 hours.

Next I tried to visualize the earth spinning on its axis and orbiting the sun, from the space perspective and from the point of view of a person standing on the ground, and to switch from one to the other.
Well it is more difficult than what is seems. After some time, I had almost a headache.

Fortunately, I was travelling with some clever people, and notably a very sharp mind. This guy is as intelligent as he is humble, which makes him a very pleasant conversation partner. So I told him about what was on my mind. Then ensued one of my best memories of this trip, and still stick in my mind today as a canonical example of a successful Socratic dialogue. New to the subject, without indication or prejudice, we gradually explored the geometry of the sun-earth system and turned the question (is there such a day ?) into another, much simpler, the answer to which would give the solution.

It might not seem very impressive but it felt fantastic to me at that time, as it was completely impromptu and we did it without pen and paper. Just bare sheer conversation. In the most ancient Greek fashion 
A glorious day, I tell you !!

As I remember now the reasoning was something like that:

• The easiest point: Standing on the North Pole, the sun stays at the same altitude in the sky throughout the day. That altitude will change with the day of the year. Its highest point is EarthTilt°, reached on the summer solstice. Its lowest point is -EarthTilt°, reached on the winter solstice. In the latter case the altitude is below zero and the sun is  invisible.
• Generalization: Wherever you stand on the planet, on a given day the angle from the earth axis to the sun is (quasi) constant. So the sun moves on a cone as seen from a ground observer, the axis of which is the earth axis. The angle of the cone depends only on the tilt of the earth so changes from 90°-EarthTilt° to 90°+EarthTilt° in the course of the year, from solstice to solstice. This is easier to realize at solar noon.
• Locally the angle from the vertical to the earth axis is simply 90°-Latitude°, and it is tilted towards the North (for the Northern hemisphere, as all I say here).
• On the summer solstice, the sun is high in the sky at noon, so the angle of the cone must be 90°-EarthTilt°.
• So the question boils down to: For a plane (the ground) and a cone (the sun path seen from the observer on the ground) defined by its peak (the observer on the ground), its axis (tilted towards the North, the angle of which to the vertical is equal to 90°-Latitude°) and its angle varying between 90°-EarthTilt° and 90°+EarthTilt°, consider the intersection of this plane and this cone. It is generally 2 lines starting from the observer (they correspond to sunrise and sunset) which are symmetrically positioned relative to the North direction from the observer. But it can be empty (days/latitudes when/where the sun stays above of below the horizon the whole days, beyond the Arctic/Antarctic circles).

Nuclide Chart

One sunday morning in October, I woke up early to go to the Hopper exhibition in the Grand Palais, Paris with a friend. I wanted to see his paintings, some of them famous, even to those who do not know who he is. I like the contrast between the reassuring gentleness of the overview, and the crudeness (sometimes cruelty) of the details.

 

But the waiting time in the line was several hours, even though we arrived at the first hour ! So instead, we ended up in the Palais de La Découverte, next door. There it was blissfully empty, at this hour. This place is welcoming and designed for children, I think, even though many adults would learn a thing or two, or more, strolling along the corridors and reading the description panels, and watching the funny experiments on display. The good old planetarium is also very nice... But there is something peculiar about the atmosphere, something difficult to describe, an impression I rarely get..  It felt that I had traveled back in time. The inside of the Museum could have been exactly the same at any point in time from 40 years back to yesterday. I felt in a time warp !

Admittedly that is rather a good sign for a Museum, but I suspect I tend to identify sciences and technology, quite naturally in my view. In the latter field however, every concept, every item carries a very clear time stamp. In the Palais de La Découverte, on the contrary, it is late 19th - early 20th century science all over the place. This period in history is arguably one of most fertile in terms of scientific progress. It was breakthrough after breakthrough, in many disciplines. The human mind was furiously mapping Nature. And science was held in very high esteem, an unequivocal source of good. As the 20th century painfully rolled out, opinions gradually changed..

So I could not help thinking that the Museum would benefit from an upgrade and a strengthening of the connection between fundamental sciences and its applications. I would also use the vast real estate more efficiently, cram more presentations/experiments, systematically put the material on line too, and direct the visitor to related web sites, display the text in several languages (at the very least English), maybe even sublet a part of the Museum to technological/scientific companies/student bodies, if only to introduce some uncontrolled element in the place.

Now, I must be unfair to some of sections of the Museum. Surely they have some exhibitions in tune with the current zeitgeist  For example, there must be a section about climate change, a relatively new concern. There are some touch screens, but very few. Alternatively, one could defend the position that this dated atmosphere is completely intended, and must be carefully maintained. One could also point out that my suggestions are utopian, and that the Museum does very well with the budget. I would still think there is room for improvement.



In spite of all the remarks above, I enjoyed it very much. Especially the 'Atomic section'. I came across an enlightening short movie introducing the Nuclide Chart, the Valley of Stability, and basic radioactivity concepts, in no more than 5mn, and it was astonishingly clear !