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).

Now with a paper and pen we can see that, if 365 is the number of days in a year, the angle of the cone is:
\[90{}^{\circ}-\text{EarthTilt${}^{\circ}$} \text{Cos}\left(2\pi  \frac{d}{365}\right)\]
Then add an atlas (the direction of the Champs Elysees is 296°, i.e. West+26°) and a computer and we can graph the cone and plane described above, for the winter solstice (21dec), the summer solstice (21jun) and an equinox (20sep). The Champs Elysees direction is in orange, and the sunrise/sunset directions are in red.

The sunset angle on the summer solstice is significantly beyond 296°, so the Schweppes ad was right. The sun does set in the Champs Elysees axis. It must be twice a year. And the same must be true for the sunrise.

This simple geometrical approach gives the following dates:
Sunset: 7 May, 5 August.
Sunrise: 4 February, 6 November.
Roughly in line with observations.

I haven't found the Schweppes commercial on youtube. But at least here is the Arc de Triomphe in the first days of August...