# Analemma of the Sun

I recently read that to date, Mount Everest has been scaled 1924 times.

Compare this figure to the number of people who have successfully captured the Analemma of the Sun. Any guesses? 1000, maybe 500? Nowhere even close. According to the founder of this website, in the entire history of mankind, not more than 20 people have managed to get it right.

This is the first ever photograph that successfully captured the Analemma. It was taken over a period of one year by Dennis di Cicco in 1978.

So what is the Analemma of the Sun? Simply put, if you look at the sun, from the same place on earth, at the exact same time everday, it’s position will shift slowly over the period of the entire year, slowly tracing out a figure-of-eight shape in the sky.

I modified part of an astro-navigation computer program I had written in 2011 to simulate an Analemma. The picture below shows what the output looked like. I forgot to the scale the axes properly and that’s why the figure looks a little distorted but as you can see, it very closely resembles a figure-of-eight.

How does it happen?

Two properties of the earth’s motion combine together to create the Analemma. These are, (a) The earth’s rotational axis is tilted 23.44 degrees with respect to it’s orbital plane and (b) the fact that the earth traces out an elliptical path (called eccentricity) as it orbits the sun.

Over a period of a full year, these two properties combine together, to trace out a figure-of-eight pattern in the sky (providing we observe the sun at the exact same time and from the exact same place).

The earth’s tilt and the eccentricity of its orbit give rise to the two factors described below that create an Analemma.

Declination of the Sun.    As it revolves around the sun, the 23.44 degree tilt of the earth’s axis causes the sun’s position (called declination) to shift north and south of the equator. This movement is extremely significant to life on earth as we know it because it is this property that gives rise to what we call ‘seasons’. Without seasons, life on earth would have been very very different.

The Equation of Time.     The earth’s orbit around the sun is not exactly circular, it is slightly elliptical in shape. According to Kepler’s 2nd Law, a planet tracing out an elliptical orbit must travel faster when it is closer to the sun and slower when it is farther away. There are some great animations on the internet that demonstrate this, so I wont go into the details here. This difference in a planet’s velocity as it orbits the sun, changes how fast the day progresses. In mathematical terms, we call this The Equation of Time. The Equation of Time gives us the difference between time measured by a sundial and time measured by a modern watch.

I wrote some very simple Octave code to visualise the Equation of Time and Declination of the Sun. Octave is great because it’s very very powerful, and it doesn’t cost a ton of money like MATLAB. Here is the octave code I wrote…..

```
%begin octave code

clear;
%calculate the equation of time for a full year
d2r = pi/180;
r2d = 180/pi;
w = 360/365.24;
d = -78:1:286;
a = w * (d + 10);
b = a +(360/pi) * 0.0167 * sin(d2r * w * (d-2));
c = (a - r2d*atan(tan(d2r*b)/cos(23.44*d2r)))/180;
eoT = 720 * (c - round(c));
figure(1);subplot(1,2,1);plot(d,eoT);
figure(1);subplot(1,2,1);xlabel('Days before and after Spring Equinox');
figure(1);subplot(1,2,1);ylabel('Minutes');
figure(1);subplot(1,2,1);title('The Equation of Time');
figure(1);subplot(1,2,1);grid on;
%declination of the sun for a full year
dec = 23.44 * sin(d/365*2*pi);
figure(1);subplot(1,2,2);plot(d,dec);
figure(1);subplot(1,2,2);xlabel('Days before and after Spring Equinox');
figure(1);subplot(1,2,2);ylabel('Degrees');
figure(1);subplot(1,2,2);title('Declination of the Sun');
figure(1);subplot(1,2,2);grid on;
%analemma = plot of eoT vs decliantion
figure(2);plot(dec,eoT);
figure(2);title('Analemma of the Sun');
figure(2);xlabel('Declination');
figure(2);ylabel('Equation of Time');

%finished octave code

```

…..and here is the output of the program. It shows how the Equation of Time and Sun’s Declination change over a period of one year. Day 0 in these graphs represents the Spring Equinox (March 20) when the Sun is directly over the Equator, and moving north.

Mathematics Rules the Universe

Now when we plot The Equation of Time against the Declination of the Sun, lo and behold……an Analemma magically appears!!!!!! Don’t you just love the maths behind how the universe works!!!

My First Analemma

Its been almost 50 weeks since I began my attempt to photograph the Analemma. I have just one exposure left to complete the sequence and I should be able to complete this project early next month, and join the ranks of the handful of people who have successfully completed this feat.

More details when the job is done……..[EDIT: This blog post outlines my Analemma attempt]

By the way, the undisputed master of the Analemma is Anthony Ayiomamtis. You will truly be amazed by his Analemma photographs.

You don’t really need expensive photographic equipment to be able to capture an Analemma. You could even mark the shadow of a stick or a pole on the ground. Over a year, these marks would also trace out a perfect Analemma! All it takes is the patience to keep going for a whole year, like these clever people did.

Maciej Zapiór and Łukasz Fajfrowski have just finished an absolutely amazing capture of the analemma. They used a pinhole camera and recorded the sun’s image directly onto photosensitive paper. See their incredible pics here. They even have a video on youtube where you can see their clever stepper motor controlled shutter in operation.