The Analemma Project

A one-year multi-exposure photograph of the sun

What is an Analemma?

Some of you may ask, "What is an analemma?" Well, if you photographed the sun once a week at the same time of day for an entire year, you would have a photo of the analemma, the sun's apparent path on the sky over the course of a year, which looks like a figure-eight. During June, the sun rises in the northeast and sets in the northwest, while in December it rises in the southeast and sets in the southwest. As most people know, this is caused by the 23.5-degree tilt of our Earth's axis, which is also responsible for the seasons and explains the north-south dimension of the analemma. But the analemma also shifts east-west, a phenomenon that should be known to people familiar with the equation of time. There are days on which the sun reaches the meridian before your wristwatch says noon (the sun "runs fast") and on other days the sun reaches the meridian later (it "runs slow"). This also explains why the earliest sunrise and latest sunset in June are before and after the summer solstice, respectively. The opposite is true around the winter solstice: the latest sunrise and earliest sunset of the year come after and before the sun turns around in December, respectively. The last part of the analemma's shape, the different sizes of the figure-8 loops, is caused by the eccentricity of the Earth's orbit.

When I began this project, there were only five successful photos that had been taken of the analemma (by photographers Dennis DiCicco, H. J. P. Arnold, Frank Zullo, Vasiliy Rumyantsev, and Jack Fishburn). These were featured in an article by Dennis DiCicco, the photographer of the first analemma photo, in the March 2000 issue of Sky & Telescope. A sixth image by Tom Fuller showed up at the beginning of August 2000 in the September issue of "Astronomy". The March 2000 S&T article inspired me to start my own photo of the analemma. Photos had been taken in the morning (the analemma leans to the left) and in the afternoon (the analemma leans to the right), but never at noon, when the figure-8 stands upright on the meridian. I had considered taking a photo like this, but it would have meant having to be at home at noon, which doesn't work since I still have to go to school. I didn't want to take the pictures each weekend because the chance of good weather is not as great; in addition, I wanted to vary the time between photos from 6 days to several weeks in order to get evenly spaced solar images. The only possible time turned out to be mid-afternoon. Other ways of photographing the analemma could not be realized, since they were not possible from my location. They included taking a picture of the analemma at sunrise or sunset, with only half of it above the horizon (only possible from the equator, it also only requires you to take photos for about 8 months, with a several-month break in between). Another view of the analemma can be seen from the southern hemisphere, where the smaller loop of the figure-8 is closer to the horizon.

This page documents the progess of my photograph, including all preparations for the project, plus an animated "picture" of the analemma should have "looked" in my camera. The image at left shows how my photo of the analemma should have looked when it was finished. I say "should have looked" because this project was full of things that can go wrong, especially when it is attempted more as an experiment than as a photo (more on this later). The image at right is the animation showing how the photo actually went along.

The Analemma Gallery

The first analemma photo, taken by Sky & Telescope associate editor Dennis DiCicco from 1978 to 1979. It followed a one-year "test exposure" (1977-78), in which Mr. DiCicco had misaligned the camera and part of the analemma was no longer on the film. Image courtesy of the photographer.

One of the latest photos of the analemma, photographed by astrophotographer Vasilij Rumyantsev of the Crimean Astrophysical Observatory. Because Mr. Rumyantsev used only a neutral density filter (filter factor 4.0), he had to stop down the lens to f/22, creating the diffraction spikes around the solar images and giving the image a dramatic touch. Image courtesy of the photographer.

The newest of the analemma photos published in the March 2000 S&T article, taken by Jack Fishburn. Image courtesy of the photographer.

Photographing the Analemma

Photographing the analemma is not as difficult as the photo might make you believe. All it requires is patience and a bit of ingenuity for devising a setup that allows you to mount a camera in place for a year and trip the shutter at the right time of day once a week. Most photographers decided on mounting the camera in a fixed place. One mounted the camera outdoors in a box which he could close to protect the camera from the elements between exposures, while three others mounted them indoors behind a window, one of them opening the window for each exposure. Rumyantsev designed a mount which could be removed and replaced, always pointing in the same direction, of course. Not wanting to have a camera permanently mounted somewhere, I decided on this option, which raised the question of how to proceed with this task. But another question had to be answered first: what camera do I use? I have a Canon EOS 50E, but this is my only camera and I need it often, meaning I had to look for another camera somewhere else. I had had some experience in modifying single-use cameras for astrophotography (making it possible to take time exposures and the like), so after time exposures, time-lapse exposures seemed to be the next logical step. The easiest camera to modify is Kodak's FunSaver Panorama (also known as Max Panorama). I stripped off the cardboard and plastic wrappers of my newly acquired single-use and went to work on adding a D5.0 solar filter to its front. This turned out to require quite a bit of fingertip work, since the camera is extremely small and I could not cover the shutter (so that I could manually reset it without winding the film). That finally done, I loaded a test film (Kodak Gold 400) and shot several test pictures, making sure the solar filter did not leak light, even when taking 100 exposures on the same frame in full sunlight. I chose ISO 400 because the Kodak Max has a shutter which stays open for about 1/125 second and its lens is at f/12. Working backwards, I used the exposure tables in Michael Covington's "Astrophotography for the Amateur" to see what kind of film this would correspond to. The tables say that for an f/11 lens with 200 ASA film, 1/125 would be right. But since I wanted to overexpose the sun images to counteract the effects that clouds or the sun's low altitude in winter would have on the its brighness, I chose 400 ASA.

>After the second test roll finally came back (mostly) positive, I loaded the actual film. I had to trim it, since 36-exposure rolls don't fit into the camera's pre-wind chamber. While working on the photo, I was not sure by how many exposures I trimmed it (I did this generously), and since I had had some problems starting the photo (see the journal below), I had to advance the film several times and did not know where I am on the roll. It is possible to expose the film leader, and if that's what I was doing right now, the photo would have been a terrible waste of time and effort.

Anyway, after advancing the film, I mounted the camera on a mount which allowed me to remove and replace the camera from its unsheltered outside spot but without disturbing its aim. This mount is designed to fit perfectly onto my window sill, allowing exactly the same view each time I replaced it. On top of this mount is another piece of wood at a 30-degree angle, allowing the camera to point up into the sky. The mount is shown in the picture above.

After attaching the camera horizontally, I wasn't sure if it should rather be attached vertically, so I sketched a test "image" of how the analemma would look (similar to the picture above left). As luck would have it, the analemma requires a vertical format, meaning I had to remove the camera from the mount (not an easy task because I had attached it with crazy glue and silicone) and change its orientation. Having finished the mount, I eagerly awaited my chance to take the first photo. That chance presented itself on Monday, March 13, and I took my photo at 14:45 LT, a time I had finally chosen because I would be home from school by then, but the sun would still be up in the winter. I live 50 degrees north of the equator, meaning that the sun sets around 15:00 LT in December and January, while in the summer it never gets dark (astronomically speaking). However, I noticed that the sun was much too far to the left at the time of the exposure, leading me to compare the direction of the sun as measured with a compass to that given by RedShift, which I used for making my calculations, including where to point the camera. Indeed the compass showed a difference of more than 10 degrees compared to the predictions. I again had to remove the camera and reglue it pointing 10 degrees further to the left.

The next day, when the silicone had dried, 14:45 came and went without a break in the clouds. One day later, though, I got my chance for the first photo, which was almost ruined by the clouds that pulled in to cover the sun only 10 seconds after my exposure.

To trip the shutter at the right time, I use a radio-controlled clock (picture below), which is accurate to less than a second. A difference of a second doesn't really matter in the final picture (it goes unnoticed), but more than a few seconds and the sun's image shifts a bit. As an example, the sun moves its own diameter (0.5 degrees) in 2 minutes.

The Schedule

To plan my exposures, I used a bit of mathematical knowledge. The following explanation is rather technical, so you might want to skip it. It is only necessary to get an even spacing of the solar images, which is a purely aesthetical matter. You can also get nice pictures of the analemma by just taking a picture once a week. However, you can also take the schedule below, begin somewhere in the middle and start from the beginning again until you reach your starting point, and you'll get a picture with the even spacing as well.

The sun moves rather slow around the solstices (the length of a day changes by a few seconds each day), while around the equinoxes it moves very fast (a day changes by a few minutes in length), making for an uneven spacing of the solar images, but I wanted them to be evenly spaced. Knowing that, like many things in nature, the speed of the sun's declination change can be represented with a sine curve, I set the length of a year (365 days) to equal the x-coordinates for a complete sine curve (360 degrees). As you can see, these are almost the same, allowing me to do this. I just had to remember to always round day lengths up to account for the extra 5 days. Then I decided on 40 exposures for my picture. As the x-coordinates progress (the days of the year), the y-coordinates (sun's declination) vary in speed. Since the sine curve moves between 1 and -1, I chose a delta-y of 0.1 for the change in the sun's declination. Using the vernal equinox (the start of my photo) as the 0-point, I took the reverse sine of 0.1, then 0.2, then 0.3 and so on to get the amount of days from the vernal equinox for an equal change in the sun's declination up to the summer solstice. This gave me the spacing (in days from the last photo) 0, 6, 6, 6, 6, 6, 7, 8, 9, 11, 26. I applied this from the vernal equinox until the summer solstice and from the autumnal equinox to the winter solstice. I reversed the numbers and applied them from the summer solstice to the autumnal equinox and from the winter solstice to the vernal equinox. I came up with the following planned exposures for the final pictures, from which I made the image at the top left of the page. The dates on the left are the planned dates, the ones on the right the actual dates. The one point that is (to me, at least) most important is the crossover point, where the two loops come together. The sun is around this position for a day or two around April 13th and August 30th, requiring only one of these exposures, and increasing the chance of good weather on at least one of the days. It seems that to the other analemma photographers, it was also the most important exposure, as they all exposed on one of these days. Of course, with this terrible weather in Germany, it was cloudy on both days, so I am now missing this point!

The Analemma Story

I started working on my mount in February 2000. Work went relatively fast, and the mount fit perfectly onto the window sill. The top of the mount had an angle of 30 degrees to the horizon, so that the center of the analemma would be in the center of the frame.

People like single-use cameras because they are so small, and that made it all the more difficult for me to make a little solar filter for mine. Kodak's "Max Panorama" has a little lever that can be cocked manually for more exposures without advancing the film, making this the camera of my choice. This lever, however, is amazingly close to the lens (since it also acts as the connection between the shutter and shutter button). Covering the lens securely with solar filter foil meant not being able to reach the lever to reset the camera. So I had to work extremely precisely in order to cover only the lens. The finished product is shown below.

I shot a first test film in mid-February, in which the solar images looked good, but as it turned out, light still leaked in from the top of the camera, where a part of the filter could not be glued down if it were not to cover the shutter lever. Therefore, I designed a mask, which would cover the camera almost completely, leaving only the filter-covered lens exposed. In this way I could reset the shutter, then cover the camera and take the picture. I shot another test film, averaging about 50-60 solar images per frame (more than for the final photo, but very small leaks would show here as well). After development, I discovered the light leak had not been completely removed, but the mask did indeed take a lot of the light. I tried scanning one of the photos and editing it with Photoshop - it turned out I could remove the light leak and easily combine the photo with another one. This would help later, since I'm taking my photo with a solar filter, meaning it will consist of the analemma on a black background. This way I could digitally composite the analemma shot with a photo of a nicer scene than the view out my window offers.

The construction was near completion, but I still had to devise a plan for the individual photos. I calculated the time span between photos for an even spacing of the solar images by taking the reverse sine of the distance, as explained above. The math is simple, but understanding the principles behind it is a bit harder, so if you want to take your own photo with even spacing, just use the dates in the table above (it doesn't matter where you start), and once you reach the end, start from the beginning until you have an entire year! Don't worry if you take your photo in a leap year. One extra day doesn't make a difference.

The open analemma camera's back. The film needs to be wound onto the spool in the dark, placed into the corresponding spots, and the back closed (still in complete darkness). As difficult as it may sound, this process is rather easy.

In early March, one month after I had started my preparation, I was ready to take the first picture. I drew a chart to correspond with the sun's alt/az coordinates on the photo dates to give me a picture of the analemma as it would look when it was finished. The distances between images turned out very nice, but I realized that the camera, which I had already glued to its mount with super glue and silicone, needed to point 15 degrees more to the right. In addition, it needed to be in the vertical format. I removed the camera (not an easy task) and reglued it.

Three days later, I took my first photo and realized that the sun was too far on the left (and that the analemma would thus go off the frame in the winter, when I would be almost finished). I double-checked the coordinates given by RedShift with a compass and noticed that the compass showed a different angle for the sun than RedShift. I had to remove the camera once again and reglued it (again!), while silicone leaked out of the tube in the process and covered my entire sweatshirt (nasty!).

The next day, when I was finally ready to take the first photo, it was cloudy (as I would have expected). One day later, though, I got my chance. About 5 seconds after the photo, a big cloud pulled in to cover the sun. I was still somewhat nervous if the analemma would fit completely onto the film frame - the sun was about 5-10 degrees to the left of the optical axis, the camera's lens allows for 25 degrees on each side, and RedShift indicated that the sun would only be 8 degrees further left at its most southerly azimuth (makes 12-17 degrees, within the margin).

After the second photo, I was still worried about the camera's aim, and even more about the fact that I didn't know at what position the film was inside the camera. I had cut the film down to about 16-20 exposures before I loaded it in the camera (it won't hold longer rolls), and I had also skipped several frames at the beginning of the project, so I was not sure how far into the film I really was. The reason for my worry was the fact that the camera would allow you to take one last photo on the film leader - if I was taking my photo on that part of the film, it would be an awful way to go! (This can be seen in the photo above right. The film leader, which hangs tight in the spool, can be exposed the way it is seen above. In addition to this being a half frame, the leader is already exposed as is.)

The photo on April 1st (and no "April fool's" here!) was the stupidest thing that happened to me yet - I forgot to push down the mount completely onto the window sill, and the solar image may be off by a few degrees. However, it was relatively dim anyway, and I could just erase it in Photoshop (is that allowed?).

On April 13th, the day on which the sun was at the crossover point, thick clouds covered the sun, preventing me from taking a phto on that day, which means taking that photo in August. Mid-April brought an incident that my boss shouldn't hear about - I closed the store (it's a photo store, coincidentally) and ran home to take a picture, then went back and opened the store again. Five minutes after I had returned, my boss stopped by for a random visit.

By this time, I had gotten used to the fact of taking a picture once a week - it had become routine! I was no longer anxious for the year to be over. Indeed, I looked forward to the fame that came with finishing the photo (would I be 6th?), but it was not like I could not sleep because I was so nervous (as it was the first few days!). In late April, I decided on starting a second photo with an older camera I had recently acquired. That way, if my rather fragile single-use failed, or the photo did not turn out for some reason or another, I would not have wasted my time, and would have a finished analemma photo only two months later.

A month before the solstice, I decided to add another one or two photos between the solstice photos and those immediately before and after them. That way, the spacing of the sun would be slightly uneven towards the solstices, hinting at the sun's slower motion at those times.

The June 7th photo was a masterpiece of stupidity: I couldn't believe that a single cloud had to move before the sun in the 30 seconds before and after the photo, and took a photo 30 seconds late! In those 30 seconds, the sun will have moved 1/4 of its own diameter - hopefully not noticeable in the final photo. In this case, I would be twice as angry at the annoying weather in Germany, which can't ever be right.

The next masterpiece of stupidity occured not much later, in fact it was the next photo that got messed up too. I was in the store on June 21st, the summer solstice, one of the most important photos in the analemma. Though I was just visiting, my boss couldn't keep up with all the customers, so he asked me to help him. This way, 15:45 came and went, and I did not take a photo. I'm still asking myself why I hang out in that store all the time, even in my free time. Anyhow, I took the photo on June 28th, since I went on a short trip on June 22nd, and thus wasn't able to catch up with the photo quickly.

The weather was terrible the entire summer, and I was glad to get enough shots to make the picture look normal. After all the setbacks that I had experienced, the analemma became more of something I just wanted to get over with instead of eagerly awaiting the day of its completion. But I couldn't know what was expecting me in autumn - weeks of cloud that did not let the sun through once! Fortunately, the weather got better shortly before the winter solstice, so I was still able to take some pictures before the sun turned around at the bottom of the analemma!

On December 13th, I didn't have to cut school - my sports teacher actually let me go home and take the picture! I left after 20 minutes of class, ran the 3.5km home in about 10 minutes, took the picture, then went back by bus to enjoy another half hour of sports. It was the second-last photo before the winter solstice, two photos I was luckily able to take after that terrible weather in the past months.

After my Christmas vacation, about two months remained to finish the analemma, and they went by rather quickly. Unfortunately, the the weather was again rather poor so I could not take the last two pictures, but all in all, I am satisfied with the result of my experiment with the single-use camera. I never started that second analemma photo I planned earlier in the year, but this was not my last analemma!

Analemma Resources

Analemma.com - the analemma explained, allows for interactive control of an imaginary analemma, where you can change a planet's eccentricity, axis tilt and other factors and see their effect on the analemma

Sky Publishing - publishers of Sky & Telescope, one of the most popular astronomy magazines in the world (and my favorite!)

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