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The solar analemma

Head of Apollo, 2d Century AD. Collection Lombry. The sculpture represent the etalon of Greek profile. The lines are pure and the surface smooth. The abundant and round-styled hair contrasts with the oval, elongated face.

Text and pictures by Anthony AYIOMAMITIS

Second Marathon Attempt (III)

Although the sun's position across the sky is virtually identical from year to year when looking at the coordinates for a specific date and time, differing by a few arc-seconds at best, a complete ephemeris was regenerated as a precautionary measure from Jan 1, 2002 to Dec 31, 2002 for each of the eleven desired analemmas. The analysis during the first run was based on the ephemeris for the 06:00:00 UT analemma as that was the first analemma undertaken. 

For this second odyssey at generating the complete set from sunrise to sunset, a more conservative approach was adopted by using the ephemeris for the perfectly vertical analemma on the meridian (10:28:16 UT) and by applying precisely the same type of analysis as the first run for the identification of the first half of each loop (the "S"-shape referred to earlier). This approach is not only more conservative but perhaps more accurate for the simple reason that the second half of the analemma schedule and each loop is immediately extracted by simply lining up target dates and matching altitudes from the first half. 

As was the case with the analysis during the first attempt, a small number of the newly scheduled imaging dates permitted for some flexibility due to the fact that the computed altitude (radix + offset) fell between two available dates (ex. July 26 or July 27, 2002, Table 2) which permitted for an extra day in honoring that particular imaging requirement. 

Furthermore, the ratio of multiple exposures between the summer and winter loops (12:28) is now in agreement with the ratio of the vertical distances (14.45░:32.41░); during the first run when the schedule was established using the ephemeris of the 06:00:00 UT analemma, the ratio was visually set to 12:28 which did not agree with the ratio of the corresponding vertical distances (8.00░:20.88░).

The analemma of 8h UT over Epha´stos' temple, Athens.

Similarly, if a multi-exposure did deviate by the self-imposed maximum of +1 day when imaging the first half of each loop, the schedule for the complementary side was adjusted accordingly due to a firm desire to simultaneously assure perfect horizontal and/or vertical symmetry. With respect to the time of exposure, the shutter was tripped at the precise second with absolutely no allowance for error.

With the guiding hand of "Απόλλων", my second attempt at generating the complete set of analemmas from sunrise to sunset was much more problem-free as I was able to eventually adhere to my regimen of shooting the sun (sort of speak) at precisely the new set of precise dates and times (Table 2) over the course of one calendar year without the slightest problem as a result of any further earthquakes or seriously adverse weather for seven of the eleven analemmas. 

Of course, the major obstacle encountered during the first run, namely the accidental film advances with the Canon AE-1 bodies, would resurface for two of my double-analemma cameras (11:00:00-12:00:00 and 13:00:00-14:00:00 UT), thus necessitating yet additional restarts three months into the project and forfeiting yet another 44 (4 analemmas and 11 exposures each) multiple exposures.  

The only deviation from the shooting schedule occurred for my two extreme analemmas which encountered a physical obstruction (a mountain range) around the winter apex point (Figures 3 and 13), thus necessitating the forced omission of several total multi-exposures. This is something which I had suspected in mid-October/2001 and a detailed analysis of the mountain's profile as a function of altitude did not permit for any creative adjustment in the shooting schedule (+1 day etc) so as to work around the obstruction, for the sun during these two dates was a mere 1.5░ below the mountain's profile with little movement in altitude during this period of the season. 

By an incredible stroke of luck, a scheduled exposure for each of these analemmas coincided with the solar disk being partially obstructed by the mountain (thank you Apollo !) which made the problem quite self-evident to the observer of the completed analemma. The local weather for both of these inflexible dates was, again fortunately, also accommodating (thank you Apollo !). Nevertheless, Murphy's Law struck a second time for these two analemmas as I would also loose the much-desired apex point (winter solstice) on the lower winter loop!

The analemma of 9h UT over Hera's temple (700 BC).

My initial plans called for an extra multi-exposure for each analemma involving the actual physical landscape as viewed through the lens. However, this was something that was eventually bypassed so as to make the creation of the composite depicted in Figure 14 possible.

Nevertheless, upon the completion of each analemma, the roll of film was advanced by one frame for the shooting of the associated landscape for future reference, and which has the added benefit of also providing a point of reference for the photo lab in the proper cutting of the film following development. The foreground images presented in the composite analemmas were shot during January and February, 2003, when the sun is relatively quite low over the horizon and a much richer shade of blue for the background sky can be acquired without having to resort to using any filters, thus making the analemma more pronounced.

Not to leave any aspect of this project untouched and bordering on something that could best be described as a Sisyphean task extraordinaire, I am also currently imaging what I believe to be the first triple analemma (129 total multi-exposures) on a single piece of film as well as the first analemma similar in appearance to a bead necklace (see Table 1).

The triple analemma is to be centered by the special case vertical analemma (10:28:16 UT) alluded to above as well as the analemmas at + 40 min (09:48:16 UT and 11:08:16 UT) which will just fit perfectly (68.3░ x 47.8░ arc coverage) on a single piece of 35mm film using my Canon FD 24mm f/2.8 lens (74░ x 53░ arc coverage in portrait mode).

My initial aspirations for the triple analemma involved the meridian case flanked by the analemmas at + 1 hr (09:28:16 and 11:28:16 UT) but this was not possible as the physical range in azimuth slightly exceeds the corresponding range available with my 24mm wide angle lens.  

A further analemma on the meridian also in progress involves imaging during every 0.6░ movement (or more) in either azimuth and/or altitude (139 total images) so as to generate a completed analemma similar in appearance to a bead necklace (the solar disk has an apparent diameter of approximately 0.5░). 

Mock-up showing the maximum variation of movements in elevation (#1) and azimut (#2).

The two points representing winter and summer solstice - Dec 21/02 and Jun 21/03, respectively - as well as the intersection of the two loops (Aug 30/03) will be included automatically. 

Ignoring these three fixed points, the greatest proportion of the remaining 136 images will be due a movement of 0.6░ or greater in altitude (#1), representing 124 out of 136 images, whereas the balance (12 out of 136) will be due to azimuth (#2). There exist no examples where both conditions (0.6░ movement in each of azimuth and altitude) occur simultaneously. 

Similarly, 86 of the 139 multi-exposures will be spaced two days apart whereas a further 27 multi-exposures will be spaced three days apart between exposures.

Yet a third analemma on the meridian in progress as well is based on a weekly shooting schedule and whose sole purpose is to illustrate the relative "slow" or "fast" movement of the sun during the different seasons. If the interval between multi-exposures is precisely the same, the relative rate of the sun's movement across the sky can be assessed more accurately. By keeping the three key reference points of the analemma fixed (summer solstice, cross-over, winter solstice), a weekly imaging schedule is easily generated without a need to adjust any of the key dates. The sole exception to the 52 planned multi-exposures is the one extra day between the winter solstice and the scheduled image immediately prior to it (Dec 13/02).

Simulating the Earth movements

Summation of the horizontal and vertical movements of the Earth axis

Previous summation as seen from the Sun. 

Documents Analemma

As described in the following section, the Canon A-1 is ideally suited for the imaging of the solar analemma as it makes this exercise completely problem-free and, to this end, two of the three above analemmas now in progress which are very aggressive in nature - 129 and 139 multiple-exposures each on a single frame of 35 mm film - involve my two faithful Canon A-1 cameras purchased over 15 years ago.

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