The
solar analemma
Text
and pictures by Anthony AYIOMAMITIS
The
challenge and the scoop (I)
Analemma
: "8"-shape figure drawed by the different positions of the sun recorded at a
certain point in time (at 24 hour intervals) and from the same
location in the course of the calendar year.
This figure is not particular to Earth and can be visible, with
other shapes (loop or teardrop) from other planets of the solar
system.
This
article is a scoop and a true challenge on the technical side as never
before such a result has been achieved.
Let's judge by yourself :
this is the first time that one recorded an analemma of the Sun extending
along a calendar year, this is the first time that one pictured
analemmas monthly during a full year, this is the first time
that one pictured an analemma at meridian, this is the first time
that one created a composite of 11 analemmas, this is the first time
that one pictured a double analemma on a single frame as this is
the first time one pictured a triple analemma, the all performance
having been executing by only one and same person ! Last
but not least, all these analemmas are symmetrical...
Consultez
cet article en français
As
amazing as it may seem, until 2002 only seven times has someone ever managed to
successfully image the solar analemma as a multiexposure on a single
piece of film and published his results. This daunting feat has been accomplished
four times in the USA, once in Russia, once in Denmark and one more time in Britain.
Since that time, other amateurs also succeeded to record it.
To
see : Analemma, the movie, on APOD
by Tom
Matheson (recorded in 2006, .WMV file of 2.1 MB)
Having read and implicitly challenged that the
analemma is one of the most difficult and demanding astronomical
phenomenon to image, I immediately set out on such a task from my
cottage villa just northeast of Athens, Greece (38.317° N, 23.800°
E). As noted by Sky & Telescope
Senior Editor Dennis di Cicco, a skilled astrophotographer, himself
author of the first analemma in 1979, "Most people say
you have to be nuts to attempt a year-long exposure of the Sun.
Those who have succeeded will probably agree" (S&T, March
2000, p135).
Although
I was eager to start on this project immediately upon reading the
article (and challenge) by di Cicco, I would have to wait until the
summer of 2001 when major construction work around my cottage would
be complete.
Regrettably,
six months and roughly 200 total multi-exposures into the project
(Dec 2001), adverse winter weather not experienced in four decades
led to my missing an exposure which brought a disappointing end to
my first attempt at replicating this unique feat.
Having
remembered that none one of the analemmas listed in the di Cicco
article had been completed over twelve consecutive months of the
same calendar year, I immediately reset my sites on a Jan/2002
restart using the trials and tribulations acquired during the first
unsuccessful attempt. With the guiding force of "Απόλλων", son of Zeus and Leto and god of light (and the sun) in
ancient Greek mythology, I was able to finally embrace the challenge
inspired by the di Cicco article eleven times over the course of
twelve consecutive months involving two Canon A-1 and five Canon
AE-1 manual focus cameras and nearly 500 final multiple-exposures of
our closest star.
Furthermore, this Herculian effort and triumph,
resembling a Sisyphean task a number of times thanks to numerous
accidental film advances, was instrumental in providing the impetus
for a return odyssey for three additional and very special analemmas
projects currently under way involving 129, 139 and 52
multi-exposures each, respectively, also on a single piece of 35-mm
film (24x36 mm).
Analytical
preparation
Waiting
that Anthony's PC is up and running, let's remind that the shape of the
analemma depends on the distance between the Earth and the Sun, the
shape of the Earth's orbit and the way that the solar time is
calculated.
Let's
take a picture of the Sun in a landscape. The vertical coordinate of the
Sun is the solar declination (in its astronomical sense), its horizontal
coordinate indicating the shift between the (apparent) solar time and the
mean solar time (read of your watch). The difference between both times is called the equation of
time and is due to the variation in the length of the synodical day
(effect of the orbital eccentricity and the irregularity of the apparent
move of the Sun across the sky). Add a correction for the mean solar time
due to the fact that one determines a fictive solar position over
the geographical equator whereas it moves along the ecliptic. It is thus necessary to
take into account of the slope of the Earth' rotation axis over the
ecliptic plane.
The
shape of the analemma changes from one planet to another but does
not depend directly on the latitude of the observer but rather of
the time of the day and the time of the year considered. However,
according to the geographical latitude of the observer, this large
loop will be differently tilted over the horizon. That said, let's return
to the work of the author.
While
planning for such an overwhelming technical and logistical exercise,
I was about to discover a number of critical issues that must be
addressed. First and foremost is the composition of the analemma
within the 35-mm frame. Using Chris Marriott's SkyMap Pro (Version 8)
sky charting and planetarium software, I was able to determine the
dates of minimum and maximum azimuth (i.e. leftmost and rightmost
analemma extrema) as well as the minimum and maximum altitude (i.e.
lowermost and uppermost extrema) so as to identify the minimum field
of view required and, therefore, maximum focal length lens to use.
These extrema are dependent on the time of day that one proposes to
shoot the time series and, as a result, we come to another key issue
that must be addressed simultaneously. This latter issue is not as
straightforward as it may seem since the rise and setting time of
the sun complicates matters. Therefore, one must seek out the latest
rise time as well as the earliest setting time and find a convenient
compromise between these two extrema. For my situation, my latest
rise time was at 05:41 UT (Jan 5 and 6, 2002) whereas my earliest
setting time was at 15:05 UT (Dec 6 and 7, 2001) which led me to
consider 06:00:00, 09:00:00 and 12:00:00 UT as initial potential
targets. Analyzing the minimum and maximum azimuth and altitude
alluded to earlier for these times over the course of twelve months
made the final selection very easy since the latter two time
candidates involved a maximum elevation of over 60° whereas the
earliest time candidate had a maximum elevation of slightly over 30°
(see Table 1). Of course, all of this proved to be immaterial a few
weeks after the initiation of my first analemma as I would return
with additional camera gear in an attempt to capture these analemmas
as well.
To
check : Time
of sunrise and sunset
Calculated
by the U.S.Naval Observatory
Having established the minimum and maximum azimuth and
elevation and, therefore, range of solar movement in azimuth (39°44'18")
and elevation (29°44'51") across the sky (Table
1), I had to identify the best focal length lens to use. Since 35-mm film is 24
mm high and 36 mm wide, a standard 50 mm lens is immediately ruled
out since its field of view is 25°x37° (my range in azimuth is
nearly 40°). In contrast, my Canon FD 24 mm f/2.8 wide angle lens
provides a field of view which is 53°x74° and, therefore, the
analemma would be occupying about 60 % of each dimension which not
only permits for some flexibility (and/or error) in the placement of
the analemma within the 35-mm frame but allows for a foreground
effect to be added once the exercise has been completed. My choice
for this lens would have a further benefit a few weeks later
following the start of my first analemma as I would very pleasantly
discover that the lens was ideal for capturing two analemmas
perfectly and simultaneously if they were spaced one hour (or less)
apart.
Analema
simulator
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A
software from Bob Urschel (248 Kb ZIP file)
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The
next issue to be addressed in this year-long exercise is
the starting date. For aesthetic purposes and using SkyMap Pro
again, I was able to determine that the precise crossover point
between the two loops would be Aug 30, 2001. I feel this
consideration is important so as not to "blur" the
crossover point when traversing from the upper to the lower loop
during the first part of the year and the return trip during the
latter part of the year. Keeping this date now fixed in any shooting
schedule I was about to implement, I directed my thoughts on the
ends of the loops and, more specifically, the dates where the sun
would be at the absolute apex (June 21, 2001 and Dec 21, 2001).
Using these three points as now fixed, I set about to establish a
shooting schedule so as to have as perfect a symmetry as possible
both vertically and horizontally.
To
read :Astronomy Software Review
For the vertical equidistance, the
ephemeris of the sun for 06:00:00 UT was used to establish the
solar altitude at the summer solstice as well as the crossover point
in the analemma; this figure was divided by 7 so as to get the
degree offset (1.15°) needed in altitude between successive solar
images for the left side of the summer loop. Similarly, for the
winter loop, the difference in altitude between the cross-over point
and winter solstice was established and the necessary divisor was
determined to also yield an offset as close to 1.15° as possible
(divisor of 18 yielding an offset of 1.16°). Using the crossover
between the two loops as a starting point and the altitude of the
sun on that day as radix, the altitude offsets were cumulatively
added (for the summer loop) or subtracted (for the winter loop) and
the corresponding day with that target altitude was identified and,
coincidentally, the first half of the analemma schedule (in the form
of an "S") had now been established.
Procedure
to simulate the analemma
with
Skymap Pro
1.
Set map date to Jan 1, 2002
2. Set map time to 12:28:16 (to simulate the meridian analemma)
3. Press "S" on left sidebar icons so as to get
the view on
the southern horizon
4. Right-click on the Sun and ... choose TRACK SUN
5. Compute position every 5 days
6. Compute 72 positions (since 1 year = 72 x 5)
7. Use alt / az (bottom option on same table)
8. Labels to be horizontal (or vertical if you prefer) |
With respect to the
horizontal symmetry between the two halves of each loop with one
side of the two loops now defined as per the above, the task at hand
was to simply identify the appropriate day that would represent a
mirror image across the minor- or major axis of the analemma. The
ephemeris for the perfectly vertical analemma on the meridian
(10:28:16 UT) was then consulted where it became a trivial exercise
of identifying dates whose altitude complemented precisely the
altitude established for the first half of the analemma.
With a
complete set of dates now identified which allow for the loops to be
evenly cut cross-sectionally in both dimensions (Table
2), a rigorous schedule between June 21, 2001 and June 05, 2002
representing 43 imaging sessions of the solar disk was carved in
stone (the intended frequency for the lower and larger loop was
adjusted downwards very slightly - 17 to 14 images per side -
following further analysis so as to make the analemma slightly more
attractive).
My initial attempts at establishing a shooting schedule to
yield perfect symmetry vertically and horizontally was based on the
calculation of the slope of the minor and major axis of the 06:00:00
UT analemma (m=-0.582879 and m=-0.817589 respectively, "m" being
the slope angle
and the use of these slopes in conjunction with the mathematical principle that
the product of two perpendicular slopes (major/minor axis vs. the
cross-sectional lines connecting mirror images) must equal -1.
However, this presupposes that azimuth and altitude are rectangular
coordinates which is not the case. Nevertheless, a rather good
approximation can be obtained !
Next chapter
Photographic
Notes
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