Variations Due to Positional Location of the Sundial.
How does the shape of the analemma change as the sundial's
location changes? What would our sundial's analemma look like at the Equator?
At the North Pole? At the South Pole?
The true (absolute) shape of our earth's analemma does not change. Never. But the
image of our analemma may change in size and shape dependent on the angle of the
surface it is projected upon. To get the most mathematically correct, undistorted
analemma possible, a sundial's projection surface would be at an angle of 90
degrees to the rays of the sun throughout the year, on average. A sundial such as
ours, with its horizontal projection surface (the floor), would possess this
90-degree angle on its standard meridian only if it were located on the earth's
equator. At any other location on the earth, the shape of our sundial's analemma
would change in the interesting ways shown in figure 5 to the left.
These changes are similar to what would happen if you were in a movie theater
and the projector were moved off to one side instead of being squarely behind you.
As the projector moved the picture on the screen would distort, stretching to one
side. The more off center, the greater the distortion.
If our sundial were on this standard meridian,
but on the 0° equator instead of 35° north - the line of the equator would
cross our analemma through its equinox points, directly beneath the gnomon - the
sun would spend half of its year north and half of its year south of the gnomon.
As the sundial is moved north or south of the equator, the analemma path lengthens.
If moved north, the north lobe of the analemma stretches out more than the south
lobe. If moved south, the south lobe stretches more. If the sundial is moved east
or west off its standard meridian, the analemma tracing moves off its noon mark
and begins to tilt.
North of the Arctic Circle and south of the Antarctic Circle, for some time
during their respective "winters", the sun does not appear in the sky. For this
reason the "winter" lobe of the far north or far south analemma remains open -
the sun's image on the floor would not be available for that portion of the loop.
The shape of the earth's analemma depends on factors other
than a sundial's location on the earth and it is constant. The only reason that
the analemma's shape changes is that the angle of its projection surface is
changed when our sundial is moved to other locations on the earth.
Many sundials are designed with horizontal floors like ours
but, if desired, one could create a sundial where the projection surface is
adjusted for its position on the earth so that it would be exactly 90 degrees to
the sun's average position and exhibit an undistorted analemma. As long as the
angle of the projection surface was adjusted for each latitude and longitude to
which the sundial was moved, the size and shape of the projected analemma would
remain constant. At the latitude of Cerrillos this would result in a floor with
an approximate 35 degree slope, which would not be too practical for standing or
walking. (Tilt is equal to the latitude plus a left-right correction for deviation
from the standard meridian.)
Note how for both the armillary and equatorial sundials their
projection surface may be tilted so that anywhere on earth it will be at 90
degrees to the sun's average position there.
The shape of the earth's analemma is determined by the day to
day changes in the sun's declination and by the equation of time, which is the
difference in the length of individual days as compared to the average length of
a day.
If you were to plot the values for the sun's
declination as "y" and the values for the
equation of time as "x" (slow times being minus "x" and fast times being
plus "x") you would end up with a figure-eight shaped plot with the northern lobe
(at the bottom of the diagram to the right) larger than the southern one. This is
the earth's analemma.
To understand this, first imagine the earth rotating in a perfect circle around
the sun with the earth's equator in the exactly same plane as its orbit.
From earth, the apparent motion of the sun would be the same throughout the year,
and every noon by our watches the sun would be in the same position overhead.
Now imagine tilting the earth's axis to 23.5 degrees from perpendicular as it is
currently in real life. This tilt is referred to as the earth's obliquity. It
will cause two things to happen:
1. First, you will get the seasons of the
year because, as the earth moves around the sun, the poles are each, in turn,
tilted toward the sun. During the part of our orbit when our northern hemisphere
is tilted toward the sun, we will have summer and the sun will appear higher in
the sky (farther north) at noon. At the opposite point in our orbit, our
hemisphere will be tilted away from the sun, it will be winter and the sun will
appear lower in the sky (farther south) at noon. The earth's tilt causes the
sun's location to be at a north or south angle to the earth's equator depending
on the time of year. This angle is called the sun's declination. It governs the
north and south movement of points along the analemma's track.
As an example, on our sundial the sun's declination determines the distance that
the sun's image will fall to the north of a point directly below our gnomon.
Toward the winter solstice, the sun's angle is more southerly, the sun appears
lower in the sky, and the declination is less. The image will fall a longer
distance north of the gnomon, nearly to the north wall. Toward the summer solstice,
the sun is higher in the sky and its image will be so close to the point directly
below the gnomon that it will fall on the window sill.
2. A second effect of the earth's tilt is
that the apparent speed (angular velocity) of the sun becomes variable. This
causes the sun to seem "fast" during some times of the year and "slow" during other
times. During "fast" times, it takes less than 24 hours for the sun to go from
its highest point as it passes over a given local meridian (local apparent noon)
to get to that same point the next day. This results in a shorter day. At other
times of the year, when the sun is "running slow," it takes a bit longer than 24
hours to get from one local apparent noon to the next with an apparently longer
day. The fast and slow times resulting from the earth's tilt accumulate each day
to equal several minutes at some times of the year. If plotted, the curve would
be bimodal, with two peaks and two valleys.
When the sun is "running slow", it will be to the east of its local meridian at
noon on the watch and it will throw an image to the west of the analemma's center
line. When the sun is "running fast", the sun will have already crossed to the
west of the local meridian at noon on the watch. Its image will then be projected
to the east of the analemma's center line. Thus, the east and west fluctuations
of an analemma are formed.
If you were to plot the north/south and
east/west shifts discussed so far, you would get a figure-eight with two equal
sized lobes. There is, however, an additional fast/slow, east/west, angular
velocity factor layered upon the tilt of the earth that gives us the different
sized lobes. This additional factor is that the earth does not move in a circular
orbit about the sun. Its orbit is elliptical and the distance from the earth to
the sun varies.
When the earth and sun are farthest from each other, a position called aphelion
which occurs during the first week of July, the sun seems to move more slowly.
When the earth and sun are closest, a position called perihelion which occurs
during the first week of January, the sun seems to move faster. If these fast and
slow times were plotted, one would obtain a graph with one peak and one valley.
When both the effects of the earth's tilt and its elliptical orbit are summed,
the two curves augment one another at some points and cancel each other out at
other points. What you end up with is the equation of time that is found in
Table of slow or fast times.
See
Positional Astronomy for a diagram of how the earth's tilt and its elliptical
orbit combine to form the equation of time. If one were to fold this diagram down
the center, turning the right side over the left, the equation of time (darker
tracing), would describe the earth's analemma.
To learn more about this topic (and others), go to:
Our noon analemma was inspired, in part, by some very accurate
sundials incorporated into structures built by prehistoric civilizations of the
Southwest. One example is the Anasazi of Chaco Canyon. For further information:
Another type of simple sundial is the noon-mark. If the sun's
image location is marked each day at local apparent noon when the sun souths, you
will eventually get a straight line running north and south. This line is commonly
referred to as a noon mark.
The sun is said to "south" when it is due south of your location, it is highest in
the sky, it is over your local meridian, and the shadows of objects are their
shortest for the day. At this time also, the shadow of a vertical stick
describes a noon mark and acts as a compass by running exactly north/south. The
sun will south and cross the noon mark at a different time each day according to
your watch because of the equation of time.
A noon mark plotted at this Information Center will not run exactly through the
center of our analemma. Note the blue line through the
analemma diagram. This is because our analemma
will be marked by our watches that are set to the Standard Meridian for Mountain
Time, a location about 63 miles east of here at 105° W longitude. It takes the
sun an additional 4 minutes and 29 seconds to appear over our local meridian
after it has reached the Mountain Time Standard Meridian. Since the sun is over
the Mountain Time Standard Meridian to our east when we mark the analemma, our
analemma marks will be shifted to the west of our local apparent noon-mark.
A small hole (aperture) in a sheet of opaque (non-see-through)
material will act as a "pin hole lens". If the sun's rays shine through it, an
upside-down image of the sun will be cast on any appropriate surface. This would
be most apparent during a solar eclipse, when the light spot on the floor of this
Information Center would appear as a crescent rather than a round disk. (There
will be a solar eclipse likely to be visible from here on August 21, 2017.)
The ideal projected light spot for marking our analemma would be sharp, small, and
bright. Our low humidity enhances the sharpness of the image. The size and
brightness of the image is affected by the distance from the aperture to the
projected image and by the size of the aperture. When the distance from gnomon to
image is increased, as happens toward the winter solstice (shortest day of the
year, when winter commences, around December 21st), the image becomes larger and
fuzzier. A smaller aperture can correct for this to a point. After that, further
lessening of the aperture size no longer decreases the image size. Instead, the
brightness of the image begins to decrease. Disks with apertures of various sizes
will be swapped out during the year to optimize the readability of the image.
Notes
Much of the information for this sundial project was obtained
from the book,
Sundials - Their Theory and Construction by Albert E.
Waugh, Dover Books ISBN 0-486-22947-5. Waugh's excellent book is very clearly
written and has detailed information on how to understand, lay out, and read many
different types of sundials.
A figure-eight representing the calculated track of our
analemma has been designed into the floor of this Information Center. Over the
next year we will be checking this theoretical track to see how well it matches
the actual track of our sun spot. It will take a year or more for us to properly
"set" our sundial.
Until the actual track of our analemma has been determined, our readings will be
marked in pen. Once it is clear where the analemma will be, the gnomon will be
adjusted to shift the analemma in such a way as to maximize the agreement between
the actual track and the calculated track. (It is considerably easier to move the
gnomon than to move the floor!) Because of variations in floor height, and other
factors, we expect there will be some small differences between the floor track
as constructed and the actual path of our sun spot. After that, placement of more
permanent markings such as tiles or other insets can begin.
Once it is calibrated, the Cerrillos Hills State Park will have a very accurate,
maintenance-free, solar-powered clock that will just keep on running and running
and running.
Over 2,000 Volunteer Hours!
From that first sparkle in Buck's eye (and the Greenway Grant
that got us going), through two years of wrestling forms, rebar, and concrete of
many colors, and now the detailed measurements and adjustments...
THIS COULD NEVER HAVE HAPPENED WITHOUT YOUR GREAT AND GENEROUS
SUPPORT. THANK YOU ALL!
This website is maintained by the Cerrillos
Hills Park Coalition
and is dedicated to the creation, enhancement and stewardship
of an historical, recreational, and cultural open space in
the
Cerrillos Hills, Santa Fe County, New Mexico, USA
