Hello Charles & all others:
You're right, the "fuzz zone" is the prenumbra, not the umbra. (mia culpa)
It seems that we are in agreement that the center of the preumbra is the
place where the shadow of the style's edge would be if the sun were a point
light source, and that this point can be estimated either by the eye or
determined exactly using a shadow sharpener (a great invention!). When
reading a large sundial this point must be determined because, in the
calculation of the hour lines, the math specified this point, right?
**** Does this mean that there is no upper limit for the size of a
sundial? *****
Assuming that the fuzziness of the shadow is not a limiting factor for
maximum size, because you can get precise readings using the fuzz zone's
center, then a huge sundial could be built that could have extremely small
time divisions. Couldn't it?
If this is true, then one second time line markings could be placed on the
dial face, couldn't they? I haven't done the math, but if the one second
lines at high noon ,when they are closest, were spaced at an easy to read
distance of about a 1/2 inch apart on a giant horizontal sundial, then the
height of the style and the diameter of the face could be determined. It
would be a large sundial indeed!
It has long been my dream to design and construct such a sundial, maybe not
with one second markings, but with 30, 20, or 10 second time lines. (What
are the time divisions on the large sundial in Japur India, does anyone
know?) I'd like to use the same basic design that I use for my horizontal
string sundials (see website). The sundial face could be located in a park
and people could walk on it. The cable style would reach way up to a pulley
attached to a building roof edge or southern wall. A very heavy
counterweight suspended from the cable would apply tension, making the cable
straight. The diameter of available stranded metal cable may be the limiting
size factor here because if the sundial were too large and the cable too
narrow then the shadow would completely disappear (like telephone lines do
on the ground).
I know that I've asked more than my fair share of questions of you all, but
of all of them, these are the questions to which I would most appreciate an
answer. I need your help on this one guys!
A dollar for your thoughts,
John Carmichael
Tucson
website: http://www.azstarnet.con/~pappas
I >At 09:23 AM 5/2/99 -0700, you wrote:
>>Hello Nit Picking Old Timers:
>>
>>Roger states what we have all thought to be true, that as the
>>style recedes from the sundial face, the shadow's edge becomes fuzzy which
>>limits the degree of precision attainable. This effect is most noticible on
>>large sundials. However, it's not as bad as we think because the eye is
>>very good at estimating where the CENTER of the fuzz zone (the umbra) is.
>
>This is just a gut level feeling, and I don't know enough math to prove or
>disprove it, but I don't think the correct point would be in the center of
>the fuzzy zone. (Am I correct in my belief that the fuzzy zone is the
>penumbra shadow?)
>
>Assuming you could safely look at the sun directly, stand so the entire
>disk of the sun is hidden behind the building in question. As you
>gradually step back, the percent of the sun's disk that is visible will
>grow at an increasing rate until half the disk is visible. After that the
>percent will continue to grow but at a decreasing rate until the entire
>disk is visible. I'm not sure this would put the correct point in the
>center of the penumbra shadow or not.
>
>As a test, you could make use of a device I read about called a "Shadow
>Sharpener", supposedly used by Chinese astronomers centuries ago.
>
>The shadow sharpener is simply a stiff sheet of opaque material with a
>clean edged round hole in the middle. I made one from the thin cardboard
>backing of a pad of paper, with a hole about half a barleycorn in diameter.
>
>To use, hold the sharpener a short distance (1 to 3 feet) away from the
>fuzzy zone, with one side facing the sun. An image of the sun will appear
>on the surface the shadow is on. Move the sharpener around until the image
>of the sun is bisected by the edge of the roof of the building, (or
>whatever comprises the gnomon). I believe then that the image of the
>gnomon will be in the exact place it would be if the sun was a point
>source. Try this and see if this point falls in the same place that your
>eye estimates it would.
>
>Charles
>
>
