After the last interesting messages on the shadows sharpeners, I try to
write some observations : as usual I hope that the English-speaking readers
excuse my inevitable errors.
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Diffraction.
John Carmichael and Gordon Uber in their last messages (4/11/2000) mention
the light diffraction to partially explain the way of working of the system
gnomon-bead built and studied from John.
I will schematize the system here as formed by a hole having the shape of a
ring and I'll call it , for brevity, "John Carmichael's System" or JCS
In my calculations I have taken the dimensions given by John in his
4/11/2000 message :
- diameter of the inside circle (that is diameter of the bead) = 1/8"
=3.175mm
- diameter of the external circle (that is of the circle in whose center the
bead is set) = 1/4" = 6.35mm
- width of the ring = 1/16" = 1.5875mm
- distance from JCS to the plane = 18" = 457.2mm
It is my opinion that in the JCS the diffraction doesn't produce any visible
and measurable effect, without special instruments, and this for different
reasons:
- the light of the Sun is white and not monochromatic : in whatever
phenomenon of diffraction the figures and the fringes due to the different
colors have the tendency to be superimposed and therefore the visible effect
decrease
- the spots and the fringes due to diffraction generally have low luminous
intensity and therefore are visible almost only in an room with a little
illumination and not certainly in a sundial in full sun
- the dimensions of the JCS are too great and the distance from the screen
(plane) is too small to give a visible diffraction
In a system as JCS the bright spot that appears in the center of the
darkened zone (in shadow) (due to the phenomenon said "Poisson's spot"
remembered by Gordon Uber) is visible with monochromatic light only if the
screen is at a distance of about 4 meters (at least).
This distance changes with the wave length w of the light : with a
yellow-green light with a wave length w = 550 nm (nanometer)=0.00055mm =
1/46180", the spot is well visible at a distance of 4.6 m
In this case the spot has a diameter smaller than 1mm and it is surrounded
by dark and clear rings.
Bringing near the screen, the figure (spot) decreases in intensity and in
diameter, the number of the external rings increases and they crowd together
; this till the image becomes confused in a neutral brightness without
sharp edges.
At the distance of the JCS the spot (always in yellow-green light) would
have a theoretical dimension smaller than 1/10 mm and the rings would have
widths inferior to 1/20 mm
When an object is illuminated and produces a shadow, some fringes of
diffraction always form at the edge of this shadow ; their maxima are at
distances about equal to the product : ( wave-length) x (distance between
shadow and object).
For ex. : with a distance = 500mm (about 20") the distances between the
maxima of the fringes are = 0.40mm in red light, 0.37mm in green light and
0.33mm in violet light (about)
With white light the maxima result less noticeable and flat and they tend to
cancel the minima; moreover the distances between the fringes are small and
the phenomenon is practically invisible.
In my opinion the phenomenon of the diffraction is not visible in common
sundials (perhaps it could be seen in sundials of big dimension in dark
room - for instance in churches)
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Shadow Sharpeners
This name is often used for different devices : for this reason it would be
opportune to find a correct definition
In John Davis Sundial Glossary there is this definition:
" various devices for sharpening the edge of a shadow, allowing more
accurate time readings to be made. Usually a mechanical addition to the
gnomon or nodus, it casts a secondary shadow, with its own penumbra, in
which the primary shadow can be located more accurately (although it may
have less contrast). The term is sometimes also used to refer to a movable
lens which produces an image of the shadow edge"
In a message of Charles ([EMAIL PROTECTED] ) (5/3/99) there is the
following description:
"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)"
Roger Bailey in a message (5/4/99) describes his experiments with this s.
sharpener.
Also in my opinion a shadow sharpener is an object, distinct from the
gnomon, that throws a shadow different from the gnomon's one and that makes
easier to localize where the penumbra begins and the gnomon's shadow
finishes
The JCS is not therefore, for me, a true shadow sharpener
I don't know s.s. that are mechanically fixed to the gnomon (as in J. Davis'
description ): if someone has some models or descriptions or images I
invite him to send this material to the Mailing List.
Personally I don't succeed in understanding how such a tool can work ,
unless it doesn't perform its task only at noon in meridian lines
The shadow sharpener described above from Charles works very well.
It is enough a hole of 3-5mm (1/5 - 1/8") ; we have to move it between the
gnomon and the screen at a distance from the gnomon shadow equal to about
100-120 times the diameter (30-50 cm = 12 - 20") ( at this distance the
image formed by the hole has no dark central zone and all the disk is in
penumbra. 1/107 = the angular diameter of the Sun in radiant)
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The JCS
The JCS is a particular object (I think very ingenious) that casts a shadow
and it's built in such a way to make sharper and more visible the central
zone of its shadow itself. For this reason it's suitable for being
employed as gnomon
I think that the dimensions used by John, I imagine gotten after numerous
tests, are those for which the contrast between the obscurity of the central
spot and the brightness of the surrounding ring is such to make appear the
central zone more sharp and better visible.
It would be interesting if someone made analogous tests using the
configurations suggested by Dave Bell
Keeping the same distance the screen :
- if we increase the diameter of the bead (central zone) then the dimension
of the dark central zone increases but the brightness of the surrounding
ring decreases
- if we decrease the diameter then the central zone becomes less dark and
the contrast decreases
Obviously the theory doesn't help to find what is the better relationship
between the diameters , since the phenomenon of the vision of objects with
different illumination and contrast is very complex.
With a program I have tried to simulate the phenomenon.
The program calculates a certain number of rays that originate from
different points of the Sun and arrive to different points of the screen and
it illuminates each of these points if the correspondent ray has not
interrupted from the gnomon.
In a message apart I send attached the images gotten in two different cases
(I send the images apart not to exceed the 25 kb !)
The curves under to the figures give the relative illuminations of the
points of a diameter.
If someone wants to verify (with this simulation) his gnomon system with
different measures (diameters, distances) or also with different forms from
JCS, I'm at disposal
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Bill Gottesman's construction
The geometric construction of the shadow of an object made by Bill is very
understandable and easy and it can be used to obtain the image produced by
an object of any form that casts its shadow on a plane (even if I don't
think that it can explain how a shadow sharpener works)
If d is the diameter of the hole, the diameter D of the image produced by
the Sun on a screen placed at the distance B is given by :
D = d + B/107 where 1/107 is the mean dimension of the Sun
in radiant ( 32' )
I suppose the screen (that is the plane) perpendicular to the ray from the
Sun
The bright intensity of the central zone of this image is
I = Io* (d*107/B)*(d*107/B) where Io it is the intensity of the
illumination in full Sun
More the hole becomes small, more the image becomes less bright.
The Sun's image is surrounded by a ring, having a width = d, in which the
Sun's imagine becomes darker and fuzzy and its bright intensity reduces to
zero.
One example: with d=1mm, B=500mm we have a Sun's image with a diameter
D=5.7mm
The inside zone, with a diameter =(5.7-2)= 3.7mm has an illumination = I
=Io/22 that reduces to 0 along the external ring having a width of 1 mm.
In the case of the JCS we can imagine the ring between the hole and the
bead as constituted by many holes (about 9, 10) with a diameter =1/16" =
1.5875mm.
Being B=18"=457.2mm we have D=5.86mm.
The spot therefore contains, and it illuminates, also the central zone.
To find the intensity of illumination and obscurity with Bill's construction
is extremely difficult, if not impossible (see the attached image)
The method therefore can be useful only to find the form of the image
(shadow and penumbra) produced by objects (gnomons) of whatever form.
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An observation on the pin-hole photography
Since the fuzzy zone has a width equal to the diameter d of the hole one
can think to decrease the diameter d to obtain more and more sharp
images.
Apart the diminution of the brightness of the image, this it is not true.
In fact, if the distance from the screen B is constant, and we decrease the
diameter d of the pin-hole , at a certain point the diffraction fringes
begin to appear and extend the image.
The pin-hole photographers use, as limit, the value of the diameter d that
produces a diffraction disk (disk of Airy) with a diameter = d.
Since the diameter of the disk of diffraction =2.44*B*w/d (formula also used
for finding the separator power of a telescope) the better diameter results
dm=SQRT(2.44*w*B) where w= wave length , B = distance from the
pin-hole and the film
Using green-yellow light with w=550nm we have dm * dm = B/745
(this formula is also used with denominator = 650, 1000)
This is the "optimal" diameter, that is the diameter which produces the
sharpest possible image
In pin-hole photography the distance B=745*d*d is called "focal length"
of the pin-hole
The ratio A = B/d = 745*d is called "Aperture" or "f-stop" of the
pin-hole
It can easily show that the exposition time to have a photo is proportional
to the square of this value; in fact the intensity of illumination of the
image is I=Io*(d*107/B)*(d*107/B) = Io*11500/(A*A)
For ex. if we want a focal = 300mm (tele) we need a pin-hole with a
diameter =0.63mm
The f-stop of this objective = 472. The required time to make a photo is
about 870 that necessary with a normal image and a f-stop = 16
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With my best wishes for a Happy Easter
Gianni Ferrari
44N37 10E54
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