John Carmichael wrote: > The design which worked the best was a 1/8 inch spherical bead, suspended by > thin brass crosswires, in the exact center of a 1/4 inch round hole. (The > style was about 24 inches from the analemma).
> A very curious thing happens with this type of style. The bead alone, by > itself, casts a shadow that was twice as big as the bead; but when the 1/8th > in. bead is in the center of a 1/4" hole, with a space of 1/16th of an inch > between the bead's edge and the hole edge, the bead's shadow miraculously > sharpens into a tight, dark shadow that is only 1/16th of an inch in > diameter, smaller than the bead itself!!!! The wires which keep the bead > suspended in the middle of the hole are so thin that they don't cast a > visible shadow onto the analemma. And Richard M. Koolish calculated: > The linear diameter of the diffraction spot (Airy disk) produced by > a pinhole of a given diameter is: > > spot = (2.44 * wavelength * focal_length) / diameter > > The optimal size is where spot = diameter, so: > > diameter * diameter = (2.44 * wavelength * focal_length) > > diameter = sqrt (2.44 * wavelength * focal_length) > > An example of a pinhole for a distance of 100 mm and a wavelength of > 550 nm is: > > diameter = sqrt (2.44 * .000550 * 100) = sqrt (.01342) = .366 mm Using a distance of 24 inches = 610 mm, this becomes 0.9 mm = 1/32 inch, still several times smaller than John's hole. I think the explanation lies in simple geometrical optics. Imagine putting your eye where the shadow is being cast and looking back toward the style and the sun. I would like to suppose that the distance to the style was something closer to 14" (subject to objection and correction from John), so that the image of the sun would be just eclipsed by the 1/4 inch bead, giving a black shadow at the center. Just a little off-center, an arc of the sun would show around the bead, so the brightness would grow, but only until the disk of the sun runs into the edge of the hole. Thereafter the brightness would decrease slowly until the sun is entirely outside the hole. This would lead to a shadow with a diameter-at-half-brightness of about 1/16 inch, within a diffuse bright field with diameter on the order of 1/4 inch. The size of the shadow is reduced at the cost of reducing the contrast with the surrounding lighted area. The principle is much the same as a sundial that images the sun through a pinhole: a sharper image is a dimmer image. --Art Carlson
