Re: Shadow Sharpener Again

[Pete Swanstrom]

Hi Bill:       To determine offset of the perceived shadow edge, I built a simple device consisting of a cardboard test pattern, a white cardboard screen, and a stick to separate them about 36".  The test pattern was two parallel strips each exactly 1" wide with a 1" gap between them.  Over a three month period or so, at different times of day, I would point this at the sun and measure the widths of the shadows and the gap I observed on the white cardboard screen with a dial caliper.  The shadows of the 1" strips each averaged about 0.72" wide and the shadow gap between the strips averaged about 1.28" wide (at a distance of 36" away from the test pattern) giving an angular displacement of arcsine(0.28"/2/36") = 0.222° for each edge.  For some real excitement, for my test pattern I could have used two parallel strips 1.28" wide with a 0.78" gap between them, and it would have cast two 1.00" wide shadows 1.00" apart!       For more info on the analemmic/equatorial dial, see http://www.swanstrom.net/sundial.   Have a great day.  Pete S.   ----- Original Message ----- From: [EMAIL PROTECTED] To: sundial@rrz.uni-koeln.de Sent: Thursday, May 30, 2002 7:04 PM Subject: Re: Shadow Sharpener Again Pete,       Thanks for clearing this up.  Amazing how our fingers do not type what we think we have said!  I think your results are in good agreement with mine. Your difference in degrees between the center of the Sun's image and the perceived edge of  the shadow of 0.222 deg corresponds to a difference in time between the passing of the center of the Sun's image and the passing of the shadow of 53 sec.             (0.222 deg)(4 min of time/deg)(60 sec/min) = 53 sec This is in good agreement with my estimate of 40 sec.         I can believe that there is a slight shift of the shadow toward the center of the Sun's image when shadows are formed when the sky is hazy.  However I have not been able to measure this shift.  I would be curious to know how you made your measurements so I could try to repeat them here.       I made a theoretical study of this phenomenon as follows.  I plotted the illumination produced by a circular object as it is progressively uncovered.  This curve, of course, is symmetrical about the center, and, to my surprise, is nearly a linear curve except near the extreme ends.  Because the eye's sensitivity is not linear, but approximately logarithmic, I then plotted the data on a logarithmic scale using EXCEL.  On such a scale you cannot start with zero when the disk of the Sun is covered, but must estimate the illumination in the shadow by the light of the sky alone.  Also the shadow will have more illumination when the sky is hazy.         I took my old SLR camera, along with a Kodak Neutral Test Card (Gray), outside and checked the exposure in the direct sun and in the shadow of my garage.       On a clear day I found the exposure had to be 5 f-stops greater in the shade than in the Sunlight (a ratio of 1 to 128, or  0.0078).  This amount was added to all the calculated values, both sunlight and shade, the resulting values converted to a percentage of the total illumination, and again plotted on a logarithmic scale.       On a hazy day I found the difference to be 4 f-stops or 1 to 32, or 0.031.  In a similar way I plotted a logarithmic curve for the change in illumination from full sun and skylight to shade with skylight only.         These curves are asymmetrical, being very steep just outside the shadow and very gradual near full sunlight. One cannot, from these curves, tell where the eye would perceive the edge of the shadow to be, but can be sure that it would be on the shadow side of the center of the Sun's image, perhaps halfway between the log of the shadow''s illumination and the log of the total illumination.  This would put the edge of the shadow 42 seconds from the middle of the Sun's image on a clear day and 35 seconds on a slightly hazy day.  If I interpret your results correctly, these figures are in rough agreement.         Would like to hear more about your analemmic/equitorial sundial.      Bill Walton      Plymouth,  MA, USA      42 N   71 W