Hello,I discussed this, using the umbra/penumbra analogy, in "The fuzziness of solar shadows," Compendium 1-3, August 1994, pp. 11-15.1) How does one size a gnomen so that the shadow will be as sharp as possible?
This obviuosly has to do with distance from dial surface, solar disk size,
etc. But also may have to do with the eye's ability to see contrast. Given
an optimum design, one could scale a dial and still have (relatively) the same
sharpness, yet the eye would need to percieve the contrast at the edge of the
shadow over a longer distance. Is there an optimal ratio for style tip size
to distance from dial face?
2) The contrast of a shadow depends on the difference in lighting in the sunnyOf course this depends upon the concentrations of a variety of atmospheric constituent particles and molecules, including water vapor and condensed water (haze and clouds) as well as dust and smoke. My article offered the following equation for the irradiance distribution from the position on the shadow receiving surface corresponding to the shadow of the center of the solar disk outward, with x being in angular units (from zero for a line through the edge of the shadow-casting object to the center of the sun's disk, in the middle of the shadow transition region) to R = 0.276 deg for the edge of the solar disk. Beyond x=R no shadowing of the direct sun occurs, ignoring refraction effects near the horizon:
part from the shadow part - how bright is the shadow? It is probably
illuminated at the intensity of scattered light in the atmosphere - what is
this? Does this vary significantly with sunrise - noon - sunset?
E(x) = Es + Eo[1 - (1/pi)cos^-1(x/R) + x SQR(R^2 - x^2)/(pi R^2)
where Eo is the solar irradiance from the direct sun on the shadow receiving surface, perpendicular to the sun's rays, and Es is the corresponding irradiance on that surface from the sky only, i.e. the irradiance in the center of a perfect shadow of a thin gnomon that is large enough to block all of the sun's disk at this point. pi = 3.1415926 in this equation. The equation is plotted in Fig. 7 of my article, for an angular displacement x, from -.3 (full shadow) to 0.3 degrees (full sun).
The next question is one of where one might obtain values for Es and Eo. Since they are varying quantities, as indicated above, one can only provide representative values. Since presumably only humans view sundials, or at least only we humans attempt to tell time from them, only the visible portion of the solar spectrum is needed, and in this case, instead of solar irradiance, we should be talking about solar illuminance, for both the direct and diffuse components.
The Illuminating Engineering Society of North America has published a "Recommended Practice for the Calculation of Daylight Availability," Jour. IES, July 1984, pp. 381-332, which offers equations for horizontal illuminance from both direct sun and diffuse sky for clear sky conditions. I seem to recall that for a relatively high sun on a clear day, the diffuse component is on the order of 15% of the total, meaning that the direct beam would be 85%. We can calculate 1 - the quotient of Ed by (Eo + Ed) to obtain a kind of contrast factor. If Ed were 0 in this case the contrast would be 1.0. If Eo is zero, as for overcast skies, the contrast would be 1 - 1 = 0. In the case just cited, with Eo = .85 and Ed = .15, relatively speaking, this would produce a contrast of 1 - .15 = .85, if my analysis is correct.
3) What "color" is a shadow? This is also probably a different spectrum of light from direct sun (different black body color temperature) - ie a different peak color. Since metals and plastics reflect color differently (one reflects its own color, the other tends to reflect the color of the incident light - I forgot which is which) - does the material on the face of the dial affect the perceived contrast?
The concept of color temperature is discussed in section 11.7 of [advert coming up] my book Introduction to Radiometry and Photometry, Artech House, Boston, 1994, pp. 364-370, web site http://www.artech-house.com. I quote the IES Lighting Handbook as saying that the color temperature of good clear sky daylight is around 7500K (p.367) while that of direct beam sunlight is 4874K (correlated color temperature, p. 369). More realistic daylight color temperature is closer to 6500K.
Your final question is a complex one, requiring a complex answer, beyond the scope of this reply. The concept of color rendering, the ability of an illuminant (incident light) to properly render or display the color intended for a reflecting surface is discussed briefly in section 11.9 of my book, and more in depth in publications on illumination engineering. Daylight is considered to be the standard, essentially having perfect color rendering. Of course this does not apply to direct beam sunlight when the sun is near the horizon, since much of the blue part of the solar spectrum has been removed by scattering in the intervening atmosphere (why the sky is blue).
5) Is it possible to have too bright a dial surface - making it difficult toOf course, a mirror surface, viewed in the specular direction would produce what might be called "terminal disability glare." Away from the specular direction of sunshine, the eye will see light reflected at the specular direction from other sources, and some may be bright enough to obscure the dial furniture, unless the important features are embossed or raised or the specularity of the surface is altered in some other way. So I would avoid shiny surfaces unless the observer was either free to move about in order to avoid glare or was restricted to a viewing direction for which little glare is possible, due mainly to a dark surface being in the specular direction relative to the observer.
look at the shadow?
6) Is a diffuse or specular surface best?Diffuse in general, unless there is a good reason for using specular reflection. Note that complete specularity is unlikely to be achieved in most ordinary dials, since the result would look like a perfect mirror. Most surfaces exhibit a combination of specular and diffuse reflection, and one in principle can choose their relative proporations by choice of materials and their surface treatments. Polished stone, for example, will have a modest amount of truly specularly reflected light and considerable more diffusely reflected light. Viewing a bright source of light reflected to the eye at the specular direction, however, produce a reflected beam strong enough to swamp the diffuse component and greatly reduce contrast of any markings imbedded in the stone.
7) There are three aspects to reading the shadow - that is the shadow, theThis is up to the designer, and much of this will depend upon his or her experience and preferences.
dial face, and the markings. What is the best marking color / material / etc?
Should the markings be raised or engraved or flush?
8) The scattered light from the atmosphere tends to e polarized - would one beIf specular reflection is partially obscuring the markings, this would help to see the markings in the shadow, but I don't think it would help much with the shadow itself. Note that essentially unpolarized direct beam sunlight is rendered partially polarized upon specular reflection, increasing with angle of incidence, so that polarized sun glasses could help remove reflected glare from the direct beam, as well as from reflected sky light in the specular direction.
better able to see a shadow wearing polarized sun glasses?
9) Can you see the shadow move - can you sense its motion? How big a dial>From my experience, you need at least a several foot shadow casting distance. In principle it does not matter, since the angular speed of movement is the same for all distances, and as you increase this distance, the shadow edges get more fuzzy and less distinct, and the relative motion with respect to the time markings is the same, but small dials cast a small shadow and the absolute movement across the dial face is not as easy to discern as with larger dials.
would you need (how long a shadow) to get a shadow speed that the eye could
sense?
Would a surface with colored inclusions (texture) help to perceive theSimilar issues, just the negative image. You don't have a point of light. I like to call it a spot of light, indicating something with some width to it.
motion? What size of texture would work best? How slow must something move
before the human eye fails to percieve its motion?10) All of these issues will be influenced by the angle of incidence of the
shadow. Perhaps also by the extremes of the day.11) Inverse problem - same issues but using a point of light rather than a
shadow to mark the time.
--
Ross McCluney, Ph.D. Principal Research Scientist
Florida Solar Energy Center, 1679 Clearlake Rd., Cocoa, FL 32922-5703
Voice: 407-638-1414 Fax: 407-638-1439 e-mail: [EMAIL PROTECTED]
Florida Solar Energy Center: http://www.fsec.ucf.edu
Sundials: http://www.sunpath-designs.com
Introduction to Radiometry and Photometry: http://www.artech-house.com
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