Hi Steve, I'm sorry I've confused you. The experiments reported in the 1941 UK paper were limited to illuminances not exceeding 10 000 lux. For that reason I had to admit that higher illuminances might well point to a different conclusion. I certainly did not intend to suggest that reflected light should be limited to 6000 lumens per square metre; simply that I have no experimental evidence handy for illuminances exceeding 10 000 lux.
Weber's Law stated that if the background luminance is Lb and the stimulus luminance is Ls, then the just-detectable luminance difference (Ls - Lb), or in the case of a sundial (Lb - Ls), is proportional to Lb. If Weber's Law were correct then the perceived contrast of the shadow on a matt sundial plate would be independent of the reflectance of the plate. Although Weber's "Law" is approximately true, for many practical purposes, over a wide range of values for Lb, it certainly fails under dim lighting and under very bright lighting, when the "Weber fraction" (Lb - Ls)/Lb rises. Ideally for a sundial plate one would aim for the value of Lb which minimises the Weber fraction. Unfortunately there is little agreement over the precise value of Lb at which this minimum occurs. It is almost certainly greater than 10 000 lumens per square meter of reflected light but, as you have observed, well below the luminance of a white surface under intense sunlight. Once again I'm sorry to have confused you. The take-home conclusion is that there is no single ideal reflectance for the plate of a sundial. It varies with the sky illuminance. When Weber's Law prevails, a reflectance of about 60 per cent is likely to be a safe bet. John Lynes PS I'm away from my computer for the coming week, so will be out of touch for some time. On 28 February 2017 at 01:48, Steve Lelievre < steve.lelievre.can...@gmail.com> wrote: > John, > > Will you clarify some things for me? > > You mention that 50 lumens per square foot is about 500 lux, and that the > cited article recommends a limit of 60% reflectance for sky illuminance of > up to 1,000 lumens per sq. ft. If I multiple all that out, it would appear > to suggest a a limit of 6,000 lux of reflected light for comfortable > viewing. Is that the case? > > Through Google, I found empirical rules for calculating the wattage of > solar radiation reaching the ground, depending on season, altitude, > declination, hour angle, and geographic elevation. I also found a > conversation factor for converting sky illumination in watts per square > metre to lux. Putting it all together, I get a figure of about 72,000 - > 91,000 lux for the incident illumination, at noon on the northern > hemisphere summer solstice at sea level, depending on latitude (and valid > for mid-latitudes only). Using the mid figure of 80,000 lux, if I want to > limit the reflected light to 6,000 lux then the reflectivity has to be less > than 22.5%, which corresponds to a lightness of only 3.5 on the Munsell > scale. > > Does this conversion make sense, or don't things work like that? > > Of course, if we've gone out in midday sun, we should be wearing > sunglasses and, again from the web, sunglasses reduce the visible light > reaching our eyes by two thirds or more. If I factor that in, my Munsell > value rises to 5.3. And, as you pointed out, when the sun isn't so high in > the sky, we can tolerate a more reflectivity on our dial face. > > Thanks for any further comments or advice, > > Steve > > P.S. Based on what I've learned so far, I'm leaning towards using a > material with a Munsell value of 6 or 7, which would correspond to the > mid-grays, tans and browns that people have been suggesting may work in > practice. It would be the number you mentioned but with the lightness > notched down a little. My design latitude of 45N is a little further south > than England (where the article's authors came from) and the summer sun is > a tad brighter. As well, I reckon a suitable colour with a number of 6 or 7 > would look OK against a lawn, flowerbed or other greenery. > > > > On 2017-02-26 7:08 AM, John Lynes wrote: > > There is no single optimum reflectance for a flat dial face. Obviously > under dim sunlight the optimum reflectance would be 100 per cent, > i.e.perfect white. > Under intense sunlight, contrast sensitivity would be optimised for a > lower value of reflectance. Thousands of papers have been written on > contrast sensitivity. One classical study is "Brightness and contrast in > illuminating engineering" by RG Hopkinson, WR Stevens and JM Waldram, > Transactions of the Illuminating Engineering Society (London), Vol 6, No 3, > pp 37-48 (1941). This indicates that when the sky illuminance on a matt > dial face is over about 50 lumens per square foot (about 500 lux) the > optimum reflectance would be about 60 per cent (a light grey, about Munsell > Value 8). Below this illuminance (which would correspond to a solar > altitude close to sunrise or sunset) the optimum reflectance would rise > quite sharply. > Note however that the maximum sky illuminance considered by the authors > was 1000 lumens per square foot (corresponding to a solar altitude of about > 20 degrees). Higher illuminances might further reduce the optimum > reflectance. > John Lynes > > >
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