If I may intersperse my comments in Carl's email: ----- Original Message ----- From: "Carl & Barbara Sabanski" <[EMAIL PROTECTED]> To: "Sundial Mailing List (E-mail)" <[email protected]> Sent: Friday, February 15, 2008 2:16 AM Subject: Monumental Sundial
> Sunny Day! > > I have been trying to understand the consequences of shadow tapering on a > monumental sundial a little better. To help visualize this I have attached > an image I modified from Wikipedia. From previous emails I come up with the > following. > > If the shadow casting object is cylindrical the resulting shadow is > theoretically as shown in the image. The diameter of the sun is about 32' > as was mentioned. The umbra will disappear at a distance "ud" equal to: > > ud = r / tan (16/60) = 214.9 x r where "r" is the radius (D/2) of the > cylinder. > > This is the maximum theoretical distance and is not reality. The umbra is > not likely to be visible at that distance just as the penumbra is not > visible as shown in the image. That is why a gnomon diameter equal to > 1/105th of the distance from the origin to the point where the readings will > occur is recommended. I don't understand why you say "..is not reality". It is reality, surely. The reason I use 105, not 107, is that at perihelion in January the sun is closer and appears larger. Its apparent radius is 16.3 arc minutes, not the 16 minutes we use as the average. This gives a distance ud=210r or 105D. > > Now at any point "P" that is a distance "d" from the cylinder the umbra has > tapered by: > > t = 2 x tan (16/60) x d x r = 0.009308 x d x r and the width of the umbra is > D - t. The equation is wrong. It should read: t = 2 x tan (16/60) x d = 0.009308 x d and the width of the umbra is D - t. > > If the object is a vertical cylinder the section that casts its shadow at > point "P" varies throughout the year as the sun's elevation changes. But as > the cylinder is vertical "d" will remain constant and so will the taper value > "t". This will also remain the same at different latitudes. > This last paragraph is correct. The next two are not. > If the cylinder is sloped the section that casts its shadow at point "P" > still varies throughout the year but now so does "d". As a result the taper > value "t" will vary. The width of the umbra at point "P" will vary > throughout the year. However, as the gnomon is cylindrical and it is the > centre of the umbra that is of interest, this change in the taper value does > not matter. No. The distance from a point to a cylinder is measured to the nearest point of the cylinder, regardless of which part is expected to be casting its shadow. So, "d" and "t" do not vary and the width of the umbra at point "P" is constant (except for the varying apparent size of the sun throughout the year, which is a small factor). This is, perhaps, counter-intuitive. (i.e. it surprised me!) Another point - it is true that the centre of the umbra is of interest, but judging the centre is easiest within the umbra ( d < 105D ) but near its tip (d=107D on average). > Now, if the wide gnomon is not a cylinder but a triangle then the style is > an edge. The shadow cast by this edge will also taper. At any given point > on the dial plate this taper will vary throughout the year because it > originates from different positions on the style. If the dial plate is not > designed to give constant values of "d" in the reading area the taper will > also vary from line to line in the reading area. The taper will also be > affected by the latitude. No. the taper of the shadow of the style edge of a conventional wedge-shaped gnomon does not vary throughout the year. Nor is it affected by latitude. However, it does indeed differ for points at different distance "d" from the style. Equatorial dials have constant "d" for all points on the dial. To achieve the same for a horizontal dial with a cylindrical gnomon, you could make the dial elliptical, so that it appears circular as viewed along the gnomon. The taper is unaffected by latitude. It is a function only of the distance from the point on the dial to the nearest point along the gnomon/style. > If I have this straight then when the hour lines are adjusted for taper this > will not be a definitive adjustment. On a smaller sundial the taper > variation will be small because of the small changes in "d". On a > monumental sundial, this variation may be significant depending on the > layout of the dial plate.. If the gnomon is a cylinder and you are reading time from the centre of its shadow the hour lines need no adjustment and there is no noon gap. If it is a conventional wedge-shaped gnomon with two style edges, then you can adjust the hour lines, by a little under a minute. This adjustment is definitive, in the sense that it is irrespective of the time of day, time of year, size of sundial, type of sundial (horizontal or vertical) or latitude. It is an angular change (about 50 seconds of time), not a linear change. It only appears smaller on a small sundial because everything is small on a small sundial. Actually, I just glossed over one small effect - the sun moves faster across the sky at the equinoxes than at the solstices (by about 8%). So, the time of year is a factor, but a pretty small one. 8% of 50 seconds is 4 seconds. I think we can ignore it. There may be some people whose judgement of the edge of a shadow is different. If they sense the slightest drop in intensity from full sun they might perceive the position quite differently, but most people seem to judge the edge very close to the dark side. Try it yourself with a shadow sharpener: punch a small hole in a piece of paper, position it near the shadow of a distant object so that you see a semicircle of light. Where you see an exact semicircle is the true position of the shadow's edge. Another factor is that the sky near the sun may be very bright due to haze, especially here in the UK (!) which, I suspect, makes you judge the shadow even further from the centre of the sun. > If a gnomon goes to a point that acts as a nodus the result of tapering will > be noticed very quickly I would think. This would be especially true at > large values of "d". A point is, by definition, vanishingly small. So, regardless of "d", its shadow is all penumbra and the estimation of its position is difficult. So a nodus should ideally be symmetrical. For instance, a sphere or X shape. The pointed end of a gnomon is a poor choice. But a W-shaped notch along a polar gnomon is fine if it is used only to judge declination. It isn't perfect, but not far off. > > If this is wrong please let me know. > > Happy Dialling! > > Carl Sabanski > www.mysundial.ca I hope this helps. Regards Chris Lusby Taylor --------------------------------------------------- https://lists.uni-koeln.de/mailman/listinfo/sundial
