Chris Lusby Taylor wrote: >Bill Maddux gives a very logical reasoning to the subject of how accurately one can read a sundial, assuming the >reading is of the centre of a symmetrical shadow line with fuzzy edges.
>I am inclined to agree that judging the centre of such a shadow must be easier than judging the position of the >shadow cast by an edge. I believe the optimum width of gnomon is such that the umbra is of zero width. Even then, >the figure of 1/60 width seems surprisingly good to me. The list is not a suitable place for full details or the diagrams, graphs, etc. needed to explore this thoroughly. When I spoke of the edge gradient of a shadow, it was with the thought that the light gradients mentioned apply to both the external boundaries of a shadow image and two the internal bounderies of an aperture's image. As for the symmetrical cut, try this little experiment: Cut a paper rectangle with a width of at least 100 mm, or draw a pair of parallel lines on a larger sheet at least 10 cm appart. Set this up at comfortable arm's length, and try to judge the center of the space between the lines or paper's edges. Mark the judged center with a fine pencil point, then locate the center by carefull measurement with a millimeter scale, and compare to determine your error. (Or for repeated trials, use a wall-mounted small slate, or "chalk board," make the judgement from a distance, and fixate on its location as you approach and make a mark. Erase and repeat this, and record and assess results by simple statistics. I think you will be surprised. Recruit "naive" observers, and see how well they do. (Also, compare the judgement of a horizontal interval vs a vertical one. I think you may find higher precision in one orientation than in the other.) Note that when centering a symmetrical object on a symmetrical target, a given error is doubled in its effect on symmetry since added distance on one side is subtracted from the other. If the centering involves placing a somewhat broadened fuzzy image relative to a double-edged defining target-interval, a comparison of the asymmetry of the edge-spacings may aid the process. I.e., if the difference in target width and image width is a small interval compared to the width of the image, decentering by a quarter of that small interval makes a three quarters discrepancy at one side while diminishing that at the other to one quarter. Thus their ratio is a very visible: three to one, instead of the one to one when centered. Also realize that psycho-physical resolution is better when distinguishing two closely spaced, extended, linear objects, than for two spots at the same separation. (This has to do with neural signal processing. and integration of adjacent receptor element responses.) If you use a circular spot, I suggest it not be too small, and that you use a target with concentric rings. You can get "moire" type effects at the arc edges that are very sensitive for alignment. (These are not diffraction effects, since the aperture acts as in a "pinhole" camera and forms an image of the sun's full half-degree wide disk.) You can learn a lot by playing with full-scale mockups in real sunlight under a variety of conditions. I recommend it over simulations, whether with artificial sources or "virtual" on a computer. It is a "fun" aspect of our dialing hobby. ========================================================== You also wrote: >3. When a sextant is used to find the sun's altitude in a known place, it is a sundial. You adjust a sextant to make >the lower 'limb' of the sun appear to touch the horizon. This can be done with great accuracy. Even my plastic >sextant is calibrated to 0.2' of arc, so I am sure the bestbrass sextants are better than that. You add on the radius >of the sun, and correct for refraction in the atmosphere, to get the true altitude of the centre. This is going to beat >any spot or shadow at times of day when the sun's altitude is changing fast. As for sextant measurement: I don't think we should consider this naked eye, not even if the sight is via a simple peep-hole, for to me the two plane-surface mirrors and required optical filters are man-made optical elements, even if no image magnification is involved. (Note that depending on the size of the eye-piece aperture and distance of the eye behind it,, you can easily get resolution limitation due to diffraction image size increase at reduced effective 1/F number, and so not have the nominal eye's 1 minute of arc best-resolution when observing contact of the arcuate limb with a visible horizon. Such uncertainty is included in full in the corrected altitude's final value.) I would , however, suggest that if you use your sextant for this purpose, set up a shallow container of water, or better oil. (Oil: motor lubricant, or my preference ---- vegetable cooking, has a higher refractive index than water, hence stronger reflection at any given angle, and oils have better viscous damping properties.) Sight the reflected solar image from the liquid's surface with the horizon mirror and bring the direct image to coincidence with it by the adjusting index mirror. Since the isostatic free surface of the liquid is spherical at earth's large radius, it is effectively a horizontal plane. As incidence and reflection angles are equal about its normal, (in this case, the local gravitational vector,) the summed- complements angle measured equals twice the elevation of the sun above the horizontal. Not only are certain errors halved when the reading is halved to get solar elevation, but the effects of ill defined temperature gradients on refractive error in a near the ground-surface long air-transmission path are avoided. The refraction error through the higher solar elevation path is smaller and easier to predict and compensate. (Of course this method also evades the very real problem of finding a terrestial view of an unobstructed "true" horizon.) (By the way, a well set up amateurs' telescope on an equatorial mount is an excellent "sundial," but great care must be taken to use a projection system, or a Herschel Wedge with filters, to avoid ever directly viewing the dangerously intense solar image. (Transmission filters used alone are NOT RECOMMENDED.) Mounting a simple gnomon and target coaxially can also serve, in place of a projection setup through the 'scope or finder. As a dialer, you may earn credit with your star-gazer friends if you instruct them in the reverse process, calculating the predicted declination and right ascension of the pre-sunset apparent sun as an aid for aligning their 'scopes when at a remote temporary site in search of better seeing and less light pollution. They often use stars for this, but surprisingly, may not think of the nearest star, and of its advantages for setting up in daylight.) Sciagraphically, Bill Maddux.
