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    Re: Shadow sights for longitude
    From: John A Brown
    Date: 2018 Jan 11, 08:03 -0800

    Hello to all.

    I hope that I will be able to explain a method of taking sun sights using shadows that I have developed over a number of years. Please refer to the images.

    IMG 1. Measuring the shadow can be done in either of two ways. The first way is static. You place the ruler so that the start of graduations is adjacent to the end of the shadow and note the time. You can leave the setup and go and make a cup of coffee and when you return the shadow will have moved off beyond the end of the ruler but the reading of the shadow's length will not have changed unless something has disturbed the setup. The length of the shadow is 24.2 mm For each sight the ruler must be repositioned against the end of the shadow hence possibly allowing the equipment to be disturbed and the level upset.

    The second method is dynamic. The gnomon is up against the ruler so that there is no gap. The shadow moves continuously along the ruler. The reading on the scale and the time of the sight must be noted simultaneously. The reading on the scale is 141.8 Note that there is a gap between the graduations on the ruler and the end of the ruler. In this case it was exactly 6 mm We can call the length of the ruler 166 mm We can call this the nominal length of the ruler. If we subtract the reading on the scale from the nominal length we get 166 -141.8 which equals 24.2 This is the same length as the shadow measured in the first method which you would expect as they were each measuring a shadow from a gnomon of equal height ie three Lego blocks. This having a nominal length for the ruler allows for easy calibration.

    IMG 2. The object of the exercise is to take a short length of ruler and turn it into a useful tool for taking sun sights. Besides the ruler, the other equipment consists of a shadow platform made from a piece of MDF and supported at one end by two small pieces of blue tack which in turn are on a short length of MDF to permit easy turning in azimuth. At the other end is a wedge made from a cut down door wedge.The spirit level consists of a vial taken from a real spirit level and glued to the top of two blocks of Lego.The join in the blocks when viewed through the bubble at an angle of about 45 degrees will show a straight line and the platform can be considered level. Not truly level of course. I have a dozen or so spirit levels and there are only one or two that show the same level.The gnomon in this case consists of 5 Lego blocks with a height of 47.6 mm When I first started shadow sights I measured all combinations of Lego blocks up to 7 high both in the up and down position. I made a list of these and have used it ever since. Also there is a watch set to UTC and a magnifying glass. The length of the shadow  has to be measured to the nearest  tenth of a millimeter.

    IMG 3.After leveling, the shadow is watched as it moves along the scale and every and every ten or fifteen seconds a reading is made and the time noted. In this case a total of nine sights were taken. At the end of this time the shadow platform was totally dismantled and put back together. Then a second set of seven sights was taken. You may have noticed that these sights were done indoors. The Australian summer can be very hot.

    IMG 4.  Now it is time to do the calculations. All sights are worked from my back yard ie a known position. For this I use a Tandy PC6 pocket computer from the 1980s. Set the second set of sights aside for the time being. I mentioned before the concept of the nominal length of a ruler. In this case the ruler had a blank bit on the end of 5 mm so the nominal length is 305 mm and the length of each shadow is found by subtracting the reading on the scale from this figure.In the figures shown the first two columns  are the times and readings on the scale. The third column is Hs but it is only necessary to record two sights whose readings on the scale vary by 0.1 mm. This will tell how much altitude will change for each 0.1  mm change in shadow length. In this case 5.6 minutes of altitude.

    The next thing to record is a list of the intercepts for each sight. It is a bit like using a slide rule ie all the intermediate steps need not be recorded. Column four is the list of intercepts. The nominal length of 305 mm was used to find the shadow length which was divided into the height of the gnomon to give tan Hs. Refraction as for stars and planets is applied before getting the intercept.This is because refraction varies with altitude whilst semi diameter is constant.

    If  carried out correctly and refraction has been allowed for as for stars and planets it should follow that the intercept is an accumulation of errors.

    1 The level is not true. However it will be the same amount out of true for each sight taken with this spirit level. It can be compensated for.

    2 Because the platform was not level the gnomon will not be vertical. It will not be vertical by the same amount for future sights using this spirit level. It can be compensated for.However the gnomon will always be at right angles to the base.

    3The semi diameter of the sun has not been allowed for but this also will be almost identical for future sights. It can be compensated for.

    4 The blank bit at the end of the ruler may have been measured incorrectly.

    We can simply allow for these errors by adjusting the figure for the nominal length. We found that each mm change in shadow length produced a change in altitude of 5.6 minutes. Call it six minutes. Divide this into the average of the intercepts which is about 23. In round figures call the answer 4. This means that we need 4 lots of 0.1 mm changes because we found that each 0.1 mm of change produces 5.6 minutes of change in altitude.

    Because the intercepts are positive we add 0.4 mm to the nominal length which now become 305.4 THIS IS THE EFFECTIVE LENGTH OF THE RULER.

    Recalculate the set again but instead of 305mm nominal length use the newly found effective length.  Only  the intercepts need be recorded. This result is shown in column 5.

    Now take the second set of sights that were put aside. Forget all about the nominal length. The ruler has been calibrated for use with the yellow and blue spirit level and the effective length of 305.4 can be applied. It is as if the problems of net being level and the semi diameter of the sun do not exist. The columns show time of sight, reading on scale, Hs, Az and intercept.

    From now on this ruler has a new designation 5DR 305.4 Y/B ie the five diamond ruler with an effective length of 305.4 mm when leveled with the yellow and blue spirit level.

    The next day two sets of sights giving good results were done and had the angle of cut been better a good position would have been obtained.

    IMG 6. Two more sets. The first with another short piece of ruler and the other with the gift card shown in my previous post.

    It must be stressed however that things do not always work out as well as this and there are many failures along the way but this all part of the learning experience.

    Kind regards,

    John A Brown.

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