A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2013 Jan 7, 10:45 -0800
Five days ago (often an eternity in NavList time),
"Naturally this is all connected with the changing semi-diameter of the Sun during the year which is about about 0.3 minutes of arc bigger than average today and will be about 0.3 minutes of arc smaller than average in July. Can you measure a change in the Sun's SD of only 0.3 minutes of arc? Try measuring the SD of the Sun today and see what you get. Do remember to use sufficient shades!"
The discussion has shifted over to a discussion of index correction since the observation in question has normally been used for that. But this misses the point.
Let's consider this observation of the Sun's diameter and why it's useful. It has no practical navigation value, and its scientific value is ancient history. But it's very similar to a short-distance "lunar" observation. It's "lunars training". We bring the limb of the Sun into contact with the opposite limb and measure the angle between them. This is very similar to bringing the limb of the Moon into contact with the limb of the Sun, both in terms of the manual operation and its expected accuracy. The biggest difference is that this angle is very small. Though lunars of less than one degree were rarely shot historically, for several reasons, there's no reason not to try them today. This operation of measuring the angular diameter of the Sun is very nearly identical to measuring a short distance lunar, such as the angle between the Moon and Jupiter when the Moon is passing close by (as will happen again in about two weeks). Measuring the Sun's diameter is effectively a "training sight" for lunars, and many of the same steps should be taken if you want to ensure accuracy.
There are several ways to do this. First, you need to decide whether you will determine the sextant's index correction separately and measure the diameter always "on-arc", like any standard sextant observation, or instead use on-arc and off-arc observations to cancel out the I.C. Note that this is a special advantage of short angle sights, and this also can apply to short distance lunars (but only under special circumstances). It's convenient to cancel out the I.C., of course, but it's also easy to screw up the off-arc readings so the benefits may not be that great.
The math of this is simple enough. You take one or more on-arc observations where you bring the Sun and its opposite limb together. Each of this will read somewhere around 32' plus or minus a bit depending on the date and the IC of the instrument. You average those. Call that sum A. If you have previously determined the IC, then the reduction is relatively simple: the diameter is A+IC. If you also take off-arc observations, you average those and call that sum B. In this case, the diameter is (A+B)/2 and, though we don't need it, the IC is then (A-B)/2.
You don't need a "super sextant" to make this observation. Obviously you will get better results with a better-quality metal sextant. If you have a higher power scope, you can expect better accuracy in direct proportion to the magnification. You may even want to try this out with different scopes and even with no scope to see how the accuracy changes. By the way, don't forget to focus your telescope! It may seem unlikely, but I have more than once watched beginning observers complain that the Sun seems fuzzy because they didn't realize that the scope could be focused.
Select the right shades for both the horizon and the reflected view. Unfortunately, some sextants do not have sufficient shades for the horizon view, and if that's the case for your instrument, there's not much you can do. The Sun, both direct and reflected should be dim enough so that it is comfortable to look at for long periods of time and yet still has clear, sharp edges.
Now there's another decision. Will you measure the Sun's horizontal diameter or vertical? Vertical is more familiar and generally easier on the arm. Visually, you will place the Sun's reflected image on top of its direct image so that they just "kiss" at the point of contact. But a vertical observation comes with an extra concern: refraction. This only matters if the Sun is below 15 degrees or so. You can avoid the whole issue by making your observations when the Sun is higher, or you can apply the refraction correction from the Nautical Almanac or similar (if the Sun is, say, 8 degrees high, find the correction for 8 and 8.5 degrees and subtract those values --that's the change in the Sun's apparent diameter due to refraction). An aside: suppose the Sun is 60 degrees high. The effect of refraction that high is very nearly 0.1' per 5 degrees in any direction at any altitude so long as both points in question are above 45 degrees. So for the Sun at 60 degrees, BOTH the horizontal and vertical diameters are reduced by 0.01'. That's well below the limits of sextant observations, but I at least think it's interesting to know that the Sun's image is reduced in both directions and stays that size for all altitudes above about 45 degrees.
Next we come to a unique concern for short angle observations. These are the only ones that are affected by so-called "side error" which is normally no "error" at all in sextant sights. When you set your sextant to zero and aim it it at, say, a distant lighthouse, you will normally see two images of the lighthouse from the direct and reflected views through the sextant. The two images will be offset a little vertically due to any residual index error and also offset side-to-side. The I.E. is correctable, as we know. The side offset or "side error" has no significance for most sextant sights and can usually be ignored, but this Sun diameter sight is an exception. Since we are bringing the images of the Sun together limb-to-limb, if they are offset side-to-side, the measurement of the Sun's diameter will be too small. We can either correct for the offset (not difficult, just plain trigonometry, details upon request), or we can adjust the sextant so that the side offset is small enough. Less than 2' is all that's needed. So aim the sextant, set to zero (corrected for IC), at the Sun. You will see two images of the Sun almost perfectly superimposed but a bit separated by side offset. The Sun's diameter is about 32', so get out the little wrench and adjust the horizon mirror until the portion that is not completely overlapped is less than one-sixteenth of the Sun's apparent diameter. A visual guess is good enough. Note that you will have to re-test your IC after you make this adjustment. You should only need to make this adjustment once (maybe once a year) so long as your sextant is stable and doesn't get bounced around too much.
If you make off-arc observations, there is an easy way to get screwed up. If the micrometer reads, let's say, 35' for an off-arc measurement, the actual angle is 60-35 or 25'. You have to decide for yourself whether you will record the angles as they are and subtract from 60 later or subtract in your head as your reading them off. But beware. This is an easy mistake to make. Reversing the micrometer tenths in your head is prone to error.
For this type of observation with a good, properly-adjusted sextant and a 7x telescope, you should expect that each observation will be within a quarter of a minute of arc of the actual diameter of the Sun a substantial majority of the time. And averaging a set of 4 to 6 observations should get you within 0.1' most of the time. Note that this is not merely observational "consistency". You're actually measuring an angle here that has a known value. I'm referring to the "accuracy" of the observations, not the precision. The actual value changes over time, but the Sun's diameter will be close to 32.5' through early February. After you've gotten yourself trained in Sun diameter observations, you can move on to real lunars and see how things change.
PS: And there's even a little statistical relevance here. The diameter of the Sun on any given day is a known quantity. If I tell a student to go outside and measure the diameter of the Sun, and that student takes a single observation (without prior knowledge!) and reports 32.4 minutes of arc when I know from almanac data that the diameter of the Sun is 32.5', then I have an immediate sense of the quality of that student's observations even from ONE observation. Yes, it could be beginner's luck, which is why we would need more observations to draw firm conclusions. But even one observation has statistical value in such a case. By contrast, if the student reports 43.5', then I can conclude from a single observation that something has probably gone badly wrong.
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