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    Re: What is the Longest Lunar Possible?
    From: Frank Reed
    Date: 2013 Jan 21, 10:19 -0800

    Sean, you asked:
    "So, may I add to Brad's question and ask, "What IS the ideal separation for a lunar?" Or are all [sufficiently large] angles equally accurate?"

    Around 70 to 100 degrees was most popular historically, and the "other body" was the Sun roughly 80% of the time. That's what I have found in early 19th century logbooks. Alex mentioned the problem of inconvenient, physically uncomfortable orientations. For a mariner at sea, you could always just wait for a more convenient situation. Lunars were never the sole method of determining longitude. Dead reckoning was the principal method through the early 19th century (until the 1830s on many American commercial vessels) and chronometers were the principal method in later decades. Lunars were used as a means of validating the longitude by DR or "by chro". Navigators would shoot a few sets around First Quarter and a few sets around Last Quarter, and, if necessary, update the longitude from those observations. The most common case that I have seen for breaking this

    Given a preferred angle around 90 degrees and considering the case of vessels in low to mid latitudes, the most common case would find the Sun in the west (or east in the morning) and the Moon in the east (or west in the morning). Since the two bodies would be on opposite sides of the sky, the orientation of the sextant was usually relatively convenient with the instrument nearly vertical. They didn't know it at the time, but lunars like these could frequently be cleared with no trig at all, just a little addition and subtraction.

    So WHY would one prefer lunars near 90 degrees? First, it's worth saying that the textbooks recommended otherwise. Bowditch, for example, suggested using shorter distances. But we know that he much preferred the stars for lunars, being a bit of a night owl and also apparently not a fan of sunlight. In practice, as seen in the primary source evidence of logbooks, navigators used the Sun most of the time and preferred angles closer to 90 degrees. There are a number of possible explanations here.

    First, it's just easier to see the Moon in daylight when it's more full. A thin crescent, when the angle is less than 50 degrees or so, can be very difficult to spot in the sky even if you know exactly where to look.

    Second, there's a convenient cancellation that occurs in the clearing of lunars that renders the Moon's altitude less important. Remember that a lunar requires measured altitudes of the Moon and the other body along with the observed distance (these can be calculated if we know the observer's approximate location but that's a lot of work if it's done by hand --historically it was much easier to measure them). The old navigation manuals were not specific on this, but it was understood that the altitudes only had to be accurate to a few minutes of arc while the distance itself had to be measured with the best possible accuracy. As it turns out mathematically (this, btw, is one of my specific personal contributions to the subject), the altitudes should be measured with an accuracy that is significantly relaxed when the objects are near 90 degrees apart. For the Sun, the altitude should be measured with an accuracy of +/-6'*sin(D)/cos(h). So if the distance, D, is 85 degrees and the altitude h is 45 degrees, the Sun's altitude can be off by as much as 8' and it will not significantly affect the clearing process. Meanwhile, the required accuracy for the Moon's altitude is even more relaxed. Its altitude should be measured with an accuracy of +/-6'*tan(D)/cos(h). So if D is 85 deg and h is, let's say, 55 degrees, then the Moon's altitude can be off by as much as 120' --fully TWO DEGREES thanks to that factor of tan(D)-- and it would not adversely affect the clearing process. I have no evidence that any navigators were aware of this relaxed requirement for accuracy in the Moon's altitude when the distance is close to 90 degrees, but it would certainly improve the accuracy of the resulting longitudes no matter what. By the way, the Sun's altitude was usually used for a time sight for local time accompanying the lunar (which gave Greenwich time) so normally a navigator would have measured the Sun's altitude more carefully anyway.

    Third, some clearing methods has corrections which were themselves proportional to 1/tan(D). Naturally those corrections would be un-necessary, saving a little work, when the distance was close to 90 degrees. This probably was more important in the early days of lunars. In the refined "cookbook" methods which were standard from about 1825 onward, it was less obvious where one could skip a step and the navigator had less incentive to break the rules of the recipe. And the methods were more efficient in any case so there really wasn't much work even with every step included.

    There's another consideration that's worth mentioning when it comes to navigational practice, and that's "navigational culture". This wasn't a science. Navigators did some things certain ways simply because that's how they were taught by other navigators. And since we're dealing with culture, the practice can vary from one segment of the world of navigation to another. Who knows... maybe there is a yet undiscovered historical tradition of Ottoman Turkish lunarians who always used short distances and always used the stars, never the Sun and never angles near 90 degrees. It's not likely, but my point is simply that it does not all come down to logic and mathematics. Navigation is a practical tool subject to cultural evolution (and devolution) as well as scientific development.

    What about modern lunars? If you're shooting lunars partly to try out the historical techniques but mostly just to see how good you are (and whether your sextant is up to the task), then very little of this matters. Any convenient angle from 0 to 120 is fine (or more than 120 if you have an instrument that can reach those angles). The problem of altitude accuracy is less important since you will almost certainly be shooting from a known location and you can calculate the altitudes (or allow my online calculator to calculate the altitudes, as most people do). You do have to worry about extreme refraction. Corrections for temperature and pressure (including altitude above sea level) are more critical when the bodies are low in the sky. With lunars you have to remember, every tenth of a minute of arc matters.

    -FER


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