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    Re: Graphs of Lunar Distances.
    From: Frank Reed
    Date: 2010 Oct 24, 18:01 -0700

    John H., you wrote:
    "An arc-minute strikes me as being quite good for a sextant."

    Well, it's not bad, but one can easily do considerably better with a sextant --as long as it's properly calibrated (more below) and as long as it is equipped with a moderately powerful telescope. Let's take Douglas Denny's OWN observations of Moon-Jupiter lunars which he posted at the end of September. Despite his protests and declarations, they clearly demonstrate considerably better than "an arc-minute" for accuracy (at the 1 s.d. level). He gave eleven sights with time and location information so it's possible to derive the actual angular error on each and every sight. The results in minutes of arc:

    Out of these eleven sights, not one shows an error as large as one minute of arc and seven out of eleven have errors smaller than 0.22 minutes of arc. That is, he was measuring angles to +/-13 seconds of arc or better 64% of the time. He had a couple of major outliers in this set which increase the standard deviation to 0.4 minutes of arc but with more consistency that shouldn't be hard to overcome. Douglas assures us that his errors are much larger than this, but that's just because there are appears to be a continuing problem with his calculations. I would imagine you know the quotation from uber-physicist Richard Feynman that goes something like: "The first principle of science is that you must not fool yourself and you are the easiest person to fool." It's just another way of reminding us that we, all of us, have a tendency to confirm our pre-conceived notions. But that's why we do experiments -- to test our ideas.

    And you wrote:
    "The current Jupiter/Moon alignment is great for lunars."

    Evening Jupiter is always nice for homework assignments. Students don't like homework that starts with "get up at 3am..."! Of course, Jupiter is good for lunars for a solid week or more out of every month for most of the year. It's just that right now, it's up in the sky during our evening "primetime". :)

    You also wrote:
    "I wish I'd had my good sextant with me. It's in my other house and the one I have lying around is a WWII vintage standard issue US Navy one it needs a lot of work. I can't touch an arc-minute with that thing."

    Yeah, I know the feeling. I suspect that there are several reasons that most modern sextant users have become accustomed to the idea that a minute of arc accuracy is about as good as it gets. First of all, for altitude sights there is an intrinsic additional source of error from the uncertainty in the actual value of the dip caused by the variability of refraction at the horizon. This apparently adds in +/-0.5' or more to the errors of altitude sights. Sights like lunars don't suffer from that problem. They can be much more accurate (in angular terms). The second issue is the age and quality of sextants available today, as you've mentioned here. Many people own sextants like your USN instrument, probably a Mk II. I own one, too, and it is desperately in need of a new correction table (one day I'll get around to it...). Folks who own these instruments often point to the calibration table and say, "look, it says the errors are all under ten seconds of arc." But if that table was prepared 65 years ago, it's probably irrelevant today. There's no getting around this with old, used instruments. Finally, most people using sextants today don't bother too much with the fine details of adjusting the mirrors and the telescope collimation. Those details really aren't necessary for common celestial navigation if errors of one to several minutes of arc are acceptable. But if a sextant is properly calibrated and adjusted using time-honored methods or modern methods, it can be considerably more accurate than ordinary expectations suggest. And I can't emphasize enough, use a 7x telescope if you have one. The resolution of the unaided human eye is about one minute of arc or slightly better (see PS). Magnify and you get better resolution in proportion.

    So what about back in the day when people shot lunars for real longitudes? What was a reasonable expectation of accuracy? Consider what David Thomson had to say when he introduced his lunar tables in 1825 (late for British shipping, right in the thick of it for American shipping):
    "By a steady perseverance, with a due attention to the directions in the Epitomes of Navigation, it will be found that the lunar distances may, in ordinary cases, be as easily observed as common altitudes, and so correctly that the longitude deduced from a set will seldom differ 10 miles from the truth. "

    Now a 10 mile (meaning minute of arc) error in longitude corresponds to just about 0.3' error in the observation and his number combined the observational error with the errors resulting from the clearing process. So his description is consistent with what I have described. Of course, it depends a lot on how you interpret his word "seldom" but if we call it as two-thirds better than that, one-third worse than that, then it's a close match for the one standard deviation error level.

    Thomson was counted as an expert on the process and practice of lunars, and his tables became very popular because they significantly shortened the tabular work. For us Bowditch-focused Americans, it's worth noting that Nathaniel Bowditch copied Thomson's long correction tables verbatim, giving Thomson's method prime place as the "second method" of clearing lunars (second only to Bowditch's own principal method and ahead of Bowditch's original method) from 1837. Since Bowditch died the next year, Thomson's method remained in the Navigator all the way up to the big overhaul that threw out most of the old material in 1881.

    PS: One might think that the exact resolution of the human eye wouldn't come up much in the wider world... And then last summer Apple introduced the display of the iPhone 4 and claimed that it had a "retina display" meaning that when it was held at a normal viewing distance, the distance between pixels was smaller than the resolution limit of human vision. You should have seen the bloggers, apple fanboys and apple skeptics, going wild debating that one! And it was all so familiar --issues that have been discussed on NavList dozens of times before. Personally, I think Steve Jobs shot himself in the foot on that one. Displays require finer resolution to eliminate the appearance of "jaggies" in diagonal lines (a hyper-resolution task for the human visual system) so at some point they are probably going to have to introduce an even finer "retina display". What do they say then... "we were just kidding last time"??

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