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    Re: lunars with and without altitudes
    From: Frank Reed CT
    Date: 2006 Nov 27, 00:21 EST
    George, regarding getting a position fix by lunar distances at known GMT,
    you wrote:
    "It's a clever, interesting, and amusing concept"
    I'm glad you like it. That's two steps forward.
    And you wrote:
    "but we would be wise not to take it too far, or too seriously, as a practical tool."
    Well, first of all, you definitely have not invested enough effort to understand it yet (evidenced by your comments with respect to the Moon's altitude --see below). More generally, as far as being a "practical tool", that's all relative today. You want practical? Buy a GPS receiver. That's practical. For anything else involving sextants and navigation, the reasons for doing it are much more diverse than mere "practicality".
    And you wrote:
    "He has claimed that a skilled observer can measure lunar distances
    reliably to 0.1 arc-minutes; claims which have been treated with
    scepticism by some contributors, me included."
    Just so we're clear here, what I claim is that a moderately careful observer with a well-adjusted metal sextant, equipped with a good telescope, who shoots four lunar distances in a row will find a standard deviation accuracy of about 0.2 minutes of arc. And when you average four of them, the standard deviation of the resulting combined observation is just about 0.1 minutes of arc, repeatedly and reliably. That's hardly a radical claim.
    And you asked:
    "Well, what precision does Frank expect to achieve, in measuring his
    lunar distances, not this time from on land, but from a yacht
    bucketing about in the Gulf Stream? Has he taken lunars under similar
    conditions, and if so, with what resulting accuracy?"
    Nope. I don't own a yacht these days. The only yacht I've ever owned was twelve feet long --it was a galley with two oarsmen, and I've never had the opportunity to "bucket about" in the Gulf Stream. I wouldn't be at all surprised if my lunar distance observations were 20% or even 30% worse while bucketing about. :-) Your results might be better or worse, but let's remember that shooting lunar distances is easier in many ways than shooting altitudes. you can sit yourself on deck very close to the vessel's center of oscillation, even below deck if you have a nice hatch, and you don't have to wait for that moment when the waves clear the horizon. A navigator with good sea legs should have no more difficulty shooting lunars except in "yacht" in the size range of my old galley.
    You asked:
    "Frank has acknowledged that a high Moon is needed to extract
    worthwhile accuracy from such a procedure. How high can the Moon be,
    from Spitzbergen? Once every 18 years (as it happens, this year) the
    Moon can reach an altitude of 40 degrees from Spitzbergen at some
    short period in each month. Most of the time, it's going to be much
    lower than that. So how does measuring a lunar distance angle to 0.1
    arc-minutes, if that can be done, result in a position to +/- 6
    nautical miles?"
    No. This is WRONG. I have said AGAIN AND AGAIN that the altitude of the Moon reduces the accuracy ONLY if the other object is more or less directly above or below the Moon (same azimuth, in other words). If the Moon and Sun (that's the only available object in the summer Arctic) are at about the same altitude, which they would be for that week or so out of each month when the Moon is visible in the high Arctic, then the accuracy of this procedure is NEARLY THE SAME as when the Moon is straight overhead in the tropics. Why is that so? Because the corresponding "cone of position" intersects the Earth's surface almost vertically when the Moon and Sun are low in the sky and at about the same altitude. [before anyone gripes, since I've said all this before in several other messages and it's been ignored, I feel justified in using CAPS for emphasis. Ok?]
    So I repeat: do a lunar distance observation (Moon-Sun) with a properly adjusted metal sextant up in the high Arctic when no horizon is visible. Wait four to six hours or so, and then do another set. If the observations have a standard deviation of 0.1 minutes of arc, the resulting position will have a standard deviation accuracy of about six nautical miles (apart from any inaccuracy arising from the dead reckoning between the sights --as with any running fix).
    George, you wrote:
    "We keep on asking Frank to provide a proper procedure for reducing
    such an observation, with a numerical example. He keeps insisting that
    he has done so, but the limited information that has been supplied is
    as yet insufficient for his steps to be followed."
    And yet D Walden followed my instructions and found his own fix by crossing lines of position from lunar distances WEEKS ago. How did he manage that if I've never posted a "proper procedure" as you claim?? What do we conclude? That Mr. Walden is more clever than Mr. Huxtable?? No. Even if true, it shouldn't be relevant for something as straight-forward as this process. No, George, I think the problem is that you're reading the posts I've made very superficially, assuming for whatever reason that there's nothing that will help you understand. So what to do? I can't make you read more carefully.
    Anyway... Here's the short version AGAIN:
    1) Measure a set of lunar distances off two bodies at two very roughly perpendicular directions from the Moon. This alignment simply guarantees that the LOPs will be roughly perpendicular to each other. As with ordinary celestial LOPs, it is not critical that the lines be exactly perpendicular, but an angle larger than 45 degrees is definitely preferred.
    2) CLEAR those lunar distances at four points evenly spaced in latitude and longitude around your estimated position and compare with the predicted distance (note: this estimated position is not strictly necessary but in the "real" world, we always have one, and in any case it's trivial to extend the procedure if your estimated position is badly wrong). NOTE: you can do this clearing using any tool that suits your fancy so long as it is an accurate approach to the problem. That means you need to have a tool that includes all the small details: temperature/pressure variations in refraction, oblateness of the Earth, refractional flattening of the Sun and Moon, parallax of the planets (if used). Many early 19th century clearing tools would not be up to this task. But if you're into the antiquarian approach, you can find plenty of methods that DID include all the fine details. Later in the century, if you insist on a paper method, you can get yourself a late 19th century Bowditch and use Chauvenet's method and its associated tables. But if you're not interested in the old calculational approach, there are a number of modern tools. For example, you could use my web site (can't take it to sea right now, but if you ask nice I can compile a stand-alone) or you could use Arthur Pearson's spreadsheet which he discussed with you in many messages on this last about four years ago. Or you could code it up yourself! It's not too terribly difficult. The IMPORTANT point here is that there is nothing new in this clearing process. It's just the ordinary process of clearing a lunar distance.
    3) Looking at those four points around your estimated position, you should find that there is a difference in sign in error across the region bounded by the points. For example, let's suppose you find that the NE corner has an error of -0.25 minutes of arc. The SE corner has an error of 0.40 minutes of arc. Suppose that the error at the NW corner is -0.40 minutes of arc. And the error at the SW corner is +0.25 minutes of arc. Clearly at some point along the east side of the box, closer to the NE corner, the error would be zero. At some point along the west side of the box, closer to the SW corner, the error would also be zero. We can easily interpolate to find those points. The two points are simply two points along a line of position. Everywhere along that LOP, the error in our observed lunar distance would be zero. In other words, at all points along the LOP our observation would be exactly correct. THEREFORE our vessel must be along that line. You repeat the procedure with the second observation. This gives a second LOP. Where they cross is your position fix. I recommend repeating the process with the distances shifted by a tenth of a minute of arc to get an error estimate for the LOP.
    3a) I should emphasize that the specific steps in part 3 can be handled in many different ways. All we're doing is generating a line of position. The exact methodology is not important. Finally, you can understand best *why* this all works by thinking in terms of "cones of position". I won't post the link to the diagrams again. It's all in the archive.
    And you wrote:
    "For example, I have pointed out how celestial positions, calculated from the Nautical
    Almanac, are insufficiently precise to provide the claimed precision,
    yet he has still not explained what the observer needs to have aboard
    to provide what's needed, and how it is to be used."
    Using the GHA and Dec from the Nautical Almanac (in its modern form) is marginally adequate and will only lead to errors larger than a tenth of a minute of arc about 20% of the time. So is all hope lost? Of course not. Today we have access to abundant astronomical ephemeris data. There are a number of sources for pre-calculated lunar distances if you insist on doing the work on paper. You could print out predicted lunar distances from my web site for any date that interests you. You could use Steve Wepster's tables. You could use Ken Muldrew's tables which he recently described on the list. Or, again, you could calculate your own. The best approach would be to load up the JPL ephemeris data. When reduced to an accuracy of one second of arc, which is more than we need, they can easily be stored in 50 kilobytes or less for one year of data (Sun, Moon, navigational planets and stars).
    Now, if you still have questions, please ask away. I would be happy to discuss it. But please, do me a favor: don't just come back saying that I haven't explained it to you.
    42.0N 87.7W, or 41.4N 72.1W.

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