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    Re: Graphs of Lunar Distances.
    From: Frank Reed
    Date: 2010 Sep 30, 17:31 -0700

    Douglas, you wrote:
    "I note you are reasonably happy with the Capella and Vega calculations where we seen to be closer to agreement, within seconds for the equivalent GMT measured anyway; but it is the Jupiter sights which trouble you. They trouble me now too."

    Well, the disagreement in the clearing for your Capella sights is smaller, yes, but it still exists, so there's a problem with the clearing there, too. For the Vega sights, since they were so "noisy" I dropped one of your sights. Because of that there's really no comparison.

    For your Jupiter sights averaged, you now get:
    "which is therefore an error of 1 minute 36.6 seconds in time; instead of my original calculation of 1min-15".8 error in time. (20 seconds worse)."

    So you definitely have a problem then. By the lunars clearing tool on my web site (which has been battle-tested by me and around a hundred other would-be lunarians), your Jupiter sights show an error of only 0.2 minutes of arc compared to the true distance (equivalent to only 22 seconds error in GMT). So this is very good. But the clearing errors are leading you astray. Something is wrong your calculations. And just to reiterate, you can expect to do even better with the observations, even as good as these are already.

    You wrote:
    "Well this _is_ the basic spherical triangle solution for clearing lunar distance"

    Well, no, not really --not literally the "basic" approach. The "basic" solution of the problem is simple, as follows: apply the ordinary spherical trig cosine formula to the triangle consisting of the zenith, Sun, and Moon (ZSM triangle), using the observed altitudes and the observed lunar distance to solve for the cosine of the angle Z at the zenith. This angle Z is the difference in azimuth between the bodies, and it is not affected by refraction and parallax in altitude, both of which act vertically. Then you take that value for cosZ and with the altitudes corrected for refraction and parallax you reverse the process and solve for the lunar distance. The result is the cleared distance. The equation that you are using can be derived from this algorithm by a sseries of simple trig identities and algebraic manipulations, and so can many, many others, but there's nothing about it that makes the one you've quoted preferred over the "basic" solution as I've just outlined. Don't get me wrong, there's nothing WRONG with this derived equation and for really primitive programmable calculators, its shortness could be useful. Beyond that, there's no reason to use it. Furthermore, there is nothing important about it historically, despite the fact that it has entered NavList lore (going all the way back to 1995) as "Young's formula" probably because it was described in Cotter (for no good reason).

    You wrote:
    "The programme necessarily adjusts for refraction of both altitudes, semi diameter and par-in-alt internally. In fact by cheating and using the almanac data for altitudes, I had to add refraction to the inputted observed altitudes, so the programme can take refraction off again!
    Unless I have gone badly wrong in the programming and have not spotted it (quite possible knowing me) then it should be giving correct results. "

    Alas, it isn't. It is NOT giving correct results. May I suggest that you take any specific lunar clearing case, like one of your Jupiter sights, work it up on your calculator, and then do it again through the clearing software on my web site. The clearing tool on my site is EXCEEDINGLY EASY to use, and it does all the work for you including calculating the altitudes if you don't have them available. You could clear your eleven Jupiter lunars in less than three minutes. Until your results match to the nearest tenth of a minute of arc most of the time, you can be sure that there is something wrong with your programmed code. Speaking from experience, you will probably discover that it is something painfully simple, like a reversed sign on refraction or something like that.

    You wrote:
    "As for Captain Cotter and his book: don't forget it was written in the 1960's with no internet access to all the wealth of information there is now; written too for those interested in 'Nautical Astronomy'- which would have been his students only at that time: a very limited audience indeed compared to today with the internet and more leisure time available"

    Yes, of course, we should give him a break and forgive this one lapse, especially since much of the rest of that book is excellent, as is his "History of the Navigator's Sextant" and also "The Complete Nautical Astronomer". Nonetheless, that specific chapter, on lunars, is a mess. You might want to say, 'it's better than nothing' but in fact it's so misleading that it may be worse than nothing for some readers. Naturally, it's not complete nonsense. About half of it is good, but how would a reader know the half that's bad? As I noted previously, it all "looks" very nice.

    You asked:
    "Can you please indicate where a 'History of Nautical Astronomy' of far better quality than this 'trash' can be found in one book, preferably too being right up to date for today's students of Celestial Navigation?"

    Wouldn't that be nice? But, nope, I can't do that. So does that mean we're compelled to read Cotter's chapter as scripture? I assume you'll say 'no'. If you want other histories, why not go back to some of the people who participated in that history? For example, you could read Mackay's "Theory and Practice of Finding the Longitude at Sea or Land" which was published in a couple of editions around 1800 and includes a very nice history of the subject up to that point (Cotter even mentions it briefly). For something a little more modern, you should read Frederic Marguet's "Histoire Generale de la navigation du XVe au XXe siecle" (General History of Navigation from the 15th to the 20th century) published in 1931, which, as you may have guessed, is in French. It's linked under "Resources" on the main NavList web page here: www.fer3.com/NavList.

    But let's think again here. Do we learn history by reading other people's summaries and then somehow averaging them to create our own interpretation of history? Certainly if a topic is vast, like the history of the Roman Empire, e.g., that may be the only practical option. But the history of celestial navigation is a rather compact thing and here we have another option that is far superior. We can turn to the real evidence and apply the methods of historical science. We can read the primary sources. There are literally thousands of logbooks available that have scarcely been touched for navigational history. This is where you will find the real history of navigation. There are newspapers and contemporary memoirs, like Basil Hall's, and letters to editors and columns in the "Nautical Magazine" and more. There are records of Congressional and Parliamentary hearings (and yes, celestial navigation was once important enough that it was discussed in public hearings). Some of these can also be found in the "Resources" section at www.fer3.com/NavList.

    By the way, I disagree with you that Cotter's audience in the 1960s was smaller than exists today for this material. Although celestial navigation was already beginning its decline by that time, celestial was actively used every day by tens of thousands of people in the 1960s and millions had used it or had been trained in it within one generation. By contrast, today, you would have to divide those numbers by a hundred. The market today for this material is somewhere around ONE PERCENT of what it was back then. It is relatively unlikely that anyone would publish a printed history of navigation or nautical astronomy in the same specialized category as Cotter's. Some general popular science book might make it, as Sobel's "Longitude" did way back in 1995. A self-published history might turn up, or Celestaire or Starpath might find a way to publish something. And of course there will be web sites and electronic publications of various sorts, but there's no market for an actual scholarly book in the style of Cotter's "History of Nautical Astronomy". His seriously flawed chapter on lunars will probably continue to mislead students of the history of navigation for decades to come.

    Of my suggestion that you can do better than you are already, you wrote:
    "Well all I can say is you are a better man than I am Gunga Din"

    No, I'm not. I have conducted a number of workshops in lunars since 2004, and when observers without previous experience in lunars shoot them with a properly adjusted sextant fitted with a decent telescope, I find that they, too, can get observations with a standard deviation of about a quarter of a minute of arc. There's no law to this, and there are plenty of ways to screw it up, but I can assure you that it's not just me. One small thing that is "just me". I recently discovered that the telescope on my sextant that I've been using for years, which is labeled as a 7x telescope, is closer to a 9x scope. Any increase in magnification in my experience directly and proportionately increases the accuracy of lunar observations. So if I get 0.2' accuracy with my sextant and that telescope, I would expect to get 0.26' accuracy with a 7x telescope (or 0.6' accuracy with a 3x scope, and so on).

    You wrote:
    "...when I find the small blurr circle of light which one views of the disk of Jupiter in the telescope, straddling the limb of the Moon and +or- half a minute of change in the micrometer drum is hardly visible at all as a change in position of that spot of light on the Moon's rim I still find it difficult to imagine anyone achieving a quarter of a minute of arc accuracy 'routinely'."

    Or there may be an issue with your sextant's telescope. Jupiter should appear not as a "blurry circle" but as a sharply defined, tiny disk.

    And then you re-stated what I consider to be your pre-conceived notion:
    "whatever you do or say, the inherent accuracy is all down to how good you can measure the movement of the Moon...and _that_ is _slow_ across the heavens, at a rate of only 30 minutes of arc in one hour: not much change going on there to measure with an instrument which is limited to around a minute of arc or slightly less in absolute terms."

    You're forgetting (or mis-reading) what I wrote to you in the last message about YOUR OWN observations. Your Jupiter set, averaged, was accurate to 22 seconds in time (equivalent to 5.4' of longitude, or 3.9 nautical miles, in your latitude). This corresponds to an error in the mean observation of the lunar distance of only 0.2'. The standard deviation of those observations is 0.4'. So more than 60% of the time, your individual Jupiter sights are at least 2.5 times more accurate than your claim here of "around a minute of arc". If I take each of your eleven Jupiter sights and work them up for error individually, I get (in order, in minutes of arc):

    0.12, 0.74, 0.12, 0.04, -0.10, -0.50, 0.16, 0.93, 0.71, 0.21, -0.16.

    NOT ONCE in that batch was your error as large as one minute of arc and SEVEN are within 0.25'. And these are only your first trials.

    You added:
    "As I said before, if the Moon travelled at twice the rate it does, then the inaccuracies would be halved and within a few seconds in time resolution would be more than possible. Lunars would then have been a good, solved, reasonably precise and practical navigation method."

    Lunars WERE "a good, solved, reasonably precise and practical navigation method." You found your position WITHIN FOUR NAUTICAL MILES using lunars. Even at the equator, taken as single observations, your position would have been accurate to WITHIN SEVEN nautical miles on seven out of eleven individual lunars. By the standards of traditional navigation, these are very good results.

    The accuracy of the observations involved in lunars was more than adequate for the task of finding longitude, as long as the observer had a properly adjusted sextant with a good telescope. Lunars were not significantly limited by the accuracy of the observations. Unlike our modern experiments, they WERE significantly limited by the accuracy of the almanac data (especially before 1800), which improved over the decades but not as rapidly as chronometers improved, and they were also significantly limited by the fact that lunars are nearly impossible for five or more days centered on New Moon every month and relatively inconvenient (and probably less accurate) for five or more days centered on Full Moon. Two or three chronometers carried aboard ship beat lunars hands down, as soon as they became cheap enough. This process of chronometers checked against each other eventually replacing lunars took three-quarters of a century to complete.


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