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    Re: Graphs of Lunar Distances.
    From: Douglas Denny
    Date: 2010 Sep 29, 14:33 -0700

    Dear Frank,

    I have looked through your critique and notes for which I thank you for going to such detail and trouble.
    Now to addressing some of the points you have made:-
    ----------

    You say:
    "...Douglas, your calculations appear to have some significant problems..."

    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.

    I have looked through the Moon/Jupiter calculation again and cannot find any problems, except in my averaged calculation where I have made a slight slip in the arithmetic mean for the eleven observation times averaged. You are of course right that it should be 22Hr-29'-11" for an Observed Lunar distance of 34deg-11'.963 (which thank goodness I got right).
    Re-caclulating the cleared distance I now get 33deg,887 470
    which is = 33deg-53'.248
    equivalent to 22Hr-30'-47".6
    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).
    --------

    You say regarding the formula:-

    ".....D = arccos(sin S sin M +(cos d - sin s sin m )(cos S cos M )/(cos s cos m)) In case anyone else is wondering, there is really no reason to prefer this formula over the usual standard direct spherical triangle solution...."


    Well this _is_ the basic spherical triangle solution for clearing lunar distance if you start with an input of Observed lunar distance; Observed Moon altitude and Observed star altitude (and I also need HP for the programme). I do not see your objection.
    The calculator programme was intended to input these quantities and work out the cleared distance, which is what it does. There is nothing wrong with the formula, it is rigorous, so if anything is wrong it is the programming.

    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. Having worked out 'manually' on the calculator some input data with this formula in 'longhand' and coming up with the same result as the programmed version I was fairly confident it was Ok. But perhaps it is not, so I shall have to work on it more to find out. I cannot explain the discrepancy.

    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; and with nothing else of like to compare with then, nor that I know of even today. You are methinks being a trifle harsh on what was a reasonable effort for the time.
    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?
    ------

    I was not deliberately reinforcing any pre-concieved ideas incidentally, when I wrote that the second set of sights confirmed my conclusions from the first set. I was going on the evidence.
    The second set of sights were not significantly increased in accuracy at all with my trying to improve in all the small ways that might add up to closer results (smaller sigma)to the absolute: using a tripod; using a set of about ten sights; careful about centring the image in the telescope; being as accurate as I thought possible doing the sights etc.

    You say:-

    I (Douglas Denny)wrote: "the inherent variability seems to be around one minute of arc measuring no matter what"

    You said:-
    ".....You should be able to do considerably better than that --at least when the other body is the Sun, Jupiter, or some of the other planets. With those objects as the "other body" in lunars, I routinely get results with a standard deviation of about a quarter of a minute of arc in individual observations and about twice as good when sets of four are averaged...."

    Well all I can say is you are a better man than I am Gunga Din; ...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'. Perhaps I need more practise or get my eyes tested.

    So, to conclude again my main theme: I think Lunars are great fun, fascinating history in their own right, and a most interesting exercise, but 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.

    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.

    It would be nice also to be able to measure with a sextant to a few seconds of arc easily, but it is not.

    These two issues are the crux of the matter.

    Douglas Denny.
    Chichester. England.

    ===============================
    Original Posting:

    Douglas, your calculations appear to have some significant problems.

    Your Jupiter sights:
    Averaging the times and observed distances, I get an observed LD of 34d 11.96' at a time of 22:29:11. Working those up with my "Lunarian Toolkit" software OR using the tools on my web site at www.HistoricalAtlas.com/lunars, the error in your longitude is only 5.4' or equivalently an error of about 22 seconds in Greenwich Time (this is VERY different from your calculations as posted). Setting aside the calculation problem for now, these are good observational results, but you can expect better with Jupiter. As for its navigational significance, clearly this is excellent. Note that in your latitude, that error in longitude corresponds to an error in the fix of only 3.4 nautical miles. It's not GPS but it's excellent by the standards of traditional navigation.

    I also worked up your Jupiter sights in batches, clearing the first four, the next four, and the last three, and then averaging the resulting errors in position in each set. Since these are smaller sets, we can expect somewhat larger errors. And they are: 4.8 n.m. error for the first four, 2.3 n.m. error for the next four, and 4.8 n.m. error again for the last three. These really demonstrate very nicely that lunars can yield very good longitudes contrary to the (mostly 20th century) historical legends about their impracticality.

    Your Capella sights:
    I again worked these up with the "Lunarian Toolkit" software. For those of you who have a copy of this software (from the Mystic Navigation Weekend), you can clear sights for stars that are not in the short list by typing the name. So where you select a star, just type "Capella" or even "Cap" (or "Zu" for "Zubenelgenubi"). In this case, the average LD is 51d 59.6' and the average time is 23:05:45. From that data, I get an error in longitude of 11.6' equivalent to 46 seconds error in GMT or equivalent to a 7 nautical mile error in position in that latitude. This is still very good. Can we really complain about SEVEN MILES?? But it's worse than your Jupiter lunars which matches my experience also. I get great results with the Sun, Jupiter, and sometimes the other bright planets, but my results with the true stars are consistently worse. Historically, the majority of lunars were Sun-Moon shots, so you should try some of those in the next few days.

    Your Vega sights:
    I've already pointed out that this is a bad choice for determining GMT directly since Vega is so far off the ecliptic, but we can still work up the sights and compare with what they should be in angular terms and from that derive an "equivalent" error in longitude "as if" a proper star had been chosen. Your Vega lunars seem to have one outlier, and, though it's a tough call, I think there's a good case here for dropping the first observation (or was the data posted incorrectly?).If we take the remaining four sights, the average observed LD is 94d 22.7' and the average time is 23:46:54. From that data, I get an equivalent error in longitude of 12.0' (under that "as if" condition above). This is not too different from your Capella sights.

    On to some other issues...

    In your post, you wrote:
    "The Moon/Vega sights were very difficult to achieve with a lunar distance of over 90 degrees even though the sextant was nearly vertical and canted to the right slightly by about 20 degrees. The swinging of the sextant to ensure correct alignment with the limb of the Moon was much more difficult than when the distance between Moon and star is only short, say 30 degrees, as the star at 90+ degrees has an unusual motion making eye/hand coordination exceedingly difficult."

    This is one case where a little practice helps a lot. Also, I have often recommended that people shooting lunars use two hands to hold the frame of the sextant. We're used to rapidly changing angles for standard LOP navigation and so we're accustomed to adjusting the sextant "live" with one hand turning the micrometer and the other holding the handle on the back of the sextant. Instead, set the angle to your best guess, and then grab the frame with your left hand while holding the handle with the right. This is much more stable. When you swing the star past the Moon, if it overlaps or misses, lower the instrument to a comfortable position and tweak the micrometer a few tenths of a minute of arc. Then raise it up again and see if it's better. Repeat as necessary. This method also helps with the general physical fatigue of shooting lunars. And incidentally, you mentioned putting the sextant on a tripod. For some purposes that may be a good idea. Another trick you can use is to sit down on the ground with a knee up and your elbow resting on that knee. This adds stability, reduces fatigue, and we know that navigators actually did things like this in the 19th century. It's a trick that would work nearly as well on the deck of a ship at sea back then as in our backyards today.

    Based on your experience with Vega, you wrote:
    "Another reason to avoid large lunar distances if at sea."

    In fact, lunars around 90 degrees, primarily Sun-Moon lunars, were PREFERRED in actual practice in the 19th century. Were you thinking of some other reason why one should "avoid large lunar distances"?

    You wrote:
    "I used altitudes obtained from computer almanac data (ICE programme) for the altitudes of the celestial bodies"

    Are you aware that the lunars tools on my web site will do all of this for you? All you have to enter is your DR, any non-standard conditions, select the body, and then enter the observed lunar distance and the time when the sight was taken. I also have a standalone version of this software (not freeware) that adds some other features.

    You wrote:
    "Clearing the distance I used direct calculation with a programmed HP50g calculator using:-
    D=arccos(sin S sin M +(cos d - sin s sin m )(cos S cos M )/(cos s cos m))"

    In case anyone else is wondering, there is really no reason to prefer this formula over the usual standard direct spherical triangle solution. Unfortunately, this trivial and historically insignificant formula was given a status it did not deserve in Cotter's "History of Nautical Astronomy" and far too many modern readers have been innocently misled by that book.

    Douglas, you wrote:
    "Cotter has a long and very interesting chapter on the lunar distance methods and tables in 'A History of Nautical Astronomy' pps 205 to 243."

    No offense to you --you didn't write it!-- but I wouldn't wish that chapter on my worst enemy. Cotter's book is loaded with interesting information in many chapters, but he was incompetent with respect to lunars, and that chapter is worthless trash. Alas, it is highly seductive. It reads like science. It reads like history. It's all beautifully typeset and it "feels" like a scholarly work. That was surely the intention --it's certainly not deliberately misleading. But it's just awful. Treat everything you read there as suspect.

    You added:
    "For the almanac Lunar distances, instead of calculating, as I do not yet have the planets almanac on my calculator..."

    ONCE MORE for the folks who don't know, you can get predicted lunar distances for the Sun, bright planets, standard lunars stars (and navigational stars if you want them), for any year from the mid 18th century through the mid 21st century (with some small uncertainty for dates after about 2020 due to the uncertainty in delta-T) from my web site at www.HistoricalAtlas.com/lunars. These are available in a variety of formats, AND they can be output for hours of Greenwich APPARENT Time instead of Mean Time which is very useful for historical work. While I'm thinking of it, these tools will be moving very soon to a new web home, but the old links will re-direct there. You won't lose them.

    By the way, I did some checking against the lunar distance tables on the lediouris site that you used. Those lunar tables are excellent, but they're not quite as accurate as the ones on my web site or in my software (just a few seconds of arc difference but it matters with lunars). Like so many others, he simply coded up the Meeus algorithms to produce his tables. That's fine for a small device with limited memory (like a programmable calculator especially) but it's rather old-fashioned otherwise, as I have described previously. The lunar distances on my web site and in my "Lunarian Toolkit" software are accurate to the nearest arcsecond with the remaining uncertainty primarily due to the high and low points in the lunar limb itself.

    You wrote:
    "I used simple interpolation between the hourly lunar distances which is quite accceptable as the second differences, representing rate of change per hour of the moon's motion, is only 0.04 moa per hour, which in 30 moa per hour Moon's motion is only 4.7 seconds in time error per hour max.
    (The early tables used three hourly Lunar Distances and the Moon's motion is not linear so prop logs were used to give greater interpolation accuracy - not needed here with hourly tabulated Lunar distances)."

    You definitely don't need second differences with those hourly tables. That's true. Note that second differences were NOT the reason why proportional logarithms were used. So-called "proportional logarithms" (PLog(x)=log(3/x) or equivalently log(3)-log(x)) were published and used for ordinary interpolation and they shortened the paperwork a bit. Today, if you're working on a calculator or computer, they're completely un-necessary. As for second difference interpolation in lunars, historically this process was easily avoided, and I have never seen a case where anyone bothered with it in actual practice. It's mentioned in textbooks of the more ponderous sort, and of course there are instructions for using second differences in the Nautical Almanacs of the era, and in the late 19th century William Thomson, Lord Kelvin, once went on about second differences in public lecture when he was trying to ridicule lunars, but in the real world, a practical lunarian could simply skip them by avoiding those geometries which would require them.

    -FER
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