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    Re: Sun Moon Lunars to 155 degrees
    From: George Huxtable
    Date: 2010 Apr 3, 00:46 +0100

    After a bit of reflection, I think I understand what Antoine is doing with 
    Bayly's observations in Uliatea. Let me try to explain it, in my own words, 
    to see if I have it right.
    At some moment three observers measure the lunar distance and the altitudes 
    of Sun and Moon. Those altitudes don't need to be known to great precision, 
    for that purpose, because they are only used to make a small correction to 
    the lunar distance.
    Then this lunar distance is compared with a modern ephemeris, via Frank 
    Reed's lunar calculator, to provide a value for GMT at the moment of 
    Then, from that modern ephemeris, at that GMT, we can predict the position, 
    in GHA and dec., of both Sun and Moon. If we then use the observed 
    altitudes of Sun and Moon, at that same moment, as the basis for a fix, 
    there are only two places on Earth where those altitudes apply, at that 
    GMT, and the other, false-fix, is thousands of miles away and we can forget 
    about it. So we have found a true position in one go, without having to 
    make prior assumptions about latitude, or anything else. And if we noted 
    the chronometer reading at the same moment, we have also determined its 
    The main inaccuracy involved is in the measurement of the lunar distance 
    itself, and each arc-minute of error in that will give rise to about 2 
    minutes in the time, or 30 arc-minutes in the resulting longitude.
    In this modern era, we can apply this technique to Bayly's data, rather 
    more precisely than he could in his own time, because we have a lunar 
    ephemeris that's free of the systematic errors that bedevilled Maskelyne's 
    almanac. Also, we know the Moon position, in dec. and GHA, much more 
    precisely than did Bayly. For him, the Moon coordinates were predicted just 
    twice a day, and the Moon shifts in the sky so curvaceously that a linear 
    interpolation within that 12-hour gap could on its own give rise to errors 
    of a couple of arc-minutes.
    That would work well from out at sea. But there's a big snag in applying it 
    to that harbour in Uliatea. The moon was somewhere to the North-West, and 
    presumably a distant horizon could be seen, from the height of the poop, 
    over the low reef, to measure the Moon altitude up from. So, no real 
    problem with the Moon. However, the Sun was in a direction of 
    East-Northeast, or thereabouts, so the near island got in the way of seeing 
    a horizon beneath it. Indeed, I have a sketch of the profile of the island, 
    drawn from a simlar anchorage-point on Cook's previous visit, four years 
    earlier. (I get more and more impressed by the standard of documentation of 
    these voyages). It's rather faded, and I doubt if it will copy clearly, but 
    if anyone asks I will be happy to post it. In the direction of the Sun at 
    that time, the shoreline was no more than about half-a-mile or so from the 
    anchorage. That would require a dip-short adjustment of about 20 
    arc-minutes, and the uncertainty in applying it would amount to several 
    arc-minutes. So Bayly didn't have a precise Sun altitude to use for fixing 
    his position. However, it was known perfectly well enough for its purpose 
    of correcting the lunar distance.
    Even with that uncertainty in the horizon under the Sun, I would estimate 
    that the dominant error will still be in the lunar distance.
    How could an observer get round that problem of the uncertainty caused by 
    the near-horizon under the Sun? He could simply wait until noon. To the 
    North, under a distant island 7 miles away, a true horizon could be seen, 
    below a noon Sun which needed no knowledge of time, and that could be 
    crossed with the morning Moon position-line. Here, I'm using concepts of 
    position-line which were not available in Bayly's time.
    Being stationary, in harbour, for a few days, and with a chronometer with a 
    known error, Sun or star altitudes could provide very precise positioning; 
    and presumably did, for Bayly.
    However, there's one aspect in which I seem to be at odds with Antoine; 
    that we have argued about before. I do not think he needs to make any 
    allowance for delta-T, between the date of the JPL prediction used by 
    Frank's lunar calculator, and 1773. I think that allowance is already made 
    within the predictions, so effectively, Antoine seems to be making it a 
    second time. We need Frank Reed to put us right about that, one way or the 
    contact George Huxtable, at  george{at}hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    contact George Huxtable, at  george{at}hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    ----- Original Message ----- 
    From: "Antoine Couette" 
    Sent: Friday, April 02, 2010 9:56 AM
    Subject: [NavList] Re: Sun Moon Lunars to 155 degrees
    Dear Dave,
    In further reference to your post :
    [NavList 12638] Re: AW: Sun Moon Lunars to 155 degrees [NavList] Re: AW: 
    Sun Moon Lunars to 155 degrees
    From: waldendand---com
    Date: 31 Mar 2010 16:39
    and to my first reply to it yesterday :
    [NavList 12651] Re: Sun Moon Lunars to 155 degrees [NavList] Re: Sun Moon 
    Lunars to 155 degrees
    From: antoine.m.couette---fr
    Date: 1 Apr 2010 14:33
    I just had the opportunity to run results through Frank's Computer.
    (hopefully no typos ...)
    For this Lunar Bayly had published the following results : S 20°49' and E 
    Here are the Lunar observation data :
    04 Aug 1773,  Moon-Sun Dist = 155°13'07" (entered as 155°13'1 into FER's 
    Computer) , SUN UL = 5°41'45" (FER : 5°41'8) , Moon UL = 10°31'52"5 (FER : 
    10°31'9 ) and T = 76° F.
    With delta T = + 16.4 s (half a second difference with the + 15.9 s you 
    used), I earlier published the following results :
    Time of the Lunar Distance UT = 15h47m58s1,
    with a position at S 20°49'29" (FER : 20°49'5 ) W 132°20'09" / E 227°39'51" 
    (FER : W 132°20'2).
    I just processed this example with Frank's Computer with the 3 sextant data 
    and the resulting position and time I derived.
    Here are the results of Frank's Computer :
    1 - With both Oblateness and Flattening : Error in Lunar -0.1' Approximate 
    Longitude Error : 3'7 , ( I almost always get a " 0.0' Error in Lunar " 
    grade in such cases, but here we have a number of standing round-off errors 
    which I am totally unable to eliminate through the data fine tuning 
    procedure recently indicated by Frank ) , and
    2 - With only Oblateness and no Flattening : Error in Lunar +0.1' 
    Approximate Longitude Error : 1.9' (no sign indicated, should be opposite 
    to the previous one) , and
    3 - With no Oblateness and only Flattening : Error in Lunar +0.0' 
    Approximate Longitude Error : 1.4' (no sign indicated, which way did it go 
    ???) , and
    4 - With no Oblateness and no Flattening : Error in Lunar +0.1' Approximate 
    Longitude Error : 4.2' (no sign indicated)
    Again, and since I always compute with Oblateness and Flattening, the 
    options of Frank's Computer are very nice to evalute the effects of both 
    Flattening and/or Oblateness.
    As we can observe, the Flattening and/or Oblateness effects account for 
    differences in Longitude well above the "One Mile" position difference 
    which you had earlier mentioned.
    Everything else being absolutely equal, if we use 2 different methods to 
    clear a Lunar, a "One Mile" position difference derived from this Lunar (at 
    least the Longitude part of it) is the result of about 1/30' (only 2" at 
    the most) difference in the resulting Geocentric Centers Distances obtained 
    once this Lunar has been cleared by these 2 methods.
    I would therefore conclude that - since in this "fast computation" you 
    elected not to take in account either Flattening or Oblateness - their 
    combined effects would most probably exceed the results of a sole 
    difference in Refraction models, and they would most probably also be the 
    main reason for this "One Mile" difference observed between our both 
    Thank you for your Kind Attention and
    Best Regards
    Antoine M. "Kermit" Couëtte
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