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    Short-cut lunars. was: Clearing lunars
    From: George Huxtable
    Date: 2010 Aug 28, 00:57 +0100

    Perhaps it's worthwhile discussing, in a bit more detail, Frank's discovery 
    that short-cut methods are possible, simplifying the clearance of lunars, 
    over quite a range of possible geometries. I think he has it right. It's 
    highly inventive. But it isn't any sort of replacement for the real thing.
    I had asked-
    "Now we come to quite different ground, a matter which Frank has raised 
    before. It is quite true that special geometries arise in which longitude 
    can be derived from a lunar distance observation, without involving trig. 
    And it's also true that the processing of lunars in that way may be 
    tolerant of quite wide deviations from those exact geometries. But how 
    tolerant? How much deviation from that geometry can be accepted, and still 
    keep within an error-band of an arc-minute or so, outside which a lunar 
    loses much of its value?"
    Frank's answer was this-
    "A whole arc-minute?? Well, quite a large number of cases! I previously 
    focused on cases where the resulting error was only a TENTH of a minute, 
    and even then there are far more cases than you would expect. It's not just 
    small, first-order deviations from the exact vertical circle case that 
    work, but much larger deviations.
    You asked:
    "So, what criteria does our navigator adopt, in order to decide whether 
    such a short-cut is acceptable, or whether a full-blown trig solution is 
    There is a very simple test. You compare the difference between the 
    observed altitude of the Moon and the adjusted altitude (adjusted to force 
    vertical circle geometry). If that difference is smaller than 
    6*tan(LD)/cos(h_moon) then any error resulting from treating it as a 
    vertical circle case is a tenth of a minute of arc or less. "
    Thanks for a direct answer to my question.
    So, let's sum up what our navigator has to do, when he decides that the 
    geometry is such that Moon and other-body appear to be somewhere near 180º 
    apart in azimuth, so a likely candidate for a short-cut, trig-free, lunar.
    Having measured sextant altitudes of moon and Sun-or-star, and lunar 
    distance between the appropriate limbs-
    Correct measured distance for index error and combined semidiameters, to 
    get apparent distance d between their centres
    Correct altitudes for index error, dip, and semidiameters, to get apparent 
    altitudes of centres above the horixontal of Moon m and Sun-or-star s.
    From refraction and parallax of other-body, where relevant, applied to s, 
    find true altitude S
    All this has to be done whether of not a short-cut procedure can be taken.
    Next comes the procedure to see if a short-cut can apply-
    Find the difference between m and the quantity 180 - s - d, the value it 
    must have if the azimuths differed by exactly 180º
    Compare that difference with the quantity 6' *tan d / cos m, by calculation 
    or by table.
    If it's more, the short-cut has to be abandoned, and normal trig clearing 
    procedure adopted.
    If it's less, the short-cut can apply. In which case-
    Discard the calculated value of m, and substitute m' = 180 - s - d.
    From refraction and parallax of Moon, applied to m', find true Moon 
    altitude M
    Then find cleared lunar distance D simply, from D = 180º - S - M.
    But if the short-cut didn't apply, then preserve the original value of m.
    From refraction and parallax of Moon, applied to m, find true Moon altitude 
    Then find cleared lunar distance D from
    cos D = (cos d - sin s sin m) (cos S cos M) / (cos s cos m) + sin S sin M
    That final step should present few difficulties to anyone with a modern 
    calculator, but was the very devil to calculate using logs, in the days 
    when it had to be done that way.
    I've written it all out to check whether I've understood the procedure. No 
    doubt Frank will put me right if I've got it wrong.
    It should be clear that the short-cut could save quite a bit of work if the 
    appropriate conditions apply. But the test is non-trivial, and if the test 
    failed often, aborting that extra work, and necessitating the full 
    procedure anyway, then overall there might not be much saving. So it 
    depends on the navigator being able to judge well in advance whether an 
    observation is a suitable candidate or not. No doubt, that skill could come 
    with experience.
    But now, consider the situation facing the navigator who has never learned 
    the full trig procedure for clearing a lunar. What happens if the test 
    fails? He has no alternative but to pack up, and wait for another day. But 
    away from the tropics, the opposite-azimuth situation will arise only once 
    a month.
    That doesn't matter, to a hobbyist, who just wishes to test  his skill when 
    the occasion arises. He could go home and try again another day. But Frank 
    was imagining himself back in the 1780s, when lunars were starting to be 
    used in earnest by professional navigators at sea. They knew how to work 
    them. They may not have understood what they were doing, but it was a 
    familiar, well-memorised routine. And when they neede a lunar, they needed 
    it there and then. Lunars were unavailable for so much of the time anyway: 
    when it was cloudy, when the Moon wasn't up, or when it was too near the 
    Sun. So such a short-cut procedure wasn't an alternative, to anyone doing 
    real navigation. The full trig procedure had to be available, at his 
    fingertips, for when it was needed.
    Maybe Frank's appearance, with his new procedure, in 1780, would have been 
    welcomed by navigators. But I have my doubts.
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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