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    Re: Latitude by Lunar Distance
    From: George Huxtable
    Date: 2006 Oct 14, 22:09 +0100

    I had charged Frank Reed with "over-egging the pudding", in claiming
    an accuracy of 6 miles either way in position-finding by his proposed
    method.
    
    His response was hardly a diplomatic one, for a list-owner addressing
    a contributor, no matter how irritating that contributor may be. It
    has given rise to a chuckle or two, here in Southmoor.
    
    Frank sneered at my use of an Ebbco sextant, which was hardly relevant
    to the question in hand. I consider that to be an appropriate
    instrument for navigating my small boat, though that does not imply
    that I am unaccustomed to other sextant types. I have no illusions
    about its imprecision, and would not dream of using it to measure a
    lunar.
    
    And he ended with-
    
    | A general comment:
    | George, we all know that you have severe 'allergic reactions' to new
    | ways of looking at celestial navigation. Basically, every time I've
    | brought up something new on this list in the past three years, you
    have
    | sunk into a state of denial, usually launched by a post saying that
    I
    | am not answering your questions the way you want them answered.
    
    I question the "usually" here, but if I that has involved a request
    for rigour in list discussions, I make no apology for it.
    
    Anyway, my back is broad. It's not the first time Frank has had a go
    at me, and with luck, if I continue in the same vein, perhaps it won't
    be the last.
    
    ======================
    
    Now, back to the matter in question, the precision of Frank's proposed
    method of position-finding without a horizon, which he claimed to be
    within 6 miles.
    
    Let's return, first, to the precision in measuring a lunar distance,
    the angle between Moon limb and star, for which he claims 0.1';
    better, at a guess, than has ever been claimed by any navigator in the
    history of navigation. This involves a subtraction of index error from
    observed angle. To get the result to 0.1', what accuracy is required
    for those two separate observations, and what accuracy does he
    achieve? Where does irradiation, at the Moon's limb, come into this,
    in determining those two quantities: does he claim immunity from its
    affects, or that somehow it doesn't apply?
    
    Let's investigate an error-budget in the overall determination. That
    was one reason why I have asked for an explanation of the procedure,
    with relevant numbers, because Frank hasn't stated where his celestial
    predictions are to come from. Is his observer, from his jungle
    clearing, privileged to obtain JPL predictions, beamed to him from a
    satellite or a convenient mobile-phone antenna? Or will he have a
    laptop with him, programmed up with JPL (or similar) algorithms? Or
    will he rely, like the rest of us do, on the Nautical Almanac, as I am
    going to assume from now on.
    
    Using the almanac, how do you find what the predicted angle should be,
    between Moon and star, at a particular moment of GMT? You will have to
    look up many quanties in the almanac, each one stated to the nearest
    0.1', a potential error of +/- 0.5'.
    
    First, to find the star position, you need-
    1. star dec
    2. star HA
    3. GHA Aries to nearest hour-point.
    4. Aries change, for part-hour
    
    and for the Moon-
    
    5. dec to nearest hour-point
    6. d correction to dec for part-hour
    7. GHA to nearest hour-point
    8. GHA change for part-hour
    9. v correction to GHA for part-hour
    
    these allow the lunar distance to be calculated, for a mythical
    observer at the centre of the Earth.
    
    9. Moon semidiameter must be allowed for
    
    The following terms vary with the altitudes of the bodies-
    10. HP cos alt (Moon parallax)
    11. star refraction
    12. temp and pressure correction to star refraction.
    13. Moon refraction.
    14. Temp and pressure correction to Moon refraction.
    
    Depending on the details of the geometry, it's possible for any one of
    those 14 terms to affect the result by up to .05', one way or another,
    simply because of the way they are tabulated in the Nautical Almanac.
    Of course, most are KNOWN to a much higher accuracy, but not from that
    Almanac.
    
    It is, of course, true that all 14 are highly unlikely to all add up
    in the same direction, to their full extent. They are also highly
    unlikely to cancel out to zero. Without making a full statistical
    analysis, it seems reasonable, to me, for the standard deviation of
    the resulting scatter to be taken as root-14 x the spread of each
    component, or 3.7 x .05', or 0.19'. If anyone can suggest a fairer way
    to combine those errors, I hope they will.
    
    So, just the uncertainty in taking numbers from the Almanac, on its
    own, has already contributed 11 miles (rms) either way to the scatter
    in the result, even before any observation errors have been accounted
    for. Above, we were analysing the error involved in one Moon-star
    position line. There will be a similar error in the other position
    line.
    
    So much for Frank's overall uncertainty of 6 miles.
    
    Does that explain why I was keen to get a full explanation of the
    procedure, with numerical examples?
    
    George.
    
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
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