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    Re: Moon Parallax, Math Trivia (was Re: venus)
    From: George Huxtable
    Date: 2004 Oct 20, 17:46 +0100

    This is one of those pernickety postings, which only interest a few
    pernickety Nav-L members. Others should press the delete button now.
    
    It seems to be worth questioning Frank Reed's contribution, which was this-
    
    ============================
    
    >Earlier we were talking about the calculation of the Moon's parallax in
    >altitude when you're given the true geocentric altitude instead of the apparent
    >altitude.
    >
    >George H wrote:
    >"To increase accuracy, we need to find a better approximation, to bring us
    >closer to-
    >OA = TA - HP cos OA, when we don't yet know what OA is.
    >My own routine, for reverse correction for parallax of the Moon's TA, takes
    >the Moon's distance D from Earth's centre in Astronomical Units (which has
    >been computed elsewhere) to determine HP.
    >The angle to be subtracted from TA to obtain OA is, in this routine-
    >Arc-tan ( cos TA / ((23455 * D) - sin TA))
    >The amount of this correction now corresponds very closely (with a change
    >of sign) to the correction that's added to OA to obtain TA. "
    >
    >I mentioned that you can also do this calculation by iteration. Now for a bit
    >of trivia: the Arctan formula, which is plenty accurate enough for practical
    >puposes, is actually an approximation. There is no closed form solution for OA
    >above. There cannot be since the equation is identical in form to Kepler's
    >Equation. It makes little difference in practical terms since HP is always a
    >small angle, but it's a bit of mathematical entertainment that the historical
    >method of entering the paper tables twice (thrice is better) is more accurate
    >than a seemingly rigorous equation. It makes not a bit of difference when
    >you can
    >throw the whole calculation on a computer --merely trivia.
    
    ===========================
    
    It's that last paragraph that I wish to question. It seems to me that the
    expression I gave is completely rigorous, so I am not sure what Frank is
    telling us. Not that it makes the slightest difference, in practice.
    
    Start from Smart's "Textbook on spherical astronomy", (mine is 5th ed.) On
    page 199 is fig. 81, which is a useful diagram to take as our starting
    point. I take it that Frank has access to a copy.
    
    For this exercise, the Earth to taken to be round.
    
    z is the true zenith angle of the Moon as seen from the Earth's centre.
    
    z' is the observed zenith angle of the Moon, as seen from a point on the
    surface.
    
    To avoid Greek letters in ASCII text, rename "rho" to be "a".
    
    Drop a perpendicular from O on to CM, at point D.
    
    Then Tan p = OD / MD
    
    or Tan p = OD / ( r - CD )
    
    clearly OD = a sin z, and CD = a cos z.
    
    In terms of true altitude TA,
    
    OD = a cos TA, CD = a sin TA.
    
    so p = atn ( cos TA / ( (r/a) - sin TA ))
    
    or observed altitude OA = TA - p = TA - atn ( cos TA / ( (r/a) - sin TA))
    
    This is purely geometry. There are no approximations that I can see. It all
    seems to be exact. So what's Frank on about?
    
    George.
    
    ================================================================
    contact George Huxtable by email at george---.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    ================================================================
    
    
    

       
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