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    Re: Frank's formulas, was: Lunars: altitude accuracy
    From: George Huxtable
    Date: 2004 Nov 3, 15:46 +0000

    Frank Reed wrote-
    >This analysis raises another entertaining issue regarding the input data for
    >clearing a lunar. We discussed the case of a lunar where the altitudes are
    >each 45 degrees on opposite sides of the zenith and the measured distance is
    >exactly 90 degrees. The difference in azimuth is 180 degrees, and the cleared
    >distance is 89d 21.3'. If I shift the Moon's altitude to 40, the difference in
    >azimuth is smaller: 147 degrees, but the cleared distance is still 89d
    >21.3'. The
    >error in altitude has no effect. But what happens if I shift the Moon's
    >altitude to 50 degrees? This is an interesting case because the
    >observation is now
    >*inconsistent*. There is no way to have a measured distance of 90 degrees when
    >the Moon is at 50 degrees and the other body is at 45 degrees. But suppose
    >that's what you've recorded. What happens? It is interesting that if you clear
    >the distance, you will *still* get the number you're looking for: 89d 21.3'.
    >But in this case, if you were to attempt to extract an actual value for the
    >difference in azimuth, you would find a meaningless number (the
    >intermediate step
    >in the calculation gives a value for cosZ of -1.19). I find it rather
    >entertaining that the clearing process is robust in this way and can handle
    >inconsistent inputs.
    Comment from George.
    Yes, I noticed the same effect when trying out examples to test Frank
    Reed's formulae for susceptibility of lunar distance correction to errors
    in altitude.
    If the true altitude of the Moon is M and the true altitude of the Sun is
    S, then the true lunar distance D can not be less than the difference
    between M and S (when the azimuth difference is zero) and cannot be greater
    than (180 - M - S), (when the azimuth difference is 180 degrees).
    For example, if the Moon's altitude is 30 deg and the Sun's is 50 deg, then
    the lunar distance MUST lie between 20 and 100 degrees. Anything else is
    If we draw a spherical triangle joining true Moon, true Sun, and Zenith,
    the lengths of the sides are-
    90-M, 90-S, and D
    we can work out the Zenith angle Z between Moon and Sun from the standard
    formula for a spherical triangle,
    cos Z = ( cos D - sin M sin S ) / (cos M cos S)    (and note that if D goes
    outside the range 20 to 100 degrees then cos Z goes outside the range ?1
    and the calculation of azimuth becomes meaningless because such azimuths
    are impossible)
    And similarly, we draw another spherical triangle which joins the apparent
    Sun, apparent Moon, and the Zenith. It's different from the first triangle
    because Moon and Sun have been displaced vertically, by the different
    effects of parallax and refraction. But because that displacement has been
    entirely vertical, towards or away from the zenith, we know that the Zenith
    angle in this triangle is exactly the same as it was before (actually,
    there can be a tiny "sideways" effect which it is usually convenient to
    So we can write, just as before,
    cos Z = ( cos d - sin m sin s ) / (cos m cos s) where the small letters
    refer to apparent angles, rather than the true angles for which we used
    Because the left hand sides of those two equations are both identical (cos
    Z) then the right hand sides must also be equal.
    ( cos D - sin M sin S ) / (cos M cos S) = ( cos d - sin m sin s ) / (cos m
    cos s)
    which we can rewrite as-
    cos D =  [(cos d - sin m sin s) (cos M cos S) / (cos m cos s)] + sin M sin S
    This is a rigorous expression giving the true lunar distance in terms of
    the apparent lunar distance and the true, and apparent, altitudes. It's the
    basis of Young's method, and several others, for clearing the lunar
    It's a standard bit of bookwork which I have spelled out to show what a
    central part the azimuth Z has played in deriving that equation.
    What has surprised me (and intrigued Frank) is that the above expression
    continues to give a value for D in circumstances that are QUITE IMPOSSIBLE,
    in that the lunar distance is such that there's no value of azimuth
    (between 0 and 180) between Sun and Moon that can accomodate such a lunar
    distance. In those circumstances, although any attempt to deduce that
    azimuth would fail, the expression for D still seems to work, and gives
    some sort of result.
    When the numbers input to the equation correspond to azimuths in range 0 to
    180, then the result D has a simple physical meaning, the true lunar
    distance D. In other situations, is there any physical meaning we can
    attach to D?
    I find it interesting that although Frank and I are intrigued about this
    matter, our resident mathematician, Alex, takes it in his stride, as only
    to be expected. I have a lot to learn, it seems.
    contact George Huxtable by email at george@huxtable.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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