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    Re: Precomputed lunar distances
    From: Frank Reed
    Date: 2005 Apr 18, 22:20 EDT
    Bill you wrote:
    "Yes, that is my image.  A two-dimensional representation of three
    dimensions.  What a camera would see."
    OK. And this can be useful so long as you remember that the sides are actually straight as an arrow.
    And you wrote:
    "What deeply confuses me is as follows. Using two hypothetical stars with
    equal declinations and an LHA between them, I calculate true separation as
    34d 27.7'.  I raise the equal Hc's of the two stars from a staring point of
    1d 36.8' in increments of 11d 02.9' (11d 02.8 for last step) and calculate
    refraction separation correction.  The results are as follows:

    Hc           Refraction   Correction
    1d 36.8'     -18.2'       -0.31796
    12d 39.7'     -4.1'       -0.57133
    23d 42.6'     -2.2'       -0.59930
    34d 45.5'     -1.4'       -0.60260
    45d 48.4'     -0.9'       -0.57422
    56d 52.3'     -0.7'       -0.64021
    67d 54.2'     -0.4'       -0.64310
    78d 57.1'     -0.2'       -0.63534
    89d 59.9'      0           0

    They do not seem to reflect refraction moving along a straight line to me,
    where I might expect the corrections to be similar to a curve derived from
    refraction values at those altitudes."
    I take it that you're suspicious of these results because it seems as if the correction is just about 0.6 minutes of arc across a wide range of altitudes. Strange, huh? Strange but true... in this case where both stars have the same altitude (and as long as the altitudes are above about 12 degrees). The correction will be 0.6 minutes at every altitude. This happens in this special case because the portion of refraction acting along the arc between the objects is increasing at a rate that exactly counter-balances the decrease in refraction at higher altitudes. At low altitudes, the arc between the stars is nearly perpendicular to the vertical arcs so only a small portion of the rather large refraction down there acts along the arc you're measuring. But consider the case where the stars are 40 degrees apart and both 70 degrees high. Now they are necessarily on opposite sides of the zenith and the great circle arc between them passes from one star through the zenith to the other. That means that all of the refraction acts 100% along the arc. It's simply an accident of Nature that the rate of change of refraction with altitude is exactly equal to the rate of change of the geometric factors with altitude. And this accident doesn't work for altitudes lower than 12 degrees (or so).
    In some old lunar methods and in some approaches to clearing star-star distances, it's been useful to pull out a specific equation for refraction instead of using a table or similar method. Above 12 to 15 degrees, refraction in minutes of arc is very nearly equal to 0.95/tan(alt). Try it and compare the results with a standard refraction table. When we use this specific form for refraction, the equation for correcting the distance between two objects for refraction simplies quite a bit. Letcher in his 1978 book "Self-Contained Celestial Navigation with H.O. 208" showed what happens under these conditions. The result for the correction to the distance is
       r = 1.90*(x - cos(D))/sin(D)
    where D is the measured distance and x is calculated from the altitudes:
      x = (sin(h1)/sin(h2) + sin(h2)/sin(h1))/2.
    And that's it. This is all you need to correct star-star distances given the conditions above. In the example you gave in your earlier message, h1 and h2 are equal in every case so x=1. And if you plug in 37 degrees for D, you will get 0.6 minutes for the refraction correction. Letcher also includes a neat little graphic trick for reading off x with a pair of dividers, but that's superfluous in these days of cheap computation. By the way, this is not a "different" way of calculating star-star distances from the one you've already seen. It's derived directly from it under the assumption of that specific law of refraction above (a law which happens to be accurate above 12 degrees or so but not below).
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