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    Re: Sun Moon Lunars to 155 degrees
    From: Frank Reed
    Date: 2010 Apr 6, 12:59 -0700

    George H, you wrote:
    "Here is a simple case, with lunar distance at 90 degrees. The Sun is at the zenith, the Moon is on the horizon. How sensitive is the cleared distance to the altitude of the Moon? And that of the Sun?"

    George, first of all you've picked a case that would never come up in the real world of lunars. Today no one would shoot lunars with altitudes below three to five degrees, and historically anything below ten degrees was considered off-limits. Second, as I've said many times, the formulas for altitude accuracy which I've given apply when the refraction is small --above ten degrees in the altitudes. The behavior when either body is below ten degrees is qualitatively similar but there are moderate quantitative changes. When both bodies are above the thick air near the horizon, it is the parallax of the Moon that drives the whole problem --EVEN with respect to the required accuracy in the other body's altitude.

    So let's try some more realistic examples that will help you see what's going on. Set yourself up a spreadsheet (or use the calculator on my web site at HistoricalAtlas.com/lunars) and try cases where the LD is near 90 degrees, and the altitudes are chosen at random between 10 and 90 subject to the condition that they are consistent with the specified LD (in other words, both altitudes can't be 80 degrees if the distance is 90). Then vary each altitude by a little bit. You will find that the accuracies required of the altitudes are consistent with these relationships:
    MoonAltError < 6'*tan(LD)/cos(hMoon)
    BodyAltError < 6'*sin(LD)/cos(hBody)
    The factor 6' is actually inversely proportional to the Moon's HP but the mean value is close enough for this issue. Note that these are derived from derivatives so that you shouldn't take them too seriously near the singular points. They are valid so long as the change in each variable is "small" in the usual calculus sense (in this case small relative to one "radian" so less than a few degrees). In other words, if you put the Moon at the zenith setting hMoon equal to 90 degrees, the formula seems to say that the Moon altitude error can be infinite but really it means you can have an error in the altitude as large as a couple of degrees.

    By the way, you may want to look through the archives at fer3.com/arc. Try searching the archive using the phrase "I am now fully convinced". Those are your words on this very topic from five years ago! You can get to that by following this link:

    This whole business of altitude accuracy in lunars and the (two different) "90 degree miracles" that result from it are matters worth discussing for the more recent lunarians in the room, so I am going to write it all up again in a different way under a new subject.


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