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    Lunars by series, quadratic correction
    From: Frank Reed
    Date: 2019 May 7, 10:15 -0700

    If the idea of a "series" solution to lunars is unfamiliar, please see my old essay on Easy Lunars here. The standard baseline quadratic correction for a series solution to the problem of clearing a lunar is:
      Q = Q0 · (1 - CCm2) / tan LD / 3438,
    where
      Q0 = dhm2 / 2.
    Here LD is the lunar distance, dhm is the altitude correction for the Moon in minutes of arc, and CCm is the "corner cosine" at the Moon in the ZSM triangle (that corner cosine is "A" in my ancient Easy Lunars essay). The result, Q, is in minutes of arc.

    In some versions of Easy Lunars, I replaced the "divide by 2" in Q0 with a factor of 0.55 because empirically that seemed to give better results. That was always a bit fishy, and just a few weeks ago I discovered a better approach which is mathematically justified: replace the Moon correction by the difference between the Moon correction and the Sun/star correction. And since these have opposite signs, this is equivalent to the sum of the absolute values of the two altitude corrections. It's fast and easy, in keeping with the original philosophy of Easy Lunars. In proper notation, use:
      Q0 = (dhm - dhs)2 / 2.
    This version of the quadratic correction significantly improves the range of validity of the many series approaches to clearing lunars, both historical and modern. Does it matter? Well, if you get a kick out of clearing lunars using a basic calculator or even using a slide rule, then yes, this will give better results.  

    But I think there's another reason to experiment with this trick. There was a bit of a "trade war" among publishers of tables in the early 19th century extending and broadening the validity of the lunars corrections tables. By normal methods, this could require a lot of work. But the approach here is simple and easily implemented. This is just the sort of trick that might have been discovered by a 19th century calculator of tables and may well have been used by some of those authors. I don't have any actual evidence of that yet; I'm just putting it out as a speculative possibility. Did any of them use it? 

    For reference, here's the complete quadratic correction (the long way around):
      Q = (1/2)(dhm2 · Sm2 / tan LD - 2dhmdh· SmSs / sin LD + dhs2 · Ss2 / tan LD) / 3438,
    where Sx = sqrt(1 - CCx2) which is equal to the sine of the corner angle at either the Moon or Sun/star. The mathematical justification for the short form above can be found by considering the case of short lunars, when LD<20° or so and when the altitudes are rather low in the sky. This is exactly the range of geometries where the usual single-term quadratic correction becomes inaccurate. Take the complete quadratic correction, apply the approximation under those circumstances, sin LD = tan LD (nearly) and also notice that in these cases the corner angles add up to nearly 180° which implies Sm =Ss (nearly). The result then follows easily.

    Practical example: suppose the altitude correction for the Moon is 50', that for the Sun is -4', the LD is 15°, and the corner cosine at the Moon is 0.3819. The standard, single-term quadratic correction would be 1.16' (squaring 50'). The improved correction (squaring 54') yields 1.35'. It's an easy way to cancel out an error of 0.2 minutes of arc.

    Frank Reed

       
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