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    Easy Lunars TYPO and an extra 10%
    From: Frank Reed CT
    Date: 2004 May 9, 17:43 EDT
    I'm resending this because it may have been missed amid the "fascinating" discussion a few days ago, and because it's important (correcting a typo, that is).

    In my original "Easy Lunars" post, I made a {at}#%&! typo. The equations for A and B have "sun" and "moon" reversed in the numerator. The correct equations are:

       corrected_LD = observed_LD + dh_Moon*A + dh_Sun*B + Q

       A = (sin(H_sun) - cos(dist)*sin(H_moon))/(cos(H_moon)*sin(dist)),
       B = (sin(H_moon) - cos(dist)*sin(H_sun))/(cos(H_sun)*sin(dist)).

       Q = 1.1 * (1/2) * (dh_Moon)^2 * cot(dist) * (1-A^2) / 3438

    You may notice a small change that I've made in the equation for Q. By straight calculus, the factor out front should be 1/2 exactly. Changing this factor to 0.55 [that's 1.1*(1/2)] makes the calculation accurate across a wider range of cases. As you may recall, this series method begins to show errors for lunar distances below 20 degrees at low altitude. I started looking at those errors a couple of days ago and noticed that they were approximately proportional to Q in the area where there's a problem. So by slightly increasing the value of Q (by 10%), the results are much better. Using 0.55 instead of 0.5 will yield cleared lunars accurate to a tenth of a minute of arc or better for all altitudes down to 15 degrees above the horizon and all lunar distances down to 10 degrees. And that covers all useful lunars. Remarkably this change does not cause any problems elsewhere, since the Q correction is smaller than a minute of arc in more typical cases.

    Hey... wait a minute, you might be saying, I like those cool lunars when the Moon is nuzzled up close to Venus or some bright star just three or four degrees away. It looks great. Can't I use those lunar distances?? Yes, you can, but it can be tricky if you intend to use any parts of the traditional methods of lunars. The problem is the interpolation step at the very end where you compare your cleared lunar with the predicted geocentric lunar distances at fixed intervals of Greenwich time.

    Usually, simple linear interpolation is completely sufficient for lunars. If your cleared lunar is 30% of the way from the lower predicted lunar to the upper predicted lunar, then the implied time is 30% of the way from the lower time to the upper time by linear interpolation. But simple geometry shows that this linear interpolation is an approximation, and for very short lunar distances, you need to go beyond it.

    When linear interpolation doesn't work, we have to work "second differences" to eliminate the error. If x is the difference in ecliptic longitude between the Moon and the other object, y is the distance of the other object off the path of the Moon across the sky, and d is the distance the Moon moves during the time between the tabulated lunars and the observed lunar, then the error in simple linear interpolation will be aproximately:
       error = 0.5*y^2*d^2/x^3.
    The main thing to notice here is that the error is inversely proportional to x cubed. That means that linear interpolation becomes rapidly worse when the Moon is close to the other object. Navigators in the late 18th through 19th century were taught to use "second differences" which is equivalent to quadratic interpolation to eliminate this error. Although this is not all that complicated, it's certainly a nuisance, and it's a good reason to avoid lunars altogether when the distance is below 10 degrees. Interpolation to "second differences" was apparently considered a good "trick question" on navigators' exams as late as 1900. Furthermore, the Nautical Almanac never tabulated lunar distances below about 15 degrees. Those short lunars that look so dramatic in the sky were never used historically.

    The error in linear interpolation is also proportional to the distance that the Moon has moved between tabulated predicted lunars. Another way to reduce the error is to use tabulated lunars that are closer together in time. Traditionally, the almanacs listed lunars for every three hours of Greenwich time. If we calculate them at hourly intervals instead, the error is reduced by a factor of 9 on average. If you want to bypass this whole problem using modern calculation methods, all that's required is a direct calculation of the Moon's and other object's positions for the exact instant when the lunar was recorded (that's what a computer-based lunars calculation should attempt to do).

    Frank E. Reed
    [X] Mystic, Connecticut
    [ ] Chicago, Illinois

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