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    Re: Lunar Distance Puzzle
    From: Dave Walden
    Date: 2011 Aug 17, 10:02 -0700

    The attached .pdf file contains 6 plotting sheets, numbered 1-6 in the upper right corner. In each, the true postion, 38.89N, 77.17W is circled. The square is the current AP. The three lines are the lunar distance LOP's based on the current AP. The cross is the fix based on the 3 LD LOP's. Notes in the plot give: the time, the "cocked hat size" (the standard deviation of the estimated position based on the Nautical Almanac method as described very nicely by Andres Ruiz), the distance from the current AP to the Fix, a variable used to search for an optimum solution which is the root sum squares of the cocked hat size and the distance AP to Fix, and finally, the distance from the fix to the true.

    Plot 1 shows the LOP's and fix for an AP at the true location and the time at the true time. Note, the cocked hat size is very small, but the Fix to True is off by 3.4 nm. To some that may seem pretty good, but I wondered why so large. It's almost entirely due to the assumption that the Moon's GHA varies linearly between 0300 and 0400.

    Plot 2 shows the case where one end of the linear interpolation for the Moon is anchored at 0345, the true time. Now, the cocked hat is even smaller and the Fix to True is on 0.24 miles. What is the source of the 0.24? The example was generated using the USNO NOVAS Fortran version 3.1 ,which uses vector astronomy (I used no polar motion, no UTC1 correction) and the JPL DE405 ephemeris. The soloution was generated using spherical astronomy and the classical models for parallax in altitude and parallax in azimuth, and 2 term expansions for geocentric latitude and rho. Close enough!

    Plot 3 (back to linear interpolation between 0300 and 0400) the result for time = 0345 but 0.2 degree offsets in the AP from the true. This is the case studied some time ago on NAVLIST. If you know the time, and have two LD's, you can calculate postion. Any pair of lines would have worked. (small cocked hat shows you that)

    Plot 4 leaves the AP at the true location but the time is advanced the .01 hr (36sec). The fix has moved, but in addition note that the cocked has has grown by an order of magnitude to .4 nm.

    Plot 5 shows the result for the initial estimate of 39N, 77W and 0330. One can now see the even larger cocked hat.

    Plot 6 show the result of starting at 5 and running the MS Excel Solver to minimize the Root Sum Sq by varying time, lat, and long. The solver has NO knowledge of the true position; there is no distance from true anywhere in the calculation, it's only on the plot. The result is right back to the answer we had for Plot 1. It works!
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