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    Re: Lunar Distance Puzzle
    From: George Brandenburg
    Date: 2011 Aug 21, 08:25 -0700

    Thank you Dave W for posting this. I've been sailing in Maine for the last two weeks so I couldn't reply earlier, but I had my iPhone along and and was fascinated following the thread. And it gave me some good mental exercise during the long hours cruising along.

    (BTW I also had a chance to try a noontime time solar sight with 2-3' seas and discovered how difficult it is in a 30' sloop... I was pleased to able to get 15' accuracy on the measurement and not get seasick!)

    Back to the puzzle I completely agree with Dave that "It REALLY does work!" and was also very impressed that Harri so quickly came up with a solution. Harri summarized the problem very nicely: "...This gives three equations in three unknowns, which can be solved numerically -- of course not always...". Each lunar distance obviously depends on GMT, and the correction to this distance depends on the altitudes of the two bodies used. Since the two altitudes in turn depend on lat, long, and GMT, we end up with one equation in GMT, lat, and long for each of the measured lunar distances. If the three lunar distances are not redundant this should result in three solvable simultaneous equations in three unknowns. (An example of a redundant case where this wouldn't work would be where all three distances were parallel.)

    The primary contribution to the distance correction is lunar parallax, and as Dave showed in a refraction-free world the problem is still solvable. In other words the inverse of the parallax correction is the main reason the distance equation depends on altitudes, hence lat and long, and not just GMT. In the real world the inverses of the refraction corrections are also needed to get an accurate answer. In any case, as I think Frank noted, this method is not particularly accurate for determining lat and long, since relatively small changes in the distance correction correspond to large variations in position.

    Finally as to Frank's comment that the third lunar distance in the problem can't contribute additional information it can only confirm the previous two... With one lunar distance and one equation you have to input values for two of the variables, namely the two altitudes (or lat and long), in order to get GMT. With two distance measurements you can input GMT and solve the two equations for lat and long, i.e. find out where the LOPs intersect.

    It is important to note that with two distance measurements if you input a different GMT the LOP intersection will shift to a different lat and long. So the role of the third lunar distance measurement will basically be to pin down GMT to the correct value. Of course all three equations contribute to finding all three unknown variables, it's not like we can say one of them determines GMT etc. Adding a fourth measurement would not provide any new information, but would over-constrain the problem and allow one to calculate a goodness of fit chisq.

    But alas I am all smoke and no fire since I do not have handy computation tools to solve the actual problem, as Dave and Harri did. But I was hoping for belated half-credit as promised.

    Cheers,
    George B
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