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    Re: Lunar Distance Puzzle
    From: Peter Hakel
    Date: 2011 Aug 15, 17:21 -0700
    I put together another "brute-force" method (Q-squared minimization by exhaustive search, see attached). The geometry is done in Cartesian coordinates which automatically handles Moon's parallax. Per Frank's suggestion, there is no refraction in this model (one could add it accompanied by computed altitudes, do the clearing and iterate). After some trial and error with the searched area boundaries I found a patch of parameter space in which the point of minimum Q-squared is located inside of it, rather than at its edge.

    Time range: 3 - 4 hours, increment 1 second
    Latitude range:  N 34.8 - 35.0 degrees, increment 0.1' = 6"
    Longitude range: W 74.5 - 75.0 degrees, increment 0.1' = 6"

    The program prints:

     Time:    3.46722223353572    
               3          28   2.00004072859883    
     
     Latitude:    34.8683325720485    
              34          52   5.99725937470794    
     
     Longitude:   -74.9216666647699    
             -74         -55  -17.9999931715429    
     
     Q2:   2.655337691841410E-010
    (square radians)

    Thus I get:
    Time = 03h 28m 02s
    Lat  = N 34d 52.1'
    Lon  = W 74d 55.3'

    So there still is a solution but it's noticeably away from those found with presented models including refraction. I wonder if this is one of those somewhat ill-posed optimization problems with a very flat Q-squared function, i.e. you can get many quite different possible solutions, all with very "acceptably-looking" (i.e. "small" or "near-zero") Q-squared values. If that is the case, then the "best" solution could change significantly (and hence be very sensitive) to errors in input data, the refraction model, or other details.


    Peter Hakel


    From: Frank Reed <FrankReed@HistoricalAtlas.com>
    To: NavList@fer3.com
    Sent: Sunday, August 14, 2011 6:36 PM
    Subject: [NavList] Re: Lunar Distance Puzzle

    Dave W., you wrote:
    "It is, as was pointed out, sensitive to refraction."
    I spent a little time trying to think this through, and I just couldn't see how it would work. I made myself a little visual model of a satellite on a known trajectory with three stars in the distance, and I could use it to get a position fix at known time or to get time given some position data (or altitudes and HP data as surrogate). But I couldn't see how you could position and time, too, as you proposed. The distance of the third star would merely be consistent with the other two. Then I tried a brute force approach generating "lunar distance lines of position" and they seemed to reach a level of closest convergence around 03:27 but there was very little change around that time. This seemed like an accidental convergence (and the LOPs did not meet presumably because I assumed it was a warm night with lower refraction). Now I see that there have been a number of comments about the sensitivity to refraction, so I have a couple of questions for you. If you set your model refraction to zero, does the solution disappear? Or in other words, is it only the variability of refraction with altitude that makes this work? Related to this, if the Moon and all three stars have altitudes above 45 degrees, is the solution worse (much more sensitive)?
    -FER
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