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    Re: Multi-Moon line exercise in 2 parts
    From: Peter Hakel
    Date: 2009 Aug 9, 09:09 -0700

    I ran your on-transit data through my rapid-fire fix program (see attached input data file) and obtained results that agree very well with those from my earlier posting (calculated by a least-squares parabolic fit).  This way I obtain:

    1900L fix:
    Latitude:     N    21 degrees  47.3'   :  standard deviation =   1.6'
    Longitude:  E  130  degrees  04.5'  :  standard deviation = 44.6'

    From looking at the azimuths (recorded to 0.1 degrees) the meridian crossing (viewed by the observer) occurred at UT=9:51:19.  The moon azimuth was changing at the rate of 0.1 degrees per 14 seconds, so this could serve as a rough error estimate, say +/- 7s.  Most likely this could be refined by taking more data points around LAM and do some kind of fitting and/or interpolation.

    My standard deviation in longitude is much larger than Antoine's so his procedure must be performing a more sophisticated statistical analysis than my simple program does.  Perhaps I am overestimating it, considering that my mean longitude value is very close to Antoine's.  There may be a rigorous way of narrowing its variance down, instead of just using the standard formula. On the other hand, my "sigma" values are consistent with the notion that meridian transits are good for calculating latitude and less good for measuring longitude.  I remember Frank mentioning in the past the notion of the "error ellipse" where you would have a smaller error along the azimuth line and a larger one along the direction perpendicular to it.  Something to think about, perhaps other list members can help me better understand this point and resolve the apparent contradiction.

    Also, while in this case I do not see any outliers in the data (all intercepts are TOWARD and generally increasing with time), there is a curious jump in value from 7.5 at 9:48:15, to 9.3 at 9:48:51.  I wonder what happened there, although it may be nothing important.

    Peter Hakel

    From: P H <pmh099@yahoo.com>
    To: NavList@fer3.com
    Sent: Friday, August 7, 2009 8:54:17 PM
    Subject: [NavList 9410] Re: Multi-Moon line exercise in 2 parts

    Backtracking the coordinates given below to UT = 10:00:00 = 1900L gives:

    N  21 degrees 47.2'
    E 130 degrees 04.9'

    The method I chose here does not include the calculation of uncertainties.  If I can find the time, I might redo this problem using the "rapid-fire fix"-type method described in my previous post.

    Peter Hakel

    From: P H <pmh099@yahoo.com>
    To: NavList@fer3.com
    Sent: Wednesday, August 5, 2009 10:01:17 AM
    Subject: [NavList 9371] Re: Multi-Moon line exercise in 2 parts


    I processed your meridional example.  From the attached spreadsheet I get the UT of transit as 9:51:33 (cell F13). The altitude 55.447 degrees (decimal, in cell F1) contains corrections for vessel motion and declination change, and pertains to the instant of your last measurement at 10:01:35.  Correcting this for index error, semidiameter, parallax, and refraction (using standard conditions) gives Ho = 55 degrees 32.2'.  From this I obtain the latitude and longitude at UT = 10:01:35 as:

    N   21 degrees 46.9'
    E 130 degrees 04.8'

    This longitude value is dead-reckoned by 10m 02s from the one extracted at the UT of transit (360 - GHA).

    This is not in the format that you asked but I'm hoping we can still compare.  I will look at this more later.

    As for accuracy, I would certainly round the result to whole minutes of arc.

    Peter Hakel

    From: "Anabasis75@aol.com" <Anabasis75@aol.com>
    To: NavList@fer3.com
    Sent: Tuesday, August 4, 2009 8:33:54 PM
    Subject: [NavList 9359] Multi-Moon line exercise in 2 parts

    I figured that I need to give some LOP data to reduce with all of these people building and using their Bygrave's slide rules, and to stir the pot a bit on the list.  So in order to kill two birds with one stone, I offer the attached Excel spreadsheet.  Lying therein are a plethora of upper limb moon LOP's which should allow eager navigators to fix the position of my ship.  There are a few features of particular note that should appeal to certain members of the list.
    1)  The given DR is a real DR tracked from the AM star fix some 14 hours earlier
    2) The moon is fairly close to the celestial equator (please note the LARGE hourly change in declination)
    3) The ship's track is at a fair clip and nearly Southerly.
    These should be considered fairly extreme conditions to work out the Latitude and Longitude of the ship; the first position from many LOP's away from upper transit, and then again around the time of transit.
    I will be most appreciative if people who solve the transit portion graphically can post an image or file of their curve and to see what the  1900L fixes is. (ZD -9)
    I offer the following challenges:
    1) what would you calculate the local zone time of transit to be?
    2) what was the 1800L fix?
    3) what was the 1900L fix?
    4) what was the actual local zone time of transit by observation?
    5) most critically, how much faith would you put in these fixes?
    I will post the actual GPS fixes and my celestial solutions for both times at a later time.
    Have fun!

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