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    Re: More about lunar observations
    From: Bill Noyce
    Date: 2002 Mar 21, 15:29 -0500

    I worked through Chuck's lunar observations, and my
    results lead to an even later time than Chuck's did.
    My calculations say the time of Chuck's middle observation
    (lunar dist = 37d 43.2) was at 00:20:30 UT, rather than
    at 00:15:27.  I doubt if this is all observational error;
    I'm pretty sure I have a lingering error in here.\
            -- Bill
    
    Here's the work.
    
    Almanac data:
                            Venus                   Moon
                      '0h' GHA    dec       '0h' GHA     dec
    19-Mar 00:00 UT   163d 28.7 +4d 26.2    125d 58.0 +15d 55.3
    19-Mar 08:00 UT                         122d 14.7 +17d 13.3
    20-Mar 00:00 UT   163d 19.8 +4d 56.7
                      ------------------    -------------------
    change in...     (24h) -8.9    +30.5  (8h) -223.3     +78.0
    change in 1 hour       -0.4     +1.3        -27.9      +9.7
    19-Mar 01:00 UT   163d 28.3 +4d 27.5    125d 30.1 +16d 05.0
    
    The '0h' GHA ignores the 15 degrees/hour of the earth's
    rotation, to make the numbers smaller for interpolation.
    It's easy enough to add in separately, but it's not needed
    for this situation, since all we care about is the difference
    in GHA for Venus and the Moon.  These results agree reasonably
    well with Chuck's -- the 01:00 moon position is off by a
    couple of tenths of a minute, probably because I had to
    interpolate where the moon's motion is changing rapidly.
    
    I computed the predicted distances using Ageton's method.
    
    First, at 00:00
        163d 28.7
      - 125d 58.0
      -----------
    LHA  37d 30.7 ->                    log csc .21544
    dec  +4d 26.2 -> log csc 1.11150    log sec .00130
                                    add --------------
     R   37d 22.8    log sec  .09984 <- log csc .21674
                 sub ---------------
     K   +5d 35.2 <- log csc 1.01166
    Lat +15d 55.3
        ---------
    dif -10d 20.1 -> log sec  .00710
       add log sec R ---------------
     Hc  51d 25.2 <- log csc  .10694
    dist = 90d - 51d 25.2 = 38d 34.8
    
    Similarly at 01:00
        163d 28.3
      - 125d 30.1
      -----------
    LHA  37d 58.2 ->                    log csc .21096
    dec  +4d 27.5 -> log csc 1.10939    log sec .00131
                                    add --------------
     R   37d 50.1    log sec  .10251 <- log csc .21227
                 sub ---------------
     K   +5d 38.9 <- log csc 1.00688
    Lat +16d  5.0
        ---------
    dif -10d 26.1 -> log sec  .00724
       add log sec R ---------------
    Hc   50d 57.5 <- log sec  .10975
    dist = 90d - 50d 57.5 = 39d 02.5
    
    Expressing Chuck's predicted distances in minutes,
    he had D1 (00:00) = 38d 34.82, D2 (01:00) = 39d 02.30
    and I got           38d 34.8,               39d 02.5
    
    Close enough here (though I had to correct some blunders
    to get this far).  I noticed that a good check on this
    manual method would be to reverse the roles of the moon
    and the other body, and compute the distance again.  Most
    of the intermediate results would be different, but of
    course the final result ought to match.  I didn't go
    through this exercise this time, though.
    
    Now for the sights.  I didn't have Chuck's Height of Eye,
    so I ignored dip.  The 3' difference in altitudes isn't
    going to lead to a significant change in Refraction or
    Parallax, as far as I can see.  The first step is to
    average the sights to bring them to a common time.
    For the distances, that's easy:
    
    00:14:59  37d 43.4
    00:15:27  37d 43.2
    00:16:01  37d 43.2
    ------------------
    00:15:29  37d 43.2  average distance
    
    Next we want the altitude of each body at 00:15:19.
    I averaged the two early sights of each body, and averaged
    the two late sights of each body, then used linear
    interpolation between them to find a 00:15:29 altitude.
    This appears to be different from what Chuck did when he
    "roughly averaged the sights".
    
          Venus                 Moon
    00:11:41  7d 50.6    00:12:48  44d 52.6
    00:14:02  7d 25.0    00:14:02  45d 25.6
    -----------------    ------------------
    00:12:53  7d 37.8    00:13:25  44d 39.1
    
    00:16:36  6d 43.0    00:17:31  43d 51.6
    00:18:09  6d 24.0    00:19:09  43d 30.0
    -----------------    ------------------
    00:17:22  6d 33.5    00:18:20  43d 40.8
    
    269 seconds -64.3    295 seconds  -58.3
    146 seconds -34.9    124 seconds  -24.5
    
    00:15:29  7d 02.9    00:15:29  44d 14.6
    
    These are the Hs of each body at the time we
    measured the lunar distance.  We need to correct
    the moon's altitude and the lunar distance for
    semidiameter (and correct both altitudes for dip,
    which I'm ignoring) to get Ha.  Then correct both
    altitudes for refraction and the moon's for
    parallax to get Ho.
    
    My tabulated SD and HP are for the middle of each day:
    12:00 UT 18-Mar  SD=14.9  HP=54.8
    12:00 UT 19-Mar  SD=15.1  HP=55.3
    
    Interpolating to 00:15 19-Mar, I get SD=15.0 and
    HP=55.1.  Because the Moon's altitude is between
    30 and 55 degrees, we can add 0.2' augmentation to
    SD.  Looking back at my work, I see I used 15.3'
    for SD - I think I just used the tabulated SD for
    the day, plus augmentation.
    
    Venus Hs 7d 02.9  Moon Hs 44d 14.6  distance 37d 43.2
                         SD      +15.3     SD       +15.3
          s  7d 02.9       m  44d 29.9        d  37d 58.5
    Refraction  -7.3  Refraction  -1.0
                      Parallax   +38.7
          S  6d 53.6       M  45d 07.6
    
    Looking back over the numbers, it seems that Parallax
    should have been closer to 39.2 -- not sure where 38.7
    came from.
    
    Now we can plug the above numbers into Borda's method.
    (Thanks to George Huxtable for writing it up.)  It can
    use the same log sec & log csc tables as Ageton's method.
    
     m  44d 29.9 -> log sec .14675
     s   7d 02.9 -> log sec .00330
     d  37d 58.5    --------------
    ------------        sum .15005
    sum 89d 33.7
    /2  44d 46.85-> log sec .14906
    -d   6d 48.35-> log sec .00306
     M  45d 07.6 -> log sec .15148
     S   6d 53.6 -> log sec .00318
    ------------    --------------
    M+S 52d 05.6        sum .30678
    /2  26d 02.8          - .15005
                    --------------
                       diff .15673
     A  33d 23.6 <-     /2  .078365
    ------------
    sum 59d 26.4 -> log csc .06495
    diff 7d 20.8 -> log csc .89323
                    --------------
                        sum .95818
    D/2 19d 22.85 <-    /2  .47909
     D  38d 45.7
    
    Converting Chuck's cleared distance to minutes,
    he got 38d 43.49 -- 1.2 minutes less than mine.
    If I had used the larger value for Parallax,
    my cleared distance would be even larger.  This
    difference looks larger than random noise, and I
    assume it's larger than Letcher's method would
    introduce, so probably I have made a mistake.
    
    But, pressing on, we have
    
    Predicted 00:00  38d 34.8  Predicted 00:00  38d 34.8
    Predicted 01:00  39d 06.7     Observed      38d 45.7
    -------------------------  -------------------------
    change in 1 hour    +31.9    observed change   +10.9
    
    10.9*60/31.9 = 20.5 minutes
    
    In other words, the observation occurred at 00:20:30 UT.
    
    Thus, my calculations say Chuck's clock is exactly 5
    minutes slow -- even worse than his own conclusion.
    
    
    

       
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