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    Running fix + lunar
    From: Paul Hirose
    Date: 2010 Nov 15, 14:14 -0800

    Observations:
    2010-10-26 14:01:25 from N40 35.1 W039 49.8 Sun alt 35 49.1
    2010-10-26 20:04:48 from N42 09.5 W038 32.9 Jup alt 20 07.6
    2010-10-27 00:36:33 from N43 08.1 W037 10.1 Moon to Jup 86 18.7
    
    Times are UTC. Sun observation is near local noon and corrected for dip,
    refraction, and semidiameter. Jupiter observation is at end of civil
    twilight and corrected for dip and refraction. Azimuth is about 116°.
    Moon (far limb) to Jupiter (center) angle includes refraction. All
    observed angles are accurate, but the times and positions have
    significant errors.
    
    The object is to correct the errors. Since there are three measurements,
    we can solve for only three unknowns. Therefore, assume all observations
    have the same time, latitude, and longitude errors.
    
    Begin with the first observation, the noon sight. At the estimated time
    and position, computed altitude is 36 38.6, which is 49.5' greater than
    the observation. One minute of time later changes computed altitude
    +1.3'. One degree north changes it -59.6'. One degree west changes it -5.8'.
    
    Jupiter computed altitude  = 20 38.4, which is 30.8' greater than the
    observation. One minute later changes computed altitude +10.0'. One
    degree north changes it -26.1'. One degree west changes it -40.2'.
    
    The computed lunar distance = 86 18.6, which is .1' less than the
    observation. One minute later changes the computed lunar distance +.5'.
    One degree north changes it -.7'. One degree west changes it by +.2'.
    
    We can now form an equation for each observation. On the left is the
    required change in the computed angle to match the observation. On the
    right are the unknown corrections in time (∆t), latitude (∆φ), and
    longitude (∆λ) to make that happen. The coefficients are the partial
    derivatives (e.g., change in altitude per minute of time) from the
    paragraphs above.
    
    -49.5 =  1.3∆t - 59.6∆φ -  5.8∆λ
    -30.8 = 10.0∆t - 26.1∆φ - 40.2∆λ
       +.1 =   .5∆t -   .7∆φ +   .2∆λ
    
    Solution:
    ∆t = +1.12 m = 1 m 7 s
    ∆φ = +.804° = N48.2'
    ∆λ = +.522° = W31.3'
    
    Apply those corrections to the time and place of each observation. In
    the table below I show the corrected values in the upper line of each
    pair, as well as the computed altitude or lunar distance. In the lower
    line are the true values for comparison.
    
    2010-10-26 14:02:32 from N41 23.3 W040 21.1 Sun alt 35 49.5
    2010-10-26 14:02:37 from N41 24.0 W040 18.0 Sun alt 35 49.1
    
    2010-10-26 20:05:55 from N42 57.7 W039 04.2 Jup alt 20 07.8
    2010-10-26 20:06:00 from N42 58.2 W039 05.5 Jup alt 20 07.6
    
    2010-10-27 00:37:40 from N43 56.3 W037 41.4 Moon to Jup 86 18.7
    2010-10-27 00:37:45 from N43 56.5 W037 30.0 Moon to Jup 86 18.7
    
    Now the observations match the computed angles within a few tenths of a
    minute. To practical accuracy we're finished, but let's repeat the 
    correction process to try for a perfect match.
    
    At the noon sight, computed altitude is .4' greater than the
    observation. One second later changes computed altitude +.02'. One
    minute north changes it -.99'. One minute west changes it -.09'.
    
    At the Jupiter sight, computed altitude is .2' greater than the
    observation. One second later changes it +.17'. One minute north changes
    it -.43'. One minute west changes it -.66'.
    
    At the lunar distance observation, computed lunar distance exactly
    matches the observation. One minute later changes the computed lunar
    distance +.01'. One minute north changes it -.01'. One minute west
    changes it .00'.
    
    The equations are:
    -.4 = +.02∆t - .99∆φ - .09∆λ
    -.2 = +.17∆t - .43∆φ - .66∆λ
    0.0 = +.01∆t - .01∆φ + .00∆λ
    
    Solution:
    ∆t = +.4 s
    ∆φ = +.4'
    ∆λ = +.1'
    
    Apply corrections:
    
    2010-10-26 14:02:32.4 from N41 23.7 W040 21.2 Sun alt 35 49.1
    2010-10-26 14:02:37.0 from N41 24.0 W040 18.0 Sun alt 35 49.1
    
    2010-10-26 20:05:55.4 from N42 58.1 W039 04.3 Jup alt 20 07.7
    2010-10-26 20:06:00.0 from N42 58.2 W039 05.5 Jup alt 20 07.6
    
    2010-10-27 00:37:40.4 from N43 56.7 W037 41.5 Moon to Jup 86 18.7
    2010-10-27 00:37:45.0 from N43 56.5 W037 30.0 Moon to Jup 86 18.7
    
    The computed angles and the observations are practically identical.
    However, it's clear there's still a substantial error in longitude at 
    the last observation. The main reason is that I had to assume all 
    observations had an identical bias in time, latititude, and longitude. 
    Most the error was of this nature. But on top of that, I injected a 3 NM 
    error in the EP for the second observation and 6 NM error in the third 
    EP (relative to the first observation EP). These are about 3% of the 
    distance run. The directions of the errors were randomly chosen, but by 
    happenstance they were at 86° and 275°, nearly reciprocal directions.
    
    The results would have been better if both altitude observations had 
    occurred at evening twilight, but I wanted to make the problem difficult.
    
    Incidentally, the three points where the observations occurred do not
    "line up" on a chart because there was a speed and course change between
    observations.
    
    I present this mostly as an interesting experiment rather than a 
    practical method for reducing sights. Nevertheless, my lunar distance 
    program uses a similar method to arrive at time and position. It's 
    unable to do what I demonstrated here, though. The program is written 
    for the classic lunar distance observation, where the altitudes and 
    separation angle of the Moon and another body are observed at 
    practically the same time and place.
    
    -- 
    I filter out messages with attachments or HTML.
    
    
    
    
    

       
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