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    Re: Captain Cook's Sep 07th, 1773 Lunar revisited
    From: Paul Hirose
    Date: 2012 Jul 20, 16:47 -0700

    Antoine Couëtte wrote:
    > From Ulietea Harbour (S 16°45'33" W151°29'48") the Moon Upper Limb Height 
    was 43°29'0 at time UT = 17h02m55s0 and the Moon was in Azimuth 312°9. I am 
    computing this way with the Moon first since most likely the Moon Height is 
    the most reliable one for refraction correction and also for the reasons 
    given in my previous post on this subject.
    
    I also computed a time sight with the Moon, at the same assumed position 
    (S16°45.55' W151°29.80' +5 meters), atmosphere conditions (78° F at 
    observer, altimeter setting 29.92), dip (3' 50"), and ∆T (+16.4 s). 
    Results (UT1):
    
    17:07:19  Couëtte lunar distance
    17:02:55  Couëtte time sight
    17:03:18  Hirose time sight
    
    At my time the observed altitude of the Sun (corrected for semidiameter, 
    refraction, etc.) is 17.1' more than the predicted altitude. The 
    observed lunar distance is .9' less than the predicted distance.
    
    The Astronomical Almanac table in Section K says ∆T = +16 at 1773.0 and 
    also at 1774.0, so +16.4 s is reasonable.
    
    My computation confirms that Antoine's assumed position, the lunar 
    distance, and the altitudes cannot all be correct:
    
    -00.004' observed - predicted lunar distance
    +41.95'  Moon altitude observed - predicted
    -39.69'  Sun altitude observed - predicted
    
    The above data are based on his assumed position and the time he 
    computed from the lunar distance (with extreme accuracy). For the Sun 
    and Moon observed altitudes I used the angles he gave, less 3' 50" for dip.
    
    I also solved for time and position:
    
    17:07:18.5 S16°45.55' W151°29.80'  initial values
    17:05:54.8 S16°30.65' W151°53.78'  Hirose solution
    17:06:04.9 S16°22.4'  W151°53.8'   Couëtte "Second Solution"
    
    Kermit's solution makes the observed and computed lunar distances 
    virtually identical. That's good. But the observed Sun altitude is 3.8' 
    less than the computed value, and the Moon altitude 3.9' less.
    
    Not that my solution is much help. The computed angles match the 
    observations to .01', but the position is tens of miles away from the 
    modern coordinates of the anchorage.
    
    
    I have followed Alex's contributions with interest. His idea that 
    perhaps the date was wrong seemed good enough to check out. But I 
    discovered the lunar rate of 12°/day removes ambiguity. Given a lunar 
    distance and approximate date, but no information on the observer's 
    position, you can still compute a date and time accurate to about 2 
    hours. Simply assume the angle is geocentric. The parallax error will be 
    1° at most.
    
    As for his theory that the altitude observations were not simultaneous 
    with the lunars, the Moon and Sun altitude rates were -10.7 and +14.1 
    '/min, respectively, and the observed - computed were +42.0' and -39.7'. 
    This implies the altitudes were shot 3 or 4 minutes before the mid time 
    of the lunars. But I wonder, were the observers that dumb? Isn't it more 
    logical, and just as easy, to shoot 5 lunars, 2 altitudes, then 5 more 
    lunars?
    
    
    Here is the full output from my lunar program, where I start with 
    Kermit's values and let it solve for time and position. Ephemeris is the 
    JPL DE422, their latest long term ephemeris, successor to DE406. IAU 
    2006 precession and 2000A nutation models. GRS80 ellipsoid.
    
    The "observed" altitudes are actually 3' 50" less than the observed 
    angles to correct for dip. My program does not automatically apply that 
    correction.
    
    
    Program Lunar2, by Paul S. Hirose.
    
    Initial conditions.
    
    estimated time:
    1773-09-07 17:07:18.52 UT1
    1773-09-07 17:07:34.88 TT (Gregorian)
    JD 2368884.5 + 0.713598 (TT)
    16.400 seconds delta T
    
    estimated position:
    -16°45.55' -151°29.80' north lat, east lon
    5 meters above ellipsoid
    
    atmosphere:
    25° C (78° F) at observer
    1013.3 mb (29.92" Hg) altimeter setting
    1012.7 mb (29.90" Hg) actual pressure
    
    Moon altitude observation:
      43°25.17' observed upper limb altitude
          0.97' refraction
         14.93' unrefracted semidiameter
      43°09.26' unrefracted altitude of center
      42°27.31' predicted altitude
       0°41.95' intercept
    311°56.18' predicted azimuth
    
    Sun altitude observation:
      12°34.57' observed lower limb altitude
          4.02' refraction
         15.89' unrefracted semidiameter
      12°46.43' unrefracted altitude of center
      13°26.13' predicted altitude
    - 0°39.69' intercept
      79°35.88' predicted azimuth
    
    Moon to Sun predicted separation angle:
      106°21.42' center to center, unrefracted
           3.58' refraction
      106°17.84' center to center, refracted
          14.93' Moon near limb refracted semidiameter
          15.84' target near limb refracted semidiameter
      105°47.07' Moon near limb to Sun near limb
      105°47.07' observed angle
    -  0°00.004' observed - predicted
    
    unrefacted separation angle rate:
    -16" per minute (topocentric)
    77% of total angular velocity
    
    --------------------
    
    Solution, after 3 iterations.
    
    corrected time:
    1773-09-07T17:05:54.84 UT1
    1773-09-07T17:06:11.24 Terrestrial Time
    16.400 seconds delta T
    
    corrected position:
    -16°30.65' -151°53.78' north lat, east lon
    53° LOP crossing angle
    
    geocentric coordinates (true equator and equinox):
      3h52m13.08s +15°39'46"  Moon RA and dec.
    14.77' apparent semidiameter
    11h05m49.32s + 5°48'19"  Sun RA and dec.
    15.89' semidiameter
    
      6h06m34.36s  local apparent sidereal time
    
    geocentric separation angle and rate:
      105°58.01' center to center
    -27.1" per minute
    85% of total angular velocity
    
    illumination conditions:
    79.9° 12.8° Sun unrefracted az, el
    306.2° Moon to Sun position angle (0 = 12 o'clock)
    73.5° Moon phase angle (0 = full, 180 = new)
    
    position angles:
    306.2° Moon to Sun
    37.1° Sun to Moon
    
    recommended limbs:
    Use Moon upper limb.
    Use Moon near limb.
    
    Moon altitude observation:
      43°25.17' observed upper limb altitude
          0.97' refraction
         14.93' unrefracted semidiameter
      43°09.26' unrefracted altitude of center
      43°09.26' predicted altitude
       0°00.00' intercept
    312°25.26' predicted azimuth
    
    Sun altitude observation:
      12°34.57' observed lower limb altitude
          4.02' refraction
         15.89' unrefracted semidiameter
      12°46.43' unrefracted altitude of center
      12°46.43' predicted altitude
       0°00.00' intercept
      79°53.91' predicted azimuth
    
    Moon to Sun predicted separation angle:
      106°21.56' center to center, unrefracted
           3.72' refraction
      106°17.84' center to center, refracted
          14.93' Moon near limb refracted semidiameter
          15.84' target near limb refracted semidiameter
      105°47.07' Moon near limb to Sun near limb
      105°47.07' observed angle
        0°00.000' observed - predicted
    
    unrefacted separation angle rate:
    -16" per minute (topocentric)
    77% of total angular velocity
    
    -- 
    I filter out messages with attachments or HTML.
    
    
    
    

       
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