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    Re: 1901 May, 22 Lunar example by French Navy Captain Arago
    From: Paul Hirose
    Date: 2010 Jan 30, 15:10 -0800

    antoine.m.couette@club-internet.fr wrote:
    > The Lunar Distance example happened on May 22, 1901 in the evening . See 
    > enclosure pages 1 to first part of page 7
    
    The French gives me some trouble, but I believe "les hauteurs vraies"
    are the unrefracted altitudes. On that basis, UT1 = 22:20:34 (10:29:55
    mean astronomical time at Paris), lat = +47°49'33", lon = -79°37'23".
    
    Arago's time nomenclature is unclear to me. Time scale A is a complete
    mystery. I suspect M is chronometer time, and Tmp the mean time at
    Paris. The PDF document says Tmp = 10:29:47, which is only 8 seconds
    from my estimate.
    
    Here's the output of my program. The initial altitude intercepts exceed
    1°, but after only 4 iterations the solution converges to .1".
    
    Initial conditions.
    
    estimated time:
    1901-05-22T22:30:00.00 UT1
    1901-05-22T22:29:59.09 Terrestrial Time
    -0.907 seconds delta T
    
    estimated position:
    +48°23'00.0" - 79°25'00.0" north lat, east lon
                  - 79°24'46.4" ephemeris east lon
    0 meters above ellipsoid
    
    atmosphere:
    26° C (79° F) at observer
    1019.9 mb (30.12" Hg) altimeter setting
    1019.9 mb (30.12" Hg) actual pressure
    
    Moon altitude observation:
      54°58'23.9" observed center altitude
          0'38.9" refraction
      54°57'45.0" unrefracted altitude of center
      53°49'43.7" predicted altitude
       1°08'01.3" intercept
    202°06'39.5" predicted azimuth
    
    Sun altitude observation:
      23°59'34.9" observed center altitude
          2'04.9" refraction
      23°57'30.0" unrefracted altitude of center
      22°18'15.3" predicted altitude
       1°39'14.7" intercept
    276°02'15.8" predicted azimuth
    
    Moon to Sun predicted separation angle:
       62°46'14.1" center to center, unrefracted
           1'45.0" refraction
       62°44'29.1" center to center, refracted
          15'51.0" Moon near limb refracted semidiameter
          15'46.5" target near limb refracted semidiameter
       62°12'51.6" Moon near limb to Sun near limb
       62°09'26.0" observed angle
    -  0°03'25.6" observed - predicted
    
    separation angle rate of change:
    +21.90" per minute (topocentric)
    82% of total angular velocity
    
    --------------------
    
    Solution, after 4 iterations.
    
    corrected time:
    1901-05-22T22:20:33.83 UT1
    1901-05-22T22:20:32.93 Terrestrial Time
    -0.907 seconds delta T
    
    corrected position:
    +47°49'33.2" - 79°37'22.8" north lat, east lon
                  - 80°36'17.6" ephemeris east lon
    76° LOP crossing angle
    
    geocentric coordinates (true equator and equinox):
      8h19m03.74s +14°28'19.1"  Moon RA and dec.
    15'38.5" apparent semidiameter
      3h55m49.26s +20°23'15.4"  Sun RA and dec.
    15'47.6" semidiameter
    
    geocentric separation angle and rate:
       62°40'47.3" center to center
    +30.80" per minute
    87% of total angular velocity
    
    illumination conditions:
    274.4° 24.0° Sun unrefracted az, el
    273° Moon to Sun position angle (0 = 12 o'clock)
    117° Moon phase angle (0 = full, 180 = new)
    
    position angles:
    273° Moon to Sun
    39° Sun to Moon
    
    recommended limbs:
    Use Moon upper limb.
    Use Moon near limb.
    
    Moon altitude observation:
      54°58'23.9" observed center altitude
          0'38.9" refraction
      54°57'45.0" unrefracted altitude of center
      54°57'45.0" predicted altitude
       0°00'00.0" intercept
    198°19'29.2" predicted azimuth
    
    Sun altitude observation:
      23°59'34.9" observed center altitude
          2'04.9" refraction
      23°57'30.0" unrefracted altitude of center
      23°57'30.0" predicted altitude
       0°00'00.0" intercept
    274°25'51.6" predicted azimuth
    
    Moon to Sun predicted separation angle:
       62°42'43.5" center to center, unrefracted
           1'39.6" refraction
       62°41'03.9" center to center, refracted
          15'51.3" Moon near limb refracted semidiameter
          15'46.6" target near limb refracted semidiameter
       62°09'26.0" Moon near limb to Sun near limb
       62°09'26.0" observed angle
    -  0°00'00.0" observed - predicted
    
    separation angle rate of change:
    +21.69" per minute (topocentric)
    82% of total angular velocity
    
    
    > The Lunar Occultation example happened on Mar 02, 1901 - NOT the same 
    > date as for the Lunar Distance - and you can find all relevant data from 
    > the middle of page all the way down to the end.
    
    I compute an occultation time 18 seconds different: 22:43:38.4 UT vs.
    22:43:56.38 (Arago). At his time, my program says the star is -6.4"
    inside the Moon's limb.
    
    My program iteratively seeks the time, latitude, and longitude where all
    three angles (separation angle and altitudes, the latter two perhaps
    calculated) match the inputs. But in an occultation, the LOPs for the
    Moon and star cross at a very small angle, so any error in altitude,
    even roundoff error, has a large effect on position. Thus the latitude
    and longitude move away from the correct values.
    
    For that reason, the program's automatic solution is inaccurate for this
    type of problem. I simply used its predicted separation angle and rate
    of change to manually solve for time. Re-running the program with the
    adjusted time gave this:
    
    estimated time:
    1901-03-02T22:43:38.40 UT1
    1901-03-02T22:43:37.17 Terrestrial Time
    -1.235 seconds delta T
    
    estimated position:
    +48°23'30.0" -  4°29'31.2" north lat, east lon
                  -  4°29'12.6" ephemeris east lon
    0 meters above ellipsoid
    
    atmosphere:
    15° C (59° F) at observer
    1013.3 mb (29.92" Hg) altimeter setting
    1013.3 mb (29.92" Hg) actual pressure
    
    Moon altitude observation:
      52°51'07.7" observed center altitude
          0'43.4" refraction
      52°50'24.3" unrefracted altitude of center
      52°50'24.3" predicted altitude
    - 0°00'00.0" intercept
    181°39'12.9" predicted azimuth
    
    kappa Cancri altitude observation:
      52°40'34.6" observed center altitude
          0'43.7" refraction
      52°39'50.9" unrefracted altitude of center
      52°39'50.9" predicted altitude
       0°00'00.0" intercept
    181°20'46.3" predicted azimuth
    
    Moon to kappa Cancri predicted separation angle:
        0°15'21.9" center to center, unrefracted
           0'00.3" refraction
        0°15'21.6" center to center, refracted
          15'21.6" Moon near limb refracted semidiameter
    -  0°00'00.0" Moon near limb to kappa Cancri center
        0°00'00.0" observed angle
        0°00'00.0" observed - predicted
    
    separation angle rate of change:
    -21.31" per minute (topocentric)
    94% of total angular velocity
    
    geocentric coordinates (true equator and equinox):
      9h01m43.73s +11°47'53.5"  Moon RA and dec.
    15'10.0" apparent semidiameter
      9h02m25.79s +11°03'44.2"  kappa Cancri RA and dec.
    0'00.0" semidiameter
    
    geocentric separation angle and rate:
        0°45'20.5" center to center
    -15.87" per minute
    51% of total angular velocity
    
    illumination conditions:
    322.5° -43.0° Sun unrefracted az, el
    275° Moon to Sun position angle (0 = 12 o'clock)
    28° Moon phase angle (0 = full, 180 = new)
    
    position angles:
    133° Moon to kappa Cancri
    314° kappa Cancri to Moon
    
    recommended limbs:
    Use Moon upper limb.
    Use Moon far limb.
    
    
    The altitude intercepts are perfect because I used the altitudes
    computed by the program as the "observations".
    
    Star position, proper motion, and parallax from the Hipparcos catalog.
    Delta T from the USNO MICA program. Sun and Moon position from the JPL
    DE406 ephemeris. Precession and nutation with the IAU 2006 model.
    
    Delta T was *negative* in 1901. That unusual circumstance revealed a bug
    in my program, which in turn was due to a bug in my positional astronomy
    DLL. It's in a feature which isn't used much, and fortunately there's an
    easy workaround. I'll update the Web site with a description of the
    workaround, and issue a fix soon.
    
    -- 
    
    
    
    
    

       
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