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    Difficult lunars from 1855
    From: Paul Hirose
    Date: 2016 Sep 13, 19:40 -0700

    The American Ephemeris and Nautical Almanac for the Year 1855 contains
    Tables for Correcting Lunar Distances by William Chauvenet. He provides
    an explanation and two examples.
    
    https://books.google.com/books?id=_yUOAAAAQAAJ&pg=RA1-PA20&focus=viewport
    
    Compare his Sun lunar solution with modern solutions:
    
    08:09:01  Chauvenet
    08:11:46  Antoine Couëtte
    08:08:30  me (full solution)
    08:11:55  me (lunar time sight)
    
    Those times are not all directly comparable. Chauvenet and I assume the
    position is approximate. The altitudes of both bodies affect the
    solution. Antoine's computation assumes position is known. I call that a
    "lunar time sight." My software allows that also, and its result is on
    the last line.
    
    However, I don't think Chauvenet's lunars are a good test for modern
    software. The main problem is his ephemeris, which differs from the JPL
    DE406 by 18 arc seconds. Therefore, I have re-computed both examples
    with JPL HORIZONS, USNO MICA, the Bennett refraction formula, the
    Nautical Almanac dip formula, and Chauvenet's own formula for refracted
    semidiameter. Those formulas are different from my software in order to
    provide an independent check.
    
    Sun lunar: 1855 September 7 08:09:01 UT1. Delta T = 7.5 seconds.
    Position N 35.5 W 030.0. Height of eye 20 feet (6.1 m), temperature 75 F
    (23.9 C), and station pressure 29.1 inches Hg (985.4 mb). Lower limb
    altitudes above the sea horizon: Sun 5.78908°, Moon 49.90027°. Lunar
    distance, near limb to near limb: 43.87438°.
    
    The solution from my lunar distance program is 08:09:10 UT1, 9 seconds
    after the correct time.
    
    Fomalhaut lunar: 1855 Aug 30 05:42:02 UT1 (delta T = 7.3). Position S
    55°20′ W 120°25′. (NOTE SOUTH LATITUDE) Height of eye 18 feet (5.5
    meters), station pressure 31 inches Hg (1049.8 mb), temperature 20 F
    (-6.7 C). Moon lower limb 6.58649 above sea horizon. Fomalhaut altitude
    52.71724. Moon far limb to Fomalhaut 46.50609.
    
    The solution from my program is 05:41:51, 11 seconds before the correct
    time.
    
    I'm satisfied with that accuracy. Altitude of the lower body in each
    case is about 6 degrees, so the solution is exceedingly sensitive to
    refraction. To create the synthetic observations I was careful to use a
    refraction formula different from my program, so a discrepancy is
    unavoidable.
    
    Of course in the real world you wouldn't shoot lunars at such low
    altitude. I suspect Chauvenet created those examples to make his method
    look good, and simpler solutions (specifically Bowditch) look bad. Even
    the atmosphere conditions are obviously non-standard: high temperature
    and low barometer in one example, opposite conditions in the other.
    That's why I call the lunars "difficult."
    
    More details on how I constructed the synthetic observations is on my
    page of lunar software tests.
    
    http://home.earthlink.net/~s543t-24dst/lunar3/tests.html
    
    

       
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