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    Refraction
    From: Dave Walden
    Date: 2008 Jul 3, 16:18 -0700
    Refraction
    Attached is a small spreadsheet that calculates the K coefficent for the horizontal mulitlayer atmosphere and the A and B coefficents for spherical layer atmosphere.  (Errors may remain.  Please comment.)  They are functions of pressure, temperature and wavelength.  Two sets of conditions can be caluclated for easy comparisons.  Those included are for the Nautical Almanac conditions and a more conventional set to conditions: 760mb, 0 deg C, lamda .54 microns.   The K =60.4 sec, should look familiar.  The Naut Alm STP yields 58.2, also familiar.  Neither is right.  They're just different.  Starting with either and adjusting to conditions for a specific observation will yield the same answer.
    The formulas can be found in Spherical Astronomy by Green.  This is the "successor" to Smart's work.  Green was coauthor with Smart on later editions and finally wrote his own.  (To my way of thinking, it's better.)
    Also attached is a plot of errors of Bennett and simpler formulas (minutes refrac vs degrees alt).  There is indeed, nothing magic about Bennett.  It get 34 min at 0 deg in case you've forgotten, but the added complication to do so only makes things worse at higher altitudes.  (the curve labeled 1term is the planar atmosphere model with just K. It's better than Bennet at all angles above 13 degrees.  The hoen-walden is a LaPlace formula with A and B. Its better at all angles above 7 degrees.  (The 1 and 2 term formulas blow up at zero, so just don't use them for very small angles.)

    The link below is supposed to take you to page 41 of Modern Astrometry by Jean Kovalevsky which shows graphically the effect of wavelength on refraction
     
    http://books.google.com/books?id=s4azHlUeIYgC&pg=PA40&lpg=PA40&dq=%22+laplace+formula%22+refraction&source=web&ots=4XjmkRNN2W&sig=AAwTVxxyNkTwP_Rm6ZVyJGbq7oQ&hl=en&sa=X&oi=book_result&resnum=4&ct=result#PPA41,M1

    ***********************************************************************************
    Kent, you wrote:
    "The factor 60,35” has been provided to me from the Observatory in Stockholm
    to be used with the true zenith altitudes for altitudes above approx. 10
    degrees. The factor is taken from modern English litterature."
    What color light? It's not often mentioned explicitly that this factor
    applies only for one specific frequency in the visible spectrum. That's one
    reason why two people can quote factors differing by an arcsecond and still
    be technically correct. It's also why we shouldn't worry too much about the
    tenths and hundredths of an arcsecond. And it's a big reason why we
    shouldn't shoot lunars when the objects are below about ten degrees: the
    stars and the limbs of the Sun and Moon are stretched out into colorful
    bands at very low altitudes. Incidentally, I use 58" usually.
    And:
    "Yes, this is correct. For altitudes below 20 degrees I use the formula
    defined by Smart."
    Since you're modeling historical methods, you could just input the whole
    refraction table as printed. Then the software would interpolate between
    values much as a 19th century navigator would have done on paper. This
    approach bypasses all this chatter about various formulae for different
    altitudes. It's also a reasonable approach for a modern analysis. Sometimes
    people doing these calculations obsess over the "Bennett" formula. This is
    because we carry around "baggage" from the calculator era and the early
    years of programming small computers. There's no real advantage to a short
    formula today (except saving a few keystrokes).
     


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