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    Re: refraction calculation
    From: George Huxtable
    Date: 2010 Apr 3, 13:44 +0100

    May I add to Antoine's words my own pleasure at seeing a contribution to 
    this list from George Bennett, long a familiar name to this list from his 
    simple and close approximation to refraction, to be found in the Almanac 
    and in Meeus.
    I have a comment on his attachment about that approximation, and its 
    development for use backwards. It concerns nothing more than two little 
    In the equation for A, why is the number 7.31 expressed as 7.31' ; and in 
    the equation for r', why is 9.48 expressed as 9.48' ?
    When I intended to try out that equation for r', seeing that the numerator 
    of the second term was expressed in arcminutes, and it would have to be 
    added to a quantity H in degrees, my first thought was to divide that 9.48 
    by 60, to put the two terms on equal terms for adding. That was wrong, I 
    soon realised. That is a numerical expression, in which the input valueas 
    are in degrees, and the result happens to come out in minutes, if we simply 
    treat those constants as numbers. So I suggest that it's misleading to 
    attach that arc-minutes label to the constant 9.48.
    In the same way, it's misleading to label the numerator of the expression 
    for A as 7.31 arc-minutes. A is to be added to h, in degrees. Meeus, in his 
    expression 16.4, simply treats 7.31 as a number, which I suggest is the 
    right thing to do.
    Bennett's simple expressions exploit the accidental fact that the 
    refraction for 45 degrees is close to 1 arc-minute. Accidental, because 
    there's no fundamental reason why degrees should be divided sexagesimally. 
    Bennett's simple expression is about the only thing I can say in favour of 
    sexagesimals: I hate them.
    There are two minor snags that might possibly, but rarely, arise when 
    implementing Bennett's formula for refraction, given as
    R = cot ( h + 7.31 / (h + 4.4�))
    One, mentioned in Meeus, is that R is not quite zero at 90�, which is 
    somewhat unphysical; symmetry says that it must be.
    Another is that some computers / calculators will not produce a cotangent 
    directly, but have to get there via calculating the tangent. And for an 
    angle within the working range, at about 89.92�, its tan becomes infinite, 
    at which point some calculators will refuse to proceed further.
    Both of these possible snags are circumvented by tinkering slightly with 
    the Bennett refraction formula, at the expense of some loss of simplicity.
    Instead of
    R = tan (90� - h - 7.31/(h+ 4.4�)), which is identical to the original 
    formula, write
    R = tan (90� - .99914h - 7.31/(h+ 4.4�))
    This sets the refraction at 90� much more closely to xero, and moves the 
    blowup angle out of any possible working range.
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. 

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