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    Re: Refraction
    From: George Huxtable
    Date: 2005 Aug 5, 22:37 +0100

    Marcel Tschudin posed an interesting question--
    >While searching with Google, I came across this mail list. May be some one
    >out here may be able to help me answering the following question:
    >How do refraction values for negative elevations have to be calculated,
    >such as e.g. the horizon from a plane? (I am interested in the range of 0?
    >to approx. ?5?.)
    >Is Bennett?s approximation also valid for negative elevations? If not,
    >what other approximation formulae should be used, or, where can one find
    >some benchmark values?
    >I am interested in formulae for both, refraction from apparent position
    >and from physical position.
    Following that, there have been several resposes from the Nav-l community.
    By the way, several of those responses from Roibert Ene were misdated to
    have a September date rather than an August one. I hope he will correct his
    calender, because my email reader, which puts correspondence into strict
    date order, keeps mis-sorting Robert's contributions.
    Marcel's question puzzled me considerably, until it emerged that was
    referring to bubble-sextant observations, with respect to the true
    horizontal, not altitudes measured up from the observed horizon with an
    ordinary sextant.
    It's not a question I am familiar with, nor is the table HO249, so I have
    little to offer in the way of a positive contribution. I've been reading
    the correspondence with interest, and have some comments to make about
    suggestions made earlier.
    Marcel wrote-
    >I also was wandering whether the approximate formulae could be used by
    >calculating the Refraction R for e.g. -2? the follwing way:
    >R(-2?)  =  R(0?)  +  (  R(0?) - R(+2?)  )
    >If this would be correct then one would not need separate formula for
    >negative elevations.
    I am sure that suggestion would not work. It would be true only if the
    refraction was linear with altitude, which is VERY far from being the case.
    Fred Hebard wrote-
    >I assume you have Meeus' formula for normal observations.
    >Meeus' formula, as transcribed by G. Huxtable, is:
    >tan(90-0.99914*S-(7.31/(S+4.4)), where S is the sextant altitude in
    >degrees.  I don't know whether this formula blows up at 0, but you
    >could try it and see whether it gives the same results as HO249's Table
    No, it doesn't blow up at 0 degrees, but it does at -4.4 degrees: This
    empirical formula doesn't pretend to give even an approximate answer for
    negative altitudes.
    By the way, that formula wasn't "transcribed" by me from Meeus, but
    tinkered-with slightly. This was to avoid an infinity arising during the
    calculation of tan, for an angle of 89.92 degrees (which is in the 'useful'
    range). The coefficient of S is adjusted to 0.99914 rather than 1.0, just
    to ensure that refraction goes to zero at an altitude of 90 degrees, as
    symmetry says it must.
    Robert Eno wrote-
    >If I had a scanner, I could scan the table for you.  Here are a few examples:
    >Height of observer: 0 feet
    >Sextant Altitude:  minus 1 degree
    >Correction: minus 35 minutes
    >Observed Altitude = minus 1 degree 35 minutes.
    That seems a bit odd, and unphysical. From a height of 0 feet, how can a
    sextant altitude possibly be -1degree? Also, the quoted value for the
    correction, at -35 minutes, is rather a surprise, considering that the
    adopted value for refraction at 0 degrees altitude is -34 minutes, and
    refraction increases very quickly as the altitude decreases towards zero.
    I can suggest a useful analysis of such low-level refractions by Andrew T
    Yoiung and George W Kattawar, titled "Sunset Science  2. A useful diagram",
    in Applied Optics , vol 37 no 18 (20 June 1998), pages 3785 to 3792. Andy
    Young is an acknowledged authority on such atmospheric optics, including
    Contact George at george@huxtable.u-net.com ,or by phone +44 1865 820222,
    or from within UK 01865 820222.
    Or by post- George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13
    5HX, UK.

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