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    Re: Refraction at the horizon.
    From: George Huxtable
    Date: 2008 Mar 19, 17:31 -0000

    Marcel wrote-
    | Yes, it means exactly this, that the powers for temperature and
    | pressure aren't any more +1 or -1. Let me explain what I did about one
    | year ago and then left unfinished because other more important work on
    | this subject turned up.
    | At that time I tried to find a transfer function for estimating
    | refraction with a fitted function for (very) high zenith distances,
    | this using the results from calculations done on the standard
    | atmosphere in a model of over 200 layers. The observer was in this
    | model at an altitude of 10km. After having found such a function, I
    | thought that different atmospheric conditions could estimated with it
    | just be scaling, using powers of +1 and -1. I was surprised to find
    | that this was only true for "smaller" ZDs. For larger ZDs, around 90
    | deg and higher, the functions for the powers would also have to be
    | found by a separate fit. I attach here a few gif files showing the
    | calculated and the fitted values:
    | Ref-ZD.gif: shows refraction as a function of ZD.
    | Ref-Temp.gif: shows how the power for the temperature changes as a
    | function of ZD when changing the temperature to 90% or 110% of the
    | initially calculated one.
    | Ref-Press.gif: shows the same thing for the pressure.
    and later added [4723]
    "... when I wrote last night the above comment related to the powers of
    temperature and pressure I should have added that I don't understand
    (yet) the reason for it. I therefore can't absolutely exclude an
    artefact of the calculation itself. However, thanks to Andrew T. Young
    the program could be tested beforehand carefully on several less and
    more critical bench mark calculations. The more critical cases were
    situations just at the edge before ducting occurs and situations with
    ducting, this for different observer positions. The results compared
    well. We found that the (small) differences we had resulted mainly
    from calculating/using gravity slightly different when calculating the
    pressure for the different layers. Andy used one same gravity value
    for all layers, calculated for an intermediate hight of the atmosphere
    (which is for his simulations absolutely sufficient) whereas in my
    model the gravity is calculated for an intermediate height within each
    When I dug again in the files which were produced one year ago I
    selected "a" set of available graphs without finding out whether they
    really corresponded to the latest results found at that time. The aim
    was to show you what I meant with exponential behaviour, not more.
    It's clear that it would be interesting to understand the reason for
    Comment from George.
    That looks distinctly odd.
    What those graphs appear to show is that changes of 10% in absolute
    temperature give rise to changes that are, at some angles, much greater than
    10% in refraction, when the angles are such that the refraction itself is
    high, and increasing rapidly with angle. But changes in pressure, which
    ought to change air density to a similar extent, have a much smaller effect
    on the refraction. That all looks rather unphysical.
    Being of a sceptical turn of mind, that would lead me to doubt the validity
    of the computer simulation. It certainly calls for closer investigation,
    before its results are accepted.
    Marcel mentions "ducting", a phenomenon that can occur in which light
    becomes trapped within an air-density contour, under certain conditions of
    temperature gradient. Indeed, when such ducting occurs, the sort of blow-up
    in refraction angle that he sees might well result. But I find it hard to
    imagine how such ducts can occur if the temperature profile is close to the
    standard-atmosphere pattern. I wonder whether a particular temperature
    profile had been deliberately chosen to show-up such a ducting situation, or
    a situation right on the edge of it.
    Gravity has reduced by only one part in 300 from the surface to 10 km up, so
    it's hard to see that having much effect, except at enormous heights, where
    no air remains.
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Navigation List archive: www.fer3.com/arc
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