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    Re: Refraction
    From: George Huxtable
    Date: 2005 Aug 14, 00:21 +0100

    Marcel Tschudin wrote- further about refraction.
    Unfortunately, there is no TRUE value for refraction at low angles of
    altitude, close above the horizon. Pulkova observatory near St Petersburg
    has been running a programme of measurements over many years, that as far
    as I'm aware still continues. Every now and then, a revised publication
    emerges with improved and updated results. Because refraction  at low
    angles varies with the local weather (and not just the air density at the
    observer), quoted values are average results, over a long time. On any day
    the actual refraction can differ, as distortions in the apparent disc of a
    low Sun clearly indicate.. Correcting for local temperature and pressure
    will do something to iron out those variations, but significant differences
    will remain.
    Bennet has provided a  formula which is an empirical attempt to fit that
    averaged data. At large angles of altitude, it becomes proportion to the
    tan of the zenith angle, as Snell's law requires. Near the horizon, where
    refraction rises sharply, the divergence from Snell's law shows up in
    correction terms which turn out to be remarkably simple. However,  I doubt
    whether those terms have any backing in terms of the physics of the
    refraction process; more likely, they are just empirical attempts to get as
    good a fit as possible, compatible with a simple calculation. It was
    devised in the days before everyone had a computer / calculator.
    So it's no surprise that tabulated refraction values agree well with
    Bennett. His formula was devised to replicate those values. In some
    publications, such as the Nautical Almanac, it appears that Bennett's
    formula itself is used as the basis for the refraction tables (though the
    constants have recently been tinkered-with a bit to improve the fit to
    recent Pulkova data) so it's not surprising that it shows good agreement.
    The almanac wisely states that-"the actual values of the dip and of the
    refraction at low altitudes may, in extreme atmospheric conditions, differ
    considerably from the mean values used in the tables".
    Bennett's empirical formula was, presumably, optimised to achieve an
    acceptable fit for positive angles of altitude and took no account of
    negative angles, and there's no reason to expect it to fit the observed
    refraction in that region.
    What I would conclude from all this is that there's no point in seeking
    extreme accuracy for such low-angle refraction predictions, positive or
    negative.. Where Marcel says-
    >In the mean time I also found the source code of a BASIC program to
    >calculate refraction by integration. The program was described in Sky &
    >Telescope of March 1989. Without having the original article, I transcribed
    >the program into the language with which I am working at the moment, i.e. in
    >Pascal/Delphi. A comparison of the refraction values, either from the table
    >6 or those from Bennett, with those of the program show that those depend
    >substantially of the selected refraction index of air. The problem of
    >calculating the refraction becomes now a problem of calculating a realistic
    >refraction index for air,  which depends on the wavelength, temperature,
    It strikes me that (as Fred Hebard has indicated) such corrections for
    wavelength and humidity are sufficiently small to be neglected, and to be
    overwhelmed, at low angles of altitude, by the unpredictable layering of
    temperature gradients in the air. Any such integration is only as good as
    the data that is available to feed into it, varying from one day to the next.
    Marcel added-
    >All this investigations done so far are for refraction values for APPARENT
    >negative altitudes. For my program I need however also the "inverse", i.e.
    >the calculation of the refraction for physical, TRUE negative
    >altitudes,which has not been tuched so far.
    Well, I touched on it, in my last posting, in quoting the refraction at the
    tangent point (which corresponds to zero degrees true altitude) to be, at
    sea level, about 29 minutes, and not 34 minutes (which is the adopted mean
    value for refraction at zero degrees apparent altitude).
    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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