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    Re: Equation for dip?
    From: Frank Reed CT
    Date: 2006 Oct 1, 01:13 EDT

    Jim Van Zandt wrote:
    " I wonder whether  that's because of the different wavelength, or different
    typical heights (so  different temperature and pressure profiles), or what?"
    It's the index of  refraction in air at visible wavelengths and the rate at
    which it changes with  atmospheric density.
    The rate of change of atmospheric density, and  therefore index of
    refraction, depends on the lapse rate (the rate of  temperature variation) in the lower
    atmosphere. If the air gets colder with  altitude much more rapidly than
    normal, the air density can actually be constant  with height. Under those
    circumstances, the air does not refract at all. The dip  equation then is exactly the
    same as the equation you would get from a straight  geometric calculation. Or
    in other words, the effective radius of the Earth is  the same as the true
    physical radius. I should say, though, that this is a  limiting case. When the
    air has that temperature profile and constant density  with height, it is
    extremely unstable and convection will set it in almost  immediately and bring the
    lapse rate closer to the normal rate.
    At the  other extreme, if the air gets warmer with height at a rather high
    rate, the  effective radius of the Earth becomes infinite --as if the Earth were
    flat-- and  the dip is exactly zero at all heights. Under these
    circumstances, the horizon  is infinitely far away and you can see across an ocean. In the
    real world,  scattering and absorption in the atmosphere makes distant objects
    disappear in a  haze before the horizon but the limiting distance may be ten
    times farther away  than a "normal" calculation would predict. And, yes, if
    the lapse rate gets even  steeper, the effective radius becomes negative and the
    sea would appear to climb  up into the sky above the true horizon.
    Generally, the normal, daily  variability of the lapse rate is equivalent to
    a much more moderate variability  in the effective radius of the Earth; one
    day 15%, another day 17%... You can  literally imagine the Earth's surface
    flexing up and down with the weather to  picture this. It works for any surface
    refraction phenomena including dip, range  of visibility, distance by height
    above the horizon, etc. All surface  (terrestrial) refraction phenomena are
    affected by this in the same way.
    42.0N 87.7W, or 41.4N 72.1W.
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