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Thanks George! That was obvious once you pointed it out and so provoked
me to write out the equations.

You wrote:

> For all angles of altitude except very low ones, it's a VERY good
> approximation to treat refraction of light in the atmosphere as though the
> earth's surface were flat, as are any boundaries or contours of pressure or
> temperature in the air above it.


I don't think that it is necessary to assume a flat Earth. The only
assumption that (I think) is needed is that all layers in the atmosphere
have plane boundaries with all of the planes parallel. Given that, the
refractive index of each intermediate layer cancels out of the
equations, leaving only the refractive index for space (unity, as you
noted) and that immediately around the observer's sextant.

The assumption of parallel layering would, as you say, become an issue
with light rays nearly tangential to the Earth's surface. It might
sometimes fail when viewing objects through a weather front too, though
cloud cover will then usually obscure astronomic bodies anyway.
Otherwise, there are local instabilities which astronomers trouble over
in terms of the "seeing" conditions at a particular time but the
resulting perturbations are (always?) too small to be of navigational
concern.

> In the plane-parallel case, however, Snell's law applies, and then the
> overall bending of light, from its original direction, as it reaches the
> observer, is determined ONLY by the angle of incidence and the difference
> between the refractive index as it comes into the atmosphere (which is
> exactly 1, for free space) and the local refractive index at the level of
> the observer. It doesn't matter a damn what happens in-between, or where it
> happens. The index in-between may vary up or down, different layers can be
> thick or thin, but nevertheless the total overall bending will be exactly
> the same. Surprising but true (well, it surprised me, at first).


It shouldn't be surprising. We are all familiar with the same phenomenon
when looking through window glass: Objects seen through a quality glass
pane appear to be displaced fractionally sideways by the refraction of
light as it passes into and out of the glass but the angle at which
those rays reach our eye is not affected by their passage through the
window pane -- and would not be whether the pane was made of low-density
acrylic or high-density lead crystal glass. Use the same glass to make
the face-plate of a SCUBA diver's mask and refraction between the water
outside and the air inside the mask has a substantial effect on vision
but the thickness (for practical purposes) and the refractive index of
the face-plate are of no consequence.

Only if the faces of the various layers (of glass or air) are either
curved or non-parallel is there any effect on the angle at which the
light reaches our eyes.


Sorry for not figuring that one out without the prompting!


Trevor Kenchington

--
Trevor J. Kenchington PhD                         Gadus@iStar.ca
Gadus Associates,                                 Office(902) 889-9250
R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555

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