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OK, Treat each individual molecule of Oxygen, Nitrogen, etc as one of a
series of plane surfaces of arbitrary orientation with which a light ray
interacts.  In order to bend away from the mean normal would still require a
cumulative decrease in the index of refraction along the entire arbitrary
path from the initial entry point of the light ray into Earth's atmosphere
to the observer's eye.

If you really want to be rigorous about it, throw out Snell's Law and use
Quantum Mechanics to model the absorption and retransmission of light from
one molecule to another.

I heartily agree that using sights lower than 15 degrees or so is
problematic, in that a lot of variation in the density of the atmosphere is
inevitable and the simplified quantitative models fall apart.  However, the
question had to do with the qualitative question of whether or not light
ever bends upwards overall.

Perhaps,  if you were in space flight, you might observe an image of a star
that was partially REFLECTED off of Earth's atmosphere.  That would be a
case where the bend would be upwards.  But if you were in space flight,
you'd have a wide selection of stars to choose from, and you could simply
avoid near eclipses like this.  Perhaps someone with knowledge/experience
with space craft guidance systems could shed some light on this, if indeed
such systems use celestial objects as references.

John


----- Original Message -----
From: "George Huxtable" 
To: 
Sent: Tuesday, July 06, 2004 3:45 AM
Subject: Refraction. was: Bubble Horizon Altitude Corrections


> Related to Fred Hebard's question, whether circumstances might occur in
> which refraction could act in the opposite direction to the usual one,
John
> Brenneise wrote-
>
> "When speaking of an index of refraction, the usual context is that of
> explaining the bending of light rays at the boundary between transmission
> media.  In our case, from the near vacuum of interplanetary space (index
of
> refraction = 1.0...) to air (at STP, index of refraction = 1.0003).
Snell's
> law describes the geometry of the bend, where:
>
>     n1 is the index of refraction in the previous media
>     theta1 is the angle of the ray in the previous media, measured from
the
> normal vector to the plane of interface.
>     n2 is the index of refraction in the new media
>     theta2 is the angle of the ray in the new media, measured from the
> normal vector to the plane of interface.
>
>     n1*sin(theta1) = n2*sin(theta2)
>
> As the density of the atmosphere increases with decreasing altitude the
> index of refraction increases.
>
> Whenever the index of refraction increases in the transition from the old
> media to the new media, the ray will bend toward the normal vector to the
> plane of the interface, making the angle of elevation grow.
>
> So, a decrease in the angle of elevation would require a decrease in the
> index of refraction.  A decrease in the index of refraction would require
an
> atmospheric event that produces a harder vacuum than interplanetary space.
> This seems awfully unlikely to me.
>
> John."
>
> =============
>
> I agree with John about the unlikelihood, but he has taken a somewhat
> oversimplified approach to a rather complex problem.
>
> His use of Snell's law is valid, but only when applied to a plane-parallel
> problem in which the contours of equal density are flat surfaces. That's a
> good approximation to refraction in the atmosphere, for light coming in
> from well above the horizontal, and would give rise to a refraction
> correction that varied smoothly with angle (with cot alt, in fact) and
> depended only on the refractive index at the observer's eye, and not at
all
> on the details of changes in atmospheric conditions on its path in.
>
> However, for light coming in from lower altitudes, other factors enter,
> that make a simple Slell's-law approach invalid. Such light rays are
> entering the atmosphere a long distance (horizontally speaking) from the
> observer, so curvature of the layers has to be taken into account. The
> surfaces of constant-density (= constant-refraction) are no longer simple
> planes, but are spheres, centred at the centre of the Earth. This effect
is
> at the root of the correction terms (from the simple cot alt) in the
> equations predicting refraction.
>
> Not only that: especially from lower altitude bodies, the light is
> travelling long distances through the layers of varying density that exist
> in the atmosphere, and making slightly different angles-of-incidence with
> each, because of the curvature. As a result, when we look at a
low-altitude
> Sun, we often see strange distortions from its expected shape: which would
> be a disc just slightly squashed in the vertical directiion. Such
> distortions would be impossible in a simple Snell's-law situation, in
which
> all that mattered was the refractive index at the observer's eye, and
> nowhere else.
>
> Now for a digression. Those distortions of the Sun's disc are of course
> nothing to do with the Sun itself. In the same atmospheric conditions, any
> other object, such as a star, would have its apparent position invisibly
> pulled-about in just the same way that the edge of the Sun disc was, but
> with no clue to the observer that that was happening. So when you see a
> distorted Sun disc next, let it remind you that any altitude observation
> might be affected in the same way, and this, together with anomalous-dip,
> may set limits on the ultimate accuracy achievable by even the best
> observer with the most expensive instrument. Such effects, as with
> anomalous dip, are entirely local and unpredictable, and there's no way to
> correct for them: the temperature and pressure corrections to refraction
> don't help here. Best you can do is to avoid low altitudes: say below 10
or
> 15deg, if you are asking for high precision. But try telling that to a
> Norwegian Winter-navigator!
>
> Back to the topic, and why I think John Brenneise's dismissal (of the
> possibility of certain conditions in the atmosphere giving rise to
reversed
> refraction) may be a touch over-hasty.
>
> As an example, consider a sudden, sharp, "front" passing the observer.
It's
> just shown as a simple line on a weather-map, separating warm and cold
> air-masses, but it has a third dimension. If we draw a vertical
> cross-section across a front, it will show a wedge of warm air overlying a
> corresponding wedge of cold air, with their junction lying at a sloping
> angle. On that junction there will be an enhanced temperature-gradient:
> cool below, warm above. How sharp  can that temperature-gradient be? I've
> no idea, but I think that in right circumstances the transition between
> warm and cold can be quite a sharp yet stable affair, which would imply a
> sharp temperature gradient along that sloping front..
>
> Now imagine that an observer is standing on the Earth's surface, somewhere
> on the line where that front meets sea-level, and sees a body in the sky,
> which just happens to be at an altitude and in a direction that the
> incoming light is subjected, along its whole path, to that enhanced
> temperature gradient, because it is passing along the boundary between
warm
> and cold air. All along its path, the gradient in the refractive index is
> causing a bending of the light. Isn't it easy to imagine, without putting
> numbers on it, how the apparent position of such a body might be greatly
> affected?
>
> You might validly object that the scenario I have imagined above, with
warm
> air overlying cold, will give rise to an enhanced temperature gradient in
> such a direction as to INCREASE the normal bending of light by refraction,
> and not counteract it as Fred Hebard's question asks. And I couldn't argue
> with that. All I am pointing out is that there can be local circumstances
> in the atmosphere which cause light refraction to behave in unexpected
> ways, yet everything remains in stable conformity with the laws of physics

> and optics. Are there circumstances in which a sufficiently stable
> configuration can occur, with an opposite temperature gradient of
> sufficient magnitude? I don't know (though rather doubt it) which is why I
> suggested that that was a question for a meteorologist.
>
> So I agree with John Brenneise that any such reversed refraction seems
> highly unlikely, but whether it's impossible is a more difficult matter.
>
> George.
>
>
>
>
>
> ================================================================
> contact George Huxtable by email at george@huxtable.u-net.com, by phone at
> 01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
> Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
> ================================================================

   

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