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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.





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