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    Refraction. was: Bubble Horizon Altitude Corrections
    From: George Huxtable
    Date: 2004 Jul 5, 22:44 +0100

    Fred Hebard has an inquiring mind, and a tendency to ask interesting
    questions. The one copied below came to me off-list
    >A private question, which you could make public if you wish.  Are there
    >ever events in the atmosphere where astronomical or distant earthly
    >objects will appear lower than they really are, rather than higher, ie.
    >events where the effective index of refraction is of opposite sign to
    >that usually encountered?  By effective index of refraction, I am
    >trying to indicate the total refraction between the object and the
    >observer rather than a local refraction.
    Atmospherics is not my specialty, though I'm as willing to pontificate
    about it as the next man. I'm posting Fred's question to the list, in the
    hope than someone will pick it up who knows more than I do.
    The quick answer is: I don't know, but think it's very unlikely.
    The direction in which light is bent depends on the density-gradient of the
    air through which it passes. Normally the density will decrease as the
    height increases, because it's proportional to pressure, and the pressure
    decreases as the height increases because there's less overburden from the
    decreasing mass of air above it. So if pressure was only factor involved,
    the density would always fall as the height increased, and any incoming
    light ray would curve downwards slightly, to be a bit nearer the vertical.
    But density (of air or any other gas) also depends on temperature. At a
    constant pressure, it's inversely proportional to the absolute temperature
    (Charles' Law, I think). The absolute temperature is measured on a acale in
    which absolute zero is at 0 degrees, not -273deg C, so that an ambient
    temp. of 10deg C is actually 283 deg absolute. If there was a temperature
    gradient in the air, the temperature falling fast enough as height
    increased, that could in theory be enough to counteract the effects of the
    falling pressure. In that case the air-density would conceivably increase,
    not decrease, as height increased, which could cause light to be curved
    upwards, not downwards; the effect that I think Fred is looking for.
    We can try to estimate the temperature gradient that would be needed.
    I will work in pounds weight and inches, as I suspect most non-scientists,
    and particularly Americans, are happier that way; and will use approximate
    At sea level, atmospheric pressure is about 14 pounds to the square inch,
    which means that a column of air of 1 square inch area, extending to the
    top of the atmosphere (wherever that might be) contains 14 pounds weight of
    Now work out the weight of air in just 1 foot height of that column. The
    volume is 1/144 cubic foot. At sea level, air has a density of .076 pounds
    per cubit foot, so the weight in that small volume is .076/144 pounds or
    .00052 pounds. This will reduce the pressure, 1 foot above sea level, by
    .00052 pounds per square inch, below its value at sea level which we took
    to be 14 pounds per square inch. So by going up a foot above sea level, the
    pressure reduces by a fraction  .00052/14, or 1 part in 27,000. If the
    temperature were constant the density of air would also reduce, by 1 part
    in 27,000 for each foot of elevation.
    Now we have to ask what temperature gradient would be needed to null out
    that density gradient. The absolute temperature would have to decrease with
    altitude by that same factor, 1 part in 27,000, for each foot of rise. At
    an ambient temperature of 283 degrees absolute, this would require an air
    temperature decrease per foot of elevation of 283/27,000, or .0105 degrees
    C per foot, or 1 deg C fall for each 95 feet of height. If the temperature
    gradient exceeds this magic figure then light will be bent in the opposite
    way to what we would normally expect.
    I hope someone will check out the numbers, and the reasoning, in that
    passage above, which takes me out of familiar territory.
    How likely is that to occur? Very likely, to the extent that we have all
    seen it happen! Driving along a road on a still, sunny, day, we are
    familiar with what appear to be blue "pools" on the tarmac surface, which
    vanish as we approach. What's happening here is that the air just above the
    road is strongly heated by the hot road surface, which in turn was heated
    by the Sun. Light rays passing through this thin layer above the road see a
    strong temperature gradient which causes a reversed bending, so a shallowly
    descending ray from the blue sky just skimming the road surface curves into
    a rising ray, so the driver sees a reflected image of a bit of the sky,
    just where he expects the road to be.
    Similar effects cause mirages and can give rise to anomalous dip, at sea.
    They can happen particularly when hot air blows from a desert coastline.
    Fred, I think, is aware of such local phenomena and asks about the
    possibility of light from astronomical or distant earthly objects being
    bent in this anomalous way. For that to happen, the temperature gradient
    over all, or most of the long light path would have to exceed the value
    calculated above. Only a meteorologist could answer that part of the
    question, but I will stick my neck out and guess that it's highly unlikely.
    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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