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Clive Sutherland, a friend and near-neighbour from the next village, has
raised some interesting questions about refraction, in a situation where
the light from the Sun arrives at an angle which grazes the horizon, near
Sunrise and Sunset.

I'll have a shot at answering, and see if he or anyone else shoots me down.

We are considering the moment when the upper limb of the Sun appears to be
sitting just on the horizon. Let's take as an example an observer with a
height-of-eye of 10 feet. His horizon would then be 3.63 miles away,
according to Norie's tables. Let's imagine that there's an assistant at
that spot on the horizon, 3.63 miles from the observer, just in line with
the Sun, and this assistant has a height-of-eye of zero feet. We can invent
a sandbank for him to stand on with his head just out of the water, poor
fellow.

The light path from the Sun to the observer can be divided into two parts:
first, from the Sun, in through the Earth's atmosphere, to reach the eye of
the assistant on the waterline. Light from the upper limb will be just
skimming the surface when it reaches him, so the apparent altitude of the
upper limb is zero degrees. Up to that point, the light will have been
deflected by refraction in the atmosphere by approximately 34 minutes of
arc (this is the value adopted in the ephemeris to use for rising and
setting predictions: we can consider its accuracy later).

The light ray then goes on, skimming the surface past the assistant, in the
second part of its path, and gradually rising above the surface until it
reaches the observer 3.63 miles further on, and 10 feet above the surface.
Clearly, there is going to be further refraction in this part of its path,
and Clive's main question appears to be: how much is it, and how do you
allow for it?

DIP.

Dip is the angle between an observer's true horizontal (90 degrees from his
zenith) and his observed horizon, and increases with his height. Mainly,
dip is just a geometrical factor, related to the curvature of the Earth's
surface. If light travelled in straight lines, dip would be very simple.
But it doesn't, not near the earth's surface. Light travelling horizontally
(or nearly so) has a small amount of curvature, about one-twelfth of the
curvature of the Earth, which alters slightly the direction of the horizon
as we see it.

Details of the effect of refraction on dip are given in W.M.Smart,
Text-book on Spherical Astronomy, C.U.P., 5th ed., 1971, p. 318.

If there was no atmosphere to refract the light, the dip would be about
one-twelfth greater than it is (and the observed horizon would be about
one-twelfth closer). To help to picture how it works, just imagine what
would happen if the Earth's atmospheric pressure (and the refraction)
happened to be twelve times greater than it is. Then horizontal light rays
would bend with a curvature that just matched the surface of the Earth, so
you would be able to see right over the horizon. The horizon wouldn't
exist.

So, with our atmosphere as it is, there's a small but significant effect,
which was taken into account when the dip table and the horizon distance
table were compiled. For a height of eye of 10 feet, the dip from the table
is 3.10 minutes of arc. If there had been no refraction, it would have been
3.36 minutes. So to estimate the additional refraction over the second part
of the Sun's light path, look up the dip (3.10 minutes) for the observer's
height of eye (10 feet), divide by 12, giving 0.26 minutes. This can be
added to the figure from the refraction tables of 34 minutes, giving a
total of 34.26, total refraction.

At the moment of Sunrise, the upper limb must appear to be in exactly the
same direction as the horizon. The zenith angle of the horizon is 90
degrees plus the dip, or 90 degrees 03.1 minutes. The apparent zenith angle
of the Sun's upper limb, as observed, must be the same, 90 degrees 03.1
minutes. The true zenith angle of the Sun's upper limb must then be, after
adding the 34.26 minutes for refraction, 90 degrees 37.36 minutes.

The next step would be to work out the position of the Sun's centre by
allowing for the semidiameter of about 16 minutes (depending on
time-of-year), which would put it at a zenith angle of 90 degrees 53.36
minutes, or an altitude of minus 53.36 minutes. A parallax correction of
0.15 minutes could also be made.

Really, what the above calculation shows is how pifflingly small is the
correction to make for the refraction over the extra path from horizon to
observer. At one-twelfth of the dip, it's far smaller than the
uncertainties in the refraction tables at low altitudes. Navigators are
often warned to keep their altitude measurements above 20 degrees or so to
avoid the large refraction corrections, and the large uncertainties in
those corrections, at low altitudes. Frequently, the temperatures of air
layers near the horizon will differ markedly from the smooth variation
presumed by the model used in calculating the refraction. This can show up
in severe distortions in the shape of the Sun disc as it rises or sets, and
sometimes in the sight of distant vessels appearing to float above and
clear of the horizon. Even if such clues don't appear, conditions of
abnormal refraction can still be present. Predicting the moment of Sunrise
or Sunset is an inaccurate business, then, and it makes no sense to go for
the ultimate in small corrections.

Clive has worries about whether the calculations would apply correctly to a
non-spherical Earth, but I can't see where any problems would arise from
its ellipsoidal shape. The corrections have already been made in our
charts, in that the spacing in miles between the latitude lines varies
slightly with latitude, to compensate.

Clive suggests that refraction in the tropics would be greater than in
higher latitudes, but why? The refraction corretions increase
proportionately with increase of pressure, and decrease by about 5% for
every increase of 10 degrees Celsius in temperature. This assumes normal
refraction conditions.

As stated above, the Ephemeris uses an "adopted" value of 34 minutes for
refraction at an apparent altitude of zero. The value given in Norie's
tables for zero degrees is 33 minutes. In my own programmes I use an
expression which I found in "Celestial Navigation by Calculator", by Keys,
pub. Stanford, page 118.
Refraction = .0162 * Tan (Alt - Arctan ((12 * Alt) + 36)) , everything in
degrees.
This curious-looking expression, which was presumably adjusted empirically
to fit the standard tables rather than derived from first principles,
mirrors the Norie tables remarkably well, and predicts a correction of 35
minutes at zero apparent altitude.

My solution to the problem of predicting just when the first flash of
Sunlight will appear above the horizon would be to send a crew member up to
the crosstrees, to shout out when he can see it.

George Huxtable.



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george@huxtable.u-net.com
George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
Tel, or fax, to 01865 820222 or (int.) +44 1865 820222.
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