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    Re: Sunrise
    From: George Huxtable
    Date: 1999 Apr 24, 6:35 PM

    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.
    
    
    
    ------------------------------
    george---.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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