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    Re: Biruni and the radius of the Earth by dip
    From: George Huxtable
    Date: 2011 Jan 7, 00:32 -0000

    Frank wrote-
    
    "I don't think there's any question that you can get excellent accuracy for
    this sort of observation, limited only by visual acuity (so roughly one
    part in 3500), and if you're doing it as a modern experiment with your
    students, you could use binoculars and get even better accuracy."
    
    What sort of observation, exactly? And one part in 3500 of of WHAT? Not, I
    suppose, in the measurement of the Earth's radius, the precision of which
    we were discussing, but is Frank perhaps suggesting he could measure the
    dip-angle of about 34 arc-minutes from  the true horizontal, from the
    mountain top to the plain below, with a precision of one part in 3500, or
    one-hundredth of a minute? I reckon not. Even if an observer could reckon
    the dip to an arc-minute, that's an accuracy of one part in 34, not 1 in
    3500.
    
    Then-
    "As you suggested in your first post on this, a water level will provide an
    excellent true horizontal level."
    
    Is Frank (and John Huth too) suggesting that a flexible pipe and a pair of
    vertical glass tubes be used, then, to provide a suitable level, to measure
    to an arc-minute? (Not, of course, that such technology was available to
    al-Biruni.). I wonder whether either of them has ever tried to make, and
    fill, and use, such a levelling device? The problem is to avoid entraining
    air-bubbles, which will then collect at any humps in the tube, unbalance
    the water-columns, and result in a false reading. If it was so simple in
    practice, buiders would use such devices in place of spirit-levels and
    theodolites. Just try it out, with a few metres of transparent plastic
    tube. Hint: use distilled, or at least boiled water, to reduce dissolved
    air.
    
    If we're still discussing Biruni, as the threadname implies, he is supposed
    to have measured first the height of a mountain by its elevation angle from
    two points on the plain below, separated by a measured distance. But that
    geometry requires those two survey points to be on the same level, and the
    profile in the Gomez paper shows that the "plain" below was anything but
    flat. And then, because he didn't have a sea-horizon, the dip to the
    visible horizon would have been  the dip to the top of one of the distant
    undulations of that plain, and the required height of the mountain was the
    height difference above that undulation, which was unknown.
    
    If an astrolabe was used for the observation, as has been suggested,
    astrolabes were divided to the nearest degree and had an observing alidade
    for measuring altitude, using a couple of pinnules. Levelling was by
    dangling from a  finger-ring or two, assuming that it the whole assembly
    had somehow been balanced to the horizontal beforehand.. All in all, I
    reckon that my assessment of the measuring precision available to Biruni
    (that he measured his 34 arc-minutes to a precision of +/-15 minutes) was
    generous. As a consequence, any similarity between his Earth-radius and its
    true value was the result of a happy accident.
    
    George
    
    contact George Huxtable, at george{at}hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
    
    
    
    If we're still discussing Biruni, as the threadname implies, he is supposed
    to have measured first the height of a mountain by its elevation angle from
    two points on the plain below, separated by a measured distance. But that
    geometry requires those two survey points to be on the same level, and the
    profile in the Gomez paper shows that the "plain" below was anything but
    flat. And then, because he didn't have a sea-horizon, the dip to the
    visible horizon would have been  the dip to the top of one of the distant
    undulations of that plain, and the required height of the mountain was the
    height difference above that undulation, which was unknown.
      Also, for a student experiment, you might want to point out that modern
    buildings are built "plumb" to fairly high accuracy which means that
    horizontal architectural lines indicate a true horizon. So for example, if
    you're on the twentieth floor of a tall building and you see another tall
    (modern) building a couple of blocks away between you and the horizon,
    angled so that you can see rows of windows "in perspective", you can cross
    the rows of window lines extended to the point at infinity and that will
    give you the true horizon. I should add that this is useful for all sorts
    of urban celestial navigation games.
    
    Refraction has a fairly simple effect on light rays in the lower couple of
    kilometers of the atmosphere so long as we assume a constant "lapse rate"
    (rate of change of air temperature with altitude). Refraction curves the
    path of a light ray into an arc of a very large circle. The radius of this
    circle can be expressed as a multiple of the radius of the Earth. Usually
    the path is much less curved than the Earth's surface so the radius of
    curvature is about six or seven times greater than the radius of the Earth.
    To put it another way, for every nautical mile that a light ray travels
    (nearly horizontal), it rotates towards the ground by 0.155 minutes of arc.
    The entertaining thing here is that we can do a transformation that makes
    the path of the light ray exactly straight and in trade makes the Earth's
    surface somewhat less curved. In other words, we change the radius of the
    Earth by 15.5% which completely accounts for refraction so then we can do
    any and all analyses (for situations involving terrestrial refraction) as
    if refraction does not exist. I've done this the long way around, working
    out the actual refraction for all of the cases important to navigation, and
    verified that it works.
    
    The cases where terrestrial refraction comes into play in navigation:
    1) dip of the horizon.
    2) dip short (dip for a visible shoreline in front of the horizon).
    3) distance to the horizon.
    4) maximum visibility distance for an object of height h seen by an
    observer at height H (really a special case of 3).
    5) distance by apparent angular height of an object of height h seen beyond
    the horizon (and partially hidden by it) by an observer at height H. This
    is table 15 in modern Bowditch (2002).
    6) distance by angle between apparent waterline of an object in front of
    the horizon and the horizon beyond.
    (there may be other cases, but these are the ones I can think of right
    now). In every one of these cases, you can work out the correct equations
    by treating it as a simple geometry problem and then replacing the true
    radius of the Earth with a "refracted radius" of the Earth at the very end.
    One potential "gotcha" in these calculations is that you have to be careful
    not to think in terms of nautical miles. Angular minutes of arc are equal
    to nautical miles only if the Earth has its true radius.
    
    But of course refraction is variable. It depends on the density of the
    atmosphere and the rate of change of the density with altitude. If you work
    out the refraction in more detail, the rate of rotation of a light ray in
    the lower part of the atmosphere (lowest few kilometers) is approximately
    equal to
     beta = alpha0*Q*Re/s
    and the equivalent "refracted radius" for the Earth is
     R = Re/(1-beta)
    where alpha0 is the the index of refraction of air minus one equal to
    0.000281, Q is just the usual temperature/pressure factor
    (=(P/1010mb)/(T/283K)), Re is the true radius of the Earth, and s is the
    scale height of the atmosphere. The scale height is the e-folding height
    for the atmospheric density in the lower part of the atmosphere which
    depends on the temperature lapse rate (note: Q is about equal to 1, s is
    usually around 9km but variable, and the product alpha0*Re is about
    1.79km). The important thing here is that this "beta" is a variable
    quantity so there is no exact and correct amount of terrestrial refraction.
    It depends on the weather. But we can always think about it as being
    equivalent to a modified radius for the Earth. Typically beta is about
    0.155 and that's how most of the tables are calculated for use in
    navigation, but it can easily be anywhere in the range from 0.10 to 0.25.
    The lowest value for beta is zero since that implies air of constant
    density which is just on the edge of instability. If there is a large
    temperature inversion (air that gets warmer at higher altitudes), then beta
    can be approach 1 which makes the Earth appear flat as far as optical
    observations are concerned (beta can even exceed one in which case the
    Earth appears bowl-shaped and the horizon becomes indistinct).
    
    Two hundred years ago, the idea that the refraction was variable and varied
    in an unpredictable way seems to have been an uncomfortable thought for the
    folks who created tables of dip, and there were some cases where authors
    claimed that their tables were better because they used the "correct" value
    for the terrestrial refraction.
    
    When comparing sources on this issue, bear in mind that different
    communities use different values for standard temperature and pressure. For
    navigation and other practical uses, it's normal to use a standard
    temperature of 10 degrees Celsius. Academic sources tend to prefer 0
    degrees Celsius.
    
    -FER
    PS: Is there a tall building in Boston with an observation deck or floor
    where you can look out to the sea horizon? If there's a large enough
    viewing area, you should be able to sight down one side of the viewing area
    and directly see the displacement of the horizon below the level.
    Photograph this a few times with different weather conditions, and you
    should be able to see the refracted horizon "breathing" up and down
    relative to the true horizon. Distant objects which just overlap on one day
    will be slightly displaced on other days.
    
    
    
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