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    Re: Biruni and the radius of the Earth by dip
    From: Frank Reed
    Date: 2011 Jan 6, 12:35 -0800

    Hi John.

    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. As you suggested in your first post on this, a water level will provide an excellent true horizontal level. 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.

    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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