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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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