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From: Bill (no email)
Date: Mon Jan 09 2006 - 23:20:04 EST
Frank wrote:
> You can calculate dip or the altitude of a tall building peeking up from
> beyond the horizon using straight Euclidean geometry and trigonometry
> ignoring
> refraction completely. Then to include refraction, you simply change the
> radius of the Earth from R to R/(1-x) where x depends on the temperature
> gradient
> of the atmosphere. On average it's about 0.15 but it can easily be anywhere
> in the range 0.13 to 0.17 and sometimes it's as low as 0 or as high as 1.0
> (temperature inversions yield higher values of x).
Ed. Frank suggest this approximates the data from Bowditch "coastal
piloting" tables.
Later Frank wrote:
"... That is, if I fire a beam of light
horizontally (or even at some significant angle away from horizontal) from
my apartment in Chicago, when it reaches an observer in Gary, Indiana 25
miles way, its direction will have rotated downward, away from a straight
line trajectory, by an angle that is directly proportional to the distance
traveled. On average, the constant of proportionality is about 0.15 minutes
of arc per nautical mile. Note that this is really a dimensionless result:
it's 0.15 arcminutes bending per 1.0 arcminute traveled as measured from
the center of the Earth...."
=====================================
This get interesting, especially the path Frank is leading me down. Using
Frank's 0.15 figure for x in R/(1-x) to approximate Bowditch, predicted lift
at 20.55 nm would be 66 feet. By my calculations with Bowditch, approx. 64
ft lift.
NOTE: I am using 21600 ft as earth's circumference, and 6076.11 ft per nm in
calculations. Also right triangles. As Frank noted, the difference is
insignificant when using a right vs oblique triangle.
Using Frank's later "real world" mean refraction of 0.15' rise per minute of
distance, predicted lift due to refraction for 20.55 nm would be 3.08
minutes or 0d 03' 05".
Using the formula for 700 nanometer light in a vacuum: Distance in meters ^2
* 2.55 * 10^-8, lift would be 121 ft.
LIFT AT 20.55nm
64 ft = 0d 01' 46"
66 ft = 0d 01' 49"
112 ft = 0d 03' 05"
121 ft = 0d 03' 20"
It would appear Bowditch significantly underestimates lift due to refraction
by today's standards.
Now the above is all well and good if we know the distance etc., but finding
distance from an object whose base is below the horizon was the initial
challenge. That leaves us with the Bowditch formula for table 15:
Distance in nm = square root ((tan A /0.0002419)^2 +((H-h) / 0.7349))) -
tan A / 0.0002419
I do not pretend to understand the geometry behind the formula at this point
(anymore than I could duplicate the Mona Lisa with cans of dayglow orange
and dayglow yellow spray paint), but looking at patterns is interesting:
Tan A / 0.0002419 = tan A * 4134 = tan A / (R/(1 - 0.1684))
Converting the second term to nautical miles where H and h were in feet:
H and h in feet: (H-h) / 0.7349
H and h in nm: (H-h) * 8268 = (H-h) * 2 * 4134
Remember good old 4134 from the first term? 3438 / (1 - 0.1864).
Coincidence? (Radius = 3438 nm)
I am most likely oversimplify the problem (when all you have is a hammer the
whole world looks like a nail), but can we bring the Bowditch formula up to
speed by solving for a new value of x so the calculation uses actual mean
refraction as we currently understand it, therefore giving a more accurate
distance?
Bill
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