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Refering a comparison of results on calculated refraction values, Frank
replied:
> Interest?? Yes, yes, yes. Can  you tell us what you found out and how you
> went about it,  please.

OK, here we go.

It seems that USNO published apparently recently new refraction values.
(That is at least what I understood from some communication done here.) The
data for altitudes between 0deg and 3deg were also communicated here. The
intention was to see how other results compare with these data, such as
Bennett's formula or results from integration using the U.S. Standard
Atmosphere or the the atmosphere models for summer and winter as described
in U.S. Standard Atmosphere Supplements, 1966 for different latitudes.

The calculations were done using the USNO standard conditions, i.e. pressure
1010hPa, temperature 10C (=283.15K), wavelength 501.69nm. The refraction
values from USNO are for 80 percent humidity. Since the contribution of the
humidity is almost neglectable, I calculated finally only the results for
the Standard Atmosphere twice, once wet and once dry. All the other
calculations were done dry. The radius of the earth was taken as 6378390m.
In order to make the results compareble to nautical observations and
experiences, a height of 10m (a.s.l.) has been chosen. (The values from USNO
and Bennett are scaled to the slightly different conditions at 10m height
than those of the standard conditions at sea level.)

For an altitude of 0 deg (slightly above the horizon) the following results
were obtained (all values in arc min.):

USNO (wet): 33.75
Bennett: 34.43
U.S. Standard Atmosphere (wet): 33.96
U.S. Standard Atmosphere (dry): 34.01
Average for January (all latitudes): 36.11
Average for July (all latitudes): 34.54

January detailed:
15 deg lat: 34.79
30 deg lat: 35.56
45 deg lat: 35.43
60 deg lat: 37.25
75 deg lat: 37.50

July detailed:
15 deg lat: 34.08
30 deg lat: 34.09
45 deg lat: 34.29
60 deg lat: 34.66
75 deg lat: 35.56

For an observer at an elevated location, the altitude of the horizon depends
on the atmosphere itself, it varies with the dip. In order to calculate the
value of the dip a regula falsi iterative procedure was implemented, which,
depending of the visibility of the light ray (light ray below or above the
radius of the earth) converges to the location of the horizon. Using this
procedure the dip and the refraction value at the dip could be calculated.
The calculated dip and the refraction at the dip allowed now also to compare
the factor in Bowditch's formula for calculating the refraction at the dip,
i.e. the value of 1.76 in the formula
Refr in arc min. = 1.76 * SQRT(of observer's height in m)
The refraction at the horizon alows also to calculate the factor K used for
the calculation of terrestrial refraction. K expresses the bending of the
light ray in multiples of earth radii. K is also hidden in the Bowditch
factor.

The results at the horizon are as follows:

Refraction
-----------
USNO (wet): not available
Bennett: 35.64 (interpolated for a dip of -0.092 arc deg
U.S. Standard Atmosphere (wet): 35.23
U.S. Standard Atmosphere (dry): 35.23
Average for January (all latitudes): 37.58
Average for July (all latitudes): 35.81

January detailed:
15 deg lat: 36.09
30 deg lat: 36.97
45 deg lat: 36.80
60 deg lat: 36.88
75 deg lat: 39.16

July detailed:
15 deg lat: 35.31
30 deg lat: 35.32
45 deg lat: 35.53
60 deg lat: 35.94
75 deg lat: 36.96

Altitude at horizon in deg arc (Dip)
-----------------------------------
USNO (wet): not available
Bennett: not available
U.S. Standard Atmosphere (wet): -0.0921
U.S. Standard Atmosphere (dry): -0.0921
Average for January (all latitudes): -0.0903
Average for July (all latitudes): -0.0917

January detailed:
15 deg lat: -0.0915
30 deg lat: -0.0907
45 deg lat: -0.0910
60 deg lat: -0.0892
75 deg lat: -0.0890

July detailed:
15 deg lat: -0.0920
30 deg lat: -0.0920
45 deg lat: -0.0919
60 deg lat: -0.0916
75 deg lat: -0.0908


Dip-factor (Bowditch = 1.76)
------------------------------
USNO (wet): not available
Bennett: not available
U.S. Standard Atmosphere (wet): 1.747
U.S. Standard Atmosphere (dry): 1.747
Average for January (all latitudes): 1.713
Average for July (all latitudes): 1.739

January detailed:
15 deg lat: 1.735
30 deg lat: 1.722
45 deg lat: 1.726
60 deg lat: 1.692
75 deg lat: 1.689

July detailed:
15 deg lat: 1.746
30 deg lat: 1.746
45 deg lat: 1.743
60 deg lat: 1.738
75 deg lat: 1.723


K-factor
---------
USNO (wet): not available
Bennett: not available
U.S. Standard Atmosphere (wet): 5.673
U.S. Standard Atmosphere (dry): 5.673
Average for January (all latitudes): 4.839
Average for July (all latitudes): 5.453

January detailed:
15 deg lat: 5.337
30 deg lat: 5.000
45 deg lat: 5.110
60 deg lat: 4.402
75 deg lat: 4.345

July detailed:
15 deg lat: 5.637
30 deg lat: 5.633
45 deg lat: 5.560
60 deg lat: 5.411
75 deg lat: 5.023

Some general observations of the results:
The refraction values fom USNO are by tendency smaller than the others.
Bennetts Formula seems to fit best to the stable atmospheric conditions of
July. It seems to be advisable to increase slightly the refraction values
from standard tables or formulae in winter and at higher latitudes.

If the observed differences are in a size where they could or should be
considered in nautical navigation, I suggest that
1) the calculations are redone and confirmed by an other person (Frank?)
2) the atmospheric models of 1966 are veryfied, this e.g. by statistical
analyses of balloon data, other measurements of the atmosphere and by
comparing the values with results of actual observations over a longer time
periode.

All input data and the results obtained are compiled in an Excel file. If
someone should be interested in it, then please let me know it. It is a way
of saying thankyou for all the help I received here.

Marcel

   

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