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[note to list members: I promise, no code in this message. there are some
conclusions relevant to navigation in the last paragraphs]

Marcel, you wrote:
"It seems that there is something wrong in my stratospheric part. May I  ask
you, Frank, to have a look at it? May be you can see what is going  wrong
there."

I am not using the analytic model from the Auer-Standish  article so I can't
help you with your code. In my opinion, the authors' choice  of an analytic
atmosphere model from 1944 was misguided. First, as I noted  previously, that
model is simplified in a fashion that was determined by the  calculational
limitations of sixty years ago. It uses the number "5" for the  polytropic index of
the troposphere simply because it's an integer. Second, the  very idea that
we need to treat "polytropic" and "exponential" sections of the  atmosphere
separately is a limitation of the analytic treatment that obscures  the
fundamental equivalence of these different sections of the atmosphere. The  whole
atmosphere (assumed to be hydrostatic and spherically symmetric) can be  specified
by giving temperature as a function of altitude based on real  empirical
observations. If I say, 'temperature falls 7.2 degrees per kilometer  up to 11km
and then remains constant above that height,' I have given enough  information
to determine the atmospheric density profile and therefore the  refraction
completely (apart from very small differences arising from changing  humidity at
low altitudes and changing composition at very high altitudes).  Equally
easily, I could feed the model a detailed temperature profile taken from  actual
weather balloon observations with a hundred different lapse rates and  altitude
limits. That's the beauty of this approach: it's completely  general.

I was able to reproduce the sample refraction tables in the  Auer-Standish
article almost perfectly. After a little more experimentation, I  found that the
best results come from setting the lapse rate to -5.692 (degrees  Celsius per
kilometer) below 11km and zero above that. Here's the zero observer  height
table:
Alt----ref"------Au-St.----diff"
0     2189.4   2189.4
1    1532.6    1532.6
2    1145.5    1145.5
3     899.2     899.2
4     732.7     732.8    0.1
5     614.5     614.6    0.1
10    330.5     330.5
15    221.5     221.5
30    104.0     104.0
45     60.1      60.2    0.1
In this table, the leftmost column is the altitude  in degrees, the second
column is the refraction in arcseconds as calculated in  my code, the third
column is the refraction listed in the Auer-Standish article  and the third column
shows the difference between the two in arcseconds (when  different from
zero). These differences are extremely small, and I consider them  insignificant.

When the observer altitude is 15,000m, the agreement is  still excellent
though the differences close to the geometric horizon are a  little larger:
Tfactor of 1.0 and Lrate=-5.692 for obs h=15000  m:
Alt----ref"------Au-St.----diff"
-3   2316.1    2316.4    0.3
-2   1188.4    1187.9   -0.5
-1    600.0     600.6    0.6
0     353.3     353.4    0.1
1     235.6     235.8    0.2
2     171.3     171.5    0.2
3     132.4     132.5    0.1
4     106.9     107.0    0.1
5       89.1     89.2     0.1
10     47.4      47.5    0.1
15      31.7     31.7
30      14.9     14.9
45       8.6      8.6
I suspect that the small variances (all  still less than 1 arcsecond!) at the
lowest altitudes are due to the small  differences in the way the troposphere
is modeled. I don't think they're  anything to worry about.

Next, the tables in the Nautical Almanac. Here  we have two versions:
pre-2004 and post 2004. They require different lapse rate  models:

Using a temperature factor of 273.15/283.15 and
changing the  lapse rate to -7.25 reproduces a refraction table
very close to the standard  (pre-2004) Nautical  Almanac:

Alt----ref'------N.A.----diff'
0.00       34.4   34.5    0.1
0.25   31.4    31.4
0.50   28.7   28.7
0.75    26.4   26.4
1.00    24.4    24.3   -0.1
1.25   22.6   22.5    -0.1
1.50   21.0   20.9    -0.1
1.75   19.6   19.5    -0.1
2.00   18.3   18.3
2.25    17.2   17.2
2.50   16.2   16.1    -0.1
2.75   15.2   15.2
3.00    14.4   14.4
3.25   13.7    13.7
3.50   13.0   13.0
3.75    12.3   12.3
4.00   11.8    11.8
4.25   11.2   11.2
4.50    10.7   10.7
4.75   10.3    10.3
5.00    9.9     9.9
6       8.5     8.5
7       7.4     7.4
8       6.6     6.6
9       5.9     5.9
10      5.3     5.3
11      4.9     4.9
12      4.5     4.5
13      4.1     4.1
14      3.8     3.8
15      3.6    3.6

The NEW  Nautical Almanac refraction table:
Using a temperature factor of  273.15/283.15 and and changing the lapse rate
as follows:
ht<3000: LRate=  -9
300013000: LRate= 0
produces a  refraction table very close to standard (post-2004) Nautical
Almanac:

Alt----ref'------N.A.----diff'
0.00    33.6   33.8   +0.2
0.25   30.8    30.9   +0.1
0.50   28.3    28.3
0.75   26.1   26.1
1.00    24.2   24.1   -0.1
1.25   22.4    22.3   -0.1
1.50   20.9   20.8    -0.1
1.75   19.5   19.4    -0.1
2.00   18.3   18.2    -0.1
2.25   17.1   17.1
2.50    16.1   16.1
2.75   15.2    15.2
3.00   14.4   14.3   -0.1

Note that  the differences in these latter tables are on the order of 0.1
arcMINUTES. Of  course since that's the precision limit of the table, it's hard
to experiment  with greater accuracy.

So which of these various lapse rate models is  correct? The short answer is
'all of them' and 'none of them'. You can find  various diagrams on the net of
"typical" temperature profiles and "standard"  atmosphere profiles, but all
of these are just long-term averages and  idealizations. There's a lot of
variability about what happens in the real world  in the lower atmosphere, and
there's no particular reason to prefer, for  example, the temperature profile that
regenerates the current almanac's  refraction table. I really do not believe
that there is any good reason to  prefer the post-2004 Nautical Almanac
refraction tables to the pre-2004 tables  (though later evidence may yet persuade
me!). The difference in the lapse rate  profile does not appear to be
significant.

Is all lost then? Is there  anything positive that comes out of this sort of
analysis?? I think there is. By  varying the temperature profiles in ways that
reflect real variability in  Nature, we can determine in an obective (albeit
theoretical) fashion just how  far we can trust the refraction tables.
Throwing in temperature inversions,  changes in the lapse rate, the location of the
lower edge of the stratosphere,  and all of that, I find that the refraction
tables are 'rock solid' above about  3.0 degrees. Almost nothing that can happen
in the weather is likely to make any  meaningful change of 0.1 minutes or
larger above that height. Even at 2.0  degrees altitude, the changes in
refraction amount to only a few tenths of a  minute. This suggests that refraction
tables can be relied upon safely down to  very low altitudes (on the other hand,
under the same conditions that would  yield unusual changes in refraction, the
'dip' of the horizon will be much more  liable to error).

I've read in various places before that the refraction  above 10 degrees or
so is almost completely insensitive to atmospheric  structure, and the
refraction close to the horizon is sensitive mostly to the  rate of change with height
of the temperature in the atmosphere. These numerical  integrations confirm
those general statements beautifully and even place  stronger limits on the
numbers. Refraction above 15 degrees is COMPLETELY  insensitive to atmospheric
structure (and for that matter, it can be calculated  easily from k*tan(z)).
Refraction between 3 and 15 degrees is nearly insensitive  to atmospheric
structure. And refraction below 3 degrees depends in detail on  the temperature
structure of the lowest layers of the troposphere. There is no  'correct'
refraction at such low altitudes. [note that all of this applies to an  observer at sea
level].

By the way, if anyone would like to experiment with these models, let me
know. I've been considering making a little web app out of  it.

-FER
42.0N 87.7W, or 41.4N  72.1W.
www.HistoricalAtlas.com/lunars

   

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