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Re: Astronomical Refraction: Computational Method for All Zenith Angles
From: Frank Reed CT
Date: 2005 Aug 19, 20:13 EDT

```Marcel you wrote:
"But to write a program correct and knowing
what  results to expect of it, depending on the data entered, is one thing,
to have  a low probability that those entered data correspond to the actual
situation  - as in the case of refraction at low altitudes - is an other. It
is for this  reason that I try to make my program as much correct as
reasonably possible  and add a warning that due to the actual conditions the
results may  differ."

Sounds good to me!

Oh, by the way, I'm Frank, not Fred. There's a Fred on the list,  too.

In another message you wrote:
" I only realised then,
that the  integration needs also to be splited into two parts, one for the
troposphere  and one for the stratosphere. "

I don't think the integration itself  needs to be split unless there's a
problem with the model atmosphere. The model  atmosphere in the Auer-Standish
article does have two pieces though: polytropic  in the troposphere and
exponential in the stratosphere and above (structure  in levels of the atmosphere above
the stratosphere are not relevant to  astronomical refraction unless the
observer is in those levels).

The  atmospheric model in the Auer-Standish article is taken from earlier
papers by  Garfinkel, but after looking at those articles, I'm a little
skeptical. The  polytropic index (the exponent "5") that Auer and Standish use is from
Garfinkel's 1944 artice but in his 1967 article he states that this integral
choice was made only to facilitate calculation on a desktop calculator. In
the  later article he gets a polytropic index closer to 4.2 (I don't remember
the  exact value). But we really don't need to bother with all of this. A simple
table of the atmosphere will suffice, and it can include details of the
atmosphere above the stratosphere easily. So that's what I used (taken from the
1976 US "standard atmosphere" for aeronautics which is widely available
online):

>>>>>>
'detailed atmosphere model  (MUmode=1):
'rho is density in kg/m^3 and ht is height in meter above sea  level.
rho(1) = 13.47: ht(1) = -1000
rho(2) = 12.25: ht(2) = 0
rho(3) =  11.12: ht(3) = 1000
rho(4) = 10.07: ht(4) = 2000
rho(5) = 9.093: ht(5) =  3000
rho(6) = 8.194: ht(6) = 4000
rho(7) = 7.364: ht(7) = 5000
rho(8) =  6.601: ht(8) = 6000
rho(9) = 5.9: ht(9) = 7000
rho(10) = 5.258: ht(10) =  8000
rho(11) = 4.671: ht(11) = 9000
rho(12) = 4.135: ht(12) =  10000
rho(13) = 1.948: ht(13) = 15000
rho(14) = .8891: ht(14) =  20000
rho(15) = .4008: ht(15) = 25000
rho(16) = .1841: ht(16) =  30000
rho(17) = .03996: ht(17) = 40000
rho(18) = .01027: ht(18) =  50000
rho(19) = .003097: ht(19) = 60000
rho(20) = .0008283: ht(20) =  70000
'values below here are extrapolations (and probably  unnecessary)
rho(21) = .0001846: ht(21) = 80000
rho(22) = .000041: ht(22)  = 90000
rho(23) = .000009: ht(23) = 100000
rho(24) = 9.7E-08: ht(24) =  130000
rho(25) = 2.5E-12: ht(25) =  200000
<<<<<

Here's the modification to  getmu:
>>>>>
h = r - REarth
SELECT CASE  MUmode
CASE 0
'Simple  exponential decay of  atmospheric density with a scale height of
9-10  km:
'I changed the scale height for the simple  model from 10km to 9.21km
density =  EXP(-h  / 9210)
CASE 1
'New model uses tabulated atmsopheric data
'this stuff with 'hinx' (a global or static var) just finds the closest
height.
DO WHILE h < ht(hinx -  1)
hinx = hinx -  1
LOOP
IF  hinx < UBOUND(ht) THEN
DO  WHILE h >  ht(hinx)
hinx =  hinx + 1
IF hinx  > UBOUND(ht) THEN EXIT DO
LOOP
END IF

IF hinx > UBOUND(ht)  THEN
density =  0
ELSE
i =  hinx
'simple interpolation into  table:
density = (rho(i - 1)  + (rho(i) - rho(i - 1)) * (h - ht(i - 1)) / (ht(i) -
ht(i - 1))) /  rho(2)
'dividing by rho(2)  normalizes densities to unit value at sea  level
END IF
END SELECT
getmu = 1 +  .000291 * density
<<<<

Using this model, I get refraction  values that seem to match the old air
tables reasonably well.

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

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