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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
    '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
    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
    DO WHILE h < ht(hinx -  1)
    hinx = hinx -  1
    IF  hinx < UBOUND(ht) THEN
    DO  WHILE h >  ht(hinx)
    hinx =  hinx + 1
    IF hinx  > UBOUND(ht) THEN EXIT DO
    END IF
    IF hinx > UBOUND(ht)  THEN
    density =  0
    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
    getmu = 1 +  .000291 * density
    Using this model, I get refraction  values that seem to match the old air
    tables reasonably well.
    42.0N 87.7W, or 41.4N  72.1W.

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