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    Re: Astronomical Refraction: Computational Method for All Zenith Angles
    From: Frank Reed CT
    Date: 2005 Aug 21, 01:32 EDT

    Marcel, you wrote (regarding splitting the  integration into two parts):
    "Does this not apply for the transition from  troposphere to stratosphere?"
    No, if you use the analytical atmosphere  model in the Auer-Standish article
    (or any other physically resonable model),  the integration is not split up.
    Instead, the "getmu" function (for getting the  index of refraction at any
    altitude) will have a condition in it: if  height<11 km, do troposphere
    calculation, else do stratosphere calculation.  This gets called at each step in the
    Today, I coded  up a much better way of dealing with all of this atmospheric
    structure. I  generate a density table directly from the temperature profile
    of the atmosphere  (this is the principal independent variable) and the
    condition of hydrostatic  equilibrium and the ideal gas law which is how atmosphere
    models are  derived usually. While these equations can be integrated
    analytically in  important special cases you then have to do a lot of work patching
    together the  pieces. But if you do the integration numerically from the ground
    up, it's  trivial to modify the temperature profile, generate new atmosphere
    data, and  then re-run the refraction tables.
    Here's the key code for  this:
    'some constans and initialization follows.  Should be
    'mostly self-explanatory.
    ht = 0
    REarth = 6378000
    dh =  5
    Rgas = 287.053
    g0 = 9.80655
    press = 101325
    Temp0 = 273.15
    Temp = Temp0
    'Build the  atmosphere from the ground up...
    'LRate is the "lapse  rate" -the rate  of change in temperature with altitude.
    'in this instance I have a temperature inversion up to 200m and  then
    'the usual decline rate of 6.5 deg C up to the tropopause  at the standard
    '11km altitude. The inversion should be deleted  for comparison with
    'normal refraction tables:
    IF ht < 200 THEN LRate = 10 ELSE LRate = -6.5
    IF ht >  11000 THEN LRate = 0
    'At each step update the temperature in  accordance with the current
    'lapse rate:
    Temp =  Temp + (LRate / 1000) * dh
    'ideal gas law gives density (rho)  from pressure and temperature:
    rho = press / (Rgas *  Temp)
    'do something with the calculated  atmosphere:
    'use hydrostatic  equilibrium to get to the next step:
    dp = -rho * g0 * dh / (1 +  ht / REarth) ^ 2
    'increment the pressure and the  height:
    press = press + dp
    ht = ht + dh
    LOOP  UNTIL ht > 80000  'nothing much left above 80km  altitude
    And that's it. This model can then be used  as input to the refraction
    integration. Using a lapse rate of 5.70 deg C per  kilometer below 11km altitude and
    a rate of zero above that altitude, I was able  to replicate almost exactly
    the refraction tables in the Auer-Standish article.  The differences were
    mostly less than one-tenth of a second of arc. This  specific lapse rate, 5.70, is
    apparently connected with the choice of an  integral polytropic index. It's
    "non-physical". There seems to be general  agreement among the atmosphere
    experts that 6.5 is a better average lapse rate  in the lower atmosphere. That's
    what I used in most of my runs.
    Switching  to the standard conditions of the Nautical Almanac, I find that I
    can reproduce  the (pre-2004) refraction table exactly down to 3 degrees
    altitude using a 6.5  lapse rate. Below three degrees there were small increasing
    differences. The  N.A. tables give 18.3, 24.3, and 34.5 minutes refraction at
    altitudes of 2, 1,  and 0 degrees, while the model I am using gives 18.4, 24.5,
    and 34.8 minutes of  refraction. I don't know whether these are important or
    The real fun  starts when we try out changes in the temperature profile. It
    is interesting to  note that the location of the tropopause makes little
    difference in the  refraction. Dropping the tropopause from 11km (typical of
    mid-latitude) to 8km  (typical of high latitudes) changes the refraction at the
    horizon by only 3  arcseconds and less than a tenth of an arcsecond at altitudes
    above ten degrees.  Similar results follow even if the observer is up at 2000m.
    If we leave the  tropopause height alone and vary the lapse rate in the
    troposphere, the changes  are much larger but still limited to the lowest angular
    Temperature inversions have a fairly dramatic effect on the refraction,  but,
    again, only very close to the horizon. I tried out a temperature increase  of
    2 deg C in the first kilometer of the atmosphere and then the usual 6.5
    decline per kilometer after that (up to the tropopause) and the results were
    impressive. Above 10 degrees altitude, there was no difference larger than an
    arcsecond from the refraction in the standard atmosphere. At 3 degrees altitude,
    the difference was only 2 arcseconds. But at altitudes of one and zero
    degrees,  the differences were 36 and 195 arcseconds respectively. Plenty big enough
    to  distort the setting Sun...
    By the way, Marcel, thanks for raising this  issue. It's led me to all sorts
    of interesting things.
    42.0N  87.7W, or 41.4N 72.1W.

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