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    Re: Modelization of Refraction
    From: George Huxtable
    Date: 2010 Mar 15, 13:47 -0000

    Andres offered the Bennett approximation for refraction-
    
    "If R is the refraction for standard conditions, (10ºC and 1010 hPa), Ha is 
    the apparent altitude and h = Ha-R, then:
    
    Bennett, G.G.: 1982, The calculation of astronomical refraction in marine 
    navigation, en Journal of the Institute of Navigation, 35, 
    255.
    R = 1/60 / TAN( Ha+7.31/(Ha+4.4) )"
    
    which gives the refraction in degrees.
    ==============
    
    This provides a remarkably good, and simple, empirical fit to refraction observations.
    
    However, it shows two minor difficulties.
    
    1. It doesn't, quite, give a refraction of zero at an altitude of 90 degrees, 
    as symmetry tells us that it must, so it's a bit 
    unphysical.
    2. There's an altitude of 89.92º, at which the denominator becomes infinite, 
    and so some computers / calculators would fail.
    
    So I usually use a slightly tinkered version, which avoids both those 
    problems, but is indistinguishable elsewhere.
    
    This is best expressed as-
    
    R in arc-minutes = tan (90 - Ha + .000861Ha - 7.31 / (Ha + 4.4) ))
    
    George.
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    ----- Original Message ----- 
    From: "Andres Ruiz" 
    To: 
    Sent: Monday, March 15, 2010 11:28 AM
    Subject: [NavList] Modelization of Refraction
    
    
    | If R is the refraction for standard conditions, (10ºC and 1010 hPa), Ha is 
    the apparent altitude and h = Ha-R, then:
    |
    |
    |
    | Bennett, G.G.: 1982, The calculation of astronomical refraction in marine 
    navigation, en Journal of the Institute of Navigation, 
    35, 255.
    |
    | R = 1/60 / TAN( Ha+7.31/(Ha+4.4) )
    |
    |
    |
    | Meeus:
    |
    | R = (58.294*TAN(90-Ha) - 0.0668*pow(TAN(90-Ha),3))/3600
    |
    | R = (58.276*TAN(90-h) - 0.0824*pow(TAN(90-h),3))/3600
    |
    |
    |
    | Saemunndsson:
    |
    | R = (1.02/(TAN(h+10.3/(h+5.11))))/60.0;
    |
    |
    |
    | Chauvenet Vol I, chapter IV gives a very good view of the general law of refraction.
    |
    |
    |
    | …
    |
    | Answering to Brad, yes the iterative process will be something like:
    |
    |
    |
    |      for( dH = 0, iter = 0;  iter < 255;  iter++ ) {
    |
    |            Hs = Hs + dH;
    |
    |            Ha = AlturaAparente( Hs,  ie,  Dip );
    |
    |            R = Refraccion( Ha, T, P );
    |
    |            dH = Hc - ( Ha - R );
    |
    |            if( fabs(dH) < 1E-8 ) break;
    |
    |            }
    |
    |
    |
    | It is more complicated if parallax, for Moon and Sun, it will be taken into account.
    |
    |
    |
    | Regards.
    |
    | ---
    |
    | Andrés Ruiz
    |
    | Navigational Algorithms
    |
    | https://sites.google.com/site/navigationalalgorithms/ 
    
    |
    |
    |
    | ________________________________
    |
    | De: navlist-bounce@fer3.com [mailto:navlist-bounce@fer3.com] En nombre de Antoine Couette
    | Enviado el: viernes, 12 de marzo de 2010 17:20
    | Para: NavList@fer3.com
    | Asunto: [NavList] Re: Star - Star Observations
    |
    | Hello to all,
    |
    | There is an EXCELLENT refraction formula which gives refraction as function 
    of unrefracted topocentric altitude.
    |
    | Its use would simply avoid the requirement for the "loop computation" you have explined.
    |
    |
    | This formula is the so-called Saemundsson's formula which can be found on 
    the Internet. Its accuracy is well under 6 arc-seconds.
    |
    | It is also quoted in M. Jean Meeus's most celebrated book "ASTRONOMICAL 
    ALGORITMS" in its Chapter on atmospheric refraction.
    |
    | Best Regards to you all
    |
    |
    | Antoine M. Couëtte
    |
    |
    |
    |
    |
    | 
    
    
    
    
    

       
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