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